Portmanteau Test

A Portmanteau Test is a class of statistical tests for a group of autocorrelations.

• Example(s):
  • Box-Pierce Test.
  • Ljung-Box Test.
• See: Unit Root Test, Dickey–Fuller Test, Phillips–Perron Test, KPSS Test, Breusch–Godfrey Test, Ljung–Box Test, Durbin–Watson Test, Augmented Dickey–Fuller Test, Time Series Analysis.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Portmanteau_test Retrieved 2016-08-07
  • A portmanteau test is a type of statistical hypothesis test in which the null hypothesis is well specified, but the alternative hypothesis is more loosely specified. Tests constructed in this context can have the property of being at least moderately powerful against a wide range of departures from the null hypothesis. Thus, in applied statistics, a portmanteau test provides a reasonable way of proceeding as a general check of a model's match to a dataset where there are many different ways in which the model may depart from the underlying data generating process. Use of such tests avoids having to be very specific about the particular type of departure being tested.

Examples: In time series analysis, two well-known versions of a portmanteau test are available for testing for autocorrelation in the residuals of a model: it tests whether any of a group of autocorrelations of the residual time series are different from zero. This test is the Ljung–Box test, which is an improved version of the Box–Pierce test,having been devised at essentially the same time; a seemingly trivial simplification (omitted in the improved test) was found to have a deleterious effect. This portmanteau test is useful in working with ARIMA models.

In the context of regression analysis, including regression analysis with time series structures, a portmanteau test has been devised, which allows a general test to be made for the possibility that a range of types nonlinear transformations of combinations of the explanatory variables should have been included in addition to a selected model structure.

• (OTexts, 2016) ⇒ https://www.otexts.org/fpp/2/62016-08-07
  • Recall that [math]\displaystyle{ r_k }[/math] is the autocorrelation for lag [math]\displaystyle{ k }[/math]. When we look at the ACF plot to see if each spike is within the required limits, we are implicitly carrying out multiple hypothesis tests, each one with a small probability of giving a false positive. When enough of these tests are done, it is likely that at least one will give a false positive and so we may conclude that the residuals have some remaining autocorrelation, when in fact they do not.

In order to overcome this problem, we test whether the first hh autocorrelations are significantly different from what would be expected from a white noise process. A test for a group of autocorrelations is called a portmanteau test, from a French word describing a suitcase containing a number of items.

One such test is the Box-Pierce test based on the following statistic

[math]\displaystyle{ Q=T\sum_{k=1}^{h}r^2_k }[/math]

where [math]\displaystyle{ h }[/math] is the maximum lag being considered and [math]\displaystyle{ T }[/math] is number of observations. If each [math]\displaystyle{ r_k }[/math] is close to zero, then [math]\displaystyle{ Q }[/math] will be small. If some [math]\displaystyle{ r_k }[/math] values are large (positive or negative), then [math]\displaystyle{ Q }[/math] will be large. We suggest using [math]\displaystyle{ h=10 }[/math] for non-seasonal data and [math]\displaystyle{ h=2m }[/math]for seasonal data, where [math]\displaystyle{ m }[/math] is the period of seasonality. However, the test is not good when [math]\displaystyle{ h }[/math] is large, so if these values are larger than [math]\displaystyle{ T/5 }[/math], then use [math]\displaystyle{ h=T/5 }[/math].

A related (and more accurate) test is the Ljung-Box test based on

[math]\displaystyle{ Q^∗=T(T+2)\sum{k=1}{h}(T−k)^−1r^2_k }[/math]

Again, large values of [math]\displaystyle{ Q^∗ }[/math] suggest that the autocorrelations do not come from a white noise series.

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