lec12 - MIT OpenCourseWare http:/ocw.mit.edu 14.384 Time...

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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 .
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Introduction 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe October 18, 2007 Lecture 12 Empirical Processes Introduction References : Hamilton ch 17, Stock HoE vol 4, Andrews HoE (more mathematical) Empirical process theory is used to study limit distributions under non-standard conditions. Applications include: 1. unit root & cointegration – e.g. y = ρy t 1 + e t , if ρ = 1, then T ( ρ ˆ - 1) ? some non-standard - distribution μ + e t t τ 2. structural breaks (testing with nuisance parameters) – e.g. y t = ( . Want to test μ + k + e t t > τ H 0 : no break k = 0. Under H 0 , τ doesn’t enter the model, but it is a nuisance parameter under the alternative. A test statistic for this hypothesis is s = max τ | t τ | where t τ is the t-statistic for testing k = 0 with the break at time τ . s will have a non-standard distribution. 3. weak instruments & weak GMM 4. Simulated GMM with non-differentiable objective function – e.g. Berry & Pakes 5. semi-parametric We will discuss 1 & 2. We will cover simulated GMM later. Functional Central Limit Theorem Let x t be a real-valued random k × 1 vector. Consider some < n valued function g t ( x t , τ ) for τ Θ, where Θ is a subset of some metric space. Let 1 ξ T ( τ ) = T X ( g t ( x t , τ ) T t =1 - Eg t ( x t , τ )) ξ T ( τ ) is a random function; it maps each τ Θ to an < n valued random variable. ξ T ( τ ) is called an empirical process. Under very general conditions, standard arguments show that ξ T ( τ ) converges pointwise, i.e. τ 2 0 Θ, ξ T ( τ 0 ) N (0 , σ ( τ 0 )). Also, standard arguments imply that on a finite collection of points,
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lec12 - MIT OpenCourseWare http:/ocw.mit.edu 14.384 Time...

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