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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?

dimitriy
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Macaulay
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  • I think the first part [here](https://stats.stackexchange.com/a/243463/7071) is what you have in mind. – dimitriy Mar 04 '22 at 19:44
  • The log example is covered [here](https://davegiles.blogspot.com/2011/03/dummies-for-dummies.html). – dimitriy Mar 04 '22 at 19:50

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