# VEC model on log prices, for random simulation?

In the context of pair trading, I’m trying to regress a VEC model on cointegrated pairs (and also a GARCH model on the residual of that VEC model).I would like to generate random réalisations of each pair, each realization having the same dynamic as its “real life” counterpart. What I did initially is that I generate random innovations first using the GARCH parameters regressed on the pair, then I use the VEC equation with the regressed parameters of the pair as well to generate the final random simulation of prices of the 2 assets.

However the problem is that doing this way leads to negative price, and a cond. variance that is the same scale when the price is close to 0 as when the price is far from 0. In the real world, prices are always positive and the cond variance decrease logarithmically when the price drops close to 0.

I’ve tried to perform the same entire process, but instead of regressing the VEC and the GARCH on prices, I regress on log prices. Then, I generate random simulations the same way as before, but I have a random sim of log prices, in log\$. Only at the end of the whole process, I put the random log-price to the exponentential and I get my real prices.

The problem of negative generated prices is fixed, and the variance of the generated prices look more like real cond variance when the price is close to 0.

Some pairs give nice results, looking like much more the shape of the real pair than before the log. The scale of the final random prices are aligned with the scale of price of the asset of the pair on which the VEC and GARCH model are regressed

However, some other pairs however show very weird result, with random prices stuck to close 0 or completely out of scale compared to the real prices of their “real” counterpart

Is the procedure that I described something that can be mathematically ok ? Is it possible to apply the VEC and GARCH models on log prices and log residual like that ?