# #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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