A Guide to Modern Econometrics / 2ed Answers to selected exercises - Chapter 5

Exercise 5.1 a. The essential conditions for b to be unbiased for are that Ef"i g = 0 and that f"1 ; :::; "N g is independent of fx1 ; :::; xN g; see Subsection 2.3.2. The essential condition for b to be consistent is Ef"i xi g = 0 (plus some regularity conditions). An estimator that is unbiased provides, in repeated sampling, an estimate that is –on average –equal to the true value. That is, we do not expect the estimator to be systematically too high or too low, no matter how big or small the sample is. An estimator is consistent if its probability limit is equal to the true value. That is, the probability that the estimator di¤ers from the true value becomes in…nitesimally small when the sample size becomes in…nitely large. While we hope that the estimator does not systematically under- or overestimate the true value when the sample size is reasonably large, this result is only true in the limit. b. We can write Ef"i xi g = 0 as x0i )xi g = 0:

Ef(yi

This is a set of moment conditions as the expression in curly brackets is a function of observable data and unknown parameters. A method of moments estimator is obtained by taking the sample equivalent of the above expectations, setting it to zero and then solving for the unknown parameters. This reproduces the OLS estimator b: c. One example is the case of endogenous regressors: the regressor xi3 is jointly determined with yi (or the same unobservables a¤ect both xi3 and yi ): An empirical example is the estimation of the returns to schooling: the level of schooling is typically related to the same unobservables (ability, intelligence) that also a¤ect a person’s wage. A second example is measurement error in the regressor xi3 (see Subsection 5.2.2). d. No. If Ef"i xi g = 6 0 the OLS estimator is biased and inconsistent, no matter what other assumptions we are making. Correcting standard errors does not solve the problem. e. An instrumental variable, zi , say, gives rise to a new moment condition that can replace the invalid one. In this case, we have the following three moment conditions: Ef(yi x0i )g = 0; Ef(yi x0i )xi2 g = 0; Ef(yi x0i )zi g = 0: 1

These moment conditions typically provide su¢ cient information to identify the three parameters in : Applying GMM results in the standard instrumental variables estimator. Essential conditions are that zi is uncorrelated to "i ; but correlated with xi3 : f. OLS minimizes the residual sum of squares and therefore maximizes the R2 : Any other estimator, including instrumental variables, results in a lower R2 : Note that we are typically not interested in obtaining an R2 that is as high a possible, but in obtaining consistent (or unbiased) estimates for the coe¢ cients of interest that are as accurate as possible. The R2 does not tell us which estimator is the preferred one. The R2 tells us how well the model …ts the data (in a given sample) and typically is only interpreted in this way when the model is estimated by ordinary least squares. g. We cannot use xi2 as instrument for xi3 because xi2 is already included in the model. The corresponding moment condition is based on Efxi2 "i g = 0 and this is already exploited. In theory, it is possible to use x2i2 as an instrument for xi3 : This produces the additional moment condition Ef(yi x0i )x2i2 g = 0: However, identi…cation is then based upon the assumption that the functional form of the model is correct and that x2i2 is appropriately excluded from the model. Economic theory rarely allows us to be that con…dent in specifying our functional form. In practice one should view a linear model as only an approximation, and employing such arti…cial instruments can be easily criticized. Exercise 5.2 a. Estimating the reduced form for schooling while including father’s and mother’s education levels (instead of the lived near college dummy) produces the following (OLS) results. Dependent Variable: ED76 Method: Least Squares Date: 01/03/06 Time: 11:30 Sample: 1 3010 Included observations: 3010 Variable

Coefficient

C AGE76 AGE76^2 BLACK SMSA76 SOUTH76 DADED MOMED

-4.832780 0.971399 -0.016765 -0.794501 0.657032 -0.262371 0.181403 0.214085

Std. Error 3.945506 0.276573 0.004800 0.109680 0.095567 0.093429 0.015551 0.017114

t-Statistic -1.224882 3.512265 -3.492362 -7.243840 6.875129 -2.808244 11.66513 12.50967

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Prob. 0.2207 0.0005 0.0005 0.0000 0.0000 0.0050 0.0000 0.0000

R-squared 0.257771 Adjusted R-squared 0.256040 S.E. of regression 2.308921 Sum squared resid 16004.01 Log likelihood -6785.708 Durbin-Watson stat 1.804996

