Module 4 — Choice and Limited Outcomes

Censoring, Truncation and Selection

Separate observed limits from sample inclusion and make selection corrections depend on credible exclusion information

Censoring, Truncation and Selection

Four zeros can mean four different things

Recorded value	Data-generating meaning	Example
true zero	outcome is genuinely zero	no paid work during year
left-censored	latent value is below a reporting limit	debt recorded as “£0–£500”
truncated	units outside range never enter data	survey samples only earners above £5,000
missing by selection	outcome exists but is unobserved	earnings absent outside linked records

Coding each as zero changes the estimand. Begin with the observation process, not the Tobit command.

Tobit is a joint model, not “OLS for many zeros”

The classical model assumes

Y_i^*=X_i'\beta+u_i,\qquad Y_i=\max(0,Y_i^*),\qquad u_i\sim N(0,\sigma^2).

The same latent index governs whether the limit is reached and the positive amount. Scholarship-related earnings may instead involve separate work-participation and wage processes. A two-part model can be more interpretable:

model $P(Y>0\mid X)$ ;
model $E[Y\mid Y>0,X]$ ;
combine them for $E[Y\mid X]$ .

Neither model automatically identifies the causal effect of scholarship receipt.

Sample selection is about unobserved outcomes

Suppose wages are observed only for employed graduates. Comparing observed wages estimates an effect among selected employed groups, whose composition may itself change with treatment.

A classical selection model writes

S_i^*=Z_i'\gamma+v_i,\quad S_i=1(S_i^*>0),

and observes $Y_i=X_i'\beta+u_i$ only when $S_i=1$ , allowing $u_i$ and $v_i$ to correlate. Heckman (1979) provides the canonical correction.

Credibility usually needs an exclusion variable in $Z$ that shifts observation or participation but not the outcome directly. Functional-form nonlinearity alone is a weak foundation.

A selection example

Offer group	Employment	Mean wage if employed
offered	80%	£31,000
not offered	60%	£32,000

The −£1,000 observed-wage gap does not show that offers reduce wages. The offer may bring lower-potential earners into employment, changing who has an observed wage. Report employment, unconditional earnings where meaningful, and bounds or sensitivity for the selected outcome.

Attrition belongs in the causal graph

If follow-up response $S$ is affected by treatment and outcome determinants, conditioning on $S=1$ can induce selection bias. A practical sequence is:

report response by assignment arm;
compare pre-treatment predictors of response;
state whether missingness is plausibly independent, conditionally independent or non-ignorable;
use weighting/imputation only under explicit assumptions;
add worst-case, Lee-style or other justified bounds where possible;
keep the full-sample ITT estimand visible.

Quick check

Every non-employed graduate is assigned wage zero. Is this wrong?

Answer

It depends on the estimand. Zero is valid for annual labour earnings if non-employment truly means no labour earnings during the window. It is invalid for the offered wage rate, which is undefined or unobserved without employment. Define the outcome before coding.

Next module: Modern Causal Analysis

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Count Outcomes

Model event rates with exposure, interpret multiplicative effects and diagnose dispersion and excess zeros

Module 5 — Modern Causal Analysis

Use flexible prediction, heterogeneity, sensitivity and transportability without weakening identification discipline