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Post-Instrument Bias in Linear Models

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Title: Post-Instrument Bias in Linear Models
Language: English
Authors: Adam N. Glynn, Miguel R. Rueda (ORCID 0000-0001-7365-7722), Julian Schuessler (ORCID 0000-0002-8604-7699)
Source: Sociological Methods & Research. 2024 53(4):1829-1845.
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: 17
Publication Date: 2024
Document Type: Journal Articles
Information Analyses
Descriptors: Causal Models, Statistical Inference, Error of Measurement, Least Squares Statistics, Statistical Bias, Mathematical Models, Comparative Analysis, Mathematical Formulas
DOI: 10.1177/00491241231156965
ISSN: 0049-1241
1552-8294
Abstract: Post-instrument covariates are often included as controls in instrumental variable (IV) analyses to address a violation of the exclusion restriction. However, we show that such analyses are subject to biases unless strong assumptions hold. Using linear constant-effects models, we present asymptotic bias formulas for three estimators (with and without measurement error): IV with post-instrument covariates, IV without post-instrument covariates, and ordinary least squares. In large samples and when the model provides a reasonable approximation, these formulas sometimes allow the analyst to bracket the parameter of interest with two estimators and allow the analyst to choose the estimator with the least asymptotic bias. We illustrate these points with a discussion of the settler mortality IV used by Acemoglu, Johnson, and Robinson.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1444065
Database: ERIC
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  Value: <anid>AN0180278657;som01nov.24;2024Oct17.05:49;v2.2.500</anid> <title id="AN0180278657-1">Post-Instrument Bias in Linear Models </title> <p>Post-instrument covariates are often included as controls in instrumental variable (IV) analyses to address a violation of the exclusion restriction. However, we show that such analyses are subject to biases unless strong assumptions hold. Using linear constant-effects models, we present asymptotic bias formulas for three estimators (with and without measurement error): IV with post-instrument covariates, IV without post-instrument covariates, and ordinary least squares. In large samples and when the model provides a reasonable approximation, these formulas sometimes allow the analyst to bracket the parameter of interest with two estimators and allow the analyst to choose the estimator with the least asymptotic bias. We illustrate these points with a discussion of the settler mortality IV used by Acemoglu, Johnson, and Robinson.</p> <p>Keywords: causal inference; instrumental variables; measurement error; covariate adjustment; linear models</p> <p>Modern methodological research on instrumental variables (IVs) has paid much attention to the fundamental role of ignorability and exclusion restriction (no direct effect) assumptions, leading to a more careful justification of the use of variables as instruments ([<reflink idref="bib7" id="ref1">7</reflink>]). While an additional focus has been on understanding IV estimators in models with heterogeneous causal effects, leading some analysts to cast doubt on the use of covariate-adjusted two-stage least-squares (2SLS) estimators ([<reflink idref="bib25" id="ref2">25</reflink>]:316), covariate adjustment in linear IV regressions remains popular. Indeed, this seems to be done in order to justify the ignorability and exclusion restriction assumptions. However, it is often unclear whether such adjustment strategies lead to valid IV estimators.</p> <p>For example, a major point of disagreement in the recent discussion between [<reflink idref="bib29" id="ref3">29</reflink>], [<reflink idref="bib30" id="ref4">30</reflink>]) and [<reflink idref="bib10" id="ref5">10</reflink>]) about an application of IV centered on this issue. One position was that adjusting for "channeling variables"—those influenced by the instrument—can be detrimental ([<reflink idref="bib30" id="ref6">30</reflink>]:448), especially when the controls could be affected by the treatment. But Chou notes that while adjusting for them may indeed lead to bias, failure to do so may in some cases lead to violations of the exclusion restrictions, so that there would be "no fix" ([<reflink idref="bib10" id="ref7">10</reflink>]:439). A leading econometrics textbook, on the other hand, seems to advocate statistical control in situations where the control is affected by the instrument ([<reflink idref="bib37" id="ref8">37</reflink>]:94), and one can find further examples of this in empirical practice (e.g., [<reflink idref="bib33" id="ref9">33</reflink>]; [<reflink idref="bib12" id="ref10">12</reflink>]; [<reflink idref="bib35" id="ref11">35</reflink>]), including one of the most cited social science papers of the last 20 years ([<reflink idref="bib1" id="ref12">1</reflink>]).</p> <p>What are the conditions under which covariate adjustments can address violations of the exclusion restriction? If such conditions are not met, how do the bias of OLS and IV estimators compare with and without the covariate adjustment? This paper provides insights into these questions focusing on the linear model with constant-effects.</p> <p>We first present large sample bias formulas for the IV and OLS estimators with the post-instrument covariate and the IV without this variable. Controlling for post-instrument covariates potentially induces a variant of "endogenous selection bias" ([<reflink idref="bib15" id="ref13">15</reflink>]). In causal graph terms, this is due to a "collider" structure. Accordingly, there is indeed a trade-off between this bias and bias from violations to the exclusion restriction when no such control is undertaken.</p> <p>However, the formulas make clear that, in some applications, the effect of interest can be bounded by its estimates from models with and without the post-instrument covariate included as control. We illustrate this point with an example based on [<reflink idref="bib1" id="ref14">1</reflink>]) (AJR) study of the effect of protection against expropriation on gross domestic product (GDP).</p> <p>We also provide general results regarding the comparison of the biases of these estimators. We find that, without additional assumptions, invariance between IV estimates (with and without this post-instrument covariate) does not imply that the exclusion restriction holds or that the estimates are not biased. This highlights the known fact that the exclusion restriction is not testable in just-identified models. Moreover, we show that the IV will have bias at least as large as OLS when (a) the instrument has the same magnitude of effect on the causal variable and the post-instrument covariate and (b) the magnitude of the unmeasured confounding is the same for the causal variable and the post-instrument covariate.</p> <p>In addition, we provide asymptotic bias formulas for the IV estimator that include the effects of measurement error in the central independent (treatment) variable and the post-instrument covariate. These bias formulas show that classical measurement error in the post-instrument covariate does not necessarily lead to attenuation, and relatedly, measurement error makes it difficult to predict the sign of the bias of the coefficient of interest.