you could estimate the model using a bivariate GARCH model with an ECM term included in the mean equation.

In my opinion, You should consider and determine the impulse response function (IRF). You can also consider both volatilities (in a wide observation window) as time series. Do they influence each other? If yes, You can easily estimate the IRF.

best regards

I would also suggest you for a GARCH model with ECM terms included in the model. It will provide you better control over your analysis.

PRE-EMPTING DATA EXAMINATION & MODEL IDENTIFICATION: The question of which model to use here seems to pre-empt the data. Generally, in research, we need to examine the data first and then ask what model is appropriate to model this type of data. It is advisable to visually examine to see the data spread in scatter plot form so one has the idea of how is distributed in space.

Secondly, if one deals with bivariates, such as X and Y, test each array whether there is a shock effect that significantly throws the series from its normal path. Then compare the response series to the explanatory series: does the response series respond to the change in X immediately? with delay? or no response at all? If that is a kink in the response series with no prior shock from the underlying predictor, what then? Preliminary examination of our data in this manner would lead to proper model identification.

If we suspect that fuel price influences food price, test it. If Y = food price and X = fuel price, run simple regression to verify if the relationship is significant before proceed to more sophisticated model. How would we discount for coincidence?

MODEL: This page link gives a good over view of when it is appropriate to apply GARCH. http://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity

ECM (Error Correction Mechanism): ECM is not a model. It is an error correcting mechanism. For a time series: Yt = B0 + B1Xt-1 where the value of the current period depends on the last period's value, we want to be sure that the data does not contain extreme value---which will throw the series out of its long-run equilibrium, i.e. is the data stationary? Stationary means that the data is mean reverting. If the data moves away from its long-rum equilibrium, we want to verify whether this is "shock effect" is permanent. if the effect of the shock is permanent, it means that we will have a new equilibrium point and the old equilibrium point would no longer be used as a reference point. Thus, we introduce ECM to correct the effect of the shock. If after ECM, the series still does not return to its original equilibrium, then we conclude that the series 'integrated" the shock into its memory and a new equilibrium point is establish. thus, ECM is not a model, it is a tool to correct error in a data in order to fit a model.

REFERENCE: For material on time series, see attached.

http://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity

http://ro.uow.edu.au/theses/3999/

https://www.jstor.org/stable/40986765?seq=1#page_scan_tab_contents

https://books.google.com.tr/books?id=xNhCDwAAQBAJ&pg=PA277&lpg=PA277&dq=model+volatility+in+an+ECM&source=bl&ots=ABWE9GyOSq&sig=-BeSo381gGBoxLjBr2T0T-qzuTs&hl=tr&sa=X&ved=0ahUKEwjqhZjb2fLYAhWHh7QKHcDlALkQ6AEIQzAD#v=onepage&q=model%20volatility%20in%20an%20ECM&f=false

https://www.omicsonline.org/open-access/effects-of-causality-and-error-correction-on-volatility-modeling-a-simulation-approach-2162-6359-1000297.php?aid=65219

I Prefer to Apply GARCH/ARCH

Hello. I hope you are doing well. It seems to me that in this case you can use NARDL midel to check the Asymmetric Impact of price volatility besides the ECM. moreover, to check the causal direction you can apply Toda Yamamato test.

Good luck