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Instrumental variables estimation

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In <a href="Statistics" title="Statistics">statistics</a>, <a href="Econometrics" title="Econometrics">econometrics</a>, <a href="Epidemiology" title="Epidemiology">epidemiology</a> and related disciplines, the method of instrumental variables (IV) is used to estimate <a href="Causal_inference" title="Causal inference">causal relationships</a> when <a href="Controlled_experiment" class="mw-redirect" title="Controlled experiment">controlled experiments</a> are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment.<a href="#cite_note-Imbens:00-1">[1]</a> Intuitively, IVs are used when an explanatory (also known as independent or predictor) variable of interest is correlated with the error term (endogenous), in which case <a href="Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> and <a href="ANOVA" class="mw-redirect" title="ANOVA">ANOVA</a> give <a href="Bias_(statistics)" title="Bias (statistics)">biased</a> results. A valid instrument induces changes in the explanatory variable (is correlated with the endogenous variable) but has no independent effect on the dependent variable and is not correlated with the error term, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.

Instrumental variable methods allow for <a href="Consistent_estimator" title="Consistent estimator">consistent</a> estimation when the <a href="Dependent_and_independent_variables" title="Dependent and independent variables">explanatory variables</a> (covariates) are <a href="Correlation" title="Correlation">correlated</a> with the <a href="Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">error terms</a> in a <a href="Regression_analysis" title="Regression analysis">regression</a> model. Such correlation may occur when:

<ol><li>changes in the dependent variable change the value of at least one of the <a href="Covariate" class="mw-redirect" title="Covariate">covariates</a> ("reverse" causation),</li>

<li>there are <a href="Omitted-variable_bias" title="Omitted-variable bias">omitted variables</a> that affect both the dependent and explanatory variables, or</li>

<li>the <a href="Errors-in-variables_models" class="mw-redirect" title="Errors-in-variables models">covariates are subject to measurement error</a>.</li></ol>

Explanatory variables that suffer from one or more of these issues in the context of a regression are sometimes referred to as <a href="Endogeneity_(econometrics)" title="Endogeneity (econometrics)">endogenous</a>. In this situation, <a href="Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a> produces biased and inconsistent estimates.<a href="#cite_note-Bullock:00-2">[2]</a> However, if an instrument is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the explanatory equation but is correlated with the <a href="Endogeneity_(econometrics)" title="Endogeneity (econometrics)">endogenous</a> explanatory variables, conditionally on the value of other covariates.

In linear models, there are two main requirements for using IVs:

<ul><li>The instrument must be correlated with the endogenous explanatory variables, conditionally on the other covariates. If this correlation is strong, then the instrument is said to have a strong first stage. A weak correlation may provide misleading inferences about parameter estimates and standard errors.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a></li>

<li>The instrument cannot be correlated with the error term in the explanatory equation, conditionally on the other covariates. In other words, the instrument cannot suffer from the same problem as the original predicting variable. If this condition is met, then the instrument is said to satisfy the exclusion restriction.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

Informally, in attempting to estimate the causal effect of some variable X ("covariate" or "explanatory variable") on another Y ("dependent variable"), an instrument is a third variable Z which affects Y only through its effect on X.

For example, suppose a researcher wishes to estimate the causal effect of smoking (X) on general health (Y).<a href="#cite_note-Leigh:00-5">[5]</a> Correlation between smoking and health does not imply that smoking causes poor health because other variables, such as depression, may affect both health and smoking, or because health may affect smoking. It is not possible to conduct controlled experiments on smoking status in the general population. The researcher may attempt to estimate the causal effect of smoking on health from observational data by using the tax rate for tobacco products (Z) as an instrument for smoking. The tax rate for tobacco products is a reasonable choice for an instrument because the researcher assumes that it can only be correlated with health through its effect on smoking. If the researcher then finds tobacco taxes and state of health to be correlated, this may be viewed as evidence that smoking causes changes in health.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The first use of an instrument variable occurred in a 1928 book by <a href="Philip_Green_Wright" title="Philip Green Wright">Philip G. Wright</a>, best known for his excellent description of the production, transport and sale of vegetable and animal oils in the early 1900s in the United States.<a href="#cite_note-6">[6]</a><a href="#cite_note-stock:trebbi03-7">[7]</a> In 1945, <a href="Olav_Reiers%C3%B8l" title="Olav Reiersøl">Olav Reiersøl</a> applied the same approach in the context of <a href="Errors-in-variables_models" class="mw-redirect" title="Errors-in-variables models">errors-in-variables models</a> in his dissertation, giving the method its name.<a href="#cite_note-8">[8]</a>

Wright attempted to determine the supply and demand for butter using <a href="Panel_data" title="Panel data">panel data</a> on prices and quantities sold in the United States. The idea was that a regression analysis could produce a demand or supply curve because they are formed by the path between prices and quantities demanded or supplied. The problem was that the observational data did not form a demand or supply curve as such, but rather a cloud of point observations that took different shapes under varying market conditions. It seemed that making deductions from the data remained elusive.

The problem was that price affected both supply and demand so that a function describing only one of the two could not be constructed directly from the observational data. Wright correctly concluded that he needed a variable that correlated with either demand or supply but not both – that is, an instrumental variable.

