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Is it valid to estimate an ARMAX model using I(1) and I(2) variables, which are made stationary after first and second differencing, respectively?

For instance, I have an I(1) stock price variable, which I difference once to make it stationary. Additionally, I have a technical indicator that is I(2), which I difference twice to achieve stationarity. I then use this differenced technical indicator as an exogenous variable in the model. Consequently, I estimate the ARMAX(1,1) model, with the stock price as the dependent variable and the differenced technical indicator as the independent exogenous variable.

Is such a model acceptable? What problems could arise with such a model and are there any related references one could look into?

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Let us call your endogenous I(1) variable $Y_t$ and the exogenous I(2) variable $X_t$. Your approach is fine unless $Y_t$ and $\Delta X_t$ are cointegrated. In the latter case, differencing $Y_t$ once and $X_t$ twice will miss the error correction term (ECT) between $Y_t$ and $\Delta X_t$.

The approach may still be acceptable if the ECT only enters the equation for $\Delta X_t$ but not $Y_t$, as then the ECT does not belong in the ARMAX model.

It may also be acceptable if your goal is forecasting and the effect of the ECT is measured rather imprecisely, so that including it in the model does not improve forecast accuracy.

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  • $\begingroup$ Thank you for your answer. Could you suggest any related sources of information, such as articles, where one could find more details on this topic? $\endgroup$
    – Pepe Frog
    Commented May 31 at 8:02
  • $\begingroup$ @PepeFrog, nothing comes to mind... $\endgroup$ Commented May 31 at 8:04

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