Auto Correlation §

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The Autocovariance Function ( ACVF ) of {}at lag h is: §

The Autocorrelation Function (ACF) of {}at lag h is: §

white noises for ACVF and ACF: §

ACVF: 0 ACF: 0

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Example 1(MA(1)):

• Mean is
• Variance is
• Autocorrelation function (ACF) is:

Suppose that an MA(1) model is , thus the coefficient . the theoretical ACF is given by

note: That the only nonzero value in the theoretical ACF is for lag 1. All other autocorrelations are 0. Thus a sample ACF with a significant autocorrelation only at lag 1 is an indicator of a possible MA(1) model. §

plot will be:

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Example 2(MA(2)):

• Mean is
• Variance is
• Autocorrelation function (ACF) is:

Suppose that an MA(2) model is , where ., thus the coefficient .Because this is an MA(2), the theoretical ACF will have nonzero values only at lags 1 and 2. the theoretical ACF is given by

and

A plot of the theoretical ACF follows:

note: The only nonzero values in the theoretical ACF are for lags 1 and 2. Autocorrelations for higher lags are 0. So, a sample ACF with significant autocorrelations at lags 1 and 2, but non-significant autocorrelations for higher lags indicates a possible MA(2) model. §

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Convariance property : §

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#timeseries#diagnois