伍德里奇計量經濟學導論第四版

1、 15CHAPTER 3TEACHING NOTESFor undergraduates, I do not work through most of the derivations in this chapter, at least not in detail. Rather, I focus on interpreting the assumptions, which mostly concern the population. Other than random sampling, the only assumption that involves more than populatio

2、nconsiderations is the assumption about no perfect collinearity, where the possibility of perfect collinearity in the sample (even if it does not occur in the population should be touched on. The more important issue is perfect collinearity in the population, but this is fairly easy to dispense with

3、 via examples. These come from my experiences with the kinds of model specification issues that beginners have trouble with.The comparison of simple and multiple regression estimates based on the particular sample at hand, as opposed to their statistical properties usually makes a strong impression.

4、 Sometimes I do not bother with the “partialling out” interpretation of multiple regression.As far as statistical properties, notice how I treat the problem of including an irrelevant variable: no separate derivation is needed, as the result follows form Theorem 3.1.I do like to derive the omitted v

5、ariable bias in the simple case. This is not much more difficult than showing unbiasedness of OLS in the simple regression case under the first four Gauss-Markov assumptions. It is important to get the students thinking about this problem early on, and before too many additional (unnecessary assumpt

6、ions have been introduced.I have intentionally kept the discussion of multicollinearity to a minimum. This partly indicates my bias, but it also reflects reality. It is, of course, very important for students to understand the potential consequences of having highly correlated independent variables.

7、 But this is often beyond our control, except that we can ask less of our multiple regression analysis. If two or more explanatory variables are highly correlated in the sample, we should not expect to precisely estimate their ceteris paribus effects in the population.I find extensive treatments of

8、multicollinearity, where one “tests” or somehow “solves” the multicollinearity problem, to be misleading, at best. Even the organization of some texts gives the impression that imperfect multicollinearity is somehow a violation of the Gauss-Markovassumptions: they include multicollinearity in a chap

9、ter or part of the book devoted to “violation of the basic assumptions,” or something like that. I have noticed that masters students who have had some undergraduate econometrics are often confused on the multicollinearity issue. It is very important that students not confuse multicollinearity among

10、 the included explanatory variables in a regression model with the bias caused by omitting an important variable.I do not prove the Gauss-Markov theorem. Instead, I emphasize its implications. Sometimes, and certainly for advanced beginners, I put a special case of Problem 3.12 on a midterm exam, wh

11、ere I make a particular choice for the function g (x . Rather than have the students directly 課后答案網ww w.kh d aw .c om 16compare the variances, they should appeal to the Gauss-Markov theorem for the superiority of OLS over any other linear, unbiased estimator.SOLUTIONS TO PROBLEMS3.1 (i Yes. Because

12、of budget constraints, it makes sense that, the more siblings there are in a family, the less education any one child in the family has. To find the increase in the number of siblings that reduces predicted education by one year, we solve 1 = .094(sibs , so sibs = 1/.094 10.6.(ii Holding sibs and fe

13、duc fixed, one more year of mothers education implies .131 years more of predicted education. So if a mother has four more years of education, her son is predicted to have about a half a year (.524 more years of education. (iii Since the number of siblings is the same, but meduc and feduc are both d

14、ifferent, the coefficients on meduc and feduc both need to be accounted for. The predicted difference in education between B and A is .131(4 + .210(4 = 1.364.3.2 (i hsperc is defined so that the smaller it is, the lower the students standing in high school. Everything else equal, the worse the stude

15、nts standing in high school, the lower is his/her expected college GPA. (ii Just plug these values into the equation:n colgpa= 1.392 .0135(20 + .00148(1050 = 2.676.(iii The difference between A and B is simply 140 times the coefficient on sat , because hsperc is the same for both students. So A is p

