My question is simple. When shall I stop when trying the value for p and q?
I have got the loglikelihood from ARCH(1) to ARCH(10). It's increasing. And then I tried GARCH(1,1), GARCH(2,1) etc. The loglikelihood is always increasing so I'm confused. Do I need to continue to apply for larger lags until the log likelihood start to decrease? What argument shall I make when I stop trying?
As the commenters point out, increasing ARCH/GARCH orders amounts to including additional degrees of freedom, so the (log) likelihood is guaranteed to increase. If you simply follow this increase in (log) likelihood, you will overfit.
One possibility would be to balance the gain in (log) likelihood against model complexity by adding a penalty term. Information criteria like the AIC or BIC are helpful here.
Here is an alternative. ARCH and GARCH are fundamentally ways to forecast future volatility. They aim at producing good density forecasts, by modeling the conditional heteroskedasticity. So one way of choosing model orders would be to fit each model, then create density forecasts for a holdout sample and assess which model gives the best density forecast. Possible tools would be the Probability Integral Transform or (proper) scoring-rules.