Assume you have a set of data points and fit a straight line y = a + b x through them the usual way. Then you can determine the standard errors of the parameters a and b as well, i.e. s(a) and s(b).

Using these fit results, you can compute a forecast y* = a + b x*. According to Gauss' error propagation, the standard error of this forecast should be

s(y*) = SQRT((dy*/da)² s(a)² + (dy*/db)² s(b)²) = SQRT(s(a)² + x* s(b)²).

OTOH, there is a 'standard error of the estimate' s(y^) = sy SQRT((n - 1)(1 - r²)/(n - 2)) with sy being the standard deviation of the y-values. This formula is said to describe the scattering of y of the data points around the fit line. Hence s(y^) is independent of x.

**What is the correct formula for the standard error of the forecast y* ?**

If s(y^) then why are you not allowed to apply Gauss here? Is it since a and b are not independent variables?

And I wonder why the error of the forecast should be constant over the entire range of x. I think it should increase when x* exceeds the range covered by the data points.

I looked through some statistics books but found noone really covering this topic (or I did miss it).

Thanks in advance for enlightenment.