伍德里奇计量经济学第六版答案Chapter 7

CHAPTER 7

TEACHING NOTES

This is a fairly standard chapter on using qualitative information in regression analysis, although I try to emphasize examples with policy relevance (and only cross-sectional applications are included.).

In allowing for different slopes, it is important, as in Chapter 6, to appropriately interpret the parameters and to decide whether they are of direct interest. For example, in the wage equation where the return to education is allowed to depend on gender, the coefficient on the female dummy variable is the wage differential between women and men at zero years of education. It is not surprising that we cannot estimate this very well, nor should we want to. In this particular example we would drop the interaction term because it is insignificant, but the issue of interpreting the parameters can arise in models where the interaction term is significant.

In discussing the Chow test, I think it is important to discuss testing for differences in slope coefficients after allowing for an intercept difference. In many applications, a significant Chow statistic simply indicates intercept differences. (See the example in Section 7.4 on student-athlete GPAs in the text.) From a practical perspective, it is important to know whether the partial effects differ across groups or whether a constant differential is sufficient.

I admit that an unconventional feature of this chapter is its introduction of the linear probability model. I cover the LPM here for several reasons. First, the LPM is being used more and more because it is easier to interpret than probit or logit models. Plus, once the proper parameter scalings are done for probit and logit, the estimated effects are often similar to the LPM partial effects near the mean or median values of the explanatory variables. The theoretical drawbacks of the LPM are often of secondary importance in practice. Computer Exercise C7.9 is a good one to illustrate that, even with over 9,000 observations, the LPM can deliver fitted values strictly between zero and one for all observations.

If the LPM is not covered, many students will never know about using econometrics to explain qualitative outcomes. This would be especially unfortunate for students who might need to read an article where an LPM is used, or who might want to estimate an LPM for a term paper or senior thesis. Once they are introduced to purpose and interpretation of the LPM, along with its shortcomings, they can tackle nonlinear models on their own or in a subsequent course.

A useful modification of the LPM estimated in equation (7.29) is to drop kidsge6 (because it is not significant) and then define two dummy variables, one for kidslt6 equal to one and the other for kidslt6 at least two. These can be included in place of kidslt6 (with no young children being the base group). This allows a diminishing marginal effect in an LPM. I was a bit surprised when a diminishing effect did not materialize.

SOLUTIONS TO PROBLEMS

7.1 (i) The coefficient on male is 87.75, so a man is estimated to sleep almost one and one-half hours more per week than a comparable woman. Further, t male = 87.75/34.33 ≈ 2.56, which is close to the 1% critical value against a two-sided alternative (about 2.58). Thus, the evidence for a gender differential is fairly strong.

(ii) The t statistic on totwrk is -.163/.018 ≈ -9.06, which is very statistically significant. The coefficient implies that one more hour of work (60 minutes) is associated with .163(60) ≈ 9.8 minutes less sleep.

(iii) To obtain 2r R , the R -squared from the restricted regression, we need to estimate the

model without age and age 2. When age and age 2 are both in the model, age has no effect only if the parameters on both terms are zero.

7.2 (i) If ?cigs = 10 then log()bwght ? = -.0044(10) = -.044, which means about a 4.4% lower birth weight.

(ii) A white child is estimated to weigh about 5.5% more, other factors in the first equation fixed. Further, t white ≈ 4.23, which is well above any commonly used critical value. Thus, the difference between white and nonwhite babies is also statistically significant.

(iii) If the mother has one more year of education, the child’s birth weight is estimated to be .3% higher. This is not a huge effect, and the t statistic is only one, so it is not statistically significant.

(iv) The two regressions use different sets of observations. The second regression uses fewer observations because motheduc or fatheduc are missing for some observations. We would have to reestimate the first equation (and obtain the R -squared) using the same observations used to estimate the second equation.

7.3 (i) The t statistic on hsize 2 is over four in absolute value, so there is very strong evidence that it belongs in the equation. We obtain this by finding the turnaround point; this is the value of

hsize that maximizes ?sat

(other things fixed): 19.3/(2?2.19) ≈ 4.41. Because hsize is measured in hundreds, the optimal size of graduating class is about 441.

(ii) This is given by the coefficient on female (since black = 0): nonblack females have SAT scores about 45 points lower than nonblack males. The t statistic is about –10.51, so the

difference is very statistically significant. (The very large sample size certainly contributes to the statistical significance.)

(iii) Because female = 0, the coefficient on black implies that a black male has an estimated SAT score almost 170 points less than a comparable nonblack male. The t statistic is over 13 in absolute value, so we easily reject the hypothesis that there is no ceteris paribus difference.

(iv) We plug in black = 1, female = 1 for black females and black = 0 and female = 1 for nonblack females. The difference is therefore –169.81 + 62.31 = -107.50. Because the estimate depends on two coefficients, we cannot construct a t statistic from the information given. The easiest approach is to define dummy variables for three of the four race/gender categories and choose nonblack females as the base group. We can then obtain the t statistic we want as the coefficient on the black female dummy variable.

7.4 (i) The approximate difference is just the coefficient on utility times 100, or –28.3%. The t statistic is -.283/.099 ≈ -2.86, which is very statistically significant.

(ii) 100?[exp(-.283) – 1) ≈ -24.7%, and so the estimate is somewhat smaller in magnitude.

(iii) The proportionate difference is .181 - .158 = .023, or about 2.3%. One equation that can be estimated to obtain the standard error of this difference is

log(salary ) = 0β + 1βlog(sales ) + 2βroe + 1δconsprod + 2δutility +3δtrans + u ,

where trans is a dummy variable for the transportation industry. Now, the base group is finance , and so the coefficient 1δ directly measures the difference between the consumer products and finance industries, and we can use the t statistic on consprod .

