<p>In principle you can just take the model coefficients and solve for <span class="math-container">$\ln x$</span>. If you have:</p>
<p><span class="math-container">$$ \ln y = \hat{\beta}_0 + \hat{\beta}_1 \ln x  \implies \ln x =  -\frac{\hat{\beta}_0}{ \hat{\beta}_1} + \frac{1}{ \hat{\beta}_1} \ln y$$</span></p>
<p>However, this being said even though you can do this mathematically the OLS assumes that the relationship is exogenous, so solving for <span class="math-container">$\ln y$</span> would not allow you to interpret the <span class="math-container">$1/ \hat{\beta}_1$</span> as a causal effect - it is just inverse of the effect of <span class="math-container">$\ln x$</span>.</p>
<p>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 <span class="math-container">$\ln x$</span> 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.</p>

CC BY-SA 4.0
2020-11-24T23:39:00.070Z
2020-11-24T23:39:00.070Z
1muflon1
<p>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?</p>

<econometrics><regression>
Log-Linear Model
Mark Winston
# Log-Linear Model

(Asked by: username_0 on 2020-11-24)

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?

**Tags:** econometrics, regression


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## 1 answers

### Answer by username_1 on 2020-11-24. Score: 1

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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