Kalman Filter Slides

# Kalman Filter Slides - NBER Summer Institute Minicourse...

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Lecture 5 1, July 21, 2008 NBER Summer Institute Minicourse – What’s New in Econometrics: Time Series Lecture 5 July 15, 2008 The Kalman filter, Nonlinear filtering, and Markov Chain Monte Carlo

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Lecture 5 2, July 21, 2008 Outline 1. Models and objects of interest 2. General Formulae 3. Special Cases 4. MCMC (Gibbs) 5. Likelihood Evaluation
Lecture 5 3, July 21, 2008 1. Models and objects of interest General Model (Nonlinear, non-Gaussian state-space model) (Kitagawa (1987), Fernandez-Villaverde and Rubio-Ramirez (2007) y t = H ( s t , ε t ) s t = F ( s t –1 , η t ) and ~ iid Example 1: Linear Gaussian Model y t = Hs t + t s t = Fs t –1 + t 0 0 ~, 0 0 t t iidN ⎛⎞ Σ ⎜⎟ Σ ⎝⎠

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Lecture 5 4, July 21, 2008 Example 2: Hamilton Regime-Switching Model y t = μ ( s t ) + σ ( s t ) ε t s t = 0 or 1 with P ( s t = i | s t –1 = j ) = p ij (using s t = F ( s t 1 , η t ) notation: s t = 1 ( t p 10 + ( p 11 p 10 ) s t 1 ), where ~ U[0,1]) Example 3: Stochastic volatility model y t = t s e t s t = + φ ( s t –1 ) + t
Lecture 5 5, July 21, 2008 Some things you might want to calculate Notation: Y t = ( y 1 , y 2 , … , y t ), S t = ( s 1 , s 2 , … , s t ), f ( . | . ) a generic density function. A. Prediction and Likelihood (i) f ( s t | Y t –1 ) (ii) f ( y t | Y t –1 ) … Note f ( Y T ) = 1 1 ( | ) T tt t fy = Y is the likelihood B. Filtering: f ( s t | Y t ) C. Smoothing: f ( s t | Y T ).

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Lecture 5 6, July 21, 2008 2. General Formulae (Kitagawa (1987)) Model: y t = H ( s t , ε t ), s t = F ( s t –1 , η t ), and ~ iid A. Prediction of s t and y t given Y t 1 . (i) 11 1 1 1 1 1 1 1 (| ) ( , | ) , ) ( | ) ) ( | ) tt t t t t t t t t t t t fs fss d s f ss f sd s fs s fs d s −− = = = YY Y (ii) ( | )( | | ) t t t fy s f s = ( “t ” component of likelihood)
Lecture 5 7, July 21, 2008 Model: y t = H ( s t , ε t ), s t = F ( s t –1 , η t ), and ~ iid B. Filtering 11 1 1 ( | ,) ( | )( | | ) (| ) , ) ) ) tt t t t t t t fy s f s s f s fs fs y −− == = YY Y C. Smoothing 1 1 1 1 1 1 1 1 ) ( , | ) , ) ( | ) ) ( | ) ( | ) ( | ) ) ) ) ( |) ) tT t t T t T t T t t t t t T t t T t t t t fss d s fs s d s fs s fs f ss f sd s f s d s ++ + + + + + + + + ⎡⎤ ⎢⎥ ⎣⎦ = ∫∫ Y Y Y Y Y Y Y Y

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Lecture 5 8, July 21, 2008 3. Special Cases Model: y t = H ( s t , ε t ), s t = F ( s t –1 , η t ), and ~ iid General Formulae depend on H , F , and densities of and . Well known special case: Linear Gaussian Model y t = Hs t + t s t = Fs t –1 + t 0 0 ~, 0 0 t t iidN ⎛⎞ Σ ⎜⎟ Σ ⎝⎠ In this case, all joint, conditional distributions and so forth are Gaussian, so that they depend only on mean and variance, and these are readily computed.
Lecture 5 9, July 21, 2008 Digression: Recall that if ~, aa a a b bb a b b a N b μ ΣΣ ⎛⎞ ⎜⎟ ⎝⎠ , then ( a | b ) ~ N ( a/b , Σ a|b ) where a | b = a + Σ ab 1 bb Σ ( b b ) and Σ a/b = Σ aa Σ ab 1 bb Σ Σ ba . Interpreting a and b appropriately yields the Kalman Filter and Kalman Smoother.

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