CHAPTER 8
SOLUTIONS TO PROBLEMS 8.1 Parts (ii) and (iii). The homoskedasticity assumption played no role in Chapter 5 in showing that OLS is consistent. But we know that heteroskedasticity causes statistical inference based on the usual t and F statistics
CHAPTER 16
SOLUTIONS TO PROBLEMS 16.1 (i) If 1 = 0 then y1 = 1z1 + u1, and so the right-hand-side depends only on the exogenous variable z1 and the error term u1. This then is the reduced form for y1. If 1 = 0, the reduced form for y1 is y1 = 2z2 + u2. (N
CHAPTER 17
SOLUTIONS TO PROBLEMS 17.1 (i) Let m0 denote the number (not the percent) correctly predicted when yi = 0 (so the prediction is also zero) and let m1 be the number correctly predicted when yi = 1. Then the proportion correctly predicted is (m0
CHAPTER 18
SOLUTIONS TO PROBLEMS 18.1 With zt1 and zt2 now in the model, we should use one lag each as instrumental variables, zt-1,1 and zt-1,2. This gives one overidentifying restriction that can be tested. 18.3 For , yt zt = yt zt + ( )zt, which is an
CHAPTER 15
SOLUTIONS TO PROBLEMS 15.1 (i) It has been fairly well established that socioeconomic status affects student performance. The error term u contains, among other things, family income, which has a positive effect on GPA and is also very likely t
CHAPTER 14
SOLUTIONS TO PROBLEMS 14.1 First, for each t > 1, Var(uit) = Var(uit ui,t-1) = Var(uit) + Var(ui,t-1) = 2 u2 , where we use the assumptions of no serial correlation in cfw_ut and constant variance. Next, we find the covariance between uit and u
CHAPTER 9
SOLUTIONS TO PROBLEMS 9.1 There is functional form misspecification if 6 0 or 7 0, where these are the population parameters on ceoten2 and comten2, respectively. Therefore, we test the joint significance of these variables using the R-squared f
CHAPTER 10
SOLUTIONS TO PROBLEMS 10.1 (i) Disagree. Most time series processes are correlated over time, and many of them strongly correlated. This means they cannot be independent across observations, which simply represent different time periods. Even s
CHAPTER 11
SOLUTIONS TO PROBLEMS 11.1 Because of covariance stationarity, 0 = Var(xt) does not depend on t, so sd(xt+h) =
0 for
any h 0. By definition, Corr(xt,xt+h) = Cov(xt,xt+h)/[sd(xt) sd(xt+h)] = h /( 0 0 ) = h / 0 . 11.3 (i) E(yt) = E(z + et) = E(z
CHAPTER 13
SOLUTIONS TO PROBLEMS 13.1 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 the increase in average education leve
CHAPTER 12
SOLUTIONS TO PROBLEMS 12.1 We can reason this from equation (12.4) because the usual OLS standard error is an estimate of / SSTx . When the dependent and independent variables are in level (or log) form, the AR(1) parameter, , tends to be posit