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Lagged Dependent Variable Predictors, Classical Measurement Error, and Path Dependency: The Conditions under Which Various Estimators Are Appropriate

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Title: Lagged Dependent Variable Predictors, Classical Measurement Error, and Path Dependency: The Conditions under Which Various Estimators Are Appropriate
Language: English
Authors: Anders Holm (ORCID 0000-0002-6641-3591), Anders Hjorth-Trolle (ORCID 0000-0003-4132-1014), Robert Andersen
Source: Sociological Methods & Research. 2025 54(1):140-172.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 33
Publication Date: 2025
Document Type: Journal Articles
Reports - Evaluative
Education Level: Elementary Education
Descriptors: Elementary Education, Scores, Error of Measurement, Predictor Variables, Markov Processes, Predictive Measurement, Least Squares Statistics, Test Reliability, Test Selection, Test Construction, Goodness of Fit, Robustness (Statistics)
DOI: 10.1177/00491241231176845
ISSN: 0049-1241
1552-8294
Abstract: Lagged dependent variables (LDVs) are often used as predictors in ordinary least squares (OLS) models in the social sciences. Although several estimators are commonly employed, little is known about their relative merits in the presence of classical measurement error and different longitudinal processes. We assess the performance of four commonly used estimators: (1) the standard OLS estimator, (2) an average of past measures (AVG), (3) an instrumental variable (IV) measured at one period previously (IV), and (4) an IV derived from information from more than one time before (IV2). We also propose a new estimator for fixed effects models--the first difference instrumental variable (FDIV) estimator. After exploring the consistency of these estimators, we demonstrate their performance using an empirical application predicting primary school test scores. Our results demonstrate that for a Markov process with classic measurement error (CME), IV and IV2 estimators are generally consistent; LDV and AVG estimators are not. For a semi-Markov process, only the IV2 estimator is consistent. On the other hand, if fixed effects are included in the model, only the FDIV estimator is consistent. We end with advice on how to select the appropriate estimator.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1457724
Database: ERIC
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  Value: <anid>AN0182244232;som01feb.25;2025Jan17.04:27;v2.2.500</anid> <title id="AN0182244232-1">Lagged Dependent Variable Predictors, Classical Measurement Error, and Path Dependency: The Conditions Under Which Various Estimators are Appropriate </title> <p>Lagged dependent variables (LDVs) are often used as predictors in ordinary least squares (OLS) models in the social sciences. Although several estimators are commonly employed, little is known about their relative merits in the presence of classical measurement error and different longitudinal processes. We assess the performance of four commonly used estimators: (<reflink idref="bib1" id="ref1">1</reflink>) the standard OLS estimator, (<reflink idref="bib2" id="ref2">2</reflink>) an average of past measures (AVG), (<reflink idref="bib3" id="ref3">3</reflink>) an instrumental variable (IV) measured at one period previously (IV), and (<reflink idref="bib4" id="ref4">4</reflink>) an IV derived from information from more than one time before (IV2). We also propose a new estimator for fixed effects models—the first difference instrumental variable (FDIV) estimator. After exploring the consistency of these estimators, we demonstrate their performance using an empirical application predicting primary school test scores. Our results demonstrate that for a Markov process with classic measurement error (CME), IV and IV2 estimators are generally consistent; LDV and AVG estimators are not. For a semi-Markov process, only the IV2 estimator is consistent. On the other hand, if fixed effects are included in the model, only the FDIV estimator is consistent. We end with advice on how to select the appropriate estimator.</p> <p>Keywords: lagged dependent variables; classical measurement error; fixed effects; instrumental variables; test scores; income mobility</p> <hd id="AN0182244232-2">Introduction</hd> <p>Longitudinal data have become increasingly important in the social sciences, especially for examining social mobility (see [<reflink idref="bib3" id="ref5">3</reflink>]; [<reflink idref="bib13" id="ref6">13</reflink>]; [<reflink idref="bib43" id="ref7">43</reflink>]), the inter-generational transmission of education ([<reflink idref="bib23" id="ref8">23</reflink>]; [<reflink idref="bib22" id="ref9">22</reflink>]; [<reflink idref="bib37" id="ref10">37</reflink>]; [<reflink idref="bib50" id="ref11">50</reflink>]; [<reflink idref="bib52" id="ref12">52</reflink>]), occupational outcomes ([<reflink idref="bib9" id="ref13">9</reflink>]), income ([<reflink idref="bib17" id="ref14">17</reflink>]; [<reflink idref="bib19" id="ref15">19</reflink>]; [<reflink idref="bib20" id="ref16">20</reflink>]; [<reflink idref="bib46" id="ref17">46</reflink>]; [<reflink idref="bib47" id="ref18">47</reflink>]; [<reflink idref="bib48" id="ref19">48</reflink>]; [<reflink idref="bib53" id="ref20">53</reflink>]; [<reflink idref="bib54" id="ref21">54</reflink>]; [<reflink idref="bib55" id="ref22">55</reflink>]), and wealth ([<reflink idref="bib16" id="ref23">16</reflink>]; [<reflink idref="bib24" id="ref24">24</reflink>]). Often these relationships are assessed using models that include previous measures of the dependent variable—either from the same individuals (e.g., [<reflink idref="bib52" id="ref25">52</reflink>]) or from generations past (e.g., [<reflink idref="bib53" id="ref26">53</reflink>])—as predictors.</p> <p>While causal interpretations are sometimes implied (see [<reflink idref="bib4" id="ref27">4</reflink>]; [<reflink idref="bib7" id="ref28">7</reflink>]; [<reflink idref="bib11" id="ref29">11</reflink>]), past outcomes are typically treated as controls or descriptive parameters ([<reflink idref="bib25" id="ref30">25</reflink>]; [<reflink idref="bib39" id="ref31">39</reflink>]; [<reflink idref="bib46" id="ref32">46</reflink>]). Regardless of which approach is taken, including a lagged dependent variable (LDV) as a predictor downwardly biases ordinary least squares (OLS) estimates of other effects ([<reflink idref="bib1" id="ref33">1</reflink>]; [<reflink idref="bib59" id="ref34">59</reflink>]; [<reflink idref="bib39" id="ref35">39</reflink>]). If these lagged predictors are measured with error, the OLS model also encounters attenuation bias ([<reflink idref="bib28" id="ref36">28</reflink>]; [<reflink idref="bib41" id="ref37">41</reflink>]; [<reflink idref="bib57" id="ref38">57</reflink>]). Less is known about the performance of lagged dependent estimators with classical measurement error (CME) under various longitudinal processes. We fill this void by providing a formal assessment of how CME affects the consistency of several widely used LDV estimators in OLS regression.</p> <p>We compare the performance of four commonly used estimators: (<reflink idref="bib1" id="ref39">1</reflink>) the simple LDV estimator, (<reflink idref="bib2" id="ref40">2</reflink>) an average of two or more past measures (AVG), (<reflink idref="bib3" id="ref41">3</reflink>) an instrumental variable (IV) constructed from information from one period before (IV), and (<reflink idref="bib4" id="ref42">4</reflink>) an IV constructed from information lagged more than one period (IV2). We also propose a new estimator—the first difference instrumental variable (FDIV) estimator—to account for CME in models with fixed effects. We explore the consistency of these five estimators for two types of longitudinal processes—Markov processes and semi-Markov processes. We also investigate how including fixed effects in the model affects their performance.</p> <p>Our findings demonstrate that the LDV and AVG estimators are inconsistent and downwardly biased for a Markov process with CME, while the IV estimator is consistent. For a semi-Markov process with CME, LDV estimates are inconsistent and biased in an unknown direction, while AVG and IV estimates are inconsistent and upwardly biased, and IV2 estimates are consistent. None of these estimators are consistent in models with fixed effects. On the other hand, our new FDIV estimator is consistent under a Markov process when fixed effects are present. Through empirical applications to models predicting elementary school students' test scores over time, we illustrate how to determine the longitudinal process and appropriate estimator.</p> <hd id="AN0182244232-3">LDV Models</hd> <p>Consider a Markov process where the response variable is dependent on its value measured at the most recent period in the past but has no relation to measures taken in periods before it. Formally, the model can be written as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub></math> </ephtml> (measured at time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> </ephtml> ) denotes the dependent variable and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> is the LDV measured at time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> </ephtml> ) and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml> is an error term independent of the LDV. From here onwards, we assume that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> in (<reflink idref="bib1" id="ref43">1</reflink>) captures the entire effect of both itself and potential confounders and mediators. Under these circumstances, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> is independent of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml> .</p> <p>This model is often used to explore the transfer of parental resources across generations or school test scores of individual students over time ([<reflink idref="bib23" id="ref44">23</reflink>]; [<reflink idref="bib37" id="ref45">37</reflink>]; [<reflink idref="bib50" id="ref46">50</reflink>]).</p> <p>For a semi-Markov process, the response depends on its most recently measured value as well as values measured further back in time, though usually to a lesser extent. The semi-Markov process is defined by: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>γ</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> is one further lagged version of the dependent variable (measured at time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mo>−</mo><mn>2</mn></math> </ephtml> ). In both circumstances, the models assume that the LDV has homogenous and linear effects and that it is independent of the error term. We later relax these assumptions but maintain them here for ease of interpretation.</p> <p>Many intergenerational processes follow a semi-Markov process characterized by a decaying influence over time. For example, the social, cultural, and financial capital of grandparents usually have less influence on children's outcomes than do the characteristics of their parents ([<reflink idref="bib34" id="ref47">34</reflink>]). Another example, which we will explore empirically later, is current school test scores being influenced both by the most recent previous score, and to a weaker extent, by scores from periods before it.</p> <hd id="AN0182244232-4">Simple LDV Estimators</hd> <p>The standard LDV estimator used for a semi-Markov process is simply the dependent variable measured in a previous period. This method is often used to analyze processes of change or to account for path dependency ([<reflink idref="bib13" id="ref48">13</reflink>]; [<reflink idref="bib37" id="ref49">37</reflink>]; [<reflink idref="bib50" id="ref50">50</reflink>]). While simple and intuitive, the method is highly vulnerable to CME and thus can result in inconsistent estimates. If the independent variable is measured with error, the dependent variable—which is simply a more recent measure of the same phenomenon—will almost surely measured with error as well. Returning to the cumulative learning process example, OLS could seriously underestimate the true association between lagged and current test scores if both are imperfect measures of underlying skills. The problem is even more severe for binary variables ([<reflink idref="bib18" id="ref51">18</reflink>]) used in linear probability models ([<reflink idref="bib45" id="ref52">45</reflink>]), which have gained increasing popularity among social researchers ([<reflink idref="bib14" id="ref53">14</reflink>]).