Worked Analysis of a Selected Wage Sample

Why a Wage Sample Needs a Selection Analysis

Hourly wages are usually recorded only for people who are working. A mean calculated from workers therefore describes the wages of workers, not automatically the wages that would be observed across a broader population of eligible people. If participation depends partly on unobserved determinants of wages, the observed workers can have systematically different wage errors from nonworkers. A wage regression that ignores this selection may then give misleading estimates of the wage equation and of population means.

This tutorial follows a hypothetical study of adults eligible for a wage survey. The target population is all eligible adults in the survey frame, whether or not they are employed. The outcome $Y_i^*$ is adult $i$'s potential hourly wage under employment, measured in dollars per hour. It is observed as $Y_i$ only when employment indicator $D_i=1$. For a nonemployed adult, $Y_i^*$ is a counterfactual wage under the stated outcome model, not a wage actually received. Thus a target-population mean of potential wages answers a different question from the mean wage among current workers.

Definition: In a sample-selection wage model, $D_i=1$ indicates that the outcome is observed, and $D_i=0$ that it is unobserved. The outcome equation describes $Y_i^*$, while the selection equation describes $D_i$. A selection-adjusted target-population mean averages the modeled potential outcome over the covariate distribution of the defined target population, rather than only over individuals with $D_i=1$.

The Model and Assumptions

Let $x_i$ be the covariate vector in the wage equation and $z_i$ the covariate vector in the employment equation; both include an intercept. Consider the latent-index selection rule and linear potential-wage equation below. The data contain $D_i$, $x_i$, and $z_i$ for everyone in the target frame, and $Y_i=Y_i^*$ only when $D_i=1$.

D_i=1\{z_i'\gamma+u_i>0\}, \qquad Y_i^*=x_i'\beta+\varepsilon_i, \qquad Y_i=Y_i^* \text{ if } D_i=1.

For the conventional parametric correction used here, assume conditional bivariate normality: given $x_i,z_i$, the errors $(\varepsilon_i,u_i)$ have a jointly normal distribution with conditional means zero, $\operatorname{Var}(u_i\mid x_i,z_i)=1$, $\operatorname{Var}(\varepsilon_i\mid x_i,z_i)=\sigma^2$, and $\operatorname{Corr}(\varepsilon_i,u_i\mid x_i,z_i)=\rho$. The variance and correlation are constant across covariate values, with $\sigma>0$ and $-1<\rho<1$. In particular, the zero conditional mean of $\varepsilon_i$ implies $E[\varepsilon_i\mid x_i]=0$, the condition needed for the target conditional mean $E[Y_i^*\mid x_i]=x_i'\beta$. These assumptions are stronger than unconditional joint normality; unconditional normality alone does not justify the conditional selection formulas below.

The selection equation implies $P(D_i=1\mid x_i,z_i)=\Phi(\eta_i)$, where $\eta_i=z_i'\gamma$ and $\Phi$ is the standard normal cumulative distribution function. For selected observations, define the inverse Mills ratio $\lambda(\eta_i)=\phi(\eta_i)/\Phi(\eta_i)$, where $\phi$ is the standard normal density. A short derivation explains why this term enters the wage equation.

E[\varepsilon_i\mid D_i=1,x_i,z_i] =\rho\sigma E[u_i\mid u_i>-\eta_i,x_i,z_i] =\rho\sigma\frac{\phi(\eta_i)}{\Phi(\eta_i)} =\rho\sigma\lambda(\eta_i).

To see the middle equality, conditional bivariate normality gives $E[\varepsilon_i\mid u_i,x_i,z_i]=\rho\sigma u_i$. Also, $u_i\mid x_i,z_i$ is standard normal, and selection is the event $u_i>-\eta_i$. For a standard normal variable $U$, integrating its density gives $\int_c^\infty u\phi(u)\,du=\phi(c)$; dividing by $P(U>c)=1-\Phi(c)$ and setting $c=-\eta_i$ yields $E[U\mid U>-\eta_i]=\phi(\eta_i)/\Phi(\eta_i)$. Consequently, the selected-sample conditional mean is $x_i'\beta+\rho\sigma\lambda(\eta_i)$. The coefficient on the correction term is $\delta=\rho\sigma$, measured in the same units as wages.

Worked Wage Analysis

Worked Example: Defining the Estimand and the Sample

Suppose a hypothetical survey samples 1,000 eligible adults. Employment and covariates are recorded for all 1,000, while hourly wages are recorded for the 600 employed adults. The covariates include education, experience, and commute cost. State the target and explain what the observed wage average estimates.

The target population is the 1,000 eligible adults represented by the survey frame, not just the 600 employed respondents. The outcome is each adult's potential hourly wage under employment, $Y_i^*$. The observed sample wage average estimates a mean among employed respondents, subject to the survey's sampling design; it is not by itself an estimate of the target-population mean of potential wages. It also does not describe wages actually received by nonemployed adults.

