IV Identification And Estimation Workshop

From an Instrument to an IV Estimate

An IV analysis is not complete when software prints a coefficient. The analyst must connect several claims: a structural parameter is defined, the instruments satisfy conditions that identify it, the sample estimator targets it, and the proposed instruments are credible in the setting. Each link matters. A nonzero first-stage relationship alone does not establish that an instrument is valid, and a precisely calculated 2SLS coefficient does not repair a failed exclusion restriction.

This tutorial develops that chain for one endogenous regressor and one or more excluded instruments, allowing for exogenous controls. The examples are hypothetical. The population discussion explains the estimand and identification; the sample calculation shows how 2SLS uses the instruments jointly.

Model and assumptions: For observations \(i=1,\ldots,n\), suppose \(Y_i=D_i\beta+X_i'\gamma+u_i\), where \(Y_i\) is the outcome, \(D_i\) is one potentially endogenous regressor, \(X_i\) is a vector of included exogenous controls, and \(u_i\) is the structural error. Let \(Z_i\) be a vector of excluded instruments. The standard linear IV conditions are: (1) the structural equation is correctly specified for the parameter of interest; (2) the instruments and included controls satisfy \(E[Z_i u_i]=0\) and \(E[X_i u_i]=0\); and (3) the instruments are relevant conditional on the controls, as expressed by the population rank condition below. These conditions identify \(\beta\) as a structural coefficient. A causal interpretation additionally requires that the structural equation and its assumptions represent the causal question of interest.

The exclusion restriction is part of the moment condition \(E[Z_i u_i]=0\): after accounting for \(D_i\) and \(X_i\), the instruments must not affect \(Y_i\) through another route represented in the error. It is not enough that an instrument predicts \(D_i\). The controls must also be exogenous as specified by \(E[X_i u_i]=0\); including a variable in \(X_i\) does not make it exogenous.

Population Identification and the Rank Condition

To express identification conditional on the controls, residualize the variables with respect to their population linear projections on \(X_i\). Write the resulting variables as \(\widetilde Y_i\), \(\widetilde D_i\), and the vector \(\widetilde Z_i\). Under the model and exogeneity conditions, the instrument moments imply

E[\widetilde Z_i\widetilde Y_i] = E[\widetilde Z_i\widetilde D_i]\beta.

There is one endogenous regressor, so the population rank condition is that \(E[\widetilde Z_i\widetilde D_i]\) is not the zero vector. It has rank one, which is the full column rank required to identify the single coefficient. If this vector is zero, the instruments provide no conditional population variation in \(D_i\), and the moment equations do not determine \(\beta\). If it is nonzero and the moment conditions hold, the coefficient is identified. This is a population condition; a nonzero sample association is not, by itself, proof of population relevance or instrument validity.

With multiple instruments, the moments may outnumber the unknown coefficient. A standard 2SLS estimator combines those moments by projecting the endogenous regressor onto the space spanned by the instruments and controls. The key identification requirement remains that this projection contains variation in the endogenous regressor after the controls have been removed.

Worked Example: Checking Population Rank

Suppose a hypothetical population has one endogenous regressor and two excluded instruments. After residualizing on the controls, the moments are \(E[\widetilde Z_i\widetilde Z_i']=I_2\), \(E[\widetilde Z_i\widetilde D_i]=(0.40,0.20)'\), and \(E[\widetilde Z_i\widetilde Y_i]=(1.12,0.56)'\). Check the rank condition and recover the coefficient.

Rank. The instrument–regressor moment vector is \((0.40,0.20)'\), which is not zero. Its rank is one, equal to the number of endogenous regressors. Thus the population rank condition holds.

Coefficient. The moment equation is \(h=g\beta\), where \(g=(0.40,0.20)'\) and \(h=(1.12,0.56)'\). Here \(h=2.8g\). Equivalently, using the identity instrument covariance as the weight,

\beta=(g'g)^{-1}g'h =\frac{0.40(1.12)+0.20(0.56)}{0.40^2+0.20^2} =\frac{0.56}{0.20} =2.8.

The calculation shows identification under the stated population moments. It does not establish that the instruments are credible in an application; the moment restrictions themselves still require substantive justification.

