The Generalized Method of Moments

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 ju and variance o2. (Hint Under the assumptions, the sample mean is a consistent estimator of //, so for purposes of deriving asymptotic results, the difference between X and // may be ignored. As such, no generality is lost by assuming the mean is zero, and proceeding from there. Obtain V, the 3x3 covariance matrix for the three moments, then use the delta method to show that the covariance matrix for the two estimators is 

Using the formula given for the moments, we obtain, p2 = o2, p3 = 0, p4 = 3a4. Insert these in the derivatives above to obtain 

Since the rows of J are orthogonal, we know that the off diagonal term in JVJ will be zero, which simplifies things a bit. Taking the parts directly, we can see that the asymptotic variance of yfb will be a Asy.Var[m3], which will be 

Inserting these parts in the expansion, multiplying it out and collecting tenns produces the lower right element equal to 24, as expected. 

Using the results in Example 18.7, estimate the asymptotic covariance matrix of the method of moments estimators of P and 7. based on m[ and tn 2. [Note You will need to use the data in Example C.l to estimate V.] 

The parts needed, using the general result given earlier, are pe 15o , p3 = 0, p2 lU = 30 . Inserting these in the parentheses and multiplying it out and collecting terms produces the upper left element of JVJ equal to 6, which is the desired result. The lower right element will be 

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