Why the autoregressive process to generate random time series?

in order to test some forecasting methods, I desire to generate random time series. I'm about to use the AR(1) model:

$X_k=\alpha X_{k-1}+ \epsilon_k$

With eventually: $\alpha>1$

How can I motivate -with scientific arguments- the choice of such a model since, I guess, other models are able to do the same ?

Is "simplicity" an available argument ? If so, this supposes the existence of complex alternatives. What are they then ? Thanks !

• I believe the key word is linearity. This is a discrete-time linear stochastic dynamical system. It stems, e.g., from a continuous-time model $dY_t=aY_tdt+dB_t$ counterpart, with $\alpha=e^aT$, where $T$ is the sampling time and $e_k$ has covariance -$(1/2)a^{-1}(1-e^{2aT})$ (if I am not missing anything). It might be an appropriate model if the underlying dynamical phenomenon is known to abide by a linear dynamical law or it is operating in a "linear regime", e.g., it is about an equilibrium state (with a small variance for the underlying noise). Mar 18, 2023 at 12:11

2 Answers

I think the proper scientific approach to modeling is to first try your best to understand the phenomenon you are studying. In particular, if a statistical model appears warranted by observed randomness, then one should first try to understand, as much as possible, the mechanism producing the apparent randomness -- and that should naturally lead to an appropriate model.

Unfortunately, this does not seem to be the way modeling is usually done. It seems that usually statisticians just choose a model or a class of models among those they are familiar with and then try to fit it to the data. At best, they would afterwards do something like what is described in Section "Evaluating the quality of forecasts" of the Wikipedia article Autoregressive model:

The predictive performance of the autoregressive model can be assessed as soon as estimation has been done if cross-validation is used. [...] if the predictive quality deteriorates out-of-sample by "not very much" (which is not precisely definable), then the forecaster may be satisfied with the performance.

There is no example in that Wikipedia article of how or where an autoregressive model may naturally arise, and I cannot imagine such an example either. So, at this point, this model seems to me entirely contrived (I would be glad to learn that I am seriously mistaken about this).

As for the simplicity criterion, I think it is applicable when one wants to choose one of a number of models with all relevant merits similar to one another and differing mainly in the degree of simplicity.

• Really appreciate your answer Losif Pinelis. Thank you ! Might I ask you whether you know other similar models (just the names) ? May 22, 2018 at 11:28
• I am not a practicing statistician and don't know much about existing statistical models; so, alas, I cannot help you in that regard. However, as I wrote, I think one should try to build a model based on the phenomenon of interest (rather than choose from a known set of models). After that model building is done, one may try to relate the model so obtained to existing ones. (Many people confuse the first letter, I, in my first name with the lower case of the letter L. My name, Iosif, is of the same root as Joseph, Josef, Iosef, Yosef, Joey, etc.) May 22, 2018 at 17:15
• Thank you Iosif (sorry for miswriting your name;)). I agree with you, that seems to be more relevant as an approach May 24, 2018 at 9:23

There isn’t any justification that would be applicable to all situations. Rather, this is a modeling choice that has to be justified in the context of your particular problem.

• Thank you for your comment. Actually, as I said before,I desire to generate time series with eventually trends. At this stage, I have not specific requirements. May 21, 2018 at 15:45