Econometric analysis Asymptotic Distribution Theory参考.ppt

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Building the Wald Statistic-1 Suppose that the same normal distribution assumptions hold, but instead of the parameter matrix ? we do the computation using a matrix Sn which has the property plim Sn = ?. The exact chi-squared result no longer holds, but the limiting distribution is the same as if the true ? were used. Building the Wald Statistic-2 Suppose the statistic is computed not with an x that has an exact normal distribution, but with an xn which has a limiting normal distribution, but whose finite sample distribution might be something else. Our earlier results for functions of random variables give us the result (xn - ?) ?Sn-1(xn - ?) ? ?2[K] (!!!)VVIR! Note that in fact, nothing in this relies on the normal distribution. What we used is consistency of a certain estimator (Sn) and the central limit theorem for xn. General Result for Wald Distance The Wald distance measure: If plim xn = ?, xn is asymptotically normally distributed with a mean of ? and variance ?, and if Sn is a consistent estimator of ?, then the Wald statistic, which is a generalized distance measure between xn converges to a chi-squared variate. (xn - ?) ?Sn-1(xn - ?) ? ?2[K] The F Statistic An application: (Familiar) Suppose bn is the least squares estimator of ? based on a sample of n observations. No assumption of normality of the disturbances or about nonstochastic regressors is made. The standard F statistic for testing the hypothesis H0: R? - q = 0 is F[J, n-K] = [(e*’e* - e’e)/J] / [e’e / (n-K)] where this is built of two sums of squared residuals. The statistic does not have an F distribution. How can we test the hypothesis? F Statistic F[J,n-K] = (1/J) ? (Rbn - q)?[R s2(X?X)-1 R’]-1 (Rbn - q). Write m = (Rbn - q). Under the hypothesis, plim m=0