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I am wondering how a logged control variable is interpreted in a linear probability model. The interpretation in the following lin-log model is clear:

(1) y = b0 + b1*log(x1)

Here, a 1% increase in x1 changes y by b1/100 units.

Now I have the following linear probability model with a logged (control) variable:

(2) Pr(y=1|X) = b0 + b1*x1 +b2*log(x2)

Is the interpretation of x2 the same as in (1), i.e. does a 1% change in x2 change the probability of y=1 by b2/100 units(?) ?

Nick Cox
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Wolfgang Schönramer
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1 Answers1

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That is correct. You can derive this by differentiating either expectation with respect to $x_2$ and solving for $b_2$ to get (approximately, after replacing $\frac{\partial y}{\partial x}$ with $\frac{\Delta y}{\Delta x}$ ):

$$\frac{b_2}{100} \approx \frac{\Delta y}{100 \cdot \frac{\Delta x_2}{x_2}}$$

The numerator is a change in the outcome, in levels for (1) or in percentage points in (2). The denominator is percentage change in $x_2$.

dimitriy
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