State Space Model, Official Statistics, Bayesian Statistics Introduction to Durbin s paper and related topics

Transcription

1 State Space Model, Official Statistics, Bayesian Statistics Introduction to Durbin s paper and related topics Yasuto Yoshizoe Aoyama Gakuin University yoshizoe@econ.aoyama.ac.jp yasuto yoshizoe@post.harvard.edu 1

2 Summary Durbin (2000) The Foreman Lecture: The State Space Approach to Time Series Analysis and its Potential for Official Statistics Durbin and Koopman Time Series Analysis by State Space Methods, Oxford Statistical Science Series 24, Oxford UP. Some related comments and/or questions Bayesian viewpoint Robustness issues Official Statistics in Japan: Guideline of seasonal adjustment Kalman filter and business cycle estimation 2

3 State Space Model and Bayeian Method Flexible: covers wide range of models, including flexible Bayesian methods Shiller s distributed lag model Seasonal adjustment: BAYSEA / DECOMP (IMS, Akaike, Kitagawa) Other models (a) Locally Linear Regression (O Hagan, Yoshizoe) y(x) = Xβ(x) + ε β(x) some process (b) Regression with time varying coefficients y t = Xβ t + ε β t N(0, ) 3

4 State Space Model and Bayeian Method Shiller s distributed lag model: y t = β 0 + L γ j x t j + ε t j=0 where γ j s follow the following prior distribution. γ j N(0, τ 2 ) Shiller model is flexible compared to other models, including polynomial lag model, Koyck lag model, etc., because it allows local linearity which is similar to state space model. It has also been practically successfully applied in various problems. 4

5 Compare with another poor Bayesian model where a polynomial of γ j holds approximately: P (γ 1,, γ L ) N(0, ) Shiller 5

6 State Space Model and Bayeian Method Seasonal adjustment: BAYSEA / DECOMP (IMS, Akaike, Kitagawa) Similar to Shiller s lag model applied to seasonal variation y t = µ + γ t + ε t where s γ t N(0, ) Shiller lag Kitagawa and Girsch smoothness prior 6

7 State Space Model and Bayeian Method Locally Linear Regression (O Hagan, Yoshizoe) y(x) = Xβ(x) + ε β(x) some process β(x) is locally almost constant: Pr{ β(x) β(x ) < K x x } = 1 ε x, x var[β(x) β(x )] is proportional to d(x, x ) 2 var[β(x)] = ξ 1 Ω (ξ 0 in the posterior) β(x) is normally distributed. 7

8 State Space Model: Local Level Model y t N(α t, σε), 2 α t α t 1 N(0, ση) 2 α Bayesian model α t N(0, σ 2 ) State Spade Model y 1 y t N(Z t α t, σε), 2 α t T t α t 1 N(0, ση) 2 8

9 Kalman Filter Filtering robustize computer intesive CPU When the model is complex, computational burden is still heavy. Moreover, there are cases where we need caution as we see in the following example. 9

10 State Space Model and Business Cycle Stock and Watson model (simplified version): Y t = β + γ C t + u t C t φ 1 C t 1 φ 2 C t 2 = δ + η t Y t is a vector of observed economic variables. C t is an unobserved business indicator. The error term in the space equation (η t, ɛ t) is normally distributed with mean 0 (n+1) 1 and variance diag(σ 2 η, σ 2 ɛ 1,, σ 2 ɛ n ) Σ u t N(0 n 1, H) and is independent of C t. 10

11 State Space Model and Business Cycle Kalman Filter or similar algorithms can be applied to extract the hidden business indicator C t, but we should be careful. The algorithms often do not converge when the dimension of Y is higher than 5, and seldom converge when the dimension is 6. The estimated parameters differ significantly depending on the initial values of the iteration. We obtain similar C t from quite different sets of parameters Occasionally, the likelihood function is not of typical shape. 11

12 State Space Model and Business Cycle Table: Estimated parameters Case 1 Case 2 Case 3 Case 4 c t φ φ y 1t γ ψ ψ y 2t σ γ ψ ψ y 3t σ γ ψ ψ y 4t σ γ ψ ψ σ Log-likelihood

13 State Space Model and Business Cycle Estimated C t for four variable model: y 1,, y Case Case Case Case

14 Shape of log-likelihood (Model 1) θ = tθ 1 + (1 t)θ 2, 1 t 2 θ = tθ 1 + (1 t)θ 3, 1 t 2 log.likelihood θ 1 θ 2 log.likelihood θ 1 θ t t θ = tθ 1 + (1 t)θ 4, 2 t 2 θ = tθ 2 + (1 t)θ 3, 1 t 2 log.likelihood θ 1 θ log.likelihood θ 2 θ t t θ = tθ 2 + (1 t)θ 4, 1 t 2 θ = tθ 3 + (1 t)θ 4, 1 t 2 log.likelihood θ 2 θ log.likelihood θ 3 θ t t 14

15 20 X-12-ARIMA (X-11, X-12-ARIMA, MITI, DECOMP) Guideline 15

16 References 1. 6 Statistical Analysis by Bayesian Method, Univ. of Tokyo Press, Bayesian Analysis of Seasonal Economic Data, International Symposium on Bayesian Statistics, Institute of Statistical Mathematics, New Development of Business Cycle Indicators, Economic and Social Resear Institute, Cabinet Office,