Probability Elasticities in Logit and Probit

Why Measure Probability Elasticity?

A marginal effect reports how much a predicted probability changes for a one-unit change in a covariate. That is useful when a unit has a clear scale, but it can be hard to compare across variables measured in different units. A probability elasticity instead describes the predicted probability’s local percentage response to a one-percent change in a covariate. It is a relative measure, and in nonlinear binary-choice models it generally depends on the covariate values at which it is calculated.

Let $Y$ be a binary outcome and let $p(x)=P(Y=1\mid x)$ denote its conditional probability. For a continuous covariate $x_j$, the calculations below hold the other covariates fixed. The covariate must be positive for the usual percentage-change interpretation of its elasticity: percentage changes in a zero or negative value are not defined in the same way.

Definition: The probability elasticity with respect to a positive, continuous covariate $x_j$ is $$ \varepsilon_{x_j} = \frac{\partial p}{\partial x_j}\frac{x_j}{p} = \frac{\partial \log p}{\partial \log x_j}, \qquad p>0. $$ It measures the local percentage change in the predicted probability associated with a one-percent change in $x_j$, holding other covariates fixed. For a small change, $dp/p \approx \varepsilon_{x_j}\,dx_j/x_j$.

An elasticity is not a probability change in percentage points. The marginal effect $\partial p/\partial x_j$ is measured in probability units per unit of $x_j$; the elasticity is dimensionless. The two are related by $\varepsilon_{x_j}=(x_j/p)(\partial p/\partial x_j)$, so knowing one does not determine the other without the covariate value and predicted probability.

General Derivation for a Binary-Choice Probability

In logit and probit models, the conditional probability is a smooth function $F$ of an index $\eta$. In a model with an additive linear index, $\eta=x'\beta$, and the probability is $p=F(\eta)$. For the covariate $x_j$, the chain rule gives $\partial p/\partial x_j=f(\eta)\beta_j$, where $f$ is the derivative of $F$. Substitution into the definition of elasticity yields the general expression below.

\varepsilon_{x_j} = \frac{x_j}{F(\eta)} \frac{\partial F(\eta)}{\partial x_j} = \frac{x_j\beta_j f(\eta)}{F(\eta)}.

This expression separates three influences: $x_j\beta_j$ describes the covariate’s contribution to change in the index, $f(\eta)$ describes how responsive the probability is to the index at that point, and division by $F(\eta)$ converts the probability change into a relative change. Because the index and probability vary with the covariates, the elasticity is generally not just a coefficient.

If a specification includes interactions or another nonlinear term in the index, the index derivative need not equal $\beta_j$. The chain-rule form remains valid with $\partial\eta/\partial x_j$ in place of $\beta_j$:

\varepsilon_{x_j} = \frac{x_j f(\eta)}{F(\eta)} \frac{\partial\eta}{\partial x_j}.

Logit Probability Elasticity

For logit, $F(\eta)=\Lambda(\eta)=1/(1+\exp(-\eta))$. Its derivative is $\Lambda(\eta)[1-\Lambda(\eta)]$. Therefore, for an additive index and $p=\Lambda(\eta)$, the general expression simplifies:

\varepsilon_{x_j} = \frac{x_j\beta_j p(1-p)}{p} = x_j\beta_j(1-p).

The logit coefficient $\beta_j$ is the change in the conditional log-odds for a one-unit increase in $x_j$, holding other covariates fixed; it is not itself a probability elasticity. The probability elasticity also depends on $x_j$ and $p$. With $x_j>0$, its sign is the sign of $\beta_j$. Even with a fixed coefficient, its magnitude can differ across observations because the predicted probability differs.

Worked Example: A Logit Elasticity at a Given Covariate Value

Suppose a hypothetical fitted logit model has index $\eta=-0.4+0.8x$. At $x=2$, calculate the predicted probability, marginal effect, and probability elasticity. Interpret the elasticity locally.

The index and predicted probability are

\eta=-0.4+0.8(2)=1.2, \qquad p=\Lambda(1.2)=\frac{1}{1+\exp(-1.2)}\approx 0.7685.

The marginal effect is $\partial p/\partial x=\beta p(1-p)$, so at this value it is

\frac{\partial p}{\partial x} = 0.8(0.7685)(1-0.7685) \approx 0.1423.

Thus, locally, a one-unit increase in $x$ is associated with an increase of about 0.1423 in the predicted probability, or 14.23 percentage points. The elasticity is

\varepsilon_x = x\beta(1-p) = 2(0.8)(1-0.7685) \approx 0.3704.

A 1% increase in $x$, near $x=2$ and with other covariates fixed, corresponds to an approximate 0.3704% increase in the predicted probability. It does not mean a 0.3704 percentage-point increase. Using the marginal effect, the same 1% increase is a change of about $0.02(0.1423)=0.002846$, or 0.285 percentage points, consistent with a relative change of approximately $0.002846/0.7685=0.370\%$.

Probit Probability Elasticity

For probit, $F(\eta)=\Phi(\eta)$, the standard normal cumulative distribution function, and its derivative is the standard normal density $\phi(\eta)$. Thus, $p=\Phi(\eta)$, and the elasticity for an additive index is

\varepsilon_{x_j} = \frac{x_j\beta_j\phi(\eta)}{\Phi(\eta)}.

Unlike the simplified logit expression, the probit elasticity retains the ratio $\phi(\eta)/\Phi(\eta)$. Both density and cumulative probability change with the index, so the elasticity depends on the index as well as on $x_j$ and its coefficient. For $x_j>0$, $\phi(\eta)>0$ and $\Phi(\eta)>0$, so the elasticity has the sign of $\beta_j$.

