State-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49

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1 State-space Model Eduardo Rossi University of Pavia November 2013 Rossi State-space Model Financial Econometrics / 49

2 Outline 1 Introduction 2 The Kalman filter 3 Forecast errors 4 State smoothing Smoothed state Smoothed state variance 5 Disturbance smoothing 6 Missing observations 7 Forecasting 8 Initialization 9 Parameter estimation 10 Linear Gaussian state space model 11 Examples of S-S representations UC models in S-S form ARIMA in S-S form 12 Kalman filter recursions Rossi State-space Model Financial Econometrics / 49

3 Introduction Basic model for representing a time series is the additive model: y t = µ t + γ t + ɛ t t = 1,..., T µ t is a slowly varying component called the trend; γ t is a periodic component of fixed period called seasonal; ɛ t is the error Rossi State-space Model Financial Econometrics / 49

4 Introduction Consider µ t = α t y t = α t + ɛ t ɛ t N(0, σ 2 ɛ) α t = α t 1 + η t η t N(0, σ 2 η) where ɛ t and η t are all mutually independent and are independent of α 1. The purpose The object of the methodology is to infer relevant properties of the α t s from a knowledge of the observations y 1,..., y T. Rossi State-space Model Financial Econometrics / 49

5 Introduction We assume initially that where a 1, P 1, σ 2 ɛ and σ 2 η are known. The model for y t is non-stationary. α 1 N (a 1, P 1 ) The local level model is an example of a linear Gaussian state space model The variable α t is called the state and is unobserved. Rossi State-space Model Financial Econometrics / 49

6 The Kalman filter The object of filtering is to update the knowledge of the system each time a new observation y t is brought in. Let and assume that Y t 1 = {y 1,..., y t 1 } α t Y t 1 N (a t, P t ) If a t and P t are known we can calculate a t+1 and P t+1 when y t is brought in. Since a t+1 = E[α t+1 Y t ] = E[α t + η t Y t ] = E[α t Y t ] P t+1 = Var[α t+1 Y t ] = Var[α t + η t Y t ] = Var[α t Y t ] + σ 2 η Rossi State-space Model Financial Econometrics / 49

7 The Kalman filter Define v t = y t a t = y t E[α t Y t 1 ] and F t = Var(v t ). Then E[v t Y t 1 ] = E[α t + ɛ t a t Y t 1 ] = a t a t = 0 thus E[v t ] = E[E[v t Y t 1 ]] = 0 and Cov[v t, y j ] = E[v t y j ] = E[E[v t Y t 1 ]y j ] = 0 so v t and y j are independent for j = 1, 2,..., t 1. When Y t is fixed Y t 1, y t are fixed v t and Y t 1 are fixed, and viceversa. Rossi State-space Model Financial Econometrics / 49

8 The Kalman filter Consequently, E[α t Y t ] = E[α t Y t 1, v t ] Var[α t Y t ] = Var[α t Y t 1, v t ] Since all variables are normally distributed, the E[α t Y t ] and Var[α t Y t ] are given by standard formulae from multivariate normal regression theory. Rossi State-space Model Financial Econometrics / 49

9 The Kalman filter Suppose that x, y and z are random vectors of arbitrary orders that are jointly normally distributed x y N (µ p, Σ) z where Σ = with Σ yz = Σ zy = 0 Σ xx Σ xy Σ xz Σ yx Σ yy Σ yz Σ zx Σ zy Σ zz = Σ xx Σ xy Σ xz Σ yx Σ yy 0 Σ zx 0 Σ zz Rossi State-space Model Financial Econometrics / 49

10 The Kalman filter Multivariate normal regression E[x y, z] = µ x + Σ xy Σ 1 yy z Var[x y, z] = Var[x y] Σ xy Σ 1 yy Σ xy Rossi State-space Model Financial Econometrics / 49

11 The Kalman filter Proof. By standard multivariate regression theory E[x w] = µ x + Σ xw Σ 1 ww (w µ w ) Var[x w] = Σ xx Σ xw Σ 1 ww Σ xw But with w = (y, z ) : Σ ww = because Σ yz = 0, and Σ xw = [Σ xy Σ xz ] [ ] Σyy 0 0 Σ zz [ Σ 1 ] [ yy 0 y µy E[x y, z] = µ x + [Σ xy Σ xz ] 0 Σ 1 zz z = µ x + Σ xy Σ 1 yy (y µ y ) + Σ xz Σ 1 zz z ] since µ z = 0. Rossi State-space Model Financial Econometrics / 49