</p> <hd id="AN0182244232-5">IV Estimators</hd> <p>IVs are another commonly used LDV estimator ([<reflink idref="bib53" id="ref54">53</reflink>]). This approach can involve utilizing data from as many as three previous periods ([<reflink idref="bib11" id="ref55">11</reflink>]). Consider a model that includes lags back one ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> ) and two ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> ) time periods. The instrument is derived from the fitted values from the regression of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml><emph><subs>.</subs></emph>: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> is the fitted values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> from its regression on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> . The Markov property for the IV is defined by: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml></p> <p>Graph</p> <p>and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>j</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn></math> </ephtml> are independent noise components and of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> is an autoregressive parameter that is less than one but larger than zero. To ease the notation, we restrict the regression coefficient, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> , to be uniform across transitions. This has no impact on our approach to deal with lags and measurement error, which applies even if there is non-constancy of parameters across transitions.</p> <p>The system of equations (<reflink idref="bib2" id="ref56">2</reflink>) and (<reflink idref="bib3" id="ref57">3</reflink>) represents a first autoregressive process, which is a special case of the Markov process for which current values depend linearly on values lagged by one period and an idiosyncratic error term but have no relation to measures before ([<reflink idref="bib38" id="ref58">38</reflink>]). It follows that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> is a valid instrument for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> because the <emph>exclusion restriction</emph> holds. In this case, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> serves as an effective instrument if it is uncorrelated with the measurement error in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> , even if it is measured with error itself ([<reflink idref="bib11" id="ref59">11</reflink>]). If a first-order autoregressive process and the exclusion restriction are untenable, the IV estimator is not just inconsistent itself; it also causes inconsistency in the estimates for other explanatory variables ([<reflink idref="bib5" id="ref60">5</reflink>]; [<reflink idref="bib6" id="ref61">6</reflink>]). Unfortunately, the exclusion restriction is often violated for social phenomena ([<reflink idref="bib5" id="ref62">5</reflink>]; [<reflink idref="bib32" id="ref63">32</reflink>]).</p> <hd id="AN0182244232-6">Extensive LDV Estimators (IV2)</hd> <p>More extensive lagged IV estimators (IV2) are often used to model a semi-Markov process ([<reflink idref="bib7" id="ref64">7</reflink>]). The IV2 estimator is an IV constructed from the fitted values of a regression of the focal independent variable lagged by more than one time on current values of the dependent variable. This can be illustrated by augmenting equations (<reflink idref="bib2" id="ref65">2</reflink>) and (<reflink idref="bib3" id="ref66">3</reflink>) with <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml> represents the dependent variable lagged by one further period and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> is an error term independent of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml> . If the exclusion restriction is violated—that is, equation (<reflink idref="bib2" id="ref67">2</reflink>) is replaced by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>γ</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml> —then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> is not a valid instrument for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> . In such cases, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> partly determines <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub></math> </ephtml> and thus provides inconsistent estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> . Using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml> would yield a consistent estimate, though it would be less efficient than the IV estimator due to the larger time gap between the instrument and the response outcome ([<reflink idref="bib4" id="ref68">4</reflink>]; [<reflink idref="bib7" id="ref69">7</reflink>]; [<reflink idref="bib11" id="ref70">11</reflink>]).</p> <hd id="AN0182244232-7">Average LDV Estimators (AVG)</hd> <p>The average LDV estimator (AVG) is sometimes employed to accommodate measurement error in LDVs for semi-Markov processes. The AVG is constructed by from the mean of the dependent variable measured at two or more previous time periods. In terms of equations (<reflink idref="bib2" id="ref71">2</reflink>) and (<reflink idref="bib3" id="ref72">3</reflink>) the AVG estimator is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mrow><mo>(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mrow><mo>(</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p>If the temporal relationship is relatively stable, using the average of repeated measures on the dependent variable can reduce measurement error ([<reflink idref="bib10" id="ref73">10</reflink>]; [<reflink idref="bib40" id="ref74">40</reflink>]; [<reflink idref="bib42" id="ref75">42</reflink>]; [<reflink idref="bib51" id="ref76">51</reflink>]). As we shall demonstrate in detail later, the estimate will also be consistent. However, if the temporal relationship is not stable and its variability results from a systematic process, the AVG estimator will be inconsistent. Using an inconsistent AVG estimator could also cause inconsistency in the estimates of other effects in the model (see [<reflink idref="bib26" id="ref77">26</reflink>]:120–23).</p> <hd id="AN0182244232-8">CME, Consistency, and Longitudinal Processes</hd> <p></p> <hd id="AN0182244232-9">Classical Measurement Error</hd> <p>The assumption that social science variables are measured without error is almost always untenable. Observational data is particularly vulnerable to measurement error. It is unrealistic, for example, to assume that a questionnaire item accurately taps the desired concept for all respondents. Likewise, administrative register data are subject to measurement error due to clerical errors or misreporting. Even experimental data are not immune to measurement error. Specifically, we cannot realistically assume that subjects always correctly interpret the instructions of the experiment, or even that subjects are always truthful. In short, there will always be at least some measurement error, even if we proceed as if it is unproblematic.</p> <p>Although it is often impossible to know if measurement error is random or systematic, the latter is particularly problematic. When the error in measurement is related either to the value of the variable itself or to an unknown omitted variable, OLS estimates of the effect of <emph>x</emph> on <emph>y</emph> will be inconsistent. While an unquestionably important issue, we do not address systematic errors of this sort in our assessment of the various estimators. Instead, we limit our focus to the impact of CME. While not as problematic as systematic measurement error, CME can still pose serious problems for OLS estimates.</p> <p>CME is independent from the true value of a stochastic variable. That is, the measurement error is random and unrelated to the value of the variable itself. Since the measurement error in <emph>y</emph> is uncorrelated with <emph>x</emph>, OLS estimates will be consistent, even if less efficient than if <emph>y</emph> was measured without error. In other words, the sampling distribution of the estimate converges on the true population parameter as the sample size increases. On the other hand, OLS estimates will be inconsistent if <emph>x</emph> is measured with error.</p> <p>The probability limits of the sample moments (assuming they exist) of two variables, <emph>x</emph> and <emph>y</emph>, both measured without error for the simple linear regression model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>β</mi><mi>x</mi><mo>+</mo><mi>u</mi></math> </ephtml> , where <emph>u</emph> is the disturbance term and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> is the regression coefficient, is given by: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>o</mi><mi>l</mi><mi>s</mi></mrow></msub><mo>=</mo><mrow><mi mathvariant="normal">plim</mi></mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mi>x</mi></mrow><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mi mathvariant="normal">plim</mi></mrow><mo stretchy="false">(</mo><mrow><msup><mi>x</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mi>y</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><mi>y</mi><mo>,</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>In this simple regression case, the OLS estimator has the desirable properties of consistency ([<reflink idref="bib29" id="ref78">29</reflink>]). These properties ensure proper inferences from the model. With repeated random sampling, the estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> will converge on the true population parameter <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> as the sample size increases. These desirable properties hold only if <emph>x</emph> and <emph>y</emph> have been measured without error. Assume that both <emph>x</emph> and <emph>y</emph> in (<reflink idref="bib4" id="ref79">4</reflink>) are measured with error such that the observed values are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mo>=</mo><mi>x</mi><mo>+</mo><msub><mi>e</mi><mi>x</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mo>=</mo><mi>y</mi><mo>+</mo><msub><mi>e</mi><mi>y</mi></msub></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mi>l</mi></msub><mo>,</mo><mi>l</mi><mo>=</mo><mi>x</mi><mo>,</mo><mi>y</mi></math> </ephtml> represents the measurement error.</p> <p>In this simple regression, measurement error in <emph>x</emph> becomes part of the error term <emph>u</emph>, resulting in: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>O</mi><mi>L</mi><mi>S</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mo>,</mo><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mo stretchy="false">(</mo><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>β</mi><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>x</mi></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>Because <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn><mo><</mo><mrow><mfrac><mrow><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>x</mi></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow><mo><</mo><mn>1</mn></math> </ephtml> , the OLS estimate is supressed toward zero. The extent of the inconsistency is determined by: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>O</mi><mi>L</mi><mi>S</mi></mrow></msub><mo>−</mo><mi>β</mi><mspace width="0.25em" /><mo>=</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></mrow><mo>)</mo></mrow><mi>β</mi></math> </ephtml></p> <p>Graph</p> <p>In other words, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>O</mi><mi>L</mi><mi>S</mi></mrow></msub></math> </ephtml> is inconsistent and downwardly biased. Because there is measurement error, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>O</mi><mi>L</mi><mi>S</mi></mrow></msub><mo>−</mo><mi>β</mi></math> </ephtml> does not converge to 0 as the sample size increases. Specifically, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><msup><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mi mathvariant="normal">′</mi></mrow></msup><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><msup><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mi mathvariant="normal">′</mi></mrow></msup><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>E</mi><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><msup><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo stretchy="false">(</mo><mrow><mi>x</mi><mi>β</mi><mo>+</mo><mi>u</mi><mo>+</mo><msub><mi>e</mi><mi>y</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mi mathvariant="normal">′</mi></mrow></msup><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></mfrac></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>β</mi><mrow><mfrac><mrow><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mo>,</mo><mi>x</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mi>v</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo><</mo><mi>β</mi></math> </ephtml> , where <emph>u</emph> is the error variance in <emph>y</emph> and assuming all variables are mean centered. In other words, measurement error in <emph>x</emph> results in OLS estimates that are inconsistent and downwardly biased. In the multiple regression case, even if only one <emph>x</emph> is measured with error—whether systematic or CME—all estimates, including those for other covariates not measured with error, are inconsistent and biased in an unknown direction ([<reflink idref="bib8" id="ref80">8</reflink>]; [<reflink idref="bib27" id="ref81">27</reflink>]).