7 Day 3: Time Varying Parameter Models

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1 7 Day 3: Time Varying Parameter Models References: 1. Durbin, J. and S.-J. Koopman (2001). Time Series Analysis by State Space Methods. Oxford University Press, Oxford 2. Koopman, S.-J., N. Shephard, and J.A. Doornik (2001). Statistical Algorithms for State Space Models Using SsfPack 2.2, Econometrics Journal, 2, Zivot, E. and J. Wang (2002). Modeling Financial Time Series with S-PLUS. Springer-Verlag, New York. 7.1 Rolling Regression For a window of width k<n<t,the rolling linear regression model is y t (n) (n 1) = X t (n) (n k) β t (n) (k 1) + ε t (n),t= n,...,t (n 1) Observations in y t (n) andx t (n) aren most recent values from times t n +1tot OLS estimates are computed for sliding windows of width n and increment m Poor man s time varying regression model Application: Simulated Data compute rolling regressions for 24-month windows incremented by 1 month Application: Exchange Rate Data compute rolling regressions for 24-month windows incremented by 1 month compute rolling regressions for 48-month windows incremented by 12 months 7.2 Time Varying Parameter Regression Model References: The most used TVP regression has the form Remarks: y t = β 0,t + β 1,t x 1t + + β k,t x kt + ν t,ν t N(0,σ 2 ν) β i,t+1 = β i,t + ξ i,t, ξ i,t N(0,σ 2 i ), i=0,...,k 30

2 Random walk specification captures variety of parameter variation Model is most conveniently estimated and analyzed using state space methods 7.3 Linear Gaussian State Space Models α t+1 m 1 y t N 1 = d t + T t α t + H t η t m 1 m m m 1 m r r 1 = c t + Z t α t + G t ε t N 1 N m m 1 N N N 1 where t =1,...,n and α 1 N(a, P), η t iid N(0, I r ), ε t iid N(0, I N ) E[ε t η 0 t]=0 Compact notation used by SsfPack αt+1 y t = δ t + Φ t α t + u t, (m+n) 1 (m+n) m m 1 (m+n) 1 α 1 N(a, P) u t iid N(0, Ω t ) where δ t = Ω t = dt, Φ c t = t Ht H 0 t 0 0 G t G 0 t Tt Ht η, u Z t = t, t G t ε t Initial value parameters P Σ = a 0 Note: For multivariate models, i.e. N>1, G t G 0 t is assumed diagonal Initial Conditions Initial state variance is assumed to be of the form P = P + κp κ =10 7 P is for stationary state components P is for non-stationary state components 31

3 7.3.2 Regression Model with Time Varying Parameters y t = β 0,t + β 1,t x t + ν t, ν t N(0,σ 2 ν) β 0,t+1 = β 0,t + ξ t,ξ t N(0,σ 2 ξ) β 1,t+1 = β 1,t + ς t,ς t N(0,σ 2 ς ) Let α t =(β 0,t,β 1,t ) 0, x t =(1,x t ) 0, H t = diag(σ ξ,σ ς ) 0 and G t = σ ν.thestate space form is αt+1 I2 Hηt = α y t + t Gε t and has parameters Φ t = I2 x 0 t x 0 t, Ω = σ 2 ξ σ 2 ς σ 2 ν The initial state matrix is Σ = Regression model with fixed parameters The regression model with fixed regressors occurs when σ 2 ξ = σ 2 ς =0 7.4 Kalman Filter and Smoother The Kalman filter is a recursive algorithm for the evaluation of moments of the normally distributed state vector α t+1 conditional on the observed data Y t =(y 1,...,y t ) and the state space model parameters. Let a t = E[α t Y t 1 ] and P t = var(α t Y t 1 ) The filtering or updating equations compute a t t = E[α t Y t ], P t t = var(α t Y t ), v t = y t c t Z t a t (prediction error), F t = var(v t ) (prediction error variance) The prediction equations of the Kalman filter compute a t+1 and P t+1 The Kalman smoothing algorithm is a backward recursion which computes the mean and variance of specific conditional distributions based on the full data set Y n =(y 1,...,y n ). 32

