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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 . Review from last time 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe October 23, 2007 Lecture 13 Unit Roots Review from last time Let y t be a random walk y t = y t + t , = 1 where t is a martingale difference sequence ( E [ t | I ] = 0), with 1 t E [ 2 I 4 t 1 ] 2 a.s. and E < K < . T Then t |- t { t } satisfy a functional central limit theorem, 1 T ( ) = [ T ] T X t t =1 W ( ) We showed last time that: 1 y T t- 1 t 1 y T 3 / 2 t- 1 1 2 R 1 W ( t ) dW ( t ) 1 2 y 2 T t 1 W ( t ) dt 1 2 R R W ( t ) 2 dt- and our OLS estimator and t-statistic have non-standard distributions: dW T - ) R W ( R W 2 dt t R W dW q R W 2 dt This is quite different from the case with | | < 1. If x t = x t- 1 + t , | | < 1 then, 1 x T t- 1 t 1 x T 3 / 2 t- 1 1 N (0 , 4 x T 2 2 t- 1 ) 1- 2 Ex t = 0 Ex 2 t = 2 1- 2 and the OLS estimate and t-stat converge to normal distributions: T ( - ) N (0 , (1- 2 )) t N (0 , 1) Adding a constant 2 Adding a constant Suppose we add a constant to our OLS regression of y t . This is equivalent to running OLS on demeaned y t . Let y m t = y t- y . Then, - 1 = y m t- 1 t ( y m t- 1 ) 2 Consider the numerator: 1 1 y T X m t- 1 t = ( T X y t- 1- y ) t 1 = y T X t- 1 t- y y = 1 y T 3 / 2 1 t- 1 t . We know that 1 T y T 3 / 2 t- 1...
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lec13 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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