#StackBounty: #regression #normal-distribution #multiple-regression #regression-coefficients Combining regression estimates by summing

Bounty: 50

I want to know if one can combine regression estimates from panel regressions when the new dependent variable is a sum of the dependent variables from previously estimated regressions.

To be concrete:

Regression 1:

$Y_{it}^1 = alpha_i + gamma_t + X_{it}beta_1 + epsilon_{it}$

Regression 2:

$Y_{it}^2 = alpha_i + gamma_t + X_{it}beta_2 + epsilon_{it}$

Regression 3:

$Y_{it}^{sum} = alpha_i + gamma_t + X_{it}beta_3 + epsilon_{it}$

where $Y_{it}^{sum} = Y_{it}^{1} + Y_{it}^{2}$ and $alpha_i$ and $gamma_t$ are person and time-fixed effects.

So, my question is: Is there a connection between $beta_3$ and $beta_1$, $beta_2$?

By definition of linear regression we should have $Y_1= mathcal{N}(Xbeta_1,sigma_1^2)$, $Y_2= mathcal{N}(Xbeta_2,sigma_2^2)$, and hence $Y_{sum}= mathcal{N}(Xbeta_1+ Xbeta_2,sigma_1^2+sigma_2^2)$.

So, then should we have $beta_3=beta_1+ beta_2$? The empirical results don’t agree with that. What am I missing? Do person and time fixed effects complicate things?

Also, how about the case when it is a poisson regression instead of a linear regression?

Any help is greatly appreciated!


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