16、redicted to have a score .00148(140 .207 higher.(iv With hsperc fixed, n colgpa = .00148sat . Now, we want to find sat such that n colgpa = .5, so .5 = .00148(sat or sat = .5/(.00148 338. Perhaps not surprisingly, a large ceteris paribus difference in SAT score almost two and one-half standard devia

17、tions is needed to obtain a predicted difference in college GPA or a half a point.3.3 (i A larger rank for a law school means that the school has less prestige; this lowers starting salaries. For example, a rank of 100 means there are 99 schools thought to be better.課后答案網ww w.kh d aw .c om 17(ii 1 &

18、gt; 0, 2 > 0. Both LSAT and GPA are measures of the quality of the entering class. No matter where better students attend law school, we expect them to earn more, on average. 3, 4 > 0. The number of volumes in the law library and the tuition cost are both measures of the school quality. (Cost

19、is less obvious than library volumes, but should reflect quality of the faculty, physical plant, and so on. (iii This is just the coefficient on GPA , multiplied by 100: 24.8%. (iv This is an elasticity: a one percent increase in library volumes implies a .095% increase in predicted median starting

20、salary, other things equal. (v It is definitely better to attend a law school with a lower rank. If law school A has a ranking 20 less than law school B, the predicted difference in starting salary is 100(.0033(20 = 6.6% higher for law school A.3.4 (i If adults trade off sleep for work, more work im

21、plies less sleep (other things equal, so 1 < 0. (ii The signs of 2 and 3 are not obvious, at least to me. One could argue that more educated people like to get more out of life, and so, other things equal, they sleep less (2 < 0. The relationship between sleeping and age is more complicated th

22、an this model suggests, and economists are not in the best position to judge such things.(iii Since totwrk is in minutes, we must convert five hours into minutes: totwrk = 5(60 = 300. Then sleep is predicted to fall by .148(300 = 44.4 minutes. For a week, 45 minutes less sleep is not an overwhelming

23、 change. (iv More education implies less predicted time sleeping, but the effect is quite small. If we assume the difference between college and high school is four years, the college graduate sleeps about 45 minutes less per week, other things equal. (v Not surprisingly, the three explanatory varia

24、bles explain only about 11.3% of the variation in sleep . One important factor in the error term is general health. Another is marital status, and whether the person has children. Health (however we measure that, marital status, and number and ages of children would generally be correlated with totw

25、rk . (For example, less healthy people would tend to work less.3.5 Conditioning on the outcomes of the explanatory variables, we have 1E( = E(1 + 2 = E(1+ E(2 = 1 + 2 = 1.3.6 (i No. By definition, study + sleep + work + leisure = 168. Therefore, if we change study , we must change at least one of th

26、e other categories so that the sum is still 168. 課后答案網ww w.kh d aw .c om 18(ii From part (i, we can write, say, study as a perfect linear function of the otherindependent variables: study = 168 sleep work leisure . This holds for every observation, so MLR.3 violated. (iii Simply drop one of the inde

27、pendent variables, say leisure :GPA = 0 + 1study + 2sleep + 3work + u .Now, for example, 1 is interpreted as the change in GPA when study increases by one hour, where sleep , work , and u are all held fixed. If we are holding sleep and work fixed but increasing study by one hour, then we must be red

28、ucing leisure by one hour. The other slope parameters have a similar interpretation.3.7 We can use Table 3.2. By definition, 2 > 0, and by assumption, Corr(x 1,x 2 < 0.Therefore, there is a negative bias in 1: E(1 < 1. This means that, on average across different random samples, the simple

29、regression estimator underestimates the effect of thetraining program. It is even possible that E(1 is negative even though 1 > 0.3.8 Only (ii, omitting an important variable, can cause bias, and this is true only when the omitted variable is correlated with the included explanatory variables. Th

30、e homoskedasticity assumption, MLR.5, played no role in showing that the OLS estimators are unbiased.(Homoskedasticity was used to obtain the usual variance formulas for the j. Further, the degree of collinearity between the explanatory variables in the sample, even if it is reflected in a correlati

