Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0
Syntax
C# |
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[SerializableAttribute] public class AutoCorrelation |
Visual Basic (Declaration) |
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<SerializableAttribute> _ Public Class AutoCorrelation |
Visual C++ |
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[SerializableAttribute] public ref class AutoCorrelation |
Remarks
AutoCorrelation estimates the autocorrelation function of a stationary time series given a sample of n observations for .
Let
be the estimate of the mean of the time series where
where K = maximumLag. Note that is an estimate of the sample variance. The autocorrelation function is estimated by
Note that by definition.
The standard errors of sample autocorrelations may be optionally computed according to the GetStandardErrors method argument stderrMethod. One method (Bartlett 1946) is based on a general asymptotic expression for the variance of the sample autocorrelation coefficient of a stationary time series with independent, identically distributed normal errors. The theoretical formula is
where assumes is unknown. For computational purposes, the autocorrelations are replaced by their estimates for , and the limits of summation are bounded because of the assumption that for all such that .
A second method (Moran 1947) utilizes an exact formula for the variance of the sample autocorrelation coefficient of a random process with independent, identically distributed normal errors. The theoretical formula is
where is assumed to be equal to zero. Note that this formula does not depend on the autocorrelation function.
The method GetPartialAutoCorrelations returns the estimated partial autocorrelations of the stationary time series given K = maximumLag sample autocorrelations for k=0,1,...,K. Consider the AR(k) process defined by
where denotes the j-th coefficient in the process. The set of estimates for k = 1, ..., K is the sample partial autocorrelation function. The autoregressive parameters for j = 1, ..., k are approximated by Yule-Walker estimates for successive AR(k) models where k = 1, ..., K. Based on the sample Yule-Walker equations a recursive relationship for k=1, ..., K was developed by Durbin (1960). The equations are given by
This procedure is sensitive to rounding error and should not be used if the parameters are near the nonstationarity boundary. A possible alternative would be to estimate for successive AR(k) models using least or maximum likelihood. Based on the hypothesis that the true process is AR(p), Box and Jenkins (1976, page 65) note
See Box and Jenkins (1976, pages 82-84) for more information concerning the partial autocorrelation function.