伍德里奇：计量经济学导论chapter14

CHAPTER 14 SOLUTIONS TO PROBLEMS 14.1 First, for each t > 1, Var(uit) = Var(uit – ui,t1) = Var(uit) + Var(ui,t1) = , where we use the assumptions of no serial correlation in {ut} and constant variance.  Next, we find the covariance between uit and ui,t+1.  Because these each have a zero mean, the covariance is E(uit ui,t+1) = E[(uit – ui,t1)(ui,t+1 – uit)] = E(uitui,t+1) – E( ) – E(ui,t1ui,t+1) + E(ui,t1uit) = E( ) = because of the no serial correlation assumption.  Because the variance is constant across t, by Problem 11.1, Corr(uit, ui,t+1) = Cov(uit, ui,t+1)/Var(?uit) =  = .5. 14.3 (i) E(eit) = E(vit  ) = E(vit)  E( ) = 0 because E(vit) = 0 for all t.  (ii) Var(vit  ) = Var(vit) + 2Var( )  2 Cov(vit, ) =  + 2 E( )  2 E(vit ).  Now, and E(vit ) =  = [  +  + + (  + ) + + ] =  + /T.  Therefore, E( ) =  =  + /T.  Now, we can collect terms: Var(vit  )  =  . Now, it is convenient to write  = 1  , where   /T and    + /T.  Then Var(vit  )  =  ( + ) 2( + /T) + 2( +  /T) =    ( + )  2(1 ) + (1 )2 =    ( + )  2 + 2 + (1 2 + /) =    ( + )  2 + 2 + (1 2 + /) =    ( + )  2 + 2 + 2 + =    ( + )  + = . This is what we wanted to show. (iii) We must show that E(eiteis) = 0 for t  s.  Now E(eiteis) = E[(vit )(vis  )] = E(vitvis)  E( vis)  E(vit ) + 2E( ) =   2(  + /T) + 2E( ) =   2(  + /T) + 2(  + /T).  The rest of the proof is very similar to part (ii):  E(eiteis)    =    2( + /T) + 2( +  /T) =    2(1 ) + (1 )2 =    2 + 2 + (1 2 + /) =    2 + 2 + (1 2 + /) =    2 + 2 + 2 + =    + = 0. 14.5 (i) For each student we have several measures of performance, typically three or four, the number of classes taken by a student that have final exams.  When we specify an equation for each standardized final exam score, the errors in the different equations for the same student are certain to be correlated: students who have more (unobserved) ability tend to do better on all tests. (ii) An unobserved effects model is scoresc  =  c + 1atndrtesc + 2majorsc + 3SATs + 4cumGPAs + as + usc, where as is the unobserved student effect.  Because SAT score and cumulative GPA depend only on the student, and not on the particular class he/she is taking, these do not have a c subscript.  The attendance rates do generally vary across class, as does the indicator for whether a class is in the student’s major.  The term c denotes different intercepts for different classes.  Unlike with a panel data set, where time is the natural ordering of the data within each cross-sectional unit, and the aggregate time effects apply to all units, intercepts for the different classes may not be needed.  If all students took the same set of classes then this is similar to a panel data set, and we would want to put in different class intercepts.  But with students taking different courses, the class we label as “1” for student A need have nothing to do with class “1” for student B.    Thus, the different class intercepts based on arbitrarily ordering the classes for each student probably are not needed.  We can replace c with 0, an intercept constant across classes. (iii) Maintaining the assumption that the idiosyncratic error, usc, is uncorrelated with all explanatory variables, we need the unobserved student heterogeneity, as, to be uncorrelated with atndrtesc.  The inclusion of SAT score and cumulative GPA should help in this regard, as as, is the part of ability that is not captured by SATs and cumGPAs.  In other words, controlling for SATs and cumGPAs could be enough to obtain the ceteris paribus effect of class attendance. (iv) If SATs and cumGPAs are not sufficient controls for student ability and motivation, as is correlated with atndrtesc, and this would cause pooled OLS to be biased and inconsistent.  We could use fixed effects instead.  Within each student we compute the demeaned data, where, for each student, the means are computed across classes.  The variables SATs and cumGPAs drop out of the analysis. SOLUTIONS TO COMPUTER EXERCISES 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) This is the only new part.  The fixed effects estimates, reported in equation form, are =    .386 y90t    +    .072 log(popit)    +    .310 log(avgincit)    +    .0112 pctstuit, (.037)        (.088)        (.066)        (.0041) N  =  64,  T  =  2. (There are N = 64 cities and T = 2 years.)  We do not report an intercept because it gets removed by the time demeaning.  The coefficient on y90t is identical to the intercept from the first difference estimation, and the slope coefficients and standard errors are identical to first differencing.  