Consider the Poisson regression model $$ Y=\exp\left( X_{1}\beta_{1}+X_{2}\beta_{2} +\varepsilon \right). $$ Is there something like the Frisch-Waugh-Lovell theorem for this model that would allow me to estimate $\beta_{2}$ by regressing $\exp (X_{2})$ on $\tilde{Y}$, where $\tilde{Y}$ are the residuals of a regression of $\exp (X_{1})$ on $Y$?
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1$\begingroup$ This is not a correct expression of Poisson regression: it is a nonlinear additive-error model. The correct expression for Poisson regression is $$E[Y] = \exp(X_1\beta_1+X_2\beta_2).$$ The importance of this is in revealing that the "residuals" have no clear definition. $\endgroup$– whuber ♦Commented Oct 9, 2019 at 13:08
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$\begingroup$ Thanks. I have corrected the expression for the Poisson regression. Please let me know if the question is still unclear. $\endgroup$– Aristide HerveCommented Oct 14, 2019 at 14:22
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$\begingroup$ You might find it instructive to contemplate what your re-expression of the model implies about the $\varepsilon$ terms. Regardless, since you now posit $Y$ is a function of $\exp(X_1),$ how could regressing $Y$ on $X_1$ possibly be relevant? $\endgroup$– whuber ♦Commented Oct 14, 2019 at 14:25
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2$\begingroup$ You're right, the questions was not well written. I hope it is clearer now. From the comments I understand that because of the non-linearity, there is probably no way we can run the regression in two steps as in the linear (OLS) case. $\endgroup$– Aristide HerveCommented Oct 15, 2019 at 7:59
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