Conditional Regime Means and Treatment Gains in Switching Models

Why Conditional Regime Means Matter

In an endogenous switching model, individuals select into one of two regimes, and each regime has its own potential outcome equation. The outcome observed for an individual is therefore determined by the chosen regime, while the outcome that would have occurred under the other regime is counterfactual. Because selection can depend on unobserved factors related to outcomes, the mean outcome among people in a regime need not equal that regime's mean outcome for people who selected elsewhere.

The bivariate-normal selection correction makes this distinction explicit. It gives conditional means for each potential outcome among people selected into either regime. Comparing the two potential outcomes within a selected group yields a conditional treatment gain: the gain for those who chose regime 1, or for those who chose regime 0. These are not generally the same as the population-average gain.

Definition: Let \(D=1\) denote selection into regime 1 and \(D=0\) selection into regime 0. Let \(Y_1\) and \(Y_0\) be an individual's potential outcomes under regimes 1 and 0, respectively. The observed outcome is \(Y=DY_1+(1-D)Y_0\). A treatment gain is the difference \(Y_1-Y_0\); its conditional mean depends on the population over which the difference is averaged.

Model and Selection Correction

Write the regime-selection rule and potential outcome equations as

D=1[z'\gamma+u>0],\qquad Y_1=x'\beta_1+\varepsilon_1,\qquad Y_0=x'\beta_0+\varepsilon_0.

Here \(x\) and \(z\) are observed covariate vectors, \(\gamma\) governs selection, and \(\beta_1,\beta_0\) are regime-specific outcome coefficients. The selection disturbance \(u\) is normalized to have variance 1, so the selection index is \(a=z'\gamma\). The model conditions on the observed covariates; for the formulas below, the disturbances have conditional mean zero before selection, and each pair \((u,\varepsilon_j)\) is jointly normal with \(\operatorname{Var}(u)=1\) and \(\operatorname{Cov}(u,\varepsilon_j)=\sigma_{ju}\), for \(j=0,1\). The outcome equations are linear in the stated covariates. These distributional assumptions imply a probit selection probability \(P(D=1\mid z)=\Phi(a)\).

Define the inverse Mills ratio \(\lambda(t)=\phi(t)/\Phi(t)\), where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution functions. The two relevant correction terms are \(\lambda(a)\) for selection into regime 1 and \(\lambda(-a)\) for selection into regime 0.

Relate outcome and selection disturbances.
Joint normality implies \(E[\varepsilon_j\mid u]=\sigma_{ju}u\), since \(\operatorname{Var}(u)=1\).

Condition on the selection event.
Regime 1 means \(u>-a\), whereas regime 0 means \(u\leq -a\). Truncating the standard normal distribution gives \(E[u\mid u>-a]=\lambda(a)\) and \(E[u\mid u\leq-a]=-\lambda(-a)\).

Obtain the outcome-error means.
Taking conditional expectations of \(E[\varepsilon_j\mid u]\) gives \(E[\varepsilon_j\mid D=1,x,z]=\sigma_{ju}\lambda(a)\) and \(E[\varepsilon_j\mid D=0,x,z]=-\sigma_{ju}\lambda(-a)\).

For completeness, the truncated-normal identities in Step 2 follow from \(\phi'(u)=-u\phi(u)\): integrating \(u\phi(u)\) from \(-a\) to infinity gives \(\phi(a)\), and dividing by \(P(u>-a)=\Phi(a)\) gives \(\lambda(a)\). Integrating from negative infinity to \(-a\) gives \(-\phi(a)\); dividing by \(P(u\leq-a)=\Phi(-a)\) gives \(-\lambda(-a)\). Thus the signs differ across selection groups.

Result: Conditional on covariates, the expected outcome under either regime and either selection status is

\begin{aligned} E[Y_1\mid D=1,x,z]&=x'\beta_1+\sigma_{1u}\lambda(a),\\ E[Y_0\mid D=1,x,z]&=x'\beta_0+\sigma_{0u}\lambda(a),\\ E[Y_1\mid D=0,x,z]&=x'\beta_1-\sigma_{1u}\lambda(-a),\\ E[Y_0\mid D=0,x,z]&=x'\beta_0-\sigma_{0u}\lambda(-a). \end{aligned}

The first and fourth expressions are observed-regime means; the second and third are counterfactual means. Each correction uses the covariance between the corresponding regime's outcome error and the selection disturbance.

