Full-Information Estimation of Endogenous Switching

Why Combine Choice and Outcome Information?

In an endogenous switching model, an observed outcome comes from the regime an individual chooses or enters. If the unobserved factors affecting regime choice are related to the factors affecting outcomes, the observed outcomes are not simply random samples from the two regime-specific outcome distributions. A likelihood for the model must therefore account for both which regime is observed and the outcome observed within that regime.

Full-information maximum likelihood (FIML) does this by constructing one joint likelihood from the regime-choice equation and the relevant outcome equation for each observation. “Full information” refers to estimating the model parameters jointly from this likelihood; it does not mean that outcomes in unchosen regimes are observed.

Here \(z_i\) contains the choice-equation covariates and \(x_{ij}\) contains the outcome covariates for regime \(j\). The two correlations \(\rho_0\) and \(\rho_1\) need not be equal. The likelihood contribution for an individual uses the error pair associated with that individual’s observed regime. A joint distribution linking the two potential-outcome errors is not needed to write this observed-data likelihood.

Deriving the Regime-Specific Contributions

Write \(\eta_i=z_i'\gamma\), and for the observed regime \(j\) define the residual and its standardised value by \(e_{ij}=Y_i-x_{ij}'\beta_j\) and \(t_{ij}=e_{ij}/\sigma_j\). The marginal outcome density at \(Y_i\) is \(\phi(t_{ij})/\sigma_j\). It must be multiplied by the probability that the individual is in the observed regime conditional on that outcome residual.

For a regime-1 observation, choice requires \(u_i>-\eta_i\). Given \(\varepsilon_{i1}=e_{i1}\), bivariate normality gives \(u_i\) a conditional mean of \(\rho_1t_{i1}\) and conditional variance \(1-\rho_1^2\). Thus the conditional choice probability is the upper-tail probability at \(-\eta_i\), which equals the standard normal CDF shown below. For a regime-0 observation, choice requires \(u_i\leq-\eta_i\), and the corresponding conditional probability is a lower-tail probability.

These are joint densities for the observed outcome and regime indicator, conditional on the covariates. For example, \(f_i(Y_i,D_i=1)\) is a density in \(Y_i\), not a probability that a continuous outcome equals one exact value. The regime indicator’s probability is incorporated through the conditional CDF factor.

Joint Estimation and a Small Likelihood Calculation

FIML estimates the choice parameters, both outcome coefficient vectors, the outcome standard deviations, and the selection-outcome correlations by maximizing \(\ell(\theta)\) jointly. The same parameters determine both the density and conditional probability in each contribution. A change in an outcome coefficient, for instance, changes the residual and can therefore change both factors. Numerical optimization is generally used, with parameterizations or constraints that preserve \(\sigma_j>0\) and \(|\rho_j|<1\).

A practical calculation proceeds observation by observation. First identify the observed regime, then compute that regime’s residual and standardized residual, evaluate its density and conditional CDF, and finally sum the log contributions. The unobserved potential outcome does not receive a separate outcome-density factor.

Why the Correlation Changes the Contribution

The correlation parameter affects how informative an observed outcome residual is about regime choice. If \(\rho_j=0\), the selection error and the outcome error in regime \(j\) are independent under the stated joint-normal assumption. The conditional regime probability then does not depend on the outcome residual. If \(\rho_j\neq0\), a residual changes the conditional distribution of the selection error and therefore changes the probability factor in the likelihood.

Optimization, Inference, and Interpretation

After maximizing the log-likelihood, uncertainty for the estimated parameters is commonly based on the curvature of the log-likelihood at its maximum. Under regularity conditions and local identification, the inverse of the negative observed Hessian provides the usual model-based covariance estimate. The resulting standard errors support inference on parameters such as \(\rho_j\), outcome coefficients, or contrasts built from them. A coefficient in an outcome equation describes the corresponding conditional outcome specification; interpretation should respect its units and the model’s assumptions.

A well-behaved numerical maximum is not enough on its own. The model requires the specified conditional distributions and the covariate variation needed to identify its parameters. In particular, the choice and outcome parameters must be distinguished by the information in the data. Exclusion restrictions and functional-form identification are important issues in that assessment, but the likelihood formula does not make weak identification disappear.

Common Pitfall

The exact factorization is \(f(Y_i\mid D_i=j,\text{covariates})P(D_i=j\mid\text{covariates})\); the conditional outcome density already accounts for selection. In general, multiplying the marginal regime-\(j\) outcome density \(f(Y_{ij}\mid x_{ij})\) by \(P(D_i=j\mid z_i)\) is incorrect when \(\rho_j\neq0\), because the outcome residual and selection error are then dependent. Equivalently, the joint contribution can be written as the marginal outcome density times \(P(D_i=j\mid Y_i,\text{covariates})\), represented by the conditional CDF in the likelihood formula. Another mistake is to include both regime-specific outcome densities for every person: only one outcome is observed, so the unchosen outcome is integrated out rather than entered as if observed.