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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?

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