How can we select the best GARCH model by carrying out likelihood ratio test? I have carried out the likelihood ratios of different GARCH models.
GARCH(1,1) and GARCH(1,0)- Rejected null hypothesis so I chose GARCH(1,1) to do more sophistication.
GARCH(3,1) and GARCH(1,1)- Failed to reject the null hypothesis
GARCH(4,1) and GARCH(1,1)- Failed to reject the null hypothesis
GARCH(2,1) and GARCH(1,1)- Failed to reject the null hypothesis
GARCH(2,2) and GARCH(1,1)- Rejected null hypothesis so then I chose GARCH(2,2) to do more sophistication.
And further, every GARCH model failed to reject the null hypothesis against GARCH(2,2). So my best model should be GARCH(2,2) or GARCH(1,1)?
 A: Your model selection strategy should be aligned with the intended use of the model. As far as I know, a sequence of likelihood ratio tests at the conventional 5% significance level is not a optimal strategy under any of the popular intended uses of the model. For a closely related explanation, see "Statistical tests for variable selection" by Rob J. Hyndman.
Among likelihood-based approaches, the prominent model selection strategies are minimization of AIC and BIC.

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*Minimization of AIC is optimal* for forecasting. The idea is that AIC estimates $-2n \ \times$ the negative expected likelihood on a new, unseen data point from the same data generating process (DGP) that generated your sample. ($n$ is the sample size, here the length of the time series.) The model with minimum AIC should therefore yield maximum likelihood. If likelihood is a relevant criterion for forecast evaluation (which it could be argued to be), go ahead with AIC minimization.

*Similarly, BIC is optimal* for finding the true DGP, or its best approximation, among the candidate models. So if you care about identifying the true DGP or its best approximation, go ahead with BIC minimization.

Alternatively, you could rely on time series cross validation and select the model that yields the most accurate out-of-sample (test-sample) forecasts.
In a cross-sectional setting, AIC and BIC have been shown to be asymptotically equivalent to different types of cross validation; see e.g. this among several related threads on Cross Validated. I suppose something similar could be (has been?) shown in the time series case as well. Of course, the precise results depend on some assumptions, but they should give a flavor of what is going on in the broader case.
*Under certain assumptions which can be found in a textbook or papers introducing AIC and/or BIC, or some threads of Cross Validated. Claeskens & Hjort "Model Selection and Model Averaging" (2008; a book) and Claeskens "Statistical model choice" (2016; an article) are good starting points. 
