Let and be two consistent, asymptotically normal estimators of the P x 1 parameter vector with Avar[n(0 - 0)] = V and Avar[n(0 - 0)] = V. Define the Q 1 param- eter vector y = g(0) for which g() is a continuously differentiable function. Show that, if is asymptotically efficient relative to 0, then = g() is asympotically efficient relative to = g(0). (Hint: first express Avar[(-y)] and Avar[n(7-7)] using the Delta Method.) This problem illustrates the ambiguity that can be caused by the invariance of the Wald statistic. Let be an asymptotically normal estimator for the scalar > 0. Let y = log(0) be an estimator of y = log(0). (a) Using the plim operator, show that is a consistent estimator of y. (b) Using the Delta Method, find the asymptotic variance of in terms of the asymptotic variance of 0. (c) Suppose that, for a sample of data, 0 = 4 and Avar[] = 4. What are the estimates and Avar[]? (d) What is the t-statistic for testing Ho : 0 = 1 given the results in (c)? (e) State the null hypothesis in (d) equivalently in terms of y, and use the values of and Avar[] given in (c) to test that null. What do you conclude?