12 The Kalman filter [ Σ 1 Var[x y, z] = Σ xx [Σ xy Σ xz ] yy 0 0 Σ 1 = ( Σ xx Σ xy Σ 1 yy Σ ) xy Σxz Σ 1 zz Σ xz = Var[x y] Σ xz Σ 1 zz Σ xz zz ] [ Σ xy Σ xz ] Rossi State-space Model Financial Econometrics / 49

13 The Kalman filter It follows that E[α t Y t ] = E[α t Y t 1, v t ] = E[α t Y t 1 ] + Cov[α t, v t ]Var[v t ] 1 v t Now, Cov[α t, v t ] = E [α t (y t a t )] = E[α t (α t + ɛ t a t )] = E[αt 2 + α t ɛ t α t a t ] = E[E[αt 2 Y t 1 ]] + E[α t ɛ t ] E[E[α t Y t 1 ]a t ] = E[E[αt 2 Y t 1 ]] E[E[α t Y t 1 ] 2 ] = E [Var[α t Y t 1 ]] = P t and F t Var[v t ] = Var[α t + ɛ t a t ] = Var[E[α t + ɛ t a t Y t 1 ]] + E[Var[α t + ɛ t a t Y t 1 ]] = E [Var[α t Y t 1 ]] + Var[ɛ t ] = P t + σɛ 2 Rossi State-space Model Financial Econometrics / 49

14 The Kalman filter Thus, since a t = E[α t Y t 1 ] E[α t Y t ] = E[α t Y t 1, v t ] = a t + P t F t v t or where is the regression coefficient of α t on v t. We have E[α t Y t ] = E[α t Y t 1, v t ] = a t + K t v t K t P t F t Var[α t Y t ] = Var[α t Y t 1, v t ] = Var[α t Y t 1 ] Cov[α t, v t ] 2 Var[v t ] 1 = P t P2 t F t = P t (1 K t ) Rossi State-space Model Financial Econometrics / 49

15 The Kalman filter The full set of relations for updating from time t to time t + 1: v t = y t a t F t = P t + σ 2 ɛ a 1 and P 1 are assumed to be known. K t = P t F t a t+1 = a t + K t v t P t+1 = P t (1 K t ) + ση 2 t = 1,..., T Rossi State-space Model Financial Econometrics / 49

16 Forecast errors The Kalman Filter (KF) residual v t = Y t a t and its variance F t are the one-step-ahead forecast error and the one-step-ahead forecast variance of y t Y t 1. The joint density of {y 1,..., y T } is p(y 1,..., y T ) = p(y 1 ) T p(y t Y t 1 ) the density of {v 1,..., v T }, provided that v t = y t a t is given by p(v 1,..., v T ) = p(y 1,..., y T ) y t v t T = p(v t ) t=1 t=2 Consequently, {v 1,..., v T } are mutually independent. Rossi State-space Model Financial Econometrics / 49

17 Forecast errors Define the state estimation error as with x t = α t a t Var(x t ) = P t the state estimation errors and the forecast errors are linear functions of the initial state error x 1 and the disturbances ɛ t and η t. v t = x t + ɛ t where x t+1 = α t+1 a t+1 = α t + η t a t K t v t = x t + η t K t (x t + ɛ t ) = L t x t + η t K t ɛ t L t = 1 K t = σ 2 ɛ/f t Rossi State-space Model Financial Econometrics / 49

18 State smoothing Smoothed state Smoothing Estimation of α 1,..., α n given the entire sample Y n = (y 1,..., y n ). The conditional density α t Y n N ( α t, V t ) where the smoothed state α t = E(α t Y n ) and the smoothed state variance V t = Var[α t Y n ]. The operation of calculating α 1, α 2,..., α n is called state smoothing. Rossi State-space Model Financial Econometrics / 49

