Consider some positive random variables $X^1, X^2$ and $Y\sim Exp(p)$ where $p=\beta_0+\beta_1X^1 + \beta_2X^2$. We have a random sample $\{X^1_i, X^2_i, Y_i\}$. Now, estimate $\beta_1, \beta_2$ is not hard, e.g. using GLM.
Now let's say that we do not observe $X^2$. Is it still possible to consistently estimate $\beta_1$ (i.e. using only a statistic consisting of $\{X^1_i, Y_i\}$)? If not, is there some assumption (normality etc...) for $X^i$ under which we can consistently estimate $\beta_1$? And some confidence intervals for $\beta_1$?
A nice method is shown by @TomChen here Beta regression, but it doesn't work for a linear $p$.