After much deliberation, Wright decided to use regional rainfall as his instrumental variable: he concluded that rainfall affected grass production and hence milk production and ultimately butter supply, but not butter demand. In this way he was able to construct a regression equation with only the instrumental variable of price and supply.<a href="#cite_note-Wool-9">[9]</a>

Formal definitions of instrumental variables, using counterfactuals and graphical criteria, were given by <a href="Judea_Pearl" title="Judea Pearl">Judea Pearl</a> in 2000.<a href="#cite_note-Pearl:00-10">[10]</a> <a href="Joshua_Angrist" title="Joshua Angrist">Angrist</a> and <a href="Alan_Krueger" title="Alan Krueger">Krueger</a> (2001) present a survey of the history and uses of instrumental variable techniques.<a href="#cite_note-angrist:00-11">[11]</a> Notions of causality in econometrics, and their relationship with instrumental variables and other methods, are discussed by <a href="James_Heckman" title="James Heckman">Heckman</a> (2008).<a href="#cite_note-Heckman:00-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2></div>

While the ideas behind IV extend to a broad class of models, a very common context for IV is in <a href="Linear_regression" title="Linear regression">linear regression</a>. Traditionally,<a href="#cite_note-bowden:turkington84-13">[13]</a> an instrumental variable is defined

as a variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> that is correlated with the independent variable

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and uncorrelated with the "error term" U in the linear equation

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<annotation encoding="application/x-tex">{\displaystyle Y=X\beta +U}</annotation>

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</math><img src="./1445a8823e8726af3b56a3209c67c2d1e5cb3166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.807ex; height:2.509ex;" alt="{\displaystyle Y=X\beta +U}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is a vector. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a matrix, usually with a column of ones and perhaps with additional columns for other covariates. Consider how an instrument allows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> to be recovered. Recall that <a href="Ordinary_least_squares" title="Ordinary least squares">OLS</a> solves for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}}</annotation>

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</math><img src="./21fd425a5a1a245a101aae3ff48df531b4dc96ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.535ex; height:3.343ex;" alt="{\displaystyle {\widehat {\beta }}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (X,{\widehat {U}})=0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X,{\widehat {U}})=0}</annotation>

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</math><img src="./1fc3464dcf9a1cbc4765255e6cf60ea5766d1b72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.289ex; height:3.343ex;" alt="{\displaystyle \operatorname {cov} (X,{\widehat {U}})=0}" loading="lazy"> (when we minimize the sum of squared errors, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{\beta }(Y-X\beta )'(Y-X\beta )}">

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<annotation encoding="application/x-tex">{\displaystyle \min _{\beta }(Y-X\beta )'(Y-X\beta )}</annotation>

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</math><img src="./225b754b244dae4ef36347b62d6de27ca2627dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.03ex; height:4.509ex;" alt="{\displaystyle \min _{\beta }(Y-X\beta )'(Y-X\beta )}" loading="lazy">, the first-order condition is exactly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'(Y-X{\widehat {\beta }})=X'{\widehat {U}}=0}">

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<annotation encoding="application/x-tex">{\displaystyle X'(Y-X{\widehat {\beta }})=X'{\widehat {U}}=0}</annotation>

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</math><img src="./217ec3b0c511c836061c8fddce15a8ae9433fa55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.424ex; height:3.509ex;" alt="{\displaystyle X'(Y-X{\widehat {\beta }})=X'{\widehat {U}}=0}" loading="lazy">. If the true model is believed to have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (X,U)\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X,U)\neq 0}</annotation>

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</math><img src="./c3146c47c4c73d487829057f6e6a88bb3fa4aa56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.289ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (X,U)\neq 0}" loading="lazy"> due to any of the reasons listed above—for example, if there is an <a href="Omitted-variable_bias" title="Omitted-variable bias">omitted variable</a> which affects both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> separately—then this <a href="Ordinary_least_squares" title="Ordinary least squares">OLS</a> procedure will not yield the causal impact of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">. OLS will simply pick the parameter that makes the resulting errors appear uncorrelated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

Consider for simplicity the single-variable case. Suppose we are considering a regression with one variable and a constant (perhaps no other covariates are necessary, or perhaps we have <a href="Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem" title="Frisch–Waugh–Lovell theorem">partialed out</a> any other relevant covariates):

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<annotation encoding="application/x-tex">{\displaystyle y=\alpha +\beta x+u}</annotation>

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</math><img src="./be1298fcd206e58a89d39d5b38c30aa553195a33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.414ex; height:2.509ex;" alt="{\displaystyle y=\alpha +\beta x+u}" loading="lazy"></dd></dl>

In this case, the coefficient on the regressor of interest is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\beta }}={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}}</annotation>

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</math><img src="./af3032973ef015f62fc685f3cc358c5245298d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.202ex; height:6.509ex;" alt="{\displaystyle {\widehat {\beta }}={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}}" loading="lazy">. Substituting for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> gives

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}={\frac {\operatorname {cov} (x,\alpha +\beta x+u)}{\operatorname {var} (x)}}\\[6pt]&={\frac {\operatorname {cov} (x,\alpha +\beta x)}{\operatorname {var} (x)}}+{\frac {\operatorname {cov} (x,u)}{\operatorname {var} (x)}}=\beta ^{*}+{\frac {\operatorname {cov} (x,u)}{\operatorname {var} (x)}},\end{aligned}}}</annotation>

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</math><img src="./c0fa71a1544a7e0714dae657ee944feb5cc4a02a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:52.127ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\widehat {\beta }}&={\frac {\operatorname {cov} (x,y)}{\operatorname {var} (x)}}={\frac {\operatorname {cov} (x,\alpha +\beta x+u)}{\operatorname {var} (x)}}\\[6pt]&={\frac {\operatorname {cov} (x,\alpha +\beta x)}{\operatorname {var} (x)}}+{\frac {\operatorname {cov} (x,u)}{\operatorname {var} (x)}}=\beta ^{*}+{\frac {\operatorname {cov} (x,u)}{\operatorname {var} (x)}},\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \beta ^{*}}</annotation>

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</math><img src="./823d440b048cd3c497f61dffcb61d897b19e96db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.391ex; height:2.676ex;" alt="{\displaystyle \beta ^{*}}" loading="lazy"> is what the estimated coefficient vector would be if

