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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) models
Vector 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
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VVs
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