VARMAXResults.test_heteroskedasticity()

statsmodels.tsa.statespace.varmax.VARMAXResults.test_heteroskedasticity

VARMAXResults.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:

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.

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 [R115] for the above formula and discussion, as well as additional details.

TODO

Allow specification of $h$

References

[R115]

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

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© 2006 Jonathan E. Taylor
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