The Kalman filter, Nonlinear filtering, and Markov Chain Monte Carlo

Transcription

1 NBER Summer Institute Minicourse What s New in Econometrics: Time Series Lecture 5 July 5, 2008 The Kalman filter, Nonlinear filtering, and Markov Chain Monte Carlo Lecture 5, July 2, 2008

2 Outline. Models and objects of interest 2. General Formulae 3. Special Cases 4. MCMC (Gibbs) 5. Likelihood Evaluation Lecture 5 2, July 2, 2008

3 . Models and objects of interest General Model (Nonlinear, non-gaussian state-space model) (Kitagawa (987), Fernandez-Villaverde and Rubio-Ramirez (2007) y t = H(s t, ε t ) s t = F(s t, η t ) ε and η ~ iid Example : Linear Gaussian Model y t = Hs t + ε t s t = Fs t + η t ε 0 0 t Σε ~ iidn, η 0 0 Σ t η Lecture 5 3, July 2, 2008

4 Example 2: Hamilton Regime-Switching Model y t = μ(s t ) + σ(s t )ε t s t = 0 or with P(s t = i s t = j) = p ij (using s t = F(s t,η t ) notation: s t = (η t p 0 + (p p 0 )s t ), where η ~ U[0,]) Example 3: Stochastic volatility model st y t = e ε t s t = μ + φ(s t μ) + η t Lecture 5 4, July 2, 2008

5 Some things you might want to calculate Notation: Y t = (y, y 2,, y t ), S t = (s, s 2,, s t ), f(.. ) a generic density function. A. Prediction and Likelihood (i) f(s t Y t ) (ii) f(y t Y t ) Note f(y T ) = f( yt Y t ) is the likelihood T t= B. Filtering: f(s t Y t ) C. Smoothing: f(s t Y T ). Lecture 5 5, July 2, 2008

6 2. General Formulae (Kitagawa (987)) Model: y t = H(s t, ε t ), s t = F(s t, η t ), ε and η ~ iid A. Prediction of s t and y t given Y t. (i) f( s Y ) = f( s, s Y ) ds t t t t t t = = f ( s s, Y ) f ( s Y ) ds t t t t t t f( s s ) f( s Y ) ds t t t t t (ii) f( yt Yt ) = f( yt st) f ( st Y t ) dst ( t component of likelihood) Lecture 5 6, July 2, 2008

7 Model: y t = H(s t, ε t ), s t = F(s t, η t ), ε and η ~ iid B. Filtering f( yt st, Yt ) f ( st Yt ) f( yt st) f ( st Yt ) f( st Yt) = f( st yt, Yt ) = = f( y Y ) f( y Y ) t t t t C. Smoothing f( s Y ) = f( s, s Y ) ds = f( s s, Y ) f( s Y ) ds t T t t+ T t+ t t+ T t+ T t+ f( s s ) f( s Y ) = f ( s s, Y ) f ( s Y ) ds = f ( s Y ) ds f( s Y ) = f( s Y ) f( s s ) ds t+ t t t t t+ t t+ T t+ t+ T t+ f( st+ Yt) t+ T t t t+ t t+ f( st+ Yt) Lecture 5 7, July 2, 2008

8 3. Special Cases Model: y t = H(s t, ε t ), s t = F(s t, η t ), ε and η ~ iid General Formulae depend on H, F, and densities of ε and η. Well known special case: Linear Gaussian Model y t = Hs t + ε t s t = Fs t + η t ε 0 0 t Σε ~ iidn, η 0 0 Σ t η 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 8, July 2, 2008

9 Digression: Recall that if a μa Σaa Σab ~ N, b μ, b Σba Σbb then (a b) ~ N(μ a/b, Σ a b ) where μ a b = μ a + Σ ab Σ (b μ b ) and Σ a/b = Σ aa Σ ab Σ Σ ba. bb bb Interpreting a and b appropriately yields the Kalman Filter and Kalman Smoother. Lecture 5 9, July 2, 2008

