Heteroskedastic Outcome Errors in Selection Models

Why the Outcome-Error Scale Matters

In a sample-selection model, outcomes are observed only for units that pass a selection rule. When the outcome error and the selection disturbance are dependent, the observed outcome errors generally have a nonzero conditional mean. The familiar inverse Mills correction describes that shift under a homoskedastic specification. If the outcome-error standard deviation varies with covariates, however, the correction must vary in scale as well as with the selection probability.

This distinction is substantive, not merely a matter of using a different standard error. A correction with the wrong scale can leave systematic differences in the adjusted outcomes. If the scale is associated with outcome regressors, those differences can affect estimated coefficients. The consequences depend on the data and covariate relationships; misspecification does not imply bias of a fixed direction in every application.

A Heteroskedastic Gaussian Selection Model

Let \(D_i\) indicate whether unit \(i\)'s outcome is observed. Write the selection rule and outcome equation as

D_i=1[z_i'\gamma+u_i>0], \qquad Y_i=x_i'\beta+\varepsilon_i \quad \text{when }D_i=1.

Here \(x_i\) and \(z_i\) are observed covariate vectors, and \(Y_i\) is observed only if \(D_i=1\). Let \(\eta_i=z_i'\gamma\). Normalize the variance of the selection disturbance to one. A particular heteroskedastic Gaussian specification is

u_i\mid x_i,z_i,w_i\sim N(0,1), \qquad \varepsilon_i=\sigma_i\left(\rho u_i+\sqrt{1-\rho^2}\,v_i\right), \qquad \sigma_i=\exp(w_i'\delta/2),

where \(v_i\mid x_i,z_i,w_i\sim N(0,1)\) is independent of \(u_i\), and \(|\rho|<1\). Thus, conditional on the covariates, \((u_i,\varepsilon_i)\) is jointly Gaussian, with

\operatorname{Var}(u_i\mid x_i,z_i,w_i)=1,\quad \operatorname{Var}(\varepsilon_i\mid x_i,z_i,w_i)=\sigma_i^2,\quad \operatorname{Cov}(\varepsilon_i,u_i\mid x_i,z_i,w_i)=\rho\sigma_i.

The exponential variance specification ensures \(\sigma_i>0\); \(w_i'\delta\) models the log variance, since \(\log(\sigma_i^2)=w_i'\delta\). The model imposes a constant conditional correlation \(\rho\), while allowing the outcome-error variance and covariance with the selection disturbance to vary with \(w_i\). Other dependence structures are possible, but they need not produce this same correction.

Definition: The inverse Mills ratio for the selected sample under this selection rule is \(\lambda(\eta)=\phi(\eta)/\Phi(\eta)\), where \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution functions. Since \(P(D_i=1\mid z_i)=\Phi(\eta_i)\), this ratio is defined for selection probability \(\Phi(\eta_i)>0\).

Deriving the Conditional Selection Correction

Selection occurs when \(u_i>-\eta_i\). Under the model, the part of \(\varepsilon_i\) involving \(v_i\) has conditional expectation zero even after conditioning on selection: \(v_i\) is independent of \(u_i\). The remaining part depends on the mean of a truncated standard normal variable. Specifically,

E(u_i\mid D_i=1,z_i) = \frac{\int_{-\eta_i}^{\infty}u\phi(u)\,du}{\Phi(\eta_i)} = \frac{\phi(\eta_i)}{\Phi(\eta_i)} = \lambda(\eta_i).

For the integral, \(\phi'(u)=-u\phi(u)\), so \(\int_{-\eta_i}^{\infty}u\phi(u)\,du=\phi(-\eta_i)=\phi(\eta_i)\). The denominator is \(P(u_i>-\eta_i)=\Phi(\eta_i)\). Taking the conditional expectation of the outcome error therefore gives the required result:

E(\varepsilon_i\mid D_i=1,x_i,z_i,w_i) = \rho\sigma_i\lambda(\eta_i), \qquad E(Y_i\mid D_i=1,x_i,z_i,w_i) = x_i'\beta+\rho\sigma_i\lambda(\eta_i).

