Discussion Score-driven models for forecasting by Siem Jan Koopman. Domenico Giannone LUISS University of Rome, ECARES, EIEF and CEPR

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1 Discussion Score-driven models for forecasting by Siem Jan Koopman Domenico Giannone LUISS University of Rome, ECARES, EIEF and CEPR 8th ECB Forecasting Workshop European Central Bank, June / 10

2 Generalise Autoregressive Score (GAS) Basic Model Specification y t : data ˆf t : time varying parameter X t : exogenous variables θ: static parameters y t ˆf t p(y t ˆf t, F t ; θ) p q ˆf t+1 = ω + A i ŝ t i+1 + i=1 s t = S t t j=1 t = log p(y t f t ; Λ) f t S t = S(t, ˆf t, F t ; θ) B j ˆft j+1 Usually: S t = αi 1 where I = E( t t) 2 / 10

3 Generalise Autoregressive Score (GAS) The origins: Creal, D., S. J. Koopman, and A. Lucas (2008). A general framework for observation driven time-varying parameter models. Discussion Paper /4, Tinbergen Institute. Harvey, A. C. and T. Chakravarty (2008). Beta-t-(E)GARCH. University of Cambridge, Faculty of Economics, Working paper CWPE The presentation Introduction to GAS (Drew, Koopman, Lucas, 2013) Optimality of score (Blasques, Koopman and Lucas, 2014) Forecasting evidence (Koopman, Lucas and Scharth, 2014) 3 / 10

4 My Quest for the Origins of GAS Siem Jan Koopman: Time series analysis by state space methods Andrew C. Harvey: Forecasting, structural time series models and the Kalman filter The discussion From State Space Models to Score Driven Models Spot the difference Dynamic Factor Models (DFM) Traditional vs Score Driven 4 / 10

5 Linear State space model y t f t N (Λf t, R) f t+1 f t N (Af t, Q) Defining ˆf t+1 := E(f t+1 y t ), we have the Kalman recursion: ˆf t+1 = Aˆf t + AG t v t where v t = (y t Λˆf t ): surprises G t = P t Λ (ΛP t Λ + R) 1 : gain Where the variance of the estimated states is computed from the recursion: E[(ˆf t+1 f t+1 )(ˆf t+1 f t+1 ) ] := P t+1 = AP t A + Q AP t K t Λ 5 / 10

6 Linear State Space model It can be shown that: y t f t N (Λf t, R) f t+1 f t N (Af t, Q) ˆf t+1 = Aˆf t + AG t v t G t v t = (Λ } R{{ 1 Λ} +Pt 1 ) 1 Λ R 1 v }{{} t I t t = log p(y t f t ; Λ) f t, I = E( t t) 6 / 10

7 Kalman and GAS Linear State Space model (Parameter-driven) y t f t N (Λf t, R) f t+1 f t N (Af t, Q) ˆf t+1 = Aˆf t + A(I t + Pt 1 ) 1 t t = log p(y t f t ; Λ) f t, I = E( t t) Generalised Autoregressive Scores (GAS) (Observation-driven) y t ˆf t p(ˆf t ; θ) ˆf t+1 = Aˆf t + α(ĩ t + 0) 1 t t = log p(y t f t ; Λ) f t, I = E( t t) 7 / 10

8 Kalman and GAS Non-Linear State space model (Parameter-driven) y t f t p y f (y t f t ; θ) f t+1 ˆf t p f+ f (f t+1 f t ; θ) Generalised Autoregressive Scores (GAS)(Observation-driven) y t ˆf t p(y t ˆf t ; θ) ˆf t+1 = Aˆf t + α( Ĩ t ) 1 t t = log p(y t f t ; Λ) f t, I = E( t t) 8 / 10

9 Dynamic Factor Model (DFM) Traditional DFM(Engle and Watson, ) y t = Λf t + e t N (0, R) f t+1 = Af t + u t N (0, Q) ˆf t+1 = Aˆf t + AG t (y t Λˆf t ) E[(ˆf t+1 f t+1 )(ˆf t+1 f t+1 ) ] := P t+1 Obs.-driven DFM (Creal, Schwaab, Koopman and Lucas, 2014) y t = Λˆf t + e t N (0, R) ˆf t+1 = Aˆf t + α(λ R 1 Λ) 1 Λ R 1 (y t Λˆf t ) = ˆf t+1 = Aˆf t + α(ˆft ˆf t ) where ˆft = (Λ R 1 Λ) 1 Λ R 1 y t if α = A then ˆf f +1 = Aˆft 9 / 10

10 Conclusions Interesting, Flexible and Useful Approximation of Linear State Space Models Interesting, Flexible and Useful Approximation of Non-Linear State Space Models Very intriguing and Interesting Research Agenda Open Questions (at least for me) Hard to interpret in some instance When is the approximation reasonable? How does it compare with other approximations (e.g. forgetting factor, Koop and Korobolis, 2013) Does it work in forecasting? (See Opschoor et al, 2014) Are computational gains going to remain relevant? Strongly suggest to read the papers 10 / 10

Generalized Autoregressive Score Models

Generalized Autoregressive Score Models by: Drew Creal, Siem Jan Koopman, André Lucas To capture the dynamic behavior of univariate and multivariate time series processes, we can allow parameters to be