Detrending a time series regression model

In my text book is says:

Regress each of $y_t, x_{t1}$, and $x_{t2}$ on a constant and the time trend $t$ and save the residuals...

What do they mean regress each on a constant? What constant specifically? This is coming from a section in the textbook (introductory econometrics to be exact) dealing with detrending a time series.

They mean the constant term in the model, i.e. they assume the model

$$y_t = \alpha t + \beta$$

The $\beta$ term is the constant they refer to. They ask you to estimate $\alpha$ and $\beta$, and save the regression residuals.

• Though I am not sure what they mean by time trend $t$, it is ambiguous. I just threw in a $t$, but it might be mentioned elsewhere in your text. Commented Dec 15, 2012 at 6:16
• I was thinking of the your answer, but I just wanted to confirm that was correct. In my text they it mentions that they regress $y_t$ on $x_{t1}, x_{t2}$, and $t$, and get the equation:$$\hat y_t=\hat\beta_0+\hat \beta_1 x_{t1}+\hat \beta_2 x_{t2}+\hat \beta_3 t$$ So I would assume you mean the constant $\beta_0$?
– Kyle
Commented Dec 15, 2012 at 6:30
• yes, though they did not say it in a very friendly way. Commented Dec 15, 2012 at 6:31
• depending on the context of what the authors are trying to explain, I'm not sure whether they mean regress each $y_t, x_{1,t}, x_{2,t}$ onto $t$ and the constant term, or the model you have above. Commented Dec 15, 2012 at 6:34
• nope they meant it in the regression above. Thanks for the clairifaction.
– Kyle
Commented Dec 15, 2012 at 7:16

Why 1 trend and/or why not incorporate level shifts as part of the pre-filtering process ? Approaches like this , geared to pulling out the "within structure" in order to assess the ralationship between y and 1 or more x's are quite presumptive and should be studiously avoided. The correct approach to form "regression type models" between time series is to determine the filter by analyzing the autocorrelative structure within each time series. The resultant series are then examined for structure using OLS methods as the x's and their lafgs have been transformed to orthogonality.