# GARCH diagnostics: autocorrelation in standardised residuals and poor results of Goodness-of-Fit Test

I am trying to fit best ARMA - GARCH model using rugarch in Python on financial data 5 min returns series. I am using last 10k observations for this purpose. The goal is to predict next return and its confidence intervals. The best fitting model according to AIC/BIC is standard GARCH ARMA(1,2)-GARCH(1,1) with Student's t distribution. When fitting the model I faced with two problems:

1. Weighted Ljung-Box Test p-values for standardised residuals below 0.05, suggesting they are correlated. No matter what I do (changing ARMA and GARCH orders, error distribution (except GED - it can't converge) and GARCH type - e.g. iGARCH, eGARCH etc.), I can't fix it. However, squared standardised residuals are not correlated, suggesting variance model captures data correctly.
2. Adjusted Pearson Goodness-of-Fit Test shows poor fit (p - values below 0.05), suggesting the chosen error distribution does not match the empirical one. But lowering sample size to 5k observations solves the problem, suggesting there was a structural break?

The questions are:

1. Would these two problems affect only forecast confidence intervals or forecast itself as well?
2. Is there any way to approach them other than discussed above?

As far as I know these two problems do not cause biased estimators, so mean model should work well, but the forecast variance becomes unreliable. I read carefully this thread, but it does not contain answer to my question.

First off, your sample is very large, so any test you conduct will likely have high power and will detect even small departures from the null hypothesis. Whether a departure is economically significant in addition to being statistically significant is another question. You may want to look at effect size (e.g. estimated autocorrelation coefficients) to evaluate that.

Would these two problems affect only forecast confidence intervals or forecast itself as well?

They would affect both the point forecast and the forecast interval.

Autocorrelation in standardized residuals suggests there is information in them that could be used for forecasting (whether using a point or an interval forecast). If you could estimate with high precision a model for the autocorrelation of the standardized residuals, you would change the point and interval forecasts of your ARMA-GARCH model accordingly. The interval forecast would have to be adjusted accordingly. In this sense the two problems you are facing indicate inadequacy of the point and interval forecasts.
Of course, it would be more natural to change the original ARMA-GARCH model than to add another model for the standardized residuals of the latter. If you were to find such a model with standardized residuals being uncorrelated and matching the distributional assumption, the point and interval forecasts from it model would most likely differ from the ones produced by your current model.

Mismatch between the hypothesized and the estimated distribution of standardized residuals would also have an effect on both the point and the interval forecast. If you were to find another distribution that yields a match, the maximum likelihood estimators of the model coefficients would be different, and so the point and the interval forecasts would be different because of their location and the shape of the distribution would introduce another change in the forecast interval.

Is there any way to approach them other than discussed above?

You can continue looking for other specifications via changing the lag orders of ARMA and GARCH, the type of GARCH and the distributional assumption, but this is what you have already tried. Another option would be to rely on a quasi MLE (QMLE) estimator based on the normal distribution. This would account for the mismatch in the distributional assumption by adjusting the standard errors of the coefficients. This would directly affect the forecast interval. The point forecast would also be affected as discussed in the paragraph above. Since the location of the forecast interval moves together with the point forecast, the former would be shifted accordingly.

[L]owering sample size to 5k observations solves the problem, suggesting there was a structural break?

This might be the case. The break might be abrupt or it may be a gradual shift (a smooth transition). It may be worth investigating, and as a result you may end up with different models for different time periods.

• Richard, thanks a lot for such a comprehensive answer! I lowered the sample size (up to the point where Nyblom stability test indicated no structural breaks), and solved second problem (matching hypothetical and actual distributions). However, problem #1 still persists. Adding ARMA terms to mean model leads to their insignificance (p values for the coefficients higher than 0.05) - can we rely in the coefficient significance test under the violated "no standardised residuals autocorrelation" (I think no)? Adding extra terms also does not lead to increase in AIC/BIC... Commented Jul 9, 2020 at 8:30
• @alex337d, Statistical significance is a poor criterion for model selection; see e.g. Hyndman "Statistical tests for variable selection", so I would not worry too much about some ARMA coefficients being insignificant. Whether these ARMA coefficients are really needed is another question. E.g. for financial returns, I would expect any autocorrelations in a given sample to be due to chance; otherwise, you could make easy money in the stock market. Commented Jul 9, 2020 at 11:21
• @alex337d, Optimization of AIC/BIC is only partly in line with removal of autocorrelation from model residuals; trying to model weak autocorrelation by making the model more complex would likely be suboptimal from the AIC/BIC perspective. Regarding significance tests, the violation of i.i.d.'ness in the form of presence of autocorrelation is a problem. Perhaps using robust standard errors may help. Intuitively, this is what these robust standard errors are for, but I am not 100% sure how they work for GARCH models. Commented Jul 9, 2020 at 11:23