31、on as high as .95, does not affect the Gauss-Markov assumptions. Only if there is a perfect linear relationship among two or more explanatory variables is MLR.3 violated.3.9 (i Because 1x is highly correlated with 2x and 3x , and these latter variables have largepartial effects on y , the simple and

32、 multiple regression coefficients on 1x can differ by largeamounts. We have not done this case explicitly, but given equation (3.46 and the discussion with a single omitted variable, the intuition is pretty straightforward.(ii Here we would expect 1 and 1 to be similar (subject, of course, to what w

33、e mean by “almost uncorrelated”. The amount of correlation between 2x and 3x does not directly effect the multiple regression estimate on 1x if 1x is essentially uncorrelated with 2x and 3x .(iii In this case we are (unnecessarily introducing multicollinearity into the regression: 2x and 3x have sma

34、ll partial effects on y and yet 2x and 3x are highly correlated with 1x . Adding2x and 3x like increases the standard error of the coefficient on 1x substantially, so se(1is likely to be much larger than se(1 . 課后答案網ww w.kh d aw .c om 19(iv In this case, adding 2x and 3x will decrease the residual v

35、ariance without causingmuch collinearity (because 1x is almost uncorrelated with 2x and 3x , so we should see se(1 smaller than se(1. The amount of correlation between 2x and 3x does not directly affect se(1.3.10 From equation (3.22 we have111211,ni ii ni i r yr =where the 1i rare defined in the pro

36、blem. As usual, we must plug in the true model for y i :1011223311211(.ni i i i ii ni i r x x x u r =+=The numerator of this expression simplifies because 11ni i r= = 0, 121ni i i r x = = 0, and 111ni i i r x = = 211ni i r =. These all follow from the fact that the 1i rare the residuals from the reg

37、ression of 1i x on 2i x : the 1i rhave zero sample average and are uncorrelated in sample with 2i x . So the numerator of 1can be expressed as2113131111.n n ni i i i i i i i rr x r u =+Putting these back over the denominator gives13111113221111.nni i ii i nni i i i r x rur r =+課后答案網ww w.kh d aw .c o

38、m 20Conditional on all sample values on x 1, x 2, and x 3, only the last term is random due to its dependence on u i . But E(u i = 0, and so131113211E(=+,ni i i ni i r xr =which is what we wanted to show. Notice that the term multiplying 3 is the regressioncoefficient from the simple regression of x

39、 i 3 on 1i r.3.11 (i 1 < 0 because more pollution can be expected to lower housing values; note that 1 isthe elasticity of price with respect to nox . 2 is probably positive because rooms roughlymeasures the size of a house. (However, it does not allow us to distinguish homes where each room is l

40、arge from homes where each room is small. (ii If we assume that rooms increases with quality of the home, then log(nox and rooms are negatively correlated when poorer neighborhoods have more pollution, something that is often true. We can use Table 3.2 to determine the direction of the bias. If 2 &g

41、t; 0 andCorr(x 1,x 2 < 0, the simple regression estimator 1has a downward bias. But because 1 < 0, this means that the simple regression, on average, overstates the importance of pollution. E(1 is more negative than 1. (iii This is what we expect from the typical sample based on our analysis i

42、n part (ii. The simple regression estimate, 1.043, is more negative (larger in magnitude than the multiple regression estimate, .718. As those estimates are only for one sample, we can never know which is closer to 1. But if this is a “typical” sample, 1 is closer to .718.3.12 (i For notational simp

43、licity, define s zx = 1(;ni i i z z x = this is not quite the samplecovariance between z and x because we do not divide by n 1, but we are only using it tosimplify notation. Then we can write 1 as11(.niii zxz z ys =This is clearly a linear function of the y i : take the weights to be w i = (z i z /s

