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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 . Local to Unity Asymptotics 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe October 25, 2007 Lecture 14 More Non-Stationarity We have seen that there’s a discrete difference between stationarity and non-stationarity. When we have a non-stationary process, limiting distributions are quite different from in the stationary case. For example, let ² be a martingale difference sequence, with E ( ² 2 | I ) = 1, E² 4 < k < ∞ . Then ξ ( τ ) = 1 t t t- 1 t T √ [ τ T ] T t =1 ² t ⇒ W ( · ). Then there is a sort of discontinuity in the limiting distribution of an AR (1) at ρ = 1: ∑ Unit Root Stationary Process y t = y t- 1 + ² t x t = ρx t- 1 + ² t Limiting distribution of ρ R T ( ρ ˆ- 1) ⇒ W dW R W 2 dt T ( ρ ˆ- ρ ) ⇒ N (0 , 1- ρ 2 ) Limiting distribution of t R t ⇒ W dW √ R W 2 dt t ⇒ N (0 , 1) In finite samples, the distribution of the t-stat is continuous in ρ ∈ [0 , 1]. However, the limit distribution is discontinuous at ρ = 1. This must mean that the convergence is not uniform. In particular, the convergence of the t-stat to a normal distribution is slower, the closer ρ is to 1. Thus, in small samples, when ρ is close to 1, the normal distribution badly approximates the unknown finite sample distribution of the...
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lec14 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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