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Lecture 14-2007

Lecture 14-2007 - Large Sample Theory Lecture XIV I Basic...

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1 Large Sample Theory Lecture XIV I. Basic Sample Theory A. The problems set up is that we want to discuss sample theory. 1. First assume that we want to make an inference, either estimation or some test, based on a sample. 2. We are interested in how well parameters or statistics based on that sample represent the parameters or statistics of the whole population. B. The complete statistical term is known as convergence. 1. Specifically, we are interested in whether or not the statistics calculated on the sample converge toward the population estimates. 2. Let n X be a sequence of samples. We want to demonstrate that statistics based on n X converge toward the population statistics for X . C. Taking a slightly different tack: The classical assumptions for ordinary least squares (OLS) as presented in White, Halbert Asymptotic Theory for Econometricians. 1. Theorem 1.1: The following are the assumptions of the classical linear model: a) The model is known to be , y X     . b) X is a nonstochastic and finite n k matrix. c) X X is nonsingular for all n k . d)   0 E   . e) 2 2 0 0 ~ 0, , N I    . 2. Given these assumptions, we can conclude that a) (Existence) Given (i.)-(iii.), n exists for all n k and is unique.

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