Why Switching Regression Needs Separate Equations
In many economic settings, a unit chooses one of two regimes, and the outcome-generating relationship differs across them. A worker may enter a training program or not; a firm may adopt a technology or retain its current process. The outcome is observed under the regime actually chosen, while the outcome that would have occurred under the other regime is unobserved. If unobserved factors that influence regime choice also affect outcomes, treating choice as unrelated to the outcome disturbances can be misleading.
An endogenous switching regression represents these features with a regime-choice equation and a separate outcome equation for each regime. The equations are specified together, but they describe distinct mechanisms: one determines which regime is selected, and the others describe outcomes under each possible regime.
The Regime-Choice and Outcome Equations
A common regime-choice specification is a latent-index model. The index combines observed characteristics \(Z_i\), a parameter vector \(\gamma\), and a disturbance \(u_i\). The observed regime indicator equals one when the index is positive.
The vectors \(Z_i\) and \(\gamma\) have matching dimensions. The latent index \(D_i^*\) is not itself observed; the data record \(D_i\). In a probit specification, \(u_i\) is standard normal, \(u_i\sim N(0,1)\). Fixing its variance to one sets the scale of the latent index: multiplying both \(Z_i'\gamma\) and \(u_i\) by the same positive constant would not change which side of zero the index falls on.
The potential outcome in each regime has its own equation. For a linear specification,
Here \(X_{1i}\) and \(X_{0i}\) are covariate vectors, and \(\beta_1\) and \(\beta_0\) are regime-specific coefficient vectors. They need not contain the same variables, and their coefficients are not required to be equal. Each coefficient describes the specified conditional relationship within its regime, subject to the model assumptions; it does not by itself establish a causal effect. The disturbances \(\varepsilon_{1i}\) and \(\varepsilon_{0i}\) represent unobserved determinants of the corresponding potential outcomes.
The notation \(\rho_j\) is the correlation between \(u_i\) and \(\varepsilon_{ji}\), for \(j\in\{0,1\}\), under this normalization. A nonzero correlation allows unobserved determinants of the outcome in a regime to be related to unobserved determinants of choosing that regime. This is the source of endogenous selection in the model. It does not mean that the observed covariates themselves are necessarily endogenous.
Worked Example: Writing the Equations for Training Choice
Suppose a hypothetical study records whether workers participate in training (\(D_i=1\)) or not (\(D_i=0\)), along with schooling, experience, and hourly earnings. Consider this specified model:
The outcome is measured in dollars per hour; schooling and experience are measured in years. For a hypothetical worker with eligibility equal to 1, 14 years of schooling, and 5 years of experience, the observed part of the choice index is \(-0.40+0.70(1)+0.05(14)=1.00\). If this worker’s disturbance is \(u_i=-0.30\), then \(D_i^*=1.00-0.30=0.70>0\), so the model assigns the worker to training, \(D_i=1\).
If the same worker has \(\varepsilon_{1i}=1.20\), the observed training-regime outcome is \(Y_i(1)=18+0.60(5)+1.20=22.20\) dollars per hour. The no-training outcome equation is still part of the model, but its outcome is not observed for this worker. The example illustrates the regime rule and the equation used for the realized outcome; it does not claim that participation causes any particular earnings difference.
How the Observed Data Are Organized
For each unit, the observed record typically contains the regime indicator \(D_i\), the outcome \(Y_i\) under that regime, and the observed covariates used in the choice and outcome equations. It does not contain both \(Y_i(1)\) and \(Y_i(0)\). The observed outcome can be written by substituting the regime equations into the observation rule:
This expression selects one outcome equation for each unit. When \(D_i=1\), the first multiplier is one and the second is zero; when \(D_i=0\), the reverse holds. The unchosen outcome is not a zero and is not an additional observed response. It is a counterfactual outcome: a model quantity for the regime the unit did not choose.
| Regime indicator | Outcome observed | Outcome not observed |
|---|---|---|
| \(D_i=1\) | \(Y_i=Y_i(1)\) | \(Y_i(0)\) |
| \(D_i=0\) | \(Y_i=Y_i(0)\) | \(Y_i(1)\) |
Worked Example: Constructing the Observed-Outcome Record
The following hypothetical values are hourly earnings in dollars. The entries in the “unobserved potential outcome” column are shown only to emphasize what is missing; they are not recorded in the observed data.
| Worker | \(D_i\) | \(Y_i(1)\) | \(Y_i(0)\) | Observed \(Y_i\) | Unobserved potential outcome |
|---|---|---|---|---|---|
| A | 1 | 22.2 | 17.8 | 22.2 | \(Y_i(0)=17.8\) |
| B | 0 | 21.0 | 19.1 | 19.1 | \(Y_i(1)=21.0\) |
| C | 1 | 24.0 | 20.5 | 24.0 | \(Y_i(0)=20.5\) |
| D | 0 | 18.6 | 17.8 | 17.8 | \(Y_i(1)=18.6\) |
For worker A, the observation rule gives \(Y_i=1(22.2)+0(17.8)=22.2\). For worker B it gives \(Y_i=0(21.0)+1(19.1)=19.1\). The same rule yields \(Y_i=24.0\) for C and \(Y_i=17.8\) for D. In an actual observed-data record, only \(D_i\) and the selected \(Y_i\) would be available, along with the covariates; neither unobserved potential outcome column would be recorded.
