MULTIVARIATE TIME SERIES & FORECASTING 1 2 Vector ARMA models - PowerPoint PPT Presentation

MULTIVARIATE TIME SERIES & FORECASTING 1

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Vector ARMA models          E 0 & Cov t t Stationarity if the roots of the equation are all greater than 1 in absolute value Then : infinite MA representation 3

Invertibility if the roots of the equation are all greater than 1 in absolute value ARMA(1,1) 4

If case of non stationarity: apply differencing of appropriate degree 5

VAR models (vector autoregressive models) are used for multivariate time series. The structure is that each variable is a linear function of past lags of itself and past lags of the other variables. 6

Vector AR (VAR) models Vector AR(p) model For a stationary vector AR process: infinite MA representation   t      Y B t          2  B I B B 1 2        1 B 7

( ) = m E Y t ( ) = 0, foranys > 0 Cov e t , Y t - s ( ) = Cov ( e t , e t ) = S Cov e t , Y t p å ( ) = Cov ( Y t - s , Y t ) = ( ) G s F i Cov Y t - s , Y t - i ' i = 1 p å ( ) F i = G s - i ' i = 1 p å ( ) = ( ) F i G 0 G i + S ' i = 1 The Yule-Walker equations can be obtained from the first p equations The autocorrelation matrix of Var(p) : decaying behavior following a mixture of exponential decay & damped sinusoid 8

Autocovariance matrix      1 1    s ' 0 V 2 V 2 V:diagonal matrix The eigenvalues of determine the behavior of the  autocorrelation matrix 9

1. Data : the pressure reading s at two ends of an industrial furnace Expected: individual time series to be autocorrelated & cross-correlated Fit a multivariate time series model to the data 2.Identify model - Sample ACF plots - Cross correlation of the time series 10

Exponential decay pattern: autoregressive model & VAR(1) or VAR(2) Or ARIMA model to individual time series & take into consideration the cross correlation of the residuals 11

VAR(1) provided a good fit 12

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ARCH/GARCH Models (autoregressive conditionally heteroscedastic) -a model for the variance of a time series. -used to describe a changing, possibly volatile variance. -most often it is used in situations in which there may be short periods of increased variation. (Gradually increasing variance connected to a gradually increasing mean level might be better handled by transforming the variable.) 14

Not constant variance Consider AR(p)= model Errors: uncorrelated, zero mean noise with changing variance 2 as an AR(l) process Model e t  white noise with zero mean & constant variance t e t : Autoregressive conditional heteroscedastic process of order l – ARCH(l) 15

Generalise ARCH model Consider the error: e t : Generalised Autoregressive conditional heteroscedastic process of order k and l – GARCH(k,l ) 16

S& P index -Initial data: non stationary -Log transformation of the data -First differences of the log data Mean stable Changes in the variance No autocorrelation left in the data 17

ACF & PACF of the squared differences: ARCH (3) model 18