A Practical Guide to State Space Modeling

Transcription

1 A Practical Guide to State Space Modeling Jin-Lung Lin Institute of Economics, Academia Sinica Department of Economics, National Chengchi University March 006 1

2 1 Introduction State Space Model (SSM) has been a very powerful framework for the analysis of dynamical systems. While linear regression models use exogenous variables to distinguish the explained variation from the unexplained variation, SSM relies the dynamics of the state variables and the linkage between the observed variables and state variables to draw statistical inference about the unobserved states. The complete path of its conditional mean and variance are the output of Kalman Filtering. SSM is particularly useful for models involving unobserved variables. Potential output, natural rate of unemployment, and common business are three handy examples in economics. Furthermore, the conventional univariate as well multivariate ARMA models, linear regression models, and spline models can be converted into a SSM where forecasting, missing value and testing structural breaks can be easily handled. The theory of SSM was proposed in 1960 s and been heavily used by economists and other social scientists for a long time. Yet, a general and easy-to-use statistical software has not been around until recently. SsfPack for Ox, to my mind, is the best software for SSM. In this note, I shall review the state space models, the Kalman Filtering, smoothing, forecasting and initialization issues. Then, a brief introduction of the SsfPack and Ox will be given. Application examples includes local trend models, airline model, structural break tests, spline, missing observations, and seasonal adjustment. Finally, I shall discuss various SSM models to estimate the potential GDP and/or NAIRU in Taiwan. In addition to this introduction, Section list an easy but important lemma for deriving the Kalman Filter. The Kalman Filter for the local level model is discussed in details in Section 3 and Section 4 summarizes the recursion equation for the general SSM. The SsfPack implementation is given in Section 5 and applications in Section 6. Section 7 concludes. An important lemma for deriving the Kalman Filter Let (x, y, z) are jointly normally distributed with µ z = 0, and Σ yz = 0. Then E(x y, z) = E(x y) + Σ xz Σ 1 zz z V ar(x y, z) = V ar(x y) Σ xz Σ 1 zz Σ zx

3 The proof of the lemma is straightforward and thus omitted. It reads as below. When a new independent piece of information is added, the conditional mean and variance can be obtained by updating the previous ones. This formula is of fundamental importance deriving the Kalman Filter. 3 Kalman Filtering for local level model The local level model described below has a simple structure and serves as an excellent framework for understanding the recursion mechanism of the Kamlan Filter. y t = α t + ε t, ε t N(0, σ t ) (1) α t+1 = α t + η t, η t N(0, η t ) () α 1 N(a 1, P 1 ) (3) In this simple model y t consists of a signal, α t, measuring the stochastic trend and a measurement error, ε t. While exogenous variables are brought in to discriminate the signal from the noise in linear regression analysis, it is the dynamics, that does the job in state space model. In other words, the different dynamics for signals and noise which latter is usually assumed to follow a white noise process enables us to decompose the observed variable into two parts: the signal related to the state variables and disturbance terms. Kalman Filter fully explores this dynamic structure for filtering, smoothing and forecasting. When up to current period (y 1,..., y t ), last period (y 1,..., y t 1 ) and whole sample (y 1,..., y n ) information are used to estimate state variables at current time t, (α t ), they are called filtering, forecasting and smoothing respectively. 3.1 Kalman filtering Let Y t = σ{y 1,, y t } and defined a t+1 = E(α t+1 Y t ), P t+1 = V ar(α t+1 Y t ). then from (1, ), we have a t+1 = E(α t y t ) P t+1 = V ar(α t y t ) + ση 3

4 Define v t = y t a t and F t = V ar(v t ). Then E(v t Y t 1 ) = 0 or v t is a martingale difference process. So where Since E(v t ) = E(E(v t Y t 1 )) = 0 Cov(v t, y j ) = E(v t y j ) = E(E(v t y t 1 )y j ) = 0, for j = 1,,, t 1. By the lemma, E(α t Y t ) = E(α t Y t 1, v t ) = E(α t Y t 1 ) + Cov(α t, v t )V ar(v t ) 1 v t Cov(α t, v t ) = E(α t (y t a t )) = E(α t (α t + ε t a t )) = E(α t (α t a t )) = V ar(α t Y t 1 ) = P t we have V ar(v t ) = F t = P t + σ ε where Further, E(α t Y t ) = a t + K t v t K t = P t F t = V ar(α t Y t ) = V ar(α t Y t 1, v t ) P t P t + σ ɛ = V ar(α t Y t 1 ) Cov(α t, v t ) V ar(v t ) 1 = P t P t F t = P t (1 K t ) 4