</p> <hd id="AN0182244232-10">Markov Process Without CME</hd> <p>We start by exploring the behavior of the four estimators for both Markov and semi-Markov processes under the perfect condition of no measurement error. If there is no measurement error, the LDV estimator is consistent: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mi>β</mi><mspace width="0.25em" /><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>The IV estimator is also consistent: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mspace width="0.25em" /><mi>β</mi><mspace width="0.25em" /><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>Moreover, the IV2 estimator is equal to the IV and the LDV estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mspace width="0.25em" /><mi>β</mi><mspace width="0.25em" /><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>On the other hand, the AVG estimator is inconsistent and downwardly biased: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mrow><mo>(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mspace width=".1em" /><mrow><mo>(</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>β</mi><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2</mn><mo fence="false" stretchy="false">{</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><mi>β</mi><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mo fence="false" stretchy="false">}</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo><</mo><mi>β</mi></mstyle></math> </ephtml> </p> <p>Graph</p> <p>where we assume a stable process over time such that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo></mrow><mn>2</mn></msubsup><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>...</mo><mo>=</mo></mrow><mn>2</mn></msubsup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mn>0</mn></msub><mo>=</mo></mrow><mn>2</mn></msubsup><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>...</mo><mo>=</mo></mrow><mn>2</mn></msubsup><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></math> </ephtml> . In summary, if there is no measurement error and the data follow a Markov process, we have <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mi>β</mi><mo>></mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>.</mo></math> </ephtml></p> <p>Graph</p> <hd id="AN0182244232-11">Semi-Markov Process Without CME</hd> <p>Under a semi-Markov process, the LDV estimator is inconsistent and upwardly biased when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>></mo><mn>0</mn></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>β</mi><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><mi>γ</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mrow><msup><mi>β</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>></mo><mi>β</mi></mstyle></math> </ephtml> </p> <p>Graph</p> <p>The IV estimator is inconsistent and upwardly biased when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>></mo><mn>0</mn></math> </ephtml> , though its unknown whether it will be larger or smaller than the LDV estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><mi>γ</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></mfrac></mrow><mo>></mo><mi>β</mi></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>The IV2 estimator is consistent: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width=".1em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mo>−</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>where the last equality follows from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mi>y</mi><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><mrow><mfrac><mi>β</mi><mrow><mn>1</mn><mo>−</mo><mi>γ</mi></mrow></mfrac></mrow><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></math> </ephtml> .</p> <p>Finally, the AVG estimator is inconsistent with an unknown bias, even if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>></mo><mi>β</mi><mo>></mo><mn>0</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>></mo><mi>γ</mi><mo>></mo><mn>0</mn></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mrow><mo>(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mrow><mo>(</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2</mn><mo fence="false" stretchy="false">{</mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>3</mn></msup><mo>+</mo><mi>γ</mi><mo>+</mo><mi>β</mi><mi>γ</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mo fence="false" stretchy="false">}</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo>≠</mo><mi>β</mi></math> </ephtml> </p> <p>Graph</p> <hd id="AN0182244232-12">Markov Process with CME</hd> <p>We now turn to the impact of CME on LDV models. Assume that some outcome, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mrow><mspace width="0.25em" /></mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mrow><mspace width="0.25em" /></mrow><msub><mi>y</mi><mn>0</mn></msub></math> </ephtml> follows a Markov process as in equations (<reflink idref="bib2" id="ref82">2</reflink>) and (<reflink idref="bib3" id="ref83">3</reflink>). Further assume that the outcomes, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mrow><mspace width="0.25em" /></mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mrow><mspace width="0.25em" /></mrow><msub><mi>y</mi><mn>0</mn></msub></math> </ephtml> are measured with error: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mn>0</mn></msub><mo>=</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>e</mi><mn>0</mn></msub></math> </ephtml> </p> <p>Graph</p> <p>Finally, assume that the measurement errors are independent of each other and uncorrelated with the true values of <emph>y</emph>.</p> <p>The probability limit of the LDV estimator predicting the outcome at time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>0</mn></msub></math> </ephtml> from information measured at <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msup><mi>β</mi><mn>3</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mrow><msup><mi>β</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>e</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>.</mo></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>If there is no CME in <emph>y</emph>, ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>=</mo><mn>0</mn></math> </ephtml> ), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mi>β</mi></math> </ephtml> ; if there is CME, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo><</mo><mi>β</mi></math> </ephtml> .</p> <p>As mentioned earlier, to account for measurement error, we might consider using the AVG estimator which uses the average of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> as regressors. The <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub></math> </ephtml> estimator provides inconsistent and downwardly bias estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mrow><mo>(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mrow><mo>(</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2</mn><mo fence="false" stretchy="false">{</mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><msup><mi>β</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mo fence="false" stretchy="false">}</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msubsup><mi>σ</mi><mi>e</mi><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>As we demonstrate in Online Appendix A1, in presence of CME (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mrow><mi mathvariant="normal">and</mi></mrow><mspace width="0.25em" /><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup></math> </ephtml> are non-zero), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>−</mo><mspace width="0.25em" /><mi>β</mi></math> </ephtml> is smaller than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>−</mo><mspace width="0.25em" /><mi>β</mi></math> </ephtml> . Put another way, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>></mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>></mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub></math> </ephtml> .</p> <p>Now consider the standard IV estimator using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> as instrument for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> for a Markov process. This estimator is consistent, making it preferable to both the AVG and the LDV estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msup><mi>β</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mrow><mi>β</mi><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>=</mo><mspace width="0.25em" /><mi>β</mi></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>The related IV estimator that uses one period further lagged value of <emph>y</emph> as instrument—that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml> —denoting it as IV2, is also consistent (see also [<reflink idref="bib49" id="ref84">49</reflink>]): <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mo>−</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>In summary, for a Markov process with CME: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mi>β</mi><mo>></mo><mspace width="0.25em" /><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>></mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>.</mo></math> </ephtml></p> <p>Graph</p> <hd id="AN0182244232-13">Semi-Markov Process with CME</hd> <p>The situation differs for a semi-Markov process. In this case, the LDV estimator is given by: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>3</mn></msup><mo>+</mo><mi>β</mi><mi>γ</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msup><mi>β</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>while the AVG estimator is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub></mtd><mtd><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mrow><mo>(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mi mathvariant="normal">var</mi></mrow><mrow><mo>(</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mstyle></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2</mn><mo fence="false" stretchy="false">{</mo><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>3</mn></msup><mo>+</mo><mi>γ</mi><mo>+</mo><mi>β</mi><mi>γ</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mo fence="false" stretchy="false">}</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>Both estimators are inconsistent in the case of a semi-Markov process ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>≠</mo><mn>0</mn></math> </ephtml> ). When <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>></mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>></mo><mn>0</mn><mo>,</mo></math> </ephtml> both estimators converge to values smaller than the true value. In many applications, it is sensible to assume that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>></mo><mn>0</mn></math> </ephtml> . For example, it could reflect the impact of a further lagged test score on current test scores, or it could represent the effect of grandparents on children.