I've read Box & Jenkins 5th edn, there is mention of ARCH and GARCH in the later chapters but they are mentioned in the model checking. Then in Hamilton, there is no mention at all. If I use an ARIMA model for hypothesis tests (assuming conditional LS and MLE for estimating the coefficients), then would heteroskedasticity be a problem, as in resulting in incorrect standard errors of the coefficients?
Yes, the lack of heteroskedasticity, i.e., a constant variance, is an assumption of the ARIMA methodology. In contrast, the ARCH methodology estimates a model in which the conditional variance varies over time. It can vary because periods of high and low volatility are bunched together, which can be modeled using any of the following methods: ARCH( the classical version), NARCH (nonlinear ARCH), and AARCH ( an ARCH specification that allows for asymmetry ). Some of my other favorites are TARCH ( a specification that permits a threshold) and EARCH. It should also be noted that it can be useful to include the structural drivers of heteroskedasticity in your model. Moreover, it is sometimes useful to specify a model in which the conditional variance is recognized as having implications for the mean.
Here is a link to a recent publication that may be useful:Article CO 2 has significant implications for hourly ambient tempera...
ARMA (and therefore ARIMA) modeling is based on white noise. White noise is a time series with independent observations with mean zero and constant variance. I would suggest that you look again at Hamilton. This point is made many times in Chapter 3 and elsewhere in Hamilton See, for example, the first sentence in Chapter 5. The white noise specification is also basic in Box, Jenkins, Reinsel, and Ljung and is used many times.
The heteroscedasticity finding indicates that a better model can be found. This may be an alternative ARIMA model or you may add an explanatory variable or perhaps model the heteroskedastic using volatility modelling. No one can advise you, at his stage, without a much greater knowledge of your series. If your chosen model has been determined using some search criterion (e.g.AIC, SBC, or ...) then you should consider revising the model.
Remember that an ARIMA model is only an approximation to the underlying model. It is not the true model. The usefulness of the model should be determined by some form of cross-validation.
For the record, estimating an ARMA or ARIMA model and not attaining white noise in the residuals is possible. What this means is that the model is not fully adequate. However, the model may nevertheless be useful.
With respect to heteroscedasticity, please keep in mind that an ARIMA model presumes that the error terms are homoscedastic.
I can add that it is possible and often useful to consider ARMA models with time-dependent coefficients and deterministic heteroscedasticity. See for example Chapter ARIMA Models with Time-Dependent Coefficients: Official Stat...
.In the latter case, robust standard errors should be used. They are based on a sandwich estimator of the asymptotic covariance matrix of the estimators, involving not only the Hessian but also on the product of gradient.
This being said, in the case of moderate heteroscedasticity, the standard errors are not necessarily bad but the problem will be with the forecast intervals because the residual variance will be artificially increased.
As I indicated earlier, heteroscedasticity is sometimes too important to ignore. While modeling the structural heteroscedasticity can be useful, it is not always sufficient.
Here is a link that may merit your consideration: https://pages.stern.nyu.edu/~churvich/Forecasting/Handouts/ARCH-M.pdf
It makes the point that "the ARCH-M (ARCH
in mean) model provides an explicit link between the risk (conditional volatility) and the best forecast of a time series. "
I am currently using this approach in estimating a model to improve the predictability of wind energy generation.
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