7.5 (i) Following the hint, colGPA = 0?β + 0?δ(1 – noPC ) + 1?βhsGPA + 2?βACT = (0?β + 0

?δ) - 0?δnoPC + 1?βhsGPA + 2?βACT . For the specific estimates in equation (7.6), 0

?β = 1.26 and 0?δ = .157, so the new intercept is 1.26 + .157 = 1.417. The coefficient on noPC is –.157.

(ii) Nothing happens to the R -squared. Using noPC in place of PC is simply a different way of including the same information on PC ownership.

(iii) It makes no sense to include both dummy variables in the regression: we cannot hold noPC fixed while changing PC . We have only two groups based on PC ownership so, in addition to the overall intercept, we need only to include one dummy variable. If we try to

include both along with an intercept we have perfect multicollinearity (the dummy variable trap).

7.6 In Section 3.3 – in particular, in the discussion surrounding Table 3.2 – we discussed how to determine the direction of bias in the OLS estimators when an important variable (ability, in this case) has been omitted from the regression. As we discussed there, Table 3.2 only strictly holds with a single explanatory variable included in the regression, but we often ignore the presence of other independent variables and use this table as a rough guide. (Or, we can use the results of Problem 3.10 for a more precise analysis.) If less able workers are more likely to receive

training, then train and u are negatively correlated. If we ignore the presence of educ and exper , or at least assume that train and u are negatively correlated after netting out educ and exper , then we can use Table 3.2: the OLS estimator of 1β (with ability in the error term) has a downward bias. Because we think 1β ≥ 0, we are less likely to conclude that the training program was

effective. Intuitively, this makes sense: if those chosen for training had not received training, they would have lowers wages, on average, than the control group.

7.7 (i) Write the population model underlying (7.29) as

inlf = 0β + 1βnwifeinc + 2βeduc + 3βexper +4βexper 2 + 5βage

+ 6βkidslt6 + 7βkidsage6 + u ,

plug in inlf = 1 – outlf , and rearrange:

1 – outlf = 0β + 1βnwifeinc + 2βeduc + 3βexper +4βexper

2 + 5βage

+ 6βkidslt6 + 7βkidsage6 + u ,

or

outlf = (1 - 0β) - 1βnwifeinc - 2βeduc - 3βexper - 4βexper 2 - 5βage - 6βkidslt6 - 7βkidsage6 - u ,

The new error term, -u , has the same properties as u . From this we see that if we regress outlf on all of the independent variables in (7.29), the new intercept is 1 - .586 = .414 and each slope coefficient takes on the opposite sign from when inlf is the dependent variable. For example, the new coefficient on educ is -.038 while the new coefficient on kidslt6 is .262.

(ii) The standard errors will not change. In the case of the slopes, changing the signs of the estimators does not change their variances, and therefore the standard errors are unchanged (but

the t statistics change sign). Also, Var(1 - 0?β) = Var(0

?β), so the standard error of the intercept is the same as before.

(iii) We know that changing the units of measurement of independent variables, or entering qualitative information using different sets of dummy variables, does not change the R -squared. But here we are changing the dependent variable. Nevertheless, the R -squareds from the regressions are still the same. To see this, part (i) suggests that the squared residuals will be identical in the two regressions. For each i the error in the equation for outlf i is just the negative of the error in the other equation for inlf i , and the same is true of the residuals. Therefore, the SSRs are the same. Further, in this case, the total sum of squares are the same. For outlf we have

SST = 2211()[(1)(1)]n n i i i i outlf outlf inlf inlf ==-=---∑∑= 2

211()()n n

i i i i inlf inlf inlf inlf ==-+=-∑∑,

which is the SST for inlf . Because R 2 = 1 – SSR/SST, the R -squared is the same in the two regressions.

7.8 (i) We want to have a constant semi-elasticity model, so a standard wage equation with marijuana usage included would be

log(wage ) = 0β + 1βusage + 2βeduc + 3βexper + 4βexper 2 + 5βfemale + u .

Then 100?1β is the approximate percentage change in wage when marijuana usage increases by one time per month.

(ii) We would add an interaction term in female and usage :

log(wage ) = 0β + 1βusage + 2βeduc + 3βexper + 4βexper 2 + 5βfemale

+ 6βfemale ?usage + u .

The null hypothesis that the effect of marijuana usage does not differ by gender is H 0: 6β = 0.

(iii) We take the base group to be nonuser. Then we need dummy variables for the other three groups: lghtuser , moduser , and hvyuser . Assuming no interactive effect with gender, the model would be

log(wage ) = 0β + 1δlghtuser + 2δmoduser + 3δhvyuser + 2βeduc + 3βexper + 4βexper 2 + 5βfemale + u .

(iv) The null hypothesis is H 0: 1δ = 0, 2δ= 0, 3δ = 0, for a total of q = 3 restrictions. If n is the sample size, the df in the unrestricted model – the denominator df in the F distribution – is n – 8. So we would obtain the critical value from the F q ,n -8 distribution.

(v) The error term could contain factors, such as family background (including parental history of drug abuse) that could directly affect wages and also be correlated with marijuana usage. We are interested in the effects of a person’s drug usage on his or her wage, so we would like to hold other confounding factors fixed. We could try to collect data on relevant background information.

7.9 (i) Plugging in u = 0 and d = 1 gives 10011()()()f z z βδβδ=+++.

