lec20

# lec20 - 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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State-Space Models 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe Novemeber 15, 2007 Lecture 20 Filtering State-Space Models The Kalman ﬁlter is widely used to compute state-space models. These often appear in macro, as well as other areas of economics. Example 1 . For example, suppose, GDP growth, y t is given by y t = μ t + ² t μ t = μ t - 1 + η t σ 2 0 where μ t is the slow moving component of GDP growth and ² t is noise with ( ² t , η t ) iid N ± 0 , ± ² 0 σ 2 η ¶¶ Example 2 . Markov Switching y t = β 0 + β 1 S t + ² t S t ∈{ 0 , 1 } P ( S t = 1 | S t - 1 = 0) =1 - q P ( S t = 1 | S t - 1 = 1) = p If y t is GDP growth, we might think of S t as representing whether or not we’re in a boom. Some questions we might want to answer in these examples include: 1. Estimate parameters: e.g. in example 1 estimate σ ² and σ η 2. Extract trend: e.g. in example 1 estimate μ t 3. Forecasting We can estimate the parameters by maximum likelihood. Often, it useful to write the joint likelihood of ( y 1 , y 2 , ..., y T ) as a product of conditional densities, T f ( y 1 , ..., y T ; θ ) = f ( y 1 ; θ ) Y f ( y t | y t - 1 , ..., y 1 ; θ ) t =2 In a state space model, we have an unobserved state variable, α t , and measurements, y t . The state variables are distributed according to a state equation,
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## lec20 - MIT OpenCourseWare http:/ocw.mit.edu 14.384 Time...

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