# How to interpret regression results when the data have been detrended?

I am planning to build a linear regression model where I explain flight ticket demand with airfares, lagged airfares, GDP etc. based on monthly data from the past 15 years. This is my first time working with time series and - if I got it right - some of my variables are trend-stationary. So my plan is to use a Hodrick-Prescott-Filter to take the trend out and just use the remaining, detrended data in my regression.

My question is: Does this change the interpretation of the results I get?

To clarify: I already took the natural logarithms of all my variables, so my regression model will look something like:

ln(ticketdemandt) = ß0 + ß1t * ln(pricet) + ß2t-k * ln(pricet-k) + ß3t * ln(GDPt) + ... + ut

(sorry about the confusing t and t-k, they are meant to be in the subscript)

So, if I found that ß1 was equal to -0.8, this would mean that for an increase in current prices of 1 %, ticket demand in the same period would decrease by 0.8 %. But what if I used detrended data for the price instead of the original ones. Would the interpretation of ß1 then stay the same or would it be somehow different?

The same question arises if I need to take the first difference (or higher differences) of data to make them stationary. How does that change the interpretation of the resulting betas?

Many thanks in advance!

• Hi: If you are going to use hodrick-prescott or differencing, then, if you want the actual predictions to reflect the model, then you are going to have to "un-do" those transformations. By "undo", I mean account for the trend or the difference when you actually predict. It's not a trivial process I don't think. Also, James Hamilton has a paper on the dangers with using Hodrick-Prescott so I would check that out before you use it. I say that because, although I'm sure Hodrick and Prescott know what they're doing, James Hamilton does also so I would see what he has to say about it. – mlofton Mar 19 at 16:17
• The title might be somewhat extreme but it's probably worth reading. nber.org/papers/w23429 – mlofton Mar 19 at 16:19
• Dear @mlofton thanks for your quick answer! I was afraid undoing would be the solution. However, I really can't imagine how to do that. (Adding a trend to the betas I get?) I couldn't find any information on that yet. Anyhow, I imagine that this must be a rather common problem that almost everybody who wants to run a regression with non-stationary data has to come across, no? Or is there a more common way to solve such an issue? – econocat Mar 19 at 17:16
• I would check out the hamilton paper first. But, as far as "undoing", it shouldn't be that horrible. You just have to keep track of what you took out by differencing or de-trending and then put it back in at the end. Not trivial but mainly book-keeping programming wise. – mlofton Mar 20 at 18:36
• Yeah, thanks for the Hamilton tipp. I had actually come across that. The reason why I was planning to use HP is that I am replicating another paper where it was also used. But I'll still consider using an alternative! – econocat Mar 21 at 9:44