Portmanteau test

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In statistics, a portmanteau test tests whether any of a group of autocorrelations of a time series are different from zero. Among portmanteau tests are both the Ljung-Box test and the (now obsolete) Box-Pierce test. The portmanteau test is useful in working with ARIMA models.

The Ljung-Box test statistic is calculated as Q=T(T+2)\sum_{k=1}^s r_k^2/(T-k).

T = number of observations
s = number of coefficients to test autocorrelation
rk = autocorrelation coefficient (for lag k)
Q = portmanteau test statistic. If the sample value of Q exceeds the critical value of a chi-square distribution with s degrees of freedom, then at least one value of r is statistically different from zero at the specified significance level. (The Null Hypothesis is that none of the autocorrelation coefficients up to lag s are different from zero.)

The Ljung-Box (1978) test is an improvement over the Box-Pierce (1970) test, whose statistic was

Q = T \sum^s_{k=1} r^2_k.

The problem with the Box-Pierce statistic was bad performance in small samples. The Ljung-Box statistic is better for all sample sizes including small ones.

Word origin. In French the word portmanteau refers to a coat rack. Just as a coat rack can hold many items of clothing (each on its own hook), a portmanteau test can be used to test multiple autocorrelation coefficients for significance.

[edit] References

  • Ljung, G. M. and Box, G. E. P., "On a measure of lack of fit in time series models." Biometrika 65 (1978): 297-303.
  • Box, G. E. P. and Pierce, D. A., "Distribution of the Autocorrelations in Autoregressive Moving Average Time Series Models", Journal of American Statistical Association, 65 (1970): 1509-1526.
  • Enders, W., "Applied econometric time series", John Wiley & Sons, 1995, p. 86-87.


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