10 (repeating) Model: y t = Hs t + ε t, s t = Fs t + η t, ε 0 0 t Σε ~ iidn, η 0 0 Σ t η Let s t/k = E(s t Y k ), P t/k = Var(s t Y k ), μ t/t = E(y t Y t ), Σ t/t- = Var(y t Y t ). Deriving Kalman Filter: Starting point: s t Y t ~ N(s t /t, P t /t ). Then st st/ t Pt/ t Pt/ t H' t ~ N, y Y t y t/ t HPt/ t HPt/ t H ' +Σ ε interpreting s t as a and y t as b yields the Kalman Filter. Lecture 5 0, July 2, 2008

11 Model: y t = Hs t + ε t, s t = Fs t + η t, Details of KF: ε 0 0 t Σε ~ iidn, η 0 0 Σ t η (i) s t/t = Fs t /t (ii) P t/t = FP t /t F + Σ η, (iii) μ t/t = Hs t/t, (iv) Σ t/t = HP t/t H + Σ ε (v) K t = P t/t H Σ tt / (vi) s t/t = s t/t + K t (y t μ t/t ) (vii) P t/t = (I K t )P t/t. Lecture 5, July 2, 2008

12 The log-likelihood is T L(Y T ) = constant 0.5 { ln Σ tt / + ( yt μtt / )' Σtt / ( yt μtt / ) } t= The Kalman Smoother (for s t/t and P t/t ) is derived in analogous fashion (see Anderson and Moore (2005 ), or Hamilton (990).) Lecture 5 2, July 2, 2008

13 5. A Stochastic Volatility Model (Linear, but non-gaussian Model) (With a slight change of notation) x t = σ t e t ln(σ t ) = ln(σ t ) + η t or, letting y t = ln( x 2 t ), s t = ln(σ t ) and ε t = ln( e 2 t ) y t = 2 s t + ε t s t = s t + η t Complication: ε t ~ ln( χ ) 2 Lecture 5 3, July 2, 2008

14 3 ways to handle the complication () Ignore it (KF is Best Linear Filter. Gaussian MLE is QMLE) Reference: Harvey, Ruiz, Shephard (994) (2) Work out analytic expressions for all the filters, etc. (Uhlig (997) does this in a VAR model with time varying coefficients and stochastic volatility. He chooses densities and priors so that the recursive formulae yield densities and posteriors in the same family.) (3) Numerical approximations to (2). Lecture 5 4, July 2, 2008

15 Numerical Approximations: A trick and a simulation method. Trick: Shephard (994), Approximate the distribution of ε by a mixture of normals, n ε = qv, where v it ~ iidn(μ i, t it it i= 2 σ i ), and P(q it =)=p i. i p i μ i σ i (numbers taken from Kim, Shephard and Chib (998)) (Note: It seems that using only n=2 does not work too poorly) Lecture 5 5, July 2, 2008

16 2 χ density and n= 7 mixture approximation (picture taken from Kim, Shephard and Chib (998)) Lecture 5 6, July 2, 2008

17 Simulation method: MCMC methods (here Gibbs Sampling) Some References: Casella and George (992), Chib (200), Geweke (2005), Koop (2003). 4. Markov Chain Monte Carlo (MCMC) methods Monte Carlo method: Let a denote a random variable with density f(a), and suppose you want to compute Eg(a) for some function g. (Mean, standard deviation, quantile, etc.) Suppose you can simulate from f(a). Then Eg ( a) = g( ai ), where a i are N i = draws from f(a). If the Monte Carlo stochastic process is sufficiently well behaved, then p Eg( a) = Eg( a) by the LLN. N Markov Chains: Methods for obtaining draws from f(a). Suppose that it is difficult to draw from f(a) directly. Choose draws a, a 2, a 3, using a Markov chain. N Lecture 5 7, July 2, 2008

