# 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.

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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.

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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. –  Cam.Davidson.Pilon Dec 15 '12 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 Dec 15 '12 at 6:30
yes, though they did not say it in a very friendly way. –  Cam.Davidson.Pilon Dec 15 '12 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. –  Cam.Davidson.Pilon Dec 15 '12 at 6:34
nope they meant it in the regression above. Thanks for the clairifaction. –  Kyle Dec 15 '12 at 7:16