5.2. a. Unobserved factors that tend to make an individual healthier also tend

Transcription

1 SOLUTIONS TO CHAPTER 5 PROBLEMS ^ ^ ^ ^ 5.1. Define x _ (z,y ) and x _ v, and let B _ (B,r ) be OLS estimator ^ ^ ^ ^ from (5.5), where B = (D,a ). Using the hint, B can also be obtained by partitioned regression: ^ (i) Regress x onto v and save the residuals, say 1 x 1. (ii) Regress y onto x. 1 1 ^ ^ But when we regress z1 onto v, the residuals are just z1 since v is N ^ orthogonal in sample to z. (More precisely, S z i1v i = 0.) Further, because i=1 ^ ^ ^ ^ we can write y = y + v, where y and v are orthogonal in sample, the ^ residuals from regressing y onto v are simply the first stage fitted values, ^ ^ y. In other words, x 1 = (z 1,y ). But the SLS estimator of B1 is obtained ^ exactly from the OLS regression y on z, y a. Unobserved factors that tend to make an individual healthier also tend to make that person exercise more. For example, if health is a cardiovascular measure, people with a history of heart problems are probably less likely to exercise. Unobserved factors such as prior health or family history are contained in u, and so we are worried about correlation between exercise and 1 u. Self self-selection into exercising predicts that the benefits of 1 exercising will be, on average, overestimated. Ideally, the amount of exercise could be randomized across a sample of people, but this can be difficult. b. If people do not systematically choose the location of their homes and jobs relative to health clubs based on unobserved health characteristics, then it is reasonable to believe that disthome and 7

2 distwork are uncorrelated with u. But the location of health clubs is not 1 necessarily exogenous. Clubs may tend to be built near neighborhoods where residents have higher income and wealth, on average, and these factors can certainly affect overall health. It may make sense to choose residents from neighborhoods with very similar characteristics but where one neighborhood is located near a health club. c. The reduced form for exercise is exercise = p 0 + p1age + pweight + p3height + p male + p work + p disthome + p distwork + u, For identification we need at least one of p and p to be different from 6 7 zero. This assumption can fail if the amount that people exercise is not systematically related to distances to the nearest health club. d. An F test of H : p = 0, p = 0 is the simplest way to test the identification assumption in part (c). As usual, it would be a good idea to compute a heteroskedasticty-robust version a. There may be unobserved health factors correlated with smoking behavior that affect infant birth weight. For example, women who smoke during pregnancy may, on average, drink more coffee or alcohol, or eat less nutritious meals. b. Basic economics says that packs should be negatively correlated with cigarette price, although the correlation might be small (especially because price is aggregated at the state level). At first glance it seems that cigarette price should be exogenous in equation (5.54), but we must be a little careful. One component of cigarette price is the state tax on cigarettes. States that have lower taxes on cigarettes may also have lower 8

3 quality of health care, on average. Quality of health care is in u, and so maybe cigarette price fails the exogeneity requirement for an IV. c. OLS is followed by SLS (IV, in this case):. reg lbwght male parity lfaminc packs Source SS df MS Number of obs = F( 4, 1383) = 1.55 Model Prob > F = Residual R-squared = Adj R-squared = 0.03 Total Root MSE = lbwght Coef. Std. Err. t P> t [95% Conf. Interval] male parity lfaminc packs _cons reg lbwght male parity lfaminc packs (male parity lfaminc cigprice) (SLS) Source SS df MS Number of obs = F( 4, 1383) =.39 Model Prob > F = Residual R-squared = Adj R-squared =. Total Root MSE =.3017 lbwght Coef. Std. Err. t P> t [95% Conf. Interval] packs male parity lfaminc _cons (Note that Stata automatically shifts endogenous explanatory variables to the beginning of the list when it reports coefficients, standard errors, and so on.) The difference between OLS and IV in the estimated effect of packs on 9

