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Most of the literature around time-series models focuses on models with AR terms. Here I have a slightly different and potentially less complicated case.

Suppose you have a time-series model with the following form:

$$ y_t = \alpha + \beta_1 X_{t} + \beta_2 X_{t-1} + ... + \epsilon_t $$ where

  • $y_t$ is a single endogenous variable
  • $X_i$ is a vector of exogenous variables at a given lag and $\beta_i$ is a vector of their coefficients. These might as well not include any lags and have just a single contemporary term $X_t$, depending on the best fit.

The reason for this form is that you need to be forecasting $y_{t+\tau}$ by receiving as inputs forecasted exogenous variables $X_{t+\tau}$. Thus even if $y_t$ has some autoregressive persistence you rely on capturing it through the relationship with the exogenous variables and their own persistence.

Lastly, you don't care so much if the individual coefficients of the exogenous variables are accurate as long as the overall forecast error is small.

Questions:

  1. Do we need stationarity of the of the endogenous and/or exogenous variables in order to get a reliable forecast in both cases where we use only contemporary exogenous terms $X_t$, or contemporary + lagged ones $X_{t-i}$?
  2. What would be the best model for estimating this form? Is it worth fitting a restricted ARIMAX or we can just use OLS, GLM, GMM?
  3. What assumptions would need to be satisfied given the form and the best model from 2. in order to get reliable forecasts (assuming that the forecasted inputs $X_{t+\tau}$ are accurate)?
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    $\begingroup$ ARIMAX and GLM are models, while OLS and GMM are estimation techniques. In that sense they are incomparable. $\endgroup$ Sep 11 at 10:27
  • $\begingroup$ Take "spurious regression" as a test-case for your question. It does not require the presence of AR terms and does require stationarity. $\endgroup$ Sep 12 at 9:08

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