(ii) Setting **01()()f z f z = gives **010011()()z z βββδβδ+=+++ or *010z δδ=+. Therefore, provided 10δ≠, we have *01/z δδ=-. Clearly, *z is positive if and only if 01/δδ is negative, which means 01 and δδ must have opposite signs.

(iii) Using part (ii) we have *.357/.03011.9totcoll == years.

(iv) The estimated years of college where women catch up to men is much too high to be practically relevant. While the estimated coefficient on female totcoll ? shows that the gap is reduced at higher levels of college, it is never closed – not even close. In fact, at four years of

college, the difference in predicted log wage is still .357.030(4).237-+=-, or about 21.1% less for women.

7.10 (i) Yes, simple regression does produce an unbiased estimator of the effect of the voucher program. Because participation was randomized, we can write

01,score voucher u ββ=++

where voucher is independent of u , that is, all other factors affecting score . Therefore, the key assumption for unbiasedness of simple regression, Assumption SLR.3, is satisfied.

(ii) No, we do not need to control for background variables. In the equation from part (i), these are factors in the error term, u . But voucher was assigned to be independent of all factors, including the listed background variables.

(iii) We should include the background variables to reduce the sampling error of the

estimated voucher effect. By pulling background variables out of the error term, we reduce the error variance – perhaps substantially. Further, we can be sure that multicollinearity is not a problem because the key variable of interest, voucher , is uncorrelated with all of the added

explanatory variables. (This zero correlation will only be approximate in any random sample, but in large samples it should be very small.) The one case where we would not add these variables – or, at least, when there is no benefit from doing so – is when the background variables

themselves have no affect on the test score. Given the list of background variables, this seems unlikely in the current application.

SOLUTIONS TO COMPUTER EXERCISES

C7.1 (i) The estimated equation is

colGPA = 1.26 + .152 PC + .450 hsGPA + .0077 ACT - .0038 mothcoll (0.34) (.059) (.094) (.0107) (.0603)

+ .0418 fathcoll (.0613)

n = 141 , R 2 = .222.

The estimated effect of PC is hardly changed from equation (7.6), and it is still very significant, with t pc ≈ 2.58.

(ii) The F test for joint significance of mothcoll and fathcoll , with 2 and 135 df , is about .24 with p -value ≈ .78; these variables are jointly very insignificant. It is not surprising the

estimates on the other coefficients do not change much when mothcoll and fathcoll are added to the regression.

(iii) When hsGPA2 is added to the regression, its coefficient is about .337 and its t statistic is about 1.56. (The coefficient on hsGPA is about –1.803.) This is a borderline case. The quadratic in hsGPA has a U-shape, and it only turns up at about hsGPA* = 2.68, which is hard to interpret. The coefficient of main interest, on PC, falls to about .140 but is still significant. Adding hsGPA2 is a simple robustness check of the main finding.

C7.2 (i) The estimated equation is

wage = 5.40 + .0654 educ + .0140 exper+ .0117 tenure

log()

(0.11) (.0063) (.0032) (.0025)

+ .199 married- .188 black- .091 south+ .184 urban

(.039) (.038) (.026) (.027)

n = 935, R2 = .253.

The coefficient on black implies that, at given levels of the other explanatory variables, black men earn about 18.8% less than nonblack men. The t statistic is about –4.95, and so it is very statistically significant.

(ii) The F statistic for joint significance of exper2 and tenure2, with 2 and 925 df, is about 1.49 with p-value≈ .226. Because the p-value is above .20, these quadratics are jointly insignificant at the 20% level.

(iii) We add the interaction black?educ to the equation in part (i). The coefficient on the interaction is about -.0226 (se≈ .0202). Therefore, the point estimate is that the return to another year of education is about 2.3 percentage points lower for black men than nonblack men. (The estimated return for nonblack men is about 6.7%.) This is nontrivial if it really reflects differences in the population. But the t statistic is only about 1.12 in absolute value, which is not enough to reject the null hypothesis that the return to education does not depend on race.

(iv) We choose the base group to be single, nonblack. Then we add dummy variables marrnonblck, singblck, and marrblck for the other three groups. The result is

log()

wage = 5.40 + .0655 educ + .0141 exper+ .0117 tenure

(0.11) (.0063) (.0032) (.0025)

- .092 south+ .184 urban+ .189 marrnonblck

(.026) (.027) (.043)

- .241 singblck+ .0094 marrblck

(.096) (.0560)

n = 935, R2 = .253.

We obtain the ceteris paribus differential between married blacks and married nonblacks by taking the difference of their coefficients: .0094 - .189 = -.1796, or about -.18. That is, a married black man earns about 18% less than a comparable, married nonblack man.

C7.3 (i) H 0:13β= 0. Using the data in MLB1.RAW gives 13?β ≈ .254, se(13

?β) ≈ .131. The t statistic is about 1.94, which gives a p -value against a two-sided alternative of just over .05. Therefore, we would reject H 0 at just about the 5% significance level. Controlling for the performance and experience variables, the estimated salary differential between catchers and outfielders is huge, on the order of 100?[exp(.254) – 1] ≈ 28.9% [using equation (7.10)].

(ii) This is a joint null, H 0: 9β = 0, 10β = 0, …, 13β = 0. The F statistic, with 5 and 339 df , is about 1.78, and its p -value is about .117. Thus, we cannot reject H 0 at the 10% level.

(iii) Parts (i) and (ii) are roughly consistent. The evidence against the joint null in part (ii) is weaker because we are testing, along with the marginally significant catcher , several other insignificant variables (especially thrdbase and shrtstop , which has absolute t statistics well below one).

