伍德里奇計量經濟學導論第四版ch13_khdaw

1、 139CHAPTER 13TEACHING NOTESWhile this chapter falls under “Advanced Topics,” most of this chapter requires no more sophistication than the previous chapters. (In fact, I would argue that, with the possible exception of Section 13.5, this material is easier than some of the time series chapters.Pool

2、ing two or more independent cross sections is a straightforward extension of cross-sectional methods. Nothing new needs to be done in stating assumptions, except possibly mentioning that random sampling in each time period is sufficient. The practically important issue is allowing for different inte

3、rcepts, and possibly different slopes, across time.The natural experiment material and extensions of the difference-in-differences estimator is widely applicable and, with the aid of the examples, easy to understand.Two years of panel data are often available, in which case differencing across time

4、is a simple way of removing g unobserved heterogeneity. If you have covered Chapter 9, you might compare this with a regression in levels using the second year of data, but where a lagged dependent variable is included. (The second approach only requires collecting information on the dependent varia

5、ble in a previous year. These often give similar answers. Two years of panel data, collected before and after a policy change, can be very powerful for policy analysis.Having more than two periods of panel data causes slight complications in that the errors in the differenced equation may be seriall

6、y correlated. (However, the traditional assumption that the errors in the original equation are serially uncorrelated is not always a good one. In other words, it is not always more appropriate to used fixed effects, as in Chapter 14, than first differencing. With large N and relatively small T , a

7、simple way to account for possible serial correlation after differencing is to compute standard errors that are robust to arbitrary serial correlation and heteroskedasticity. Econometrics packages that do cluster analysis (such as Stata often allow this by specifying each cross-sectional unit as its

8、 own cluster.課后答案網 w w w .k h d a w .c o m 140SOLUTIONS TO PROBLEMS13.1 Without changes in the averages of any explanatory variables, the average fertility rate fell by .545 between 1972 and 1984; this is simply the coefficient on y84. To account for theincrease in average education levels, we obtai

9、n an additional effect: .128(13.3 12.2 .141. So the drop in average fertility if the average education level increased by 1.1 is .545 + .141 = .686, or roughly two-thirds of a child per woman.13.2 The first equation omits the 1981 year dummy variable, y81, and so does not allow any appreciation in n

10、ominal housing prices over the three year period in the absence of anincinerator. The interaction term in this case is simply picking up the fact that even homes that are near the incinerator site have appreciated in value over the three years. This equation suffers from omitted variable bias.The se

11、cond equation omits the dummy variable for being near the incinerator site, nearinc , which means it does not allow for systematic differences in homes near and far from the site before the site was built. If, as seems to be the case, the incinerator was located closer to less valuable homes, then o

12、mitting nearinc attributes lower housing prices too much to theincinerator effect. Again, we have an omitted variable problem. This is why equation (13.9 (or, even better, the equation that adds a full set of controls, is preferred.13.3 We do not have repeated observations on the same cross-sectiona

13、l units in each time period, and so it makes no sense to look for pairs to difference. For example, in Example 13.1, it is very unlikely that the same woman appears in more than one year, as new random samples are obtained in each year. In Example 13.3, some houses may appear in the sample for both

14、1978 and 1981, but the overlap is usually too small to do a true panel data analysis.13.4 The sign of 1 does not affect the direction of bias in the OLS estimator of 1, but only whether we underestimate or overestimate the effect of interest. If we write crmrte i = 0 + 1unem i + u i , where u i and

15、unem i are negatively correlated, then there is a downward bias in the OLS estimator of 1. Because 1 > 0, we will tend to underestimate the effect of unemployment on crime.13.5 No, we cannot include age as an explanatory variable in the original model. Each person in the panel data set is exactly

16、 two years older on January 31, 1992 than on January 31, 1990. This means that age i = 2 for all i . But the equation we would estimate is of the formsaving i = 0 + 1age i + ,where 0 is the coefficient the year dummy for 1992 in the original model. As we know, when we have an intercept in the model