19 State smoothing Smoothed state The forecast errors v 1,..., v n are mutually independent and are a linear transformation of y 1,..., y n the errors (v t,..., v n ) are independent of (y 1,..., y t 1 ) with zero means. when y 1,..., y n are fixed Y t 1, v t,..., v n are fixed and vice versa. E[x y, z] = E[x y] + Σ xz Σ 1 zz z α t = E[α t Y n ] = E[α t Y t 1, v t,..., v n ] = E[α t Y t 1 ] + Cov[α t, (v t,..., v n ) ]Var[(v t,..., v n ) ] 1 (v t,..., v n ) 1 Cov(α t, v t ) F t 0 v t = a t Cov(α t, v n ) 0 F n v n n = a t + Cov(α t, v j )F 1 j v j j=t Rossi State-space Model Financial Econometrics / 49

20 State smoothing Smoothed state Cov(α t, v j ) = Cov(x t, v j ) for j = t,..., n and Similarly, Cov(x t, v t ) = E[x t (x t + ɛ t )] = Var(x t ) = P t Cov(x t, v t+1 ) = E[x t (x t+1 + ɛ t+1 )] = E[x t (L t x t + η t K t ɛ t )] = P t L t Cov(x t, v t+2 ) = P t L t L t+1 Cov(x t, v n ) = P t L t L t+1... L n 1. Rossi State-space Model Financial Econometrics / 49

21 State smoothing Smoothed state Substituting in α t : α t = v t v t+1 v t+2 a t + P t + P t L t + P t L t L t F t F t+1 F t+2 = a t + P t r t 1 where r t 1 = v t F t + L t v t+1 F t+1 + L t L t+1 v t+2 F t L t L t+1... L n 1 v n F n is a weighted sum of innovations after t 1. r n = 0 since no observations are available after time n. Rossi State-space Model Financial Econometrics / 49

22 State smoothing Smoothed state The value of r t 1 can be evaluated using the backward recursion r t 1 = v t F t + L t r t t = n, n 1,..., 1 The smoothed stare can be calculated by the backwards recursion (smoothing state recursion) with r n = 0. r t 1 = v t F t + L t r t α t = a t + P t r t 1 t = n, n 1,..., 1 Rossi State-space Model Financial Econometrics / 49

23 State smoothing Smoothed state variance The error variance of the smoothed state V t = Var[α t Y n ] = Var[α t Y t 1, v t,..., v n ] = Var[α t Y t 1 ] Cov[α t, (v t,..., v n ) ]Var[(v t,..., v n ) ] 1 Cov[α t, (v t,..., v n ) ] 1 Cov(α t, v t ) F t 0 Cov(α t, v t ) = P t..... Cov(α t, v n ) 0 F n Cov(α t, v n ) n = P t Cov[(α t, v j )] 2 F 1 j j=1 Given that Cov(α t, v j ) = Cov(x t, v j ) for j = t,..., n and Cov(x t, v t ) = P t Cov(x t, v t+1 ) = P t L t Rossi State-space Model Financial Econometrics / 49

24 State smoothing Smoothed state variance We can obtain V t = P t Pt 2 N t 1 where N t 1 = 1 + L 2 1 t + L 2 t L 2 1 t L 2 t L 2 t+1... L 2 1 n 1 F t F t+1 F t+2 N t = 1 + L 2 1 t+1 + L 2 F t+1 F t+1l 2 1 t L 2 t+2 F t+1l 2 t+2... L 2 1 n 1 t+3 N n = 0. N t 1 can be calculated using the backward recursion F n F n N t 1 = 1 F t + L 2 t N t N t = Var(r t ) since the forecast errors v t are independent. Rossi State-space Model Financial Econometrics / 49

25 State smoothing Smoothed state variance The error variance of the smoothed state can be calculated by the backwards recursion N t 1 = 1 F t + L 2 t N t V t = P t P 2 t N t 1 t = n,..., 1 From the standard error V t of α t we can construct confidence intervals for α t for t = 1,..., n. It is also possible to derive the smoothed covariances between the states Cov(α s, α t Y n ), t s. Rossi State-space Model Financial Econometrics / 49