For a population target mean, the wage model must be averaged over the target population's $x_i$ values. The selection model is needed because the wage errors among employed adults may have a nonzero conditional mean. The target definition and the missing-outcome mechanism are therefore part of the analysis, not details to decide after fitting the regression.

Worked Example: Selection Probability and Correction Term

For illustration, suppose a probit selection equation estimated on the full frame is $\widehat{\eta}_i=-0.50+0.08\,\text{education}_i-0.04\,\text{commute cost}_i$. Education is measured in years and commute cost in dollars per week. Calculate the fitted selection probability and correction term for an adult with 12 years of education and a weekly commute cost of $10.

Substitution gives $\widehat{\eta}_i=-0.50+0.08(12)-0.04(10)=0.06$. Thus the fitted employment probability is $\Phi(0.06)\approx0.5239$. The inverse Mills ratio is $\widehat{\lambda}_i=\phi(0.06)/\Phi(0.06)\approx0.3982/0.5239=0.7601$. This term is dimensionless. It is calculated for every person in the target frame, although the wage equation can be estimated only for people whose wages are observed.

The commute-cost variable is a candidate exclusion: it appears in the selection equation but not the wage equation. That exclusion is useful only if, conditional on wage covariates, commute cost affects employment but does not directly affect potential wages. The equation's estimated coefficient does not establish that substantive claim.

Identification and Estimation

A practical analysis first estimates the selection equation, typically by probit using the full eligible frame. It then calculates $\widehat{\lambda}_i$ for the selected observations and estimates the wage equation among them, including that generated regressor. If $\delta=\rho\sigma$, the second-stage conditional-mean equation is $E[Y_i\mid D_i=1,x_i,z_i]=x_i'\beta+\delta\lambda(\eta_i)$. The coefficient $\delta$ captures selection dependence under the model: its sign is the sign of $\rho$, since $\sigma>0$. A positive value means that, at the same covariates, selected people have positive average wage errors under the assumed model.

The model needs credible identification, not merely a computed correction term. An excluded variable in $z_i$ that is absent from $x_i$ can provide variation in selection probability without directly shifting the conditional wage mean. Its exclusion must be justified using the setting and relevant knowledge. The selection index must also vary enough that the correction term is not nearly a linear combination of the wage covariates. Without a valid exclusion, identification may rest heavily on the probit and normality functional forms; a numerically estimated coefficient is not proof that the parameters are well determined. These concerns are especially important when the exclusion is weak or the correction term has little independent variation.

In practice, the fitted second-stage equation is a convenient way to illustrate the conditional mean, but ordinary least-squares standard errors that treat $\widehat{\lambda}_i$ as fixed generally omit first-stage estimation uncertainty. Inference should account for the generated regressor, for example through an appropriate two-step variance estimator or a full-information method. The numerical examples below focus on fitted means, not hypothesis tests.

Worked Example: Fitting and Interpreting the Wage Equation

Suppose the hypothetical two-step fit is $\widehat{Y}_i=8.00+1.20\,\text{education}_i+0.150\,\text{experience}_i+3.50\,\widehat{\lambda}_i$, with wages in dollars per hour and experience in years. Interpret the correction coefficient and calculate the model-implied conditional wage for a selected adult with 12 years of education, 8 years of experience, and $\widehat{\lambda}_i=0.7601$.

Substituting the values gives $\widehat{Y}_i=8.00+1.20(12)+0.150(8)+3.50(0.7601)$. The terms are $8.00, $14.40, $1.20, and approximately $2.660 per hour, respectively, so $\widehat{Y}_i\approx\$26.26$ per hour. This is a fitted conditional mean for a selected adult with these covariates, not a prediction guaranteed to equal that person's wage.

The coefficient 3.50 is an estimated $\widehat{\delta}$, in dollars per hour per unit of the dimensionless inverse Mills ratio. Under the stated model it equals $\rho\sigma$; it is not itself the correlation $\rho$. A positive estimate indicates positive selection dependence under the model, but its magnitude alone does not diagnose whether the exclusion or distributional assumptions are credible.

From Selected Wages to a Target-Population Mean

For any target-population member with observed covariates, the fitted mean potential wage under the maintained zero-conditional-mean assumption is $x_i'\widehat{\beta}$. The target mean estimate averages these fitted values over the target population's covariate distribution. In contrast, the model-implied selected mean averages $x_i'\widehat{\beta}+\widehat{\delta}\widehat{\lambda}_i$ over selected people. These averages differ for two reasons: the selected and target populations can have different $x_i$ distributions, and the selected conditional error mean can be nonzero.

