Interpreting regression results for different units of measure

Question

Could you kindly help me with interpreting the results from the Probit model for different units of measure of the covariates?

Consider the Probit Model $$ Y=1\{X\beta+\epsilon \geq 0\} $$ with $\epsilon\perp X$ and $\epsilon \sim N(0,1)$.

Let $\hat{\beta}$ be the MLE estimator of $\beta$.

Suppose $X$ is continuous. Then, the estimated marginal effect of $X$ on $\mathbb{P}(Y=1|X=x)$ when $x=\bar{x}_n$, where $\bar{x}_n$ is the sample average, is $$ \frac{\partial \hat{\mathbb{P}}(Y=1|X=x)}{\partial x} \Big|_{x=\bar{x}_n}=\hat{\beta}_{\text{MLE}} \times \phi(\bar{x}_n\hat{\beta}_{\text{MLE}}) $$ where $\phi$ is the standard normal pdf.

Suppose you get $\frac{\partial \hat{\mathbb{P}}(Y=1|X=x)}{\partial x}=0.2$ and $Y$ is employment status (working/not working). I believe that, for any unit of measure of $x$, $0.2$ means that: a marginal increase of $x$ increases the conditional probability for an individual featuring the values of $X$ at the sample mean to be employed by $0.2$ (or, equivalently, by $20$ percentage points).

Now, I want to ask whether we can (and in positive case, how) refine the interpretation of $0.2$ if

• $X_k$ is income measured in $1,000\$$?

• $X_k$ is the logarithm of the income measured in $1,000\$$? (here I guess there is some percentage variation to use?)

Please forgive the possible no-sense of the variable meaning and/or the number $0.2$.

Answer 1

When you are trying to find the appropriate interpretation of parameters in a regression model, it is best to work with the original model form, and not complicate this by bringing in consideration of the estimators. Now, under your model you have $P(x) \equiv \mathbb{P}(Y=1|X=x) = \Phi ( x \beta )$, which gives:

$$\begin{equation} \begin{aligned} P'(x) &= \frac{\beta}{\sqrt{2 \pi}} \cdot \exp \Big( -\frac{\beta^2}{2} x^2 \Big), \\[10pt] P''(x) &= - \frac{x \beta^3}{\sqrt{2 \pi}} \cdot \exp \Big( -\frac{\beta^2}{2} x^2 \Big). \end{aligned} \end{equation}$$

So you have:

$$\frac{P''(x)}{P'(x)} = - \frac{x \beta^3}{\sqrt{2 \pi}} \Big/\frac{\beta}{\sqrt{2 \pi}} = - x \beta^2.$$

Re-arranging yields the parameter of interest:

$$\beta = \sqrt{- \frac{1}{x} \cdot \frac{P''(x)}{P'(x)}}.$$

From this equation we see that the interpretation of $\beta$ is quite complicated --- it can be interpreted as a quantity pertaining to the rates-of-change of the conditional probability of a positive response. Contrary to the assertion in your question, the model (and hence the interpretation of the parameter) is not invariant to changes in the scale of $x$.