I have two Poisson regression models using same single feature. Model 1 is
$$E[\log(y)| x] = a_1 + b_1 \log(x)$$
and Model 2 is
$$E[\log(z)|x] = a_2 + b_2 \log(x) $$
When I build Model 3 on $y+z$ like
$$E[\log(y+z)|x] = a_3 + b_3 log(x)$$
is there any relationship between $a_3$, and $b_3$ with $a_1$, $b_1$ and $a_2$, $b_2$?
Thanks a lot!
A math fact is that, for two Poisson processes $Y, Z$ with rates $\lambda_1, \lambda_2$, their sum also follows a Poisson process with rate $\lambda_1 + \lambda_2$.
The rate for $Y$ is $\lambda_1(X) = \exp(a_1 + b_1 \log X)$ and for $Z$ is $\lambda_2(X) = \exp(a_2 + b_2 \log X)$.
When you add these together, the sum process does not follow a linear model, so the model for the $Y+Z$ rate/intensity of $\exp(a_3 + b_3 \log X)$ is misspecified. The actual calculated values of $a_3$ and $b_3$ will vary depending on the actual distribution of $X$ in the sample. It is, however, easy to obtain those values numerically.
If you fit the linear model 3, as you call it, the predicted values with approximate the actual process to varying degrees of accuracy. For values of $X$ very close to 0, the linear approximation to the exponential is good and the parameters are approximately $a_3 \approx a_1 + a_2$ and $b_3 \approx b_1 + b_2$.