26 Disturbance smoothing Smoothed observation disturbance ɛ t = E[ɛ t Y n ] = y t α t smoothed state disturbance η t = E[η t Y n ] = α t+1 α t the estimates of ɛ t and η t are useful for detecting outliers and structural breaks. Rossi State-space Model Financial Econometrics / 49

27 Missing observations Missing observations are very easy to handle in Kalman filtering. Suppose the observations y j, j = τ,..., τ 1 are missing for 1 < τ < τ n. The most ideal way to deal with it is to define a new series y t = y t t = 1,..., τ 1 y t = y t+τ τ t = τ,..., n n = n (τ τ) The model is the same local level model with y t = y t except that α τ = α τ 1 + η τ 1 where η τ 1 N (0, (τ τ)σ 2 η) Rossi State-space Model Financial Econometrics / 49

28 Missing observations Filtering at time t = τ,..., τ 1 t 1 E[α t Y t 1 ] = E[α t Y τ 1 ] = E α τ + η j Y τ 1 = a τ and t 1 Var[α t Y t 1 ] = Var[α t Y τ 1 ] = Var α τ + η j Y τ 1 = P τ + (t τ)ση 2 giving j=τ j=τ a t+1 = a t P t+1 = P t + σ 2 η t = τ,..., τ 1 We can use the original KF for all t, by taking v t = 0 and K t = 0 at the missing time points. Rossi State-space Model Financial Econometrics / 49

29 Forecasting Let y n+j be the minimum MSE forecast given the time series {y 1,..., y n }, j = 1, 2,..., J, with J > 0. Then y n+j = E[y n+j Y n ] The theory of forecasting for the local level model: we regard forecasting as filtering the observations (y 1,..., y n, y n+1,..., y n+j ) using KF and treating the last J observations y n+1,..., y n+j as missing. Letting a n+j = E[α n+j Y n ] P n+j+1 = P n+j + σ 2 η j = 1,..., J 1 with a n+1 = a n+1 and P n+1 = P n+1 obtained from the KF. Rossi State-space Model Financial Econometrics / 49

30 Forecasting The forecast of y: for j = 1,..., J. y n+j = E[y n+j Y n ] = E[α n+j Y n ] + E[ɛ n+j Y n ] = a n+j F n+j = Var[y n+j Y n ] + Var[ɛ n+j Y n ] = P n+j + σ 2 ɛ The KF can be applied for t = 1,..., n + J where the observations at times n + 1,..., n + J are treated as missing. Rossi State-space Model Financial Econometrics / 49

31 Initialization How to start up the filter when nothing is known about the distribution of α 1. Diffuse prior: α 1 N(a 1, P 1 ) where a 1 is set at an arbitrary value and P 1. v 1 = y 1 a 1 substituting into the equations for a 2 and P 2 : F 1 = P 1 + σ 2 ɛ a 2 = a 1 + P 1 P 1 + σɛ 2 (y 1 a 1 ) ( P 2 = P P 1 = P 1 P 1 + σ 2 ɛ P 1 + σɛ 2 σɛ 2 + ση 2 ) + σ 2 η letting P 1, we obtain a 2 = y 1, P 2 = σ 2 ɛ + σ 2 η, then we proceed with the KF (diffuse KF). Rossi State-space Model Financial Econometrics / 49

32 Parameter estimation Since p(y 1,..., y n ) = p(y t 1 )p(y t Y t 1 ) the joint density of y 1,..., y n can be expressed as t = 2,..., n n p(y) = p(y t Y t 1 ) t=1 where p(y 1 Y 0 ) = p(y 1 ). p(y t Y t 1 ) = N (a t, F t ) v t = y t a t the log-likelihood is given by log L = log p(y) = n 2 log (2π) 1 2 n t=1 ( log F t + v t 2 ) F t with v t and F t from the KF. This the prediction error decomposition of the likelihood. Estimation proceeds by numerically maximizing log L. Rossi State-space Model Financial Econometrics / 49

33 Parameter estimation The log-likelihood in the diffuse case. All terms in the log-lik expression remain finite as P 1 with y fixed except the term for t = 1. To remove the influence of P 1 define the diffuse log-likelihood as: ( log L d = lim log L + 1 ) P 1 2 log P 1 ) = 1 2 lim P 1 n 2 log (2π) 1 2 ( log F 1 P 1 + v 2 1 n t=2 F 1 ( log F t + v 2 t F t since F 1 /P 1 1 and v 2 1 /F 1 0 as P 1. P 1 does not depend on σ 2 ɛ and σ 2 η, the values of σ 2 ɛ and σ 2 η that maximize log L are identical to the values that maximize log L log P 1. ) Rossi State-space Model Financial Econometrics / 49

