12 Distribution of the OLS Estimator

# 12 Distribution of the OLS Estimator - Economics 241B Large...

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Unformatted text preview: Economics 241B Large Sample Distribution of the OLS Estimator The importance of the OLS procedure, originally developed for the classic re- gression model, is that it has good asymptotic properties for a family of models, distinct from the classic model, that are useful in economics. Of the models in the family, the model presented here has the widest range of application. No speci&c distribution assumption for the error (such as the Gaussian assump- tion) is required to derive an asymptotic distribution for the OLSE. The require- ment in &nite sample theory that the regressors be strictly exogenous (or "&xed"), is replaced by a much weaker assumption that the regressors be predetermined. The Model We use the term data generating process (DGP) for the stochastic process that generated the &nite sample ( Y;X ) . Therefore, if we specify the DGP, then the &nite sample distributions of ( Y;X ) can be determined. For &nite sample theory, in which the number of distributions was &xed and &nite, the model is de&ned as a set of &nite sample distributions. In contrast, for asymptotic theory the model is stated as a set of DGP¡s. The model that we study is the set of DGP¡s that satisfy the following set of assumptions. Assumption 2.1 (linearity): Y t = X t & + U t ( t = 1 ;:::;n; ) where X t is a K-dimensional vector of regressors, & is a K-dimensional vector of coe¢ cients and U t is the latent error. Assumption 2.2 (ergodic stationarity): The ( K + 1)-dimensional vector stochastic process f Y t ;X t g is jointly stationary and ergodic. Assumption 2.3 (predetermined regressors): All regressors are predeter- mined, in the sense that they are orthogonal to the contemporaneous error: E ( X tk U t ) = for all t and k (= 1 ; 2 ;:::;K ) . This can be written as E ( g t ) = 0 where g t & X t ¡ U t : Assumption 2.4 (rank condition): The K ¢ K matrix E ( X t X t ) is nonsin- gular (and hence &nite). We denote this matrix by & XX . Assumption 2.5 ( g t is a martingale di/erence sequence with &nite second moments): f g t g is a martingale di/erence sequence (so by de&nition E ( g t ) = ). The K & K matrix of cross moments, E ( g t g t ) , is nonsingular. Let S denote Avar (& g ) (the variance of the asymptotic distribution of p n & g , where & g = 1 n P t g t ). By Assumption 2.2 and the Ergodic Stationary Martingale Di/erences CLT, S = E ( g t g t ) . Assumption 2.1 follows directly from the classic regression model. The remaining assumptions require comment. ¡ (Ergodic stationarity) We in no way rule out cross-section data, as a trivial, but important special case is a random sample (that is, f Y t ;X t g is i.i.d.). Indeed, once independence has been assumed, large sample results can be derived for the case where f Y t ;X t g is independently but not identically dis- tributed (i.n.i.d.) provided that some conditions on the higher moments of ( U t ;X t ) are satis&ed....
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## This note was uploaded on 12/26/2011 for the course ECON 241b taught by Professor Staff during the Fall '08 term at UCSB.

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12 Distribution of the OLS Estimator - Economics 241B Large...

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