# What's the definition of "Dynamic Regression Models"?

I am trying to learn about Dynamic Regression models. However, the sources on the topic is (relatively) few compared to other TS topics, and so I cannot really get a grasp of where to start. I really want to get a specific definition of what a dynamic regression model is. What for example is the difference between an ARIMA model with external regressors and a dynamic regression models?

Is it simply a AR distributed lag model with ARMA errors?

• Hi: Andrew Harvey's "Econometric Analysis of Time Series" is a great place to start. Nerlove's "Economic Analysis of Time Series" is probably the next one to go to after that. You are correct that an ARIMA model with external regressor is a dynamic regression model. So, are auto-regressive distributed lag models ( which are discussed in any decent econometric text). I also agree that the names and terminology can be confusing. Pankratz's text discusses them from the ARIMA viewpoint that you mentioned but I prefer Harvey's and Nerlove's discussions. Good luck and I hope this helps some. Aug 14, 2018 at 19:22
• Hey mlofton. Really appreciate the answer. Will try and start with the Harvey book and attack it from there. Aug 15, 2018 at 6:58
• You're welcome. Harvey's style can be terse and diffcult to digest but he has a few chapters-discussions that are the best I've seen. Aug 15, 2018 at 9:43
• I haven't got access to Harvey's book unfortunately. Would you be able to provide a general definition of what constitues a DRM? Aug 16, 2018 at 19:33

No problem but I'd definitely try to get my hands on some kind of textbook or lecture notes. The following is not a formal definition but here goes my attempt: To me, a DRM is any model which is time dependent, in the sense that, the next value of an observation (on the right hand side of the model ), at the next point in time, changes the model forecast. So, suppose you are sitting at time $t$, and you have a model say $y_{t+1} = \alpha y_{t} + \beta_{1} x_{t} + \beta_2 x_{t-1} + \epsilon_{t+1}$.

Now, depending on the convention, this model would probably be referred to as an ARDL(1,1) which is a specific case of a class of dynamic regression models called auto-regressive distributed lag models. Note that, when sitting at time $t$, this model ( after some algebraic manipulation and assumptions about $x_{t}$ and $\epsilon_{t+1}$ ) can provide a forecast for say $y_{t+h}$ where $h$ is the forecast horizon. My point here is that that $h$ does not necessarily need to be assumed to be one. Next , suppose one unit of time passes so that a new observation, $x_{t+1}$ arrives at time $t+1$. This means that the new model forecast of $y_{t+h+1}$ will get updated to reflect this new observation. Note also (and this can be crucial) , the error terms can play a big role also ( be say MA(1) or AR(1) rather than just IID-normal ) so, even if the new observation of $x$ at time $t+1$ was the same as the observations at time $t$ and time $t-1$, and there was no lagged dependent variable, $y_{t-1}$ like there is above, the forecast at time $t+h+1$ could still change !!!!!! This is a powerful concept and would not be the case in a more "static" type of regression model. Essentially, this is what is meant by the term dynamic, namely that, because of the lagged dependent variable on the RHS and or the error terms, the new $x_t$ observations that arrive in the future could stay constant for many periods and the model forecasts could still change.

I hope this helped a little. It isn't really an answer but I put it in the answer section because there's more space. I happen to have a lot of experience with dynamic regression models so I've collected many lecture notes and texts over the years. When I get a chance this weekend, I'll try to send some useful links but the dynamic concept is pretty much as simple as what I explained. There's nothing terribly complex regarding the term "dynamic" but it can sound intimidating. Estimation of parameters can be a problem sometimes because of multiple parameters, multi-collinearity etc. Stability of parameters over time is another possibly problematic issue. Remind me if I forget to post links.

https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_3.pdf

http://www-personal.umich.edu/~franzese/DeBoefKeele.2008.TakingTimeSeriously.pdf

https://www.nuff.ox.ac.uk/politics/papers/2005/Keele%20DeBoef%20ECM%20041213.pdf

https://pdfs.semanticscholar.org/presentation/fd49/2458fbe607f3bf19ff28aa872b4980ebd629.pdf