</p> <p>We continue to re-analyze AJR to illustrate how these points can inform researchers in an applied setting. AJR's main conclusion is that better protection of property rights has a positive causal effect on economic development. Our model without measurement error confirms AJRs prediction that the IV estimated effect of the protection against expropriations index on GDP is understated due to unmeasured common causes of ethnic fractionalization (a post-instrument variable) and GDP. However, we show that if ethnic fractionalization is measured with error, the IV estimate will be biased upwards.</p> <p>We also illustrate how our results can be used to understand the sources of bias. In AJR, the authors offer as one potential explanation for the difference between their OLS and IV results attenuation bias in the OLS estimates caused by classical measurement error. Using AJRs IV estimate in their baseline specification as the true effect and the fact that the OLS estimate is smaller than that of the IV, our analysis implies that at least 35% of the variance in the expropriation variable must be due to measurement error to rationalize their findings.</p> <p>Finally, we comment on the implications of the paper's results. Our goal is to provide advice to practitioners facing different choices regarding which variables should be added as controls in their analysis and, in particular, we discuss reporting standards that should be upheld when addressing a violation of the exclusion restriction by adding a covariate is deemed worthwhile. To conclude, we highlight the fact that all of our results are in the context of linear constant-effects models and point readers to relevant literature that deals with heterogeneous effects (e.g., [<reflink idref="bib18" id="ref15">18</reflink>]; [<reflink idref="bib22" id="ref16">22</reflink>]; [<reflink idref="bib32" id="ref17">32</reflink>]). We use constant-effect models not because we believe that such parametric models are necessarily realistic, but because they allow for transparent and constructive derivations of biases ([<reflink idref="bib28" id="ref18">28</reflink>]; [<reflink idref="bib16" id="ref19">16</reflink>]; [<reflink idref="bib17" id="ref20">17</reflink>]) and they naturally lead to estimation routines based on OLS or 2SLS.</p> <p>Our paper belongs to the literature that studies the appropriate set of controls to achieve identification of causal effects. The consequences of adjusting for variables affected by the treatment has received the most attention ([<reflink idref="bib31" id="ref21">31</reflink>]; [<reflink idref="bib4" id="ref22">4</reflink>]; [<reflink idref="bib24" id="ref23">24</reflink>]). Although this work highlights the risks linked to conditioning on post-treatment variables, recent findings identify conditions under which such adjustments could be bias-reducing ([<reflink idref="bib17" id="ref24">17</reflink>]). Less attention has been given to the role of covariate adjustments in IVs analysis. [<reflink idref="bib34" id="ref25">34</reflink>]), [<reflink idref="bib9" id="ref26">9</reflink>]), and [<reflink idref="bib16" id="ref27">16</reflink>]) study bias induced by controlling for a post-treatment covariate and sample selection in IV settings, but do not study situations in which covariates are included as a way to address a violation of the exclusion restriction. [<reflink idref="bib13" id="ref28">13</reflink>]) explore situations in which an instrument affects more than one variable that has effects on an outcome and where researchers control for one of them. [<reflink idref="bib20" id="ref29">20</reflink>]) study several models where the instrument influences sample selection and there are variables that affect sample inclusion that could be adjusted for. We complement these analyses by providing large sample bias formulas for the IV and OLS estimators with the post-instrument covariate and analyze the adjustment when such variable is measured with error. Finally, our paper also contributes to work that explores violations of exclusion restrictions and sensitivity analyses to such violations (e.g. [<reflink idref="bib11" id="ref30">11</reflink>]; [<reflink idref="bib5" id="ref31">5</reflink>]).</p> <hd id="AN0180278657-2">Models for Bias Formulas and Comparison</hd> <p>We are interested in situations in which a researcher wants to estimate the effect of an explanatory variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> on a dependent variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> but worries about an unmeasured common cause of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> or classical measurement error in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> . Suppose we have a linear model <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></math> </ephtml> , where the latter expression might be due to an omitted variable or measurement error. To address the endogeneity problem, the researcher considers using a variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> as an instrument. For an IV regression to give consistent estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> , three conditions must hold: The model must be correct—specifically, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> cannot enter the equation for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> (exclusion restriction), the instrument must be related to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></math> </ephtml> ), and it must not be related to other determinants of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> ). Unfortunately, the researcher is concerned that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> violates the first condition, by having an effect on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> through <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , an observed variable available to the researcher. Our goal is to determine the consequences of including <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> as a control in the IV regression.</p> <p>As our running example, we consider [<reflink idref="bib1" id="ref32">1</reflink>]). In this paper, the authors want to estimate the effect of protection of property rights (measured with an index of protection against expropriation), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> , on GDP per capita, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> . An analysis based on OLS regressions will not give accurate estimates, since: It is difficult to account for all common causes of economic institutions and economic performance, and any index against expropriation is measured with error and cannot capture all institutional arrangements that lead to property right protection. To address these issues, AJR propose as an instrument of the index of expropriation the mortality rates of settlers in the colonization period. AJR argue that in places were Europeans faced higher mortality rates, they could not settle in mass and were more likely to impose extractive economic institutions—those that give property right protections only to the elites. The fact that economic and political elites have an advantaged position in society to maintain the rules that favor them explains why the settler mortality in the colonial period would impact current indexes of expropriation.</p> <p>AJR are aware of potential violations of the exclusion restriction and present as robustness tests results in which they control for variables that capture alternative links between the settler mortality rate and GDP per capita. One of them is ethnolinguistic fragmentation, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> . Because ethnolinguistic fragmentation has been linked to the provision of public goods, incidence of conflict, social capital, and directly to economic growth ([<reflink idref="bib3" id="ref33">3</reflink>]; [<reflink idref="bib23" id="ref34">23</reflink>]; [<reflink idref="bib6" id="ref35">6</reflink>]), a potential impact of settler mortality on fragmentation could bias the effect of interest. The question that arises is whether the authors should control for measures of ethnolinguistic fragmentation to alleviate a violation of the exclusion restriction.