ARIMAX: Time series modeling with X and D (exogenous qttative and qualitative variables)

Question

I need to fit a ARIMAX model (in R, library: TSA), with variables like

$Y = $ time series that I want to predict/forecast

$X_{i} =$ exogenous quantitative variables

$D_{j} =$ qualitative variables (dummies)

As far as I know, the ARIMA parts of ARIMAX function only refers to $Y$, and adding $X$ and $D$ captures just a "one day" effect.

Is it right?

Is there any way to surpass that issue?

Exemple:

Imagine that

$D_{2.k} = 1$ if event $A$ happened;

$D_{2.k} = 0$ otherwise; $\forall k$.

I know that when $A$ happened, at day $t$, $Y$ could have been affected from $t$ to $t+p$; i. e., a day $t$ event splashes $(t+1), (t+2), \dots, (t+m)$.

Same thing about all $X$ and $D$.

Answer 1

If your predictor variable is stochastic then you can employ Transfer Function Identification in order to form and encode a lead , contemporaneous or lag effect of one or more a predictor series.

This is largely correct except for suggestions regarding model identification.

Also see https://onlinecourses.science.psu.edu/stat510/node/75/ and @forecaster's excellent comments here Transfer function in forecasting models - interpretation and a lengthy discussion here steps to time series analysis on my data

If your predictor variable is not stochastic then all I can suggest is a trial and error approach incorporating different dummies (0/1) .