I am learning about residual diagnostics for MA/ARIMA models. I have learnt from the following post that the Ljung-Box test cannot be used for models with lagged components of the dependent variable (i.e. when there is weak exogeneity in the model). It is preferred to use the Breusch-Godfrey test in this case. But, I am confused with the usage of Breusch-Godfrey test when there is simply MA components or a combination of both, i.e., ARIMA models.
- Would the auxiliary equation be like: $w_t = \alpha_0 + ( \sum_{i=1}^{k} \alpha_i y_{t-i} ) + ( \sum_{i=1}^{p} \rho_i w_{t-i} ) + e_t$, where $y_{t-i}$ denote the lagged dependent variables. If yes, then I am guessing this will be generic form for ARIMA models (where $k$ would denote the
pvalue inARIMA(p, q)) and in case of MA models, $k$ would be0. The variablepin the auxiliary regression equation would denote the maximum lags for which the test is being performed. Please correct me if this understanding is incorrect. - We know that when there are AR terms, Ljung-Box test does not hold good. However, how can we mathematically compare it with Breusch-Godfrey test when there is just the MA terms? Logically, I am thinking both would be equally preferable, but Breusch-Godfrey test would stand-out as it comes with weaker assumptions and does not depend upon absence of autocorrelation to test its presence in the first place, like L-B test. Any good pointers would really be helpful to understand this assumption of mine.