17、we cannot include an explanatory variable that is constant across i; this violates Assumption MLR.3. Intuitively, since age changes by the same amount for everyone, we cannot distinguish the effect of age from the aggregate time effect.課后答案網 w w w .k h d a w .c o m 14113.6 (i Let FL be a binary vari

18、able equal to one if a person lives in Florida, and zero otherwise. Let y90 be a year dummy variable for 1990. Then, from equation (13.10, we have the linear probability modelarrest = 0 + 0y90 + 1FL + 1y90FL + u .The effect of the law is measured by 1, which is the change in the probability of drunk

19、 driving arrest due to the new law in Florida. Including y90 allows for aggregate trends in drunk driving arrests that would affect both states; including FL allows for systematic differences between Florida and Georgia in either drunk driving behavior or law enforcement.(ii It could be that the pop

20、ulations of drivers in the two states change in different ways over time. For example, age, race, or gender distributions may have changed. The levels of education across the two states may have changed. As these factors might affect whether someone isarrested for drunk driving, it could be importan

21、t to control for them. At a minimum, there is the possibility of obtaining a more precise estimator of 1 by reducing the error variance. Essentially, any explanatory variable that affects arrest can be used for this purpose. (See Section 6.3 for discussion.13.7 (i It is not surprising that the coeff

22、icient on the interaction term changes little whenafchnge is dropped from the equation because the coefficient on afchnge in (3.12 is only .0077 (and its t statistic is very small. The increase from .191 to .198 is easily explained by sampling error.(ii If highearn is dropped from the equation so th

23、at 10= in (3.10, then we are assuming that, prior to the change in policy, there is no difference in average duration between high earners and low earners. But the very large (.256, highly statistically significant estimate on highearn in (3.12 shows this presumption to be false. Prior to the policy

24、 change, the high earning group spent about 29.2% exp(.2561.292 longer on unemployment compensation than the low earning group. By dropping highearn from the regression, we attribute to the policy change the difference between the two groups that would be observed without any intervention.SOLUTIONS

25、TO COMPUTER EXERCISESC13.1 (i The F statistic (with 4 and 1,111 df is about 1.16 and p -value .328, which shows that the living environment variables are jointly insignificant.(ii The F statistic (with 3 and 1,111 df is about 3.01 and p -value .029, and so the region dummy variables are jointly sign

26、ificant at the 5% level.(iii After obtaining the OLS residuals, u, from estimating the model in Table 13.1, we run the regression 2uon y74, y76, , y84 using all 1,129 observations. The null hypothesis of homoskedasticity is H 0: 1 = 0, 2 = 0, , 6 = 0. So we just use the usual F statistic for joint 課

27、后答案網 w w w .k h d a w .c o m 142significance of the year dummies. The R -squared is about .0153 and F 2.90; with 6 and 1,122 df , the p -value is about .0082. So there is evidence of heteroskedasticity that is a function of time at the 1% significance level. This suggests that, at a minimum, we shou

28、ld compute heteroskedasticity-robust standard errors, t statistics, and F statistics. We could also useweighted least squares (although the form of heteroskedasticity used here may not be sufficient; it does not depend on educ , age , and so on.(iv Adding y74educ , , y84educ allows the relationship

29、between fertility and education to be different in each year; remember, the coefficient on the interaction gets added to the coefficient on educ to get the slope for the appropriate year. When these interaction terms are added to the equation, R 2 .137. The F statistic for joint significance (with 6

30、 and 1,105 df is about 1.48 with p -value .18. Thus, the interactions are not jointly significant at even the 10% level. This is a bit misleading, however. An abbreviated equation (which just shows the coefficients on the terms involving educ iskids = 8.48 .023 educ + .056 y74educ .092 y76educ(3.13

31、(.054 (.073 (.071 .152 y78educ .098 y80educ .139 y82educ .176 y84educ . (.075 (.070 (.068 (.070Three of the interaction terms, y78educ , y82educ , and y84educ are statistically significant at the 5% level against a two-sided alternative, with the p -value on the latter being about .012. The coeffici