18 Draw a i+ from a conditional distribution, say h(a i+ a i ), where h has the following properties: () f(a) is the invariant distribution associated with the Markov chain. (That is, if a i is draw from f, then a i+ a i is a draw from f.) (2) Draws can t be too dependent (or else Eg ( a) = g( ai ) will not be a N i = good estimator of Eg(a).) N Markov chain theory (see refs above) yields sufficient conditions on h that imply consistency and asymptotic normality of Eg ( a ). In practice, diagnostics are used on the MC draws to see if there are problems. Lecture 5 8, July 2, 2008

19 How can h(a i+ a i ) be constructed so that f is invariant distribution. Gibbs sampling is one way. (Others ) Gibbs idea: partition a as a = (a, a 2 ). Then f(a, a 2 ) = f(a 2 a )f(a ). This suggests the following: given the i th draw of a, say a i = ( i generate a i+ in two steps: a, a ), 2 i (i) draw (ii) draw a + from f(a i a + from f(a 2 2 i a ) 2 i a + ) i Gibbs sampling is convenient when draws from f(a 2 a i ) and f(a 2 a i + ) are easy. Lecture 5 9, July 2, 2008

20 Issues: When will this work (or when will it fail) draws are too correlated (requiring too many Gibbs draws for accurate Monte Carlo sample averages). Examples (i) Bimodality: Lecture 5 20, July 2, 2008

21 (i) Absorbing point at ( a, 2 a ): Prob(a = a a 2 = 2 a ) = Prob(a 2 = 2 a a = a ) = Lecture 5 2, July 2, 2008

22 N ( ) = g( ai ) Checking quality of approximation: Eg a N( Eg ( a) Eg( a)) N(0, V) d N i = () 95% CI for Eg(a) = ± ˆ Eg( a).96 V / N (2) Multiple runs from different starting values (should not differ significantly from one another) (3) Compare ( ) Eg a based on N first draws and last N last draws (say first /3 and last /3 middle /3 left out). The estimates should not differ significantly from one another. Lecture 5 22, July 2, 2008

23 Returning to the Stochastic Volatility Model x t = σ t e t, ln(σ t ) = ln(σ t ) + η t or y t = 2 s t + ε t, s t = s t + η t n y t = ln( x t ), ε t = ln( χ ) qv it it, where v it ~ iidn(μ i, σ i ), and P(q it =)=p i. i= Smoothing Problem: E(σ t Y T ) = E(g(s t ) Y T ) with g(s) = e s : ( 7, T ) t t= it i=, t= T Let a { s},{ q } = = (a, a 2 ) Jargon: Data Augmentation add a 2 to problem even though it is not of direct interest.) Lecture 5 23, July 2, 2008

24 Model: y t = 2 s t + n qv it it, s t = s t + η t, v it ~ iidn(μ i, i= 2 σ i ), and P(q it =)=p i. Gibbs Draws (throughout condition on Y T ) T (i) (a a 2 ): { s t} t= { q } 7, T it i=, t= With { q } 7, T it i=, t= varying system matrices). known, this is a linear Gaussian model (with known time T { s t} ( t= { } 7, T it i, t q = =,Y T) is normal with mean and variance easily determined by formulae analogous to Kalman-filter (see Carter, C.K. and R. Kohn (994)). Lecture 5 24, July 2, 2008

25 (ii) (a 2 a ): { } 7, T q T it { s } i=, t= t t= With s t known, ε t = y t 2s t can be calculated. So T Prob(q it = { t} s =,Y T) = 7 t f j = ( ε ) p i t i f ( ε ) p j t j 2 where f i is the N(μ i, σ i ) density. Lecture 5 25, July 2, 2008

