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econ583asymptoticsprimer

econ583asymptoticsprimer - A Primer on Asymptotics Eric...

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A Primer on Asymptotics Eric Zivot Department of Economics University of Washington September 30, 2003 Revised: October 7, 2009 1 Introduction The two main concepts in asymptotic theory covered in these notes are Consistency Asymptotic Normality Intuition consistency: as we get more and more data, we eventually know the truth asymptotic normality: as we get more and more data, averages of random variables behave like normally distributed random variables 1.1 Motivating Example Let 1  denote an independent and identically distributed (iid ) random sam- ple with [ ]= and var ( )= 2 We don’t know the probability density function (pdf) ( θ ) but we know the value of 2 The goal is to estimate the mean value from the random sample of data. A natural estimate is the sample mean ˆ = 1 X =1 = ¯  Using the iid assumption, straightforward calculation show that  var(ˆ 2 1

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S incewedon ’tknow ( θ ) we don’t know the pdf of ˆ  All we know about the pdf of is that ]= and var )= 2 However, as →∞ var 2 0 and the pdf of ˆ collapses at  Intuitively, as ˆ converges in some sense to  In other words, the estimator ˆ is consistent for  Furthermore, consider the standardized random variable = ˆ p var(ˆ ) = ˆ q 2 = μ ˆ For any value of   [ ]=0 ( )=1 butwedon’tknowthepdfof since we don’t know (  ) Asymptotic normality says that as gets large, the pdf of is well approximated by the standard normal density. We use the short-hand notation = μ ˆ (0 1) (1) to represent this approximation. The symbol “ ” denotes “asymptotically distrib- uted as”, and represents the asymptotic normality approximation. Dividing both sides of (1) by  and adding  the asymptotic approximation may be re-written as ˆ =  + μ  2 (2) The above is interpreted as follows: the pdf of the estimate ˆ is asymptotically distributed as a normal random variable with mean and variance 2 The quantity 2 iso ftenre ferredtoasthe asymptotic variance of ˆ  and is denoted avar ) The square root of avar ) is called the asymptotic standard error of ˆ and is denoted ASE( ˆ ) With this notation, (2) may be re-expressed as ˆ (  avar(ˆ )) The quantity 2 is sometimes referred to as the asymptotic variance of ) The asymptotic normality result (2) is commonly used to constuct a con f dence interval for  For example, an asymptotic 95% con f dence interval for has the form ˆ ± 1 96 × p avar(ˆ ) This con f
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econ583asymptoticsprimer - A Primer on Asymptotics Eric...

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