32、ents are large in magnitude as well. The coefficient on educ which is for the base year, 1972 is small and insignificant, suggesting little if any relationship between fertility andeducation in the early seventies. The estimates above are consistent with fertility becoming more linked to education a

33、s the years pass. The F statistic is insignificant because we are testing some insignificant coefficients along with some significant ones.C13.2 (i The coefficient on y85 is roughly the proportionate change in wage for a male(female = 0 with zero years of education (educ = 0. This is not especially

34、useful because the U.S. working population without any education is a small group; such people are in no way “typical.”(ii What we want to estimate is 0 = 0 + 121; this is the change in the intercept for a male with 12 years of education, where we also hold other factors fixed. If we write 0 = 0 121

35、, plug this into (13.1, and rearrange, we getlog(wage = 0 + 0y85 + 1educ + 1y85(educ 12 + 2exper + 3exper 2+4union + 5female + 5y85female + u .Therefore, we simply replace y85educ with y85(educ 12, and then the coefficient and standard error we want is on y85. These turn out to be 0 = .339 and se(0

36、= .034. Roughly, the nominal increase in wage is 33.9%, and the 95% confidence interval is 33.9 ± 1.96(3.4, or 課后答案網 w w w .k h d a w .c o m 143about 27.2% to 40.6%. (Because the proportionate change is large, we could use equation(7.10, which implies the point estimate 40.4%; but obtaining the

37、 standard error of this estimate is harder.(iii Only the coefficient on y85 differs from equation (13.2. The new coefficient is about .383 (se .124. This shows that real wages have fallen over the seven year period, although less so for the more educated. For example, the proportionate change for a

38、male with 12 years of education is .383 + .0185(12 = .161, or a fall of about 16.1%. For a male with 20 years of education there has been almost no change .383 + .0185(20 = .013.(iv The R -squared when log(rwage is the dependent variable is .356, as compared with .426 when log(wage is the dependent

39、variable. If the SSRs from the regressions are the same, but the R -squareds are not, then the total sum of squares must be different. This is the case, as the dependent variables in the two equations are different.(v In 1978, about 30.6% of workers in the sample belonged to a union. In 1985, only a

40、bout 18% belonged to a union. Therefore, over the seven-year period, there was a notable fall in union membership.(vi When y85union is added to the equation, its coefficient and standard error are about .00040 (se .06104. This is practically very small and the t statistic is almost zero. There has b

41、een no change in the union wage premium over time.(vii Parts (v and (vi are not at odds. They imply that while the economic return to union membership has not changed (assuming we think we have estimated a causal effect, the fraction of people reaping those benefits has fallen.C13.3 (i Other things

42、equal, homes farther from the incinerator should be worth more, so 1 > 0. If 1 > 0, then the incinerator was located farther away from more expensive homes.(ii The estimated equation islog(price = 8.06 .011 y81 + .317 log(dist + .048 y81log(dist (0.51 (.805 (.052 (.082 n = 321, R 2 = .396, 2R

43、= .390.While 1 = .048 is the expected sign, it is not statistically significant (t statistic .59.(iii When we add the list of housing characteristics to the regression, the coefficient on y81log(dist becomes .062 (se = .050. So the estimated effect is larger the elasticity of price with respect to d

44、ist is .062 after the incinerator site was chosen but its t statistic is only 1.24. The p -value for the one-sided alternative H 1: 1 > 0 is about .108, which is close to being significant at the 10% level.課后答案網 w w w .k h d a w .c o m144C13.4 (i In addition to male and married , we add the varia

45、bles head , neck , upextr , trunk , lowback , lowextr , and occdis for injury type, and manuf and construc for industry. The coefficient on afchnge highearn becomes .231 (se .070, and so the estimated effect and t statistic are now larger than when we omitted the control variables. The estimate .231