</p> <p>The fundamental problem with this approach is the possibility that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> is itself affected by the error term ([<reflink idref="bib8" id="ref36">8</reflink>]). Our interpretation of the situation studied by AJR is presented by the causal graph ([<reflink idref="bib27" id="ref37">27</reflink>]) in Figure 1.[<reflink idref="bib6" id="ref38">6</reflink>] The relationship between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> is represented by the dashed arrow. This is because, arguably, ethnolinguistic fragmentation is also influenced by cultural, geographical, and historical factors that are unobserved determinants of GDP per capita.</p> <p>Graph: Figure 1. Model's graphical representation</p> <p>Note that we do not have an arrow from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> . If we did, the graph would be cyclic, further complicating identification. The existing arrow from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> can capture how high ethnolinguistic fragmentation could lead to violations of property rights of minorities in cases where there is ethnic conflict.</p> <p>In order to assess the relative value of "fixing" the IV estimation by controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , we also consider as an alternative OLS estimates. In accordance with Figure 1, the new causal model for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> that serves as the basis for both the fixed IV and the OLS approach is then <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><msub><mi>β</mi><mi>x</mi></msub><mi>x</mi><mo>+</mo><msub><mi>β</mi><mi>w</mi></msub><mi>w</mi><mo>+</mo><mi>ϵ</mi><mo>.</mo></math> </ephtml></p> <p>Graph</p> <p>Note that while this structural model makes a strong constant effects assumption, we do not make any assumptions about the shape of the distributions of the involved variables (including <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math> </ephtml> ). Furthermore, our analysis does not make any assumptions on the functional form of the causal models for the other observed variables ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>w</mi></math> </ephtml> ). Indeed, the consistency of 2SLS estimation does not rely on a correctly specified first-stage model ([<reflink idref="bib36" id="ref39">36</reflink>]:Proposition 3). If one introduces stronger linearity assumptions for the other observed variables, one can also derive our results directly from the causal graph using the rules of path-tracing ([<reflink idref="bib28" id="ref40">28</reflink>]).</p> <p>Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>i</mi></msub></math> </ephtml> denote the standard deviation of variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> </ephtml> denote the correlation between variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mo fence="false" stretchy="false">{</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo fence="false" stretchy="false">}</mo></math> </ephtml> . Note that in Figure 1, correlations are induced by open paths between variables ([<reflink idref="bib14" id="ref41">14</reflink>]). The next result allows us to compare the probability limits of the estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> . All derivations are in the appendix.</p> <p>The probability limits of IV estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> with and without <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and the OLS estimates with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> for the model in Figure 1 are: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><mspace width=".1em" /><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext>no \,w</mtext></mrow></msubsup></mtd><mtd><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mi>w</mi></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><msub><mi>β</mi><mi>w</mi></msub></mrow><mrow><msub><mi>σ</mi><mi>x</mi></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></mrow></mfrac></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></mtd><mtd><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mo>−</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mi>ϵ</mi></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></mrow><mrow><msub><mi>σ</mi><mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>O</mi><mi>L</mi><mi>S</mi><mtext> \;w</mtext></mrow></msubsup></mtd><mtd><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mi>ϵ</mi></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>σ</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>.</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>According to this result, there are two conditions that make <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></math> </ephtml> consistent for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> : If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math> </ephtml> are uncorrelated ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub><mo>=</mo><mn>0</mn></math> </ephtml> ), or if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> are uncorrelated ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo>=</mo><mn>0</mn></math> </ephtml> ). While the first of them is hypothetically possible, the second is ruled out by assumption, as the researcher thinks the instrument is impacting the outcome through <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> . If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> is affected by the error ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub><mo>≠</mo><mn>0</mn></math> </ephtml> ), the result tells us it is possible to obtain worse estimates (compared to OLS) by running an IV regression even when the instrument is not related to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> through different channels than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> .</p> <p>This is a variant of endogenous selection bias ([<reflink idref="bib15" id="ref42">15</reflink>]): <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> is a collider on the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mspace width=".1em" /><mo>→</mo><mspace width=".1em" /><mi>w</mi><mspace width=".1em" /><mo>←</mo><mspace width=".1em" /><mi>ϵ</mi><mspace width=".1em" /><mo>→</mo><mspace width=".1em" /><mi>y</mi></math> </ephtml> . Controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> does block direct effects of the instrument on the outcome, but at the same time opens up this path, leading to an alternative non-causal correlation (see also [<reflink idref="bib20" id="ref43">20</reflink>]) and [<reflink idref="bib16" id="ref44">16</reflink>]) for similar cases where sample selection impacted by the instrument violates the exclusion restriction). Along these lines, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> in Figure 1 does not d-separate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> in the graph that removes the arrow from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> , which is one of the sufficient conditions for identification of the effect in linear models ([<reflink idref="bib8" id="ref45">8</reflink>]).