Type:
Solution Manual
Statistics and Econometrics Resource: Methods and Applications Edition:
1st Edition
Author(s):
Orley Ashenfelter David J. Zimmerman Phillip B. Levine
SOLUTIONS MANUAL
To
Statistics and Econometrics: Theory and Applications
by Orley Ashenfelter, Phillip B. Levine, and David J. Zimmerman
CONTENTS INSTRUCTOR’S NOTES SOLUTIONS TO CHAPTER PROBLEMS CHAPTER 2 BASIC PROBABILITY THEORY CHAPTER 3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS CHAPTER 4 MATHEMATICS OF EXPECTATIONS CHAPTER 5 MULTIVARIATE DISTRIBUTIONS CHAPTER 6 SAMPLING AND SAMPLING DISTRIBUTIONS CHAPTER 7 ESTIMATION CHAPTER 8 INTERVAL ESTIMATION AND HYPOTHESIS TESTING CHAPTER 9 SIMPLE LINEAR REGRESSION CHAPTER 10 INFERENCES IN SIMPLE LINEAR REGRESSION CHAPTER 11 MULTIPLE REGRESSION CHAPTER 12 EXTENDING THE MULTIPLE REGRESSION MODEL CHAPTER 13 SPECIFICATION ERROR, MULTICOLLINEARITY, AND MEASUREMENT ERROR CHAPTER 14 HETEROSKEDASTICITY AND SERIAL CORRELATION CHAPTER 15 SIMULTANEOUS EQUATIONS CHAPTER 16 QUALITATIVE CHOICE AND LIMITED DEPENDENT VARIABLE MODELS CHAPTER 17 LINEAR TIME SERIES MODELS CHAPTER 18 PANEL DATA MODELS
INSTRUCTOR’S NOTES
Although the market for introductory econometrics books is rather well-developed, we had never been completely satisfied with the choices available for our classes. As empirical labor economists, we found that we were unable to locate books that were able to stimulate in students our interest in conducting econometric exercises. We hope to further that goal in this book by providing boxes in each chapters that contain summaries of published research that address important real world questions. In our own research, we have relied heavily on econometric tools to address such questions, which are unable to be answered by other means. For instance, economic analysis is often capable of providing theoretical predictions regarding the direction of a particular relationship but it cannot provide a magnitude of that relationship. Is the demand curve steep or flat? To what extent do wages rise with education? For other questions, economics cannot even tell us the direction of the relationship. Does labor supply rise with an increase in the wage? Other questions outside the purview of economic analysis still require estimates of the impact of one variable on another. Does student performance improve with smaller class sizes? It is the desire to answer questions like these that have always motivated us to conduct econometric analyses and we have written this book in an attempt to motivate a future generation. But this focus on real world questions does not diminish the need to understand the technical intricacies of a sophisticated econometric analysis. The credibility of econometric research exclusively relies upon whether or not it was conducted in an appropriate manner. Research that addresses an important topic but that does not display a mastery of the technical components of the analysis is of little value. Therefore, we also seek to present a rigorous introduction to econometric techniques. In the development of this book, there were two fundamental decisions that we made regarding the coverage of material. First, we had to make a choice between a book that covered all potentially relevant material or a book that focused just on the topics we considered most important. Some books on the market treat many greater topics than we do and serve a useful purpose as a reference. We have chosen to limit the topics covered to those that we address in our courses. From our experiences in teaching econometrics, we have found that those books which provide encyclopedic coverage of the field typically overwhelm students. When they go to read the book, they find so many topics that were not covered in lecture, that they tend not to use the book at all. On the other hand, those books may be invaluable for the relatively few students who pursue further study in econometrics. The book that we have written has largely been developed straight from our lecture notes, so we teach virtually all the material in it. We sought to present the material in such a way that faculty can cover a particular topic and simply assign to students to “read all of chapter 12.” Similarly, problems at the end of chapters are designed more like problem sets rather than an exhaustive coverage of all the topics in the chapter. The second important decision we had to make was where to begin and end our coverage of material. Some schools teach a one-semester course in econometrics that begins with basic
probability theory and ends with regression analysis. The curriculum at other institutions includes a two-semester sequence. Probability and statistics is covered in the first semester and regression analysis, including more sophisticated topics like time series and panel data methods, are covered in the second semester. We have chosen to write a book that is flexible enough to cover both types of instruction. For those who teach all the material in one semester, coverage could begin from the beginning of the book and end wherever the instructor makes it (perhaps through multiple regression in chapter 12). Alternatively, in a two-semester sequence, the first half of the book (through, say, chapters 9 or 10 on simple linear regression) could be covered in the first semester and the remainder in the second semester. Of course, the second semester would have to review the basics of probability and statistics, so we have provided Appendix A1 to facilitate that process. In some institutions, that first semester course would be taught by, say, the math department, which would not be likely to adopt this book. Nevertheless, instructors could assign Appendix A1 followed by the material in Chapters 9 through 18 for a self-contained treatment of the second semester’s material. For those who choose to use this book for both semesters of the two-semester sequence, students will appreciate the ability to buy a single book.
