1 Economics 435 Spring 1998 D. Maki Midterm Examination Instructions: There are three parts to this examination, each with two questions. Answer one question from each part (a total of three questions) Questions are equally weighted. Total time allowed = three hours. This is a closed book examination. Throughout the examination, I am not looking for SHAZAM commands as answers. Part I: Answer one of the following two questions. 1. Are the following statements correct - sometimes, always or never? Explain, defining terms carefully. a. OLS estimates of the slope are more precisely estimated if the X values are closer to their sample mean. b. If Xt and ut are correlated, estimates can still be unbiased. c. Estimators cannot be BLUE unless the u's are Normally distributed. d. If the error terms are not Normally distributed, then the t- and F-tests cannot be performed. e. If the variance of u is larger, then confidence intervals for the estimates will be wider. f. If you choose a higher level of significance, a regression coefficient is more likely to be significant. g. The p-value is the probability that the null hypothesis is true. h. If you add a regressor to a model, the R2 must increase. 2. Given a (true) population model: Y=á1+á2X1+á3X2+u, a. Explain in detail how you would use the MacKinnon-White- Davidson test to determine whether LINLIN or LOGLOG were appropriate functional forms. b. If you erroneously estimated the model: Y=á1+á2X2+v, would á-hat2 be biased?. If so, derive an expression for the bias. c. If you erroneously estimated the model: Y=á1+á2X2+á3X3+á4X4+v, would á-hat2 be biased? If so, derive an expression for the bias. Part II. Answer one of the following two questions. 3. a. A Cobb-Douglas production function can be written: ln(Q)=á1+á2ln(L)+á3ln(K)+u for the two-factor case, where Q is output, and L and K are labour and capital inputs. A test for constant returns to scale (testing whether output doubles when both inputs double) consists of testing whether á2+á3=1. Explain in detail how you would do this. 2 b. A trancendental production function (TPF) can be written: ln(Q)=á1+á2ln(L)+á3ln(K)+á4L+á5K+u. Explain in detail how you would test whether the TPF reduces to a Cobb-Douglas in a given application. c. Explain in detail how you would test the joint hypotheses that the TPF reduces to a Cobb-Douglas and displays constant returns to scale. 4. Assume you have quarterly time series data covering 1960.I through 1989.IV, which is used to estimate a bivariate regression equation. a. Explain how you would conduct a Chow test of parameter stability which breaks the sample in half. b. Explain how you would test whether the coefficients differ in the 1970's, assuming they were the same in the 1960's as they were in the 1980's. c. Explain how you would test whether the coefficients differ by decade, that is, that the 1960's, 1970's and 1980's all have different coefficients. d. Explain how you would test whether the relationship between Y and X should be nonlinear. Part III. Answer one of the following two questions. 5. Using 40 quarters of data, someone regressed NUMCARS=number of new car sales per 1,000 population on PRICE=new car price index, INCOME=per-capita real disposable income in nominal dollars, INTRATE=an interest rate and UNEMP=an unemployment rate. The following table gives the coefficient estimates and standard errors (in parentheses) for three models. Variables Model A Model B Model C CONSTANT -7.453352 -10.554074 15.238094 (13.5782) (4.621104) (1.167467) PRICE -0.071391 -0.079392 -0.024883 (0.034730) (0.011022) (0.007366) INCOME 0.003159 0.00356 (0.001763) (0.0006266) INTRATE -0.153699 -0.146651 -0.204769 (0.049190) (0.039229) (0.051442) UNEMP -0.072547 (0.298195) -------------------------------------------------------- RSS 23.510464 23.550222 44.65914 Adjusted R2 0.758 0.764 0.565 a. Which coefficients have theoretically justifiable signs? 3 b. Which coefficients are statistically significant (use t- value > 2, one-tailed test)? c. Is there anything in these results which must be a typo (impossible result)? Explain. d. How do you explain the change in sign for the intercept in moving from Model B to Model C? Does this represent a problem? Explain. e. Which of the three models do you consider "best"? Define your criteria and explain how you applied them. f. Calculate the test statistic for the ANOVA test for Ho:R2=0 for Model C. 6. Answer all (unrelated) parts of this question. a. Explain how you would determine an appropriate polynomial approximation to a nonlinear function. b. If you were asked to estimate a function: Y=á1+exp(á2X), where both parameters are expected to be positive, what would the function look like if graphed in Y-X space? How would you go about estimating the parameters? c. Do the maximum likelihood estimators of á1 and á2 have smaller variances than the corresponding least squares estimators for the bivariate regression model? Explain. d. Explain how you would construct a confidence interval for a forecast value of Y for X=Xo given a bivariate model.