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (x,u)=0}</annotation>

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</math><img src="./75bbbbe23a56f637c789bd97f96d3259e9602ef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.186ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (x,u)=0}" loading="lazy">. In this case, it can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \beta ^{*}}</annotation>

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</math><img src="./823d440b048cd3c497f61dffcb61d897b19e96db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.391ex; height:2.676ex;" alt="{\displaystyle \beta ^{*}}" loading="lazy"> is an unbiased estimator of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (x,u)\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (x,u)\neq 0}</annotation>

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</math><img src="./8a2a0c3673401dc30a93056f242ccc8504463349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.186ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (x,u)\neq 0}" loading="lazy"> in the underlying model that we believe, then <a href="Ordinary_least_squares" title="Ordinary least squares">OLS</a> gives an inconsistent estimate which does not reflect the underlying causal effect of interest. IV helps to fix this problem by identifying the parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\beta }}">

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<annotation encoding="application/x-tex">{\displaystyle {\beta }}</annotation>

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</math><img src="./4f08e5bb6adcac8ac464df79e6a2e43779898ab7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle {\beta }}" loading="lazy"> not based on whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is uncorrelated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy">, but based on whether another variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> is uncorrelated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy">. If theory suggests that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> is related to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> (the first stage) but uncorrelated with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> (the exclusion restriction), then IV may identify the causal parameter of interest where OLS fails. Because there are multiple specific ways of using and deriving IV estimators even in just the linear case (IV, 2SLS, GMM), we save further discussion for the <a href="#Estimation">Estimation</a> section below.

<div class="mw-heading mw-heading2"><h2 id="Graphical_definition">Graphical definition</h2></div>

IV techniques have been developed among a much broader class of non-linear models. General definitions of instrumental variables, using counterfactual and graphical formalism, were given by Pearl (2000; p. 248).<a href="#cite_note-Pearl:00-10">[10]</a> The graphical definition requires that Z satisfy the following conditions:

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<annotation encoding="application/x-tex">{\displaystyle (Z\perp \!\!\!\perp Y)_{G_{\overline {X}}}\qquad (Z\not \!\!{\perp \!\!\!\perp }X)_{G}}</annotation>

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</math><img src="./1e933eb12404270d9193060e66fff55e6eb010f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:26.557ex; height:3.509ex;" alt="{\displaystyle (Z\perp \!\!\!\perp Y)_{G_{\overline {X}}}\qquad (Z\not \!\!{\perp \!\!\!\perp }X)_{G}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \perp \!\!\!\perp }">

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<annotation encoding="application/x-tex">{\displaystyle \perp \!\!\!\perp }</annotation>

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</math><img src="./54a92a12159150196642b9b59e5363d496521d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \perp \!\!\!\perp }" loading="lazy"> stands for <a href="Bayesian_network#d-separation" title="Bayesian network">d-separation</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\overline {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\overline {X}}}</annotation>

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</math><img src="./224530d558d9ff9ac4537789f05df3c90cd7f9e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.578ex; height:3.009ex;" alt="{\displaystyle G_{\overline {X}}}" loading="lazy"> stands for the <a href="Bayesian_network" title="Bayesian network">graph</a> in which all arrows entering X are cut off.

The counterfactual definition requires that Z satisfies

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<annotation encoding="application/x-tex">{\displaystyle (Z\perp \!\!\!\perp Y_{x})\qquad (Z\not \!\!{\perp \!\!\!\perp }X)}</annotation>

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</math><img src="./0873a84054c16fcf7b1e05cf14e92c1ecd4edcc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.844ex; height:2.843ex;" alt="{\displaystyle (Z\perp \!\!\!\perp Y_{x})\qquad (Z\not \!\!{\perp \!\!\!\perp }X)}" loading="lazy"></dd></dl>

where Yx stands for the value that Y would attain had X been x and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \perp \!\!\!\perp }">

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<annotation encoding="application/x-tex">{\displaystyle \perp \!\!\!\perp }</annotation>

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</math><img src="./54a92a12159150196642b9b59e5363d496521d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.176ex;" alt="{\displaystyle \perp \!\!\!\perp }" loading="lazy"> stands for independence.

If there are additional covariates W then the above definitions are modified so that Z qualifies as an instrument if the given criteria hold conditional on W.

The essence of Pearl's definition is:

<ol><li>The equations of interest are "structural", not "regression".</li>

<li>The error term U stands for all exogenous factors that affect Y when X is held constant.</li>

<li>The instrument Z should be independent of U.</li>

<li>The instrument Z should not affect Y when X is held constant (exclusion restriction).</li>

<li>The instrument Z should not be independent of X.</li></ol>

These conditions do not rely on specific functional

form of the equations and are applicable therefore to

nonlinear equations, where U can be non-additive

(see Non-parametric analysis). They are also applicable to a system of multiple

equations, in which X (and other factors) affect Y through

several intermediate variables. An instrumental variable need not be

a cause of X; a proxy of such cause may also be

used, if it satisfies conditions 1–5.<a href="#cite_note-Pearl:00-10">[10]</a> The exclusion restriction (condition 4) is redundant; it follows from conditions 2 and 3.

<div class="mw-heading mw-heading3"><h3 id="Selecting_suitable_instruments">Selecting suitable instruments</h3></div>

Since U is unobserved, the requirement that Z be independent of U cannot be inferred from data and must instead be determined from the model structure, i.e., the data-generating process. <a href="Causal_graphs" class="mw-redirect" title="Causal graphs">Causal graphs</a> are a representation of this structure, and the graphical definition given above can be used to quickly determine whether a variable Z qualifies as an instrumental variable given a set of covariates W. To see how, consider the following example.