34 Linear Gaussian state space model Linear Gaussian state space model is defined in three parts: State equation: Observation equation: Initial state distribution: α t+1 = T t α t + R t ζ t ζ t i.i.d.n (0, Q t ) y t = Z t α t + ɛ t ɛ t i.i.d.n (0, H t ) α 1 N (a 1, P 1 ) The matrices Z t, T t, R t, H t, Q t are independent of {ɛ 1,..., ɛ n } and {η 1,..., η n }. In many applications R t = I m, the theory remains valid if R t (m r). Rossi State-space Model Financial Econometrics / 49

35 Linear Gaussian state space model The idea underlying the model is that the development of the system over time is determined by α t according to the state equation. Because α t cannot be observed directly we must base the analysis on observations y t. The matrices Z t, T t, R t, H t and Q t are initially assumed to be known and the error terms ɛ t and η t are assumed to be serially independent and independent of each other at all time points. Matrices Z t, and T t 1, are permitted to depend on y 1,..., y t 1 The initial state vector is assumed to be N (a 1, P 1 ) independently of ɛ 1,..., ɛ n and η 1,..., η n. a 1 and P 1 are assumed to be known. Rossi State-space Model Financial Econometrics / 49

36 Linear Gaussian state space model State space model is linear and Gaussian: therefore properties and results of multivariate normal distribution apply; State vector α t evolves as a VAR(1) process; System matrices usually contain unknown parameters; Estimation has therefore two aspects: 1 measuring the unobservable state (prediction, filtering and smoothing); 2 estimation of unknown parameters (maximum likelihood estimation); State space methods offer a unified approach to a wide range of models and techniques: dynamic regression, ARIMA, UC models, latent variable models, spline-fitting and many ad-hoc filters; Rossi State-space Model Financial Econometrics / 49

37 Examples of S-S representations Regression with Time Varying Coefficients: regressors in Z t = X t T t = I R t = I regression model with coefficient α t following a random walk. Rossi State-space Model Financial Econometrics / 49

38 Examples of S-S representations UC models in S-S form Local level model: y t = µ t + ɛ t ɛ t N(0, σ 2 ɛ) µ t+1 = µ t + η t η t N(0, σ 2 η) State equation: α t+1 = T t α t + R t ζ t ζ t i.i.d.n (0, Q t ) Observation equation: α t = µ t T t = 1 R t = 1 Q t = σ 2 η y t = Z t α t + ɛ t ɛ t i.i.d.n (0, H t ) Z t = 1 H t = σ 2 ɛ Rossi State-space Model Financial Econometrics / 49

39 Examples of S-S representations UC models in S-S form Local linear trend model y t = µ t + ɛ t ɛ t N(0, σɛ) 2 µ t+1 = µ t + ν t + η t η t N(0, ση) 2 ν t+1 = ν t + ξ t ξ t N(0, σξ) 2 If ξ t = η t = 0 this entails ν t+1 = ν t = ν µ t+1 = µ t + ν y t = µ t + ɛ t µ t+1 = µ t + ν deterministic trend plus noise. Rossi State-space Model Financial Econometrics / 49

40 Examples of S-S representations UC models in S-S form State equation: α t = α t+1 = T t α t + R t ζ t ζ t i.i.d.n (0, Q t ) ] [ ] [ ] [ ] σ 2 T ν t = R t 0 1 t = Q 0 1 t = η 0 0 σξ 2 [ µt Observation equation: y t = Z t α t + ɛ t ɛ t i.i.d.n (0, H t ) Z t = [ 1 0 ] H t = σ 2 ɛ Rossi State-space Model Financial Econometrics / 49