Sample 2SLS After Partialling Out Controls

For a sample calculation, let \(Y\), \(D\), \(X\), and \(Z\) collect the observations. Define \(M_X=I-X(X'X)^{-1}X'\), assuming \(X\) has full column rank. Then \(\widetilde Y=M_XY\), \(\widetilde D=M_XD\), and \(\widetilde Z=M_XZ\) are the sample residuals after linear projection on the included controls. Let \(P_{\widetilde Z}=\widetilde Z(\widetilde Z'\widetilde Z)^{-1}\widetilde Z'\), when the residualized instruments have full column rank.

The 2SLS coefficient on \(D\), after accounting for \(X\), is

\widehat{\beta}_{2SLS} = (\widetilde D'P_{\widetilde Z}\widetilde D)^{-1} \widetilde D'P_{\widetilde Z}\widetilde Y.

This formula makes the role of the instruments explicit: \(P_{\widetilde Z}\widetilde D\) is the part of the control-adjusted \(D\) explained by the control-adjusted instruments. The denominator must be nonzero. With one endogenous regressor, that is the sample counterpart of having instrument-predicted variation in \(D\) after controlling for \(X\).

Worked Example: Calculating 2SLS With Two Instruments

Consider hypothetical data on hourly earnings \(Y\), years of schooling \(D\), and two excluded instruments, with exogenous controls already partialled out. Suppose the residualized cross-products are \(\widetilde Z'\widetilde Z=\begin{pmatrix}8&0\\0&8\end{pmatrix}\), \(\widetilde Z'\widetilde D=(8,4)'\), and \(\widetilde Z'\widetilde Y=(20,16)'\). The sample size and cross-products are stipulated summaries of this hypothetical data; calculate the 2SLS coefficient.

The residualized instrument matrix is full rank because its diagonal entries are positive. Also, \(\widetilde Z'\widetilde D\ne 0\), so the instruments produce a nonzero fitted component of schooling. Since \((\widetilde Z'\widetilde Z)^{-1}=\begin{pmatrix}1/8&0\\0&1/8\end{pmatrix}\), the denominator is

\widetilde D'P_{\widetilde Z}\widetilde D = (\widetilde Z'\widetilde D)'(\widetilde Z'\widetilde Z)^{-1} (\widetilde Z'\widetilde D) = (8,4) \begin{pmatrix}1/8&0\\0&1/8\end{pmatrix} \begin{pmatrix}8\\4\end{pmatrix} =8+2=10.

The numerator is

\widetilde D'P_{\widetilde Z}\widetilde Y = (\widetilde Z'\widetilde D)'(\widetilde Z'\widetilde Z)^{-1} (\widetilde Z'\widetilde Y) = (8,4) \begin{pmatrix}1/8&0\\0&1/8\end{pmatrix} \begin{pmatrix}20\\16\end{pmatrix} =20+8=28.

Therefore \(\widehat{\beta}_{2SLS}=28/10=2.8\). Under the structural assumptions, this estimate represents an increase of 2.8 dollars in hourly earnings per additional year of schooling. The units follow from dollars per hour divided by years of schooling. The computation itself does not verify the exclusion restriction or justify a causal interpretation.

Interpreting the Coefficient and Assessing Credibility

The coefficient is a change in the outcome associated with a one-unit change in the endogenous regressor, conditional on the controls, under the stated structural model. To read it causally, the analyst must defend the assumptions that make the instrument moments valid. Under a constant linear effect, a credible IV estimate can be interpreted as that effect. If effects differ across individuals, a single linear IV coefficient need not equal the population average effect; its interpretation depends on the instrument and on the heterogeneity structure.

Instrument credibility is an argument about the data-generating process, not a label attached to a variable. Separate three questions:

Relevance: does the instrument predict the endogenous regressor conditional on the included controls? The population rank condition states the identification requirement; sample relevance is assessed from the observed first stage.
Independence or exogeneity: is the instrument unrelated to unobserved determinants of the outcome, conditional on the controls? Random assignment can support this claim, but implementation, attrition, or post-assignment selection may undermine it.
Exclusion: does the instrument affect the outcome only through the endogenous regressor, conditional on the controls? A plausible story must address alternative pathways, not merely say that the instrument was assigned externally.

These claims are logically distinct. Evidence that an instrument predicts \(D\) concerns relevance; it does not establish independence or exclusion. Likewise, failure to find an association between an instrument and a measured baseline variable does not prove independence from unmeasured determinants of the outcome.