C7.4 (i) The two signs that are pretty clear are 3β< 0 (because hsperc is defined so that the smaller the number the better the student) and 4β> 0. The effect of size of graduating class is not clear. It is also unclear whether males and females have systematically different GPAs. We may think that 6β< 0, that is, athletes do worse than other students with comparable

characteristics. But remember, we are controlling for ability to some degree with hsperc and sat .

(ii) The estimated equation is

colgpa = 1.241 - .0569 hsize + .00468 hsize 2 - .0132 hsperc (0.079) (.0164) (.00225) (.0006)

+ .00165 sat + .155 female + .169 athlete

(.00007) (.018) (.042)

n = 4,137, R 2 = .293.

Holding other factors fixed, an athlete is predicted to have a GPA about .169 points higher than a nonathlete. The t statistic .169/.042 ≈ 4.02, which is very significant.

(iii) With sat dropped from the model, the coefficient on athlete becomes about .0054 (se ≈ .0448), which is practically and statistically not different from zero. This happens because we do not control for SAT scores, and athletes score lower on average than nonathletes. Part (ii) shows that, once we account for SAT differences, athletes do better than nonathletes. Even if we do not control for SAT score, there is no difference.

(iv) To facilitate testing the hypothesis that there is no difference between women athletes and women nonathletes, we should choose one of these as the base group. We choose female nonathletes. The estimated equation is

colgpa= 1.396 - .0568 hsize+ .00467 hsize2- .0132 hsperc

(0.076) (.0164) (.00225) (.0006)

+ .00165 sat+ .175 femath+ .013 maleath- .155 malenonath

(.00007) (.084) (.049) (.018)

n = 4,137, R2 = .293.

The coefficient on femath = female?athlete shows that colgpa is predicted to be about .175 points higher for a female athlete than a female nonathlete, other variables in the equation fixed. The hypothesis that there is no difference between female athletes and female nonathletes is testing by using the t statistic on femath. In this case, t = 2.08, which is statistically significant at the 5% level against a two-sided alternative.

(v) Whether we add the interaction female?sat to the equation in part (ii) or part (iv), the outcome is practically the same. For example, when female?sat is added to the equation in part (ii), its coefficient is about .000051 and its t statistic is about .40. There is very little evidence that the effect of sat differs by gender.

C7.5 The estimated equation is

salary= 4.30 + .288 log(sales) + .0167 roe- .226 rosneg

log()

(0.29) (.034) (.0040) (.109)

n = 209, R2 = .297, 2R= .286.

The coefficient on rosneg implies that if the CEO’s firm had a negative return on its stock over the 1988 to 1990 period, the CEO salary was predicted to be about 22.6% lower, for given levels of sales and roe. The t statistic is about –2.07, which is significant at the 5% level against a two-sided alternative.

C7.6 (i) The estimated equation for men is

sleep= 3,648.2 - .182 totwrk- 13.05 educ+ 7.16 age-.0448 age2+ 60.38 yngkid (310.0) (.024) (7.41) (14.32) (.1684) (59.02)

n = 400, R2 = .156

and the estimated equation for women is

sleep= 4,238.7 - .140 totwrk- 10.21 educ- 30.36 age- .368 age2- 118.28 yngkid

(384.9) (.028) (9.59) (18.53) (.223) (93.19) n = 306, R 2 = .098.

There are certainly notable differences in the point estimates. For example, having a young child in the household leads to less sleep for women (about two hours a week) while men are

estimated to sleep about an hour more. The quadratic in age is a hump-shape for men but a U-shape for women. The intercepts for men and women are also notably different.

(ii) The F statistic (with 6 and 694 df ) is about 2.12 with p -value ≈ .05, and so we reject the null that the sleep equations are the same at the 5% level.

(iii) If we leave the coefficient on male unspecified under H 0, and test only the five interaction terms, male ?totwrk , male ?educ , male ?age , male ?age 2, and male ?yngkid , the F statistic (with 5 and 694 df ) is about 1.26 and p -value ≈ .28.

(iv) The outcome of the test in part (iii) shows that, once an intercept difference is allowed, there is not strong evidence of slope differences between men and women. This is one of those cases where the practically important differences in estimates for women and men in part (i) do not translate into statistically significant differences. We need a larger sample size to confidently determine whether there are differences in slopes. For the purposes of studying the sleep-work tradeoff, the original model with male added as an explanatory variable seems sufficient.

C7.7 (i) When educ = 12.5, the approximate proportionate difference in estimated wage between women and men is -.227 - .0056(12.5) = -.297. When educ = 0, the difference is -.227. So the differential at 12.5 years of education is about 7 percentage points greater.

(ii) We can write the model underlying (7.18) as

log(wage ) = 0β + 0δ female + 1βeduc + 1δ female ?educ + other factors

= 0β + (0δ + 12.51δ) female + 1βeduc + 1δ female ?(educ – 12.5)

+ other factors

≡ 0β + 0θ female + 1βeduc + 1δ female ?(educ – 12.5) + other factors ,

where 0θ ≡ 0δ + 12.51δ is the gender differential at 12.5 years of education. When we run this regression we obtain about –.294 as the coefficient on female (which differs from –.297 due to rounding error). Its standard error is about .036.

(iii) The t statistic on female from part (ii) is about –8.17, which is very significant. This is because we are estimating the gender differential at a reasonable number of years of education, 12.5, which is close to the average. In equation (7.18), the coefficient on female is the gender differential when educ = 0. There are no people of either gender with close to zero years of

education, and so we cannot hope – nor do we want to – to estimate the gender differential at educ = 0.

