Lecture+14+Multiple+Regression+Analysis+-+Asymptotics

# Lecture+14+Multiple+Regression+Analysis+-+Asymptotics -...

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Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-1 Lecture 14 Multiple Regression Analysis: OLS Asymptotics C In Chapters 3 and 4, we covered what are called finite (also small or exact ) sample properties of the OLS estimators in the population model: B Unbiasedness B BLUE (Gauss Markov Theorem) B Confidence intervals B Hypothesis tests C The first requires MLR.l - MLR.4; the second also requires MLR.5; the last two require MLR.6 as well.

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Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-2 C From the frequentist/objectivist standpoint, probabilistic assumptions MLR.1 - MLR.5 are assertions about properties of reality. B Can these assumptions be weakened without losing these results? No. B How robust are these results in cases where these assumptions are false? Not very. < If the error is not normally distributed, the distribution of a t- statistic is not exactly t, and an F statistic does not have an exact F distribution for any sample size.
Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-3 B Can additional assumptions be added and the previous assumptions weakened without losing these results? Yes, is we are willing to accept asymptotic results as n 6 4 . < OLS has satisfactory large sample properties. < Without the normality assumption (Assumption MLR.6), t and F statistics have approximately t and F distributions, at least in large sample sizes.

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Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-4 5.1 Consistency C Unbiasedness of estimators, although important, cannot always be achieved. For example, the standard error of the regression, is not an unbiased estimator for F , the standard deviation of the error u in a multiple regression model. C Although the OLS estimators are unbiased under MLR.1 - MLR.4, in Chapter 11 we will find that there are time series regressions where the OLS estimators are not unbiased.
Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-5 Definition: Based on a sample of size n, the estimator W n is a consistent estimator of 2 iff for every , > 0, as n 6 4 , i.e., the probability that the estimator W n is within a small interval of 2 approaches one as n 6 4 . C Although not all useful estimators are unbiased, virtually all economists agree that consistency is a minimal requirement for an estimator. B Nobel laureate Clive W. J. Granger once remarked, “If you can’t get it right as n goes to infinity, you shouldn’t be in this business.” B The implication is that, if your estimator of a particular population parameter is not consistent, then you are wasting your time. (C.7)

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Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-6 Theorem: According to the LLN the sample mean is a consistent estimator of the population mean, i.e., Property PLIM.1: Consider a continuous function ( = g( 2 ). Suppose plim(W n ) = 2 . Define an estimator of ( by G n = g(W n ). Then or equivalently, (C.8) (C.9) (C.10)
Lecture 14, ECON 123A, Fall 2011 Dale J. Poirier 14-7 Property PLIM.2: If plim (T n ) = " and plim (U n ) = $, then (i) plim (T n + U n ) = " +$ (ii) plim (T n U n ) = "$(iii) plim (T n /U n ) = " /$ provided \$ 0

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## This note was uploaded on 12/13/2011 for the course ECON 123a taught by Professor Staff during the Fall '08 term at UC Irvine.

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Lecture+14+Multiple+Regression+Analysis+-+Asymptotics -...

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