Worked Example: Evaluating a Training Voucher Instrument

A hypothetical city randomly offers some unemployed residents a training voucher. Let \(Z\) indicate an offer, \(D\) indicate training participation, and \(Y\) be monthly earnings one year later. Suppose the offer raises participation by 0.18, and the average earnings difference between those offered and not offered is 360 dollars per month. The proposed IV analysis uses the offer as an instrument for participation.

Numerical estimate. In the simple binary-instrument setup, the reduced-form difference divided by the first-stage difference is

\widehat{\beta}_{IV} = \frac{360\ \text{dollars per month}}{0.18\ \text{participation probability}} = 2{,}000\ \text{dollars per month}.

This is the IV ratio estimate, in monthly dollars per unit change in participation. The first stage is nonzero in the hypothetical sample, so the offer predicts participation. Under a constant effect and valid instrument conditions, the estimate would represent the effect of participation on monthly earnings.

Credibility assessment. Random assignment supports independence of the offer from pre-existing determinants of earnings, provided assignment was implemented as planned and the analysis does not introduce selection that breaks that comparison. Exclusion is less automatic: if receiving a voucher also provides job-search services, transportation assistance, or a direct financial benefit, the offer could affect earnings without changing training participation. Those pathways would violate the proposed exclusion restriction unless adequately represented in the model. The estimate is therefore numerically defined, but its causal interpretation depends on whether such alternative pathways are absent or otherwise addressed.

Derivation: Why the Sample Formula Uses Projected Regressors

The 2SLS procedure can be understood from the joint regression of \(Y\) on \(D\) and \(X\) using the fitted component of \(D\) explained by \(Z\) and \(X\). After partialling out \(X\), that fitted component is \(\widehat{\widetilde D}=P_{\widetilde Z}\widetilde D\). The coefficient from regressing \(\widetilde Y\) on \(\widehat{\widetilde D}\) is

\frac{\widehat{\widetilde D}'\widetilde Y} {\widehat{\widetilde D}'\widehat{\widetilde D}} = \frac{\widetilde D'P_{\widetilde Z}'\widetilde Y} {\widetilde D'P_{\widetilde Z}'P_{\widetilde Z}\widetilde D} = \frac{\widetilde D'P_{\widetilde Z}\widetilde Y} {\widetilde D'P_{\widetilde Z}\widetilde D}.

The second equality uses the projection properties \(P_{\widetilde Z}'=P_{\widetilde Z}\) and \(P_{\widetilde Z}^2=P_{\widetilde Z}\). This gives the stated 2SLS formula. The derivation also explains a common error: using ordinary least squares standard errors from a regression on the fitted regressor as though it were observed data does not account for the IV estimation procedure. The coefficient formula and its appropriate inference are separate matters.

Common Mistakes and a Practical Summary

“The instrument predicts treatment, so it is valid.” Prediction addresses relevance only. Independence and exclusion require separate arguments grounded in the setting.
“The sample rank condition proves the instrument is strong and credible.” Rank is an identification condition, not a complete assessment of finite-sample information or validity.
“A 2SLS coefficient is automatically a causal effect.” The coefficient has a causal interpretation only if the structural model and instrument assumptions support it.
“A coefficient of 2.8 means 2.8 percent.” Units must be calculated from the outcome and regressor. If the outcome is dollars per hour and the regressor is years, the coefficient is dollars per hour per year.

A complete IV analysis therefore defines the structural coefficient, states the instrument and control assumptions, checks population identification, computes the estimator appropriate to the sample design, and evaluates the substantive case for instrument credibility. No single numerical diagnostic can substitute for this chain of reasoning.

Check Your Understanding

Use the model and reasoning in this tutorial to answer the following questions.

With one endogenous regressor and three excluded instruments, what population rank condition is required after residualizing on the controls?
Why does a nonzero sample first-stage association not establish the exclusion restriction?
In the 2SLS example, which cross-products determine the numerator and denominator of the coefficient?
An estimated coefficient is 4.5 when the outcome is measured in dollars per day and the endogenous regressor in treatment days. State its units and explain what additional assumptions are needed for a causal interpretation.
In the voucher example, name one feature that supports independence and one possible pathway that threatens exclusion.