</p> <p>The IV estimator is now defined by: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><mi>γ</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>Hence, when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>></mo><mn>0</mn></math> </ephtml> , the IV estimator is larger than the true parameter. If period <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mspace width="0.25em" /><mo>=</mo><mo>−</mo><mn>2</mn></math> </ephtml> of the dependent variable is used as an instrument for period <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mspace width="0.25em" /><mo>=</mo><mo>−</mo><mn>1</mn></math> </ephtml> , the IV estimator is inconsistent and too large because it absorbs the effect of the second-order lag. On the other hand, the IV2 estimator is consistent: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width=".1em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width="0.25em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width=".1em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mo>−</mo><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width=".1em" /><mrow><mi mathvariant="normal">cov</mi></mrow><mspace width=".1em" /><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>Now consider situations when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mi>β</mi></math> </ephtml> (i.e., the two period lagged effects are as large as one period lagged effects). For both the IV estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo></mrow><mi>β</mi></mfrac></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mi>β</mi><mo>></mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>and the AVG estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>=</mo><mn>2</mn><mi>β</mi><mo>></mo><mi>β</mi><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>the estimates are inconsistent and upwardly biased. While the IV2 estimator is less efficient than the single period lagged IV estimator (see Online Appendix A2), it is a consistent estimator for both Markov and semi-Markov processes. For a semi-Markov process, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>></mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mi>β</mi><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>and for a Markov process, <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mi>β</mi><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>In short, when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>,</mo><mi>β</mi><mo>></mo><mn>0</mn></math> </ephtml> and CME is present: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>></mo><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>></mo><mspace width="0.25em" /><mi>β</mi><mo>=</mo><mrow><mi mathvariant="normal">plim</mi></mrow><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>In the case of CME (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo>></mo><mn>0</mn></math> </ephtml> ) with a semi-Markovian process, the LDV estimator is also inconsistent, though its sampling mean may be larger or smaller than the true parameter: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mi>β</mi><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msup><mi>β</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>≠</mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>If measurement error is large and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> is small, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo><</mo><mi>β</mi></math> </ephtml> . The true effect is likely, though not necessarily, bounded between the LDV and IV estimator. If the LDV estimator differs from zero, the lagged value has an effect, regardless of the value of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> </ephtml> , that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>></mo><mn>0</mn><mo stretchy="false">⇔</mo><mi>β</mi><mo>></mo><mn>0</mn></math> </ephtml> .</p> <p>In sum, the IV estimator is consistent, while the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub></math> </ephtml> are inconsistent and downwardly biased. For a Markov process, both the IV and IV2 estimators are consistent. When the process is semi-Markovian, only the IV2 is consistent. Conveniently, a Hausman test for equality of the IV and IV2 estimators—or equivalently, a Sargan test of over-identification ([<reflink idref="bib21" id="ref85">21</reflink>])—implicitly provides a test of the null hypothesis that there is a Markov process.</p> <hd id="AN0182244232-14">Summary of the Performance of the Estimators</hd> <p>The relative performance of the four estimators under different circumstances is summarized in Table 1. The relative size of the IV2 estimator is key to identifying the type of dynamic process. If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>></mo><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo>></mo><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub></math> </ephtml> , three scenarios are possible: (<reflink idref="bib1" id="ref86">1</reflink>) a Markov process with CME, (<reflink idref="bib2" id="ref87">2</reflink>) a semi-Markov process with CME, or (<reflink idref="bib3" id="ref88">3</reflink>) a semi-Markov process without CME. When <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>></mo><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub></math> </ephtml> a longitudinal process with a longer memory than a semi-Markov process is indicated. In this scenario, the IV absorbs effects from the second order term, while the IV2 estimator includes effects from further lags. If the process or path dependency has a longer history than what is implied by the semi-Markov process, all the estimators are inconsistent. This process is identified by longer lags having less impact than shorter ones. If we can assume that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>></mo><mi>γ</mi></math> </ephtml> , which is typical when studying intergenerational transmission (parents, ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> ), matter more than grandparents, ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> </ephtml> ), conditional on parents) or skills (last year's skill level matters more than the skill level of two years ago, conditional on last year), the IV2 estimator will be larger than the IV estimator. The results in Table 1 are under the assumptions that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>></mo><mi>β</mi><mo>></mo><mn>0</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>></mo><mi>γ</mi><mo>></mo><mn>0</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo>≥</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mn>0</mn></msub></mrow><mn>2</mn></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mn>0</mn></msub></mrow><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mn>1</mn></msub></mrow><mn>2</mn></msubsup><mo>.</mo></math> </ephtml></p> <p>Table 1. Summarizing Results for Estimators Under Different Conditions.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="left" /></colgroup><thead><tr><th align="left" /><th align="center">Classical Measurement Error</th><th align="center">No Measurement Error</th></tr></thead><tbody><tr><td>Markov process</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo xmlns="">=</mo><mrow xmlns=""><mi mathvariant="bold">p</mi><mi mathvariant="bold">l</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">m</mi></mrow><mrow xmlns=""><mspace width="0.25em" /></mrow><msub xmlns=""><mrow><mover><mrow><mi mathvariant="bold-italic">β</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mrow><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">V</mi></mrow><mn>2</mn></mrow></msub><mo xmlns="">=</mo><mi xmlns="">β</mi><mo xmlns="">></mo><mrow xmlns=""><mspace width="0.25em" /><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo xmlns="">></mo><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi mathvariant="bold">p</mi><mi mathvariant="bold">l</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">m</mi></mrow><mrow xmlns=""><mspace width="0.25em" /></mrow><msub xmlns=""><mrow><mover><mrow><mi mathvariant="bold-italic">β</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></mtd></mtr><mtr><mtd /></mtr></mtable></mrow></msub><mo xmlns="">=</mo><mrow xmlns=""><mi mathvariant="bold">p</mi><mi mathvariant="bold">l</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">m</mi></mrow><mrow xmlns=""><mspace width="0.25em" /></mrow><msub xmlns=""><mrow><mover><mrow><mi mathvariant="bold-italic">β</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mrow><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">V</mi></mrow></mrow></msub><mo xmlns="">=</mo><mrow xmlns=""><mi mathvariant="bold">p</mi><mi mathvariant="bold">l</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">m</mi></mrow><mrow xmlns=""><mspace width="0.25em" /></mrow><msub xmlns=""><mrow><mover><mrow><mi mathvariant="bold-italic">β</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mrow><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">V</mi></mrow><mn>2</mn></mrow></msub><mo xmlns="">=</mo><mi xmlns="">β</mi><mo xmlns="">></mo><mspace width="0.25em" xmlns="" /><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub></math></p></td></tr><tr><td>Semi-Markov process</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo xmlns="">></mo><mspace width="0.25em" xmlns="" /><mrow xmlns=""><mi mathvariant="bold">p</mi><mi mathvariant="bold">l</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">m</mi></mrow><mrow xmlns=""><mspace width="0.25em" /></mrow><msub xmlns=""><mrow><mover><mrow><mi mathvariant="bold-italic">β</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mrow><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">V</mi></mrow><mn>2</mn></mrow></msub><mo xmlns="">=</mo><mi xmlns="">β</mi><mo xmlns="">></mo><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub><mo xmlns="">≠</mo><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub></math></p></td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo xmlns="">></mo><mrow xmlns=""><mi mathvariant="bold">p</mi><mi mathvariant="bold">l</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">m</mi></mrow><mrow xmlns=""><mspace width="0.25em" /></mrow><msub xmlns=""><mrow><mover><mrow><mi mathvariant="bold-italic">β</mi></mrow><mo stretchy="false">^</mo></mover></mrow><mrow><mrow><mi mathvariant="bold-italic">I</mi><mi mathvariant="bold-italic">V</mi></mrow><mn>2</mn></mrow></msub><mo xmlns="">=</mo><mi xmlns="">β</mi><mo xmlns="">≠</mo><mspace width="0.25em" xmlns="" /><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo xmlns="">,</mo><mspace width="0.25em" xmlns="" /><mrow xmlns=""><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" xmlns="" /><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>A</mi><mi>V</mi><mi>G</mi></mrow></msub></math></p></td></tr></tbody></table> </ephtml> </p> <p>1 Note. Consistent estimators are denoted in bold font.</p> <hd id="AN0182244232-15">Heterogenous Effects and Nonlinearity</hd> <p>Until now we have assumed that the basic LDV predictor model, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mn>0</mn></msub></math> </ephtml> operates under the conditions of homogeneity (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> is constant across observations) and linearity. It has been demonstrated by [<reflink idref="bib30" id="ref89">30</reflink>] that IVs identify the parameters in nonlinear models measured with error. Less is known about the identification of the parameters in nonlinear models in the face of both heterogenous effects and measurement error. We now turn our attention to the conditions under which the IV estimator identifies the parameters in the presence of these circumstances. We restrict our discussion to the IV estimator, though our results also apply to the IV2 estimator under no further assumptions. Both the LDV and AVG estimators are always biased even with linearity and homogenous parameters and hence receive no further discussion.</p> <p>To simplify matters, we consider a quadratic polynomial relationship: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>=</mo><msub><mi>β</mi><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>β</mi><mrow><mn>2</mn><mi>i</mi></mrow></msub><msubsup><mi>y</mi><mrow><mo>−</mo><mn>1</mn><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mi>u</mi><mrow><mn>0</mn><mi>i</mi></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>where subscript <emph>i</emph> indicates individuals. In this model, effects are allowed to correlate across the linear and quadratic terms of the LDV. Rather than derive the theoretical properties of the various estimators, which is beyond the scope of this paper, we use simulations to explore the impact of heterogeneous effects on the LDV estimators when the variables are measured with and without error. We explore the results when the lagged independent variables (both the linear and quadratic terms) are instrumented with variables lagged oner period further, which we denote as the IV estimator in the case of nonlinearity. We also investigate what happens in the linear case with heterogeneous effects.