Conditional Treatment Gains

Subtract the conditional mean under regime 0 from that under regime 1 within the same selection group. For people selected into regime 1, the treatment gain is

E[Y_1-Y_0\mid D=1,x,z] =x'(\beta_1-\beta_0)+(\sigma_{1u}-\sigma_{0u})\lambda(a).

For those selected into regime 0, it is

E[Y_1-Y_0\mid D=0,x,z] =x'(\beta_1-\beta_0)-(\sigma_{1u}-\sigma_{0u})\lambda(-a).

These are commonly called the treatment effect on the treated (ATT) and the treatment effect on the untreated (ATU), respectively, when regime 1 is the treatment. More precisely, the displayed expressions are conditional on \(x,z\) as well as selection status. To obtain an ATT or ATU for a group with varying covariates, average the corresponding conditional gain over that group's covariate distribution.

The population-average gain conditional on \(x,z\) is different. Before conditioning on \(D\), both outcome errors have mean zero, so

E[Y_1-Y_0\mid x,z]=x'(\beta_1-\beta_0).

This is a model-based average over both selection groups at the given covariates. It does not require \(\sigma_{1u}=\sigma_{0u}\). Instead, the selection corrections cancel when the group-specific gains are weighted by their selection probabilities.

Worked Example: Four Means and Two Conditional Gains

Consider hypothetical conditional predictions at covariates \(x,z\). Suppose \(a=0.4\), \(x'\beta_1=15\) outcome units, \(x'\beta_0=12\) outcome units, \(\sigma_{1u}=1.00\), and \(\sigma_{0u}=0.50\) outcome units. Standard-normal values give \(\Phi(0.4)=0.6554\), \(\phi(0.4)=0.3683\), \(\lambda(0.4)=0.5619\), and \(\lambda(-0.4)=1.0688\). The selection probability for regime 1 is therefore 0.6554.

Means among those selected into regime 1. The observed-regime mean is \(E[Y_1\mid D=1,x,z]=15+(1.00)(0.5619)=15.5619\) outcome units. Their counterfactual mean under regime 0 is \(E[Y_0\mid D=1,x,z]=12+(0.50)(0.5619)=12.28095\), or 12.2810 outcome units. The conditional gain is \(15.5619-12.28095=3.28095\), or 3.281 outcome units.

Means among those selected into regime 0. Their counterfactual mean under regime 1 is \(E[Y_1\mid D=0,x,z]=15-(1.00)(1.0688)=13.9312\) outcome units. Their observed-regime mean is \(E[Y_0\mid D=0,x,z]=12-(0.50)(1.0688)=11.4656\) outcome units. The gain for this group is \(13.9312-11.4656=2.4656\) outcome units.

The gains also follow directly from the formulas: \(3+(1.00-0.50)(0.5619)=3.28095\) for \(D=1\), and \(3-(1.00-0.50)(1.0688)=2.4656\) for \(D=0\). The baseline regime difference is 3 outcome units, but selection changes each group's conditional gain.

Worked Example: Checking the Population-Average Gain

Use the same hypothetical covariate values and parameters. The probability of regime 1 is \(p=\Phi(0.4)=0.6554\), and the probability of regime 0 is \(1-p=0.3446\). The conditional gains are approximately 3.281 outcome units for \(D=1\) and 2.466 outcome units for \(D=0\).

Weighting the gains by their selection probabilities gives \(0.6554(3.28095)+0.3446(2.4656)\), which is approximately \(2.1503+0.8497=3.000\) outcome units. This equals \(x'(\beta_1-\beta_0)=15-12=3\) outcome units.

The equality is not accidental. Since \(p\lambda(a)=\phi(a)\) and \((1-p)\lambda(-a)=\phi(a)\), the weighted correction terms are

(\sigma_{1u}-\sigma_{0u}) \{p\lambda(a)-(1-p)\lambda(-a)\} =(\sigma_{1u}-\sigma_{0u})\{\phi(a)-\phi(a)\}=0.

Thus, at fixed \(x,z\), the group-specific gains average to the population-average gain when weighted by the model's regime probabilities. This identity does not mean that either selected group's gain equals the population-average gain.

