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Suppose we have estimated parameters $\hat{\beta} = [\hat{\beta}_{0}, \hat{\beta}_{1}, \hat{\beta}_{2}]$ from a level-level regression: $$\hat{y} = \hat{\beta}_{0} + \hat{\beta}_{1}X_{1} + \hat{\beta}_{2}X_{2}$$ I want to use these estimates to set a prior on $\gamma_{1}$ in a Bayesian log-level regression: $$\log(y) = \gamma_{0} + \gamma_{1}X_{1} + \gamma_{2}X_{2} + e$$

Is it possible to derive an expression for $\gamma_{1}$ from the $\hat{\beta}$s?

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  • $\begingroup$ I think the first part here is what you have in mind. $\endgroup$
    – dimitriy
    Mar 4, 2022 at 19:44
  • $\begingroup$ The log example is covered here. $\endgroup$
    – dimitriy
    Mar 4, 2022 at 19:50
  • $\begingroup$ So gamma_1 = dy/dx * 1/mean(y_hat) = beta_1 / mean(y_hat)? $\endgroup$
    – Macaulay
    Mar 11, 2022 at 1:19
  • $\begingroup$ It won't be exact, only approximate. $\endgroup$
    – dimitriy
    Mar 11, 2022 at 3:03
  • $\begingroup$ Where does the approximation error come from? $\endgroup$
    – Macaulay
    Mar 11, 2022 at 22:30

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