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MULTIVARIATE TIME SERIES & FORECASTING
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MULTIVARIATE TIME SERIES & FORECASTING 1 2 Vector ARMA models - - PowerPoint PPT Presentation
MULTIVARIATE TIME SERIES & FORECASTING 1 2 Vector ARMA models - - PowerPoint PPT Presentation
MULTIVARIATE TIME SERIES & FORECASTING 1 2 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
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Vector ARMA models
t t
Cov E &
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
t
t
B Y
1 2 2 1
B B B I B
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E Yt
( ) = m
Cov et,Yt-s
( ) = 0, foranys > 0
Cov et,Yt
( ) = Cov(et,et) = S
G s
( ) = Cov(Yt-s,Yt) =
Cov Yt-s,Yt-i
( )
i=1 p
å
Fi
'
= G s -i
( )Fi
' i=1 p
å
G 0
( ) =
G i
( )Fi
' i=1 p
å
+ S
The Yule-Walker equations can be obtained from the first p equations The autocorrelation matrix of Var(p) : decaying behavior following a mixture
- f exponential decay & damped sinusoid
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Autocovariance matrix
2 1 ' 2 1
V V
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
- 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
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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 et
2 as an AR(l) process
white noise with zero mean & constant variance
t
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 stable Changes in the variance No autocorrelation left in the data
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