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

13.26346 2.676913 4.514092 4.530064 148.9389 0.000000

The t-statistics on the two parents’education variables indicate that these two variables are highly signi…cant. This indicates that father’s and mother’s education exhibit signi…cant correlation with schooling, after correcting for the impact of age, black, smsa and south. Also note that the R2 of this reduced form (0.2578) is substantially higher than the one reported for the speci…cation in Table 5.2 (0.1185). This is a good signal because instruments have to be correlated with the variable they are supposed to instrument. It does not, however, indicate that the instruments are valid in the sense that they are uncorrelated with the wage equation’s error term. b. We estimate the wage equation, using age and age squared as instruments for experience and its square, and the two parents’education variables as instruments for schooling. The results are as follows (Eviews 3.1): Dependent Variable: LWAGE76 Method: Two-Stage Least Squares Date: 01/03/06 Time: 11:50 Sample: 1 3010 Included observations: 3010 Instrument list: C MOMED DADED AGE76 AGE76^2 BLACK SMSA76 SOUTH76 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C 4.650018 ED76 0.082404 EXP76 0.076783 EXP762 -0.001889 BLACK -0.177202 SMSA76 0.154015 SOUTH76 -0.121326

0.119828 0.007137 0.016321 0.000819 0.019844 0.016672 0.015502

38.80571 11.54523 4.704667 -2.307644 -8.929747 9.237944 -7.826282

R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic)

0.287993 0.286571 0.374852 176.9404 0.000000

Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat

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0.0000 0.0000 0.0000 0.0211 0.0000 0.0000 0.0000 6.261832 0.443798 421.9647 1.864384

According to this speci…cation, the returns to schooling are estimated at somewhat more than 8% per year of schooling. This is less than the estimate obtained in Table 5.3, using lived near college as an instrument, and actually quite close to the OLS estimate on the returns to schooling reported in Table 5.1 (0.0740). c. A simple way to compute the overidentifying restrictions test is obtained from the auxiliary regression of the IV residuals upon the full set of instruments (see p. 147). The results of this are as follows: Dependent Variable: RES Method: Least Squares Date: 01/03/06 Time: 12:08 Sample: 1 3010 Included observations: 3010 Variable C MOMED DADED AGE76 AGE76^2 BLACK SMSA76 SOUTH76

Coefficient 0.098033 0.003529 -0.003222 -0.007205 0.000124 0.000848 0.000725 0.000219

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Std. Error 0.640424 0.002778 0.002524 0.044893 0.000779 0.017803 0.015512 0.015165

0.000731 -0.001599 0.374778 421.6562 -1312.920 1.865270

t-Statistic 0.153075 1.270502 -1.276635 -0.160492 0.159465 0.047622 0.046728 0.014469

Prob.

0.8783 0.2040 0.2018 0.8725 0.8733 0.9620 0.9627 0.9885

Mean dependent var -8.07E-15 S.D. dependent var 0.374479 Akaike info criterion 0.877688 Schwarz criterion 0.893660 F-statistic 0.313789 Prob(F-statistic) 0.948102

Note that the R2 of this regression would be zero by construction in the exactly identi…ed case. In the present situation, we have one overidentifying restriction and the overidentifying restrictions test is obtained as N R2 of this regression. This results in a value of 2:20. For a Chi-squared distribution with one degree of freedom, this does not imply a rejection. While the overidentifying restriction is not rejected, this does not necessarily imply that both instruments are valid. d. We add the instrument “lived near college”to the model estimated under b. and obtain the following results: Dependent Variable: LWAGE76 4

Method: Two-Stage Least Squares Date: 01/03/06 Time: 12:17 Sample: 1 3010 Included observations: 3010 Instrument list: C MOMED DADED AGE76 AGE76^2 BLACK SMSA76 SOUTH76 NEARC4 Variable

Coefficient

Std. Error

C 4.637434 ED76 0.083228 EXP76 0.077208 EXP762 -0.001910 BLACK -0.176023 SMSA76 0.153210 SOUTH76 -0.120878

0.119328 0.007101 0.016321 0.000819 0.019822 0.016661 0.015502

R-squared Adjusted R-squared S.E. of regression F-statistic Prob(F-statistic)

0.287572 0.286148 0.374964 177.2298 0.000000

t-Statistic

38.86294 11.72066 4.730572 -2.333005 -8.880406 9.195623 -7.797699

Prob.

0.0000 0.0000 0.0000 0.0197 0.0000 0.0000 0.0000

Mean dependent var S.D. dependent var Sum squared resid Durbin-Watson stat

6.261832 0.443798 422.2148 1.864548

When we test the two overidentifying restrictions in a similar way as under c., we obtain the test statistic 3010 R2 = 3010 0:001142 = 3:4374; which does not allow us to reject the null hypothesis. Accordingly, if we would be con…dent that lived near college is a valid instrument, there is no evidence that parents’education is inappropriate as an (additional) instrument. e. The estimates for the returns to schooling are 0.133 (0.051) if we use lived near college as an instrument, 0.082 (0.007) if we use father’s and mother’s education, and 0.083 (0.007) if we use all three variables as instruments. Note that using parents’ education substantially improves the precision of the IV estimates. This is not surprising given the signi…cance in the reduced form of the extra instruments (cf. Table 5.2, and part a.). Statistically, we do not …nd any evidence that would lead us to reject the validity of parents’education as instruments. However, some may argue that parent’s education is partly determined by the same unobservables that determine a kid’s schooling.

c 2006, John Wiley and Sons

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