# REGRESSION :A log-linear interaction term

How do I interpret a log-linear interaction term, is it possible?

My model: $$Y= B_1 + B_2\log X_1 + B_3X_2 + B_4(\log X_1 X_2) + u$$

• A true "interaction term" would be $X_2\log(X_1)$ rather than $\log X_1X_2 = \log(X_1) + \log(X_2)$ (when the $X_i$ are both positive, anyway). Which one did you mean? – whuber Apr 8 '19 at 18:42

Correct Model

As @whuber suggested, your interaction term should consist of the product of $$\log X_1$$ and $$X_2$$. That means your model should be specified as:

$$Y= B_1 + B_2\log X_1 + B_3X_2 + B_4(\log X_1)(X_2) + u$$

In this model, the effect of $$\log X_1$$ on $$Y$$ is assumed to depend on $$X_2$$. Conversely, the effect of $$X_2$$ on $$Y$$ is assumed to depend on $$\log X_1$$.

What does the effect of of $$\log X_1$$ on $$Y$$ look like, given $$X_2$$?

To see how the effect of $$\log X_1$$ on $$Y$$ depends on $$X_2$$, you can re-write the model as follows:

$$Y= B_1 + (B_2 + B_4X_2)\log X_1 + B_3X_2 + u$$

From this new expression of the model, you can see that the effect of $$\log X_1$$ on $$Y$$ is quantified by the slope $$B_2 + B_4X_2$$. In particular, given $$X_2$$, every 1-unit increase in the value of $$\log X_1$$ is associated with a change of $$B_2 + B_4X_2$$ units in the expected value of $$Y$$. This change clearly depends on $$X_2$$.

What does the effect of of $$X_2$$ on $$Y$$ look like, given $$X_1$$?

To see how the effect of $$X_2$$ on $$Y$$ depends on $$\log X_1$$, you can re-write the model as follows:

$$Y= B_1 + B_2\log X_1 + (B_3 + B_4 \log X_1)X_2 + u$$

From this alternative formulation of the original model, you can surmise that the effect of $$X_2$$ on $$Y$$ is quantified by the slope $$B_3 + B_4 \log X_1$$. Thus, given $$X_1$$, every 1-unit increase in the value of $$X_2$$ is associated with a change of $$B_3 + B_4 \log X_1$$ units in the expected value of $$Y$$.