</p> <p>However, note that with regards to the estimators that adjust for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , the only parameters that cannot be estimated are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>ϵ</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> . Among them, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>ϵ</mi></msub></math> </ephtml> appears in the numerator of the bias terms of the estimates that include <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> . So, if a researcher is interested in assessing relative bias of the OLS and IV estimators when including <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , she only needs to provide a statement about the relative values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> (as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>ϵ</mi></msub></math> </ephtml> cancels). Equipped with sample estimates of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub></math> </ephtml> , a researcher could determine combinations of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> that would make one estimator preferred to the other one. Clearly, substantive knowledge regarding the sources of endogeneity and, in particular, the sign of the unknown correlations could further limit the range of the values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> that are relevant for this comparison, as well as bounds for the ratio of the biases.</p> <p>Similarly, the expressions in Proposition 1 allow for a comparison of the biases between the IV adjusted and unadjusted models. This comparison only depends on the relative sizes of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>w</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>ϵ</mi></msub><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> after replacing the rest of parameters with their sample analogs.</p> <p>Also note that knowledge of the sign of the correlations in the expressions of Proposition 1 and the fact that they are in the interval <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> allow the researcher to determine asymptotic bounds of the effect of interest. To illustrate, we can return to the settler mortality example. Recall that in this case, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> captures GDP per capita, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> is colonial settler mortality, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> is an index of protection against expropriation, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> is a measure of ethnolinguistic fragmentation. If higher settler mortality induces a more hierarchical social structure separating natives, slaves, and Europeans, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo>></mo><mn>0</mn></math> </ephtml> . Because ethnolinguistic fragmentation has been linked with more conflict and less social capital <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub><mo><</mo><mn>0</mn></math> </ephtml> and, possibly, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>w</mi></msub><mo><</mo><mn>0</mn></math> </ephtml> . In addition, if the correlations between fragmentation and protection against expropriation ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub></math> </ephtml> ) and between fragmentation and settler mortality ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub></math> </ephtml> ) are weaker than the correlation of settler mortality and protection against expropriation ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml> )—as they are in AJR data—, the IV coefficient on protection against expropriation in the model that controls for fragmentation would understate the effect. That is, the IV estimate with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> would (asymptotically) bound the true effect from below, while the IV estimate without <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> would bound it from above.</p> <p>More generally, the following corollary indicates when the adjusted IV estimates with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and without it can bound the effect of interest.</p> <p>If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>g</mi><mi>n</mi><mo stretchy="false">(</mo><mrow><mfrac><msub><mi>β</mi><mi>w</mi></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></mfrac></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mi>g</mi><mi>n</mi><mo stretchy="false">(</mo><mrow><mfrac><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub><mrow><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub></mrow></mfrac></mrow><mo stretchy="false">)</mo></math> </ephtml> , then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> is asymptotically bounded by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext>no \,w</mtext></mrow></msubsup></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></math> </ephtml> .</p> <p>In our discussion of the AJR example, we have assumed that all variables are measured <emph>without error</emph> to illustrate how a researcher could use Proposition 1. In the next section, we examine what happens if the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> are measured with error, and return to the settler mortality example.</p> <p>Proposition 1 also highlights that finding an equality of adjusted and unadjusted estimates is not necessarily informative. It turns out that when the scaled effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> equals the negative of the scaled confounding of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> , IV with and without <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> produces similar estimates. Therefore, seeing that the IV estimate does not change after controlling for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> does not mean estimates are consistent nor that the concerns about violating the exclusion restriction are unimportant. The result is inline with the fact that it is not possible to test the exclusion restriction in just-identified models without further assumptions.</p> <p>If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msub><mi>σ</mi><mi>w</mi></msub><msub><mi>β</mi><mi>w</mi></msub></mrow><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></mfrac></mrow><mo>=</mo><mo>−</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mi>ϵ</mi></msub><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>w</mi></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml> , then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext>no \,w</mtext></mrow></msubsup><mo>=</mo><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></math> </ephtml> .</p> <p>We can also consider the simple case where we remove any arrows between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , confounding is equally bad for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> is an equally strong instrument for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> . Under those conditions, we can prove that OLS has less large-sample bias than IV.</p> <p>If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><mo>=</mo><mo fence="false" stretchy="false">|</mo><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub><mo fence="false" stretchy="false">|</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><mo>=</mo><mo fence="false" stretchy="false">|</mo><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub><mo fence="false" stretchy="false">|</mo></math> </ephtml> then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mi>B</mi><mi>i</mi><mi>a</mi><mi>s</mi><mo stretchy="false">(</mo><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>O</mi><mi>L</mi><mi>S</mi></mrow></msubsup><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><mo>≤</mo><mo fence="false" stretchy="false">|</mo><mi>B</mi><mi>i</mi><mi>a</mi><mi>s</mi><mo stretchy="false">(</mo><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi></mrow></msubsup><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo></math> </ephtml> .