E460
PROBLEM SET 1
In answering the following questions, do not restrict yourself to finding answers that are natural numbers. Instead, provide decimal answers where you compute them to be so, even though it may lack some intuition. 1. A firm with
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. E
PROBLEM SET 1 ECONOMICS 418 UNIVERSITY OF ARIZONA
(1) Explain why the following claims might not imply causal relationships and outline how you would test for a causal eect using an experimental methodology with a treatment and control group (a) Peo
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
ECON 418 Introduction to Econometrics
Anna Breman
Univeristy of Arizona
Part 3: Regression analysis Multiple Regression Analysis with Qualitative Information: Dummy Variables. Chapter 7
Anna Breman (Univeristy of Arizona)
ECON 418
Fall 2008
1 /
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, wh
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 + u
ECON 418 Introduction to Econometrics
Anna Breman
Univeristy of Arizona
Part 3: Regression analysis Chapter 8: Heteroskedasticity
Anna Breman (Univeristy of Arizona)
ECON 418
Fall 2008
1 / 33
Chapter 8 I
Heteroskedasticity
Outline
1 2
What is
ECON 418 Introduction to Econometrics
Anna Breman
Univeristy of Arizona
Part 4: Advanced Regression analysis Chapter 15: Instrumental Variables
Anna Breman (Univeristy of Arizona)
ECON 418
Fall 2008
1 / 19
Chapter 15 I
Instrumental Variables
E
ECON 418 Introduction to Econometrics
Anna Breman
Univeristy of Arizona
Part 3: Regression analysis Endogeneity
Anna Breman (Univeristy of Arizona)
ECON 418
Fall 2008
1 / 32
Endogeneity I
Endogeneity occurs whenever the error term, u, is corre
APPENDIX A
SOLUTIONS TO PROBLEMS A.1 (i) $566. (ii) The two middle numbers are 480 and 530; when these are averaged, we obtain 505, or $505. (iii) 5.66 and 5.05, respectively. (iv) The average increases to $586 while the median is unchanged ($505). A
-log: C:\Documents and Settings\SCLab42\My Documents\assignment5.log log type: text opened on: 30 Nov 2009, 17:14:35 . tsset t time variable: t, 1 to 108
. regress totacc feb mar apr may jun jul aug sep oct nov dec spdlaw beltlaw t Source | SS df MS -+-Mo
-log: C:\Documents and Settings\SCLab42\My Documents\assignment5_1.log log type: text opened on: 30 Nov 2009, 19:00:15 . drop if hours = 0 ( (325 observations deleted) . gen lnhours = log(hours) . gen lnwage = log(wage) . gen hearn = huswage*hushrs . hen
Economics 418
Solon
Exercise 1
Consider the random variable Y u where is an unknown parameter and u is a random variable
with mean zero.
a.
What is the expected value of Y?
b.
Suppose you have a random sample Y1 , Y2 ,., YN drawn from the distribution for
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 lik
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 {ut} and constant variance. Next, we find the covariance between uit a
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
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 ahe
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 = , w
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
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 y1 y2 y n
Therefore, XX =
xx
t =1 t
n
t
and Xy =
x y
CHAPTER 1
SOLUTIONS TO PROBLEMS 1.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each student is assigned a different class size without regard to any student characteristics such as ability and family backgr
CHAPTER 2
SOLUTIONS TO PROBLEMS 2.1 (i) Income, age, and family background (such as number of siblings) are just a few possibilities. It seems that each of these could be correlated with years of education. (Income and education are probably positive
CHAPTER 3
SOLUTIONS TO PROBLEMS 3.1 (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
CHAPTER 4
SOLUTIONS TO PROBLEMS 4.1 (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 con
CHAPTER 5
SOLUTIONS TO PROBLEMS 5.1 Write y = 0 + 1 x1 + u, and take the expected value: E(y) = 0 + 1 E(x1) + E(u), or y =
0 + 1 x since E(u) = 0, where y = E(y) and x = E(x1). We can rewrite this as 0 = y 1 x. Now, 0 = y 1 x1 . Taking the
CHAPTER 6
SOLUTIONS TO PROBLEMS 6.1 The generality is not necessary. The t statistic on roe2 is only about .30, which shows that roe2 is very statistically insignificant. Plus, having the squared term has only a minor effect on the slope even for lar
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