# Week 6 lab: training access and career risk sets ## Economic question and learning objectives Can a training or mobility evaluation separate greater capability from changing access, selection and opportunity supply? You will reconstruct an assignment-based assessment effect, build linked worker/opportunity/event ledgers, distinguish baseline cohorts from conditional career risk sets, and propose a design that separates at least two mechanisms. Adhvaryu–Kala–Nyshadham (2023) is the primary paper anchor. Its training-access design motivates distinct skill, output, career, pay and retention outcomes. Haegele (2026), Benson–Li–Shue (2019) and Bianchi et al. (2023) motivate application access, promotion assignment and scarce opportunities. These are conceptual anchors, not claims that the artificial lottery reproduces any paper's institutional protocol or findings. ## Setup and evidence boundary From the lab root: ```bash ENV_NAME=research bash smoke.sh ``` The executed route is **original synthetic design reconstruction**, with `empirical_replication: false` and marker **`SYNTHETIC_TEACHING_CONSTRUCTION_NOT_PUBLISHED_REPLICATION`**. All workers, outcomes, lotteries and transition rules are invented. Original archive files/licenses were not inspected here; no original data or code were downloaded or executed. The optional empirical route was omitted. The generator needs an empty or nonexistent output root. The wrapper stages a fresh run and installs a fixed inventory without replacing the authored README or source notes. Smoke executes real code into fresh directories, verifies all 35 starter assertions and independent calculations, and tests exit-code-2 overwrite refusal. No internet, credentials, GPU or installation is needed; significance is not a validation criterion. ## Reproduce: a stratified training-offer lottery ### Read the assignment and observation ledgers `data/teaching/workers.csv` contains eighteen initially eligible workers, six per department in three departments. Exactly three workers per department receive offers. All potential independent-assessment outcomes, latent completion behavior and career draws are generated first; a separate random-number stream then draws the lottery. The learner table excludes those known potentials and behavioral draws. The panel covers months −1, 0, 1, 2, 3 and 4, giving 108 worker-months. Independent baseline assessment occurs at month 0, training offer at 1, possible completion at 2, application/selection at 3 and endline assessment at 4. The fixed assignment label appears on every panel row, but no actual offer event precedes month 1. Completion and offer are distinct variables. Every baseline worker has an independent month-4 assessment, even after leaving. This artificial follow-up guarantee isolates the skill identification exercise; it is not a claim that an original field dataset has complete post-exit outcomes. Routine workplace output is a different index and becomes missing, not zero, at exit. Read `teaching-schema.csv` before merging. The worker table is unique by worker; the panel by worker/month; the opportunity table by opening; the event ledger permits several ordered events in a worker-month. Check merge cardinality before joining events to the panel so that multiple events do not duplicate outcomes. ### Estimate the offer effect and first stage Subtract baseline from endline assessment for each worker. Within each department, subtract the unoffered mean change from the offered mean change, then average the three department contrasts. Equal department sizes make this the baseline-population weighted estimate: ```{math} \widehat\tau=\frac{1}{3}\sum_{d=1}^{3} \left(\overline{\Delta A}_{d,Z=1}-\overline{\Delta A}_{d,Z=0}\right). ``` Reproduce `output/reproduced/assignment_effect.json`, including the completion first stage. The estimand concerns a training **offer**, not a comparison of completers. The saved completer-versus-noncompleter endline difference is explicitly noncausal. An IV interpretation for receipt would need relevance, exclusion, monotonicity and a compatible interference assumption; an offer can communicate support or alter access without completion. Compute the stratified Neyman variance estimate by summing the squared stratum weights times the treated and control sample-variance/size terms. With weights $1/3$ and three observations per arm within department: ```{math} \widehat V=\sum_{d=1}^{3}\left(\frac13\right)^2 \left(\frac{s_{d,1}^2}{3}+\frac{s_{d,0}^2}{3}\right). ``` Under this fixed-potential-outcome lottery it estimates an upper bound on finite-population randomization variance. It does not guarantee that a normal interval is exact in this small sample. Repeated monthly rows do not create additional independent lotteries. ### Keep two different enumerations separate There are $\binom63^3=8000$ equally likely permitted assignments. The **Fisher sharp-null reference** holds the observed assessment changes fixed and enumerates all assignments with exactly three offers in every department. Its two-sided p-value is the fraction whose absolute statistic is at least as large as observed, including ties. Inspect `sharp_null_distribution.csv`. No Monte Carlo correction is needed for exhaustive enumeration. Do not permute across departments, change treatment counts, or describe this as a general weak-null test with unrestricted heterogeneous effects. The **instructor truth calculation** instead changes each worker's outcome according to the alternate offer before computing the statistic. `data/instructor_only/potential-outcome-distribution.csv` saves those 8,000 statistics, and `truth_check.json` verifies that their mean equals the known finite-population offer ATE, 2.4. This expectation is not the null reference distribution. Do not add these potential outcomes or latent compliance types to learner predictors. The skill endpoint deliberately depends only on a worker's own offer through the invented completion rule. That no-interference condition applies to this endpoint. It does not carry over to the competitive career module. ## Build a career and opportunity ledger `opportunities.csv` records three openings, one per department, announced at month 1 and filled at month 3. Every worker begins eligible. Two candidates per department are guaranteed to remain through selection and apply, ensuring that the toy slots fill. Other applications depend on a pre-generated draw and offer; some workers exit at month 2 or 4. In `events.csv`, month-3 application precedes selection. Month-2 completion precedes any end-of-month exit. In `career_panel.csv`, first promotion and first exit are mutually exclusive **absorbing first-event states**. Later exits after promotion, re-entry, repeat courses and multi-grade careers are outside this exercise. Keep