β> 0 signals discrimination against C7.8 (i) If the appropriate factors have been controlled for,

1

minorities: a white person has a greater chance of having a loan approved, other relevant factors fixed.

(ii) The simple regression results are

approve= .708 + .201 white

(.018) (.020)

n = 1,989, R2 = .049.

The coefficient on white means that, in the sample of 1,989 loan applications, an application submitted by a white application was 20.1% more likely to be approved than that of a nonwhite applicant. This is a practically large difference and the t statistic is about 10. (We have a large sample size, so standard errors are pretty small.)

?β≈ .129, (iii) When we add the other explanatory variables as controls, we obtain

1

se(1?β) ≈ .020. The coefficient has fallen by some margin because we are now controlling for factors that should affect loan approval rates, and some of these clearly differ by race. (On average, white people have financial characteristics – such as higher incomes and stronger credit histories – that make them better loan risks.) But the race effect is still strong and very significant (t statistic ≈ 6.45).

(iv) When we add the interaction white?obrat to the regression, its coefficient and t statistic are about .0081 and 3.53, respectively. Therefore, there is an interactive effect: a white applicant is penalized less than a nonwhite applicant for having other obligations as a larger percent of income.

(v) The trick should be familiar by now. Replace white?obrat with white?(obrat– 32); the coefficient on white is now the race differential when obrat = 32. We obtain about .113 and se ≈ .020. So the 95% confidence interval is about .113 ± 1.96(.020) or about .074 to .152. Clearly, this interval excludes zero, so at the average obrat there is evidence of discrimination (or, at least loan approval rates that differ by race for some other reason that is not captured by the control variables).

C7.9 (i) About .392, or 39.2%.

(ii) The estimated equation is

e k = -.506 + .0124 inc-.000062 inc2 + .0265 age- .00031 age2-.0035 male

401

(.081) (.0006) (.000005) (.0039) (.00005) (.0121)

n = 9,275, R2 = .094.

(iii) 401(k) eligibility clearly depends on income and age in part (ii). Each of the four terms involving inc and age have very significant t statistics. On the other hand, once income and age are controlled for, there seems to be no difference in eligibility by gender. The coefficient on male is very small – at given income and age, males are estimated to have a .0035 lower probability of being 401(k) eligible – and it has a very small t statistic.

(iv) Somewhat surprisingly, out of 9,275 fitted values, none is outside the interval [0,1]. The smallest fitted value is about .030 and the largest is about .697. This means one theoretical problem with the LPM – the possibility of generating silly probability estimates – does not materialize in this application.

(v) Using the given rule, 2,460 families are predicted to be eligible for a 401(k) plan.

(vi) Of the 5,638 families actually ineligible for a 401(k) plan, about 81.7 are correctly predicted not to be eligible. Of the 3,637 families actually eligible, only 39.3 percent are correctly predicted to be eligible.

(vii) The overall percent correctly predicted is a weighted average of the two percentages obtained in part (vi). As we saw there, the model does a good job of predicting when a family is ineligible. Unfortunately, it does less well – predicting correctly less than 40% of the time – in predicting that a family is eligible for a 401(k).

(viii) The estimated equation is

e k = -.502 + .0123 inc-.000061 inc2 + .0265 age- .00031 age2

401

(.081) (.0006) (.000005) (.0039) (.00005)

-.0038 male + .0198 pira

(.0121) (.0122)

n = 9,275, R2 = .095.

The coefficient on pira means that, other things equal, IRA ownership is associated with about

a .02 higher probability of being eligible for a 401(k) plan. However, the t statistic is only about

1.62, which gives a two-sided p-value = .105. So pira is not significant at the 10% level against

a two-sided alternative.

C7.10 (i) The estimated equation is

points = 4.76 + 1.28 exper-.072 exper2+ 2.31 guard+ 1.54 forward

(1.18) (.33) (.024) (1.00) (1.00)

n = 269, R2 = .091, 2R = .077.

(ii) Including all three position dummy variables would be redundant, and result in the

dummy variable trap. Each player falls into one of the three categories, and the overall intercept is the intercept for centers.

(iii) A guard is estimated to score about 2.3 points more per game, holding experience fixed. The t statistic is 2.31, so the difference is statistically different from zero at the 5% level, against a two-sided alternative.

(iv) When marr is added to the regression, its coefficient is about .584 (se = .740). Therefore, a married player is estimated to score just over half a point more per game (experience and position held fixed), but the estimate is not statistically different from zero (p -value = .43). So, based on points per game, we cannot conclude married players are more productive.

(v) Adding the terms 2 and marr exper marr exper ?? leads to complicated signs on the three terms involving marr . The F test for their joint significance, with 3 and 261 df , gives F = 1.44 and p -value = .23. Therefore, there is not very strong evidence that marital status has any partial effect on points scored.

(vi) If in the regression from part (iv) we use assists as the dependent variable, the coefficient on marr becomes .322 (se = .222). Therefore, holding experience and position fixed, a married man has almost one-third more assist per game. The p -value against a two-sided alternative is about .15, which is stronger, but not overwhelming, evidence that married men are more productive when it comes to assists.

C7.11 (i) The average is 19.072, the standard deviation is 63.964, the smallest value is –502.302, and the largest value is 1,536.798. Remember, these are in thousands of dollars.

(ii) This can be easily done by regressing nettfa on e401k and doing a t test on ?e401k β; the

estimate is the average difference in nettfa for those eligible for a 401(k) and those not eligible.