SOLUTIONS TO CHAPTER 2
1) Yes. P( A B / B) =
P( A B B) P( A B) = = P( A / B) P( B) P( B)
2) a) S={Heads, Tails}. The probability for each is .5 b) S={all of the cards in the deck}. The odds of even A occurring is ¼. c) The cast of two unbiased coins, where event A occurs if the roll is less than 5. S={(1,1),(1,2),(1,3),...,(6,5),(6,6)}, i.e. list of all 36 possible combinations Pr(A)=Pr{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)} = 6/36 = 1/6
3) S{all of the cards in the deck} a) P(A)=12/52 b) P(A and B) = 2/52 c) P(A or B)=16/52 d) P(C or D)=1 e) P(C and D)=0 4)
Prob(Win/Three Point attempt) = .3 Prob(Win/Two Point attemp)=P(make two pointer and win in overtime) = .5*.5 = .25 (assuming independence) So, go for it! Try the three pointer. 5) P(good shot)=P( good / correct ) P(correct ) + P( good / correct ') P(correct ') = (1/3)*(1/4) +(1/5)(3/4) = 7/30=.2333 and P(correct ' good ) P( good ) P( good correct ') P ( good / correct ') P (correct ') = = P( good ) P ( good )
P(correct '/ good ) =
=(1/5)(3/4)/.233 = .643
6) Let: H=hired I=Interviewed Given: P(H)=.04 so P(~H)=.96 P(I/H)=.98 P(I/~H)=.01 We want to know P(H/I).
P( H I ) P( I H ) = P( I ) P( I ) P( I / H ) P( H ) = P( I / H ) P( H ) + P ( I / H ') P( H ')
P( H / I ) =
= (.98)*(.04) / ( (.98*.04) + .01*.96)) = .803. Thus better than an 80% chance.
7) The possibilities are (let A, B, C indicate acceptance at that school): Outcome Prob note: (P(~A)=1-P(A)) Also, we assume independence. ABC
.2*.4*.6=.048
i.e. got in to all three
ABC’ AB’C AB’C’ A’BC A’BC’ A’B’C A’B’C’
.2*.4*.4=.032 .2*.6*.6=.072 .2*.6*.4=.048 .8*.4*.6=.192 .8*.4*.4=.128 .8*.6*.6=.288 .8*.6*.4
i.e. got in to A and B but not C
=.192
i.e. didn’t get in to any
This is the whole distribution. This is a simple way to solve the problem. The odds you get rejected at all three is .192 or 19.2%. The odds you get accepted to at least one is simply 1 minus the odds you don’t get into any – which is 1-.192=.808. 8)
Q1 = Use marijuana on the job ? Q 2 = Wish you win the lottery ? P( yes ) = .44 P(Q1) = 0.7 P(Q 2) = 0.3 Want : P( yes / Q1) P( yes ) = P ( yes Q1) + P ( yes Q 2) = P( yes / Q1) P (Q1) + P ( yes / Q 2) P (Q 2) we assume that P ( yes / Q 2) = 1, and solving for P ( yes / Q1), we get : P( yes ) − P ( yes / Q 2) P (Q 2) P( yes / Q1) = P(Q1) (.44) − (1)(0.3) = (0.7) = 0.2
9) Fred should switch doors after a booby prize has been shown behind one of the other doors. There are a few different ways to reason the answer. Suppose that Fred initially choose door #1. Then Fred has a 1/3 probabilit of being correct. This also means there is a 2/3 probability the car is behind either door #2 or door #3. Now suppose Monty opens