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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Introduction 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe October 2, 2007 Lecture 8 Bootstrap A good reference is Horowitz (2004) Introduction We have a sample z = { z i , i = 1 , ..., n } ∼ F from a distribution F . We have a statistic of interest, T ( z ) whose distribution we want to compute (perhaps because we want to do some hypothesis tests). Let P ( T ( z ) ≤ t ) = G n ( t, F ) be the cdf of T ( z ) from a sample of size n . G n is some complicated function of F , which we do not know. We want to approximate G n . One way to do this would be to use the asymptotic distribution of G n . Let’s say G n ( t, F ) → G ( t, F ∞ ). Example 1 . z i iid with Ez 2 2 2 i = μ , Ez i = σ . Suppose we know σ and we’re estimating the mean by T ( z ) = 1 z n ∑ i . We know √ n ( T ( z )- μ ) ⇒ N (0 , σ 2 ) so we can use the normal distribution to compute p-values for hypothesis testing and to compute confidence intervals. For example, a 95% confidence interval would be [ T ( z ) √- n Z 1- α/ 2 , T ( z ) √- n 1 α/ 2 ], where α is the α quantile of the normal distribution N (0 , σ 2 Z- Z ). The bootstrap is another approach to approximating G ( t, F ∞ ). Bootstrap ˆ Instead of using the asymptotic distribution to approximate G n ( · , F ), we use G n ( · , F ) ≈ G n ( · , F n ) where ˆ F ( t ) = 1 n ∑ n 1 i =1 ( x ≤ t ) is the empirical cdf. In practice, we simulate to compute G n ( · ˆ , F n ). A general n algorithm for the bootstrap with iid data is { } ˆ 1. Generate bootstrap sample z b * = z 1 * b , ..., z nb * independently drawn from F n for b = 1 ..B . By this, we....
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lec8 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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