Using the 9,275 observations gives ?18.858 and 14.01.e401k e401k t β== Therefore, we strongly

reject the null hypothesis that there is no difference in the averages. The coefficient implies that, on average, a family eligible for a 401(k) plan has $18,858 more on net total financial assets.

(iii) The equation estimated by OLS is

nettfa = 23.09 + 9.705 e401k - .278 inc + .0103 inc 2 - 1.972 age + .0348 age 2 (9.96) (1.277) (.075) (.0006) (.483) (.0055)

n = 9,275, R 2 = .202

Now, holding income and age fixed, a 401(k)-eligible family is estimated to have $9,705 more in wealth than a non-eligible family. This is just more than half of what is obtained by simply comparing averages.

(iv) Only the interaction e401k?(age- 41) is significant. Its coefficient is .654 (t = 4.98). It shows that the effect of 401(k) eligibility on financial wealth increases with age. Another way to think about it is that age has a stronger positive effect on nettfa for those with 401(k) eligibility. The coefficient on e401k?(age- 41)2 is -.0038 (t statistic = -.33), so we could drop this term.

(v) The effect of e401k in part (iii) is the same for all ages, 9.705. For the regression in part (iv), the coefficient on e401k from part (iv) is about 9.960, which is the effect at the average age, age = 41. Including the interactions increases the estimated effect of e401k, but only by $255. If we evaluate the effect in part (iv) at a wide range of ages, we would see more dramatic differences.

(vi) I chose fsize1 as the base group. The estimated equation is

nettfa = 16.34 + 9.455 e401k-.240 inc + .0100 inc2- 1.495 age + .0290 age2

(10.12) (1.278) (.075) (.0006) (.483) (.0055)

-.859 fsize2 - 4.665 fsize3- 6.314 fsize4-7.361 fsize5

(1.818) (1.877) (1.868) (2.101)

n = 9,275, R2 = .204, SSR= 30,215,207.5

The F statistic for joint significance of the four family size dummies is about 5.44. With 4 and 9,265 df, this gives p-value = .0002. So the family size dummies are jointly significant.

(vii) The SSR for the restricted model is from part (vi): SSR r = 30,215,207.5. The SSR for the unrestricted model is obtained by adding the SSRs for the five separate family size regressions. I get SSR ur = 29,985,400. The Chow statistic is F = [(30,215,207.5 - 29,985,400)/ 29,985,400]*(9245/20) ≈ 3.54. With 20 and 9,245 df, the p-value is essentially zero. In this case, there is strong evidence that the slopes change across family size. Allowing for intercept changes alone is not sufficient. (If you look at the individual regressions, you will see that the signs on the income variables actually change across family size.)

C7.12 (i) For women, the fraction rated as having above average looks is about .33; for men, it

is .29. The proportion of women rated as having below average looks is only .135; for men, it is even lower at about .117.

(ii) The difference is about .04, that is, the percent rated as having above average looks is about four percentage points higher for women than men. A simple way to test whether the difference is statistically significant is to run a simple regression of abvavg on female and do a t test (which is asymptotically valid). The t statistic is about 1.48 with two-sided p-value = .14. Therefore, there is not strong evidence against the null that the population fractions are the same, but there is some evidence.

(iii) The regression for men is

log()wage = 1.884 - .199 belavg - .044 abvavg (0.024) (.060) (.042)

n = 824 R 2 = .013

and the regression for women is

log()wage = 1.309 - .138 belavg + .034 abvavg (0.034) (.076) (.055)

n = 436 R 2 = .011.

Using the standard approximation, a man with below average looks earns almost 20% less than a man of average looks, and a woman with below average looks earns about 13.8% less than a woman with average looks. (The more accurate estimates are about 18% and 12.9%,

respectively.) The null hypothesis H 0: 10β= against H 1: 10β< means that the null is that people with below average looks earn the same, on average, as people with average looks; the alternative is that people with below average looks earn less than people with average looks (in the population). The one-sided p -value for men is .0005 and for women it is .036. We reject H 0 more strongly for men because the estimate is larger in magnitude and the estimate has less sampling variation (as measured by the standard error).

(iv) Women with above average looks are estimated to earn about 3.4% more, on average, than women with average looks. But the one-sided p -value is .272, and this provides very little evidence against H 0: 2β = 0.

(v) Given the number of added controls, with many of them very statistically significant, the coefficients on the looks variables do not change by much. For men, the coefficient on belavg becomes -.143 (t = -2.80) and the coefficient on abvavg becomes -.001 (t = -.03). For women, the changes in magnitude are similar: the coefficient on belavg becomes -.115 (t = -1.75) and the coefficient on abvavg becomes .058 (t = 1.18). In both cases, the estimates on belavg move closer to zero but are still reasonably large.

(vi) The SSR for women is 83.6933, for men it is 166.0841, and so the unrestricted SSR is 249.7774 (rounded to four decimal places). The SSR obtained by adding the dummy variable female to the regression in part (v) is 261.1771. Therefore, the F statistic (Chow statistic) is 261.1771249.7774)249.((1260214) 4.7773313

4F -?≈-=?

With 13 and 1,232 df the p -value is essentially zero. So we strongly reject the common slopes formulation even if we allow for a different intercept. A look at the estimated regression functions for men and women shows that some of the slopes are quite different.

C7.13 (i) 412/660 ≈ .624.

