Modelling stationary and integrated time series in one system

Question

I am currently investigating commodities and their impact on the oil price.

I have 8 variables of different stationarities

1. $y$ = dependent variable (oil price) is non-stationary I(1);
2. three independent variables $x_1,x_2,x_3$ are stationary;
3. the rest $x_4,x_5,x_6,x_7$ are non-stationary I(1).

Objective: Find long/short run relationship between $x_1,x_2,\dotsc,x_7$ on $y$. Not the other way around.

Model 1: Take the first difference of $y$, $x_4,x_5,x_6,x_7$ (they are stationary at first differences) and utilize Vector Autoregression (VAR). However, if I take the first difference, the long-run relationship might disappear; so the model is useless?

Is there a better way to do this? I'm thinking of doing cointegration of the nonstationary variables, then using a vector error correction model (VECM) to find long-run relationship and then taking the first differences of the same variables to use in a VAR together with the stationary variables.

Model 2: Use cointegration on non-stationary variables to find the number of cointegrated equations defined as N. Then use a VECM with N cointegrations and all the variables included.

Is this a valid approach? As far as I know VECM only works for non-stationary variables.

Answer 1

A detailed treatment of an analogous question is given here. In your case, you are interested in only one equation of the model, i.e. the one where $y$ is the dependent variable.

Regarding the specific aspects of your post:

However, if I take the first difference, the long-run relationship might disappear; so the model is useless?

Stating that the model is useless is quite harsh, but certainly the issue you indicate above is a valid point of concern. Just differencing cointegrated variables makes you lose the error correction term that should actually be in the model. Hence, you should examine presence of cointegration and resort to differencing only in absense thereof.

I'm thinking of doing cointegration of the nonstationary variables, then using a vector error correction model (VECM) to find long-run relationship and then taking the first differences of the same variables to use in a VAR together with the stationary variables.

I think you could include the stationary variables in the VECM as indicated in the answer linked above. That would allow assessing the short term impact of each of the $x$s on $y$ as well as keeping the long-term effect via the error correction term.

Model 2: Use cointegration on non-stationary variables to find the number of cointegrated equations defined as N. Then use a VECM with N cointegrations and all the variables included. Is this a valid approach? As far as I know VECM only works for non-stationary variables.

As becomes clear from this and your comment below, building Model 2 includes forming a VECM that treats the stationary variables on the same terms as the integrated variables, which is not right. Even if you inted to use only one of the model equations, the rest of the model being problematic might lead, for example, to inferior lag order selection for the whole model and thus implicitly for the equation of interest.