Tests of Selection Dependence and Model Restrictions

Why Test Restrictions in a Selection Model?

In a sample-selection model, an outcome is observed only for units that satisfy a selection rule. If unobserved factors affecting the outcome are correlated with unobserved factors affecting selection, the observed outcomes may not represent the outcome equation’s errors as if they had been sampled independently. A central diagnostic question is therefore whether the outcome and selection errors are correlated. Likelihood-based tests also let us assess joint restrictions on several model parameters rather than considering one coefficient at a time.

A test of zero error correlation is informative only relative to the maintained model: its functional form, distributional assumptions, covariates, and identification. Rejection is evidence against the specified restriction within that model; it does not, by itself, identify why the restriction fails.

Model and Restrictions

For unit \(i\), let \(d_i^*=z_i'\gamma+u_i\) be a latent selection index, and observe \(d_i=1\) when \(d_i^*>0\). The outcome \(y_i^*=x_i'\beta+\varepsilon_i\) is observed only when \(d_i=1\). Use the Gaussian sample-selection model with independent observations and

Here \(x_i\) and \(z_i\) are observed covariate vectors, \(\beta\) and \(\gamma\) are coefficient vectors, and the selection-error variance is normalized to one. The parameter \(\rho\) is the correlation between the outcome and selection errors. The null \(H_0:\rho=0\) says those errors are uncorrelated; under the joint normality assumption, it implies their independence. Other restrictions can involve components of \(\beta\) or \(\gamma\), or several parameters jointly.

To make the likelihood explicit, define \(e_i=(y_i-x_i'\beta)/\sigma\) for selected units, and let \(\phi\) and \(\Phi\) denote the standard normal density and distribution functions. Conditional on covariates, the likelihood contribution is

The first contribution is the joint density of an observed outcome and selection; the second is the probability of nonselection. Let \(\ell(\theta)\) be the sum of the log contributions, where \(\theta\) collects all free parameters. The unrestricted fit maximizes \(\ell(\theta)\) over the full parameter space. A restricted fit maximizes it subject to the null restriction. Both fits must use the same observations and likelihood specification.

Three Tests of Restrictions

Write a set of \(q\) restrictions as \(H_0:g(\theta)=0\), where \(g\) is a vector-valued function. The alternatives allow at least one restriction to fail. Each test below has a large-sample chi-square reference distribution with \(q\) degrees of freedom when the model is correctly specified, the restrictions are locally identified, and the usual likelihood regularity conditions hold. In particular, the null should not place a parameter on a boundary.

For a scalar restriction \(H_0:\rho=0\), this reduces to \(W=(\hat\rho/SE(\hat\rho))^2\), with one degree of freedom. For a vector of coefficients set to zero, \(g(\theta)\) is that coefficient vector and the covariance matrix must include its off-diagonal elements. The test is not generally the sum of separate squared \(t\)-statistics.

The unrestricted maximum cannot have a lower log-likelihood than the restricted maximum, so \(LR\) is nonnegative. This comparison tests the restrictions jointly and measures the log-likelihood cost of imposing them.

The nuisance adjustment matters: the raw score for a restricted parameter can overlap with score directions for parameters that remain free. One precise formulation uses a locally equivalent parameterization \((\lambda,\eta)\), where \(\lambda=0\) imposes the null and \(\eta\) collects nuisance parameters. Partition the score and expected information at the restricted fit as \(U=(U_\lambda,U_\eta)\) and

The efficient score and its information are

Thus the score statistic is \(LM=U_{\lambda\cdot\eta}'I_{\lambda\lambda\cdot\eta}^{-1}U_{\lambda\cdot\eta}\). In a correctly optimized restricted fit, \(U_\eta=0\) up to numerical tolerance, but the efficient-information adjustment still accounts for nuisance estimation. Unlike Wald and likelihood-ratio tests, the score test does not require estimating the unrestricted model.

Why the Three Statistics Agree Asymptotically

Under the null and regularity conditions, the log-likelihood near the restricted estimate has a quadratic approximation. In the \(q\) directions that release the restrictions, let \(\delta\) denote a small parameter change, \(U\) the efficient score, and \(I\) its efficient information. The change in log-likelihood is approximately

Maximizing this expression gives \(\hat\delta=I^{-1}U\). Substitution gives a maximized log-likelihood gain of \(\frac{1}{2}U'I^{-1}U\); multiplying by two yields the score statistic. The same local quadratic approximation makes the likelihood-ratio and Wald statistics asymptotically equivalent. They can nevertheless differ in finite samples because they use different aspects of the likelihood: the unrestricted estimate, the two maximized likelihoods, or the slope at the restricted estimate.

Worked Tests

Conditions, Interpretation, and Common Pitfalls

The chi-square approximations are large-sample results, not guarantees for every data set. They rely on an interior null, a well-behaved likelihood, valid covariance or information estimates, and sufficient identification of the parameters being tested. In the selection model, identification must be supported by the design and model assumptions; a numerical test statistic cannot repair a lack of information about the parameters. If a restriction lies on a boundary, the usual chi-square reference distribution may not apply. The reference distribution also does not protect against incorrect joint normality, a misspecified selection rule, or an inappropriate variance structure.

The three tests have practical trade-offs. A Wald test is convenient when the unrestricted covariance matrix is available, but can be sensitive to parameterization and finite-sample estimates. A likelihood-ratio test requires both restricted and unrestricted fits, but compares their maximized likelihoods directly. A score test is useful when the restricted model is easier to fit; its calculation still requires a valid score and information matrix. For a regular, well-identified model with a large sample, the tests should lead to similar conclusions, but their finite-sample statistics and p-values need not match.

• Common mistake: “A significant test proves selection bias is present.” Rejection of \(H_0:\rho=0\) is evidence against zero error correlation within the maintained Gaussian model. It may also reflect misspecification; it does not isolate the source of rejection.
• Common mistake: “A non-significant test proves selection is ignorable.” Failure to reject is not proof of zero correlation. Limited precision, weak identification, or small samples can make meaningful departures difficult to detect.
• Common mistake: “The joint test is equivalent to checking each coefficient separately.” A joint test uses the covariance among restrictions and tests their conjunction. Its result need not match the pattern of separate tests.
• Common mistake: “Any chi-square reference is valid.” Verify the number of independent restrictions and the regularity conditions. For example, a boundary null may require a different reference distribution.