</p> <p>We analyze 500 simulated datasets, each of 5000 observations, that draw the heterogenous coefficients from a joint normal distribution with means: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">¯</mo></mover></mrow><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mover><mi>β</mi><mo stretchy="false">¯</mo></mover></mrow><mn>1</mn></msub><mo>=</mo><mn>0.1</mn></math> </ephtml> , where the correlation between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>2</mn></msub></math> </ephtml> equals <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.5</mn></math> </ephtml> . We then simulate the process of generating <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mspace width="0.25em" /></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> according to (<reflink idref="bib3" id="ref90">3</reflink>) where the <emph>u</emph>'s are serially independent and uncorrelated with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>t</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>2</mn></msub></math> </ephtml> . Finally, we ensure that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> </ephtml> percent of the variance in each of the observed <emph>y</emph> variables is measurement error. In other words, the correlation between the "true" values of the <emph>y</emph>'s and those measured with error is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.5</mn></math> </ephtml> . We then apply IV estimation to (<reflink idref="bib5" id="ref91">5</reflink>) such that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> is instrumented with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow><mn>2</mn></msubsup></math> </ephtml> . Results from these simulations are reported in Table 2. Notice that the LDV without measurement error captures the true average coefficients, both in the linear and nonlinear models. When there is measurement error, the LDV underestimates the true effect in both the linear and nonlinear models. On the other hand, even with measurement error, the IV performs extremely well in both the linear and nonlinear models.</p> <p>Table 2. Simulation Results From Estimating a Linear and a Quadradic Model With Correlated Heterogenous Effects.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="char" char="(" /><col align="char" char="(" /><col align="char" char="(" /></colgroup><thead><tr><th align="left" /><th align="left">True Value</th><th align="center">LDV With Heterogenous Effects</th><th align="center">LDV With Heterogenous Effects and Measurement Error</th><th align="center">IV With Heterogenous Effects and Measurement Error</th></tr></thead><tbody><tr><td colspan="5">Linear model</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">¯</mo></mover></mrow><mn>1</mn></msub></math></p></td><td>1</td><td align="char" char=".">1.000 (0.001)</td><td align="char" char=".">0.429 (0.001)</td><td align="char" char=".">1.003 (0.005)</td></tr><tr><td colspan="5">Quadratic model</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">¯</mo></mover></mrow><mn>1</mn></msub></math></p></td><td align="char" char=".">1</td><td align="char" char=".">0.983 (0.040)</td><td align="char" char=".">0.588 (0.025)</td><td align="char" char=".">0.970 (0.049)</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mover><mi>β</mi><mo stretchy="false">¯</mo></mover></mrow><mn>2</mn></msub></math></p></td><td align="char" char=".">0.1</td><td align="char" char=".">0.102 (0.014)</td><td align="char" char=".">0.086 (0.011)</td><td align="char" char=".">0.103 (0.013)</td></tr></tbody></table> </ephtml> </p> <hd id="AN0182244232-16">Confounding Variables</hd> <p>The negative impact of CME is attenuated if present for several covariates ([<reflink idref="bib27" id="ref92">27</reflink>]). It is important, then, to explore how the IV estimator performs when both the LDV and a potential confounder is measured with error. We perform simulations in the same way as above, though in this case, we use samples of 50,000 observations as well as 5,000. The model takes the using the following model: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>a</mi><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>0</mn></msub><mo>=</mo><mi>a</mi><mo>+</mo><msub><mi>e</mi><mn>0</mn></msub></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>a</mi><mo>+</mo><msub><mi>u</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub><mo>=</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>c</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mn>0</mn></msub></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><msub><mi>e</mi><mn>1</mn></msub></math> </ephtml> </p> <p>Graph</p> <p>where <emph>a</emph>, <emph>e<subs>j</subs></emph>, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mspace width="0.25em" /><mo>=</mo><mspace width="0.25em" /><mo>−</mo><mn>1</mn></math> </ephtml> , 0, and <emph>u<subs>j</subs></emph><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>,</mo><mspace width="0.25em" /><mi>j</mi><mspace width="0.25em" /><mo>=</mo><mspace width="0.25em" /><mo>−</mo><mn>1</mn><mo>,</mo><mspace width="0.25em" /><mn>0</mn><mo>,</mo><mspace width="0.25em" /><mn>1</mn></math> </ephtml> are standard normal errors, which implies that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">cor</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>,</mo><mspace width="0.25em" /><msub><mi>c</mi><mi>t</mi></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /><mo>=</mo><mspace width="0.25em" /><mn>0.5</mn><mo>,</mo><mspace width="0.25em" /><mi>t</mi><mspace width="0.25em" /><mo>=</mo><mspace width="0.25em" /><mo>−</mo><mn>1</mn><mo>,</mo><mspace width="0.25em" /><mn>0</mn></math> </ephtml> , such that <emph>c<subs>t</subs></emph>, <emph>t</emph><emph>=</emph> −1, 0, is a confounder of <emph>y<subs>t</subs></emph>, <emph>t</emph><emph>=</emph><emph>0, 1,</emph> when predicting <emph>y<subs>t</subs></emph><emph><subs>+</subs></emph><emph><subs>1</subs></emph>. Because they are dependent on a common factor <emph>a,</emph> the confounders ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>0</mn></msub></math> </ephtml> ) are correlated with each other and the first period LDV ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> ). In turn, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>0</mn></msub></math> </ephtml> is correlated with the confounder <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>c</mi><mn>0</mn></msub></math> </ephtml> because <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">cor</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>,</mo><mspace width="0.25em" /><msub><mi>c</mi><mi>t</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mspace width="0.25em" /><mn>0.5</mn><mo>,</mo><mspace width="0.25em" /><mi>t</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>,</mo><mspace width="0.25em" /><mn>0</mn></math> </ephtml> . For our simulations, the LDV (measured with error) is instrumented with a further lag (as outlined previously); the confounders (also measured with error) are instrumented with the lagged confounder measured with error.</p> <p>The results of our simulations are shown in Table 3. Notice that the OLS regression precisely estimates the true direct and indirect effects when there is no measurement error. However, both the direct and indirect effects are downwardly inconsistent in the face of measurement error. Finally, the IV estimator performs better than the OLS estimator, both with and without the presence of measurement error.</p> <p>Table 3. Simulation Results from Estimating a Model with a Confounder. The Standard Errors for all Estimates are <<0.001.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="left" /><th align="left">True Value</th><th align="left">OLS</th><th align="left">OLS with Measurement Error</th><th align="left">IV with Measurement Error</th></tr></thead><tbody><tr><td colspan="5">5,000 observations</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">β</mi></math></p> Total effect</td><td>1.333</td><td>1.286</td><td>1.126</td><td>1.335</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">β</mi></math></p> Direct effect</td><td>1</td><td>1.001</td><td>0.951</td><td>1.001</td></tr><tr><td colspan="6">50,000 observations</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">β</mi></math></p> Total effect</td><td>1.333</td><td>1.286</td><td>1.125</td><td>1.333</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">β</mi></math></p> Direct effect</td><td>1</td><td>1.000</td><td>0.950</td><td>1.000</td></tr></tbody></table> </ephtml> </p> <hd id="AN0182244232-17">Fixed Effects in LDV Models with Measurement Error</hd> <p>Applying fixed effects can provide significant insight about within individual dynamics when the appropriate data are available. But this does not come without challenges. Specific to the problem at hand, including fixed effects further exacerbates the problems of measurement error (see [<reflink idref="bib44" id="ref93">44</reflink>], [<reflink idref="bib56" id="ref94">56</reflink>]), resulting in inconsistent estimates. We now explore how the various estimators perform under these conditions. We exclude the AVG estimator from this exercise because its derivation is overly cumbersome and, more importantly, our previous results demonstrate that it is not the best estimator even under ideal conditions.</p> <p>The fixed effect model extends from equation (<reflink idref="bib1" id="ref95">1</reflink>) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><mi>a</mi><mo>+</mo><mi>β</mi><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mi>t</mi></msub><mo>;</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mspace width="0.25em" /><mo>−</mo><mn>3</mn></math> </ephtml></p> <p>Graph</p> <p>and <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub><mo>=</mo><mi>a</mi><mo>+</mo><msub><mi>u</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></math> </ephtml></p> <p>Graph</p> <p>where <emph>a</emph> represents the individual fixed effect. In this case, the LDV estimator is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>L</mi><mi>D</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>≠</mo><mi>β</mi><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>the IV estimator is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>≠</mo><mi>β</mi><mo>,</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>and the IV2 estimator is: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>I</mi><mi>V</mi><mn>2</mn></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>3</mn></msup></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><mi>β</mi><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>≠</mo><mi>β</mi></mstyle></math> </ephtml></p> <p>Graph</p> <p>In other words, all three estimators—the LDV, IV, and IV2—are inconsistent in models with fixed effects and CME. The IV and IV2 estimators have a sampling mean that is larger than the true parameter; whether the sampling mean for the LDV is smaller or larger than the true population mean depends on the variance in the fixed effects and the amount of measurement error.</p> <p>With CME, the traditional fixed effect estimator—which is equivalent to the first difference estimator for short panel datasets—is also inconsistent ([<reflink idref="bib44" id="ref96">44</reflink>]; [<reflink idref="bib56" id="ref97">56</reflink>]): <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd /><mtd><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow><mrow><mi mathvariant="normal">plim</mi></mrow><mspace width="0.25em" /><msub><mrow><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>F</mi><mi>D</mi></mrow></msub></mrow><mo>=</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr></mtable></mtd></mtr><mtr><mtd /><mtd><mspace width="2em" /><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><msub><mi>e</mi><mn>0</mn></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd /><mtd><mspace width="2em" /><mo>=</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><msub><mi>y</mi><mn>0</mn></msub><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>−</mo><msub><mi>σ</mi><mrow><msub><mi>y</mi><mn>0</mn></msub><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub><mo>−</mo><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub><mo>−</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub></mrow><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>.</mo></mstyle></mrow></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>If the process is stable—that is, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><msub><mi>y</mi><mi>t</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mi>σ</mi><mrow><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mi>t</mi></msub></mrow><mn>2</mn></msubsup><mo>=</mo><msubsup><mi>σ</mi><mrow><msub><mi>e</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></msubsup></math> </ephtml> = <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mi>e</mi><mn>2</mn></msubsup></math> </ephtml> —this can be reduced to <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2</mn><msub><mi>σ</mi><mrow><msub><mi>y</mi><mi>t</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>−</mo><msub><mi>σ</mi><mrow><msub><mi>y</mi><mi>t</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub><mo>−</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>σ</mi><mi>e</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>−</mo><mn>2</mn><msub><mi>σ</mi><mrow><msub><mi>y</mi><mi>t</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub></mrow></mfrac></mrow><mo>.