Two-Step and Full-Information Estimators Under Misspecification

Why Misspecification Changes the Comparison

Heckman two-step estimation and full-information maximum likelihood (FIML) use different amounts of structure. Two-step estimation first models selection, then uses an estimated correction in an outcome regression. FIML specifies and estimates a joint model for selection and outcome. Under a correctly specified model, FIML can use more information and is generally asymptotically more efficient. That comparison does not mean FIML is automatically more reliable when assumptions fail: it relies on a more detailed distributional specification, and misspecification can change the parameter it estimates.

To make the comparison precise, consider an outcome \(Y_i^*\) that is observed only when \(S_i=1\):

S_i=1\{Z_i'\gamma+U_i>0\}, \qquad Y_i^*=X_i'\beta+\varepsilon_i.

Here \(X_i\) and \(Z_i\) are observed covariates, \(U_i\) is the selection error, and \(\varepsilon_i\) is the outcome error. Write \(\eta_i=Z_i'\gamma\). In the selected sample, \(Y_i=Y_i^*\); when \(S_i=0\), the outcome is unobserved. The target \(\beta\) describes the outcome equation, not the outcome values missing from the data by themselves.

Benchmark assumptions: Conditional on \((X_i,Z_i)\), the errors have a bivariate normal distribution with mean zero, \(\operatorname{Var}(U_i\mid X_i,Z_i)=1\), \(\operatorname{Var}(\varepsilon_i\mid X_i,Z_i)=\sigma^2\), and \(\operatorname{Cov}(U_i,\varepsilon_i\mid X_i,Z_i)=\rho\sigma\), where \(|\rho|<1\). The same conditional distribution applies across observations. The selection rule and outcome equation are correctly specified, and the covariates and variation in selection provide identification. These conditional-distribution assumptions are essential: normality of the errors alone, without specifying their relationship to the covariates, does not imply the needed conditional formulas.

What the Two-Step Correction Requires

Under the benchmark assumptions, the selection probability is \(P(S_i=1\mid X_i,Z_i)=\Phi(\eta_i)\). Let \(\phi\) and \(\Phi\) denote the standard normal density and distribution functions, and define the inverse Mills ratio by \(\lambda(\eta)=\phi(\eta)/\Phi(\eta)\). Among selected observations, the outcome error has conditional mean \(\rho\sigma\lambda(\eta_i)\). Thus the selected-sample conditional mean is

E[Y_i\mid S_i=1,X_i,Z_i] =X_i'\beta+\rho\sigma\lambda(\eta_i).

The two-step procedure estimates the selection equation by probit using the observed selection indicators, calculates \(\widehat{\lambda}_i=\lambda(Z_i'\widehat{\gamma})\), and regresses selected outcomes on \(X_i\) and \(\widehat{\lambda}_i\). The coefficient on the correction estimates \(\rho\sigma\). Inference must account for estimating the first-stage parameters and generating the correction; ordinary second-stage standard errors alone generally do not do so.

When the two-step correction is valid: The first-stage estimate must recover the relevant selection index or selection probabilities, and the outcome error's conditional mean among selected observations must have the specified correction form (or a correctly specified alternative). The outcome conditional mean must be correctly specified, and the regressors in the selected-sample regression must identify its coefficients. Joint normality is a sufficient basis for the inverse Mills ratio form under the stated conditional assumptions; it is not necessary if that conditional-mean form is justified by other assumptions. A correct probit equation alone does not guarantee a correct outcome correction.

Worked Example: Calculating the Benchmark Correction

Suppose a hypothetical worker has \(\eta_i=0.5\), and the benchmark model has \(\sigma=2\) outcome units and \(\rho=-0.4\). Calculate the expected outcome error among workers with these covariates who are selected.

The conditional normal assumptions imply \(E[\varepsilon_i\mid S_i=1,X_i,Z_i]=\rho\sigma\lambda(\eta_i)\). At \(\eta_i=0.5\), \(\phi(0.5)\approx0.3521\) and \(\Phi(0.5)\approx0.6915\), so

\lambda(0.5)=\frac{\phi(0.5)}{\Phi(0.5)} \approx\frac{0.3521}{0.6915}=0.5092.

The covariance parameter gives \(\rho\sigma=(-0.4)(2)=-0.8\) outcome units. Therefore,

E[\varepsilon_i\mid S_i=1,X_i,Z_i] =(-0.8)(0.5092)=-0.4074 \quad\text{outcome units}.