26 More Complicated Examples: TVP-VAR-SV Model: p (e t ~ SV) y = Φ y + e t t t i t i= (VAR) Cogley and Sargent (2005), Uhlig (997), (SVAR) Primiceri (2005), (Markov Switching VAR) Sims and Zha (2006). Simple univariate version: SW (2002) y t is quarterly GDP growth rates. Compute model s implied SD of y t + y t + y t 2 + y t 3 = Annual growth rate. Lecture 5 26, July 2, 2008

27 Lecture 5 27, July 2, 2008

28 UC-SV: Y t = τ t + ε t, τ t = τ t + η t (ε t and η t ~ SV) Stock and Watson (2007), Note: ΔY t = η t + ε t ε t, so with constant volatility Y t ~ IMA(,), and SV yields a time varying MA coefficient. Lecture 5 28, July 2, 2008

29 Y t = τ t + ε t, τ t = τ t + η t 7 2 ln( εt ) 2ln( σ ε, t) q ε, it, v ε, it, i= = +, 7 2 ln( ηt ) = 2ln( σ η, t) + q η, it, v η, it, i= ln(σ ε,t ) = ln(σ ε,t ) + υ ε,t, ln(σ η,t ) = ln(σ η,t ) + υ η,t, ( ) a = { t},{ ε, t, η, t},{ qε,, i t, qη,, i t} τ σ σ = (a, a 2, a 3 ) Gibbs Draws: {τ t } {σ ε,t, σ η,t }, {q ε,i,t, q η,i,t }, Y T : Kalman filter UC Model {σ ε,t, σ η,t } {τ t }, {q ε,i,t, q η,i,t }, Y T : Kalman filter SV (as above) {q ε,i,t, q η,i,t } {τ t },{σ ε,t, σ η,t }, Y T : Mixture indicator draws as above Lecture 5 29, July 2, 2008

30 Inflation (GDP Deflator) and smoothed estimate of τ (N =0,000, burnin = 000) Lecture 5 30, July 2, 2008

31 Estimates of τ from two independent sets of draws Lecture 5 3, July 2, 2008

32 Estimates of σ η from two independent sets of draws Lecture 5 32, July 2, 2008

33 N Eg( a) g( a ) = ; N i = i N ( Eg ( a ) Eg ( a )) N (0, V ) d Average values over all dates Serial Correlation in g(a i ) V / N V / n Eg ( a) τ % σ η % Lecture 5 33, July 2, 2008

34 What can go wrong (2): Absorbing Barrier (or just getting stuck ) Y t = τ t + ε t, τ t = τ t + η t 7 2 ln( εt ) 2ln( σ ε, t) q ε, it, v ε, it, i= = +, 7 2 ln( ηt ) = 2ln( σ η, t) + q η, it, v η, it, i= ln(σ ε,t ) = ln(σ ε,t ) + υ ε,t, ln(σ η,t ) = ln(σ η,t ) + υ η,t, ( ) a = { t},{ ε, t, η, t},{ qε,, i t, qη,, i t} τ σ σ = (a, a 2, a 3 ) Cecchetti, Hooper, Kasman, Schoenholtz and Watson (2007) What happens if σ η,t gets very small? Lecture 5 34, July 2, 2008

35 Computing the likelihood: Particle filtering Model: y t = H(s t, ε t ), s t = F(s t, η t ), ε and η ~ iid The t th component of likelihood: f( y Y ) = f( y s ) f ( s Y ) ds t t t t t t t Often f(y t s t ) is known, and the challenge is f(s t Y t ). Particle filters use simulation methods to draw samples from f(s t Y t ), say (s t, s 2t, s nt ), where s it is a called a particle. The t th component of the likelihood can then be approximated as t f( yt Y ) = f( yt sit). n = Methods for computing draws utilize the structure of the particular problem under study. Useful references include Kim, Shephard and Chib (998), Chib, Nardari and Shephard (2002), Pitt and Shephard (999), and Fernandez-Villaverde and Rubio-Ramirez (2007). i n Lecture 5 35, July 2, 2008