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Econometrics--Econ 388 Winter 2010, Richard Butler Final Exam your name_________________________________________________ Section Problem Points Possible I 1-20 3 points each II 21 22 23 24 25 26 15 points 10 points 10 points 10 points 10 points 10 points III 27 30 points IV 30 points 15 points 28 29 1 I. Define or explain the following terms: 1. inverse of a matrix-2. for the regression Yi 0 1Wi 2 X i 3 Zi i Write out the STATA code to do a Hausman test for endogeneity of W on the right hand side of the equation (X and Z are exogeneous, as are the excluded variables T, U). 3. Write out the STATA code to estimate the model in question 2 with two-stage least squares estimators using exogeneous variables X, Z, T, U: 4. write out the STATA code for second version of the White test (where predicted variances are checked against the predicted value of Y and predicted value of Y-squared using a LM test)- 5. STATA code for the linear probability model where I regress a married dummy on age, educ, and male, controlling for heteroskedasticity - 6. law of large numbers- 7. cointegration of two time series, wt and vt-- 8. dummy variable trap- 9. give the reason “iterations” are printed out for probit estimates but not regular regression estimates- 10. method of moment estimators – 2 11. maximum likelihood estimation criterion - 12. logistic regression model - 13. random walk - 14. structural vs. reduced form equations - 15. Breusch-Pagan test- 16. t-ratio or t-statistic for an estimated regression beta- 17. type I vs. type II errors in statistics - 18. Durbin-Watson test - 19. orthogonal projection - 20. formula for the 95% prediction interval for yT (value of yt next year) when the true model is yt xt t when the usual assumptions hold-3 II. Some Concepts 21. Suppose that the joint distribution for random variables x, y is given as f ( x, y) .6 x .41 x .3 y .521 y 2 xy for values x=0, 1 and y=0, 1. A. What are the joint probabilities f(x=0,y=0), f(x=0,y=1), f(x=1,y=0), and f(x=1,y=1)? B. calculate the marginal probability densities f(x) and f(y) C. Calculate E(x) and V(x) (no credit unless you show the right formulas). D. Calculate the conditional probability density f(y|x=0) (again, no credit unless you show the right formulas) E. Are x and y independent? Why or why not? 4 22. Peter Trueheart regressed wages on SEX (1=male, 2=female) and got the results 𝑊𝑎𝑔𝑒𝑠𝑖 = 𝛽̂0 + 𝛽̂1 𝑆𝐸𝑋𝑖 + 𝜇̂ 𝑖 , when he realized his error and replaced SEX with a dummy variable for FEMALE (1=female, 0=male), he got the new estimates 𝑊𝑎𝑔𝑒𝑠𝑖 = 𝛼̂0 + 𝛼̂1 𝐹𝐸𝑀𝐴𝐿𝐸𝑖 + 𝜀̂𝑖 What is the exact mathematical relationship between the alphas and the betas, and how is 𝜇̂ 𝑖 mathematically related to 𝜀̂𝑖 ? 23. Calculate the expenditure (quarterly expenses are given by E) elasticity with respect to income (annual consumer income is given by I) when the coefficients from the expenditure regression are given by 𝐸𝑖 = 1256 + .065 𝐼𝑖 + 𝜇̂ 𝑖 And where average quarterly expenditures are $2000 and average annual income is $20,000. 5 24. Is there a correlated regressor problem (a right hand side variable violates regression assumption number III) when we leave out an important determinant of Y (say a Z variable) that is uncorrelated with all the included right hand side variables (the Xs, so Z is uncorrelated with the Xs)? Why or why not? 25. Explain what the following test does in this panel data set that relates individual wages to their socio-demographic information, occupation, and year wages were observed: *nr = equals the unique identifier for each individual in this panel data set; ********* need to avoid lagging between people **************************; ** so the bysort only generates it for those with the same id, namely *******; ** the same nr values. *******************************************; bysort nr: gen laglwage = lwage[_n-1]; regress lwage laglwage black hisp educ exper expersq married union occ1 occ2 occ4 occ5 occ6 occ7 occ8 occ9 d82 d83 d84 d85 d86 ; predict resids, residuals; bysort nr: gen lag_resids = resids[_n-1]; *need to avoid lagging between people; list nr year resids lag_resids; regress lwage laglwage black hisp educ exper expersq married union occ1 occ2 occ4 occ5 occ6 occ7 occ8 occ9 d82 d83 d84 d85 d86 lag_resids; (what does lag_resids test do in this regression)? 6 26. Let { et : t , -1, 0, 1, 2, …}be a sequence of independent, identically distributed random variables with mean zero and variance 2 . Define a stochastic process by yt et (1/ 3)et 1 (1/ 2)et 2 for t= 1, 2, 3, …. a) find the mean, variance, and covariances (find covariances for Cov(yt, yt-h) for h=1,2,3 and 4) for the process yt. b) Is this process stationary? explain c) Is this process weakly dependent? explain 7 III. Some Bigger Proofs. 27. Corrections for a) Heteroskedasticity, b) First Order Autoregressive Correlation, or c) Measurement Error (using Instrumental Variables), can all be viewed as regressing TY on TX for the following linear model: Y X In each of the three cases, indicate what the T matrix is that is pre-multiplied through the model to make the respective correction (make clear whatever additional variables or parameters you introduce represent). 8 28. For the heteroskedasticity model (and more generally, the generalized least squares): where Y X ~ N (0, ) the generalized least squares (weighted least squares) estimator is ˆGLS ( X ' 1 X )1 X ' 1Y prove that it is the best (minimum variance in the matrix sense), linear, unbiased estimator among the class of all linear, unbiased estimators for this model (it is the case, and you may assume that for any nxk matrix N, the matrix is N ' N positive definitive). 9 29. Given the usual assumptions about the n by k matrix of instrumental variables, Z, prove that the instrumental variable estimator is consistent: Z 'X i.e., prove that for ˆIV (Z ' X )1 Z 'Y that plim ˆIV . (You may assume that is a n positive definite matrix with finite elements for any value of n, the sample size). 10