Murphy-Topel Inference for the Generated Selection Correction

Why Second-Stage OLS Uncertainty Is Not the Whole Story

In Heckman two-step estimation, the second-stage regression includes an estimated inverse Mills ratio. It is therefore not an ordinary regression with a fixed design matrix: the correction term changes when the first-stage probit estimate changes. Treating the fitted correction as known omits a source of sampling variation and can make standard errors too small or otherwise inaccurate.

The Murphy–Topel approach accounts for this dependence by treating the two stages as one stacked estimation problem. It combines the second-stage estimating equation with the first-stage probit score, then derives the variance of the resulting joint influence function. The method is useful both for a model-based covariance calculation and for a sandwich estimate that allows more general variation in the estimating contributions.

Model, Notation, and Conditions

For each observation \(i\), let \(D_i\) indicate whether the outcome is observed, let \(z_i\) be the selection covariates, and let \(x_i\) be the outcome covariates. The selection equation is \(D_i=1[z_i'\gamma+u_i>0]\). The outcome equation is \(y_i=x_i'\beta+\epsilon_i\), observed only when \(D_i=1\). Under the Gaussian sample-selection model, \(u_i\) has a standard normal distribution, the outcome error has standard deviation \(\sigma\), and their correlation is \(\rho\). The conditional mean of the outcome error among selected observations is \(\rho\sigma\lambda(z_i'\gamma)\), where \(\lambda(t)=\phi(t)/\Phi(t)\), \(\phi\) is the standard normal density, and \(\Phi\) is its cumulative distribution function.

Write \(\delta=\rho\sigma\), \(\eta_i=z_i'\gamma\), and \(w_i(\gamma)=(x_i',\lambda(\eta_i))'\). The second-stage parameter is \(\theta=(\beta',\delta)'\). Thus, among selected observations, the conditional-mean regression is \(y_i=w_i(\gamma)'\theta+r_i\), with \(E[r_i\mid x_i,z_i,D_i=1]=0\) under the stated model. The two-step procedure estimates \(\gamma\) by probit on the full sample, computes \(\widehat\lambda_i=\lambda(z_i'\hat\gamma)\), and regresses selected outcomes on \(x_i\) and \(\widehat\lambda_i\).

Use full-sample normalization for both estimating equations. This makes the selected-outcome equation an average over all \(n\) observations, with unselected observations contributing zero. Its root is exactly the selected-sample OLS estimate, because multiplying that equation by \(n\) gives the usual selected-sample normal equations.

Stacking the Two Estimating Equations

Let \(P_i=\Phi(\eta_i)\). The probit score contribution for observation \(i\) is

The second-stage contribution is \(g_i(\theta,\gamma)=D_iw_i(\gamma)\{y_i-w_i(\gamma)'\theta\}\). The two estimates solve the sample averages of \(s_i(\gamma)\) and \(g_i(\theta,\gamma)\) equal to zero. Define the population matrices

Under a correctly specified probit, \(I\) is both the expected negative derivative of the score and the score variance. \(Q\) is the second-stage design matrix in full-sample normalization. Both are required to be nonsingular for the corresponding parameters to be locally identified by these equations.

The key extra derivative is the effect of changing \(\gamma\) on the second-stage equation. Since \(\lambda\) is a generated regressor, its derivative must be included. Differentiating the ratio \(\phi(t)/\Phi(t)\) gives

Let \(e\) be a vector of zeros with a one in its last position, corresponding to the inverse Mills ratio in \(w_i\). For \(r_i=y_i-w_i'\theta\), the derivative matrix of the second-stage equation with respect to \(\gamma\) is

The first term differentiates the \(w_i\) multiplying the residual; the second differentiates the residual itself, because its fitted value includes \(\delta\lambda(\eta_i)\). Omitting either term does not give the derivative of the estimating equation. Under the correctly specified conditional-mean model, \(E[r_i\mid x_i,z_i,D_i=1]=0\), so the expectation of the first term is zero. Keeping it in the general derivative makes clear where that simplification comes from.

