Detrending Data

[econmiller]

I think I'm overlooking something very obvious, but I can't figure this out. I'm trying to show different ways of detrending data by using the EViews forecast function. From the outset, I expect that regressing the dependent variable in differences and forecasting the levels will give an identical trend as regressing the dependent variable in levels against @trend. That is d(Y) = c should give the same result as Y = c + b*@trend. But as the attached graph shows, I'm getting very different results. Any thoughts would be greatly appreciated.

Steve

[startz]

There's a big difference in what the two models imply for residuals. The first model is "difference stationary," the second is "trend stationary."

[econmiller]

Thanks Startz,

I had to temporarily divert my attention to other issues. I'm not sure this helps me with an explanation as to why the two approaches result in different slopes. Maybe I'm not fully appreciating the role of stationarity in estimating the linear growth. Detrending from levels appears to generate a preferable estimate of the trend over averaging over the trend stationary data. But conceptually, am I not doing measuring the same thing - the average change, period over period?

Many thanks,
Steve

[startz]

If there were no errors, the two methods would be the same. (One form is like a differential equation and the other is like the solution to that differential equation.) That's why your intuition seems reasonable.

But there are residuals. If the residual in the linear trend equation is u, then the residual in the differenced equation is u-u(-1). So least squares is minimizing different objective functions.

[econmiller]

Ah, I think I'm seeing your point Startz. In essence, since the least squares objective function is measuring two different sets of values, we would not expect the two trend estimates to be equal and that the values that correspond to E[error]=0 and not comparable. Let me play around with this a bit. Thanks Startz!
Steve