</p> <p>This result gives a way to compare relative biases in large samples when assuming the absence of any causal relationships between the treatment and the post-instrument covariate is plausible. The result implies that even in linear constant-effects models, in order for IV with a post-instrument covariate to be preferred to OLS, one would need to estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>x</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mi>w</mi></mrow></msub></math> </ephtml> , and then argue that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> was sufficiently small vis-a-vis <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>x</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> .</p> <hd id="AN0180278657-3">Measurement Error</hd> <p>One reason why researchers employ IVs is to address concerns regarding measurement error in their treatment variable and the biases that this induces in their estimates. We now explore situations in which measurement error affects the explanatory variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> while maintaining all other relationships between the variables as described by the model in Figure 1.</p> <p>Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> denote the observed covariates with <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>u</mi><mi>x</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow><mo>=</mo><mi>w</mi><mo>+</mo><msub><mi>u</mi><mi>w</mi></msub></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>w</mi></msub></math> </ephtml> are zero-mean measurement errors with variances <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>x</mi></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>u</mi><mi>w</mi></msub></mrow><mn>2</mn></msubsup></math> </ephtml> . We focus on the case of classical measurement error, in which the measurement errors are not related to the true value of the explanatory variables. We further assume that the measurement error terms are not correlated with the error term in the population model, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math> </ephtml> , the instrument, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> </ephtml> , nor each other.</p> <p>It is well known that IVs regression with a valid instrument can give us consistent estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> even when we only observe <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> . We are now interested in studying the question of how the IV estimates perform when there is a violation of the exclusion restriction and we control for an additional variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> that captures the alternative link (other than through <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> </ephtml> ) between the instrument and the outcome. The following proposition gives expressions characterizing large sample bias of such an approach.</p> <p>If <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>x</mi><msub><mi>u</mi><mi>x</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>x</mi><msub><mi>u</mi><mi>w</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>w</mi><msub><mi>u</mi><mi>x</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>w</mi><msub><mi>u</mi><mi>w</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mi>u</mi><mi>x</mi></msub><msub><mi>u</mi><mi>w</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mi>u</mi><mi>x</mi></msub><mi>ϵ</mi><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mi>u</mi><mi>w</mi></msub><mi>ϵ</mi><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>z</mi><msub><mi>u</mi><mi>w</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>z</mi><msub><mi>u</mi><mi>x</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></math> </ephtml> , the probability limits of IV estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> with and without <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and the OLS estimates with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> are: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><mspace width=".1em" /><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext>no \;w</mtext></mrow></msubsup></mtd><mtd><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mo>+</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>β</mi><mi>w</mi></msub></mrow><mrow><msub><mi>σ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub></mrow></mfrac></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>I</mi><mi>V</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></mtd><mtd><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mo>+</mo><mrow><mfrac><mrow><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>w</mi></msub></mrow><mn>2</mn></msubsup><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>β</mi><mi>w</mi></msub></mrow><mrow><msub><mi>σ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>σ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>−</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mi>ϵ</mi></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub></mrow><mrow><msub><mi>σ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>.</mo></mtd></mtr><mtr><mtd><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mover><mi>β</mi><mo>^</mo></mover></mrow><mi>x</mi><mrow><mi>O</mi><mi>L</mi><mi>S</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></mtd><mtd><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mrow><mfrac><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>x</mi></msub></mrow><mn>2</mn></msubsup><mrow><msubsup><mi>σ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mfrac><mrow><msub><mi>σ</mi><mi>ϵ</mi></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>σ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mrow><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>w</mi></msub></mrow><mn>2</mn></msubsup><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>β</mi><mi>w</mi></msub></mrow><mrow><msub><mi>σ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>σ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>,</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>It is known that OLS estimates will not be necessarily biased towards zero when other mismeasured variables are adjusted for (e.g, [<reflink idref="bib19" id="ref46">19</reflink>]). More interestingly, we also see that including <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> in the IV regression adds a term in the bias expression that is proportional to the variance of the measurement error of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> and to its effect on the outcome. This makes clear that including <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> will not necessarily bias the IV estimate toward zero either. Note also that even if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϵ</mi></math> </ephtml> are uncorrelated, adjusting for this noisy covariate would not fully intercept the exclusion violation. This leads to an inconsistent estimator where the asymptotic bias will depend on the relationships between the instrument and the post-instrument variable, instrument and treatment, and treatment and post-instrument variable. In sum, the proposition highlights that a researcher trying to address a violation of the exclusion restriction by adding a regressor should not only be concerned about potential unmeasured common causes of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> , but also about measurement error in the added regressor.