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I am trying to know what is the most simple model for time series data, with exogenous variables. What is the most simple framework I can use ? Is it possible to build a model more simple than ARIMAX, ECM, VARX or ARDL, and, specifically, a model without any AR or MA component ?

I imagine something just as simple as :

$$Y_{t} = \alpha_{1}X_{1,t} + \alpha_{2}X_{2,t} + u_{t}$$

with most probably some time-series-specific properties (all series stationary, stationary residual, etc.).

Is it possible ?

Sorry if I talk complete non-sense, but I can’t wrap my head around an AR or a MA component being mandatory while I’m interested by which exogenous variable is useful to predict $Y_t$.

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    $\begingroup$ Hi: yes. what you have is fine ( no ARIMA terms necessary ) but three things to consider: 1) should the X's be lagged by one time unit ? or do they really occur so that the current X's influence the current $Y$ ? 2) You may want to add a lagged dependent variable to the RHS so say $\alpha_0 Y_{t-1}$. If you choose not to, then your assumption is that any effects due to $X_{i}$ are instantaneous and don't carry over to future time periods. 3) Think about the error term being independent ? Does that seems reasonable for your model ? $\endgroup$
    – mlofton
    Commented Dec 1, 2023 at 4:10
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    $\begingroup$ @mlofton, your comment answers the question. Why not post it as an answer? $\endgroup$
    – Richard Hardy
    Commented Dec 1, 2023 at 10:59
  • $\begingroup$ okay. I'll cut and paste. good idea. $\endgroup$
    – mlofton
    Commented Dec 1, 2023 at 16:38

1 Answer 1

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what you have is fine ( no ARIMA terms necessary ) but three things to consider:

  1. should the X's be lagged by one time unit ? or do they really occur so that the current X's influence the current 𝑌 ?

  2. You may want to add a lagged dependent variable to the RHS so say $\alpha 𝑌_{t-1}$. If you choose not to, then your assumption is that any effects due to 𝑋 are instantaneous and don't carry over to future time periods.

  3. Think about the error term being independent ? Does that seems reasonable for your model ?

#=============================================================== #12-05-2023 #===============================================================

ADDENDUM: To clarify the third point. Take a simple AR(1).

$y_t = \rho y_{t-1} + \epsilon_t $

Even though $\epsilon_t$ is independent, if we write the model using the lagged operator, we end up with:

$y_t = \sum_{i=0}^{t} \rho^{i} \epsilon_{t-i}$

So each $y_t$ has all the previous $\epsilon_t$ contained in it so the $y_t$ are not independent, even though $\epsilon_t$ is independent. So, the point is that the error term can matter in that, even when it looks independent, it has dependency implications that may not be obvious.

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  • $\begingroup$ Hi mlofton, thank you for your detailed answer. I have market it as solving the question. I will definitely consider option 1) as I am not positive yet about whether the effect of X on Y will be contemporary or not. Option 2) is probably what I need, i.e. I will probably include an AR term anyway. I have a concrete model to build but my question was also kind of a theoretical, i.e. I wanted to understand if the existence of such model with minimal requirements was even theoretically possible. $\endgroup$
    – Johannes Konrad
    Commented Dec 4, 2023 at 16:23
  • $\begingroup$ I definitely lack some knowledge or understanding for your point 3). I have always assumed that dependant errors (which, as far as I know, is equivalent to autocorrelated errors) will make problems for the model. Therefore, I never wonder if independent errors make sense, I just assume that's just the way it should be. I always assume that independance/no auto-correlation is needed. Can you tell me what I am missing ? $\endgroup$
    – Johannes Konrad
    Commented Dec 5, 2023 at 10:37
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    $\begingroup$ Hi Johannes: I've been at this stuff a while so no problem explaining. I added some material at the bottom of my answer. I recommend Andrew Harvey's text: "the econometric analysis of time series" in order to see how the independent-dependency of error terms can be quite subtle. He writes in a way that can be difficult at first reading ( very terse ) but, if you read him a few times, you can eventually see that there is a lot of gold in what he says. This is actually true of all his books but check out that one first. $\endgroup$
    – mlofton
    Commented Dec 5, 2023 at 17:58
  • $\begingroup$ Hi, thank you for the additional elements on your response ! I will give a try to this textbook, to see if I can understand this point a little better. If I'm not too busy I will try to comment further this thread in order to give solution to whoever is intersted. $\endgroup$
    – Johannes Konrad
    Commented Dec 6, 2023 at 15:38
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    $\begingroup$ Hi: it's more of a time-series book and, not easy, especially because it's your first introduction to Harvey's style. But, like I said, Harvey's books are deceiving. They are not easy to read but you read them a couple of times ( or more ), let them sink in and, eventually, you realize that there is a lot of unique info in his writings. $\endgroup$
    – mlofton
    Commented Dec 7, 2023 at 16:56

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