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Time Series Analysis - ARIMA Models - Inverse Autocorrelations

[Home] [Up] [Basics] [AR(1) process] [AR(2) process] [AR(p) process] [MA(1) process] [MA(2) process] [MA(q) process] [ARMA(1,1) process] [ARMA(p,q) process] [Wold's decomp.] [Non stationarity] [Differencing] [Behavior] [Unit Root Tests] [Inverse Autocorr.]

n. Inverse autocorrelations

The inverse autocorrelations were introduced by Cleveland (Cleveland 1972) and are important in ARIMA model identification and estimation.

Cleveland's original definition was related to the frequency domain. Let W(t) be a stationary time series with spectral density s. It is supposed that 1/s is integratable on [0,1].

The autocovariance and autocorrelation function of W(t) are given by


The inverse autocovariance and autocorrelation function is defined by


which is the same as the autocovariance and autocorrelation function of the model


The use of this inverse autocorrelation function (IACF) for identification has advantages over the traditional PACF in the case that W(t) has an MA structure. In fact the IACF can be interpreted just as a PACF, except that it (sometimes) gives a much clearer picture for diagnostic purposes.

Note that the (I)ACF and (P)ACF are not defined for continuous processes due to the fact that 1/s cannot be integrated. In other words: since the (I)ACF and (P)ACF are dependent on the sampling rate of the time series (generated by a continuous process), they are somewhat artificial in nature.

Cleveland proposed two estimation procedures. The first algorithm involves the estimation the parameters in the K-th order autoregressive regression


for which the spectral densities are



The second algorithm involves the use of smoothed periodogram estimation of the IACF which is beyond the scope of this work. In modern literature, many other estimation algorithms have been suggested but since they do not enhance the understanding of the IACF, they are omitted.

The "enlightening" work of Chatfield enhances the understanding of the IACF by using the Mohr-equation


where G*(z) is the inverse autocovariance generating function (IACVGF), G(z) is the autocovariance generating function


The inverse autocorrelation generating function (IACGF) is


whereas its the autocorrelation generating function (ACGF) is given by


It is obvious to see that (V.I.1-248) multiplied with (V.I.1-247) reduces to


but surprisingly it can be found that


except if W(t) is a white noise process.

This is perhaps not really surprising, since if

would be true in general, there would be no gain in using the IACF in stead of the PACF.

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AR(1) process
AR(2) process
AR(p) process
MA(1) process
MA(2) process
MA(q) process
ARMA(1,1) process
ARMA(p,q) process
Wold's decomp.
Non stationarity
Unit Root Tests
Inverse Autocorr.
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