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Unformatted text preview: Variance of each term ( ) is '. Variance of /n. Var n Based on the CLT, n N[ , ] σ → σ → σ → σ d x x x w Q w Q w Q Part 12: Asymptotics for the Regression Model Asymptotic Distribution ˜˜™™™ ™ 13/38 11 211 21 21 21 Limiting distribution of ' n( ) n n n is the same as that of n . 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 Q Q w Q Q Q Q b Q b ε β β 21 , ( n) ] σ / Q Part 12: Asymptotics for the Regression Model Asymptotic Properties p Probability Limit and Consistency p Asymptotic Variance p Asymptotic Distribution ˜˜™™™ ™ 14/38 Part 12: Asymptotics for the Regression Model Root n Consistency p How ‘fast’ does b β? p Asy.Var[ b ] =σ2/n Q1 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 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) Q1 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 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,nK] = (1/J)( Rb  q )’[ R s2( XX )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 Part 12: Asymptotics for the Regression Model Full Rank Quadratic Form A crucial distributional result (exact): If the random vector x has a Kvariate normal distribution with mean vector and covariance matrix , then the random variable W = ( x ) 1 ( x ) has a chisquared distribution with K degrees of freedom. (See Section 5.4.2 in the text.) ˜˜˜™™ ™ 18/38 Part 12: Asymptotics for the Regression Model Building the Wald Statistic1 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...
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 Fall '10
 H.Bierens
 Econometrics, Least Squares, Regression Analysis, Variance, regression model, Wald

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