I know that the log-linear model shows the percent change in y if there is a one-unit change in x but how would you solve for it the other way to show a percent change in x if there is a one-unit change in y?


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In principle you can just take the model coefficients and solve for $\ln x$. If you have:

$$ \ln y = \hat{\beta}_0 + \hat{\beta}_1 \ln x \implies \ln x = -\frac{\hat{\beta}_0}{ \hat{\beta}_1} + \frac{1}{ \hat{\beta}_1} \ln y$$

However, this being said even though you can do this mathematically the OLS assumes that the relationship is exogenous, so solving for $\ln y$ would not allow you to interpret the $1/ \hat{\beta}_1$ as a causal effect - it is just inverse of the effect of $\ln x$.

If you believe that there is relationship that goes both ways you should be using some model that allows for that. A vector autoregression (VAR) would be one example of such model. VAR would estimate the relationship going both ways giving you separate set of coefficients when $\ln x$ is the dependent variable. However, this is just one example, there is a cornucopia of models that can help deal with endogeneity. For overview of them you can see Verbeek (2008) A Guide to Modern Econometrics, 4th ed.


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