Regression - Interpretation of coefficents I'm a bit stucked with a problem where I suppose the answer is rather simple.
I have a very simple regression model:
$x_{it} = x_{t-1} + \beta_i P_i(d_{it-1} - x_{it-1}) + \epsilon_{it}$
where $\beta$ basically measures the adjustment from the last observed value $x_{t-1}$ towards another observed value $d_{t-1}$ and $P_i$ is a dummy for different products.
In my case, $\beta$ typically lies between 0 and 1.
Now I have several products $i = 1,...,N$ I can compute the adjustment coefficient $\beta_i$ for and my goal is to explain how the different adjustment parameters depend on some product properties, $m_1,m_2$ for example. So easily spoken something like:
$\beta_i = \delta_0 + \delta_1m_{1i} + \delta_2m_{2i} + \epsilon_{i}$ 
From my understanding I can not regress on a regression coefficient, that just doesn't sound right. The most obvious would be to include interaction terms:
$x_{it} = x_{t-1} + (\delta_0 + \delta_1m_{1i} + \delta_2m_{2i} 
)P_i(d_{it-1} - x_{it-1}) + \epsilon_{it}$.
Is this the right approach meaning do the deltas explain the variation among the betas? Are there actually methods, with which I can actually estimate the deltas? FE/RE for example would estimate deltas for each product i, which is not what I want.
Any help is highly appreciated.
Thanks a lot, 
Chris
 A: You need to ask yourself the following two questions
A) The error term is $\varepsilon_{i}$ in both equations. I presume that's just a typo?
B) Why do you need $P_{i}$ if everything is already indexed by ${i}$?
In order to go from the first two equations to the third you need to assume that the error term in the second equation is zero - that is, the adjustment term $\beta_{i}$ depends non-stochastically on the product characteristics, which is definitely what I would do (if you don't, the model gets harder to identify and estimate and I don't know how much better it would fit your deta).
In terms of whether you can estimate the $\delta$s directly, the answer is definitely yes. In terms of how to do it, I only know how to do it in STATA, which has an excellent Maximum Likelihood engine (command mlexp) and that is how I would do it. If you choose to go down that route, you will need to make a distributional assumption on $\varepsilon_{it}$, and write its likelihood function in terms of the parameters and the observed variables.
Hope that helps