41 Examples of S-S representations ARIMA in S-S form ARIMA(p,d,q): All ARIMA(p,d,q) models have a (non-unique) state space representation. y t = d x t α t = r = max (p, q + 1) y t = φ(l)y t = θ(l)η t r r 1 φ j y t j + η t + θ j η t j j=1 j=1 Z t = [1, 0, 0,..., 0] y t φ 2 y t φ r y t r+1 + θ 1 η t θ r 1 η t r+2 φ 3 y t φ r y t r+2 + θ 1 η t θ r 1 η t r+3. φ r y t 1 + θ r 1 η t Rossi State-space Model Financial Econometrics / 49

42 Examples of S-S representations ARIMA in S-S form T t = φ φ r φ r R t = R = 1 θ 1. θ r 1 Observation equation: H t = 0 ɛ t = 0. ζ t = η t+1 y t = Z t α t Rossi State-space Model Financial Econometrics / 49

43 Examples of S-S representations ARIMA in S-S form MA(1) model: y t = η t + θ 1 η t 1 Observation equation: y t = Z t α t Z t = [ 1 0 ] H t = 0 State equation: [ yt+1 θ 1 η t+1 ] = [ ] [ yt θ 1 η t ] [ 1 + θ 1 ] η t+1 Q t = σ 2 η Rossi State-space Model Financial Econometrics / 49

44 Examples of S-S representations ARIMA in S-S form ARMA(2,1), r = 2, y t = φ 1 y t 1 + φ 2 y t 2 + η t + θ 1 η t 1 Observation equation: State equation: [ ] [ y t+1 φ1 1 = φ 2 y t + θ 1 η t+1 φ 2 0 y t = Z t α t Z t = [ 1 0 ] H t = 0 ] [ ] [ y t 1 + φ 2 y t 1 + θ 1 η t θ1 ] η t+1 y t+1 = φ 1 y t + φ 2 y t 1 + θ 1 η t + η t+1 φ 2 y t + θ 1 η t+1 = φ 2 y t + θ 1 η t+1 Rossi State-space Model Financial Econometrics / 49

45 Examples of S-S representations ARIMA in S-S form ARIMA(2,1,1): Observation equation: State equation: State vector: y t = φ 1 y t 1 + φ 2 y t 2 + η t + θ 1 η t 1 α t+1 = y t = [1, 1, 0] α t φ φ 2 0 α t = α t θ 1 y t 1 y t φ 2 y t 1 + θ 1 η t η t+1 Rossi State-space Model Financial Econometrics / 49

46 Examples of S-S representations ARIMA in S-S form 1st equation 2nd equation 3rd equation y t = y t 1 + y t y t+1 = φ 1 y t + φ 2 y t 1 + η t+1 + θ 1 η t φ 2 y t + θ 1 η t+1 = φ 2 y t + θ 1 η t+1 Rossi State-space Model Financial Econometrics / 49

47 Examples of S-S representations ARIMA in S-S form ARIMA(2,2,1): 3rd equation 2 y t = φ 1 2 y t 1 + φ 2 2 y t 2 + η t + θ 1 η t α t+1 = φ 1 1 α t η t φ 2 0 α t = y t 1 y t 1 2 y t φ 2 2 y t 1 + θ 1 η t θ 1 2 y t+1 = φ 1 2 y t + φ 2 2 y t 1 + η t+1 + θ 1 η t The need to facilitate the initialization explains why the S-S model is set in this form. The elements y 0 and y 0 are treated as diffuse random elements while the other elements, including 2 y t are stationary which have proper unconditional means and variances. Rossi State-space Model Financial Econometrics / 49

48 Kalman filter recursions The unobserved state α t can be estimated from the observations with the Kalman filter: v t = y t z T α T F t = Z t P t Z t + H t K t = T t P t Z t F 1 t a t+1 = T t a t + K t v t L t = T t K t Z t P t+1 = T t P t L t + R t Q t R t for t = 1,..., n and starting with given values for a 1 and P 1. Rossi State-space Model Financial Econometrics / 49

49 Kalman filter recursions The contemporaneous filtering equations incorporate the computation of the state vector estimator a t t E[α t Y t ] and its variance P t t : M t = P t Z t a t t = a t + M t Ft 1 v t P t t = P t M t Ft 1 M t a t+1 = T t a t t P t+1 = T t P t t T t + R t Q t R t Rossi State-space Model Financial Econometrics / 49