statsmodels.tsa.exponential_smoothing.ets.ETSResults.test_heteroskedasticity¶

ETSResults.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{‘breakvar’, 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.

alternativestr, ‘increasing’, ‘decreasing’ or ‘two-sided’

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

use_fbool, 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:¶

outputndarray

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 \(d\) = max(loglikelihood_burn, nobs_diffuse)` (usually corresponding to diffuse initialization under either the approximate or exact approach).

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

TODO

• Allow specification of \(h\)

References

[1]

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