Marginal Effects in Truncated Regression

Why Truncation Changes a Marginal Effect

In a truncated regression, outcomes below a threshold are not recorded in the sample at all. The regression may describe a latent outcome for everyone in the target population, but the observed data describe only those who cross the threshold. As a result, a covariate can affect the mean of the latent outcome and also change which outcomes enter the observed sample. The mean among selected observations therefore need not have the same slope as the latent mean.

This tutorial considers selection above a fixed threshold. It derives the selected-sample conditional mean and its marginal effect for a continuous covariate, then contrasts those quantities with the regression’s latent mean and slope. The distinction is about the population being described: neither effect should be interpreted as causal without additional assumptions that justify a causal interpretation.

The normality, constant-variance, and fixed-threshold assumptions matter for the formulas below. In particular, if the threshold varies with covariates, or the conditional variance depends on covariates, differentiating the conditional mean requires additional terms. The earlier material on the truncated regression likelihood addresses how the selection rule enters estimation; here the focus is the interpretation and calculation of marginal effects.

The Conditional Mean Above the Threshold

Let \(\phi\) and \(\Phi\) denote the standard normal density and cumulative distribution functions, and let \(Q(a)=1-\Phi(a)\) be the upper-tail probability. For a given covariate vector \(x\), define the standardized threshold and the upper inverse Mills ratio by

The selection probability at \(x\) is \(P(Y^*>c\mid x)=Q(a(x))\). The selected-sample mean is the expected latent outcome conditional on passing that threshold.

The added term is positive: the selected outcomes have, on average, been drawn from the upper part of the conditional outcome distribution. Its size depends on how restrictive the threshold is relative to \(\mu(x)\). When the threshold is far below the latent mean, most outcomes are selected and the adjustment is small. When it is high relative to that mean, selection is more restrictive and the adjustment is larger.

Deriving the Mean and Its Marginal Effect

The conditional-mean formula follows by standardizing the outcome. Given \(x\), write \(Z=(Y^*-\mu(x))/\sigma\), so \(Z\sim N(0,1)\) and selection is equivalent to \(Z>a(x)\). Then

For the last equality, use \(\phi'(z)=-z\phi(z)\). Thus \(\int_a^\infty z\phi(z)\,dz=[-\phi(z)]_a^\infty=\phi(a)\), since \(\phi(z)\) tends to zero as \(z\) tends to infinity. Dividing by \(Q(a)\) conditions on being above the threshold.

Now consider a continuous covariate \(x_j\), holding the other covariates fixed. Since \(\partial\mu(x)/\partial x_j=\beta_j\), the standardized threshold changes at rate \(\partial a(x)/\partial x_j=-\beta_j/\sigma\). To differentiate the selection adjustment, the derivative of the inverse Mills ratio is

Applying the chain rule gives the marginal effect in the selected sample:

The latent conditional mean has marginal effect \(\partial\mu(x)/\partial x_j=\beta_j\). The selected-sample effect multiplies that slope by a factor that depends on the threshold’s position relative to \(\mu(x)\). This factor is the variance of a standard normal variable conditional on exceeding \(a(x)\): indeed, \(E[Z\mid Z>a]=\lambda(a)\), \(E[Z^2\mid Z>a]=1+a\lambda(a)\), and subtracting the squared conditional mean gives \(1+a\lambda(a)-\lambda(a)^2\). It is positive and less than one. Thus, under this model, the selected-sample effect has the same sign as \(\beta_j\) but a smaller magnitude. Its size varies across covariate values even though the latent-mean slope is constant.

Worked Calculations

Reporting and Interpreting Marginal Effects

For a continuous covariate, the formula gives a local effect at a particular \(x\). Since \(a(x)\) varies across observations, reporting only the effect at the covariate means need not describe the typical effect. An average marginal effect can instead be calculated by evaluating the effect at each covariate vector in a specified target set and averaging:

Here \(T\) denotes the observations used to define the target covariate distribution, and \(n_T\) is their number. The target should be stated: averaging over the full intended population’s covariates and averaging over observed selected cases answer different descriptive questions. Uncertainty for estimated marginal effects can be calculated with the delta method, using the estimated parameter covariance matrix, or with a suitable bootstrap procedure. The resulting interval concerns the specified marginal effect, not the probability that a parameter is inside the interval.

For a binary or other discrete covariate, report a finite change in the conditional mean, as in the third example. The derivative formula is for a continuous covariate and need not equal a finite difference for a discrete change. In either case, label the result clearly as an effect on the latent conditional mean or as a contrast in the selected-sample conditional mean. Unless the design and identifying assumptions support causal claims, describe these as model-based associations or conditional-mean changes rather than causal effects.

Common Pitfall

A related interpretive error is to say that the selected-sample effect is “the effect for people who would be observed.” The formula describes how the conditional mean among those above the threshold changes as \(x\) changes. Because the set passing the threshold can also change, it is not generally a comparison of an unchanged group of individuals.