</mo></mstyle></math> </ephtml></p> <p>Graph</p> <p>Accordingly, the FD estimator can even be negative despite <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>></mo><mn>0</mn></math> </ephtml> , when the persistence across periods is small enough to be offset by the variance in outcomes and measurement error (i.e., when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>σ</mi><mrow><msub><mi>y</mi><mi>t</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo><</mo><msub><mi>σ</mi><mrow><msub><mi>y</mi><mi>t</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub><mo>+</mo><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>e</mi><mn>2</mn></msubsup></math> </ephtml> ).</p> <p>Influenced by [<reflink idref="bib7" id="ref98">7</reflink>], we propose a new estimator to tackle the limitations discussed above. As we shall demonstrate, our FDIV estimator provides consistent estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> even with measurement error in previous lags and fixed effects. To limit the influence of measurement error in previous lags, the FDIV estimator extends back several periods to instrument differences in more recent lags. As is obvious below, the application of the FDIV requires a least five waves of data: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.25em" /><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>F</mi><mi>D</mi><mi>I</mi><mi>V</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>=</mo><mspace width="0.25em" /><mi>a</mi><mo>+</mo><mi>β</mi><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub><mo>=</mo><mspace width="0.25em" /><mi>a</mi><mo>+</mo><msub><mi>u</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></math> </ephtml> </p> <p>Graph</p> <p>Taking the probability limit results in: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mrow><mrow><mi mathvariant="normal">plim</mi></mrow><msub><mrow><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow></mrow><mrow><mi>F</mi><mi>D</mi><mi>I</mi><mi>V</mi></mrow></msub></mrow></mtd><mtd><mo>=</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>0</mn></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mn>0</mn></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow><mrow><mrow><mi mathvariant="normal">cov</mi></mrow><mo stretchy="false">(</mo><mrow><msub><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mo>−</mo><mn>4</mn></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msup><mi>β</mi><mn>5</mn></msup><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>−</mo><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>4</mn></msup><mo>−</mo><msup><mi>β</mi><mn>3</mn></msup></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>3</mn></msup><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow><mrow><msup><mi>β</mi><mn>4</mn></msup><msubsup><mi>σ</mi><mi>a</mi><mn>2</mn></msubsup><mo>−</mo><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>3</mn></msup><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>−</mo><mi>β</mi></mrow><mo stretchy="false">)</mo><msubsup><mi>σ</mi><mi>u</mi><mn>2</mn></msubsup></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>For a Markov process, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>F</mi><mi>D</mi><mi>I</mi><mi>V</mi></mrow></msub></math> </ephtml> is consistent. When there is no measurement error, the estimator reduces to the standard FD estimator. Without fixed effects, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>F</mi><mi>D</mi><mi>I</mi><mi>V</mi></mrow></msub></math> </ephtml> reduces to the IV estimator. Finally, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>β</mi><mo stretchy="false">^</mo></mover></mrow><mrow><mi>F</mi><mi>D</mi><mi>I</mi><mi>V</mi></mrow></msub></math> </ephtml> is equal to the LDV estimator when there is both no measurement error and no fixed effects. The FDIV estimator differs significantly from these other measures, however, in that it exploits only variation associated with change, making it is less efficient than both the LDV and the IV estimator. In the case of a semi-Markov process, the FDIV estimator is typically upwardly inconsistent (when the second-order term is positive) as it captures the effect of the second-order term. In this case, a further lagged difference for instruments is required, which now requires data from at least six repeated measures. In other words, the FIDV has utility for extensive panels but has only limited usefulness for panels with few waves.</p> <hd id="AN0182244232-18">Choosing the Appropriate Estimator: An Empirical Application</hd> <p>Before demonstrating our approach with empirical examples, we summarize the behavior of the various estimators. For this exercise, we assume linearity and homogeneity of the coefficients. As previously, we also interpret the effect of the LDV as a total effect, which justifies the assumption that the error term is independent of the LDV. Further, we refer to our simulation results on homogenous effects and suggest that our results may be robust certain versions of heterogenous effects. We do not test for nonlinearity for simplicity.</p> <p>To assess which estimator to employ, we suggest taking the following steps:</p> <p></p> <ulist> <item> Fit four models to the same data, each model including only one of the estimators (the models are otherwise identical),</item> <p></p> <item> Formally test for difference in the estimates between the four different estimators,</item> <p></p> <item> Compare both the fit and the predictive performance of the four models. The relative magnitude of the estimates and the overall fit and predictive power of the models provides guidance on the magnitude of CME and the longitudinal process under investigation. This information should guide the decision on the proper estimator to employ.</item> </ulist> <p>The problems associated with LDV predictors are especially relevant in the study of intergenerational transmission of educational attainment and income ([<reflink idref="bib36" id="ref99">36</reflink>]; [<reflink idref="bib53" id="ref100">53</reflink>]; [<reflink idref="bib54" id="ref101">54</reflink>]; [<reflink idref="bib55" id="ref102">55</reflink>]) and in models of learning where student grades are available across more than two years ([<reflink idref="bib23" id="ref103">23</reflink>]; [<reflink idref="bib37" id="ref104">37</reflink>]; [<reflink idref="bib50" id="ref105">50</reflink>]). We thus explore empirical examples from this area of research. As well as demonstrating the performance of the four estimators, we show how their presence affects other estimates.</p> <hd id="AN0182244232-19">Models Without Fixed Effects</hd> <p>We first consider the conditions that identify a Markov model without CME. In these cases, estimates for the LDV, IV, and IV2 should be statistically equivalent and larger than the AVG which is inconsistent because it fails to capture the true dynamics over time. The LDV should be preferred because of its relative simplicity and statistical efficiency.</p> <p>A Markov process with CME is identified by (<reflink idref="bib1" id="ref106">1</reflink>) a failure to reject the null hypothesis that the IV and IV2 estimates are equal and (<reflink idref="bib2" id="ref107">2</reflink>) these two estimators are significantly larger than the LDV and AVG estimates. The model fit (e.g., as measured by the residual sum of squares or the AIC or BIC) and predictive power (e.g., the <emph>R<sups>2</sups></emph>) will be higher for the AVG model than for the LDV model because the former contains more information on the dependent variable, and both the IV and IV2 will have less predictive power and statistical efficiency ([<reflink idref="bib33" id="ref108">33</reflink>]). Under these conditions, the IV and IV2 estimators are consistent but the LDV and AVG estimates are downwardly inconsistent, though even more so for the LDV. For a Markov process with CME, then, the IV estimator should be preferred.</p> <p>A semi-Markov process is recognized by the IV2 estimate being smaller than the IV estimate. In this case, the IV estimator is inflated because it fails to identify the important longer historical process. On the other hand, the IV2 estimator performs identifies the longer historical dynamic process. Still, we have no way of knowing which estimator to prefer because we cannot distinguish the effect of measurement error from the effect of the (long) trajectory of the data process. This distinction is only possible with IV estimators lagged back further in time. In contrast to the Markovian case, then, the AVG model will not necessarily have more predictive power over the LDV model.</p> <p>If the AVG estimator is misspecified, it will incorrectly force equal weight to the first-order and the second-order lags. Similarly, if the LDV estimator is misspecified (over and above what is implied from CME), it gives full weight to the first-order lag and no weight whatsoever to the second-order lag despite its importance. Still, if the first lag is much more important than the second lag, the LDV model will yield a better fit than the AVG model. Regardless, both estimates will be downwardly inconsistent in the case of CME. In sum, in the case of a semi-Markov process (assuming no fixed effects), only the IV2 estimator is consistent. This situation can only be verified using data that allow instruments created from lagged variables back at least three periods.</p> <p>Turing to our empirical example, our goal is to predict educational test scores from past test scores. We apply the four estimators to identical models (interchanging only the LDV-related measures) to two datasets: 1) a subset of the Danish administrative register data (provided by Statistics Denmark) and 2) the National Longitudinal Survey of Youth 1979–Children and Young Adults (NLSY-CYA) in the US ([<reflink idref="bib58" id="ref109">58</reflink>]). The Danish register data include results from the Danish National Tests, a standardized national program that tests children in primary education in the Danish language in the 2nd, 4th, 6th, and 8th test scores. The US NLSY-CYA data contain information reading scores for US children at ages 8, 10, 12, and 14, which corresponds roughly to the ages of students in the Danish data. To facilitate comparability, we use only observations with valid information in all four periods. The analytical sample sizes are 110,329 individuals for the Danish registry data and 4,498 individuals for the NLSY-CYA.</p> <p>Table 4 displays the coefficients for the four estimates from OLS regression models. Current performance measured in 8th grade, lagged test scores are measured in grades 6, 4, and 2. The response variable in all models is performance measured in grade eight (at age 14 for the US data). The IV estimator is constructed from the fitted values from an OLS regression of scores in grade 6 on scores in grade 4—which is the most recent measurement before the final test score—regressed on current score. The IV2 estimator is constructed from the fitted values of scores in grade 6 regressed on scores in grade 2, the second to last time that performance was measured before grade eight. Finally, the AVG estimator is simply the mean score in grades four and six. Control variables are excluded from the models to allow direct comparison of cross-national differences in the total effect of past performance on current performance.</p> <p>Table 4. Estimators Predicting Test Scores on Previous Test Scores Using Danish Register Data and NLSY Data.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="center" colspan="5">Danish Register Data</th><th align="left" colspan="4">NLSY</th></tr><tr><th align="left" /><th align="center">LDV</th><th align="center">AVG</th><th align="center">IV</th><th align="center">IV2</th><th align="center">LDV</th><th align="center">AVG</th><th align="center">IV</th><th align="center">IV2</th></tr></thead><tbody><tr><td>Constant term</td><td align="char" char=".">0.000(0.002)</td><td align="char" char=".">0.000(0.002)</td><td align="char" char=".">0.000(0.002)</td><td align="char" char=".">0.000(0.002)</td><td align="char" char=".">0.000(0.011)</td><td align="char" char=".">0.000(0.010)</td><td align="char" char=".">0.000(0.011)</td><td align="char" char=".">0.000(0.011)</td></tr><tr><td>Lagged test scores</td><td align="char" char=".">0.696***(0.002)</td><td align="char" char=".">0.775***(0.002)</td><td align="char" char=".">0.881***(0.003)</td><td align="char" char=".">0.885***(0.004)</td><td align="char" char=".">0.689***(0.011)</td><td align="char" char=".">0.782***(0.011)</td><td align="char" char=".">0.916***(0.016)</td><td align="char" char=".">0.862***(0.019)</td></tr><tr><td><italic>R</italic>2</td><td align="char" char=".">0.485</td><td align="char" char=".">0.524</td><td align="char" char=".">0.451</td><td align="char" char=".">0.450</td><td align="char" char=".">0.475</td><td align="char" char=".">0.517</td><td align="char" char=".">0.424</td><td align="char" char=".">0.445</td></tr><tr><td><italic>N</italic></td><td align="char" char=".">110,329</td><td align="char" char=".">110,329</td><td align="char" char=".">110,329</td><td align="char" char=".">110,329</td><td align="char" char=".">4,498</td><td align="char" char=".">4,498</td><td align="char" char=".">4,498</td><td align="char" char=".">4,498</td></tr></tbody></table> </ephtml> </p> <p>2 Note. *<emph>p</emph> > 0.05, ***<emph>p</emph> > 0.01, ***<emph>p</emph> > 0.001.