Why Correlated Disturbances Matter
The role of correlated errors can be seen directly from the joint-normal assumption. For either regime \(j\), the pair \((u_i,\varepsilon_{ji})\) is jointly normal, with \(\operatorname{Var}(u_i)=1\) and \(\operatorname{Cov}(u_i,\varepsilon_{ji})=\rho_j\sigma_j\). The conditional expectation of one component of a bivariate normal pair given the other is linear:
This follows by substituting the assumed covariance and the normalized variance into the conditional-mean formula. If \(\rho_j\ne 0\), units with different values of the choice disturbance also have different conditional averages of the outcome disturbance in regime \(j\). Since the choice rule depends on \(u_i\), the disturbance relevant to an outcome can be systematically related to who enters a regime. An outcome regression that ignores this dependence may therefore mistake selection-related differences for differences in the outcome relationship.
Worked Example: Interpreting a Choice–Outcome Correlation
Suppose the choice disturbance has variance 1. In a hypothetical model, the regime-1 outcome disturbance has standard deviation \(\sigma_1=3.00\) dollars per hour and correlation \(\rho_1=0.25\) with \(u_i\). The covariance is \(\operatorname{Cov}(u_i,\varepsilon_{1i})=\rho_1\sigma_1=0.25(3.00)=0.750\) dollars per hour.
Under joint normality, at a specified value \(u_i=0.80\), the conditional mean of the regime-1 outcome disturbance is \(E(\varepsilon_{1i}\mid u_i=0.80)=0.25(3.00)(0.80)=0.600\) dollars per hour. This is a conditional expectation given the latent disturbance’s value, not a claim that every worker with that value has a disturbance of 0.600. It illustrates how the outcome disturbance can move with the unobserved factor that enters the regime-choice rule.
The correlation is not the effect of training on earnings, nor is it a difference between regime-specific coefficients. It is a feature of the assumed joint distribution of unobserved disturbances. The sign and size of \(\rho_1\) alone do not identify a causal effect.
What the Observed Data Can and Cannot Show
The model’s observed-data structure has an important consequence: the data reveal the outcome disturbance in only the chosen regime for each unit. They do not reveal both regime-specific disturbances for that unit. A joint distribution for \((\varepsilon_{0i},\varepsilon_{1i})\) may be included in a broader parametric specification, but a cross-regime covariance is not directly observed from paired outcomes, because the two outcomes are never jointly recorded for a unit. The choice–outcome dependence parameters concern the relationship between \(u_i\) and the disturbance for the outcome that may be observed.
The model specification also does not, by itself, guarantee that its parameters are identified. Identification depends on the variation and support of the observed covariates and on the restrictions imposed on the model. The earlier material on exclusion restrictions and functional-form identification addresses those issues. Here the key task is to define the equations, errors, and observation rule without treating an unobserved potential outcome as observed data.
Common Mistakes
- Filling in the unchosen outcome with zero. Zero would be a substantive outcome value, not a missing counterfactual. Record the outcome only under the chosen regime.
- Using one outcome equation for both regimes by default. Switching regression allows \(\beta_1\) and \(\beta_0\) to differ; equality would be an additional restriction, not part of the general setup.
- Assuming regime choice is exogenous without stating it. If the model allows \(\rho_j\ne 0\), the outcome disturbance may be associated with the choice disturbance. That dependence must be represented rather than silently ignored.
- Calling an observed regime difference a causal effect. Outcomes are observed for different chosen groups. Their raw difference does not, on its own, compare both potential outcomes for the same units.
- Confusing the latent index with the observed choice. \(D_i^*\) is a model quantity; the recorded indicator is \(D_i\), determined by whether \(D_i^*>0\).
Check Your Understanding
Answer each question using the model definitions above.
- Write the observation rule for a unit with \(D_i=0\). Which potential outcome is observed?
- Why is the variance of the probit choice disturbance normalized to one?
- In a regime-1 outcome equation, what does \(\rho_1=0\) imply about the choice and outcome disturbances under the stated joint-normal assumptions?
- Explain why the unchosen potential outcome should not be entered as a zero in the observed dataset.
- If \(Y_i(1)=X_{1i}'\beta_1+\varepsilon_{1i}\), which components are observed for a unit who chooses regime 0?