计量经济学导论及习题答案等教辅资源 Chapter 11.docx

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1、Introduction to Econometrics, 3e (Stock)Chapter 11 Regression with a Binary Dependent Variable11.1 Multiple Choice1) The binary dependent variable model is an example of aA) regression model, which has as a regressor, among others, a binary variable.B) model that cannot be estimated by OLS.C) limite

2、d dependent variable model.D) model where the left-hand variable is measured in base 2.Answer: C2) (Requires Appendix material) The following are examples of limited dependent variables, with the exception ofA) binary dependent variable.B) log-log specification.C) truncated regression model.D) discr

3、ete choice model.Answer: B3) In the binary dependent variable model, a predicted value of 0.6 means thatA) the most likely value the dependent variable will take on is 60 percent.B) given the values for the explanatory variables, there is a 60 percent probability that the dependent variable will equ

4、al one.C) the model makes little sense, since the dependent variable can only be 0 or 1.D) given the values for the explanatory variables, there is a 40 percent probability that the dependent variable will equal one.Answer: B4) E(Y | X1,Xk) = Pr(Y = 1 | X,XQ means thatA) for a binary variable model,

5、 the predicted value from the population regression is the probability that Y=l, given X.B) dividing Y by the Xs is the same as the probability of Y being the inverse of the sum of the Xs.C) the exponential of Y is the same as the probability of Y happening.D) you are pretty certain that Y takes on

6、a value of 1 given the Xs.Answer: A5) The linear probability model isA) the application of the multiple regression model with a continuous left-hand side variable and a binary variable as at least one of the regressors.B) an example of probit estimation.C) another word for logit estimation.D) the ap

7、plication of the linear multiple regression model to a binary dependent variable.Answer: D(f) The predicted number of failures from this regression is 5.7.Predicted Number of O-Ring FailuresNoOFailsa)-n=eL jo-aqulnN3) A study tried to find the determinants of the increase in the number of households

8、 headed by a female. Using 1940 and 1960 historical census data, a logit model was estimated to predict whether a woman is the head of a household (living on her own) or whether she is living within anothers household. The limited dependent variable takes on a value of one if the female lives on her

9、 own and is zero if she shares housing. The results for 1960 using 6,051 observations on prime-age whites and 1,294 on nonvvhites were as shown in the table:Regression(1) White(2) NonwhiteRegression modelLogitLogitConstant1.459(0.685)-2.874(1.423)Age-0.275(0.037)0.084 (0.068)age squared0.00463 (0.00

10、044)0.00021(0.00081)education-0.171(0.026)-0.127(0.038)farm status-0.687(0.173)-0.498(0.346)South0.376 (0.098)-0.520(0.180)expected family earnings0.0018 (0.00019)0.0011 (0.00024)family composition4.123 (0.294)2.751 (0.345)Pseudo-R-0.2660.189Percent Correctly Predicted82.083.4where age is measured i

11、n years, education is years of schooling of the family head, farm status is a binary variable taking the value of one if the family head lived on a farm, south is a binary variable for living in a certain region of the country, expected family earnings was generated from a separate OLS regression to

12、 predict earnings from a set of regressors, and family composition refers to the number of family members under the age of 18 divided by the total number in the family.The mean values for the variables were as shown in the table.Variable(1) White mean(2) Nonwhite meanage46.142.9age squared2,263.51,9

13、65.6education12.610.4farm status0.030.02south0.30.5expected family earnings2,336.41,507.3family composition0.20.3(a) Interpret the results. Do the coefficients have the expected signs? Why do you think age was entered both in levels and in squares?(b) Calculate the difference in the predicted probab

14、ility between whites and nonvvhites at the sample mean values of the explanatory variables. Why do you think the study did not combine the observations and allowed for a non white binary variable to enter?(c) What would be the effect on the probability of a nonwhite woman living on her own, if educa

15、tion and family composition were changed from their current mean to the mean of whites, while all other variables were left unchanged at the nonwhite mean values?Answer:(a) Since these are logit estimates, the value of the coefficients cannot be interpreted easily. However, statements can be made ab

