asymptoticsprimerslides

asymptoticsprimerslides - Economics 583: Econometric Theory...

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Economics 583: Econometric Theory I A Primer on Asymptotics Eric Zivot October 7, 2009
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The two main concepts in asymptotic theory that we will use 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
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Motivating Example: Let X 1 ,...,X n denote an independent and identically distributed (iid ) random sample with E [ X i ]= μ and var ( X i )= σ 2 . We don’t know the probability density function (pdf) f ( X i , θ ) , but we know the value of σ 2 . Goal: Estimate the mean value μ from the random sample of data
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Remarks 1. A natural question is: how large does n have to be in order for the asymp- totic distribution to be accurate? 2. We can use Monte Carlo simulation experiments to evaluate the asymptotic approximations for particular cases. 3. We can often use bootstrap techniques to provide numerical estimates for avar(ˆ μ ) and con f dence intervals. These are alternatives to the analytic formulas derived from asymptotic theory. 4. If we don’t know σ 2 , we have to estimate avar μ ) .
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Probab i l ityTheo ryToo ls The probability theory tools (theorems) for establishing consistency of estima- tors are Laws of Large Numbers (LLNs). The tools (theorems) for establishing asymptotic normality are Central Limit Theorems (CLTs). A comprehensive reference is White (1994), Asymptotic Theory for Econome- tricians , Academic Press.
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Laws of Large Numbers Let X 1 ,...,X n be a iid random variables with pdf f ( X, θ ) . Forag ivenfunc- tion g, de f ne the sequence of random variables based on the sample Y 1 = g ( X 1 ) Y 2 = g ( X 1 ,X 2 ) . . . Y n = g ( X 1 n ) Example: Let X N ( μ, σ 2 ) so that θ =( μ, σ 2 ) and de f ne Y n = ¯ X n = 1 n n X i =1 X i . Hence, sample statistics may be treated as a sequence of random variables.
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De f nition 1 Convergence in Probability Let Y 1 ,...,Y n be a sequence of random variables. We say that Y n converges in probability to c, which may be a constant or a random variable, and write Y n p c or p lim n →∞ Y n = c if ε> 0 , lim n →∞ Pr( | Y n c | )=0
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Remarks 1. Y n p c isthesameas Y n c p 0 2 .Fo ravec to rp roce s s , Y n =( Y n 1 ,...,Y nk ) 0 , Y n p c if Y ni p c i for i = i,. ..,n.
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De f nition 2 Consistent Estimator If ˆ θ is an estimator of the scalar parameter θ, then ˆ θ is consistent for θ if ˆ θ p θ If ˆ θ is an estimator of the n × 1 vector θ , then ˆ θ is consistent for θ if ˆ θ i p θ i for i =1 ,...,n. Remark: All consistency proofs are based on a particular LLN. A LLN is a result that states the conditions under which a sample average of random variables con- verges to a population expectation. There are many LLN results. The most straightforward is the LLN due to Chebychev.
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Chebychev’s LLN Theorem 1 Chebychev’s LLN Let X 1 ,...,X n be iid random variables with E [ X i ]= μ< and var ( X i )= σ 2 < . Then ¯ X = 1 n n X i =1 X i p E [ X i μ The proof is based on the famous Chebychev’s inequality.
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Lemma 2
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asymptoticsprimerslides - Economics 583: Econometric Theory...

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