Reading the Corrections and Their Limits

The sign and size of the selection correction depend on both the inverse Mills ratio and the covariance \(\sigma_{ju}\). If a high value of the selection disturbance tends to accompany a high outcome disturbance in regime \(j\), then \(\sigma_{ju}>0\). The conditional mean of that outcome error is consequently positive among those selected into regime 1 and negative among those selected into regime 0. A negative covariance reverses these directions. The size of a correction is in outcome units because \(\lambda(\cdot)\) is dimensionless and \(\sigma_{ju}\) has the units of the outcome.

The counterfactual mean is a model-based expectation, not a directly observed average. Its interpretation relies on the outcome equations, the selection rule, the joint-normality assumptions, and adequate covariate support. If the intended interpretation is causal, the potential-outcome setup also requires consistency and no interference: each individual's observed outcome equals the potential outcome for the regime actually received, and one individual's regime does not change another individual's outcome. A calculated conditional gain is not, by itself, proof that assigning a regime would cause that gain.

In empirical work, the model parameters and selection index are estimated. Substituting estimates into the formulas gives predicted conditional means and gains. Their uncertainty must account for estimation of both the outcome and selection components; a plug-in point estimate alone does not express that uncertainty.

Worked Example: Selection Can Separate the Two Group Gains

Consider a second hypothetical covariate profile with \(a=-0.5\), \(x'\beta_1=20\) outcome units, \(x'\beta_0=18\) outcome units, \(\sigma_{1u}=0.60\), and \(\sigma_{0u}=-0.20\) outcome units. Here \(\phi(0.5)=0.3521\), \(\Phi(-0.5)=0.3085\), and \(\Phi(0.5)=0.6915\). Therefore \(\lambda(a)=\lambda(-0.5)=1.1411\), \(\lambda(-a)=\lambda(0.5)=0.5092\), and the probability of regime 1 is 0.3085.

Among those selected into regime 1, \(E[Y_1\mid D=1,x,z]=20+(0.60)(1.1411)=20.6847\) outcome units, while \(E[Y_0\mid D=1,x,z]=18+(-0.20)(1.1411)=17.7718\) outcome units. Their gain is \(20.6847-17.7718=2.9129\) outcome units, equivalently \(2+(0.60-(-0.20))(1.1411)=2.9129\).

Among those selected into regime 0, \(E[Y_1\mid D=0,x,z]=20-(0.60)(0.5092)=19.6945\) outcome units and \(E[Y_0\mid D=0,x,z]=18-(-0.20)(0.5092)=18.1018\) outcome units. Their gain is \(19.6945-18.1018=1.5927\) outcome units, equivalently \(2-(0.60-(-0.20))(0.5092)=1.5927\). Both group gains exceed zero here, but they differ because the outcome errors have different covariances with selection.

Common Pitfalls

Using the wrong sign for regime 0. The truncated mean of \(u\) among \(D=0\) is \(-\lambda(-a)\), not \(+\lambda(-a)\). That sign carries through to both regime-specific outcome means.
Using one correction for both outcome equations. The correction for \(Y_1\) uses \(\sigma_{1u}\); the correction for \(Y_0\) uses \(\sigma_{0u}\). Their difference determines how selection changes conditional treatment gains.
Calling the observed outcome mean a counterfactual mean. Within \(D=1\), \(Y_1\) is observed and \(Y_0\) is counterfactual; within \(D=0\), \(Y_0\) is observed and \(Y_1\) is counterfactual.
Equating the ATT, ATU, and population-average gain. These average over different groups. They coincide only under special parameter or selection conditions, not as a general property of switching models.
Interpreting a model-based contrast as automatically causal. Causal language requires the potential-outcome and assignment assumptions relevant to the application, in addition to the model's distributional assumptions.

Key takeaway: Compute each potential-outcome mean within each selection group using that outcome equation's own selection covariance and the appropriate inverse Mills ratio. Then subtract within a group to obtain its conditional gain; average across groups only with the population and weights that match the estimand.

Check Your Understanding

Answer each question using the model and assumptions stated above.

For \(a=z'\gamma\), which inverse Mills ratio appears in the conditional outcome means for individuals selected into regime 0, and what is the sign of its correction?
Why can the counterfactual mean \(E[Y_0\mid D=1,x,z]\) differ from the regime-0 mean \(x'\beta_0\)?
Write the conditional gain for individuals selected into regime 1. Which covariance difference determines its selection correction?
At fixed \(x,z\), explain why the selection corrections cancel when the two selection-specific gains are weighted by their selection probabilities.
What additional assumptions are needed before describing a model-based conditional gain as a causal treatment effect?