The adjustment is observation-specific: it is the product of the dependence parameter, the outcome-error standard deviation, and the inverse Mills ratio. In the homoskedastic special case, \(\sigma_i=\sigma\) for every unit, so the factor \(\rho\sigma\) is constant and the correction reduces to a constant coefficient multiplying \(\lambda(\eta_i)\). With heteroskedasticity, that simplification generally fails.

Worked Example: Different Error Scales, Different Corrections

Consider two hypothetical covariate profiles with \(\eta=0\), \(\rho=0.4\), and a variance model \(\log(\sigma_i^2)=\delta_0+\delta_1w_i\), where \(\delta_0=0\), \(\delta_1=\log(4)\), and \(w_i\) is either 0 or 1. Both profiles have selection probability \(\Phi(0)=0.5\), but their outcome-error scales differ.

At \(w_i=0\), \(\sigma_i=\exp(0/2)=1\). Using \(\lambda(0)=\phi(0)/\Phi(0)\approx0.7979\), the correction is

\rho\sigma_i\lambda(\eta_i) =0.4(1)(0.7979) \approx 0.3192.

At \(w_i=1\), \(\sigma_i=\exp(\log(4)/2)=2\). The correction is now

0.4(2)(0.7979)\approx0.6383.

The selection probability and inverse Mills ratio are identical for the two profiles, yet the second profile's conditional outcome-error shift is twice as large because its outcome-error standard deviation is twice as large. A constant-scale correction calibrated to the first profile would not capture that difference.

What the Correction Does to a Fitted Outcome Equation

The conditional mean equation suggests the form of a selection-adjusted regression among selected units. If the model is correctly specified, its conditional mean includes \(\rho\exp(w_i'\delta/2)\lambda(z_i'\gamma)\), not just a constant times \(\lambda(z_i'\gamma)\). In estimation, the selection index, variance parameters, and dependence parameter are unknown and must be estimated. A two-step procedure can estimate the selection equation, construct the corresponding correction using estimated parameters, and then estimate the outcome equation with the heteroskedastic correction. Full-information estimation is another approach.

These are model-based procedures: their interpretation depends on the selection rule, Gaussian dependence specification, and variance function being suitable. Identification also matters. Excluded variables and support in the selection covariates can help distinguish the selection correction from the outcome equation; a flexible variance model does not replace that information.

Worked Example: A Constant-Scale Correction Can Change a Slope

To isolate the arithmetic, consider a hypothetical constructed set of selected outcomes with \(x_i=0,1,2\), \(\eta_i=0\) for every unit, \(\rho=0.5\), and known standard deviations \(\sigma_i=1,2,3\), respectively. Suppose the outcomes equal their conditional means, with underlying outcome equation \(x_i'\beta=10+2x_i\). This constructed illustration has no additional outcome residuals.

Since \(\lambda(0)\approx0.7979\), the correct corrections are \(0.5\sigma_i(0.7979)\), or approximately \(0.3989,\ 0.7979,\ 1.1968\). The resulting outcomes and correctly adjusted values are:

\begin{aligned} Y_1&=10+0.3989=10.3989, &Y_1-0.3989&=10,\\ Y_2&=12+0.7979=12.7979, &Y_2-0.7979&=12,\\ Y_3&=14+1.1968=15.1968, &Y_3-1.1968&=14. \end{aligned}

The correctly adjusted values lie on the line with slope \((14-10)/(2-0)=2\). Now suppose an analyst incorrectly uses the \(w\)-profile scale \(\sigma=1\) for every unit. The analyst subtracts \(0.5(1)(0.7979)=0.3989\) from each outcome, obtaining approximately \(10,\ 12.3990,\ 14.7979\). The fitted slope is then

\frac{14.7979-10}{2-0}\approx2.399.

In this deliberately simple construction, the scale error changes the slope because the omitted part of the correction increases with \(x_i\). In other data, it could primarily affect an intercept or other coefficients, or the difference could be small. The example shows why a constant-scale correction cannot be presumed harmless; it does not imply that every homoskedastic correction produces a slope bias of this sign or size.

