by anna
Last Updated May 16, 2018 14:19 PM

If we have the following general regression:

$$ln(Y)=\beta_0+\beta_1 X_1$$

Then it can be interpreted as an increase of 1 unit in $X_1$ will increase $Y$ by $100 \times \beta_1\%$.

But what if the increase in $X_1$ is not 1 unit but say 0.5. Would it then be interpreted as "an increase in $X_1$ by 0.5 unit will increase $Y$ by $(100\beta_1 \times 0.5)%$?

An additive increase in $X_1$ by $\Delta$ (i.e. $X_1' = X_1 + \Delta$) leads to an additive increase of $\ln(Y)$ by $\beta_1 \Delta$: $$\ln(Y') = \ln(Y) + \beta_1 \Delta$$

This is the same as a **multiplicative** increase of $Y$ by $\exp(\beta_1 \Delta)$:
$$\begin{aligned}
Y' &= \exp(\ln(Y')) \\
&= \exp(\ln(Y) + \beta_1 \Delta) \\
&= \exp(\ln(Y)) \times \exp(\beta_1 \Delta) \\
&= Y \times \exp(\beta_1 \Delta)
\end{aligned}$$

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