Instrumental Variable (IV) Regression ECO 311- Class Notes Instrumental Variable (IV) Regression Prof. Erdinç American University in Bulgaria

Motivation: When is it useful? Intuition Instrumental Variables Motivation: When is it useful? Intuition Use: What are the requirements for good instruments?

Suppose you are interested in the wage returns to education. Motivation Suppose you are interested in the wage returns to education. Is it reasonable to assume and ? Are educ and exp independent of the error term? Are the unobserved determinants of wages uncorrelated with education and exp? Are factors which might influence wages and how much schooling one gets? --intelligence: it might influence wages and educ (so in error term it hides, and makes educ to be correlated with u)  

Motivation --persistence -- good health? –good social network ---need to control for these.. Endogeneity issue: educ and u are contemporanously correlated! (intelligence in u and educ)

all regressors, including educ must have Basic Idea If we do not control for these potential confounders, our OLS estimators will be biased. (see the exogeneity requirement for OLS consistency: all regressors, including educ must have We would like to use the variation in education that is not correlated with the error term, u. (potential instruments!) An instrument is a variable that determines the endogenous regressor but affects the dependent variable only through its effect on the independent variables. It can not have a direct effect on wages.  

Potential instruments for education: Basic Idea Potential instruments for education: -- Distance to nearest college (Card, 1995) --Quarter of birth (Angrist & Krueger, 1991) Call the instrument, Z. We would like the isntrument to be independent of the error term. But It must be sufficiently correlated with what it is instrumenting..  

Two Stage Least Squares First Stage: Regress educ on all regressors inc. Z Get the estimated regression (first stage) educ^ is an unbiased estimate of educ. Why? (Z, exp are uncorrelated with u in the wage equation, so its linear combination is also uncorrelated with u. Hence, educ^ is also uncorrelated with u. Therefore, it can be used in the wage equation. This is the second stage equation. Now, beta 1 hat will be an unbiased estimate of beta.

Weak Instruments Problem Correlation of instrument and the endogenous variable must be sufficiently strong, i.e. Rule of Thumb: Do an F test on the null hypothesis that the coefficients on the instruments in the first stage are zero. (F should be at least 10; preferably 12.) We must reject the null (Stata: estat firststage command gives Partial Rsq, and F value along with the critical values) IV estimates are as good as the instruments they use. This is relatively easier to assess than the next issue: The validity of the instruments…  

Validity of the Instruments If an instrument has a direct causal effect on the dependent variable, it is correlated with the errors in the equation and it is not valid (it should only influence, be correlated with what it is instrumenting, e.g. educ), so we must have We must have another requirement in place for a valid instrument which is But this is much more difficult to test empirically. Exception: If you have more than one plausible instrument, you can see if both give you the same answers in the first stage (Sargan test) The IV estimator is asymptotically consistent but biased towards OLS in finite samples. The size of the bias is inversely related to the size of the sample but positively related to the weakness of the instruments.  

Tests in Stata After ivregress 2sls Y X (endo_var Z) Conduct estat endog (Durbin-Wu test for endogenity) estat firststage gives you the partial Rsq and F test for the strength of the instruments. estat overid gives you the test for the validity of the instruments (provided you have more than 1 instruments for each endogenous variable): Sargan Test  

IV and Simultaneous Equations Consider the two simultaneous equations (structural equations): Why are y1 and y2 endogenous? (1) (2) First, find the reduced form for this model (express both y1 and y2 in terms of the exogenous variables z1 and z2) and estimate these two (first stage) by OLS. Second, Use the estimated y1^ and y2^ from the first stage reduced form in the second stage structural equations by replacing y2 with y2^ in the (1) equation and by replacing y1 with y1^ in the (2) equation.