44、 zx . To show unbiasedness, as usual we plug y i = 0 + 1x i + u i into this equation, and simplify: 課后答案網ww w.kh daw .c om 21111011111(nii i i zxnni zx i ii i zxniii zxz z x u s z z s z z u s zz u s =+=+=+where we use the fact that 1(ni i z z = = 0 always. Now s zx is a function of the z i and x i a

45、nd theexpected value of each u i is zero conditional on all z i and x i in the sample. Therefore, conditional on these values,1111(E(E(niii zxz z u s =+=because E(u i = 0 for all i . (ii From the fourth equation in part (i we have (again conditional on the z i and x i in the sample,2111222212Var (Va

46、r(Var(n ni i i i i i zx zxnii zxz z u z z u s s z z s =because of the homoskedasticity assumption Var(u i = 2 for all i . Given the definition of s zx , this is what we wanted to show.課后答案網ww w.kh d aw .c om 22(iii We know that Var(1 = 2/21(.ni i x x = Now we can rearrange the inequality in the hint

47、, drop x from the sample covariance, and cancel n -1everywhere, to get 221(/ni zx i z z s = 211/(.ni i x x = When we multiply through by 2 we get Var(1 Var(1, which is what we wanted to show.3.13 (i The shares, by definition, add to one. If we do not omit one of the shares then the equation would su

48、ffer from perfect multicollinearity. The parameters would not have a ceteris paribus interpretation, as it is impossible to change one share while holding all of the other shares fixed. (ii Because each share is a proportion (and can be at most one, when all other shares are zero, it makes little se

49、nse to increase share p by one unit. If share p increases by .01 which is equivalent to a one percentage point increase in the share of property taxes in total revenue holding share I , share S , and the other factors fixed, then growth increases by 1(.01. With the other shares fixed, the excluded s

50、hare, share F , must fall by .01 when share p increases by .01.SOLUTIONS TO COMPUTER EXERCISESC3.1 (i Probably 2 > 0, as more income typically means better nutrition for the mother and better prenatal care. (ii On the one hand, an increase in income generally increases the consumption of a good,

51、and cigs and faminc could be positively correlated. On the other, family incomes are also higher for families with more education, and more education and cigarette smoking tend to benegatively correlated. The sample correlation between cigs and faminc is about .173, indicating a negative correlation

52、.(iii The regressions without and with faminc aren 119.77.514bwghtcigs =21,388,.023n R =and n 116.97.463.093bwghtcigs faminc =+21,388,.030.n R =課后答案網ww w.kh d aw .c om 23The effect of cigarette smoking is slightly smaller when faminc is added to the regression, but the difference is not great. This

53、is due to the fact that cigs and faminc are not very correlated, and the coefficient on faminc is practically small. (The variable faminc is measured in thousands, so $10,000 more in 1988 income increases predicted birth weight by only .93 ounces.C3.2 (i The estimated equation isn 19.32.12815.20pric

54、e sqrft bdrms =+288,.632n R =(ii Holding square footage constant, n price = 15.20 ,bdrms and so n price increases by 15.20, which means $15,200.(iii Now n price = .128sqrft + 15.20bdrms = .128(140 + 15.20 = 33.12, or $33,120. Because the size of the house is increasing, this is a much larger effect

55、than in (ii. (iv About 63.2%. (v The predicted price is 19.32 + .128(2,438 + 15.20(4 = 353.544, or $353,544. (vi From part (v, the estimated value of the home based only on square footage and number of bedrooms is $353,544. The actual selling price was $300,000, which suggests the buyer underpaid by

56、 some margin. But, of course, there are many other features of a house (some that we cannot even measure that affect price, and we have not controlled for these.C3.3 (i The constant elasticity equation isn log( 4.62.162log(.107log(salary sales mktval =+ 2177,.299.n R =(ii We cannot include profits in logarithmic form because profits are negative for nine of the companies in the sample. When we add it in levels form we getn log( 4.69.161log(.098log(.000036salary sales mktval profits =+2177,.299.n R =The coefficient on profits is very small. Here, profits are measure