Suppose that we wish to estimate the effect of a university tutoring program on grade point average (<a href="Grading_in_education" title="Grading in education">GPA</a>). The relationship between attending the tutoring program and GPA may be confounded by a number of factors. Students who attend the tutoring program may care more about their grades or may be struggling with their work. This confounding is depicted in the Figures 1–3 on the right through the bidirected arc between Tutoring Program and GPA. If students are assigned to dormitories at random, the proximity of the student's dorm to the tutoring program is a natural candidate for being an instrumental variable.

<div class="gallerytext">Figure 1: Proximity qualifies as an instrumental variable given Library Hours</div>

</li>

<div class="gallerytext">Figure 2: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\overline {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\overline {X}}}</annotation>

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</math><img src="./224530d558d9ff9ac4537789f05df3c90cd7f9e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.578ex; height:3.009ex;" alt="{\displaystyle G_{\overline {X}}}" loading="lazy">, which is used to determine whether Proximity is an instrumental variable.</div>

</li>

<div class="gallerytext">Figure 3: Proximity does not qualify as an instrumental variable given Library Hours</div>

</li>

<div class="gallerytext">Figure 4: Proximity qualifies as an instrumental variable, as long as we do not include Library Hours as a covariate.</div>

</li>

</ul>

However, what if the tutoring program is located in the college library? In that case, Proximity may also cause students to spend more time at the library, which in turn improves their GPA (see Figure 1). Using the causal graph depicted in the Figure 2, we see that Proximity does not qualify as an instrumental variable because it is connected to GPA through the path Proximity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

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<annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation>

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</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }" loading="lazy"> Library Hours <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

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<annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation>

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</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }" loading="lazy"> GPA in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\overline {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\overline {X}}}</annotation>

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</math><img src="./224530d558d9ff9ac4537789f05df3c90cd7f9e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.578ex; height:3.009ex;" alt="{\displaystyle G_{\overline {X}}}" loading="lazy">. However, if we control for Library Hours by adding it as a covariate then Proximity becomes an instrumental variable, since Proximity is separated from GPA given Library Hours in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\overline {X}}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{\overline {X}}}</annotation>

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</math><img src="./224530d558d9ff9ac4537789f05df3c90cd7f9e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.578ex; height:3.009ex;" alt="{\displaystyle G_{\overline {X}}}" loading="lazy">.

Now, suppose that we notice that a student's "natural ability" affects his or her number of hours in the library as well as his or her GPA, as in Figure 3. Using the causal graph, we see that Library Hours is a collider and conditioning on it opens the path Proximity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

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<annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation>

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</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \rightarrow }" loading="lazy"> Library Hours <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }">

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<annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation>

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</math><img src="./046b918c43e05caf6624fe9b676c69ec9cd6b892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }" loading="lazy"> GPA. As a result, Proximity cannot be used as an instrumental variable.

Finally, suppose that Library Hours does not actually affect GPA because students who do not study in the library simply study elsewhere, as in Figure 4. In this case, controlling for Library Hours still opens a spurious path from Proximity to GPA. However, if we do not control for Library Hours and remove it as a covariate then Proximity can again be used an instrumental variable.

<div class="mw-heading mw-heading2"><h2 id="Estimation">Estimation</h2></div>

We now revisit and expand upon the mechanics of IV in greater detail. Suppose the data are generated by a process of the form

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<annotation encoding="application/x-tex">{\displaystyle y_{i}=X_{i}\beta +e_{i},}</annotation>

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</math><img src="./8c47e20a501fd80b4293570c186f40e544afff9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.464ex; height:2.509ex;" alt="{\displaystyle y_{i}=X_{i}\beta +e_{i},}" loading="lazy"></dd></dl>

where

<ul><li>i indexes observations,</li>

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> is the i-th value of the dependent variable,</li>

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> is a vector of the i-th values of the independent variable(s) and a constant,</li>

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<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

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</math><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"> is the i-th value of an unobserved error term representing all causes of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy">, and</li>

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is an unobserved parameter vector.</li></ul>

The parameter vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is the causal effect on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> of a one unit change in each element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy">, holding all other causes of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> constant. The econometric goal is to estimate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">. For simplicity's sake assume the draws of e are uncorrelated and that they are drawn from distributions with the same <a href="Variance" title="Variance">variance</a> (that is, that the errors are serially uncorrelated and <a href="Homoskedastic" class="mw-redirect" title="Homoskedastic">homoskedastic</a>).

Suppose also that a regression model of nominally the same form is proposed. Given a random sample of T observations from this process, the <a href="Least_squares" title="Least squares">ordinary least squares</a> estimator is

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{\mathrm {OLS} }=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }y=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }(X\beta +e)=\beta +(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }e}</annotation>

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</math><img src="./3d46c5a989004fdedaf80d83555302250decb47c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.581ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{\mathrm {OLS} }=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }y=(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }(X\beta +e)=\beta +(X^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }e}" loading="lazy"></dd></dl>

where X, y and e denote column vectors of length T. This equation is similar to the equation involving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cov} (X,y)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X,y)}</annotation>

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</math><img src="./de7916cfb59a9327b6cf917e77c648dc737bed3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.401ex; height:2.843ex;" alt="{\displaystyle \operatorname {cov} (X,y)}" loading="lazy"> in the introduction (this is the matrix version of that equation). When X and e are <a href="Correlation" title="Correlation">uncorrelated</a>, under certain regularity conditions the second term has an expected value conditional on X of zero and converges to zero in the limit, so the estimator is <a href="Estimator_bias" class="mw-redirect" title="Estimator bias">unbiased</a> and consistent. When X and the other unmeasured, causal variables collapsed into the e term are correlated, however, the OLS estimator is generally biased and inconsistent for β. In this case, it is valid to use the estimates to predict values of y given values of X, but the estimate does not recover the causal effect of X on y.