4 bwght is huge. With the OLS estimate, one more pack of cigarettes is estimated to reduce bwght by about 8.4%, and is statistically significant. The IV estimate has the opposite sign, is huge in magnitude, and is not statistically significant. The sign and size of the smoking effect are not realistic. d. We can see the problem with IV by estimating the reduced form for packs:. reg packs male parity lfaminc cigprice Source SS df MS Number of obs = F( 4, 1383) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.9448 packs Coef. Std. Err. t P> t [95% Conf. Interval] male parity lfaminc cigprice _cons The reduced form estimates show that cigprice does not significantly affect packs; in fact, the coefficient on cigprice is not the sign we expect. Thus, cigprice fails as an IV for packs because cigprice is not partially correlated with packs (with a sensible sign for the correlation). This is separate from the problem that cigprice may not truly be exogenous in the birth weight equation a. Here are the OLS results:. reg lwage educ exper expersq black south smsa reg661-reg668 smsa66 30

5 Source SS df MS Number of obs = F( 15, 994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.378 lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black south smsa reg reg reg reg reg reg reg reg smsa _cons The estimated return to education is about 7.5%, with a very large t statistic. These reproduce the estimates from Table, Column () in Card (1995). b. The reduced form for educ is. reg educ exper expersq black south smsa reg661-reg668 smsa66 nearc4 Source SS df MS Number of obs = F( 15, 994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = educ Coef. Std. Err. t P> t [95% Conf. Interval] exper expersq

6 black south smsa reg reg reg reg reg reg reg reg smsa nearc _cons The important coefficient is on nearc4. Statistically, educ and nearc4 are partially correlated, in a sensible way: holding other factors in the reduced form fixed, someone living near a four-year college at age 16 has, on average, almost one-third a year more education than a person not near a four-year college at age 16. This is not trivial effect, so nearc4 passes the requirement that it is partially correlated with educ. c. Here are the IV estimates:. reg lwage educ exper expersq black south smsa reg661-reg668 smsa66 (nearc4 exper expersq black south smsa reg661-reg668 smsa66) (SLS) Source SS df MS Number of obs = F( 15, 994) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black south smsa reg reg

7 reg reg reg reg reg reg smsa _cons The estimated return to education has increased to about 13.%, but notice how wide the 95% confidence interval is:.4% to 3.9%. By contrast, the OLS confidence interval is about 6.8% to 8.%, which is much tighter. Of course, OLS could be inconsistent, in which case a tighter CI is of little value. But the estimated return to education is higher with IV, something that seems a bit counterintuitive. One possible explanation is that educ suffers from classical errors-in-variables. Therefore, while OLS would tend to overestimate the return to schooling because of omitted "ability," the measurement error in educ leads to an attenuation bias. Measurement error may contribute to the larger IV estimate, but it is not especially convincing. It seems unlikely that educ satisfies the CEV assumptions. For example, if we think the measurement error is due to truncation -- people are asked about highest grade completed, not actually years of schooling -- then educ is always less than or equal to educ. The measurement error could not be independent of educ, either. If we think it is unobserved quality of schooling, then it seems likely that quality of schooling -- part of the measurement error -- is positively correlated with actual amount of schooling. This, too, violates the CEV assumptions. Another possibility for the much higher IV estimate comes out of the recent treatment effect literature, which is covered in Section Of course, we must also remember that the point estimates -- more precisly, the IV estimate -- is subject to substantial 33

8 sampling variation. At this point, we do not even know of OLS and IV are statistically different from each other. See Problem 6.1. d. When nearc is added to the reduced form of educ it has a coefficient (standard error) of.13 (.077), compared with.31 (.089) for nearc4. Therefore, nearc4 has a much stronger ceteris paribus relationship with educ; nearc is only marginally statistically significant once nearc4 has been included. The SLS estimate of the return to education becomes about 15.7%, with 95% CI given by 5.4% to 6%. The CI is still very wide Under the null hypothesis that q and z are uncorrelated, z and z are 1 exogenous in (5.55) because each is uncorrelated with u. Unfortunately, y 1 is correlated with u, and so the regression of y on z, y, z does not produce a consistent estimator of 0 on z even when E(z q) = 0. We could find ^ that J1 from this regression is statistically different from zero even when q and z are uncorrelated -- in which case we would incorrectly conclude that z is not a valid IV candidate. Or, we might fail to reject H 0: J 1 = 0 when z and q are correlated -- in which case we incorrectly conclude that the elements in z are valid as instruments. The point of this exercise is that one cannot simply add instrumental variable candidates in the structural equation and then test for significance of these variables using OLS. This is the sense in which identification cannot be tested. With a single endogenous variable, we must take a stand that at least one element of z is uncorrelated with q a. By definition, the reduced form is the linear projection 34

