Negative values in predictions for an always-positive response variable in linear regression I'm trying to predict a response variable in linear regression that should be always positive (cost per click). It's a monetary amount. In adwords, you pay google for clicks on your ads, and a negative number would mean that google pays you when people clicked :P
The predictors are all continuous values. The Rsquared and RMSE are decent when compared to other models, even out-of-sample:
  RMSE        Rsquared 
1.4141477     0.8207303

I cannot rescale the predictions, because it's money, so even a small rescaling factor could change costs significantly.
As far as I understand, for the regression model there's nothing special about zero and negative numbers, so it finds the best regression hyperplane no matter whether the output is partly negative.
This is a very first attempt, using all variables I have. So there's room for refinement.
Is there any way to tell the model that the output cannot be negative?
 A: I assume that you are using the OLS estimator on this linear regression model. You can use the inequality constrained least-squares estimator, which will be the solution to a minimization problem under inequality constraints. Using standard matrix notation (vectors are column vectors) the minimization problem is stated as
$$\min_{\beta} (\mathbf y-\mathbf X\beta)'(\mathbf y-\mathbf X\beta) \\s.t.-\mathbf Z\beta \le \mathbf 0 $$
...where $\mathbf y$ is $n \times 1$ , $\mathbf X$ is $n\times k$, $\beta$ is $k\times 1$ and $\mathbf Z$ is the $m \times k$ matrix containing the out-of-sample regressor series of length $m$ that are used for prediction. We have $m$ linear inequality constraints (and the objective function is convex, so the first order conditions are sufficient for a minimum).
The Lagrangean of this problem is
$$L =  (\mathbf y-\mathbf X\beta)'(\mathbf y-\mathbf X\beta) -\lambda'\mathbf Z\beta = \mathbf y'\mathbf y-\mathbf y'\mathbf X\beta -  \beta'\mathbf X'\mathbf y+ \beta'\mathbf X'\mathbf X\beta-\lambda'\mathbf Z\beta$$
$$= \mathbf y'\mathbf y -  2\beta'\mathbf X'\mathbf y+ \beta'\mathbf X'\mathbf X\beta-\lambda'\mathbf Z\beta $$
where $\lambda$ is a $m \times 1$ column vector of non-negative Karush -Kuhn -Tucker multipliers. The first order conditions are (you may want to review rules for matrix and vector differentiation)
$$\frac {\partial L}{\partial \beta}= \mathbb 0\Rightarrow  -  2\mathbf X'\mathbf y +2\mathbf X'\mathbf X\beta - \mathbf Z'\lambda  $$
$$\Rightarrow \hat \beta_R = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf y + \frac 12\left(\mathbf X'\mathbf X\right)^{-1}\mathbf Z'\lambda = \hat \beta_{OLS}+ \left(\mathbf X'\mathbf X\right)^{-1}\mathbf Z'\xi \qquad [1]$$
...where $\xi = \frac 12 \lambda$, for convenience, and $\hat \beta_{OLS}$ is the estimator we would obtain from ordinary least squares estimation.
The method is fully elaborated in Liew (1976).
