I can't find an adequate conditional model for this time series I have the European TTF GAS spot Price time series from 31/12/1990 to 31/10/2022:
https://docs.google.com/spreadsheets/d/1Iu84-oFtv3-ybmp72IJ_s1DfcTG_TDu7/edit?usp=sharing&ouid=117791817408001078818&rtpof=true&sd=true
I downloaded it from Word Bank:
https://www.worldbank.org/en/research/commodity-markets
I would like to model TTS price or return dynamics by a conditional model for the mean and a conditional model for the variance. I found many papers that fit an ARMA(1,1)-GARCH(1,1) to the raw returns, I tried with multiple ARMA(p,q)-GARCH(r,s)-types models with both a Student-t and a Gaussian distributional assumption for the innovations (of course also comparing models according to BIC and AIC), but every times standardized residuals and/or their squared values present some high degree of autocorrelation. For example ARMA(1,0)-GARCH(1,1) assuming Student-t distribution for the innovations gives:

In the figure the second row are standardized residuals and the third are squared standardized residuals.
Then I assumed that autocorrelation may be imputable to the seasonal nature of Natural GAS, therefore I deseasonalized log(P) without the dummy-variables approach, but nothing changed. How can Natural GAS TTF spot price be modeled?
Thank you in advance.

Some thoughts:

*

*Statistically significant (partial) autocorrelations at high-order lags that are not associated with seasonal frequencies (12, 24, 36, ...) may be due to chance.

*Structural breaks may induce persistence and autocorrelations at high-order lags and otherwise mess a lot up. Instead of modelling untransformed series with ARMA-GARCH, modelling log-returns or other transformations that involve differencing may alleviate the problem.* (David Hendry, Jennifer Castle and "robust forecasting devices" are some relevant names and keywords; an example of a relevant paper is Castle et al. "Robust Approaches to Forecasting" (2014).) But it seems you are already doing that, so it has not worked as well as one might have hoped.

*Indeed, if you plot the data you will notice that at least the last 18 months show quite different behaviour than the preceding 30 years. I doubt the same model that would fit the 1991-2020 period nicely would fit the 2021-2022 data reasonably well. Out of curiosity, would ARMA(1,1)-GARCH(1,1) with some seasonal adjustment work OK for 1991-2020?

*30+ years is a while. In addition to potential structural breaks, there may be some gradual changes in the data generating process. Using the same time-constant model for the entire sample might therefore produce some undesirable artefacts.

*You diagnostic QQ plot shows violations of normality. However, that need not be a problem, as you said your model assumes a Student-$t$ distribution. Try using the appropriate QQ plot instead.

*What is the difference between the second and third rows of plots? Are raw standardized residuals used in the second row while squared ones in the third row?

*This may come at a cost. Generally, differencing a time series that is not integrated leads to overdifferencing and its problems (e.g. a unit root moving-average component is introduced).