5 Eqs. (4) and (4) say that the forecast error, v t is used to update the estimate of mean and variance of α 5 when it becomes available. The complete updating equations are: 3. Initial conditions v t = y t a t F t = P t + σ ε a t+1 = a t + K t v t P t+1 = P t (1 K t ) + σ η K t = P t F t Initial condition has to be given to complete the recursion and diffuse prior is the most commonly used one. Let a 1 be any number, and P 1 as in α 1 N(a 1, P 1 ). In other words, we do not have any information on α 1. It is easy to see: a = a 1 + P 1 (y P 1 + σε 1 a 1 ) y 1 P 1 σε P = + σ P 1 + σε η σε + ση The diffuse prior is equivalent to starting the recursion from t = and using y 1 as initial conditions. 3.3 Kalman smoothing We now turn to state smoothing. Let y = σ(y 1,, y n ). That is, y consists the whole sample information. â t = E(α t y) = E(α t Y t 1, v t,, v n ) where = a t + P t r t 1 r t 1 = v t F t + L t r t L t = 1 K t = σ ɛ F t 5

6 with r n = 0 For the conditional variance, V t = V ar(α t y) = V ar(α t Y t 1, v t,, v n ) = P t P t N t 1 where N t 1 = 1 F t + L t N t with N n = 0. The Kalman Filter can be summarized as below. Starting with initial condition (a 1, P 1 ), (a, P, F ) are computed. When y becomes available, v = y a is used to update the estimate of conditional mean and variance of α. Also, a 3, P 3 are computed and then updated when y 3 comes in. The process is repeated until at the end of the sample period. a n, P n, F n are computed and updated. Now, using the smoothing recursion, â n 1, V n 1 are computed. With the latter, â n, V n can be computed. The process is repeated until at the beginning of the period and â 1, V 1 are computed. 3.4 Missing observations Missing observations can be easily handled. Let y t, j = τ,, τ 1, are missing. Then for t = τ,, τ 1 E(α t Y t 1 ) = E(α t Y τ 1 ) t 1 = E(α i + η τ Y τ 1 ) j=i = a τ V ar(α t Y t 1 ) = V ar(α t Y τ 1 ) t 1 = V ar(α τ + η j Y τ 1 ) j=τ = P τ + (t τ)σ η As no new information is available during the missing periods, v t = 0, conditional mean remains unchanged and conditional variance increases linearly with time. 3.5 Forecasting Forecasting future values, y n+1,..., y n+h is equivalent to treating y n+1,, y n+h as missing and using the Kalman Filtering. 6

7 3.6 Likelihood function and parameter estimation Log-likelihood can be obtained as a side product of Kalman filtering. P (y 1,, y t ) = P (y t Y t 1 )P (Y t 1 ), we have with P (y 1 y 0 ) = P (y 1 ) and P (y) = P (y 1,, y t ) = Π n t=1p (y t Y t 1 ) logl = logp (y) = n log(π) 1 n (log F t + v t ) t=1 F t Since To achieve computation efficiency, we can substitute (σ η = qσ ε) and the system becomes y t = α t + ε t, ε t N(0, σ ε) α t+1 = α t + η t, η(t) N(0, qσ ε) The concentrated log likelihood becomes P t = P t σ ε, F t = F t v t = y t a t, F t σ ε = P t + 1 a t+1 = a t + K t v t, P t+1 = P t (1 K t ) + q K t = P t Ft = P t F t logl d = n log(π) n 1 logσε 1 ˆσ ε 1 n vt = n 1 Ft t= logl dc = n log(π) n Diagnostic checking n (logft + t=1 n 1 logˆσ ε 1 n t= v t σ εf t logf t Diagnostic checking is based upon the assumption that v t iid.n(0, F t ). Thus, e t = v t N(0, 1) Ft ) 7

8 Normality can be checked by examining the skewness and kurtosis Normality: skewness and kurtosis S = m 3 m 3 K = m 4 m N = { S 6 N(0, 6 n ) N(3, 4 n ) (K 3) + } χ () 4 The Ljung-Box Q statistics is useful for checking serial correlation and there are a bundle of statistics for heteroscedasticity test. 4 Kalman Filtering for general models The general state space model can be written as: y t = Z t α t + ε t, ε t N(0, H t ) α t+1 = T t α t + R t η t, η t N(0, Q t ) α 1 N(a 1, P 1 ) 8

9 Table 1: Dimensions for state space models vector matrix y t p 1 Z t p m α t m 1 T t m m ε t p 1 H t p p η t r 1 R t m r a 1 m 1 Q t r r v t p 1 P 1 m m a t m 1 P t m m 4.1 Initial conditions Diffuse prior P = P + κp P is m m matrix with zeros and ones. κ is large, say κ = Filter recursion As before, define The recursions become a t+1 = E(α t+1 Y t ) P t+1 = V ar(α t+1 y t ) v t = y t Z t a t F t = Z t P t Z t + H t K t = T t P t Z tf t 1 L t = T t K t Z t a t+1 = T t a t + K t v t P t+1 = T t P t L t + P t Q t P t Denote a t t = E(α t Y t ), P t t = V ar(α t Y t ), we have a t t = a t + M t Ft 1 v t 9

