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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

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Maybe I can give some directions.

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.

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  • $\begingroup$ 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 $\endgroup$ – RamenZzz Apr 17 at 14:42

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