Solved – Stationary of exogenous variables in Dynamic Regression with SARIMA errors

Question

I want to create a dynamic regression model with ARIMA-errors. What I am trying to figure out is if the exogenous variable, x_t and the variable I want to predict, y_t need to have the exact same number of differencing, or if them both being stationary is enough.

For example, if one exogenous variable, x_t, needs to be differenced one time to be made stationary while the y_t needs to be differenced two times to be made stationary. Is one then forced to difference the x_t a second time even though it is already stationary?

Answer 1

You need the dependent variable and the independent variable to have the same order of integration, otherwise they would diverge from each other asymptotically, invalidating both the intuitive or subject-matter explanation and statistical properties of the estimators.

• If $\Delta y_t$ and $x_t$ are not cointegrated, use $\Delta^2 y_t$ and $\Delta x_t$.
• If $\Delta y_t$ and $x_t$ are cointegrated, use $\Delta^2 y_t$ and $\Delta x_t$ and include the error correction term as a regressor (unless $\Delta y_t$ does not correct towards equilibrium so that the loading of the error correction term is zero; this could be tested by including the error correction term and checking whether it is significantly different from zero).