ARE 212 - Problem Set 5

ARE 212 - Problem Set 5
Due May 1st
Part I: Theory (Optional)
1. Show that the parameter estimates for b2SLS and bIV are equivalent if we have a model with one endogenous
variable and one instrumental variable.
2. Prove that for E[Z，X] to be of full column rank, at least one of the θj in the linear projection xk = δo +δ1x1 +δ2x2 + . . . + δk1xk1 + θ1z1 + θ2z2 + . . . + θMzM + η has to be different from zero.
Part II (Applied): Instrumental Variables
1. Let’s revisit the model and data from problem set 4. We would like to estimate the model:log(wage) = βo+exper·β1+tenure·β2+married·β3+south·β4+urban·β5+black·β6+educ·β7+abil·γ+ε (1)
One of the big problems in the labor literature is that we do not (as econometricians) observe ability. If ability
is not correlated with any of the right hand side variables, we can include it in the disturbance and nothing is
lost by not observing it. If, however, it is correlated with one or more of the right hand side variables, OLS is
no longer unbiased or consistent. Assume that ability is correlated with education and none of the other right
hand side variables.
(a) Derive the bias of β7 and show what direction the bias goes in depending on whether the correlation
between ability and education is positive or negative.
(b) You showed in the first part that we can derive the sign/direction of the bias. One approach that has
been take in the literature is using a ”proxy” variable for the unobservable ability. We will use IQ here to
proxy for ability. Estimate the model above excluding ability, record your parameter estimates, standard
errors and R2.
(c) Estimate the model including IQ as a proxy, record your parameter estimates, standard errors and R2.
(d) What happens to returns to schooling? Does this result confirm your suspicion of how ability and schooling
are expected to be correlated?
2. This problem from Wooldridge asks you to try and recreate some of the results in Card(1995), which is on
reserve and on the website. Use the dataset card.raw on the website.
(a) Read the data in your favorite statistical program. Plot the series make sure your data are read in
correctly.

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(b) Estimate a log(wage) regression via Least Squares with educ, exper, exper2, black, south, smsa, reg661
through reg668 and smsa66 on the right hand side. Check your results against Table2, column 5.
(c) Estimate a reduced form equation for educ containing all of the explanatory variables and the dummy
variable nearc4. Is the partial correlation between nearc4 and educ statistically significant?
(d) Estimate the log(wage) equation by instrumental variables, using nearc4 as an instrument for educ.
Compare the 95% confidence interval for the return to educutioan to that obtained from the Least Squares
regression above.
(e) Now use multiple instruments. Use nearc2 and nearc4 as instruments for educ. Comment on the signifi-
cance of the partial correlations of both instruments in the reduced form. Show your standard errors from
the second stage and compare them to the correct standard errors.
(f) Conduct a Hausman test for endogeneity of educ. Report your test statistic, critical value and p-value.

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