# 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? Jul 17 '18 at 20:24
• @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. 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. Oct 15 '19 at 3:04
• This sounds more like a semi-elasticity to me rather than an elasticity. I also think you might have to divide by 100, like I did here. Oct 15 '19 at 3:59