</p> <p>The formula for the OLS asymptotic bias also makes clear that unlike in the case without measurement error, a comparison of the relative biases with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> </ephtml> included between estimators is not straightforward. This is because the ratio of such biases depends on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>w</mi></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><msub><mi>u</mi><mi>w</mi></msub></mrow></msub></math> </ephtml> , in addition to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>ϵ</mi></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub></math> </ephtml> . This further highlights the difficulties of attempting to address a violation of the exclusion restriction by adding a post-instrument covariate when measurement error is an additional concern.</p> <hd id="AN0180278657-4">Application</hd> <p>To illustrate how these results can be applied, we return to the AJR's example. We now allow for the protection against expropriation and ethnolinguistic fragmentation variables to have an additive error as described in the previous section and label them <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> , respectively. This is consistent with AJR's own motivation for using an IV strategy and the difficulties of measuring ethnic diversity and fragmentation ([<reflink idref="bib26" id="ref47">26</reflink>]). For ease of exposition, we reproduce AJR's estimates from their Tables 4 and 6 and Online Appendix A in Table 1, focusing on the regressions which include as a control the measure of ethnolinguistic fragmentation.[<reflink idref="bib7" id="ref48">7</reflink>]</p> <p>Table 1. Point Estimates of the Effects of Institutions on Economic Performance.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="left" /><th align="left">OLS</th><th align="left" colspan="2">Instrumental variable (IV)</th></tr><tr><th align="left">Explanatory variables</th><th align="left">(1)</th><th align="left">(2)</th><th align="left">(3)</th></tr></thead><tbody><tr><td>Protection against expropriation (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></math></p>)</td><td>0.46</td><td>0.94</td><td>0.74</td></tr><tr><td>Ethnolinguistic fragmentation (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math></p>)</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>1.3</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>1.02</td></tr><tr><td colspan="4"><italic>First stage results</italic></td></tr><tr><td>Log European settler mortality (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">z</mi></math></p>)</td><td /><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.61</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.64</td></tr></tbody></table> </ephtml> </p> <p>1 Outcome, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> </ephtml> , is GDP per capita. Column (<reflink idref="bib1" id="ref49">1</reflink>) corresponds to Model (<reflink idref="bib7" id="ref50">7</reflink>) Table 6 Panel C in AJR, column (<reflink idref="bib2" id="ref51">2</reflink>) corresponds to Model (<reflink idref="bib1" id="ref52">1</reflink>) Table 4 Panels A and B in AJR, and column (<reflink idref="bib3" id="ref53">3</reflink>) corresponds to Model (<reflink idref="bib7" id="ref54">7</reflink>) from Table 6 Panels A and B in AJR.</p> <p>Using their data ([<reflink idref="bib2" id="ref55">2</reflink>]), the sample estimates of the correlations are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>=</mo><mn>0.49</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>=</mo><mo>−</mo><mn>0.52</mn></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>=</mo><mo>−</mo><mn>0.22</mn></math> </ephtml> . This implies that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><msub><mi>ρ</mi><mrow><mi>z</mi><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>=</mo><mo>−</mo><mn>0.41</mn></math> </ephtml> .[<reflink idref="bib8" id="ref56">8</reflink>] Therefore, if we assume based on the literature findings (and as in AJR) that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub><mo><</mo><mn>0</mn></math> </ephtml> and that there is no measurement error in the added regressor ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><msub><mi>u</mi><mi>w</mi></msub></mrow></msub><mo>=</mo><mn>0</mn></math> </ephtml> ), Proposition 2 concurs with the AJR analysis that the IV analysis including ethnolinguistic fragmentation has negative bias.[<reflink idref="bib9" id="ref57">9</reflink>]</p> <p>However, with measurement error in the post-instrument covariate ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><msub><mi>u</mi><mi>w</mi></msub></mrow></msub><mo>≠</mo><mn>0</mn></math> </ephtml> ), Proposition 2 indicates that when ethnolinguistic fragmentation has a negative effect on GDP (as suggested by the literature), the second term of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi></math> </ephtml> of the IV with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></math> </ephtml> will be positive. Hence it is not easy to determine the sign of the bias as it depends on the fragmentation effect ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>w</mi></msub></math> </ephtml> ), the confounding ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mi>w</mi><mi>ϵ</mi></mrow></msub></math> </ephtml> ), and the variance of the error ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>w</mi></msub></mrow><mn>2</mn></msubsup></math> </ephtml> ).</p> <p>Our results allow us to explain possible sources of differences between OLS and IV estimates. AJR's OLS estimate is smaller than the IV estimate that conditions on ethnic fractionalization and also than that of the model that does not. AJR explain this difference as a result of attenuation bias caused by measurement error in the institutions variable with OLS. Using Proposition 2, we examine under what conditions measurement error is consistent with those differences. An inspection of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi></math> </ephtml> for OLS in Proposition 2 together with our computed correlations indicates that the third term in that expression is positive. Moreover, the second term in the expression of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi></math> </ephtml> for OLS is likely positive as well, since <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub></math> </ephtml> will be positive and possibly larger than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mover><mi>x</mi><mo stretchy="false">~</mo></mover></mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow></mrow></msub><mo>⋅</mo><msub><mi>ρ</mi><mrow><mrow><mover><mi>w</mi><mo stretchy="false">~</mo></mover></mrow><mi>ϵ</mi></mrow></msub></math> </ephtml> . What would be the minimum variance in the measurement error of the institutional variable that is consistent with the observed differences between the OLS and IV estimates?</p> <p>To answer that question, we set the two last (positive) terms in the expression of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi></math> </ephtml> for OLS to zero. We then use AJR's IV estimate in the model that includes ethnolinguistic fragmentation in place of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> , 0.74, and solve for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>x</mi></msub></mrow><mn>2</mn></msubsup></math> </ephtml> to obtain <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.77</mn></math> </ephtml> .