To recover the underlying parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">, we introduce a set of variables Z that is highly correlated with each <a href="Endogeneity_(econometrics)" title="Endogeneity (econometrics)">endogenous</a> component of X but (in our underlying model) is not correlated with e. For simplicity, one might consider X to be a T × 2 matrix composed of a column of constants and one endogenous variable, and Z to be a T × 2 consisting of a column of constants and one instrumental variable. However, this technique generalizes to X being a matrix of a constant and, say, 5 endogenous variables, with Z being a matrix composed of a constant and 5 instruments. In the discussion that follows, we will assume that X is a T × K matrix and leave this value K unspecified. An estimator in which X and Z are both T × K matrices is referred to as <a href="Identifiability" title="Identifiability">just-identified</a> .

Suppose that the relationship between each endogenous component xi and the instruments is given by

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=Z_{i}\gamma +v_{i},}</annotation>

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The most common IV specification uses the following estimator:

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{\mathrm {IV} }=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y}</annotation>

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</math><img src="./b9f0a29cbf570049a748a54d13439a26da9bb480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.205ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{\mathrm {IV} }=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y}" loading="lazy"></dd></dl>

This specification approaches the true parameter as the sample gets large, so long as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z^{\mathrm {T} }e=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{\mathrm {IV} }=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }X\beta +(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }e\rightarrow \beta }</annotation>

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</math><img src="./285991e47b18f7e0de8746cca51d10101036619e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.239ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{\mathrm {IV} }=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }X\beta +(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }e\rightarrow \beta }" loading="lazy"></dd></dl>

As long as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z^{\mathrm {T} }e=0}">

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<annotation encoding="application/x-tex">{\displaystyle Z^{\mathrm {T} }e=0}</annotation>

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</math><img src="./a3b0813c7aa06507e6a077afcd1e2c5295067607.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.472ex; height:2.676ex;" alt="{\displaystyle Z^{\mathrm {T} }e=0}" loading="lazy"> in the underlying process which generates the data, the appropriate use of the IV estimator will identify this parameter. This works because IV solves for the unique parameter that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z^{\mathrm {T} }e=0}">

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Now an extension: suppose that there are more instruments than there are covariates in the equation of interest, so that Z is a T × M matrix with M > K. This is often called the over-identified case. In this case, the <a href="Generalized_method_of_moments" title="Generalized method of moments">generalized method of moments</a> (GMM) can be used. The GMM IV estimator is

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{\mathrm {GMM} }=(X^{\mathrm {T} }P_{Z}X)^{-1}X^{\mathrm {T} }P_{Z}y,}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{Z}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{Z}=Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }}</annotation>

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This expression collapses to the first when the number of instruments is equal to the number of covariates in the equation of interest. The over-identified IV is therefore a generalization of the just-identified IV.

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</style><div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px solid #aaa"><div class="hidden-title skin-nightmode-reset-color" style=""><div class="center">Proof that βGMM collapses to βIV in the just-identified case</div></div><div class="hidden-content mw-collapsible-content" style="">

Developing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{\text{GMM}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{\mathrm {GMM} }=(X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y}</annotation>

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</math><img src="./1252f96fc89a8c52eff282fbd9ec2608780d2528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.774ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{\mathrm {GMM} }=(X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }X)^{-1}X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y}" loading="lazy"></dd></dl>

In the just-identified case, we have as many instruments as covariates, so that the dimension of X is the same as that of Z. Hence, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{\mathrm {T} }Z,Z^{\mathrm {T} }Z}">

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</math><img src="./43d0bd582115a54d95372d4f751312df463d35fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.107ex; height:2.676ex;" alt="{\displaystyle Z^{\mathrm {T} }X}" loading="lazy"> are all squared matrices of the same dimension. We can expand the inverse, using the fact that, for any invertible n-by-n matrices A and B, (AB)−1 = B−1A−1 (see <a href="Invertible_matrix#Properties" title="Invertible matrix">Invertible matrix#Properties</a>):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\beta }}_{\mathrm {GMM} }&=(Z^{\mathrm {T} }X)^{-1}(Z^{\mathrm {T} }Z)(X^{\mathrm {T} }Z)^{-1}X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y\\&=(Z^{\mathrm {T} }X)^{-1}(Z^{\mathrm {T} }Z)(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y\\&=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y\\&={\widehat {\beta }}_{\mathrm {IV} }\end{aligned}}}</annotation>

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</math><img src="./3b4fd4f8a2574538396b89f2bb2083a9c3b437c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:53.335ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}{\widehat {\beta }}_{\mathrm {GMM} }&=(Z^{\mathrm {T} }X)^{-1}(Z^{\mathrm {T} }Z)(X^{\mathrm {T} }Z)^{-1}X^{\mathrm {T} }Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y\\&=(Z^{\mathrm {T} }X)^{-1}(Z^{\mathrm {T} }Z)(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }y\\&=(Z^{\mathrm {T} }X)^{-1}Z^{\mathrm {T} }y\\&={\widehat {\beta }}_{\mathrm {IV} }\end{aligned}}}" loading="lazy"></dd></dl>

Reference: see Davidson and Mackinnnon (1993)<a href="#cite_note-14">[14]</a>: 218

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There is an equivalent <a href="Parameter_identification_problem" title="Parameter identification problem">under-identified</a> estimator for the case where m < k. Since the parameters are the solutions to a set of linear equations, an under-identified model using the set of equations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z'v=0}">

