N n therefore n n0 n conclude n n approximately n σ

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n N[ , ] Therefore, n N[0,   ( ] N[ ,   ] Conclude : n( ) N[ ,   ] Approximately : N[ - - = ÷ ÷ → σ → σ = σ - → σ → d d d a X'X X b Q w w 0 Q Q w Q Q Q 0 Q b 0 Q b 0 ε β β 2 -1 ,  ( n) ] σ / Q
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Part 12: Asymptotics for the Regression Model Asymptotic Properties p Probability Limit and Consistency p Asymptotic Variance p Asymptotic Distribution ™  14/38
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Part 12: Asymptotics for the Regression Model Root n Consistency p How ‘fast’ does b  β? p Asy.Var[ b ] =σ2/n Q -1 is O(1/n) n Convergence is at the rate of 1/n n n b has variance of O(1) p Is there any other kind of convergence? n x1,…,xn = a sample from exponential population; min has variance O(1/n2). This is ‘n – convergent’ n Certain nonparametric estimators have variances that are O(1/n2/3). Less than root n convergent. n Kernel density estimators converge slower than n ™  15/38
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Part 12: Asymptotics for the Regression Model Asymptotic Results p Distribution of b does not depend on normality of ε p Estimator of the asymptotic variance (σ2/n) Q -1 is (s2/n) ( X’X /n)-1. (Degrees of freedom corrections are irrelevant but conventional.) p Slutsky theorem and the delta method apply to functions of b . ™  16/38
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Part 12: Asymptotics for the Regression Model Test Statistics We have established the asymptotic distribution of b . We now turn to the construction of test statistics. In particular, we based tests on the Wald statistic F [J,n-K] = (1/J)( Rb - q )’[ R s2( XX )-1 R ]-1( Rb - q ) This is the usual test statistic for testing linear hypotheses in the linear regression model, distributed exactly as F if the disturbances are normally distributed. We now obtain some general results that will let us construct test statistics in more general situations. ™  17/38
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Part 12: Asymptotics for the Regression Model Full Rank Quadratic Form A crucial distributional result (exact): If the random vector x has a K-variate normal distribution with mean vector and covariance matrix , then the random variable W = ( x - ) -1 ( x - ) has a chi-squared distribution with K degrees of freedom. (See Section 5.4.2 in the text.) ™  18/38
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Part 12: Asymptotics for the Regression Model 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 S n which has the property plim S n = . The exact chi-squared result no longer holds, but the limiting distribution is the same as if the true were used. ™  19/38
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Part 12: Asymptotics for the Regression Model Building the Wald Statistic-2 Suppose the statistic is computed not with an x that has an exact normal distribution, but with an x n 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 ( x n - ) 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 ( S n) and the central limit theorem for x n.
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