Portmanteau test

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: "r"_"k" = 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 test 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.Fact|date=July 2008

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.