ADDITIONAL PROBLEMS FOR SECOND TEST
ADDITIONAL PROBLEMS FOR SECOND TEST ECON 8840
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Date Created: 03/28/16
Chapter 13 13.3 We do not have repeated observations on the same cross-sectional 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 1978 and 1981, but the overlap is usually too small to do a true panel data analysis. 13.5 No, we cannot include age as an explanatory variable in the original model. Each person in the panel data set is exactly two years older on January 31, 1992 than on January 31, 1990. This means that ∆age = 2 for ali i. But the equation we would estimate is of the form saving = + age + , i 0 1 i … where i0 the coefficient the year dummy for 1992 in the original model. As we know, when we have an intercept in the model we cannot include an explanatory variable that is constant across i. C13.2 (i) The coefficient on y85 is reflects the change in wage for a male (female = 0) with zero years of education (educ = 0). This is not especially 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 = +012 0 this i1 the change in the intercept for a male with 12 years of education, where we also hold other factors fixed. If we write 0 012 , p1ug this into (13.1), and rearrange, we get 0 log(wage)= + 0 y85 + e1uc + y851(educ – 12) + exper + 2 3exper 2 + 4nion + fe5ale + y85 5emale + u. Therefore, we simply replace y85 educ with y85 (educ – 12), and then the 0 coefficient and standard error we want is on y85. These turn out to be = .339 ˆ 0 and se( ) = .034. Roughly, the nominal increase in wage is 33.9%, and the 95% confidence interval is 33.9 1.96(3.4), or about 27.2% to 40.6%. (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 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 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 about 18% belonged to a union. Therefore, over the seven-year period, there was a notable fall in union membership. (vi) When y85 union 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 been 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. Chapter 14 C14.1 (i) This is done in Computer Exercise 13.5(i). (ii) See Computer Exercise 13.5(ii). (iii) See Computer Exercise 13.5(iii). (iv) The fixed effects estimates, reported in equation form, are . xtreg lrent y90 lpop lavginc pctstu, i(city) fe Fixed-effects (within) regression Number of obs = 128 Group variable: city Number of groups = 64 R-sq: Obs per group: within = 0.9765 min = 2 between = 0.2173 avg = 2.0 overall = 0.7597 max = 2 F(4,60) = 624.15 corr(u_i, Xb) = -0.1297 Prob > F = 0.0000 lrent Coef. Std. Err. t P>|t| [95% Conf. Interval] y90 .3855214 .0368245 10.47 0.000 .3118615 .4591813 lpop .0722456 .0883426 0.82 0.417 -.104466 .2489571 lavginc .3099605 .0664771 4.66 0.000 .1769865 .4429346 pctstu .0112033 .0041319 2.71 0.009 .0029382 .0194684 _cons 1.409384 1.167238 1.21 0.232 -.9254394 3.744208 sigma_u .15905877 sigma_e .06372873 rho .8616755 (fraction of variance due to u_i) F test that all u_i=0: F(63, 60) = 6.67 Prob > F = 0.0000 There are N = 64 cities and T = 2 years. The coefficient on y90 is identical totthe intercept from the first difference estimation, and the slope coefficients and standard errors are identical to first differencing. C14.2 (i) We report the fixed effects estimates in equation form as . xtreg lcrmrte d82 d83 d84 d85 d86 d87 lprbarr lprbconv lprbpris lavgsen lpolpc, i(county) fe Fixed-effects (within) regression Number of obs = 630 Group variable: county Number of groups = 90 R-sq: Obs per group: within = 0.4342 min = 7 between = 0.4066 avg = 7.0 overall = 0.4042 max = 7 F(11,529) = 36.91 corr(u_i, Xb) = 0.2068 Prob > F = 0.0000 lcrmrte Coef. Std. Err. t P>|t| [95% Conf. Interval] d82 .0125802 .0215416 0.58 0.559 -.0297373 .0548977 d83 -.0792813 .0213399 -3.72 0.000 -.1212027 -.0373598 d84 -.1177281 .0216145 -5.45 0.000 -.1601888 -.0752673 d85 -.1119561 .0218459 -5.12 0.000 -.1548716 -.0690407 d86 -.0818268 .0214266 -3.82 0.000 -.1239185 -.0397352 d87 -.0404704 .0210392 -1.92 0.055 -.0818011 .0008602 lprbarr -.3597944 .0324192 -11.10 0.000 -.4234806 -.2961082 lprbconv -.2858733 .0212173 -13.47 0.000 -.3275538 -.2441928 lprbpris -.1827812 .0324611 -5.63 0.000 -.2465496 -.1190127 lavgsen -.0044879 .0264471 -0.17 0.865 -.0564421 .0474663 lpolpc .4241142 .0263661 16.09 0.000 .3723191 .4759093 _cons -1.604135 .1685739 -9.52 0.000 -1.935292 -1.272979 sigma_u .43487416 sigma_e .13871215 rho .90765322 (fraction of variance due to u_i) F test that all u_i=0: F(89, 529) = 45.87 Prob > F = 0.0000 . The coefficients on the year dummies are not directly comparable with those in the first-differenced equation because we did not difference the year dummies in (13.33). The fixed effects estimates are unbiased estimators of the parameters on the time dummies in the original model. The first-difference and fixed effects slope estimates are broadly consistent. The variables that are significant with first differencing are significant in the FE estimation, and the signs are all the same. The magnitudes are also similar, although, with the exception of the insignificant variable log(avgsen), the FE estimates are all larger in absolute value. But we conclude that the estimates across the two methods paint a similar picture. (ii) When the nine log wage variables are added and the equation is estimated by fixed effects, very little of importance changes on the criminal justice variables. The following table contains the new estimates and standard errors. Independent Standard Variable Coefficient Error log(prbarr) –.356 .032 log(prbconv) –.286 .021 log(prbpris) –.175 .032 log(avgsen) –.0029 .026 log(polpc) .423 .026 The changes in these estimates are minor, even though the wage variables are jointly significant. The F statistic, with 9 and N(T – 1) – k = 90(6) – 20 = 520 df, is F 2.47 with p-value .0090. C14.7 (Parts (i)-(iii) only (i) If there is a deterrent effect then 1 < 0. The sign of 2s not entirely obvious, although one possibility is that a better economy means less crime in general, including violent crime (such as drug dealing) that would lead to fewer murders. This would imply >20. (ii) The pooled OLS estimates using 1990 and 1993 are ^ mrdte it = 5.28 2.07 d93 + t .128 exec it+ 2.53 unem it (4.43) (2.14) (.263) (0.78) 2 N = 51, T = 2, R = .102 There is no evidence of a deterrent effect, as the coefficient on exec is actually positive (though not statistically significant). (iii) The first-differenced equation is ^ mrdte it = .413 .104 exec i .067 unem i (.209) (.043) (.159) n = 51, R = .110 Now, there is a statistically significant deterrent effect: 10 more executions is estimated to reduce the murder rate by 1.04, or one murder per 100,000 people. Is this a large effect? Executions are relatively rare in most states, but murder rates are relatively low on average, too. In 1993, the average murder rate was about 8.7; a reduction of one would be nontrivial. For the (unknown) people whose lives might be saved via a deterrent effect, it would seem important.
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