# When is there no point in using regression?

Under what circumstances would using regression with two given variables not increase accuracy of prediction?

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please be more specific. Since you do not put any conditions on the model or what are you trying to predict. Take for example two independent random variables $X$ and $Y$, regression will not increase accuracy of prediction of $Y$ using $X$, since they are not related. But that is probably not the answer you are looking for. –  mpiktas Apr 19 '11 at 9:34
"Not increase accuracy of prediction?" Compared to what alternative? –  user603 Apr 19 '11 at 19:06

Considering the OLS case

$$Y_{i}=\alpha+\beta X_{i}$$

One case is when you try to predict using values of $X_{i}$ outside your sample range (extrapolation). Say if your data had $1<X_{i}<10$ in the sample, and you try to predict for when a new value is $X=100$.

In OLS you have a prediction interval for a new value $X_{p}$ of:

$$\hat{Y}_{p}=\overline{Y}+\hat{\beta}(X_{p}-\overline{X})\pm t_{\alpha/2}^{(n-2)}\hat{\sigma}_{Y|X}\sqrt{1+\frac{1}{n}+\frac{X_{p}-\overline{X}}{\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}}$$

Compare this to a prediction interval without using $X$

$$\hat{Y}_{p}=\overline{Y}\pm t_{\alpha/2}^{(n-1)}\hat{\sigma}_{Y}\sqrt{1+\frac{1}{n}}$$

on comparing the two, if the new covariate $X_{p}$ is far enough away from the mean value $\overline{X}$ the OLS prediction interval will get wider and wider, whereas the unconditional interval remains constant. Further if the $X$ variable is uncorrelated with $Y$ so that $\hat{\beta}=0$, then $\hat{\sigma}_{Y|X}\geq \hat{\sigma}_{Y}$ and the prediction interval is always wider.

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+1 extrapolation and prediction intervals is an excellent point! For regression of one scalar onto another, plotting the prediction intervals ought to be standard operating procedure. –  Dikran Marsupial Apr 19 '11 at 14:17