exited workers in the baseline cohort even after they leave the current risk set. The instructor's `candidate_score` is a **latent month-3 selection index sharing components with later skill**, plus pre-generated ranking noise. It is not an observed month-4 test available prospectively at month 3. The supplied DGP uses that shared component to determine the winner; it is not a recommended forecasting feature or a causal mechanism estimate. Routine output also uses a simple interpolation and a mechanical role component; only months 0 and 4 have independent tests, so the interpolated index is not evidence about the precise timing of skill acquisition. For each offer group, report applications and promotions per baseline worker and promotions per applicant. Inspect `career_funnel.csv` and show total slots and filled positions. These are realized allocation summaries with competition: training or application by one worker can displace another. Do not attach the assessment ITT's own-treatment-only interpretation to a promotion contrast. Next calculate cause-specific hazards among workers at risk at the beginning of each month. Accumulate promotion and exit incidence using survival just before the month; verify that survival plus the two incidences equals one. Inspect `competing_risks.csv`. With three filled slots and no censoring, final first-promotion incidence is $3/18$. Correct risk-set accounting is descriptive; it does not make endogenous promotion or training timing causal. ## Diagnose: known-answer constructions All following examples are saved in `output/transfer/diagnostic_examples.json`. They are separate from the realized eighteen-worker lottery where indicated. ### 1. A denominator reversal Two groups begin with 100 eligible workers each. The reform group has 40 applicants and 10 promotions; comparison has 20 applicants and 8 promotions. Advancement per eligible worker is 10% versus 8%, while advancement per applicant is 25% versus 40%. Both calculations are correct and answer different questions. Explain why conditioning on an intervention-generated applicant pool changes the comparison. The actual eighteen-worker funnel need not show this reversal; never substitute the illustrative counts for its realized rates. ### 2. Competing exits versus promotion-only survival Of 100 initial workers, 40 exit in month 1, then 30 of the remaining 60 promote in month 2. First-promotion cumulative incidence is $(1-0.4)(30/60)=0.30$. A promotion-only Kaplan–Meier calculation censoring exits gives 0.50. Explain why observed promotion before exit differs from a hypothetical promotion-only risk process. The latter is not the observed cohort advancement probability; neither calculation is automatically a causal effect. Connect this example to the separate simulated panel without mixing their denominators. ### 3. Current and next-role comparative advantage A produces 12 in the current role and 13 as manager; B produces 10 and 18. Promoting A yields $13+10=23$, versus $18+12=30$ when B promotes. Incremental contributions are 1 and 8. This static comparison holds effort incentives, wages, training and acceptance fixed; it is not an estimated universal cost of the Peter Principle or a complete optimal promotion rule. ### 4. A developmental assignment A move costs 3 now and gains 5 next period. With discount 0.9, net value is 1.5; if only half the future benefit is captured, it is −0.75. State the fixed-gain and capture assumptions. Do not confuse organizational capture with worker welfare or a full dynamic assignment model. ### 5. Training finance The invented cost is 70, annual production gain 80, annual extra pay 25, discount 0.95 and survival 0.9 per period over two future periods. Verify firm NPV 17.231375, extra pay while employed 39.650625 and combined net production 56.882. The first two sum to the third by accounting. Fixed retention and omitted post-exit returns limit the calculation; additional pay is not complete worker welfare or an estimate from the training paper. ### 6. Fixed slots and observation monotonicity A single vacancy switches from A to B. The total selected remains one, while A's observation indicator falls and B's rises. Equal aggregate selection therefore does not establish monotone individual observation for Lee-type bounds. Neither equal selection nor a selected sample of future managers reveals every candidate's counterfactual productivity. ### 7. Assignment and disclosure without skill gains Hold a common test at 50 while routine output moves 50→57 through assignment. Separately, hold a hidden test at 50 while a supervisor rating moves 3→4 through disclosure. These deliberately constructed examples refute the mechanical equation of recorded improvement with learning. They do not estimate realistic magnitudes or prove those mechanisms explain a real program. ### 8. Local incentives to block a move Destination gain 8 minus origin loss 3 minus moving cost 1 gives firm net 4. A locally rewarded origin manager bears −3 without credit; a transfer credit of 4 changes that payoff to +1. Explain which gaming, replacement, future relationship and worker-preference margins are absent before recommending a release or credit policy. ## Transfer: a career-transition panel design Complete `transfer-memo-template.md`: two pages plus a denominator table. Choose training access, certification disclosure, confidential applications, developmental rotations or opportunity supply. Define the organizational boundary, baseline cohort, eligible opportunities and information available at each decision date. Separate training offer from promotion offer, manager release and actual start. Specify two competing mechanisms and a discriminating prediction, independent common-task/quality measurement, worker welfare, opportunity counts, displacement and possible peers. Name the comparison, inference unit, observation horizon and largest remaining threat. Event dates and correct cumulative-incidence estimates do not supply an identification argument by themselves; modern absorbing-treatment DiD is not a turnkey estimator for repeated courses or promotions. Classify proposed data access honestly. Public NLS, O*NET or LEHD documentation is not a firm's internal-applicant ledger. No external contact, restricted-data application or new original-data collection is required. State one conclusion the proposed design cannot establish. ## Submission Submit the assignment/observation dictionary, assessment ITT and first stage, an explanation of the two enumerations, career denominators and competing-risk calculations, the short diagnostics, the proposal and a run log with seed and changes. Every figure or table must disclose its synthetic status. An attractive promotion regression cannot replace the missing eligibility and opportunity ledgers.