Influence Function and Murphy–Topel Variance

A first-order expansion of the probit equation gives \(\sqrt n(\hat\gamma-\gamma)=I^{-1}n^{-1/2}\sum_i s_i+o_p(1)\). Expanding the outcome equation jointly in \(\theta\) and \(\gamma\) gives a second contribution: changing \(\gamma\) changes the fitted inverse Mills ratios and hence the second-stage equation. Since the derivative with respect to \(\theta\) is \(-Q\), the resulting expansion is

This is the Murphy–Topel adjustment in influence-function form. The term \(g_i\) is the contribution one would use if the generated correction were fixed. The term \(C I^{-1}s_i\) is the contribution transmitted from first-stage estimation. Because the same observations inform both stages, the two terms need not be independent; their covariance belongs in the variance calculation.

The matrices \(Q\) and \(I\) are symmetric here, so their inverses are also symmetric. Expanding the middle variance shows the contributions explicitly:

For the correctly specified probit, \(\operatorname{Var}(s_i)=I\). If, in addition, the conditional outcome-mean assumption makes \(\operatorname{Cov}(g_i,s_i)=0\), the expression simplifies to \(\Omega=\operatorname{Var}(g_i)+C I^{-1}C'\). The general formula is safer to present and implement: it retains the empirical cross-covariance and does not require setting it to zero by assumption.

Practical Calculation and Worked Examples

In practice, estimate the matrices and covariance from fitted contributions. Evaluate \(\hat s_i\), \(\hat g_i\), and the sample analogues \(\hat I\), \(\hat Q\), and \(\hat C\) at \((\hat\gamma,\hat\theta)\). Form the adjusted contribution \(\hat v_i=\hat g_i+\hat C\hat I^{-1}\hat s_i\). The sample covariance of these contributions, combined with \(\hat Q^{-1}\) on both sides and the \(1/n\) scaling, gives a sandwich estimate. Centering the contributions by their sample mean is a standard finite-sample implementation; at an exact solution their relevant averages are zero up to numerical precision. This calculation captures heteroskedasticity in the contributions as well as the generated-regressor adjustment.

Why Naive Standard Errors Can Mislead

Conventional second-stage OLS standard errors condition on the values of \(\widehat\lambda_i\) as if they were fixed. A heteroskedasticity-robust OLS sandwich improves the estimate of the second-stage contribution’s variance, but it still treats the generated correction as fixed. Neither calculation, by itself, includes \(C I^{-1}s_i\) or its covariance with \(g_i\). A second-stage robust covariance matrix is therefore not automatically a Murphy–Topel covariance matrix.

Under the conditional-mean model, \(g_i\) can be uncorrelated with the probit score because the corrected outcome residual has conditional mean zero given the covariates and selection. Even in that case, first-stage uncertainty generally remains: the term \(C I^{-1}C'\) is present when the score variance equals \(I\). If the first-stage score and outcome contribution are not uncorrelated, the cross terms must also be retained.

The distinction matters most when the generated correction is sensitive to the first-stage parameters, when first-stage uncertainty is substantial, or when the second-stage sample is limited. The correction is also relevant for standard errors of linear contrasts: for a contrast \(a'\theta\), use \(a'\widehat{\operatorname{Var}}(\hat\theta)a\), not a variance that ignores the generated regressor.

Common Mistakes

• Treating the fitted inverse Mills ratio as observed data. This ignores its dependence on \(\hat\gamma\). Include the derivative of the second-stage equation with respect to \(\gamma\).
• Assuming robust OLS standard errors solve the generated-regressor problem. They address heteroskedasticity in the second-stage contribution, but not uncertainty transmitted from the probit. Use the stacked sandwich or an equivalent Murphy–Topel calculation.
• Dropping cross-covariances without justification. The two equations use the same observations. Keep the covariance terms in the general estimator unless the model and conditions justify their removal.
• Using the selected-sample size inconsistently. With full-sample normalization, \(Q\) and the influence contributions are population averages over all observations, with \(D_i=0\) contributions set to zero. Selected-sample normalization is also possible, but its scaling must be carried through consistently.
• Calling the correction a causal adjustment. It corrects the conditional mean under the selection-model assumptions; it does not, by itself, establish a causal interpretation for an outcome coefficient.