9 L(q 1,x,q ) = p + xp + p q, and we want to show that P 1 = 0 when q is uncorrelated with x. Now, because q is a linear function of q and a, and a is uncorrelated with x, q is uncorrelated with x if and only if q is uncorrelated with x. Assuming then that q and x are uncorrelated, q is also uncorrelated with x. A basic fact 1 about linear projections is that, because q and q are each uncorrelated with 1 the vector x, P = 0. This follows from Property LP.7: P can be obtained by 1 1 first projecting x on 1, q and obtaining the population residuals, say r. Then, project q onto r. But since x and q are orthogonal, r = x - M. 1 x Projecting q on (x - M ) just gives the zero vector because E[(x - M ) q ] = 1 x x 1 0. Therefore, P = 0. 1 b. If q and x are correlated then P $ 0, and x appears in the reduced 1 form for q. It is not realistic to assume that q and x are uncorrelated. 1 Under the multiple indicator assumptions, assuming x and q are uncorrelated is the same as assuming q and x are uncorrelated. If we believe q and x are uncorrelated then there is no need to collect indicators on q to consistently estimate B: we could simply put q into the error term and estimate B from an OLS regression of y on 1, x a. If we plug q = (1/d )q - (1/d )a into equation (5.45) we get y = b + b x b x + h q + v - h a, (5.56) K K where h 1 _ (1/d 1). Now, since the zh are redundant in (5.45), they are uncorrelated with the structural error, v (by definition of redundancy). Further, we have assumed that the z are uncorrelated with a. Since each x h 1 j is also uncorrelated with v - h a, we can estimate (5.56) by SLS using 1 1 instruments (1,x,...,x,z,z,...,z ) to get consistent of the b and h. 1 K 1 M j 1 35

10 Given all of the zero correlation assumptions, what we need for identification is that at least one of the z appears in the reduced form for h q. More formally, in the linear projection 1 q = p + p x p x + p z p z + r, K K K+1 1 K+M M 1 at least one of p,..., p must be different from zero. K+1 K+M b. We need family background variables to be redundant in the log(wage) equation once ability (and other factors, such as educ and exper), have been controlled for. The idea here is that family background may influence ability but should have no partial effect on log(wage) once ability has been accounted for. For the rank condition to hold, we need family background variables to be correlated with the indicator, q, say IQ, once the x have been netted 1 j out. This is likely to be true if we think that family background and ability are (partially) correlated. c. Applying the procedure to the data set in NLS80.RAW gives the following results:. reg lwage exper tenure educ married south urban black iq (exper tenure educ married south urban black meduc feduc sibs) Instrumental variables (SLS) regression Source SS df MS Number of obs = F( 8, 713) = 5.81 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] iq tenure educ married south urban

11 black exper _cons reg lwage exper tenure educ married south urban black kww (exper tenure educ married south urban black meduc feduc sibs) Instrumental variables (SLS) regression Source SS df MS Number of obs = F( 8, 713) = 5.70 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] kww tenure educ married south urban black exper _cons Even though there are 935 men in the sample, only 7 are used for the estimation, because data are missing on meduc and feduc. What we could do is define binary indicators for whether the corresponding variable is missing, set the missing values to zero, and then use the binary indicators as instruments along with meduc, feduc, and sibs. This would allow us to use all 935 observations. The return to education is estimated to be small and insignificant whether IQ or KWW used is used as the indicator. This could be because family background variables do not satisfy the appropriate redundancy condition, or they might be correlated with a. (In both first-stage regressions, the F 1 37

