statsmodels.tsa.statespace.sarimax.SARIMAXResults.test_heteroskedasticity

SARIMAXResults.test_heteroskedasticity(method, alternative='two-sided', use_f=True)

Test for heteroskedasticity of standardized residuals

Tests whether the sum-of-squares in the first third of the sample is significantly different than the sum-of-squares in the last third of the sample. Analogous to a Goldfeld-Quandt test.

Parameters:

method : string {‘breakvar’} or None

The statistical test for heteroskedasticity. Must be ‘breakvar’ for test of a break in the variance. If None, an attempt is made to select an appropriate test.

alternative : string, ‘increasing’, ‘decreasing’ or ‘two-sided’

This specifies the alternative for the p-value calculation. Default is two-sided.

use_f : boolean, optional

Whether or not to compare against the asymptotic distribution (chi-squared) or the approximate small-sample distribution (F). Default is True (i.e. default is to compare against an F distribution).
Returns:

output : array

An array with (test_statistic, pvalue) for each endogenous variable. The array is then sized (k_endog, 2). If the method is called as het = res.test_heteroskedasticity(), then het[0] is an array of size 2 corresponding to the first endogenous variable, where het[0][0] is the test statistic, and het[0][1] is the p-value.

Notes

The null hypothesis is of no heteroskedasticity. That means different things depending on which alternative is selected:

  • Increasing: Null hypothesis is that the variance is not increasing throughout the sample; that the sum-of-squares in the later subsample is not greater than the sum-of-squares in the earlier subsample.
  • Decreasing: Null hypothesis is that the variance is not decreasing throughout the sample; that the sum-of-squares in the earlier subsample is not greater than the sum-of-squares in the later subsample.
  • Two-sided: Null hypothesis is that the variance is not changing throughout the sample. Both that the sum-of-squares in the earlier subsample is not greater than the sum-of-squares in the later subsample and that the sum-of-squares in the later subsample is not greater than the sum-of-squares in the earlier subsample.

For h = [T/3], the test statistic is:

H(h) = \sum_{t=T-h+1}^T \tilde v_t^2 \Bigg / \sum_{t=d+1}^{d+1+h} \tilde v_t^2

where d is the number of periods in which the loglikelihood was burned in the parent model (usually corresponding to diffuse initialization).

This statistic can be tested against an F(h,h) distribution. Alternatively, h H(h) is asymptotically distributed according to \chi_h^2; this second test can be applied by passing asymptotic=True as an argument.

See section 5.4 of [R89] for the above formula and discussion, as well as additional details.

TODO

  • Allow specification of h

References

[R89](1, 2) Harvey, Andrew C. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.