16、out the direction of the relationship between the dependent variable and the regressors. There is a decrease in the probability of females of living on their own with an increase in years of education. Not living on a farm also lowers the probability. These results hold both for whites and non white

17、s. In addition, for whites the probability of living on her own increases up to a point with age, but then decreases. This is the result of age entering as a level and the square of age. This relationship with regard to age is not statistically significant for nonwhites. In the south, white females

18、are more likely to live on their own, but non whites are not. An increase in expected family earnings and family composition increase the probability of females living on their own.(b) For whites, the probability is 0.90, while for nonvvhites, it is 0.88. In the above approach, all coefficients are

19、allowed to vary, whereas in a combined sample, the coefficients on the variables other than the binary race variable would have to be identical.(c) The probability would increase to 0.81.4) A study investigated the impact of house price appreciation on household mobility. The underlying idea was tha

20、t if a house were viewed as one part of the households portfolio, then changes in the value of the house, relative to other portfolio items, should result in investment decisions altering the current portfolio. Using 5,162 observations, the logit equation was estimated as shown in the table, where t

21、he limited dependent variable is one if the household moved in 1978 and is zero if the household did not move:Regression modelLogitconstant-3.323(0.180)Male-0.567(0.421)Black-0.954(0.515)Married780.054 (0.412)marriage change0.764(0.416)A7983-0257 (0.921)PURN-4.545(3.354)Pseudo 陵0.016where male, blac

22、k, married78, and marriage change arc binary variables. They indicate, respectively, if the entity was a male-headed household, a black household, was married, and whether a change in marital status occurred between 1977 and 1978. A7983 is the appreciation rate for each house from 1979 to 1983 minus

23、 the SMSA-vvide rate of appreciation for the same time period, and PNRN is a predicted appreciation rate for the unit minus the national average rate.(a) Interpret the results. Comment on the statistical significance of the coefficients. Do the slope coefficients lend themselves to easy interpretati

24、on?(b) The mean values for the regressors are as shown in the accompanying table.VariableMeanmale0.82black0.09married780.78marriage change0.03A79830.003PNRN0.007Taking the coefficients at face value and using the sample means, calculate the probability of a household moving.(c) Given this probabilit

25、y, what would be the effect of a decrease in the predicted appreciation rate of 20 percent, that is A7983 = -0.20?Answer:(d) Since the logit model is nonlinear, the slope coefficients cannot be easily interpreted. However, the signs of the coefficients indicate the direction of the relationship betw

26、een the regressors and the binary dependent variable. Accordingly, being married or having experienced a marriage change increases the probability of moving. A male-headed household or a black household is less likely to move. If the predicted appreciation rate relative to the national average incre

27、ased, then the household is less likely to move. The same holds for the actual appreciation rate from 1979 to 1983. None of the slope coefficients are statistically significant with the exception of the black household and marriage change coefficients. The two t-statistics are -1.85 and 1.84 respect

28、ively. These would be statistically significant at the 5% level of a one-sided hypothesis test.(e) The probability is 0.021.(f) The resulting probability would be 0.051, i.e., more than twice the value in the previous result.5) A study analyzed the probability of Major League Baseball (MLB) players

29、to survive for another season, or, in other words, to play one more season. The researchers had a sample of 4,728 hitters and 3,803 pitchers for the years 1901-1999. All explanatory variables are standardized. The probit estimation yielded the results as shown in the table:Regression(1) Hitters Pitc

30、hersRegression modelprobitprobitconstant2.010 (0.030)1.625 (0.031)munber of seasons played-0.058(0.004)-0.031(0.005)performance0.794(0.025)0.677(0.026)average performance0.022 (0.033)0.100 (0.036)where the limited dependent variable takes on a value of one if the player had one more season (a minimu

31、m of 50 at bats or 25 innings pitched), number of seasons played is measured in years, performance is the batting average for hitters and the earned run average for pitchers, and average performance refers to performance over the career.(a) Interpret the two probit equations and calculate survival p

32、robabilities for hitters and pitchers at the sample mean. Why are these so high?(b) Calculate the change in the survival probability for a player who has a very bad year by performing two standard deviations below the average (assume also that this player has been in the majors for many years so tha