12 statistic for joint significance of meduc, feduc, and sibs have p-values below.00, so it seems the family background variables are sufficiently partially correlated with the ability indicators.) 5.8. a. Plug in the indicator q for q and the measurement x for x, being 1 K K sure to keep track of the errors: y = g + b x b x + g q + v - b e + g a, K K 1 1 K K 1 1 _ g 0 + b1x bkx K + g1q 1 + u where g 1 = 1/d 1. Now, if the variables z 1,..., zm are redundant in the structural equation (so they are uncorrelated with v), and uncorrelated with the measurement error e and the indicator error a, we can use these as IVs K 1 for x and q in SLS. We need M > because we have two explanatory K 1 variables, x and q, that are possibly correlated with the composite error u. K 1 b. The Stata results are. reg lwage exper tenure married south urban black educ iq (exper tenure married south urban black kww meduc feduc sibs) (SLS) Source SS df MS Number of obs = F( 8, 713) = Model Prob > F = Residual R-squared = Adj R-squared =. Total Root MSE =.4 lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ iq exper tenure married south urban black _cons

13 The estimated return to education is very large, but imprecisely estimated. The 95% confidence interval is very wide, and easily includes zero. Interestingly, the the coefficient on iq is actually negative, and not statistically different from zero. The large IV estimate of the return to education and the insignificant ability indicator lend some support to the idea that omitted ability is less of a problem than schooling measurement error in the standard log-wage model estimated by OLS. But the evidence is not very convincing given the very wide confidence interval for the educ coefficient Define q = b - b, so that b = b + q. Plugging this expression into the equation and rearranging gives log(wage) = b 0 + b1exper + bexper + b 3(twoyr + fouryr) + q4fouryr + u = b + b exper + b exper + b totcoll + q fouryr + u, where totcoll = twoyr + fouryr. Now, just estimate the latter equation by SLS using exper, exper, distyr and dist4yr as the full set of instruments. ^ We can use the t statistic on q to test H : q = 0 against H : q > ^ a. For b, the lower right hand element in the general formula (5.4) 1 with x = (1,x) and z = (1,z) is s [Cov(z,x) /Var(z)] ^ Alternatively, you can derive this formula directly by writing rn(b1 - b 1) = # -1/ N ( -1 N )$ N S (zi - z)u i/ N S (zi - z)(xi - x). Now, r zx = Cov(z,x) /(szs x), i=1 i=1 so simple algebra shows that the asymptotic variance is s /(rzxs x). The asymptotic variance for the OLS estimator is s /s. Thus, the difference is x 39

14 the presence of rzx in the denominator of the IV asymptotic variance. b. Naturally, as the error variance s increases so does the asymptotic variance of the IV estimator. More variation in x in the population is better: as sx increases the asymptotic variance decreases. These effects are identical to those for OLS. A larger correlation between z and x reduces the asymptotic variance of the IV estimator. As rzx L 0 the asymptotic variance increases without bound. This is why an instrument that is only weakly correlated with x can lead to very imprecise IV estimators Following the hint, let y be the linear projection of y on z, let a be the projection error, and assume that L is known. (The results on generated regressors in Section show that the argument carries over to 0 the case when L is estimated.) Plugging in y = y + a gives 0 y = z D + a y + a a + u Effectively, we regress y on z, y. The key consistency condition is that 1 1 each explanatory is orthogonal to the composite error, a a + u. By assumption, E(z u ) = 0. Further, E(y a ) = 0 by construction. The problem 1 is that E(z a ) $ 0 necessarily because z was not included in the linear 1 1 projection for y. Therefore, OLS will be inconsistent for all parameters in general. Contrast this with SLS when y is the projection on z and z : y 1 = y + r = zp + r, where E(z r ) = 0. The second step regression (assuming that P is known) is essentially y = z D + a y + a r + u Now, r is uncorrelated with z, and so E(z r ) = 0 and E(y r ) = 0. The 1 lesson is that one must be very careful if manually carrying out SLS by explicitly doing the first- and second-stage regressions. 40

