# How to estimate model coefficients for option-like response: ie. response is related to variables in the form y = max( formula, 0)

Rewrite

Hopefully someone can point me to a resource on how to estimate the parameters I'm trying to model. I've had trouble giving a title to my question and googling for resources.

Suppose $$y$$ represents the value of something over time. In an ideal setting it can be modeling as the difference between two values:

1. $$x_{1}(r,T,t,M)$$ which behaves like $$M(1-e^{r(T-t)})/(1-e^{rT})$$ which represent how much you have of something
2. $$x_{2}(r,t,N)$$ which behaves like $$N*(0.9)^{t}$$ which represent how much the thing you have is worth
3. $$r,T,t,M,N$$ are all values associated with a single data point
4. $$y = max( x_{1} - x_{2}, 0 )$$

The response observed response $$y$$ is truncated at zero, so that it is a positive continuous variable with a mass at zero. From my understanding this kind of looks like modeling an option-like response.

The formula $$y = max( x_{1} - x_{2}, 0 )$$ should work in theory. In practice it does, the difference between the response and the formula value has a mean and median near zero, but is very heavy tailed. I've applied linear models tried to see if my other variables could explain the residuals (difference between y and my model). But that doesn't work all that well, and it doesn't make as much theoretical sense.

Instead what I'd like to do is adjust $$x_{2}$$ or $$x_{1}$$ according to other variables in the data. For instance, suppose $$x_{3}$$ is a categorical variable, and that for certain values of $$x_{3}$$ we'd expect $$x_{2}$$ to be smaller at any given point in time by a fixed proportion. That means, for certain values of $$x_{3}$$ we'd expect y to look like $$y = max( x_{1} - \beta_{2}*x_{2}, 0 )$$ where $$\beta_{2} < 1$$

My question is how to go about estimating $$\beta_{2}$$? In the more general setting, how do you estimate coefficients $$y = max( \beta_{1}x_{4}x_{1} - \beta_{2}*x_{3}x_{2}, 0 )$$

My initial reaction was to try to use a sort of glm technique to estimate the coefficents for this, but the function isn't expressible linearly. And that is, at the moment, the extent of the techniques I know how to apply. However, I do have a formula that relates my data to the response that works in theory. I think this situation does come up in financial applications so there must be a resource that can guide me. My statistical know-how is limited (I've read Casella and Berger but i'm no expert). Frankly, I just don't know what to do in this situation and were to look. Any help would be appreciated

First you can write $$z_1 = x_1 * x_3$$ and $$z_2 = x_2 * x_4$$. Then your model is: $$y = \max(\beta_1z_1 + \beta_2z_2, 0)$$ where I incorporated the minus sign into $$\beta_2$$.
Now, for the values of $$y > 0$$ the formula reduces to $$y = \beta_1z_1 + \beta_2z_2$$ which is a usual linear model. This means that you could try estimating your parameters from the $$y > 0$$ sub-population.
This leaves you with the problem of integrating the $$y = 0$$ sub-population. I would suggest taking a look at inflated and zero-inflated distributions.
• FYI I will rewrite my question with more detail to be clearer (I can tag it better too). I don't think this fits my situtation. The region where y>0 is not "fixed" by the equation. For certain values $x_{i}$ the equation will be zero, and be underestimating y. Adjusting beta2 down for for certain factors would increase the formula. Its the level of adjustment I want to model. This may become clearer in my rewrite – RamenZzz Apr 17 at 14:42