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<div class="mw-heading mw-heading2"><h2 id="Interpretation_as_two-stage_least_squares">Interpretation as two-stage least squares</h2></div>

One computational method which can be used to calculate IV estimates is two-stage least squares (2SLS or TSLS). In the first stage, each explanatory variable that is an endogenous covariate in the equation of interest is regressed on all of the exogenous variables in the model, including both exogenous covariates in the equation of interest and the excluded instruments. The predicted values from these regressions are obtained:

Stage 1: Regress each column of X on Z, (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Z\delta +{\text{errors}}}">

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and save the predicted values:

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {X}}=Z{\widehat {\delta }}={\color {ProcessBlue}Z(Z^{\mathrm {T} }Z)^{-1}Z^{\mathrm {T} }}X={\color {ProcessBlue}P_{Z}}X.\,}</annotation>

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In the second stage, the regression of interest is estimated as usual, except that in this stage each endogenous covariate is replaced with the predicted values from the first stage:

Stage 2: Regress Y on the predicted values from the first stage:

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which gives

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<annotation encoding="application/x-tex">{\displaystyle \beta _{\text{2SLS}}=\left(X^{\mathrm {T} }{\color {ProcessBlue}P_{Z}}X\right)^{-1}X^{\mathrm {T} }{\color {ProcessBlue}P_{Z}}Y.}</annotation>

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</math><img src="./31e98cd3c352311c9d791409295d09d6544b78aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.843ex; height:3.843ex;" alt="{\displaystyle \beta _{\text{2SLS}}=\left(X^{\mathrm {T} }{\color {ProcessBlue}P_{Z}}X\right)^{-1}X^{\mathrm {T} }{\color {ProcessBlue}P_{Z}}Y.}" loading="lazy"></dd></dl>

This method is only valid in linear models. For categorical endogenous covariates, one might be tempted to use a different first stage than ordinary least squares, such as a <a href="Probit_model" title="Probit model">probit model</a> for the first stage followed by OLS for the second. This is commonly known in the econometric literature as the forbidden regression,<a href="#cite_note-15">[15]</a> because second-stage IV parameter estimates are consistent only in special cases.<a href="#cite_note-16">[16]</a>

<div class="hidden-begin mw-collapsible mw-collapsed" style="border:1px solid #aaa"><div class="hidden-title skin-nightmode-reset-color" style=""><div class="center">Proof: computation of the 2SLS estimator</div></div><div class="hidden-content mw-collapsible-content" style="">

The usual OLS estimator is: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\widehat {X}}^{\mathrm {T} }{\widehat {X}})^{-1}{\widehat {X}}^{\mathrm {T} }Y}">

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Replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {X}}=P_{Z}X}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {X}}=P_{Z}X}</annotation>

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</math><img src="./3b34ef9e4026170e4e54b3a69ecfe22d522874f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.545ex; height:3.176ex;" alt="{\displaystyle {\widehat {X}}=P_{Z}X}" loading="lazy"> and noting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{Z}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{Z}}</annotation>

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</math><img src="./6f42051aac26076a25d473a58b68606e9c40a0ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.913ex; height:2.509ex;" alt="{\displaystyle P_{Z}}" loading="lazy"> is a symmetric and <a href="Idempotence" title="Idempotence">idempotent</a> matrix, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{Z}^{\mathrm {T} }P_{Z}=P_{Z}P_{Z}=P_{Z}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{Z}^{\mathrm {T} }P_{Z}=P_{Z}P_{Z}=P_{Z}}</annotation>

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</math><img src="./dff576d931c2614f91187ba0dc96fe1ca5eafd4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.088ex; height:3.176ex;" alt="{\displaystyle P_{Z}^{\mathrm {T} }P_{Z}=P_{Z}P_{Z}=P_{Z}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \beta _{\text{2SLS}}=({\widehat {X}}^{\mathrm {T} }{\widehat {X}})^{-1}{\widehat {X}}^{\mathrm {T} }Y=\left(X^{\mathrm {T} }P_{Z}^{\mathrm {T} }P_{Z}X\right)^{-1}X^{\mathrm {T} }P_{Z}^{\mathrm {T} }Y=\left(X^{\mathrm {T} }P_{Z}X\right)^{-1}X^{\mathrm {T} }P_{Z}Y.}</annotation>

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</math><img src="./c6ac72cc53233b8330db8dc82ce60734317dd5db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:76.894ex; height:4.009ex;" alt="{\displaystyle \beta _{\text{2SLS}}=({\widehat {X}}^{\mathrm {T} }{\widehat {X}})^{-1}{\widehat {X}}^{\mathrm {T} }Y=\left(X^{\mathrm {T} }P_{Z}^{\mathrm {T} }P_{Z}X\right)^{-1}X^{\mathrm {T} }P_{Z}^{\mathrm {T} }Y=\left(X^{\mathrm {T} }P_{Z}X\right)^{-1}X^{\mathrm {T} }P_{Z}Y.}" loading="lazy"></dd></dl>

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The resulting estimator of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is numerically identical to the expression displayed above. A small correction must be made to the sum-of-squared residuals in the second-stage fitted model in order that the covariance matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is calculated correctly.

