Econometrics-I-12

Variance of each term(| is variance of/n var n based

Info iconThis preview shows pages 13–21. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 1-1 2-1-1 2-1 2-1 2-1 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 2-1 , ( 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 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 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 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( 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 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 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...
View Full Document

{[ snackBarMessage ]}

Page13 / 39

Variance of each term(| is Variance of/n Var n Based on the...

This preview shows document pages 13 - 21. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online