4 The smoothed estimates of the state vector α t and its variance matrix are denoted ˆα t = a t n = E[α t Y n ] P t n = var(ˆα t Y n ) The smoothed estimate ˆα t is the optimal estimate of α t using all available information Y n. Thesmoothedestimateoftheresponsey t and its variance are computed using ŷ t = c t +Z t ˆα t var(ŷ t Y n ) = Z t var(ˆα t Y n )Z 0 t The smoothed disturbance estimates are the estimates ε t and η t based on all available information Y n, and are denoted ˆε t = ε t n = E[ε t Y n ] ˆη t = η t n = E[η t Y n ] Remarks Recursions are easy to code up in matrix programming languages like GAUSS, MATLAB, OX, S-PLUS, R SsfPack by Siem-Jan Koopman is a suite of C functions to efficiently implement the Kalman Filter and related algorithms. SsfPack has implementations in OX and S-PLUS. Eviews also implements the algorithms of SsfPack 7.5 Prediction Error Decomposition of Log-Likelihood The prediction error decomposition (PED) of the log-likelihood function for the unknown parameters ϕ of a state space model is ln L(ϕ Y n ) = nx ln f(y t Y t 1 ; ϕ) t=1 = nn 2 ln(2π) 1 2 nx t=1 ln Ft + v 0 tf 1 t v t where f(y t Y t 1 ; ϕ) is a conditional Gaussian density implied by the state space model The vector of prediction errors v t and prediction error variance matrices F t are computed from the Kalman filter recursions. 33

5 The state-space model parameters ϕ may be estimated by maximum likelihood using ln L(ϕ Y n ) computed from the PED. Remarks Care must be used to ensure that ϕ is identified Parameter transformations are often used to simplify estimation Use ϕ =exp(σ 2 ) to ensure positive variance Use ϕ = exp(p)/(1+exp(p)) to ensure probabilities lie between 0 and 1 Exploit invariance property of MLE Use delta method to compute asymptotic variances of un-transformed parameters 7.6 Example: Simulated Random Walk Slope Data The estimated model assumes random intercept and slope y t = α t + β t x t + ε t x t iid N(0, 1) ε t iid N(0,σ 2 ε) α t = α t 1 + ξ t,ξ t iid N(0.σ 2 ξ) β t = β t 1 + η t,η t iid N(0.σ 2 η) Thetruevaluesare σ ε =0.5,σ ξ =0,σ η = Parameter transformations To ensure positive variances, the log-likelihood is constructed using the parameterization ϕ 1 = ln(σ 2 ξ) σ 2 ξ =exp(ϕ 1 ) ϕ 2 = ln(σ 2 η) σ 2 η =exp(ϕ 2 ) ϕ 3 = ln(σ 2 ε) σ 2 ε =exp(ϕ 3 ) Note that <ϕ i < 34

6 7.6.2 Estimation results Estimation is performed using the SsfPack functions in S+FinMetrics. MLE of ϕ is The ϕ ϕ ϕ BytheinvariancepropertyofMLE,theestimatesofσ =exp( 1 2 ϕ)are Delta Method σ ξ σ η σ ε Let ˆϕ be an estimator such that n(ˆϕ ϕ) N(0, V) Let g(ϕ) be a continuous and differentiable function, independent of n. Then n(g(ˆϕ) g(ϕ)) N(0,GVG 0 ) G = g(ϕ) ϕ 0 For example, let ϕ = (ϕ 1,ϕ 2,ϕ 3 ) 0 g(ϕ) = (exp(ϕ 1 /2), exp(ϕ 2 /2), exp(ϕ 3 /2)) 0 = (g 1 (ϕ),g 2 (ϕ),g 3 (ϕ)) 0 Then G = = g 1 (ϕ) ϕ 1 g 2 (ϕ) ϕ 1 g 3 (ϕ) ϕ 1 g 1 (ϕ) ϕ 2 g 2 (ϕ) ϕ 2 g 3 (ϕ) ϕ 2 g 1 (ϕ) ϕ 3 g 2 (ϕ) ϕ 3 g 3 (ϕ) ϕ 3 exp(ϕ 1/2)/ exp(ϕ 2 /2)/ exp(ϕ 3 /2)/2 35

7 7.7 Example: Exchangeratedata TVP AR(1) model for forward discount The MLE of ϕ is f t s t = α t + β t (f t 1 s t 1 )+ε t ϕ ϕ ϕ BytheinvariancepropertyofMLE,theestimatesofσ =exp( 1 2 ϕ)are σ ξ σ η σ ε TVP regression model for differences regression The MLE of ϕ is s t+1 = α t + β t (f t s t )+ε t ϕ NA NA ϕ NA NA ϕ NA NA Note: Hessian fails to invert at MLE. BytheinvariancepropertyofMLE,theestimatesofσ =exp( 1 2 ϕ)are σ ξ NA NA σ η NA NA σ ε NA NA 36