lec16 - 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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Multi-dimensional Random Walk 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 1, 2007 Lecture 16 Cointegration We think, or at least we cannot reject the null hypothesis, that many macro series have unit roots. For example, log consumption and log output are both non-stationary, but log consumption - log output is stationary. This situation is called cointegration. The practical problem is that when we have cointegration, asymptotics change completely. Furthermore, we really do not have enough data to definitively tell whether or not we have cointegrated series. Multi-dimensional Random Walk Let ² t be k × 1 with E [ ² t | ² t , ... ] = 0, E [ ² ² 0 ² , ... ] = I , and finite fourth moments. Then, - 1 t t | t - 1 k 1 ξ T ( τ ) = [ τ T ] T X ² t t =1 W where W is a k -dimensional Brownian motion. We also want to allow for serial correlation, so we need to look at the behavior of v t = F ( L ) ² t , X i | F i | < The longterm variance of v t will be F (1) F (1) 0 . Finally, let y t = y t - 1 + v t We know the following results: 1 [ T τ ] T X v t t =1 F (1) W ( τ ) 1 1 y T X t - 1 ² t F (1) Z W ( t ) dW ( t ) 0 1 y T 3 / 2 X t F (1) Z W dt 1 y T 2 X t y t 0 F (1) Z W W 0 dt Each of these are just multi-dimensional variants of single dimension results that we’ve already seen.
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Regression 2 Regression Let y t = βx t + e t where e t is stationary, and Dx t = z t , where z t = [ z 1 t z 2 t z 3 t z 4 t ] where z 1 t is zero-mean and stationary, z 2 t is a constant,
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lec16 - MIT OpenCourseWare http:/ocw.mit.edu 14.384 Time...

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