# linear probability model interpretation

I have a question regarding the interpretation of a log independent variable in a linear-probability model.

For example: I have $\log(GDP)$ as my independent variable and the coefficient is 0.35. Can I then interpret the result as follows: A one UNIT increase in the log of GDP is associated with a 3.5% increase in the probability that $Y = 1$?

I know that this question was answered elsewhere, however, people talked about a PERCENT instead of unit increase and this confused me.

If your $$y$$ variable is binary, i.e. 0 or 1, then one interpretation of your coefficient can be is as follows: a one unit increase in log GDP would increase $$y$$ by .35.

But let's clarify your other area of concern. Let's express this model as follows $$y_{i} = \beta_{0} + \beta_{1}log(GDP_{i}) + u_{i}.$$

If $$y_{i} \in [0,1]$$, then we can think about the dependent variable as measured in terms of percentage. Right? Recall $$\Delta{log(x)} \approx \%\Delta x$$. What happens when we add a little extra to GDP? Let's call the new dependent variable $$\tilde{y}_{i}$$. Hence, we can write the following $$\tilde{y}_{i} - y_{i} = \beta_{1}\left\{log(GDP_{i} + \delta) - log(GDP_{i})\right\}.$$

Hence,

$$\beta_{1} = \frac{\tilde{y}_{i} - y_{i}}{log(GDP_{i} + \delta) - log(GDP_{i})}$$.

Therefore, if the change in $$y$$ is very small, we can think of numerator as a percentage point change in $$y$$. Thus, it has the interpretation of elasticity, i.e. $$\beta_{1}$$ is the percent point by which the dependent variable responds to a given percentage change in the independent variable. Hopefully that clarifies both areas of confusion for you!

• Thank you very much for your answer. Just to get that right. Since I have a binary outcome in my linear proability model I can interpret the coefficients (i.e 0.35) as 0.35/100 increase in the probability of Y=1. Moreover, when I have the log of an independent variable, I also talk about percent increase instead of unit increase when it comes to the independent variable. Hence, a one percent increase in the log of GDP is associated with a 0.35/100 unit increase in the probability that Y=1? Is that correct? – Julian Jul 17 '18 at 20:24
• @Julian please don't use answers for commenting. – Tim Jul 17 '18 at 20:30
• @Julian not quite. Please carefully read my answer. $\hat{\beta}_{1}$ in this case is the elasticity measure in itself. Hence, a 1% increase in GDP elicits a 0.35% change in $y$. The other interpretation is a one unit increase in log of GDP increases $y$ by 0.35. – JuliusBilly Jul 18 '18 at 1:30
• Can you clarify why the dependent variable is measured in terms of percentage when it is binary? The numerator seems to be in percentage points, rather then percentage. – Dimitriy V. Masterov Oct 15 '19 at 3:04
• @DimitriyV.Masterov very, very good catch. I updated the response. – JuliusBilly Oct 15 '19 at 3:54