All questions are compulsory. Part one should be answered, very briefly, in a maximum of two sheets of paper. PART ONE 1. Suppose that a score on a final exam, score, depends on classes attended, attend, and unobserved factors that affect exam performance (such as student ability) score = β0 + β1 attend + u When would you expect this model tosatisfy E( u | attend ) = 0? 2. In the following example, the OLS fitted line explaining college GPA, colGPA, in terms of high school GPA, hsGPA, and ACT score, ACT, is
ˆ colGPA = 1.29 + .453 hsGPA + .0094 ACT If the average high school GPA in the sample is about 3.4 and the average ACT score in the sample is about 24.2, what is the average college GPA in the sample? (Hint: Use the algebraicproperty concerning sample averages of all the variables and the OLS regression line) 3.
ˆ Suppose you estimate a regression model and obtain β1 = .56 and p-value = .086 for testing H0: β1 = 0 against H1: β1 ≠ 0. What is the p-value for testing H0: β1 = 0 against H1: β1 > 0?
In a regression model with a large sample size (n→∞), what is an approximate ˆ 95% confidence interval for β j underMLR.1 through MLR.5 (Gauss-Markov assumptions)? We call this an asymptotic confidence interval.
ˆ Explain why choosing a model by maximizing R 2 or minimizing σ (the standard error of the regression) is the same thing.
Let y, p and m denote the real output in 1989 pesetas, the price level and the quantity of money, respectively, all measured in natural logs, in thefollowing models:
y = β0 + β1 m + u p = β0 + β1 m + w
a) Using model (1), state the null hypothesis that money does not affect real output, against the alternative that if money increases, real output increases. b) Using model (2), state the null hypothesis that the price level is strictly proportional to the quantity of money, against the alternative that if money increases, theincrease in the price level is less than proportional. c) Let
ˆ p = 0.003 + 0.971 m (0.015) (0.017)
be the estimated model (2) for a sample of 62 observations, standard errors in brackets. Test the null hypothesis of question b) above, at the 5% and 10% significance levels. d) If real output were measured in €, how are the estimates and t-statistics of model (1) affected? (Hint: Remember thatall the variables are in natural logs.) 7. Let W, educ, and exper be the nominal monthly wage, years of education and years of experience, respectively. Let male be a dummy variable, which takes value one when the individual is male, and zero otherwise. Using data from a recent survey, the following models were estimated:
Dependent Variable: W Method: Least Squares Date: 01/30/06 Time: 08:47Sample: 1 500 Included observations: 500 Variable C MALE EDUC EDUC*MALE EXPER EXPER*MALE R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient 106.8320 14.92081 24.93752 15.19665 34.63248 15.41967 0.986585 0.986449 49.85851 1228020. -2661.046 1.959561 Std. Error 7.522148 10.86735 0.950112 1.392210 0.346800 0.501065 Equation 1
t-Statistic14.20233 1.372994 26.24692 10.91549 99.86303 30.77379
Prob. 0.0000 0.1704 0.0000 0.0000 0.0000 0.0000 861.9887 428.3035 10.66818 10.71876 7265.904 0.000000
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
Dependent Variable: W Method: Least Squares Date: 01/30/06 Time: 08:48 Sample: 1 500 Included observations: 500 Variable CEDUC EXPER R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient 106.8655 32.51901 42.51501 0.830946 0.830265 176.4561 15474974 -3294.501 2.034541 Std. Error 19.21086 2.456772 0.885262
t-Statistic 5.562766 13.23648 48.02531
Prob. 0.0000 0.0000 0.0000 861.9887 428.3035 13.19000 13.21529 1221.442 0.000000