When will a non-lagged regression term, in a forecasting algorithm, outperform an algorithm that doesn't require the regression term?

Question

I am struggling to understand when a regression variable that is non-lagged would be beneficial to a forecasting algorithm.

I have been investigating the unobserved component model algorithm. I am finding that even when the non-lagged regression variable is linearly correlated with the response, a better model (lower cross-validated MASE) does not use the regression variable. It seems a 'slope' term within UCM could just as easily be used as the regression variable.

For UCM terms: https://www.lexjansen.com/nesug/nesug04/an/an03.pdf

I am also investigating ARIMA models. I am finding again that if the non-lagged regression variable is linearly correlated with the response that this same trend can be estimated just with an ARIMA model. For example, I could use a difference to de-trend the model so it's now stationary. A transfer function utilizing the correlated regression variable is no longer significant.

For ARIMA terms: https://v8doc.sas.com/sashtml/ets/chap7/sect8.htm#:~:text=Thus%2C%20the%20general%20notation%20for,order%20of%20the%20seasonal%20part.

ARIMA Model:

I guess the question boils down to, When is the regression variable explaining variation in the response that couldn't easily be estimated by terms expressed in lags of the response?".

Is there a simple guide during EDA of the y & x that I could use to try & identify patterns that a simple lagged y model could not explain?

Thank you in advance for your help, narnia649

Answer 1

Hi: I glanced at the paper and that model has a lot of terms in it so I won't even try to go into all of them. I'll just focus on one issue since it's related to your question. The thing to realize is that, when you use a lagged response variable on the RHS of a model, you really are using past $X_t$, it's just implicit. Take the simplest case, namely, ARDL(1,0). ( this is called an autogressive distributed lag in the econometrics literature ).

$Y_t = \alpha Y_{t-1} + \beta X_{t} + \epsilon_t$.

This can be re-written as:

$Y_{t}(1 - \alpha L) = \beta X_{t} + \epsilon_t$.

Then, dividing both sides by $(1 - \alpha L)$, ( and assuming that $\alpha$ is between 0 and 1 so that the infinite series converges ) results in:

$Y_{t} = \beta \sum_{i=0}^{\infty} \alpha^{i} X_{t-i} + \sum_{i=0}^{\infty} \alpha^{i} \epsilon_{t-i}$.

So, in the simplest case ( I imagine this carries over to the more complicated case where you have multiple lagged terms but i won't try to show it ), it's really just decorative in that it is possible to get rid of the lagged dependent variable. The problem with doing that is that the series is infinite and you'll have difficulties estimating it if the $X_{t-i}$ are highly correlated.

The model I just described is has another name called the "koyck distributed lag" in case you want to find it in the literature. But the point I'm trying to make is that, when you see a lagged dependent variable in a model, it's really just a simpler way of expressing the fact that the response is based on regressor variables and error terms terms that go way back in time.