# How to identify the ARCH and GARCH lag length in dynamic conditional correlation GARCH model?

I just follow the Stata manual for DCC GARCH model. This model contains ARCH(1) and GARCH(1) terms. But my question is, on what basis and how we can can select appropriate ARCH and GARCH oder to run the DCC GARCH model.

DCC-GARCH model is built in a stepwise fashion.

1. First, individual (G)ARCH models (that is, either ARCH(q) or GARCH(p,q) or some alternative models) are fitted for all the univariate time series in the system. Each time series is then scaled by dividing each observation by the corresponding fitted conditional standard deviation from the individual (G)ARCH model.
2. Second, a DCC model is fitted on the scaled time series.

Think about the time-varying conditional covariance matrix. First, the diagonal terms are modelled by (G)ARCH models, and these terms are used to scale the data. Second, the off-diagonal terms are modelled using a DCC model.

Since the first step of fitting an individual (G)ARCH model for each univariate time series is carried out independently of the second step, you can forget about DCC for now and focus on (G)ARCH modelling of each univariate time series.

Now GARCH(1,1) is perhaps the most popular (G)ARCH specification and sort of a benchmark model. People sometimes (often?) just impose the order (1,1) without clarification as GARCH(1,1) is supposed to fit reasonably well in many situations. But that need not be a convincing argument.

Once you have fitted a GARCH(1,1) model you can test for remaining ARCH effects using Li-Mak test. If the model does not pass the test, GARCH(1,1) has not done its direct job; then you may want to consider other model specifications. There are other tests, too, but Li-Mak test checks out how well the model performed at its central task of fitting the conditional variance.

Speaking of other models when you are not satisfied with GARCH(1,1), you may want to consider a bunch of other models (like ARCH(1), ARCH(2), GARCH(1,1), GARCH(1,2) etc) and select the best one from them. One way of doing that is to pick the model that has the lowest AIC or BIC value. However, model selection is a broad topic and suggestions do vary. You may find numerous discussions of the topic here at Cross Validated by checking out the relevant tag.

• HEllo Richard Hardy, thank you so much for your valuable information. Let me know, in case of multivariate GARCH modles (i.e. CCC, DCC, VCC etc.) can we use suggested lad order based on fitting an individual (G)ARCH model for each univariate time series? – Shafaqat Mehmood Feb 14 '15 at 14:56
• I don't quite get your question. If you are asking whether the logic I laid out for DCC also holds for CCC and VCC, the answer is yes (although it's been some time since I read about VCC, as far as I remember it should hold). – Richard Hardy Feb 14 '15 at 15:05