Appropriate regression-like model where the response is on half-integers 
What is an appropriate model for the above scatter plot? I am not fully satisfied with a simple linear regression model. Any suggestions? Y in this problem is discrete in nature. It only increments by 0.5.
 A: You could modify your linear function with applying floor/ceiling on your dependent variable (see Wikipedia).
This way you would get a discrete step function that mimics your data (see image below).

Given your regression function $y = 0.009x - 0.002$, you have to transform it in the following way: $y = ⌊0.009(x+offset) - 0.002⌋$ and select the $offset$ in a way that maximizes your fit, probably somewhere around $50$.
A more general (and complicated) solution would be to define constant linear functions for every possible Y-value. Then, you have to select intervals on the X-scale to specify where you use each of your constant functions in order to maximize your fit. Obviously, not all constant functions will be used.
Given your plot, I would select something like this (as a rule of thumb):
$$
f(x) = \left\{ 
  \begin{array}{l l}
    -2 & \quad \text{x < -150} \\
    -1 & \quad \text{-150 <= x < -25} \\
     0 & \quad \text{-25 <= x < 50} \\
     1 & \quad \text{50 <= x < 125} \\
     2 & \quad \text{x >= 125} \\
  \end{array} \right.
$$