Under this model and for these covariates, selected workers' mean outcome error is about \(-0.4074\) units. The result follows from the conditional distribution given the covariates as well as the normal error model; it cannot be inferred from joint normality stated without that conditioning assumption.

What FIML Adds—and What It Assumes

FIML estimates the selection and outcome parameters together from likelihood contributions for both selected and unselected observations. Under the benchmark model, those contributions use the specified selection probability and the joint distribution of the observed outcome and selection event. This can use information that the two-step procedure does not use as efficiently. But the likelihood also commits to the selection equation, the outcome distribution, and the dependence structure between \(U_i\) and \(\varepsilon_i\). Correct specification of the outcome equation by itself is not enough to establish that the full likelihood consistently estimates \(\beta\).

If the full likelihood is correctly specified, the model is identified, and usual regularity conditions hold, FIML is consistent and asymptotically efficient within that parametric model. Two-step estimation can also be consistent under the benchmark assumptions, but it generally does not attain the same asymptotic efficiency. The difference is an asymptotic one; it does not guarantee a smaller error in every finite sample.

Under misspecification, maximum likelihood generally converges to a pseudo-true parameter: the parameter value that best fits the specified likelihood within its restricted family, in the relevant large-sample sense. That value need not equal the target \(\beta\). A robust or sandwich standard error may improve inference about the pseudo-true parameter, but it does not make that parameter equal the desired outcome coefficient.

Worked Example: A Misspecified Selection Link

Consider a hypothetical selection equation whose true disturbance \(U_i\) follows a logistic distribution with variance one, rather than a standard normal distribution. A logistic variable with variance one has scale \(s=\sqrt{3}/\pi\approx0.5513\). Suppose its true index is \(\eta_i=0.5\). Under this logistic selection model,

P(S_i=1\mid X_i,Z_i) =P(U_i>-0.5) =\frac{1}{1+\exp(-0.5/s)} \approx\frac{1}{1+\exp(-0.9069)} \approx0.7124.

A standard normal probit selection equation at the same index instead gives \(\Phi(0.5)\approx0.6915\). The difference is about \(0.7124-0.6915=0.0209\), or 2.09 percentage points. These are probabilities evaluated at the stated index; they are not a claim about the fitted probability after estimating a misspecified probit in a particular sample.

If the true selection link is logistic, a probit first stage need not recover the true selection probabilities or index. The standard normal inverse Mills ratio calculated from that probit therefore does not, in general, equal the actual conditional-mean correction. A two-step outcome regression using it may be inconsistent. FIML based on the same incorrect normal selection link is also misspecified; using a joint likelihood does not repair the link. Either estimator could still be consistent in a special case—for example, if selection is unrelated to the outcome error and the outcome regression remains valid—but that conclusion would require a separate justification.

Misspecification: Which Method Is Vulnerable?

The relevant question is not simply whether a model is “parametric” or “two-step.” It is which conditional restrictions the estimator needs and whether those restrictions hold.

Possible failure	Implication for two-step estimation	Implication for FIML
Incorrect selection equation or link	The estimated index and correction may be wrong, so the selected-sample regression need not recover \(\beta\).	The selection likelihood is misspecified; its joint estimates need not recover \(\beta\).
Correct selection probabilities, but incorrect joint error distribution	May remain consistent if the outcome conditional mean correction is nevertheless correctly specified and other regression conditions hold.	The joint likelihood is misspecified; its efficiency guarantee under the benchmark model no longer applies, and consistency is not automatic.
Correct outcome mean, but incorrect selected-sample correction	The second-stage conditional mean is wrong, so the estimated outcome coefficients may be inconsistent.	The specified joint model can also be wrong; a correct marginal outcome equation does not by itself ensure consistent FIML.
Correct joint parametric model and identification	Can be consistent, with inference adjusted for the generated correction; generally less efficient than FIML.	Can be consistent and asymptotically efficient under regularity conditions.

This comparison explains why two-step estimation is sometimes described as less dependent on the full joint distribution. That description is only relative: the standard two-step procedure still needs a valid selection model and a valid conditional-mean correction. If the entire joint distribution is wrong but the selection probability and correction form remain correct, two-step estimation may retain consistency where normal FIML does not. If the correction form is wrong too, that potential advantage disappears.

