# Linear regression with logarithmic dependent variable

I have the following regression:

$$\ln(1 + \text{hours}) = \beta_0 + \beta_1 \text{ educ} + \beta_{2-x} \text{ other independent variables} + \varepsilon$$

$\beta_2$ for educ is $0.0642$. How would one interpret this coefficient?

It's fairly straight forward. This is a log-linear model. A one-unit change in $x_i$ is associated with a $100 * \beta_i$ percent change in $y$.

$100\% * 0.0642 = 6.42\%$

We can now interpret it in percentages. For each one unit increase in educ, there is a 6.42% increase in your outcome variable.

ETA in light of whuber's comment:

To be more precise, the relationship between the percent change in $y$ and change in $x$ is:

$100\% * (e^{\beta_i} - 1)$

So,

$100\% * (e^{0.0642} -1) = 6.630564\%$

A whuber states, this is important as the absolute value of $\beta_i$ grows larger than 0.1. The simple $\beta_i * 100$ rule serves as an easy approximation when $|\beta_i| < 0.1$.

• This is only approximately correct. The approximation starts to break down noticeably when $|\beta_i| \gt 0.1$. However, since $y=1+\text{hours}$ in the question, the interpretation ought to concern the original response variable $\text{hours}$ rather than $y$ itself--and now it's no longer such a simple relationship. – whuber Oct 27 '16 at 18:24