4. (6 marks) Consider the regression model y = Xb + u , where the Gauss-Markov assumptions hold.Let be the OLS estimator of b , Z = G(X) be an n ´(k +1) matrix function of X and Z!X be a(k +1) ´(k +1) non-singular matrix.
a. Prove that the estimator b! = (Z!X)-1 Z!y is unbiased.
b. State var(b! ) as a function of Z , X and s 2 .
c. Briefly explain which estimator is better: bˆ or b! .
5. (6 marks) Consider the regression model y = b0 + d0 d + b1x + d1 (d × x) + u , where d is a dummy variable and x is a quantitative variable. For simplicity, assume that u = 0 .
a. Calculate the unique (strictly positive) value of x for which yˆ yˆ .=d=0 d=1
b. State the necessary parameter restriction(s) for part (a) to be true.
c. State a test statistic that tests whether there is a structural difference between the two cases represented by the dummy variable.
6. (10 marks) Consider the regression model y = Xb + u
Heteroscedasticity will not cause the OLS estimators to be biased or inconsistent, but it will make problems for standard errors, t-statistics and various significance tests. To fix it, a“transformation matrix”, P , is needed, such that the transformed error vector
a. Derive a diagonal matrix, P , that satisfies PP! = V-1 .
b. Restate the model using the transformation matrix.
u! º Pu , satisfies
c. State the OLS estimator, bˆ , for the transformed model in terms of X , y and V .
d. Prove that bˆis unbiased.
e. State the variance-covariance matrix in terms of X and V .
7. (6 marks) Consider the regression model y = b0 + b1x + u , where x is unobservable. For each yi , there are m measures on x: zh = x + eh for h = 1,..., m . Assume that x is uncorrelated with u, e1 , and em and that the measurement errors are pairwise uncorrelated with the same variance, s > 0 .
Let w = 1 (z +... + z) be the average of the measures on x so that, for each observation i,
Let be the OLS estimator from the regression of y on a constant and w,
b. Prove that b1 is a biased estimator.
c. Briefly explain the sign of the bias and why it disappears as m increases.
8. (4 marks) Consider the regression yt = a0 + d0 xt + d1xt-1 + d2 xt-2 + ut . If xt increases one unit, it will have a long-run effect on y equal to q º d0 + d1 + d2 . A limitation of this model is that the standard error of this long-run effect cannot be obtained easily.
a. Eliminate d0 from the model and isolate q .
b. To ensure that q has economic meaning, briefly explain the relationship between Dxt , Dxt-1 and Dxt-2 .
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