33、t his average performance is hardly affected). How does this change the survival probability when compared to the answer in (a)?(c) Since the results seem similar, the researcher could consider combining the two samples. Explain in some detail how this could be done and how you could test the hypoth

34、esis that the coefficients are the same.Answer:(a) Note that all variables are standardized, so that the mean is zero. This results in a survival probability of 0.997 for hitters and 0.991 for pitchers. These results are so high because there is a high probability, in general, for a player to return

35、 the following season.(b) Since the variables are standardized, this implies a change of two for the performance variable. The result for hitters is a lowering of the survival probability to 0.65, and for pitchers to 0.633(c) After combining the sample for hitters and pitchers, you would allow for a

36、 different intercept and slopes by introducing a binary variable for pitchers if hitters are the default. This binary variable would be introduced by itself and in combination with each of the above variables, thereby allowing all coefficients to differ. You could then conduct an F-test for the join

37、t hypothesis that all coefficients involving the binary variables are zero. If the hypothesis cannot be rejected, then there is no difference between the coefficients for hitters and pitchers.6) The logit regression (11.10) on page 393 of your textbook reads:PKdeny =11 P/Imtio,blndO = F(-4.13 + 5.37

38、 P/lratio + 1.27 black)(a) Using a spreadsheet program such as Excel, plot the following logistic regression function with aAJAAAsingle X, Y； = *kk where Bn = -4.13, Bi = 5.37, 02 = L27. Enter values for Xin thel+e-(p()+PiXii+P2X2i)first column starting from 0 and then increment these by 0.1 until y

39、ou reach 2.0. Let X2 be 0 at first. Then enter the logistic function formula in the next column. Next allow X2 to be 1 and calculate the new values for the logistic function in the third column. Finally produce the predicted probabilities for both blacks and whites, connecting the predicted values w

40、ith a line.(b) Using the same spreadsheet calculations, list how the probability increases for blacks and for whites as the P/l ratio increases from 0.5 to 0.6.(c) What is the difference in the rejection probability between blacks and whites for a P/I ratio of 0.5 and for 0.9? Why is the difference

41、smaller for the higher value here?(d) Table 11.2 on page 401 of your textbook lists logit regressions (column 2) with further explanatory variables. Given that you can only produce simple plots in two dimensions, how would you proceed in (a) above if there were more than a single explanatory variabl

42、e?Answer:(e) The increase in the deny probability increases by 9.7 percentage points for whites, and by 13.3 percentage points for blacks.(f) At a P/I value of 0.5z the difference is approximately 30%, while it is 20% for the higher value. As the ratio increases, the probability that everyone gets r

43、ejected increases and approaches 1, regardless of race.(g) In that case you would have to hold the other explanatory variables constant. A simple solution would be to set all of these to zero. A more reasonable approach would be to set them to their sample average if they are continuous variables, a

44、nd to set them either to 0 or 1 for binary variables.7) Equation (11.3) in your textbook presents the regression results for the linear probability model.a. Using a spreadsheet program such as Excel, plot the fitted values for whites and blacks in the same graph, for P/I ratios ranging from 0 to 1 (

45、use 0.05 increments).b. Explain some of the strengths and shortcomings of the linear probability model using this graph.Answer:c. The strength is that the regression line is easy to interpret once you realize that the fitted values are probabilities of being denied a loan: increases in the P/I ratio

46、 of 10 percentage points increase the probability of being denied by roughly 6 percentage points. The role of the binary variable for blacks also becomes clear: blacks have a roughly 18 percentage point higher probability of being rejected for a loan when compared to whites, at any given level of a

47、P/I ratio. As for shortcomings, it becomes clear that this model cannot be used to calculate the probability of rejection for whites with a P/I ratio less than approximately 20 percent. In that case, the predicted probability would be negative. Similarly, you would expect the probability increase fo

48、r a given change in the P/I ratio to change as the P/I ratio becomes larger; this is not the case for the linear probability model. Furthermore, you will find values larger than 1 for the P/I ratio in the data set used for Chapter 11. As a result, the predicted probability of being rejected for a loan would be above 1 for some individuals, which does not make sense.8) Equation (11.3) in your textb