中级计量经济学2班10期末试题

Intermediate Econometrics Final Exam Spring 2010 Note: (1) You have two hours to complete the exam. (2) You can answer in either English or Chinese. 1 True or False (20 points) Indicate true or false for the following statements. Explain only if the statement is false. 2 points each. 1. Instrumental variable regression is needed when the residuals are correlated with the regressors. 2. Consider the regression model ̂ln(consumption) = 0.8 (0.2) + 3.2 (1.1) education + 1.4 (0.5) education ∗ f emale + 0.9 (0.4) ln(income), where education measures years of schooling and female is the dummy if one is a female, and values in brackets are standard errors. This model shows that on average an additional year of education shall increase consumption by 3.2% on average, and this result is significant at 5% significance level. 3. Consider y = xβ + u, V(βˆIV) using z as an instrument is always larger than V ( βˆOLS ) , so we should avoid using IV whenever x and u are independent. 4. Consider the model y1 = β0 + β1y2 + β2z1 + u1 (1) y2 = µ + δ1z1 + δ2q1 + δ3q2 + v, (2) where cov(y2, u1) , 0, z1, q1, q2 are exogenous variables. Then equation (1) is called the reduced form equation. 5. The weakest consistency condition for FGLS for AR(1) is E (ut) = 0 and E[xk,t−1ut+ xktut−1] = 0 for t = 1, ...,T, k = 1, ...,K. 6. The GLS estimators for correcting heteroskedasticity are one kind of weighted least squares estimators. 1 7. When serial correlation exists in the error term, the OLS variance formula σ2/S STx overstates the true variance of the OLS estimator. 8. When the error term is AR(1) with the correlation coefficient ρ > 0, the t statistics ignoring this correlation tends to be smaller than that taking into account of the serial correlation. 9. If ̂log (y) = xβˆ, then the predicted value of y is yˆ = exp ( ̂log (y)) . 10. The heteroskedasticity-robust standard errors are always bigger than the usual stan- dard errors for the linear probability model. 2 Multiple Choice Questions (50 points) One of the choices is correct. Choose the correct one. 5 points for each question. 1. Consider the following four tests: White test, Breusch Pagan test, Chow test, DW test. How many of them can be used to test for heteroskedasticity? (a) one b) two c) three (d) four 2. Which of the following statements is wrong? (a) When the hetereskedasticity is of unknown form, the feasible generalized least square estimator can be unbiased. (b) The FGLS estimator is consistent and is always asymptotically more efficient than OLS. (c) For large sample sizes, FGLS is an attractive alternative to OLS when there is evidence of heteroskedasticity. (d) We use the FGLS estimates in place of the OLS estimates under heteroskadas- ticity, because they are more efficient and have associated test statistics with the usual t and F distributions. 3. Which of the following statements is not true? (a) It is impossible to see serial correlation in cross-sectional data. (b) The R-squared is still an appropriate measure when there exists serial correla- tion for stationary series. 2 (c) A t test can be used to test for serial correlation when used appropriately. (d) DW test is used to test whether the error terms are correlated. 4. Which of the following is not true? (a) The multiple linear regression with a binary dependent variable is called the linear probability model. (b) The fitted values from linear probability model are called predicted probabili- ties. (c) OLS is the efficient estimator for linear probability model. (d) Linear probability model assumes the probability is linearly related to the in- dependent variables. 5. In class we discussed the application of Granger Causality on chicken and eggs. Based on those discussions, how many of the following statement(s) is (are) wrong? (a) zero (b) one (c) two (d) three (1) The evidence shows that the lagged values of chickens can predict current values of eggs. (2) If the lagged values of eggs can help predict current values of chickens, then we conclude that in the beginning eggs appeared before chickens. (3) If the lagged values of chickens can help predict current values of eggs, then we conclude that in the beginning chickens appeared before eggs. 6. Following our discussions in class about the impact of education on wage, which of the following has NOT been used in the literature as an instrument? (a) draft lottery number b) birth quarter c)income d)mother’s education 7. Which of the following statements about dealing with endogeneity is WRONG? (a) Let z2 be the instrument for y2, then one can test endogeneity of y2 through testing H0: δ1 = 0 in y1 = β0 + β1y2 + β2z1 + δ1yˆ2 + error, where yˆ2 = pi0 + pi1z1 + pi2z2. (b) If we can find suitable proxy variable for the endogenous variable, we can at least mitigate the endogeneity problem. 3 (c) Instrumental variable estimation is one of the methods to deal with endogeneity caused by omitted-variable bias. (d) If we cannot find good instruments, have panel data at hand, and there is no appropriate proxy, we can use the imperfect OLS estimator but acknowledging the problem. 8. Consider two models yi1 = β0 + β1xi1 + β2xi2 + ui1 y˜i2 = µ0 + δ1 x˜i1 + δ2 x˜i2 + ui2, where y˜i2 = 10yi1, x˜i1 = 2xi1,x˜i2 = 5xi2.OLS regression for yi1 equation gives βˆ1 = 2.5, βˆ2 = 3.Which of the following about the OLS regression of the y˜i2 equation is true? (a) δˆ1 = 12.5, δˆ2 = 6;b).δˆ1 = 0.5, δˆ2 = 1.5 c).δˆ1 = 2.5, δˆ2 = 6 d).δˆ1 = 2.5, δˆ2 = 3 9. Consider the model that evaluate how marriage decisions are affected by various factors, where the dependent variable is a dummy that equals to one when one is married. marriedi = β0 + β1xi1 + β2xi2 + ui1, An OLS regression is done on this equation, then which of the following statement is true? (a) We can assume that ui1 is i.i.d.with mean zero and variance σ2. (b) Because OLS regression usually does not put constraints on the dependent variables, the predicted values are always meaningful. (c) The usual F statistic for this model no longer has F distribution. (d) It is meaningless to run OLS regressions since the dependent variable is not continuous. 10. Which of the following statement is true? (a) Because adjusted R squared are positive and don’t increase with additional explanatory variables, it is a better measure for goodness of fit. (b) The higher the R squared, the better the fit of the regression. (c) We can use adjusted R squared to choose between non-nested models. (d) A higher R squared after estimating the model using 2SLS means better good- ness of fit. 4 3 Analytical Questions (30 points) 1. A researcher uses the following equation to study return to education ln(wage) = µ + β1edu + β2edu2 + β3rural ∗ edu + β4 exp er + β5 exp er2 + u (3) where the natural log of hourly wage depends on one’s own education (edu), whether he is from rural area (rural), and his/her working experience, u is the error term. (a) (6 points) The researcher starts with a simple form of regression ln(wage) = µ + β ∗ rural + u with the following summary statistics: ln(wage) mean s.d. rural = 1 1.5 0.8 rural = 0 2.4 1.6 Derive the OLS estimated parameter µ and β. (b) (4 points) If all explanatory variables satisfy the exogeneity assumption, pro- vide the BLUE estimator for (3) . (c) (5 points) This researcher’s work is criticized as he ignores that one’s innate ability may be correlated with education. List the requirements for valid in- struments (3). (d) (5 points) Based on our discussions in class, list instruments that has been used in the literature for the identification of return to education equation, and discuss when the instruments can be valid. (e) (5 points) Suppose your listed instruments are all valid. Write out the steps you used to get the consistent estimates for return to education. 5