46、 implies a substantial response of durat to the change in the cap for high-earnings workers.(ii The R -squared is about .041, which means we are explaining only a 4.1% of the variation in log(durat . This means that there are some very important factors that affect log(durat that we are not controll

47、ing for. While this means that predicting log(durat would be very difficult for a particular individual, it does not mean that there is anything biased about 1: it could still be an unbiased estimator of the causal effect of changing the earnings cap for workers compensation.(iii The estimated equat

48、ion using the Michigan data islog(durat = 1.413 + .097 afchnge + .169 highearn + .192 afchnge highearn (0.057 (.085 (.106 (.154n = 1,524, R 2 = .012.The estimate of 1, .192, is remarkably close to the estimate obtained for Kentucky (.191. However, the standard error for the Michigan estimate is much

49、 higher (.154 compared with .069. The estimate for Michigan is not statistically significant at even the 10% level against 1 > 0. Even though we have over 1,500 observations, we cannot get a very precise estimate. (For Kentucky, we have over 5,600 observations.C13.5 (i Using pooled OLS we obtainl

50、og(rent = .569 + .262 d90 + .041 log(pop + .571 log(avginc + .0050 pctstu (.535 (.035 (.023 (.053 (.0010 n = 128, R 2 = .861.The positive and very significant coefficient on d90 simply means that, other things in the equation fixed, nominal rents grew by over 26% over the 10 year period. The coeffic

51、ient onpctstu means that a one percentage point increase in pctstu increases rent by half a percent (.5%. The t statistic of five shows that, at least based on the usual analysis, pctstu is very statistically significant.(ii The standard errors from part (i are not valid, unless we thing a i does no

52、t really appear in the equation. If a i is in the error term, the errors across the two time periods for each city are positively correlated, and this invalidates the usual OLS standard errors and t statistics.(iii The equation estimated in differences is課后答案網 w w w .k h d a w .c o m145 log(rent = .

53、386 + .072 log(pop + .310 log(avginc + .0112 pctstu(.037 (.088 (.066 (.0041 n = 64, R 2 = .322.Interestingly, the effect of pctstu is over twice as large as we estimated in the pooled OLSequation. Now, a one percentage point increase in pctstu is estimated to increase rental rates by about 1.1%. Not

54、 surprisingly, we obtain a much less precise estimate when we difference (although the OLS standard errors from part (i are likely to be much too small because of the positive serial correlation in the errors within each city. While we have differenced away a i , there may be other unobservables tha

55、t change over time and are correlated with pctstu .(iv The heteroskedasticity-robust standard error on pctstu is about .0028, which is actually much smaller than the usual OLS standard error. This only makes pctstu even more significant (robust t statistic 4. Note that serial correlation is no longe

56、r an issue because we have no time component in the first-differenced equation.C13.6 (i You may use an econometrics software package that directly tests restrictions such as H 0: 1 = 2 after estimating the unrestricted model in (13.22. But, as we have seen many times, we can simply rewrite the equat

57、ion to test this using any regression software. Write the differenced equation aslog(crime = 0 + 1clrprc -1 + 2clrprc -2 + u .Following the hint, we define 1 = 1 2, and then write 1 = 1 + 2. Plugging this into the differenced equation and rearranging giveslog(crime = 0 + 1clrprc -1 + 2(clrprc -1 + c

58、lrprc -2 + u .Estimating this equation by OLS gives 1= .0091, se(1 = .0085. The t statistic for H 0: 1 = 2 is .0091/.0085 1.07, which is not statistically significant.(ii With1 = 2 the equation becomes (without the i subscriptlog(crime = 0 + 1(clrprc -1 + clrprc -2 + u= 0 + 1(clrprc -1 + clrprc -2/2 + u ,where 1 = 21. But (clrprc -1 + clrprc -2/2 = avgclr .(iii The estimated equation is課后答案網 w w w .k h d a w .c o m 146 log(crime .099 .0167 avgclr(.063 (.0051 n = 53, R 2 = .175, R = .159. Si