Worked Example: A Probit Elasticity and Its Marginal Effect

Consider a hypothetical fitted probit model with index $\eta=-0.5+0.5x$. At $x=1.5$, calculate the predicted probability, marginal effect, and elasticity.

First, the index is $\eta=-0.5+0.5(1.5)=0.25$. The standard normal values are $\Phi(0.25)\approx0.5987$ and $\phi(0.25)\approx0.3867$. The predicted probability is therefore approximately 0.5987.

The probit marginal effect is $\partial p/\partial x=\beta\phi(\eta)$. Substituting the values gives

\frac{\partial p}{\partial x} = 0.5\phi(0.25) \approx 0.5(0.3867) \approx 0.1933.

The elasticity is

\varepsilon_x = x\beta\frac{\phi(\eta)}{\Phi(\eta)} = 0.5(1.5)\frac{\phi(0.25)}{\Phi(0.25)} \approx 0.75\frac{0.3867}{0.5987} \approx 0.4844.

Near $x=1.5$, a 1% increase in $x$ corresponds to an approximate 0.4844% increase in the predicted probability, holding other covariates fixed. The marginal-effect calculation gives the change in probability for a 1% increase in $x$, which is $0.015(0.1933)\approx0.002900$, or about 0.290 percentage points. Relative to the probability 0.5987, that is approximately $0.002900/0.5987=0.4844\%$.

Why Elasticities Depend on Covariate Values

A coefficient determines how a continuous covariate changes the index in an additive model, but the probability response to an index change depends on the current index. Consequently, the marginal effect and the elasticity can vary across observations even when the fitted coefficient is the same. The following logit calculation illustrates both kinds of variation.

Worked Example: Comparing Elasticities Across Covariate Values

Use the hypothetical logit model $\eta=-2+x$, with $x>0$. Calculate the predicted probability, marginal effect, and elasticity at $x=1$, $x=2$, and $x=3$.

At $x=1$, $\eta=-1$, so $p=\Lambda(-1)\approx0.2689$. The marginal effect is $p(1-p)\approx0.1966$ per unit of $x$, and the elasticity is $x(1-p)=1(0.7311)\approx0.7311$.

At $x=2$, $\eta=0$, so $p=0.5000$. The marginal effect is $0.5(0.5)=0.2500$ per unit of $x$, and the elasticity is $2(1-0.5)=1.000$.

At $x=3$, $\eta=1$, so $p=\Lambda(1)\approx0.7311$. The marginal effect is $0.7311(0.2689)\approx0.1966$ per unit of $x$, and the elasticity is $3(1-0.7311)\approx0.8068$.

The marginal effect is largest at $x=2$ among these three values, while the elasticity is largest there as well but differs in scale and interpretation. The estimates demonstrate why a coefficient alone cannot provide an observation-specific probability elasticity: the relevant covariate value and predicted probability are also needed. The calculations are pointwise; they do not describe an average effect over a sample.

Interpreting and Reporting an Elasticity

A clear report should identify the model, the covariate, the evaluation point, and what is held fixed. For example: “At $x=2$, with the other covariates fixed, the fitted logit probability elasticity is 0.3704; a 1% increase in $x$ is associated locally with an approximately 0.3704% increase in the predicted probability.” If the elasticity is negative, the predicted probability falls locally as the covariate rises by a percentage. If the elasticity is near zero, the relative probability response to a small percentage change is correspondingly small at that evaluation point.

Elasticity is a local derivative, not an exact prediction for a large change. For a finite change, the probability should be recalculated at the new covariate value using the fitted model; multiplying an elasticity by a large percentage change need not give an accurate result. Also, the probability in the denominator matters: the same absolute probability change represents a larger relative change when the baseline probability is smaller.

These formulas describe fitted probability effects at specified covariate values. They do not by themselves quantify sampling uncertainty in an estimated elasticity. A numerical elasticity from estimated coefficients is a plug-in estimate; inferential intervals or tests require accounting for uncertainty in the estimated parameters.

Common Mistakes

Treating a coefficient as the probability elasticity. In logit and probit, the coefficient describes index change (and in logit, log-odds change). The elasticity also depends on the covariate value and the predicted probability.
Confusing relative percent change with percentage points. An elasticity of 0.5 means that a 1% covariate increase corresponds locally to about a 0.5% relative increase in probability, not a 0.5 percentage-point increase.
Using the elasticity for a large finite change as if it were exact. The derivative is local. Recompute the model probability at the new covariate value for a finite change.
Applying the standard percentage interpretation when the covariate is zero or negative. The usual percent change in $x_j$ is not defined at zero and is not generally meaningful for a negative value. Interpret the derivative or choose a suitable positive scale instead.
Ignoring interactions in the index. If the index derivative with respect to a covariate depends on other regressors, use $\partial\eta/\partial x_j$, not just the standalone coefficient, in the chain-rule formula.

Key takeaway: For a positive continuous covariate in a binary-choice model, the probability elasticity is the marginal effect multiplied by $x_j/p$. In an additive logit model it is $x_j\beta_j(1-p)$; in an additive probit model it is $x_j\beta_j\phi(\eta)/\Phi(\eta)$. Both are point-specific relative responses, not constant probability effects.

Check Your Understanding

Use the definitions and formulas above to answer these questions.

In an additive logit model, what additional quantities besides $\beta_j$ are required to calculate the elasticity at an observation?
A logit elasticity is 0.8 at a specified point. What does this say about a 1% increase in the covariate, and what does it not say?
For a probit model, write the elasticity formula in terms of $x_j$, $\beta_j$, $\eta$, $\phi$, and $\Phi$.
Why can two observations with the same fitted coefficient have different probability elasticities?
If a model’s index contains an interaction involving $x_j$, what derivative should replace $\beta_j$ in the general chain-rule expression?