# Can I use breaking series as a regressor in a VAR model?

Context:

• I have two series, price and sales.
• Sales is mean-reverting stationary but price is stationary only after controlling for an intercept break.
• I want to set up a 2-equation VAR model and the research interest is to estimate the cumulative effect of price on sales through impulse response function.

My question: Is the impulse-response function (irf) of a price shock on sales still biased even after I include the break dummy as a regressor in the two equations? Say the VAR model is: \begin{aligned} Sales_t &= \beta_{10} + \beta_{11}Sales_{t-1} + \beta_{12}Sales_{t-2} + \beta_{13}Price_{t-1} + \beta_{14}Price_{t-2} + \beta_{15}D_t + e_t \\ Price_t &= \beta_{20} + \beta_{21}Sales_{t-1} + \beta_{22}Sales_{t-2} + \beta_{23}Price_{t-1} + \beta_{24}Price_{t-2} + \beta_{25}D_t + e_t \end{aligned}

My answer is yes, irf is still biased because the regressors $$Price_{t-1}$$ and $$Price_{t-2}$$ are still nonstationary.

My solution: include both equations the interaction terms: $$\beta_{6}Price_{t-1}\times D_t$$ and $$\beta_{7}Price_{t-2}\times D_t$$.

Would you please assess the above my question, my answer, and my solution?