# Risks involved in using Linear Regression model when the residuals are autocorrelated

I am working on building a model which contains time series input variables. I build the data around a Linear Regression model and found that the residuals are autocorrelated. Promptly shifted to using a ARIMA model after some research. My question is what is the risk involved here. As for the comparison done on a set of data the outputs are pretty much similar between both the models

• What is your goal? Inference or prediction? Commented Jan 12 at 12:36
• It is more of an inference rather than prediction I would say here. Not related to your question but, also to point I found a way to generate upper/lower bounds of value through Margin of Error of the model. Does this help in any way with the inference of the data Commented Jan 12 at 12:39
• Also, can you please be more specific as to what "shifted to using a ARIMA" means in contrast to linear regression? There are multiple very different way of marrying predictors and ARIMA. Commented Jan 12 at 12:39
• Yes, I tried testing the data using the ARIMA model here. Commented Jan 12 at 12:43

One risk is spurious regression/correlation. There are lots of humorous examples of this, many of them involve time series. Whenever two variables are both related to time, those two variables will be related to each other. So, for instance, shark attacks are related to ice cream sales, the number of people with PhDs is related to box office sales, and so on.

In real data, though, the silliness may not be so apparent, and one way of detecting it is that there is autocorrelation.

I side step that issue. I'm retired now, but, when I was a consultant, if a client had a time series problem, I farmed it out to someone else. (I did do a lot of multilevel models on longitudinal data, but not things like ARIMA).

• It might be better to think of these examples of spurious regression as arising not from autocorrelation, but from a common "trend" -- an underlying variable. When the autocorrelation is short-range compared to the ordered explanatory variable (such as time), or when the autocorrelation is negative, ignoring it can still give excellent parameter estimates. It would be a mistake, though, to interpret the usual estimate of the error variance in standard ways: all p-value will be wrong, perhaps hugely so.
– whuber
Commented Jan 12 at 13:01

OLS assumes independent errors (read residuals), so if your residuals are autocorrelated this is an issue for interpretation (but not necessarily for prediction). If the OLS model gives comparable results to ARIMA this might indicate that ARIMA did not capture the autocorrelation either, so you might need to rethink your models. As for the risks these are usually over/under-estimation of variances and hence incorrect significances (in case of positive autocorrelation this would usually result in too narrow variances).

• Autocorrelation indicates that predictions can be improved slightly in the case of estimating the conditional mean, which could refer to past, present, future or purely hypothetical instance. But when prediction refers to a future time, especially one step ahead, predictions can be improved greatly by incorporating autocorrelation. Commented Jan 12 at 16:10