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jordan 11 for sale ri ′ui
of ii on 1, xi,jordan 11 for sale, i = 1, 2,..., N. Equation (9.19) makes it clear that the statistic is based on the sample covariance between ii and .i. From (9.19), we see that the asymptotic distribution depends on the asymptotic distribution of i =1 i N .12 Σ/N i′u..ii i =1 As shown in Wooldridge (1990a, 1991a), under H0 (either (9.6) or (9.8)), NN .12 .12N / i′u. = N /r′u + o () (9.20)Σ ii Σ ii p 1 i=1 i =1 where ri ≡ gi .π0 . xiΠ are the population residuals from the population regression of gi on 1, xi. Under H0, E(ui|ri) = 0 (since ri is either a function of xi or of (xi, zi)), and so E(ri ′ui) = 0. It follows that the second term in (9.20) has a limiting q-variate normal distribution. Therefore, whether LM has a limiting chi-square .1 N i= iidistribution under H0 hinges on whether .2(N Σ 1rr′ ) is a consistent estimator of var(ri ′ui) = E(u2 i ri ′ri). By the law of iterated expectations, E(u2i r′ i ri) = E[E(u2i |ri)|r′ i ri] =σ2E(r′ i ri), (9.21) where the second equality holds provided E(u2i |ri) = E(u2i ) =σ2. For testing (9.8) under (9.17),jordan 8 aquas, E(u2i |xi, zi) = var(ui|xi, zi) =σ2, and ri is a function of (xi,jordans for sale online, zi), so E(u2i |ri) =σ2. If we only assume E(r′ iui) = 0 – for example, E(u) = 0,jordan 7 craptors, E(x′u) = 0, and E(z′u) = 0 in (9.9) – then E(u2|r) and var(u|r) are not necessarily the same, and (9.17) is no longer enough to ensure that LM has an asymptotic chi-square distribution. We can also use (9.19) and (9.20) to see how LM behaves when the conditional mean null hypothesis holds but the auxiliary homoskedasticity assumption does not. An important point is that the representation in (9.20) holds with or without homoskedasticity, which implies that LM has a well-defined limiting distribution even if the conditional variance is not constant. Therefore,jordans for sale, the rejection frequency tends to some number strictly less than one (typically, substantially below one), which means that a diagnostic test for the conditional mean has no systematic power to detect heteroskedasticity. Intuitively,
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