[<reflink idref="bib10" id="ref58">10</reflink>] This means that measurement error must account for at least 35% ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mrow><mfrac><mn>0.77</mn><msup><mn>1.47</mn><mn>2</mn></msup></mfrac></mrow></math> </ephtml> ) of the variation in the observed institutions variable to rationalize the difference between IV and the OLS estimates. Clearly, we could have chosen a different estimate of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> to compute that minimum measurement error variance. If we use instead AJR's baseline estimate of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.94</mn></math> </ephtml> , the measurement error in the institutions variable must make up nearly 50% ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mrow><mfrac><mn>1.05</mn><msup><mn>1.47</mn><mn>2</mn></msup></mfrac></mrow></math> </ephtml> ) of the total variance.</p> <p>It is important to note that the previous analysis relies on assumptions highlighted in Proposition 2 that are in line with classical measurement error. It also relies on the assumptions used in the linear constant-effect models of AJR and this paper. We do believe, however, that these assumptions provide a first step to analyze this application in a way that is consistent with reasonable scenarios for the situation of interest.</p> <hd id="AN0180278657-5">Discussion and Conclusion</hd> <p>Instrumental variables regression methods allow researchers to address estimation challenges like unobserved heterogeneity and classical measurement error. Whether the method delivers accurate results depends on the tenability of its assumptions. Here, we have studied one way in which researchers have dealt with potential violations of one of them, the exclusion restriction. We find that although it is possible for researchers to fix violations of the exclusion restriction by adding controls, doing so requires a number of strong assumptions. When these do not hold, the IV estimates can be worse than what the researcher would obtain running an OLS regression.</p> <p>Our findings have a number of implications for practice. First, even when the linear constant-effects model is used, the inclusion of a post-instrument covariate in 2SLS estimation requires strong theoretical assumptions, and therefore, researchers may want to use alternative approaches. One of them is to focus on the estimated effect of the instrument (the "reduced form effect") which will not be invalidated by an exclusion restriction violation. In many cases, when the IV analysis is questionable, this effect will have some theoretical or policy relevance that could be emphasized. Another approach is to conduct a sensitivity analysis with respect to the exclusion restriction (e.g. see [<reflink idref="bib11" id="ref59">11</reflink>])), as the results might be robust to these violations.</p> <p>Second, if conditioning on post-instrument covariates is conducted, the corresponding OLS analysis should be reported and the measurement error and unmeasured common causes in both approaches should be discussed in concert. The formulas presented here should help with this discussion and depending on the application of interest, might give bounds for the effect of interest.</p> <p>Finally, we note that the entirety of this paper relies on the linear constant-effects model. If the constant-effects model is not a reasonable approximation, then there are many potential parameters of interest from an IVs analysis ([<reflink idref="bib21" id="ref60">21</reflink>]), and assessment of exclusion restriction violations becomes more complicated. [<reflink idref="bib18" id="ref61">18</reflink>]) and [<reflink idref="bib22" id="ref62">22</reflink>]) provide some strategies in this context and also surveys of the literature. Finally, in [<reflink idref="bib32" id="ref63">32</reflink>]) we provide a nonparametric analysis of the problem and develop a semi-parametric sensitivity analysis.</p> <hd id="AN0180278657-6">Supplemental Material</hd> <p>Graph: Supplemental material, sj-pdf-1-smr-10.1177_00491241231156965 for Post-Instrument Bias in Linear Models by Adam N. Glynn, Miguel R. Rueda, and Julian Schuessler in Sociological Methods & Research</p> <ref id="AN0180278657-7"> <title> References </title> <blist> <bibl id="bib1" idref="ref12" type="bt">1</bibl> <bibtext> Acemoglu Daron, Johnson Simon, Robinson James A. 2001a. " The Colonial Origins of Comparative Development: An Empirical Investigation." The American Economic Review. 91(5):1369‐1401.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref51" type="bt">2</bibl> <bibtext> Acemoglu Daron, Johnson Simon, Robinson James A. 2001b. 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Cambridge, MA: The MIT Press.</bibtext> </blist> </ref> <ref id="AN0180278657-8"> <title> Footnotes </title> <blist> <bibtext> Data and code to replicate all empirical results found in this paper can be found under <ulink href="http://dx.doi.org/10.17605/OSF.IO/Z38RX">http://dx.doi.org/10.17605/OSF.IO/Z38RX</ulink>.</bibtext> </blist> <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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Danish National Research Foundation (DNRF144).</bibtext> </blist> <blist> <bibtext> Miguel R. Rueda https://orcid.org/0000-0001-7365-7722 Julian Schuessler https://orcid.org/0000-0002-8604-7699</bibtext> </blist> <blist> <bibtext> The supplemental material for this article is available online.</bibtext> </blist> <blist> <bibtext> AJR do not provide a model's graphical representation.</bibtext> </blist> <blist> <bibtext> Unlike many other papers that add post-instrument controls, AJR's report OLS estimates as well as IV results for all specifications, discuss whether the added regressors can be correlated to the error term in the main model, and make available their data and code for replication.</bibtext> </blist> <blist> <bibtext> The computed values of the standard deviations are: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mrow><mi>x</mi><mo stretchy="false">~</mo></mrow></mrow></msub><mo>=</mo>1.47</math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mrow><mi>w</mi><mo stretchy="false">~</mo></mrow></mrow></msub><mo>=</mo>0.32</math> </ephtml> .</bibtext> </blist> <blist> <bibtext> Their analysis, included in Appendix A page 1396, differs from ours as it treats the protection against expropriation as exogenous while the added covariate (ethnolinguistic fragmentation) is correlated with the error term from the second stage.</bibtext> </blist> <blist> <bibtext> That is, we solve for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>x</mi></msub></mrow>2</msubsup></math> </ephtml> in expression <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mi>β</mi><mo>^</mo></mrow><mi>x</mi><mrow><mi>O</mi><mi>L</mi><mi>S</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup><mo>=</mo><msub><mi>β</mi><mi>x</mi></msub><mrow><mo>(</mo>1<mo>−</mo><mrow><mfrac><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>x</mi></msub></mrow>2</msubsup><mrow><msubsup><mi>σ</mi><mrow><mrow><mi>x</mi><mo stretchy="false">~</mo></mrow></mrow>2</msubsup><mo stretchy="false">(</mo>1<mo>−</mo><msubsup><mi>ρ</mi><mrow><mrow><mi>x</mi><mo stretchy="false">~</mo></mrow><mrow><mi>w</mi><mo stretchy="false">~</mo></mrow></mrow>2</msubsup><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></math> </ephtml></bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext>where we assume <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi><mi>i</mi><mi>m</mi><mspace width=".1em" /><msubsup><mrow><mi>β</mi><mo>^</mo></mrow><mi>x</mi><mrow><mi>O</mi><mi>L</mi><mi>S</mi><mspace width="0.25em" /><mtext> w</mtext></mrow></msubsup></math> </ephtml> is 0.46 (their OLS estimate that adjust for fractionalization), replace <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mi>x</mi></msub></math> </ephtml> by 0.74, and use the estimated correlation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mrow><mrow><mi>x</mi><mo stretchy="false">~</mo></mrow><mrow><mi>w</mi><mo stretchy="false">~</mo></mrow></mrow></msub><mo>=</mo><mo>−</mo>0.22</math> </ephtml> and variance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow><mrow><mi>x</mi><mo stretchy="false">~</mo></mrow></mrow></msub><mo>=</mo>1.47</math> </ephtml> .