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Extra info for A Guide to Modern Econometrics (2nd Edition)
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Note that this result does not hold for unbiasedness, as E{s}2 = E{s 2 } (see Appendix B). 16 The non-singularity of xx requires that, asymptotically, there is no multicollinearity. The requirement that the limit is finite is a ‘regularity’ condition, which will be satisfied in most empirical applications. A sufficient condition is that the x-variables are independent drawings from the same distribution with a finite variance. Violations typically occur in time series contexts where one or more of the x-variables may be trended.
64)). 76). In particular, we can relax assumption (A2) to xi and εi are independent. (A8) 36 AN INTRODUCTION TO LINEAR REGRESSION This condition does not rule out dependence between xi and εj for i = j , which is of interest for models with lagged dependent variables. Note that (A8) implies (A7). Further discussion on the asymptotic distribution of the OLS estimator and how it can be estimated is provided in Chapters 4 and 5. 3 Small Samples and Asymptotic Theory The linear regression model under the Gauss–Markov conditions is one of the very few cases in econometrics in which the finite sample properties of the estimator and test statistics are known.
Rb − q) (R(X X)−1 R )−1 (Rb − q) ∼ χJ2 . 63) As σ 2 is unknown we have to replace it by its estimate s 2 . There are two ways to continue. 15 Often, one refers to this as a Wald test. e. 64) which, under H0 , follows an F distribution with J and N − K degrees of freedom. As before, large values of f lead to rejection of the null. 59) given above. It is simply a matter of computational ease which one to use. Note that in large samples f ≈ ξ /J . 14 15 Full row rank implies that the restrictions do not exhibit any linear dependencies.