# What is wrong with lagged regressor in time series regression?

I am trying to explain a time series with the help of other related series. I really get nice fits using a standard LM approach with NeweyWest VC matrix. The fit even increases drastically when I replace some of the explanatories by higher lags (lag 10) of these explanatories (quarterly series) to the mix.

Is it ok, to use such high lags? The length of the series is about 120. What's the risk of using high lags? At least it seems uncommon to me to use it.

• Verify that the residuals of the model are uncorrelated and homoskedastic. All time-series auto-regressive models incorproate lags. It's also not clear that the series you are using is stationary. – Ram Ahluwalia Sep 15 '11 at 14:00

It's not "wrong," but it's probably a bad idea. One problem is that you don't have lags for your first ten observations, so you can't use those in your analysis, effectively making your data set smaller.

There are certain lags that we think make sense intuitively: Last period probably effects this period, this time this year is probably related to this time next year due to some seasonal variation patterns. One lag and four lags for you would make sense. Having two years out influence what happens today would be surprising and 2.5 years (or 10 quarters) seems stranger still.

I would chalk up a significant lag at quarter 10 to chance, rather than a good model. Including this lag can lead to overfitting. If you overfit, you will have trouble with out-of-sample forecasting. As a test on this, you might run the model with the 10 quarter lag on the first and second halves of your data to see if you still get a significant/similar result.

Lastly, except in the cases of intuition, I don't like to include one lag, then skip a bunch, then include another. For example, including lags 1 and 4 makes sense intuitively, so that's fine, but adding lags 1, 4, and 10 just seems strange.

Time series is as much art as science, so it does take some playing around.

What you are doing with using lags of explanatories is in my opinion a very bad idea if the explanatory variable is auto-correlated with itself. The lags are correlated thus identification will probably fail as you are trying to use ESTIMATION to perform IDENTIFICATION. Box and Jenkins and a host of others suggest pre-whitening to identify the correct X structure in an ARMAX model. Art becomes science when you understand what to do. Time series is not art but intelligent analytics done artfully make the science a lot easier. In my opinion the art "nearly vanishes" as you hone the science.

• When you say "suggest pre-whitening to identify the correct X structure in an ARMAX model" are you suggesting it is appropriate to identify the auto-correlation structure of each individual variable, and this somehow guides you for what exogenous factors should be included in the ARMAX model? – Andy W Sep 15 '11 at 21:30
• :Andy Exactly !the process is filter/convert X to x where x is white noise.Apply this filter to Y to get y;use cross-corr on these suitably stationary/transformed series to IDENTIFY a suitable model between X and Y.This is done separately for each X in model on the premise that the x's are independent of each other.Careful model RE-IDENTIFICATION enables one to "correct" the form of the transfer between and X and Y. Ultimately one must identifu both the ARMA component in the TF model reflecting omitted Stochastic Series and the Pulses/level shifts/trends refectimg omitted deterministic series. – IrishStat Sep 15 '11 at 23:00