</p> <p>There are several noteworthy findings in Table 4. First, the estimates for the Danish data and the US data are strikingly similar. Despite large differences in the schooling systems of the two countries, skills appear to beget skills in a similar way. In both countries, if we know how students perform in grades four and six, we can make quite good predictions about how they will perform in grade eight. Second, the effect of past performance is very strong regardless of the estimator employed. Third, the patterns across models suggest a Markov process for the Danish data and a semi-Markov process for the U.S. data.</p> <p>For both the Danish and the US data, the LDV and AVG estimates are smaller than the IV and IV2 estimates, and the IV and IV2 estimates are similar. Hausmann tests of equality of the IV and the IV2 coefficient confirm a Markov process for the Danish data (the test statistic is 2.286 with a <emph>p</emph>-value of 0.131); For the US data, the hypothesis of equality of the two estimators is rejected (the test statistic is 27,771 with a <emph>p</emph>-value of 0.00001). Moreover, the explanatory power (<emph>R</emph><sups>2</sups>) of the model is the largest for the AVG estimator, indicating that further lagged values of the dependent variable carry predictive information over and above information from the first lag. We conclude from the IV estimates that test scores evolve according to a Markov process for the Danish data and thus prefer either the IV or IV2 estimators on the grounds they are consistent. On the other hand, for the US data, there is weak evidence of a semi-Markov process because the IV is larger than the IV2. In this case, the IV2 should be preferred.</p> <p>To demonstrate how the estimated effects of other explanatory variables can differ by the choice of estimator, the models in Table 5 build on those in Table 4 by controlling for mother's education. If the LDV is measured with error and correlated with other independent variables, estimates of the other independent variables can be conflated. We focus on the effect of mother's education for substantive reasons—it is well-known that maternal education influences student performance ([<reflink idref="bib15" id="ref110">15</reflink>]; [<reflink idref="bib31" id="ref111">31</reflink>])—and because it tends to be reliably measured and varies little over time (i.e., it is usually stable by the time children attend school; [<reflink idref="bib15" id="ref112">15</reflink>]).</p> <p>Table 5. Comparing Estimators Predicting Test Scores on Previous Test Scores Using Danish Register Data and NLSY Data, Controlling for Mother's Education.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="center" colspan="5">Danish Register Data</th><th align="center" colspan="4">NLSY</th></tr><tr><th align="center" /><th align="center">LDV</th><th align="center">AVG</th><th align="center">IV</th><th align="center">IV2</th><th align="center">LDV</th><th align="center">AVG</th><th align="center">IV</th><th align="center">IV2</th></tr></thead><tbody><tr><td>Constant</td><td align="char" char=".">−0.529***(0.013)</td><td align="char" char=".">−0.425***(0.013)</td><td align="char" char=".">−0.200***(0.003)</td><td align="char" char=".">−0.196***(0.015)</td><td align="char" char=".">−0.326***(0.059)</td><td align="char" char=".">−0.257***(0.057)</td><td align="char" char=".">−0.023(0.064)</td><td align="char" char=".">−0.097(0.064)</td></tr><tr><td>Lagged test scores</td><td align="char" char=".">0.671***(0.002)</td><td align="char" char=".">0.752***(0.002)</td><td align="char" char=".">0.869***(0.003)</td><td align="char" char=".">0.872***(0.004)</td><td align="char" char=".">0.674***(0.011)</td><td align="char" char=".">0.767***(0.012)</td><td align="char" char=".">0.914***(0.017)</td><td align="char" char=".">0.856***(0.021)</td></tr><tr><td>Mother's education (years)</td><td align="char" char=".">0.040***(0.001)</td><td align="char" char=".">0.032***(0.001)</td><td align="char" char=".">0.015***(0.001)</td><td align="char" char=".">0.015***(0.001)</td><td align="char" char=".">0.031***(0.006)</td><td align="char" char=".">0.025***(0.005)</td><td align="char" char=".">0.002(0.006)</td><td align="char" char=".">0.009(0.006)</td></tr><tr><td><italic>R</italic>2</td><td align="char" char=".">0.493</td><td align="char" char=".">0.528</td><td align="char" char=".">0.457</td><td align="char" char=".">0.456</td><td align="char" char=".">0.479</td><td align="char" char=".">0.519</td><td align="char" char=".">0.424</td><td align="char" char=".">0.448</td></tr><tr><td><italic>N</italic></td><td align="char" char=".">110,329</td><td align="char" char=".">110,329</td><td align="char" char=".">110,329</td><td align="char" char=".">110,329</td><td align="char" char=".">4,498</td><td align="char" char=".">4,498</td><td align="char" char=".">4,498</td><td align="char" char=".">4,498</td></tr></tbody></table> </ephtml> </p> <p>3 Note. ***<emph>p</emph> < 0.001.</p> <p>The results in Table 5 clearly demonstrate that the estimated effect of maternal education is highly sensitive to the estimator employed. For both the Danish and US data, the LDV estimator leaves substantial variance in educational performance for maternal education unexplained. The AVG estimator also leaves explanatory power to maternal education, albeit somewhat less than does the LDV estimator, indicating that measurement error in test scores inflates the effect of maternal education. For both the IV and IV2 estimators, the effect of maternal education completely disappears for the NLSY data and dramatically decreases for the Danish register data. Most striking, for the NLSY, the impact of mother's education is more than 10 times larger when using the AVG estimator compared to when using the IV estimator. These findings have substantive implications regarding the role of academic achievement as a mediator of the social origin and destination ([<reflink idref="bib35" id="ref113">35</reflink>]). When ability or achievement is measured with error, the effect of social origin—which is correlated with ability and achievement—could be overestimated, and the effects of ability and achievement are likely to be underestimated.</p> <hd id="AN0182244232-20">Example 2: the FDIV Estimator and Models with Fixed Effects</hd> <p>Using data from five waves of the STAR project, we now compare the FDIV estimate to the other four estimators when fixed effects are specified. Collected in Tennessee, USA in the years 1982 to 1985, the STAR data contain a long time series of repeated measures ([<reflink idref="bib12" id="ref114">12</reflink>]). We focus on math scores measured from grade 1 through grade 5 that form part of the Stanford Achievement Test (SAT). Current performance measured in 5th grade, lagged test scores measured in grades 4, 3, 2, and 1. All models specify fixed effects representing the students. In addition to using <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub><mo>−</mo><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msub></math> </ephtml> as an instrument, we also use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>y</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msub></math> </ephtml> , ([<reflink idref="bib7" id="ref115">7</reflink>]) to improve the efficiency of the FDIV estimator.</p> <p>Table 6 shows the results for all five estimators. Both the AVG and the IV estimators are substantially larger than the LDV estimator, indicating that the latter underestimates the association between current and lagged test scores. CME is apparent from the fact that the IV is significantly larger than the AVG estimator. Moreover, the IV and the IV2 estimates are not statistically significantly different from each other, indicating a Markov process for test scores across grades. Most importantly, the FDIV estimator is much smaller than the IV and IV2 estimators, suggesting that a large proportion of the transmission of test scores across grades is due to a latent ability variable. These results support our earlier assertion that the FDIV should be preferred if there are fixed effects and CME. The limitation, of course, is that the FDIV is only suitable for data that include measurements in at least five periods.</p> <p>Table 6. LDV, AVG, IV, IV2, and FDIV Estimates Predicting 5th Grade Math Test Scores for the STAR Data.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="center" /></colgroup><thead><tr><th align="center" colspan="6">STAR math test score data</th></tr><tr><th align="center" /><th align="center">LDV</th><th align="center">AVG</th><th align="center">IV</th><th align="center">IV2</th><th align="center">FDIV</th></tr></thead><tbody><tr><td>Constant term</td><td align="char" char=".">268.69***(12.08)</td><td align="char" char=".">201.43***(11.90)</td><td align="char" char=".">44.40(22.84)</td><td align="char" char=".">19.92(26.62)</td><td align="char" char=".">17.43(23.03)</td></tr><tr><td>Lagged test scores</td><td align="char" char=".">0.655***(0.017)</td><td align="char" char=".">0.797***(0.018)</td><td align="char" char=".">0.968***(0.032)</td><td align="char" char=".">1.003***(0.037)</td><td align="char" char=".">0.229*(0.095)</td></tr><tr><td><italic>R</italic>2</td><td align="char" char=".">0.48</td><td align="char" char=".">0.55</td><td align="char" char=".">0.37</td><td align="char" char=".">0.34</td><td align="char" char=".">.</td></tr><tr><td><italic>N</italic></td><td align="center">1,657</td><td align="center">1,657</td><td align="center">1,657</td><td align="center">1,657</td><td align="center">1,657</td></tr></tbody></table> </ephtml> </p> <p>4 Note. *<emph>p</emph> < 0.05; ***<emph>p</emph> < 0.001.</p> <hd id="AN0182244232-21">Conclusion</hd> <p>Our main goal was to provide a formal assessment of how four commonly used estimators derived from LDVs are affected by CME, both when fixed effects are present and not present in the model. We also propose a new IV estimator, which we call the FDIV estimator, that can accommodate fixed effects. Our analyses have demonstrated some important insights with practical implications on how to choose the appropriate estimator.</p> <p>First, if the one-period and two-period lagged IV estimators are equal, this lends strong support to the Markovian case. Under a Markov process, IV and IV2 estimators are consistent and the LDV estimators and AVG estimators based on the average previous lagged variables are downwardly inconsistent. Given its simplicity, the IV estimator is preferred in these situations.</p> <p>Second, if the IV and IV2 estimators differ, this implies that at least a semi-Markovian process. In this case, the IV and the AVG estimators are upwardly inconsistent; the LDV estimator is inconsistent in an unknown direction. Only the IV2 estimator, which uses an instrument based on values of the dependent variable from two periods back (two-period lagged), is consistent. Moreover, when the LDV estimate is different from both the AVG and the IV estimate, there is likely substantial measurement error in the data. In short, if the data follow a semi-Markov process, the IV2 estimator is generally preferred.</p> <p>Choosing the wrong estimator will not only result in inadequate descriptions of dynamics over time; it will also impact the estimates for other important predictors. Our empirical example demonstrated that using the wrong estimator exaggerated the effects of mother's education of children's outcomes. That is, the LDV and AVG models provided much larger estimates for the effects of mother's education than did the IV and IVs models. This further highlights the importance of testing for the most suitable LDV estimator.