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Time Series Analysis - ARIMA models - Model Checking

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V.I.3 Univariate Stochastic ARIMA Model Checking

The techniques used in model checking are not different from those used in model identification. In fact, the estimated residual series (from an estimated ARIMA model) is analyzed: it is investigated whether an ARMA pattern can be recognized in the residual ACF and PACF. In case the residuals contain an ARMA structure, the model should be modified accordingly: otherwise the assumption that

Time Series Analysis - ARIMA models - Model Checking

is violated.

In order to check whether the ACF and PACF spikes are significantly different from zero, several Chi-square-based tests exist. In the recent literature many likewise, and alternate test procedures have been proposed:

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the reversed residuals approach (LAWRANCE and LEWIS 1990);

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the score or Lagrange Multiplier tests (LJUNG 1988);

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the approximation of simultaneous confidence intervals for the ACF (HOSKING and RAVISHANKER 1990); etc...

Another approach, is to use the standardized cumulative periodogram of the residuals, due to Bartlett.

An unbiased estimate of the cumulative periodogram is given by

(V.I.3-1)

Other diagnostic checks could be considered, such as the SMP for residual heteroskedasticity checking, histograms (or suspended root displays; see MILLS 1990) for normality checks, R-square for interpolation performance, the Durbin-Watson statistic for first order autocorrelation, the residual mean and its standard error for testing whether E(e) = 0, runs-tests for autocorrelation, and many others.
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