<div class="mw-heading mw-heading2"><h2 id="Non-parametric_analysis">Non-parametric analysis</h2></div>

When the form of the structural equations is unknown, an instrumental variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> can still be defined through the equations:

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<annotation encoding="application/x-tex">{\displaystyle x=g(z,u)\,}</annotation>

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</math><img src="./77f55fb91847c6ffdd1a13d5f3922b7617fa82e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.192ex; height:2.843ex;" alt="{\displaystyle x=g(z,u)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y=f(x,u)\,}</annotation>

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</math><img src="./6459e1b51dbcd56bb974effc1b9eb22bdfd39786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.422ex; height:2.843ex;" alt="{\displaystyle y=f(x,u)\,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> are two arbitrary functions and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Unlike linear models, however, measurements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z,X}">

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<annotation encoding="application/x-tex">{\displaystyle Z,X}</annotation>

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</math><img src="./182d238268dd4d88800e4aafc4b7eff929d82692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.694ex; height:2.509ex;" alt="{\displaystyle Z,X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> do not allow for the identification of the average causal effect of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, denoted ACE

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<annotation encoding="application/x-tex">{\displaystyle {\text{ACE}}=\Pr(y\mid {\text{do}}(x))=\operatorname {E} _{u}[f(x,u)].}</annotation>

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</math><img src="./e6ea36b8f79d0df3bb0e66e1b9273b9c9ae67edb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.668ex; height:2.843ex;" alt="{\displaystyle {\text{ACE}}=\Pr(y\mid {\text{do}}(x))=\operatorname {E} _{u}[f(x,u)].}" loading="lazy"></dd></dl>

Balke and Pearl [1997] derived tight bounds on ACE and showed that these can provide valuable information on the sign and size of ACE.<a href="#cite_note-balke:pearl97-17">[17]</a>

In linear analysis, there is no test to falsify the assumption the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> is instrumental relative to the pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (X,Y)}</annotation>

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</math><img src="./41f29b9537685f499713112d6802e811cbf51bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}" loading="lazy">. This is not the case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is discrete. Pearl (2000) has shown that, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, the following constraint, called "Instrumental Inequality" must hold whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

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</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> satisfies the two equations above:<a href="#cite_note-Pearl:00-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle \max _{x}\sum _{y}[\max _{z}\Pr(y,x\mid z)]\leq 1.}</annotation>

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</math><img src="./b089badc50cd6306cc6c673f45027813e1ebc23e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.83ex; height:5.843ex;" alt="{\displaystyle \max _{x}\sum _{y}[\max _{z}\Pr(y,x\mid z)]\leq 1.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Interpretation_under_treatment_effect_heterogeneity">Interpretation under treatment effect heterogeneity</h2></div>

The exposition above assumes that the causal effect of interest does not vary across observations, that is, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> is a constant. Generally, different subjects will respond in different ways to changes in the "treatment" x. When this possibility is recognized, the average effect in the population of a change in x on y may differ from the effect in a given subpopulation. For example, the average effect of a job training program may substantially differ across the group of people who actually receive the training and the group which chooses not to receive training. For these reasons, IV methods invoke implicit assumptions on behavioral response, or more generally assumptions over the correlation between the response to treatment and propensity to receive treatment.<a href="#cite_note-18">[18]</a>

The standard IV estimator can recover <a href="Local_average_treatment_effect" title="Local average treatment effect">local average treatment effects</a> (LATE) rather than <a href="Average_treatment_effects" class="mw-redirect" title="Average treatment effects">average treatment effects</a> (ATE).<a href="#cite_note-Imbens:00-1">[1]</a> Imbens and Angrist (1994) demonstrate that the linear IV estimate can be interpreted under weak conditions as a weighted average of local average treatment effects, where the weights depend on the elasticity of the endogenous regressor to changes in the instrumental variables. Roughly, that means that the effect of a variable is only revealed for the subpopulations affected by the observed changes in the instruments, and that subpopulations which respond most to changes in the instruments will have the largest effects on the magnitude of the IV estimate.

For example, if a researcher uses presence of a land-grant college as an instrument for college education in an earnings regression, she identifies the effect of college on earnings in the subpopulation which would obtain a college degree if a college is present but which would not obtain a degree if a college is not present. This empirical approach does not, without further assumptions, tell the researcher anything about the effect of college among people who would either always or never get a college degree regardless of whether a local college exists.

<div class="mw-heading mw-heading2"><h2 id="Weak_instruments_problem">Weak instruments problem</h2></div>

As Bound, <a href="David_A._Jaeger" title="David A. Jaeger">Jaeger</a>, and Baker (1995) note, a problem is caused by the selection of "weak" instruments, instruments that are poor predictors of the endogenous question predictor in the first-stage equation.<a href="#cite_note-19">[19]</a> In this case, the prediction of the question predictor by the instrument will be poor and the predicted values will have very little variation. Consequently, they are unlikely to have much success in predicting the ultimate outcome when they are used to replace the question predictor in the second-stage equation.

In the context of the smoking and health example discussed above, tobacco taxes are weak instruments for smoking if smoking status is largely unresponsive to changes in taxes. If higher taxes do not induce people to quit smoking (or not start smoking), then variation in tax rates tells us nothing about the effect of smoking on health. If taxes affect health through channels other than through their effect on smoking, then the instruments are invalid and the instrumental variables approach may yield misleading results. For example, places and times with relatively health-conscious populations may both implement high tobacco taxes and exhibit better health even holding smoking rates constant, so we would observe a correlation between health and tobacco taxes even if it were the case that smoking has no effect on health. In this case, we would be mistaken to infer a causal effect of smoking on health from the observed correlation between tobacco taxes and health.

<div class="mw-heading mw-heading3"><h3 id="Testing_for_weak_instruments">Testing for weak instruments</h3></div>

The strength of the instruments can be directly assessed because both the endogenous covariates and the instruments are observable.<a href="#cite_note-Stock:00-20">[20]</a> A common rule of thumb for models with one endogenous regressor is: the <a href="F-test" title="F-test">F-statistic</a> against the <a href="Null_hypothesis" title="Null hypothesis">null</a> that the excluded instruments are irrelevant in the first-stage regression should be larger than 10.

