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MULTIVARIATE TIME SERIES & FORECASTING

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s : autocovariance function of the individual time series sii

´

´

21

21

ss

ss

VsVs

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Vector ARMA models

tt CovE &0

if the roots of the equation

are all greater than 1 in absolute value

Then : infinite MA representation

Stationarity

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Invertibility

if the roots of the equation

are all greater than 1 in absolute value

ARMA(1,1)

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If case of non stationarity: apply differencing of appropriate degree

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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.

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Vector AR (VAR) modelsVector AR(p) model

For a stationary vector AR process: infinite MA representation

tt BY

1

221

B

BBIB

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The Yule-Walker equations can be obtained from the first p equationsThe autocorrelation matrix of Var(p) : decaying behavior following a mixture of exponential decay & damped sinusoid

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Autocovariance matrix

21'2

10 VV

s

V:diagonal matrix

The eigenvalues of determine the behavior of the autocorrelation matrix

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2.Identify model - Sample ACF plots- Cross correlation of the time series

Example

1. Data : the pressure reading s at two ends of an industrial furnaceExpected: individual time series to be autocorrelated & cross-correlatedFit a multivariate time series model to the data

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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

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VAR(1) provided a good fit

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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.)

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Not constant variance

Consider AR(p)= model

Errors: uncorrelated, zero mean noise with changing variance

Model et2 as an AR(l) process

white noise with zero mean & constant variancet

et: Autoregressive conditional heteroscedastic process of order l –ARCH(l)

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Generalise ARCH model

Consider the error:

et: Generalised Autoregressive conditional heteroscedastic process of order k and l–GARCH(k,l)

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S& P index-Initial data: non stationary-Log transformation of the data-First differences of the log data

Mean stableChanges in the variance

ACF& PACF:No autocorrelation left in the data

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