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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 $0.2$.

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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 $x$.

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