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Unformatted text preview: Economics 241B Modes of Convergence We have concentrated on the (exact) &nite-sample distribution for the OLS es- timator. The &nite-sample theory breaks down if one of the following three assumptions is violated: 1) the exogeneity of the regressors, 2) the normality of the error term, 3) the linearity of the regression. We now develop an alterna- tive approach, which requires only the third assumption. The approach, termed asymptotic or large-sample theory, derives an approximation to the distribution of the estimator and its associated test statistics, assuming the sample size is su ciently large. Rather than making assumptions on the sample of a given size, large-sample theory makes assumptions on the stochastic process that generates the sample. We study the limiting behavior of a sequence of random variables & & Y 1 ; & Y 2 ;::: , which we denote by & Y n . 1 Modes of Convergence Proof that an estimator is consistent requires that we construct a limit argument. One might naturally ask, what is the limit of & Y n as n ! 1 ? Because & Y n is a random variable, the (deterministic) limit does not exist. To understand the point, recall the following De&nition. A sequence of real numbers f & 1 ;& 2 ;::: g is said to converge to a real number & if, for any > there exists an integer N such that for all n > N : j & n & & j < : We express the convergence as & n ! & as n ! 1 or (more precisely) as lim n !1 & n = & . The de&nition applies to vectors. Let & n 2 R k . If lim n !1 & n;i = & i for each i = 1 ;:::;k , then lim n !1 & n = & . Example. Let & n = 1 & 1 n . For any > there exists an integer N such that for all n > N : j & n & 1 j < , so lim n !1 & n = 1 . (For example, N = [ 1 & ] , where [ ] yields the nearest integer that is at least as large as 1 & .) 1 A sequence is the most common mathematical construct that may possess a limit, but certainly not the only one. A sequence is a countably in&nite collection of numbers (vectors, matrices). A sequence must embody a rule that de&nes it and is ordered: The rule generates the n th member of the sequence, for any positive integer n . Example. Let & n = ( & 1) n . For values of > 1 there exists an integer N (for example N = 1 ) such that for all n > N : j & n & j < . For values of j j 1 and for every value of N , j & n & j = 1 > for all n > N . Because a similar argument holds for any proposed constant limit value, the limit does not exist. The de&nition of a deterministic limit requires that j & n & & j < for all values of n > N , not merely for mostor severalvalues of n > N . Because & Y n is a random variable, we cannot be sure that & & & Y n & & & < for all values of n > N ....
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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.
- Fall '08