/Statsmodels

# statsmodels.tsa.statespace.structural.UnobservedComponentsResults.test_heteroskedasticity

UnobservedComponentsResults.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. The null hypothesis is of no heteroskedasticity.

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). output – 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 is an array of size 2 corresponding to the first endogenous variable, where het is the test statistic, and het is the p-value. array

#### 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  for the above formula and discussion, as well as additional details.

TODO

• Allow specification of $$h$$

#### References

  Harvey, Andrew C. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.