# The relationship between autoregressive model and distributed-lag model

The autoregressive models (koyck model, adaptive expectation model, potential adjustment model) I have learned so far are all derived from distributed lag models. And intuitively it makes sense since how could an outcome variable effect itself? the outcome should only be affected by the lagged effect of the external variable X. (correct me if I am wrong).

Therefore, can we say that all autoregressive models are derived from distributed-lag models?

And is it true that whether the constructed autoregressive models violate the OLS assumptions depend on the procedure that the autoregressive model is derived?

Thanks

• Hi: AR models are very different from distributed lag models because distributed lag models have an exogenous regressor $X_t$. OTOH, AR models don't have the exogenous regressor so the LHS truly is a function of the previous value of itself. This is not the case in a distributed lag because it's really an exogenous variable that is causing the relationship between the LHS and the previous value of itself. Your confusion is that in the ADL, it looks like there is a relationship between the LHS and the previous value of itself but there really isn't. Feb 14, 2020 at 5:08
• There might be some confusion in terminology. If autoregression and autoregressive model are taken to mean different things, then Yuan might be right. However, I think it is common among statisticians and econometricians to interpret both terms as autoregression where there are no exogenous variables, only a single dependent variable and its lags. Feb 14, 2020 at 11:43
• @Richard hi Richard, so is it correct to say that autoregressive model is derived from the distributed lag model? And is autoregressive model a time series model or multiple linear regression model? Thanks!
– Yuan
Feb 15, 2020 at 15:50
• I don't know. I use the term for autoregression just like mlofton does. An autoregression is not derived from a distributed lag model. An autoregression is both a multiple linear regression and a time series model. These are not mutually exclusive categories. Feb 15, 2020 at 15:58

There wasn't enough space in my comment to explain it clearly but this should clarify. Take the koyck distributed lag:

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

Now, using the lag operator, this can be re-written as

$$y_{t} = \sum_{i=0}^\infty \rho^{i}x_{t-i} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i}$$

Notice that, in the immediately previous equation, there is no longer a relationship between the LHS and the previous value of itself. It's an illusion that's only caused by the exogenous regressor having a lagged effect.

• Thanks, in the book (himayatullah.weebly.com/uploads/5/3/4/0/53400977/…) that I read, the autoregressive model is defined as models that include the lagged value of the dependent variable among its explanatory variables: eg Yt = a + βXt + rYt-1 + μt. So it should be allowed to have exogenous regressors
– Yuan
Feb 14, 2020 at 5:37
• Maybe it's semantics- terminology but, to me, what you wrote there is a koyck disributed lag with a trend term or AR(1) with exogenous regressor and trend term. If you want A REALLY GREAT explanation of distributed lag models, I strongly recommend Andrew Hervey's "economic analysis of time series". It's a tough read ( his style is very terse ) but, over time, you eventually realize that it's the best text around for distributed lags and their subtleties compared to straight time series models like the AR, MA etc. Feb 15, 2020 at 16:28
• Hi: Since you included the gujarati link, check out chapter 17 if you haven't already. It's also quite nice as far as explaining ADL's. It's not quite as good as Harvey's ( gujarati's book is a lot more general so that's understandable ) but still good. Feb 15, 2020 at 16:34