15 5.1. This problem is essentially proven by the hint. Given the description of ^, the only way the K columns of ^ can be linearly dependent is if the last column can be written as a linear combination of the first K - 1 columns. This is true if and only if each qj is zero. Thus, if at least one qj is different from zero, rank ^ = K a. In a simple regression model with a single IV, the IV estimate of the ^ & N & N slope can be written as b 1 = S (zi - z)(yi - y) / S (zi - z)(xi - x) = 7i=1 8 7i=1 8 & N & N S z i(yi - y) / S z i(xi - x). Now the numerator can be written as 7i=1 8 7i=1 8 N N & N S z i(yi - y) = S ziyi - S zi y = N1y1 - N1y = N 1(y1 - y). i=1 i=1 7i=1 8 N where N 1 = S zi is the number of observations in the sample with z i = 1 and i= y1 is the average of the yi over the observations with z i = 1. Next, write y as a weighted average: y = (N 0/N)y 0 + (N 1/N)y 1, where the notation should be clear. Straightforward algebra shows that y1 - y = [(N - N 1)/N]y1 - (N 0/N)y = (N 0/N)(y1 - y 0). So the numerator of the IV estimate is (N0N 1/N)(y1 - y 0) The same argument shows that the denominator is (N0N 1/N)(x1 - x 0). Taking the ratio proves the result b. If x is also binary -- representing some "treatment" -- x1 is the fraction of observations receiving treatment when z i = 1 and x0 is the fraction receiving treatment when z i = 0. So, suppose x i = 1 if person i participates in a job training program, and let z = 1 if person i is eligible i - for participation in the program. Then x1 is the fraction of people participating in the program out of those made eligibile, and x0 is the fraction of people participating who are not eligible. (When eligibility is necessary for participation, x = 0.) Generally, x - x is the difference in

16 participation rates when z = 1 and z = 0. So the difference in the mean response between the z = 1 and z = 0 groups gets divided by the difference in participation rates across the two groups a. Taking the linear projection of (5.1) under the assumption that (x,...,x,z,...,z ) are uncorrelated with u gives 1 K-1 1 M L(y z) = b + b x b x + b L(x z) + L(u z) K-1 K-1 K K since L(u z) = 0. = b + b x b x + b x K-1 K-1 K K b. By the law of iterated projections, L(y 1,x,...,x,x ) = b + b x 1 K-1 K b x + b x. Consistency of OLS for the b from the regression y K-1 K-1 K K j on 1, x,..., x, x follows immediately from our treatment of OLS from 1 K-1 K Chapter 4: OLS consistently estimates the parameters in a linear projection provided we can assume no perfect collinearity in (1,x,...,x,x ). 1 K-1 K c. I should have said explicitly to assume E(z z) is nonsingular -- that is, SLS.a holds. Then, x is not a perfect linear combination of K (x,...,x ) if and only if at least one element of z,..., z has a nonzero 1 K-1 1 M coefficient in L(x 1,x,...,x,z,...,z ). In the model with a single K 1 K-1 1 M endogenous explanatory variable, we know this condition is equivalent to Assumption SLS.b, the standard rank condition. ( ) ^ In L(x z) = z^, we can write ^ =, where I is the K x K ^1 IK 9 0 K identity matrix, 0 is the L x K zero matrix, ^ is L x K, and ^ is K x K. As in Problem 5.1, the rank condition holds if and only if rank(^) = K. 1 a. If for some x, the vector z does not appear in L(x z), then ^ has j 1 j 11 4

17 a column which is entirely zeros. But then that column of ^ can be written as a linear combination of the last K elements of ^, which means rank(^) < K. Therefore, a necessary condition for the rank condition is that no columns of ^11 be exactly zero, which means that at least one zh must appear in the reduced form of each x, j = 1,...,K. j 1 b. Suppose K = and L =, where z appears in the reduced form form both x and x, but z appears in neither reduced form. Then the x matrix 1 ^11 has zeros in its second row, which means that the second row of ^ is all zeros. It cannot have rank K, in that case. Intuitively, while we began with two instruments, only one of them turned out to be partially correlated with x and x. 1 c. Without loss of generality, we assume that z appears in the reduced j form for x ; we can simply reorder the elements of z to ensure this is the j 1 case. Then ^11 is a K1 x K1 diagonal matrix with nonzero diagonal elements. ( ) ^11 0 Looking at ^ =, we see that if ^11 is diagonal with all nonzero ^1 I 9 K0 diagonals then ^ is lower triangular with all nonzero diagonal elements. Therefore, rank ^ = K. 43