I want to fit a SARIMA model on a time series, daily sampled.
The time series has a 7 days seasonality and, over a moving 7 days window, the variance increases with the mean.
With a hyperparameters grid-search I defined the best model as ARIMA(1, 1, 2)(1, 0, 1)[7], where d=1 is adequate to deal with the non linear trend.
On original data, both Breusch-Pagan test and Goldfeld-Quandt test (between first third and last third of sample, see https://www.statsmodels.org/stable/generated/statsmodels.tsa.statespace.mlemodel.MLEResults.test_heteroskedasticity.html#statsmodels.tsa.statespace.mlemodel.MLEResults.test_heteroskedasticity) return p-values far less then 0.01
Taking the log of observations, there is a very weak correlation between moving average and variance and Breusch-Pagan test returns a p-value far greater than 0.1 but Goldfeld-Quandt test is still
In general the Goldfeld-Quandt (GQ) test is a very old fashion one (think that nowadays most of econometrics textbooks do not treat it anymore) so the Breush-Pagan (BP) is a better alternative. Moreover, the BP test is also Better suited for time series data.
Max -
In an update to https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, I noted that hypothesis testing doesn't help you deal with heteroscedasticity for finite populations, and in another update I noted that transformations are not ideal either. They can damage interpretability, and generally do not completely resolve the issue. In finite populations we should expect heteroscedasticity (see the "Essential Heteroscedasticity" paper for that project linked above) and can measure it ("Estimating the Coefficient of Heteroscedasticity," using the Excel tool also provided). Time series would be different and more complex, more often involving autocorrelation. Anyway, I would rather model the actual situation than try to transform and then guess from a nebulous hypothesis test if I've now got something close enough.
Best wishes - Jim
James R Knaub really interesting thank you. I've downloaded and I'll carefully read your papers. From preliminary tests, I indeed found that (in this specific case) taking the log of the observations, to try to solve heteroscedasticity, increases both the prediction errors for higher values and confidence intervals along the whole time series. So maybe it's actually better to use the original observations rather than transform them.
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