\widehat{\mu}_{\mathrm{target}} =\frac{1}{N}\sum_{i=1}^{N}x_i'\widehat{\beta}, \qquad \widehat{\mu}_{\mathrm{selected}} =\frac{1}{n_1}\sum_{i:D_i=1} \left(x_i'\widehat{\beta}+\widehat{\delta}\widehat{\lambda}_i\right),

Here $N$ is the number of people in the target frame and $n_1$ is the number selected. If the target is a weighted population, the averages must instead use weights appropriate to that target. In either case, averaging predictions over selected covariates alone answers a selected-population question, not the target-population question.

Worked Example: Comparing the Two Mean Wages

Use the fitted wage equation above. Suppose the full hypothetical target frame has mean education 12 years and mean experience 8 years. Among the 600 employed adults, suppose mean education is 13 years, mean experience is 9 years, and the arithmetic mean of their individual correction terms is 0.700. Calculate the two model-implied means.

For the target frame, average $x_i'\widehat{\beta}$. Because the wage equation is linear, the stated covariate means give $\widehat{\mu}_{\mathrm{target}}=8.00+1.20(12)+0.150(8)=\$23.60$ per hour. This is the estimated mean potential wage under employment for the defined eligible population, under the model assumptions.

For employed adults, use their covariate means and average correction term: $\widehat{\mu}_{\mathrm{selected}}=8.00+1.20(13)+0.150(9)+3.50(0.700)$. The calculation is $8.00+15.60+1.35+2.45=\$27.40$ per hour. The $3.50 correction contribution reflects the model-implied positive mean wage error among selected adults at their observed selection indices. The difference between $27.40 and $23.60 is not solely a selection-error effect: the groups also have different average education and experience.

The calculation uses the average of individual inverse Mills ratios, as specified. In general, $\lambda(\eta)$ is nonlinear, so $\frac{1}{n_1}\sum_i\lambda(\eta_i)$ need not equal $\lambda$ evaluated at the average selection index.

Sensitivity Analysis and Common Pitfalls

The correction relies on the conditional distributional assumptions and on the exclusion restriction, so conclusions should be assessed against plausible alternatives. One transparent calculation holds the fitted wage coefficients and covariate summaries fixed, varies the assumed dependence parameter, and recomputes the selected mean. This isolates the correction's mechanical sensitivity; a full sensitivity analysis should refit the wage equation under each alternative because changing assumptions can also change $\widehat{\beta}$.

Worked Example: Varying Selection Dependence

For the selected group in the preceding calculation, suppose the outcome-error standard deviation is held at $\sigma=5.00$ dollars per hour and the average correction term is 0.700. Retain the selected group's average $x_i'\widehat{\beta}=\$24.95$ per hour. Calculate the model-implied selected mean at $\rho=0.70$, $0$, and $-0.20$, holding those other quantities fixed.

Since $\delta=\rho\sigma$, at $\rho=0.70$ the correction is $0.70(5.00)(0.700)=\$2.45$ per hour, giving a mean of $24.95+2.45=\$27.40$. At $\rho=0$, the correction is zero and the mean is $24.95 per hour. At $\rho=-0.20$, it is $(-0.20)(5.00)(0.700)=-\$0.700$ per hour, giving $24.25 per hour.

These are conditional sensitivity calculations, not three separately estimated results. They show that the selected mean depends on the assumed selection dependence. In an empirical analysis, report which assumptions change, whether the exclusion remains credible, and how refitting affects the target-population mean as well as the selected mean. A target mean is not automatically invariant when the model is refitted under alternative assumptions.

Common mistake: A researcher estimates the wage equation on workers, obtains an average fitted wage, and labels it the mean potential wage for all eligible adults. This ignores both the different covariate composition and the conditional wage-error shift induced by selection. The correction does not justify the target label by itself: define the target, average $x_i'\widehat{\beta}$ over its covariate distribution for the potential-wage mean, and state the assumptions that make that extrapolation meaningful. Nor should a positive correction coefficient be described as proof of selection bias; it is evidence about dependence only conditional on the maintained model and identification assumptions.

Key takeaway: A defensible selected-wage analysis links the estimand, conditional error assumptions, selection equation, wage equation, and target covariate distribution. Report selected and target means as distinct quantities, and show how key assumptions affect the conclusions.

Check Your Understanding

Work through these questions using the model and definitions above.

Why does unconditional joint normality of the errors alone fail to establish the selection probability conditional on $x_i,z_i$?
For a selected observation with $\eta_i=0$, write the selection probability and inverse Mills ratio using $\Phi$ and $\phi$.
What substantive claim must hold for commute cost to be a valid exclusion from the wage equation?
Explain why the target-population mean is averaged over all eligible adults' wage covariates, while the selected mean includes the correction term.
If $\sigma>0$ and the estimated correction coefficient is negative, what sign of $\rho$ does the maintained model imply? What does that conclusion depend on?