CHAPTER 4
SOLUTIONS TO PROBLEMS 4.2 (i) and (iii) generally cause the t statistics not to have a t distribution under H0. Homoskedasticity is one of the CLM assumptions. An important omitted variable violates Assumption MLR.3. The CLM assumptions contain
APPENDIX B
SOLUTIONS TO PROBLEMS
B.1 Before the student takes the SAT exam, we do not know nor can we predict with certainty
what the score will be. The actual score depends on numerous factors, many of which we
cannot even list, let alone know ahead of
STUDENT SOLUTIONS MANUAL
Jeffrey M. Wooldridge
Introductory Econometrics: A Modern Approach, 4e
CONTENTS
Preface
iv
Chapter 1
Introduction
1
Chapter 2
The Simple Regression Model
3
Chapter 3
Multiple Regression Analysis: Estimation
9
Chapter 4
Multiple Re
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
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 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 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 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 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 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 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
APPENDIX D
SOLUTIONS TO PROBLEMS
0 1 6
2 1 7
20
D.1 (i) AB =
1 8 0 =
4 5 0
5
3 0 0
6
12
36 24
(ii) BA does not exist because B is 3 3 and A is 2 3.
D.3 Using the basic rules for transpose, ( XX) = ( X)( X) = XX , which is what we wanted to
sho
APPENDIX E
SOLUTIONS TO PROBLEMS
E.1 This follows directly from partitioned matrix multiplication in Appendix D. Write
x1
x
2
X = 2 , X = ( x1 x x ), and y =
n
x
n
Therefore, XX =
n
xx
t =1
t
t
and Xy =
n
x y
t =1
t
t
y1
y2
y
n
. An equivalent
CHAPTER 1
SOLUTIONS TO PROBLEMS 1.1 It does not make sense to pose the question in terms of causality. Economists would assume that students choose a mix of studying and working (and other activities, such as attending class, leisure, and sleeping) based
CHAPTER 2
SOLUTIONS TO PROBLEMS 2.2 (i) Let yi = GPAi, xi = ACTi, and n = 8. Then x = 25.875, y = 3.2125, (xi x )(yi y ) =
i =1 n
5.8125, and (xi x )2 = 56.875. From equation (2.9), we obtain the slope as 1 =
i =1
n
5.8125/56.875 .1022, rounded to four
CHAPTER 3
SOLUTIONS TO PROBLEMS 3.2 (i) hsperc is defined so that the smaller it is, the lower the students standing in high school. Everything else equal, the worse the students standing in high school, the lower is his/her expected college GPA. (ii) Jus
CHAPTER 5
SOLUTIONS TO PROBLEMS 5.2 The variable cigs has nothing close to a normal distribution in the population. Most people do not smoke, so cigs = 0 for over half of the population. A normally distributed random variable takes on no particular value
CHAPTER 6
SOLUTIONS TO PROBLEMS 6.1 This would make little sense. Performances on math and science exams are measures of outputs of the educational process, and we would like to know how various educational inputs and school characteristics affect math an
CHAPTER 7
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, tmale = 87.75/34.33 2.56, which is close to the 1% critical value agains
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
APPENDIX C
SOLUTIONS TO PROBLEMS
C.1 (i) This is just a special case of what we covered in the text, with n = 4: E( Y ) = and
Var( Y ) = 2/4.
(ii) E(W) = E(Y1)/8 + E(Y2)/8 + E(Y3)/4 + E(Y4)/2 = [(1/8) + (1/8) + (1/4) + (1/2)] = (1 +
1 + 2 + 4)/8 = , which