(ii) The OLS estimates of the LPM are

ecobuy = .424 -.803 ecoprc + .719 regprc + .00055 faminc + .024 hhsize (.165) (.109) (.132) (.00053) (.013)

+ .025 educ -.00050 age

(.008) (.00125)

n = 660, R2 = .110

If ecoprc increases by, say, 10 cents (.10), then the probability of buying eco-labeled apples falls by about .080. If regprc increases by 10 cents, the probability of buying eco-labeled apples increases by about .072. (Of course, we are assuming that the probabilities are not close to the boundaries of zero and one, respectively.)

(iii) The F test, with 4 and 653 df, is 4.43, with p-value = .0015. Thus, based on the usual F test, the four non-price variables are jointly very significant. Of the four variables, educ appears to have the most important effect. For example, a difference of four years of education implies an increase of .025(4) = .10 in the estimated probability of buying eco-labeled apples. This suggests that more highly educated people are more open to buying produce that is environmentally friendly, which is perhaps expected. Household size (hhsize) also has an effect. Comparing a couple with two children to one that has no children – other factors equal – the couple with two children has a .048 higher probability of buying eco-labeled apples.

(iv) The model with log(faminc) fits the data slightly better: the R-squared increases to about .112. (We would not expect a large increase in R-squared from a simple change in the functional form.) The coefficient on log(faminc) is about .045 (t = 1.55). If log(faminc) increases by .10, which means roughly a 10% increase in faminc, then P(ecobuy = 1) is estimated to increase by about .0045, a pretty small effect.

(v) The fitted probabilities range from about .185 to 1.051, so none are negative. There are two fitted probabilities above 1, which is not a source of concern with 660 observations.

(vi) Using the standard prediction rule – predict one when .5

ecobuy≥ and zero otherwise –

i

gives the fraction correctly predicted for ecobuy = 0 as 102/248 ≈ .411, so about 41.1%. For ecobuy = 1, the fraction correctly predicted is 340/412 ≈ .825, or 82.5%. With the usual prediction rule, the model does a much better job predicting the decision to buy eco-labeled apples. (The overall percent correctly predicted is about 67%.)

C7.14 (i) The estimated LPM is

respond = .282 + .344 resplast + .00015 avggift

(.009) (.015) (.00009)

n = 4,268, R2 = .110

Holding the average gift fixed, the probability of a current response is estimated to be .344 higher if the person responded most recently.

(ii) Once we control for responding most recently, the effect of of avggift is very small. Even if avggift is 100 guilders more (the mean is about 18.2 with standard deviation 78.7), the probability of responding this period is only .015 higher. Plus, the t statistic has a two-sided p-value of about .09, so it is only marginally statistically significant.

(iii) The coefficient on propresp is about .747 (standard error = .034). If propresp increases by .1 (for example, from .4 to .5), the probability of responding is about .075 higher.

(iv) When propresp is added to the regression, the coefficient on resplast falls to about .095 (although it is still very statistically significant). This makes sense, because the relationship between responding currently and responding most recently should be weaker once the average response is controlled for. Certainly resplast and propresp are positively correlated.

(v) The coefficient on mailsyear is about .062 (t = 6.18). This is a reasonably large effect: each new mailing is estimated to increase the probability of responding by .062. Unfortunately, we do not know how the charitable organization determines the mailings sent. To the extent that it depends only on past gift giving, as controlled for by the average gift, the most recent response, and the response rate, the estimate could be a good (consistent) estimate of the causal effect. But if mailings are determined by other factors that are necessarily in the error term – such as income – then the estimate would be systematically biased. If, say, more mailings are sent to people with higher incomes, and higher income people are more likely to respond, then the regression that omits income produces an upward bias for the mailsyear coefficient.

C7.15 (i) The smallest and largest values of children are 0 and 13. The average value is about 2.27. Naturally, no woman has 2.27 children.

(ii) Of the 4,358 women for whom we have information on electricity recorded, 611, or

14.02 percent, have electricity in the home.

(iii) Naturally, we must exclude the three women for whom electric is missing. The average for women without electricity is about 2.33 and the average for women with electricity is about 1.90. Regressing children on electric gives a coefficient on electric which is the difference in average children between women with and without electricity. We already know the estimate is about -.43. The simple regression gives us the t statistic, -4.44, which is very significant.

(iv) We cannot infer causality because there can be many confounding factors that are correlated with fertility and the presence of electricity. Income is an important possibility, as are education levels of the woman and spouse.

(v) When regressing children on electric, age, age2, urban, spirit, protest, and catholic, the coefficient on electric becomes -.306 (se = .069). The effect is somewhat smaller than in part (iii), but it is still on the order of almost one-third of a child (on average). The t statistic has barely changed, -4.43, and so it is still very statistically significant.

? is -.022 and its t statistic is -1.31 (two-(vi) The coefficient on the interaction electric educ

sided p-value = .19). Thus, it is not statistically significant. The coefficient on electric has become much smaller in magnitude and statistically insignificant. But one must interpret this coefficient with caution: it is now the effect of having electricity on the subpopulation with educ = 0. This is a nontrivial part of the population (almost 21 percent in the sample), but it is not the entire story.

(vii) If we use (7)

?- instead, we force the coefficient on electric to be the effect electric educ

of electric on the subpopulation with educ = 7 – both the modal and median value. The coefficient on electric becomes -.280 (t = -3.90), which is much different from part (vi). In fact, the effect is pretty close to what was obtained in part (v).