</bibtext> </blist> </ref> <aug> <p>By Adam N. Glynn; Miguel R. Rueda and Julian Schuessler</p> <p>Reported by Author; Author; Author</p> <p></p> <p>Adam N. Glynn is Professor of Political Science and Quantitative Theory and Methods at Emory University. His research focuses on causal inference and measurement in the social sciences.</p> <p>Miguel R. Rueda is an Associate Professor in the Department of Political Science at Emory University. His research interests include electoral manipulation, civil conflict, electoral systems, and political methodology.</p> <p>Julian Schuessler is a Post-Doc at the Centre for the Experimental-Philosophical Study of Discrimination (CEPDISC), Department of Political Science, Aarhus University, Denmark. His research focuses on using directed acyclic graphs for causal and statistical inference as well as empirical analyses of discrimination.</p> </aug> <nolink nlid="nl1" bibid="bib25" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib29" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib30" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib10" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib37" firstref="ref8"></nolink> <nolink nlid="nl6" bibid="bib33" firstref="ref9"></nolink> <nolink nlid="nl7" bibid="bib12" firstref="ref10"></nolink> <nolink nlid="nl8" bibid="bib35" firstref="ref11"></nolink> <nolink nlid="nl9" bibid="bib15" firstref="ref13"></nolink> <nolink nlid="nl10" bibid="bib18" firstref="ref15"></nolink> <nolink nlid="nl11" bibid="bib22" firstref="ref16"></nolink> <nolink nlid="nl12" bibid="bib32" firstref="ref17"></nolink> <nolink nlid="nl13" bibid="bib28" firstref="ref18"></nolink> <nolink nlid="nl14" bibid="bib16" firstref="ref19"></nolink> <nolink nlid="nl15" bibid="bib17" firstref="ref20"></nolink> <nolink nlid="nl16" bibid="bib31" firstref="ref21"></nolink> <nolink nlid="nl17" bibid="bib24" firstref="ref23"></nolink> <nolink nlid="nl18" bibid="bib34" firstref="ref25"></nolink> <nolink nlid="nl19" bibid="bib13" firstref="ref28"></nolink> <nolink nlid="nl20" bibid="bib20" firstref="ref29"></nolink> <nolink nlid="nl21" bibid="bib11" firstref="ref30"></nolink> <nolink nlid="nl22" bibid="bib23" firstref="ref34"></nolink> <nolink nlid="nl23" bibid="bib27" firstref="ref37"></nolink> <nolink nlid="nl24" bibid="bib36" firstref="ref39"></nolink> <nolink nlid="nl25" bibid="bib14" firstref="ref41"></nolink> <nolink nlid="nl26" bibid="bib19" firstref="ref46"></nolink> <nolink nlid="nl27" bibid="bib26" firstref="ref47"></nolink> <nolink nlid="nl28" bibid="bib21" firstref="ref60"></nolink>
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  Label: Title
  Group: Ti
  Data: Post-Instrument Bias in Linear Models
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Adam+N%2E+Glynn%22">Adam N. Glynn</searchLink><br /><searchLink fieldCode="AR" term="%22Miguel+R%2E+Rueda%22">Miguel R. Rueda</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-7365-7722">0000-0001-7365-7722</externalLink>)<br /><searchLink fieldCode="AR" term="%22Julian+Schuessler%22">Julian Schuessler</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-8604-7699">0000-0002-8604-7699</externalLink>)
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  Label: Source
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2024 53(4):1829-1845.
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  Label: Availability
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  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
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  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 17
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2024
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Information Analyses
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Causal+Models%22">Causal Models</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Inference%22">Statistical Inference</searchLink><br /><searchLink fieldCode="DE" term="%22Error+of+Measurement%22">Error of Measurement</searchLink><br /><searchLink fieldCode="DE" term="%22Least+Squares+Statistics%22">Least Squares Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Bias%22">Statistical Bias</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Models%22">Mathematical Models</searchLink><br /><searchLink fieldCode="DE" term="%22Comparative+Analysis%22">Comparative Analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Formulas%22">Mathematical Formulas</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/00491241231156965
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0049-1241<br />1552-8294
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Post-instrument covariates are often included as controls in instrumental variable (IV) analyses to address a violation of the exclusion restriction. However, we show that such analyses are subject to biases unless strong assumptions hold. Using linear constant-effects models, we present asymptotic bias formulas for three estimators (with and without measurement error): IV with post-instrument covariates, IV without post-instrument covariates, and ordinary least squares. In large samples and when the model provides a reasonable approximation, these formulas sometimes allow the analyst to bracket the parameter of interest with two estimators and allow the analyst to choose the estimator with the least asymptotic bias. We illustrate these points with a discussion of the settler mortality IV used by Acemoglu, Johnson, and Robinson.
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  Data: As Provided
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  Data: 2024
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  Data: EJ1444065
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1444065
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        Value: 10.1177/00491241231156965
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 17
        StartPage: 1829
    Subjects:
      – SubjectFull: Causal Models
        Type: general
      – SubjectFull: Statistical Inference
        Type: general
      – SubjectFull: Error of Measurement
        Type: general
      – SubjectFull: Least Squares Statistics
        Type: general
      – SubjectFull: Statistical Bias
        Type: general
      – SubjectFull: Mathematical Models
        Type: general
      – SubjectFull: Comparative Analysis
        Type: general
      – SubjectFull: Mathematical Formulas
        Type: general
    Titles:
      – TitleFull: Post-Instrument Bias in Linear Models
        Type: main
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      – PersonEntity:
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            NameFull: Adam N. Glynn
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            NameFull: Miguel R. Rueda
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            NameFull: Julian Schuessler
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          Dates:
            – D: 01
              M: 11
              Type: published
              Y: 2024
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              Value: 53
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          Titles:
            – TitleFull: Sociological Methods & Research
              Type: main
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