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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 . State-Space Models 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 20, 2007 Lecture 21 Maximum Likelihood and Kalman Filter State-Space Models Last time we introduced a state-space model, y t = Z t t + S t t t = T t t- 1 + R t t Q with t iid N , . We saw how we can use the Kalman filter to construct the distribution t 1 H of y t |Y t and t |Y t 1 . We also saw how to construct the distribution of given all observations of y , -- t |Y T . We can write many models in state-space form: Example 1 . AR ( p ): P y t = X i y t 1 + - t i =1 Then our state equation is 1 t y t y t- = . . = 1 ... p t 1 1 + . . t- . . . y t- p +1 . 1 . The measurement equation is y t = [1 0 ... 0] t For an AR model, it is straightforward to write down the likelihood directly, so there is no need to write down the state-space form or use the Kalman filter. However, for MA and ARMA models, the likelihood is very difficult to evaluate without using the Kalman filter. Example 2 . MA Model: y t = t + t- 1 State equation: t = t t 1 = 1 t + t- 1- Measurement: y t = [1 ] t Error-in-variables 2 Example 3 . ARMA Model: y t = 1 y t- 1 + 2 y t- 2 + t + t- 1 State equation: t = y t 1 y 2 y t 1 + t = 1 t- 1 1 + t 2 2 y t- 2 + - t- 1 Measurement: y t = [1 0] t Error-in-variables...
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lec21 - MIT OpenCourseWare http://ocw.mit.edu 14.384 Time...

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