I am currently investigating commodities and their impact on the oil price.
I have 8 variables of different stationarities
- $y$ = dependent variable (oil price) is non-stationary I(1);
- three independent variables $x_1,x_2,x_3$ are stationary;
- 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.