Functional Form and Structural Change
1. In Solow's classic 1957 study of technical change in the U.S. Economy, he suggests the following aggregate production function q t A t f k t where q t is aggregate output per manhour, k t is the aggregate capital labor ratio, and A t is the technology index. Solow considered four static models, q A a plnk, q A a - p k, ln q A a plnk, ln q A a - p k. He also estimated a dynamic model, q t A t - q t-1 A t-1 a pk. b Solow's data for the years 1909 to 1949 are listed in Table A8.1 Op. cit., page...
Maximum Likelihood Estimation
1. Assume the distribution of x is fx 1 8, 0 lt x lt 8. In random sampling from this distribution, prove that the sample maximum is a consistent estimator of 8. Note you can prove that the maximum is the maximum likelihood estimator of 8. But, the usual properties do not apply here. Why not Hint Attempt to verify that the expected first derivative of the log-likelihood with respect to 8 is zero. Using the result of the previous problem, the density of the maximum is n z 8 n-1 1 8 , 0 lt z lt 8....
The Generalized Method of Moments
1. For the normal distribution li2k o2k 2k k 2k and 2k i 0, k 0, 1, . . . . Use this result to analyze the two estimators m A U m4 1 - and b2 f. m2 m2 where mk i2i 1 xi - x k. The following result will be useful Asy.Cov fnm ,4nink k - LL fihLj-Lk-i - jLL1-iL i - kL-L i- Use the delta method to obtain the asymptotic variances and covariance of these two functions assuming the data are drawn from a normal distribution with mean and variance o2. Hint Under the assumptions, the sample mean is a...
FiniteSample Properties of the Least Squares Estimator
1. Suppose you have two independent unbiased estimators of the same parameter, 8, say 9iand 8 2, with different variances, v1 and v2. What linear combination, 8 c181 c2 8 2 is the minimum variance unbiased estimator of 8 Consider the optimization problem of minimizing the variance of the weighted estimator. If the estimate is to be unbiased, it must be of the form c181 c2 82 where c1 and c2 sum to 1. Thus, c2 1 - c1. The function to minimize is Min L c12v1 1 - c1 2v2. The necessary condition is...
Autocorrelation
1. Does first differencing reduce autocorrelation Consider the models yt P'xt St, where St pSt-i ut and st ut - Xut-i. Compare the autocorrelation of St in the original model to that of vt in yt - yt-i P' xt - xt-i vt where vt St - sh. For the first order autoregressive model, the autocorrelation is p. Consider the first difference, vt St - St-i which has Var vt 2Var st - 2Cov st,St-i 2ct 2 1 1 - p2 - p 1 - p2 2ct 2 1 p and Cov vt,vt_i 2Cov st,St-i - Var st - Cov st,st-i a 2 i i - p2 2p - i -...
Nonspherical Disturbances The Generalized Regression Model
1. What is the covariance matrix, Cov p,p-b , of the GLS estimator pp X'Q-1X -1X'Q-1yand the difference between it and the OLS estimator, b X' X 1X' y The result plays a pivotal role in the development of specification tests in Hausman 1978 . Write the two estimators as p p X'Q-1X -1X'Q-1s and b p X'X -1X's. Then, p- b X'Q-1X -1X'Q-1 - X' X -1X' s has E p- b 0 since both estimators are unbiased. Therefore, Cov p, p- b E p- p p- b ' . E X'Q-1X -1X'Q-1ss' X'Q-1X -1X'Q-1 - X'X -1X' ' X'Q-1X -1X'...
Inference and Prediction
1. A multiple regression of y on a constant, x1, and x2 produces the results below y 4 ,4xi .9x2, R 8 60, e' e 520, n 29, X'X The estimated covariance matrix for the least squares estimates is the test may be based on t .4 .9 - 1 .410 .256 - 2 .051 1 2 .399. This is smaller than the critical value of 2.056, so we would not reject the hypothesis. 2. . Using the results in Exercise 1, test the hypothesis that the slope on x1 is zero by running the restricted regression and comparing the two sums...
Least Squares
. . The normal equations are given by 3-12 , X'e 0, hence for each of the columns of X, xk we know that xk'e 0. This implies that . ei 0 and . xiei 0 . b Use ei 0 to conclude from the first normal equation that a y bx . c Know that jei 0 and j,xiei 0 . It follows then that x x 0 . Further, the latter implies x x a bxt 0 or x x yi y b xi x 0 from which the result follows. 2. Suppose b is the least squares coefficient vector in the regression of y on X and c is any other Kx1 vector. Prove that...