We do not report an R-squared because none is comparable to the R-squared obtained from first differencing. C14.3 (i) 135 firms are used in the FE estimation.  Because there are three years, we would have a total of 405 observations if each firm had data on all variables for all three years.  Instead, due to missing data, we can use only 390 observations in the FE estimation.  The fixed effects estimates are =    1.10 d88t    +    4.09 d89t    +    34.23 grantit (1.98)        (2.48)        (2.86) +    .504 granti,t-1        .176 log(employit) (4.127)                (4.288) n  =  390,  N  =  135,  T  =  3. (ii) The coefficient on grant means that if a firm received a grant for the current year, it trained each worker an average of 34.2 hours more than it would have otherwise.  This is a practically large effect, and the t statistic is very large. (iii) Since a grant last year was used to pay for training last year, it is perhaps not surprising that the grants does not carry over into more training this year.  It would if inertia played a role in training workers. (iv) The coefficient on the employees variable is very small:  a 10% increase in employ increases predicted hours per employee by only about .018.  [Recall:  (.176/100) (%employ).]  This is very small, and the t statistic is practically zero. C14.5 (i) Different occupations are unionized at different rates, and wages also differ by occupation.  Therefore, if we omit binary indicators for occupation, the union wage differential may simply be picking up wage differences across occupations.  Because some people change occupation over the period, we should include these in our analysis. (ii) Because the nine occupational categories (occ1 through occ9) are exhaustive, we must choose one as the base group.  Of course the group we choose does not affect the estimated union wage differential.  The fixed effect estimate on union, to four decimal places, is .0804 with standard error = .0194.  There is practically no difference between this estimate and standard error and the estimate and standard error without the occupational controls ( = .0800, se = .0193). C14.7 (i) If there is a deterrent effect then 1 < 0.  The sign of 2 is 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 2 > 0. (ii) The pooled OLS estimates using 1990 and 1993 are =     5.28        2.07 d93t  +     .128 execit  +      2.53 unemit (4.43)    (2.14)    (.263)    (0.78) N = 51,  T = 2,  R2 = .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 =     .413      .104 execi        .067 unemi (.209)    (.043)    (.159) n = 51,  R2 = .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. (iv) The heteroskedasticity-robust standard error for execi is .017.  Somewhat surprisingly, this is well below the nonrobust standard error.  If we use the robust standard error, the statistical evidence for the deterrent effect is quite strong (t 6.1).  See also Computer Exercise 13.12. (v) Texas had by far the largest value of exec, 34.  The next highest state was Virginia, with 11.  These are three-year totals. (vi) Without Texas in the estimation, we get the following, with heteroskedasticity-robust standard errors in []: =     .413      .067 execi        .070 unemi (.211)    (.105)    (.160) [.200]    [.079]    [.146] n = 50,  R2 = .013 Now the estimated deterrent effect is smaller.  Perhaps more importantly, the standard error on execi  has increased by a substantial amount.  This happens because when we drop Texas, we lose much of the variation in the key explanatory variable, execi. (vii) When we apply fixed effects using all three years of data and all states we get =     1.73 d90t  +      1.70 d93t        .054 execit  +      .395 unemit (.75)    (.71)    (.160)    (.285) N = 51,  T = 3,  R2 = .068 The size of the deterrent effect is only about half as big as when 1987 is not used.  Plus, the t statistic, about .34, is very small.  The earlier finding of a deterrent effect is not robust to the time period used.  Oddly, adding another year of data causes the standard error on the exec coefficient to markedly increase. C14.9 (i) The OLS estimates are 128.54  +      11.74 choice  +      14.34 prftshr  +     1.45 female      1.50 age (55.17)    (6.23)    (7.23)    (6.77)    (.78) +    .70 educ        15.29 finc25  +      .19 finc35        3.86 finc50                          (1.20)    (14.23)    (14.69)    (14.55)              13.75 finc75       2.69 finc100    25.05 finc101    .0026 wealth89 (16.02)    (15.72)    (17.80)    (.0128)