Regression model when under-estimations costs us more than over-estimations

Question

We have a factory and we are planning how many items produce in 2014. During the learning process we minimize the mean squared error. But, under-estimations costs us more than over-estimations. Let's say we estimated and produced $p$ items, but we have to sell $d$ items.

• In case of $p > d$, getting rid of 1 surplussed item costs us 30\$.
• In case of $p < d$, buying 1 item (from friendly factory) costs us 200\$.

Can this additional data can be used to formulate better objective function? If yes, what it will be?

Answer 1

Here's how. Suppose your regression model takes the form $y_i=x_i'\beta+u_i$. Then let $\hat\beta$ be the minimizer of

$$ \Biggl[200 \sum_{i=1}^n \max(y_i-x_i'\tilde\beta,0) - 30 \sum_{i=1}^n \min(y_i-x_i'\tilde\beta,0)\Biggr]$$

with respect to $\tilde\beta$.

Answer 2

Your function is telling you that if you underestimate, it is costlier to buy than overestimating and get rid of surplus. You need to consider this in your decision. You need some certainty that you overproduce. This is different than utility. You need to add an optimum based on utility. In both cases maybe you gain money but less utility if you underestimate or grossly overestimate.