Multivariate forecasting with VAR models

Transcription

1 Multivariate forecasting with VAR models Franz Eigner University of Vienna UK Econometric Forecasting Prof. Robert Kunst 16th June 2009

2 Overview Vector autoregressive model univariate forecasting multivariate forecasting model-free: models: feedback? no yes smoothing filtering ARIMA, GARCH open loop system (single equation) closed loop system (multiple equation) multiple regression transfer function stationary: VAR,SVAR nonstationary: VEC,SVEC

3 Definition of VAR(p) Stationary vector autoregressive process Definition of VAR Forecasting Extensions of the VAR A VAR consists of a set of K-endogenous variables y t = (y 1t,...,y kt,...,y Kt ) for k = 1,...,K A VAR(p) process is defined as y t = Φ 1 y t Φ p y t p + u t where Φ i are (KxK) coefficient matrices for i = 1,...,p and u t is K-dimensional white noise. The VAR(p) process is stable (stationary series), if det(i K Φ 1 z... Φ p z p ) 0 for z 1 (1)

4 Definition of bivariate VAR(1) Stationary bivariate vector autoregressive process Definition of VAR Forecasting Extensions of the VAR Bivariate VAR(1) process y t = [ Φ 1 y t 1 + u t ] with ut T φ11 φ = (u 1t,u 2t ) and Φ 1 = 12 φ 21 φ 22 consists of two equations: y 1t = φ 11 y 1,t 1 + φ 12 y 2,t 1 + u 1t y 2t = φ 21 y 1,t 1 + φ 22 y 2,t 1 + u 2t!11 Y1,t-1 Y1,t... Y2,t-1!12!22!21 Y2,t... Concept of Granger causality

5 Definition of VAR Forecasting Extensions of the VAR Specification, estimation and structural analysis Finding optimal time lag p information criterion Model specification pitfall Number of parameters increases tremendously with more lags Coefficients are estimated by OLS on each equation Structural analyses 1 Granger causality 2 Impulse response analysis 3 Forecast error variance decomposition

6 Forecasting Vector autoregressive model Definition of VAR Forecasting Extensions of the VAR naive forecast: Minimum mean square error (MMSE) For a VAR(1) process y t = Φ 1 y t 1 + u t one-step-ahead forecast: ŷ N (1) = ˆΦ 1 y N two-step ahead forecast: ŷ N (2) = ˆΦ 1 ŷ N (1) = ˆΦ 2 1 y N Forecasts for horizons h are therefore obtained with ŷ N (h) = ˆΦ h 1y N

7 Extensions Vector autoregressive model Definition of VAR Forecasting Extensions of the VAR VARMA, VMA VARX, VARMAX (including exogenous variables) imposing more restrictions: (model reduction) Structural VAR (SVAR) Bayesian VAR (BVAR)

8 VAR model and Cointegration VAR model and cointegration before: stationary time series ( stability condition) now: nonstationary data one could difference the data, but not adequate in the presence of cointegration. Cointegration y t I (d) is cointegrated, if there exists a kx1 fixed vector β 0, so that βy t is integrated of order < d. Assume: y 1t and y 2t are I (1). They are cointegrated when a linear combination of y 1t and y 2t exists with (y 1t βy 2t ) I (0)

9 Bivariate cointegrated VAR(2) VAR model and cointegration Consider the bivariate VAR(2) y t = Φ 1 y t 1 + Φ 2 y t 2 + u t with the matrix polynomial for z=1 ( stability condition (1)) Φ(1) = (I Φ 1 Φ 2 ) = Π rank(π) equals the cointegration rank of the system y t 0... no cointegration ( difference VAR) 1... one cointegrating vector ( VECM) 2... process is stable ( VAR)

10 VAR model and cointegration (VECM) Implementing cointegration in a VAR(2) using the VECM form: y t = y t y t 1 = Πy t 1 + Γ 1 y t 1 + u t where Γ 1 = Φ 2 is the transition matrix and Π = α β holds α as the»loading matrix«(speed of adjustment) β consisting the independent cointegrating vector βy t 1 as the lagged disequilibrium error Πy t 1 as the error correction term (long-run part) (to catch the idea: consider bivariate VAR(1) equation: y 1t = α 1 (y 1,t 1 βy 2,t 1 ) + u 1t with long-run equilibrium y 1t = βy 2t ) estimation by reduced rank regression and forecast as in VAR

11 Applications of VAR/VEC VAR model and cointegration VAR economics and finance growth rates of macroeconomic time series and some asset returns VEC economics - Money demand: money, income, prices and interest rates - Growth theory: income, consumption and investment - Fisher equation: nominal interest rates and inflation finance cointegration between prices of the same asset trading on different markets due to arbitrage