Worked Example: Interpreting an Efficiency Comparison

Suppose, purely hypothetically, that in one correctly specified and identified design, the calculated asymptotic variance for a particular coefficient of \(\beta\) is \(0.0400\) for FIML and \(0.0625\) for two-step estimation. Both variances are in squared outcome-coefficient units. The corresponding asymptotic standard errors are

SE_{\text{FIML}}=\sqrt{0.0400}=0.2000, \qquad SE_{\text{two-step}}=\sqrt{0.0625}=0.2500.

The variance ratio is

\frac{\operatorname{Var}(\widehat{\beta}_{\text{two-step}})} {\operatorname{Var}(\widehat{\beta}_{\text{FIML}})} =\frac{0.0625}{0.0400}=1.5625.

In this hypothetical correctly specified design, the two-step estimator's asymptotic variance is 56.25% larger for this coefficient. This calculation illustrates a possible efficiency comparison; it is not a universal ratio. If either estimator is inconsistent under misspecification, comparing its variance with that of a consistent estimator does not establish that it is preferable. Variability around the wrong limiting value is not efficiency for estimating the target.

A Short Derivation of the Correction

The correction's normal-theory basis can be derived conditional on covariates. Under the benchmark assumptions, \(E[\varepsilon_i\mid U_i,X_i,Z_i]=\rho\sigma U_i\). Selection is the event \(U_i>-\eta_i\), so

E[\varepsilon_i\mid S_i=1,X_i,Z_i] =\rho\sigma E[U_i\mid U_i>-\eta_i].

For a standard normal \(U_i\), its density satisfies \(\phi'(u)=-u\phi(u)\). Therefore,

E[U_i\mid U_i>-\eta_i] =\frac{\int_{-\eta_i}^{\infty}u\phi(u)\,du}{P(U_i>-\eta_i)} =\frac{\phi(-\eta_i)}{1-\Phi(-\eta_i)} =\frac{\phi(\eta_i)}{\Phi(\eta_i)} =\lambda(\eta_i).

The numerator equality follows by integrating \(-\phi'(u)\); the last equality uses symmetry of the standard normal distribution. This establishes the correction under the stated conditional Gaussian model. If the conditional distribution of \(U_i\) changes with covariates, or if the joint conditional distribution no longer gives \(E[\varepsilon_i\mid U_i,X_i,Z_i]=\rho\sigma U_i\), this derivation does not establish the same formula.

Practical Interpretation and Common Pitfall

Before choosing between the methods, state the intended outcome parameter and separate three questions: Is the selection equation correctly specified? Is the conditional outcome mean among selected observations correctly specified? Is the full joint distribution used by FIML correctly specified? The answers can differ. Exclusion restrictions and sufficient variation in the selection index support identification, but they do not establish the distributional assumptions or conditional-mean restrictions.

Common mistake: “FIML uses the full data and is more efficient, so it must be more trustworthy when the error distribution is wrong.”
Correction: The efficiency claim applies under the correctly specified, identified likelihood and regularity conditions. With a wrong likelihood, FIML can converge to a pseudo-true value rather than \(\beta\). Two-step estimation may be consistent under weaker distributional assumptions only when its selection and conditional-mean requirements still hold.
Another mistake: Treating a correct probit selection equation as sufficient for a valid second-stage correction.
Correction: The correction must represent the conditional mean of the outcome error among selected observations. The selection probability alone does not establish that relationship.

Key takeaway: Under a correctly specified and identified Gaussian selection model, both approaches can be consistent, and FIML is generally asymptotically more efficient. Under misspecification, neither method has a blanket guarantee: assess the selection equation, the selected-sample outcome mean, and—especially for FIML—the full joint distribution separately.

Check Your Understanding

Answer each question by identifying the assumption that supports the conclusion.

Why does joint normality of \(U_i\) and \(\varepsilon_i\) alone not establish the inverse Mills ratio conditional mean given \(X_i\) and \(Z_i\)?
If the selection probability model is correct but the joint error distribution is not Gaussian, what additional condition could allow two-step estimation to remain consistent?
Why does a sandwich standard error not correct inconsistency caused by a misspecified FIML likelihood?
In what setting is the standard claim that FIML is asymptotically more efficient than two-step estimation justified?
If a probit selection equation is misspecified, why might the standard inverse Mills ratio fail to correct the selected-sample outcome regression?