Portmanteau test

Portmanteau tests are statistical tests that can be used to test several autocorrelation coefficients to determine whether they are significantly different from zero. This is especially important when checking the autocorrelation of the residuals as part of the diagnostic phase of a time series analysis .

Portmanteau tests are purely significance tests. You are not testing against a clearly formulated counter-hypothesis .

The test statistic is called the Q statistic .

Box / Pierce

The original version of the test is from Box / Pierce (1970).

The hypotheses for this test are:

{\ displaystyle H_ {0} \ colon \ rho _ {1} ({\ hat {Z_ {t}}}) = \ dots = \ rho _ {K} ({\ hat {Z_ {t}}}) = 0 \,}

and

{\ displaystyle H_ {1} \ colon \ rho _ {l} ({\ hat {Z_ {t}}}) \ neq 0 \,}

applies to at least one l .

It is the (empirical) of the auto-correlation series to lag (the time shift) and the number of test autocorrelations. ${\ displaystyle \ rho _ {l} ({\ hat {Z_ {t}}})}$ ${\ displaystyle ({\ hat {Z_ {t}}})}$ ${\ displaystyle l}$ ${\ displaystyle K}$

The test statistic is here

{\ displaystyle Q_ {BP} = T \ sum _ {l = 1} ^ {K} \ rho _ {l} ^ {2} ({\ hat {Z_ {t}}}) \ ,,}

where is the scope of the data set. ${\ displaystyle T}$

This test variable is distributed with degrees of freedom under the null hypothesis χ ² ; can therefore be discarded if ${\ displaystyle \ nu = Kpq}$ ${\ displaystyle H_ {0}}$

{\ displaystyle Q_ {BP}> \ chi _ {\ nu; 1- \ alpha} ^ {2} \ ,.}

Choosing a suitable value for is problematic. If it is too low, the asymptotics of the approximation does not apply. Even too big will have undesirable effects. The following rule of thumb can be used to determine: ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle \ chi ^ {2}}$ ${\ displaystyle K}$ ${\ displaystyle K}$

{\ displaystyle K \ approx {2} {\ sqrt {T}} \ ,.}

Ljung / Box

Since the Box-Pierce test only works satisfactorily for long time series with more than 100 time series values, a modified test statistic is used by Ljung / Box (1978). Here, T through T (T + 2) / (TK) replaced. The test statistic results:

${\ displaystyle Q_ {LB} = T (T + 2) \ sum _ {l = 1} ^ {K} {\ frac {1} {Tl}} \ rho _ {l} ^ {2} ({\ hat {Z_ {t}}}).}$

literature

Box, GEP ; Pierce, DA : Distribution of Residual Correlations in Autoregressive-Integrated Moving Average Time Series Models; Journal of the American Statistical Association , Vol. 65, 1509-1526, 1970, doi : 10.1080 / 01621459.1970.10481180 , JSTOR 2284333 .
Ljung, GM ; Box, GEP: On a Measure of Lack of Fit in Time Series Models; Biometrika 65, No. 2, 297-303, 1978, doi : 10.1093 / biomet / 65.2.297 .