Suggested Answers Problem set 4 ECON 60303

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1 Suggested Answers Problem set 4 ECON Bill Evans Spring 04. A program that answers part A) is on the web page and is named psid_iv_comparison.do. Below are some key results and a summary table is included. The standard errors are so much smaller in the fixedeffect estimate because the RMSE is ½ the size the fixed-effects explain a large fraction of the variance in outcomes.. * get fixed effects estimates;. areg wagel union tenure, absorb(id); Linear regression, absorbing indicators Number of obs = 3945 F(, 354) = 36.3 Prob > F = R-squared = Adj R-squared = Root MSE =.3987 wagel Coef. Std. Err. t P> t [95% Conf. Interval] union tenure _cons id F(788, 354) = (789 categories). * check that whin panel means have zero mean;. sum uniond tenured; Variable Obs Mean Std. Dev. Min Max uniond e tenured e * run sls;. reg wagel union tenure (uniond tenured); Instrumental variables (SLS) regression Source SS df MS Number of obs = F(, 394) = 7.37 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.5345 wagel Coef. Std. Err. t P> t [95% Conf. Interval] union tenure _cons

2 Parameter Estimates and Standard errors Fixed Covariates Effect Union (0.040) Tenure 0.0 (0.005) SLS (0.0534) 0.0 (0.003). a) It is easy to show that the instrument is uncorrelated wh u i.. The correlation coefficient between u i and X for the sample is by definion n T ( X )( ) i t X ui u. Working wh the n summation terms, note that we can drop one of the means so drop u and note that by construction X =0 so n T Xu i t i n. Because u i does not vary whin t, we can wre this as n T T u i i X t n. Note that by construction, X 0 t so by construction 0. b) Sort the data by person then year. Let Z X and because the data is sorted by person then year Z X MX where M inm tand Mt It '. The IV estimate for is t ( Z ' X ) Z ' Y [( MX )' X ] [( MX )' Y] [ X ' MX ] [ X ' MY ] which is by definion the fixed effect estimate. 3. Let var( ) and var( ). 0 0 var Var g ' g were g and therefore Var 4 t( ) t( ) t( ) 4 4

3 The reason why this result is important is that consider a well-specified SLS model wh a strong first stage. This means that if t( ) is large then 0. We can then see that t( ) t( ) t( ). Therefore, wh a strong first stage, the t-statistic on the SLS estimate will be t( ) functionally the same as the t-statistic on the reduced form. 4. We answered this in class. They key point of the monotonicy assumption is that the upper right corner box in the x box is the empty set. Suppose there are two types of families: those that prefer a mix so they try for more kids when endowed wh two boys or two girls, and families that prefer more boys than girls, so when they are endowed wh a mix of boys and girls, they try for a third. In this case, the numerator and denominator in the Wald estimate will contain a mixture of two groups of people and as a result, is not clear for what group the IV estimate is measuring. m 5. It is easiest to first show why the model works when we use z0) in h ( z i). To make things easy, assume that ρ= and hence, in the first stage, h ( z ) D ( z z ) ( D ) ( z z ). i i i 0 i i 0 In this case x D w D ( z z ) ( D ) ( z z ) v and i 0 i i i i 0 i i 0 i E[ x D ] w D ( z z ) i i 0 i i i 0 E[ x D 0] w ( D ) ( z z ) i i 0 i i i 0 ATT = E[ x D ] E[ x D 0] D ( z z ) ( D ) ( z z ) i i i i i i 0 i i 0 For people that are just below the cutoff, regardless of the, ( Di) z0) 0because z0) 0. For people right above the cutoff, Di z0) 0 again because z0) 0. Therefore ATT collapses to ATT E[ xi Di ] E[ xi Di 0] When we so not rescale the running variable x D w D z ( D ) z v and i 0 i i i i i i i hence E[ xi Di ] E[ xi Di 0] Di zi ( Di ) zi. Now, the running variable terms do not drop out even though they are equal in value (z). Empirical portion The program that answers questions -7 is called twinsta.do and the program for question 8 is called twinst_random_instrument.do.. Although twins by construction increase the number of children in the house by when they are first born, the coefficient on twinst in the first stage is only 0.7. The coefficient is < because the constraint 3

4 generated by twins may not bind over time. Suppose a woman was planning on having two kids: one at age and another at age 5. If she had a twin at age, the coefficient on winst in the st stage at that age would be however, by age 6, the coefficient would have fallen to zero because the mother would have had the second child anyway. Note the R in the first stage is 0.5 and the ratio OLS SLS Var( ) / Var( ) for weeks worked is (0.5745) /(.4756) =0.5. Note that in 5 of 6 cases, there is a statistically significant in the exogenous covariates difference between mothers wh and whout twins. Although twins are thought to be random, the correlation between twin status and observed characteristics gives one pause for concern that the results are subjected to omted variables bias. 3. Note that there is a slight different in the SLS estimates in the case of the Wald estimate and the one controlling for covariates. This is because the control variables are correlated wh the twin instrument. 4. The correlaton coefficient is 0.39, so he correlation between z and ε must be about 0.4 the rate of x and ε in order for the SLS estimate to be more consistent that OLS. 5. Note that the st stage coefficients increase as we interact age at first birth wh twin. In general, the twin birth should be more of a shock to fertily the older the mother which is exactly the case. The first-stage F statistic indicates there is no concern about fine sample bias. 6. The p-value on the test of over-identifying restrictions indicates that we cannot reject the null the model is appropriately specified. Note the similary of the estimates between the over-identified model and the just identified model in the previous table. 7. Results from the simulation are in the table below. The key aspect of the simulation is that as you add junk to the model in the form of 5, 0 and 30 instruments, the first stage F plummets. Notice as well that as this F falls, the SLS estimate systematically moves towards the OLS estimate. Therefore, adding useless instruments to the model does come at a cost in that the model approaches the OLS estimates. Answers for Question st stage Reduced-form Wald Outcome: Second weeks worked Weeks Worked Second (0.407) Twinst (0.009) (.476) (0.03) R Answers for Problem 3 Coefficient on twinst and (standard error) educ agefts agem Whe black other_race 0.7 (0.045) (0.064) 0.5 (0.087) (0.009) 4

5 Answers for Question 4 Controlling for covariates st stage OLS SLS Outcome: Second weeks worked Weeks Worked Second (0.577) (0.03) (.40) (0.030) Twinst 0.83 st stage F 57 R Correlation coefficient (second, twinst)=0.39 Answers for question 6/7 Controlling for covariates st stage OLS Wald Outcome: Second Weeks worked Weeks worked Second (0.577) Twinst x agefst< (0.008) Twinst x 0 agefst< (0.008) Twinst x Agefst> (0.0) (0.03) (.370) (0.030) st stage F Overid test (p-value).90 (0.39).9 (0.38) R Answers for question 8 Estimation Method Instruments st stage F (actual or average) OLS or SLS (actual or average) OLS SLS Mysteryz SLS 000 replications, mysterz and random instruments SLS 000 replications, mysterz and random instruments SLS 000 replications, mysterz and random instruments SLS 000 replications, 0 random instruments