</p> <p>We also demonstrated the performance of our new FDIV estimator, which limits problems with measurement error and hence may bring the researcher closer to the causal mechanisms behind estimated effects of lagged variables in models with fixed effects. Applying the standard fixed effects estimator in the presence of measurement error bias the estimator toward zero. On the other hand, the FDIV estimator is consistent even in the presence of CME and fixed effects. The only limitation to the FDIV estimator is that it requires data from at least six repeated measures. Nonetheless, the FDIV could be applied in settings with a shorter time series—for example, to study intergenerational transmission of education or income—using IVs aside from lagged differences ([<reflink idref="bib7" id="ref116">7</reflink>]) such as information on siblings or cousins.</p> <p>Although most of our discussion pertained to linear relationships, we demonstrated that extension to nonlinear relationships is relatively straightforward. We focused on linear relationships as they allow for relatively simple explanations and because of their widespread presence in the social science literature. Like all statistical models, it is important that nonlinear relationships are dealt with by specifying the appropriate transformations, interaction effects, or other nonlinear specifications before applying any of the estimators that we recommend ([<reflink idref="bib2" id="ref117">2</reflink>]). Our findings are not robust to improper specification of the functional form.</p> <p>Our contribution could be extended in several ways. First, our tests could be applied to cross-sectional data as well as longitudinal data. For example, data from siblings could be used to create IVs when assessing various parent-child correlations, such as income or educational attainment. If information on cousins exists, we could also test whether siblings influence each other—a type of violation of the exclusion restriction—by instrumenting sibling information from information on cousins. Similarly, grandparents' income could be used as an IV assessing the intergenerational transmission of income. Caution is needed when taking this approach, however. For example, the IV estimate will be upwardly inconsistent if grandparents' income is used as an instrument and yet it has a true positive effect of its own.</p> <p>We found important differences in how the various estimators performed when only CME is present. Although we have explicitly addressed the impact of CME, this does not limit the applicability of our findings. Given that CME is all but certain for most social science data, our findings are relevant to any study using LDVs as predictors. The impact of even more malicious forms of measurement error is almost certainly greater. If severe, an unknown systematic measurement error would completely invalidate the estimates altogether, regardless of their sampling distributions.</p> <hd id="AN0182244232-22">Author's Note</hd> <p>All files used for simulations and all data from STAR and programs using STAR are available through the project "lagged depended variables" (osf.io/z89ug/) on center of open source (OSF). 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Introductory Econometrics: A Modern Approach. South-Western.</bibtext> </blist> </ref> <ref id="AN0182244232-24"> <title> Footnotes </title> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> The author(s) received no financial support for the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> Anders Holm https://orcid.org/0000-0002-6641-3591 Anders Hjorth-Trolle https://orcid.org/0000-0003-4132-1014</bibtext> </blist> </ref> <aug> <p>By Anders Holm; Anders Hjorth-Trolle and Robert Andersen</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Anders Holm is Professor in Sociology. His research interests are in social stratification and quantitative methodology. His publications have appeared in Annual Review of Sociology, Sociological Methodology and Journal of Marriage and Family.</p> <p>Robert Andersen is Professor of Business, Economics and Public Policy, and Professor of Strategy. His research interest are in inequality, poverty, political attitudes, and quantitative methods. His publications have appeared in American Sociological Review, American Journal of Political Science and Annual Review of Sociology.</p> </aug> <nolink nlid="nl1" bibid="bib13" firstref="ref6"></nolink> <nolink nlid="nl2" bibid="bib43" firstref="ref7"></nolink> <nolink nlid="nl3" bibid="bib23" firstref="ref8"></nolink> <nolink nlid="nl4" bibid="bib22" firstref="ref9"></nolink> <nolink nlid="nl5" bibid="bib37" firstref="ref10"></nolink> <nolink nlid="nl6" bibid="bib50" firstref="ref11"></nolink> <nolink nlid="nl7" bibid="bib52" firstref="ref12"></nolink> <nolink nlid="nl8" bibid="bib17" firstref="ref14"></nolink> <nolink nlid="nl9" bibid="bib19" firstref="ref15"></nolink> <nolink nlid="nl10" bibid="bib20" firstref="ref16"></nolink> <nolink nlid="nl11" bibid="bib46" firstref="ref17"></nolink> <nolink nlid="nl12" bibid="bib47" firstref="ref18"></nolink> <nolink nlid="nl13" bibid="bib48" firstref="ref19"></nolink> <nolink nlid="nl14" bibid="bib53" firstref="ref20"></nolink> <nolink nlid="nl15" bibid="bib54" firstref="ref21"></nolink> <nolink nlid="nl16" bibid="bib55" firstref="ref22"></nolink> <nolink nlid="nl17" bibid="bib16" firstref="ref23"></nolink> <nolink nlid="nl18" bibid="bib24" firstref="ref24"></nolink> <nolink nlid="nl19" bibid="bib11" firstref="ref29"></nolink> <nolink nlid="nl20" bibid="bib25" firstref="ref30"></nolink> <nolink nlid="nl21" bibid="bib39" firstref="ref31"></nolink> <nolink nlid="nl22" bibid="bib59" firstref="ref34"></nolink> <nolink nlid="nl23" bibid="bib28" firstref="ref36"></nolink> <nolink nlid="nl24" bibid="bib41" firstref="ref37"></nolink> <nolink nlid="nl25" bibid="bib57" firstref="ref38"></nolink> <nolink nlid="nl26" bibid="bib34" firstref="ref47"></nolink> <nolink nlid="nl27" bibid="bib18" firstref="ref51"></nolink> <nolink nlid="nl28" bibid="bib45" firstref="ref52"></nolink> <nolink nlid="nl29" bibid="bib14" firstref="ref53"></nolink> <nolink nlid="nl30" bibid="bib38" firstref="ref58"></nolink> <nolink nlid="nl31" bibid="bib32" firstref="ref63"></nolink> <nolink nlid="nl32" bibid="bib10" firstref="ref73"></nolink> <nolink nlid="nl33" bibid="bib40" firstref="ref74"></nolink> <nolink nlid="nl34" bibid="bib42" firstref="ref75"></nolink> <nolink nlid="nl35" bibid="bib51" firstref="ref76"></nolink> <nolink nlid="nl36" bibid="bib26" firstref="ref77"></nolink> <nolink nlid="nl37" bibid="bib29" firstref="ref78"></nolink> <nolink nlid="nl38" bibid="bib27" firstref="ref81"></nolink> <nolink nlid="nl39" bibid="bib49" firstref="ref84"></nolink> <nolink nlid="nl40" bibid="bib21" firstref="ref85"></nolink> <nolink nlid="nl41" bibid="bib30" firstref="ref89"></nolink> <nolink nlid="nl42" bibid="bib44" firstref="ref93"></nolink> <nolink nlid="nl43" bibid="bib56" firstref="ref94"></nolink> <nolink nlid="nl44" bibid="bib36" firstref="ref99"></nolink> <nolink nlid="nl45" bibid="bib33" firstref="ref108"></nolink> <nolink nlid="nl46" bibid="bib58" firstref="ref109"></nolink> <nolink nlid="nl47" bibid="bib15" firstref="ref110"></nolink> <nolink nlid="nl48" bibid="bib31" firstref="ref111"></nolink> <nolink nlid="nl49" bibid="bib35" firstref="ref113"></nolink> <nolink nlid="nl50" bibid="bib12" firstref="ref114"></nolink>
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  Label: Title
  Group: Ti
  Data: Lagged Dependent Variable Predictors, Classical Measurement Error, and Path Dependency: The Conditions under Which Various Estimators Are Appropriate
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Anders+Holm%22">Anders Holm</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-6641-3591">0000-0002-6641-3591</externalLink>)<br /><searchLink fieldCode="AR" term="%22Anders+Hjorth-Trolle%22">Anders Hjorth-Trolle</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-4132-1014">0000-0003-4132-1014</externalLink>)<br /><searchLink fieldCode="AR" term="%22Robert+Andersen%22">Robert Andersen</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2025 54(1):140-172.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 33
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Evaluative
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="DE" term="%22Scores%22">Scores</searchLink><br /><searchLink fieldCode="DE" term="%22Error+of+Measurement%22">Error of Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink><br /><searchLink fieldCode="DE" term="%22Markov+Processes%22">Markov Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Predictive+Measurement%22">Predictive Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Least+Squares+Statistics%22">Least Squares Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Reliability%22">Test Reliability</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Selection%22">Test Selection</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Construction%22">Test Construction</searchLink><br /><searchLink fieldCode="DE" term="%22Goodness+of+Fit%22">Goodness of Fit</searchLink><br /><searchLink fieldCode="DE" term="%22Robustness+%28Statistics%29%22">Robustness (Statistics)</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00491241231176845
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0049-1241<br />1552-8294
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Lagged dependent variables (LDVs) are often used as predictors in ordinary least squares (OLS) models in the social sciences. Although several estimators are commonly employed, little is known about their relative merits in the presence of classical measurement error and different longitudinal processes. We assess the performance of four commonly used estimators: (1) the standard OLS estimator, (2) an average of past measures (AVG), (3) an instrumental variable (IV) measured at one period previously (IV), and (4) an IV derived from information from more than one time before (IV2). We also propose a new estimator for fixed effects models--the first difference instrumental variable (FDIV) estimator. After exploring the consistency of these estimators, we demonstrate their performance using an empirical application predicting primary school test scores. Our results demonstrate that for a Markov process with classic measurement error (CME), IV and IV2 estimators are generally consistent; LDV and AVG estimators are not. For a semi-Markov process, only the IV2 estimator is consistent. On the other hand, if fixed effects are included in the model, only the FDIV estimator is consistent. We end with advice on how to select the appropriate estimator.
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  Data: As Provided
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  Label: Entry Date
  Group: Date
  Data: 2025
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  Label: Accession Number
  Group: ID
  Data: EJ1457724
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1457724
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      – Type: doi
        Value: 10.1177/00491241231176845
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 33
        StartPage: 140
    Subjects:
      – SubjectFull: Elementary Education
        Type: general
      – SubjectFull: Scores
        Type: general
      – SubjectFull: Error of Measurement
        Type: general
      – SubjectFull: Predictor Variables
        Type: general
      – SubjectFull: Markov Processes
        Type: general
      – SubjectFull: Predictive Measurement
        Type: general
      – SubjectFull: Least Squares Statistics
        Type: general
      – SubjectFull: Test Reliability
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      – SubjectFull: Test Selection
        Type: general
      – SubjectFull: Test Construction
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      – SubjectFull: Goodness of Fit
        Type: general
      – SubjectFull: Robustness (Statistics)
        Type: general
    Titles:
      – TitleFull: Lagged Dependent Variable Predictors, Classical Measurement Error, and Path Dependency: The Conditions under Which Various Estimators Are Appropriate
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            NameFull: Anders Holm
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            NameFull: Anders Hjorth-Trolle
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            NameFull: Robert Andersen
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              M: 02
              Type: published
              Y: 2025
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            – TitleFull: Sociological Methods & Research
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