Selection Also Changes the Conditional Variance

The heteroskedastic specification affects more than the conditional mean. Under the same Gaussian model, the selected-sample variance of the outcome error is

\operatorname{Var}(\varepsilon_i\mid D_i=1,x_i,z_i,w_i) = \sigma_i^2\left[1-\rho^2\lambda(\eta_i)\{\eta_i+\lambda(\eta_i)\}\right].

To see the source of this expression, the truncated-normal variance is \(\operatorname{Var}(u_i\mid D_i=1,z_i)=1-\eta_i\lambda(\eta_i)-\lambda(\eta_i)^2\). The independent \(v_i\) retains variance one, so substituting these component variances into \(\varepsilon_i=\sigma_i(\rho u_i+\sqrt{1-\rho^2}v_i)\) gives the displayed result. The selected-sample variance therefore depends on both \(\sigma_i^2\) and the selection index.

Worked Example: Conditional Variance at the Selection Threshold

Suppose \(\eta_i=0\), \(\rho=0.5\), and \(\sigma_i=2\). Since \(\lambda(0)\approx0.7979\), the selected-sample variance is

2^2\left[1-(0.5)^2(0.7979)(0+0.7979)\right] \approx 4(1-0.1592) \approx3.363.

The unconditional outcome-error variance at this covariate profile is \(2^2=4\); selection reduces it here because the errors are correlated with the selection disturbance and the disturbance is truncated. The conditional mean correction and variance formula are distinct implications of the joint model. Neither justifies treating selected outcomes as an ordinary homoskedastic regression.

Estimation and Interpretation

A practical specification can let \(w_i\) include variables that predict the outcome-error variance, which may overlap with \(x_i\) or \(z_i\). After estimating the selection equation, the correction must combine the estimated selection index with the estimated scale for each observation. For example, the generated term has the form \(\hat{\rho}\exp(w_i'\hat{\delta}/2)\lambda(z_i'\hat{\gamma})\). A fitted correction is not itself evidence that the Gaussian variance model is correct; the functional form and covariate support still need substantive and diagnostic scrutiny.

Inference must reflect estimation of the selection equation and the generated correction. Treating that estimated correction as fixed generally gives incorrect standard errors and can understate or overstate uncertainty, depending on the model and covariance terms. Use a suitable two-step variance correction or a bootstrap that re-estimates all stages in each resample. Robust standard errors for the outcome regression alone do not automatically account for first-stage estimation.

Common Mistakes

Using only the inverse Mills ratio. The correction is not generally a constant times \(\lambda(\eta_i)\) when \(\sigma_i\) varies. Under the stated model it is \(\rho\sigma_i\lambda(\eta_i)\).
Equating heteroskedasticity with a changing correlation. In this specification \(\rho\) is constant, but the covariance \(\rho\sigma_i\) varies with the outcome-error scale. Other models could instead let the correlation vary.
Claiming a predetermined direction of coefficient bias. A wrong correction can alter fitted coefficients, especially when its omitted variation aligns with outcome regressors, but its direction and magnitude depend on the covariates and model.
Ignoring generated-regressor uncertainty. The estimated correction is not known data; standard errors need to account for estimation in the selection and variance components.

Key takeaway: Under a joint Gaussian model with \(\operatorname{Var}(\varepsilon_i\mid x_i,z_i,w_i)=\sigma_i^2\) and constant conditional correlation \(\rho\), the selection-induced conditional mean shift is \(\rho\sigma_i\lambda(\eta_i)\). Heteroskedasticity changes the scale of the correction unit by unit, so a constant-scale adjustment is generally a different model.

Check Your Understanding

Use the stated heteroskedastic Gaussian model to answer the following.

If \(\eta_i=0\), \(\rho=0.3\), and \(\sigma_i=1.5\), calculate the conditional mean selection correction using \(\lambda(0)\approx0.7979\).
Explain why two observations with the same \(\eta_i\) can have different corrections under this model.
State the variance function implied by \(\sigma_i=\exp(w_i'\delta/2)\).
When might applying one constant-scale correction affect an estimated outcome slope rather than only its intercept?
Why do ordinary outcome-regression standard errors generally fail to reflect all uncertainty in a two-step procedure with an estimated correction?