<div class="mw-heading mw-heading2"><h2 id="Statistical_inference_and_hypothesis_testing">Statistical inference and hypothesis testing</h2></div>

When the covariates are exogenous, the small-sample properties of the OLS estimator can be derived in a straightforward manner by calculating moments of the estimator conditional on X. When some of the covariates are endogenous so that instrumental variables estimation is implemented, simple expressions for the moments of the estimator cannot be so obtained. Generally, instrumental variables estimators only have desirable asymptotic, not finite sample, properties, and inference is based on asymptotic approximations to the <a href="Sampling_distribution" title="Sampling distribution">sampling distribution</a> of the estimator. Even when the instruments are uncorrelated with the error in the equation of interest and when the instruments are not weak, the finite sample properties of the instrumental variables estimator may be poor. For example, exactly identified models produce finite sample estimators with no moments, so the estimator can be said to be neither biased nor unbiased, the nominal size of test statistics may be substantially distorted, and the estimates may commonly be far away from the true value of the parameter.<a href="#cite_note-Nelson_1990:00-21">[21]</a>

<div class="mw-heading mw-heading2"><h2 id="Testing_the_exclusion_restriction">Testing the exclusion restriction</h2></div>

The assumption that the instruments are not correlated with the error term in the equation of interest is not testable in exactly identified models. If the model is overidentified, there is information available which may be used to test this assumption. The most common test of these overidentifying restrictions, called the <a href="Sargan%E2%80%93Hansen_test" title="Sargan–Hansen test">Sargan–Hansen test</a>, is based on the observation that the residuals should be uncorrelated with the set of exogenous variables if the instruments are truly exogenous.<a href="#cite_note-22">[22]</a> The Sargan–Hansen <a href="Test_statistic" title="Test statistic">test statistic</a> can be calculated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TR^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle TR^{2}}</annotation>

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</math><img src="./692d761134f5a3a6deeac953745d1be7bfe851fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.455ex; height:2.676ex;" alt="{\displaystyle TR^{2}}" loading="lazy"> (the number of observations multiplied by the <a href="Coefficient_of_determination" title="Coefficient of determination">coefficient of determination</a>) from the OLS regression of the residuals onto the set of exogenous variables. This statistic will be asymptotically chi-squared with m − k degrees of freedom under the null that the error term is uncorrelated with the instruments.

<ul><li><a href="Control_function_(econometrics)" title="Control function (econometrics)">Control function (econometrics)</a> – Statistical methods to correct for endogeneity problems</li>

<li><a href="Optimal_instruments" title="Optimal instruments">Optimal instruments</a> – Technique for improving the efficiency of estimators in conditional moment models</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFGreene2008" class="citation book cs1"><a href="William_Greene_(economist)" title="William Greene (economist)">Greene, William H.</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/econometricanaly00gree_641">Econometric Analysis</a> (Sixth ed.). Upper Saddle River: Pearson Prentice-Hall. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/econometricanaly00gree_641/page/n351">314</a>–353. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-600383-0</bdi>.</cite></li>

<li><cite id="CITEREFGujaratiPorter2009" class="citation book cs1"><a href="Damodar_N._Gujarati" title="Damodar N. Gujarati">Gujarati, Damodar N.</a>; <a href="Dawn_C._Porter" title="Dawn C. Porter">Porter, Dawn C.</a> (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/basiceconometric05edguja">Basic Econometrics</a> (Fifth ed.). New York: McGraw-Hill Irwin. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/basiceconometric05edguja/page/711">711</a>–736. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-337577-9</bdi>.</cite></li>

<li>Keane, Michael P.; Neal, Timothy (2024). "<a href="https://doi.org/10.1146/annurev-economics-092123-111021" class="extiw external" title="doi:10.1146/annurev-economics-092123-111021">A Practical Guide to Weak Instruments</a>". Annual Review of Economics. 16: 185–212.</li>

<li><cite id="CITEREFSargan1988" class="citation book cs1"><a href="Denis_Sargan" title="Denis Sargan">Sargan, Denis</a> (1988). Lectures on Advanced Econometric Theory. Oxford: Basil Blackwell. pp. 42–67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-631-14956-9</bdi>.</cite></li>

<li><cite id="CITEREFWooldridge2013" class="citation book cs1"><a href="Jeffrey_Wooldridge" title="Jeffrey Wooldridge">Wooldridge, Jeffrey M.</a> (2013). Introductory Econometrics: A Modern Approach (Fifth international ed.). Mason, OH: South-Western. pp. 490–528. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-111-53439-4</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li>Wooldridge, J. (1997): Quasi-Likelihood Methods for Count Data, Handbook of Applied Econometrics, Volume 2, ed. M. H. Pesaran and P. Schmidt, Oxford, Blackwell, pp. 352–406</li>

<li>Terza, J. V. (1998): "Estimating Count Models with Endogenous Switching: Sample Selection and Endogenous Treatment Effects." Journal of Econometrics (84), pp. 129–154</li>

<li>Wooldridge, J. (2002): "Econometric Analysis of Cross Section and Panel Data", MIT Press, Cambridge, Massachusetts.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://emlab.berkeley.edu/users/mcfadden/e240b_f01/ch4.pdf">Chapter</a> from <a href="Daniel_McFadden" title="Daniel McFadden">Daniel McFadden</a>'s textbook</li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Kb4LvSguwjg&list=PLD15D38DC7AA3B737&index=12">Econometrics lecture (topic: instrumental variable)</a> on <a href="YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> by <a href="Mark_Thoma" title="Mark Thoma">Mark Thoma</a>.</li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=D5lt9bhOshc&list=PLD15D38DC7AA3B737&index=15#t=54m09s">Econometrics lecture (topic: two-stages least square)</a> on <a href="YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> by Mark Thoma</li></ul>

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