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伍德里奇计量经济学第六版答案Appendix-E

271 APPENDIX E SOLUTIONS TO PROBLEMS E.1 This follows directly from partitioned matrix multiplication in Appendix D. Write X = 12n ?? ? ? ? ? ???x x x , X ' = (1'x 2'x n 'x ), and y = 12n ?? ? ? ? ? ??? y y y Therefore, X 'X = 1 n t t t ='∑x x and X 'y = 1 n t t t ='∑x y . An equivalent expression for ?β is ?β = 1 11n t t t n --=??' ???∑x x 11n t t t n y -=??' ??? ∑x which, when we plug in y t = x t β + u t for each t and do some algebra, can be written as ?β= β + 1 11n t t t n --=??' ???∑x x 11n t t t n u -=??' ??? ∑x . As shown in Section E.4, this expression is the basis for the asymptotic analysis of OLS using matrices. E.2 (i) Following the hint, we have SSR(b ) = (y – Xb )'(y – Xb ) = [?u + X (?β – b )]'[ ?u + X (?β – b )] = ?u '?u + ?u 'X (?β – b ) + (?β – b )'X '?u + (?β – b )'X 'X (?β – b ). But by the first order conditions for OLS, X '?u = 0, and so (X '?u )' = ?u 'X = 0. But then SSR(b ) = ?u '?u + (?β – b )'X 'X (?β – b ), which is what we wanted to show. (ii) If X has a rank k then X 'X is positive definite, which implies that (?β – b ) 'X 'X (?β – b ) > 0 for all b ≠ ?β . The term ?u '?u does not depend on b , and so SSR(b ) – SSR(?β) = (?β– b ) 'X 'X (?β – b ) > 0 for b ≠?β. E.3 (i) We use the placeholder feature of the OLS formulas. By definition, β = (Z 'Z )-1Z 'y = [(XA )' (XA )]-1(XA )'y = [A '(X 'X )A ]-1A 'X 'y = A -1(X 'X )-1(A ')-1A 'X 'y = A -1(X 'X )-1X 'y = A -1?β . (ii) By definition of the fitted values, ?t y = ?t x β and t y = t z β. Plugging z t and β into the second equation gives t y = (x t A )(A -1?β ) = ?t x β = ?t y . (iii) The estimated variance matrix from the regression of y and Z is 2σ(Z 'Z )-1 where 2σ is the error variance estimate from this regression. From part (ii), the fitted values from the two

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伍德里奇《计量经济学导论》(第6版)复习笔记和课后习题详解-第二篇(第10~12章)【圣才出品】

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1．高斯-马尔可夫定理假设（见表10-2） 表10-2 高斯-马尔可夫定理假设

2．OLS估计量的性质与高斯-马尔可夫定理（见表10-3）

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本例，相应自由度就是多少？(4)EＶiews给出得相应概率就是0、８９,试判断原回归式误差项中就是否存在异方差。 5．假设上市公司绩效值（ＮEＲ)服从正态分布，模型满足同方差假定条件。（１)作为样本，739个上市公司绩效值得（ＮER)分布得均值与方差就是多少？当基金持股比例（RAＴE)为0、４０时,上市公司绩效值条件分布得均值与方差就是多少？(方差写出公式即可) 四、我们想要研究国内生产总值（GDP）、平均国外生产总值（FＧDP)与实际有效汇率指数(REER）对出口贸易额(EX）得影响，建立线性模型： 样本区间为1979年-２0０2年,GDP与FＧDP均以亿美元为计量单位.用普通最小二乘法估计上述模型,回归结果如下（括号内得数字为回归系数估计量得标准差）: = —2200、９0 ＋0、02＊GDＰ＋１、02＊FGDP ＋9、49*R ＥＥＲ （8３0、52）（0、002６）（0、3895)（3、431５） R２=0、９８, DW=0、50 whｉtｅ检验(有交叉)得统计量为：T*Ｒ2=2０、96；ＧDP、FGＤP =０、８7，ｒGDP,REＥ与REＥR之间得相关系数分别为：ｒG ＤP,ＦＧDP R= —０、24，ｒFGDＰ,REEＲ= —0、2８ 1。判断上述模型就是否满足经典假定条件；如果不满足，简要写出修正方法(15分) 2．检验原假设：与(）(５分) 3．检验整个方程得显著性（）（6分） 4．解释回归参数估计值=0、０２得经济意义（4）

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计学(第六版)第七章课后练习答案

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（2）已知：96.1%,951,12,81,1002/05.0==-===z s x n α。 由于100=n 为大样本，所以总体均值μ的95%的置信区间为： 352.281100 1296.1812 /±=? ±=±n s z x α 即)352.83,648.78( （3）已知：58.2%,991,12,81,1002/05.0==-===z s x n α。 由于100=n 为大样本，所以总体均值μ的99%的置信区间为： 096.381100 1258.2812 /±=? ±=±n s z x α 即)096.84,940.77( 7.5 （1）已知：96.1%,951,5.3,25,602/05.0==-===z x n ασ。 由于总体标准差已知，所以总体均值μ的95%的置信区间为： 89.02560 5.39 6.1252 /±=? ±=±n z x σ α 即)89.25,11.24( （2）已知：33.2%,981,89.23,6.119,752/02.0==-===z s x n α。 由于75=n 为大样本，所以总体均值μ的98%的置信区间为： 43.66.11975 89.2333.26.1192 /±=? ±=±n s z x α 即)03.126,17.113( （3）已知：645.1%,901,974.0,419.3,322/1.0==-===z s x n α。 由于32=n 为大样本，所以总体均值μ的90%的置信区间为： 283.0419.332 974.0645.1419.32 /±=? ±=±n s z x α 即)702.3,136.3(

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