10 4.3 State smoothing Let y = σ(y 1,..., y n ). Then, a t+1 = T t a t t F t = Z t P t Z t + H t M t = P t Z t P t t = P t M t Ft 1 M t P t+1 = T t P t t T t + R t Q t R t ˆα t = E(α t y) = a t t + P t t T t Pt+1(ˆα 1 t+1 a t+1 ) = a t + P t Z tf t 1 v t + P t L tpt+1(ˆα 1 t+1 a t+1 ) Let then with r n = 0. For the variance, 5 SsfPack notations r t = P 1 t+1(ˆα t+1 a t+1 ) r t 1 = Z tf t 1 v t + L tr t V t = V ar(α t y) = P t P t N t 1 P t N t 1 = Z tf t 1 Z t + L tn t L t r t 1 = Z tf t 1 v t + L tt t ˆα t = a t + P t r t 1 α t+1 = d t + T t α t + H t εt Q t = c t + Z t α t 10

11 ( at+1 y t = Q t + G t ε ) t y t δ t = Ω t = = δ t + Φ t α t + u t ( ) dt c t u t = [ Ht H t H t G t G t H t G t G t α 1 N(a 1, P 1 ) Σ = ( ) Ht ε t G t ] Φ t = [ ] P1 a 1 [ Tt Z t ] 11

12 Table : Dimension of SSM matrices α t+1, d t, a : m 1, y t, Q t, c t : N 1 T t, P : m m, Z t : N m H t : m r, G t : N r Φ : (m + N) m, δ : (m + N) 1 Ω : (m + N) (m + N), Σ : (m + 1) 1 Four possible model specifications: nphi momega mphi momega msigma mphi momega msigma mdelta mphi momega msigma mdelta mj-phi mj-omega mj-delta mxt where the last specification works for time-varying Z t and T t with mj-phi, mj- Omega, mj-delta giving the corresponding row numbers in mxt, the data matrix. What is more, exogenous regressors can be added to the model. Below are summarized modules in SsfPack. Model in static space form AddSsfreg: GetSsfARMA: GetSsfspline: GetSsfstsm: SsfCombine: SsfCombineSym: add regressor effect put ARMA in static space. put regression in state space put cubic spline in state space combine two model Combine two symmetric models General state space algorithm KalmanFil: KalmanSmo: SimSmoDraw: SimSmoWgt: Kalman Filter Smoothing simulation smoother Covariance output of simulation smoother 1

13 Ready-to-use Function SsfConDens: mean of a draw of conditional density SsfLik: log-likelihood function SsfLikeConc: Profile log-likelihood function SsfLikeSco: Score vector SsfMomentEst: predictor, forecasting and smoothing SsfRecursion: State space recursion 6 Applications:estimating potential GPP and/or NAIRU Potential output and non-accelerating inflation rate unemployment (NAIRU) are defined as the level of output and unemployment rate consistent with a stable rate of inflation. Here, I discuss three models to estimate potential output and/or NAIRU. They are Watson (1986), Kuttner (1994), and Apel & Jansson(1998). Watson (1986) y t = y p t + z t y p t = y p t 1 + µ y + e yt, e pt N(0, σp) z t = φ 1 z t 1 + φ z t + e zt, e zt N(0, σ z) where y t : observed output, and y p t : potential GDP. In term of Ox notations. the model is : y p t+1 µ y z t+1 z t = φ 1 φ y p 1 0 ( ) t 0 1 ept z t e z zt y t t δ = µ y Φ = φ 1 φ

14 Ω = Kuttner (1994) y p t = y p t 1 + µ y + e yt y t = y p t + z t π t = µ π + r y t 1 + βz t 1 + v t + δ 1 v t 1 + δ v t + δ 3 v t 3 Apel and Jansson (1999) y p t = α + y p t 1 + ε p t u n t = u n t 1 + ε n t y t = y p t + φ 0 (u t u n t ) + φ 1 (u t 1 u n t 1) + ε y t u t u n t = δ(u t 1 u n t 1) + ε c t π t = ρ 1 π t 1 + η 1 (u t u n t ) + ωx t + ε τ 7 Conclusions State space models are very useful in econometric modeling, especially when unobserved variables are involved. SsfPack is a very general, easy-to-use and efficient package for SSM computation. 14

15 References Durbin, J. and S.J. Koopman (001), Time Series Analysis by State Space Methods, New York: Oxford University Press Koopman, S.J., N. Shephard, and J. A. Doornik (1998), Statistical algorithms for models in state space model using SsfPack., Econometric Journals,, Ox is an object-oriented statistical system. Console Ox (no graphics) can be obtained freely via SsfPack is a suite of C routines for carrying out computations involving the statistical analysis of univariate and multivariate models in state space form. The fully implemented link is to Ox and can be obtained freely at GnuDraw is an Ox package meant for creating GnuPlot graphics. Both can be obtained at cbos/index.html?content=/ cbos/gnudraw.html Professional Ox is bundled of Console Ox and GiveWin provides complete support for graphical environment. It can purchased at 15