WHY MATHEMATICS?

Why Mathematics? | Quantile Treatment Effects in Regression Discontinuity Designs: Local CDFs, Cutoff Jumps, Fuzzy Assignment and Distributional Effects

How potential-outcome distributions can be estimated near a policy threshold, why effects may differ across quantiles, and which diagnostics support a local causal reading.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Define the running variable, cutoff and assignment mechanism
Estimate local potential-outcome CDFs on both sides
Invert the CDFs to obtain local quantile treatment effects
Audit manipulation, balance, bandwidth, first stage and support

Chapter map

1. Running variable
2. Assignment
3. Cutoff
4. Local sample
5. Left CDF
6. Right CDF
7. Median effect
8. Lower quantile
9. Upper quantile
10. Distribution regression
11. Inversion
12. Fuzzy first stage
13. Manipulation
14. Balance
15. Sensitivity
16. Claim

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

1. Running variable

A small example makes the mechanism visible. Define score and units.

A concrete situation

Centre cutoff at zero.

What the mathematics does not license

Data-driven recentering can manipulate design. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Freeze rule. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

2. Assignment

This is where modelling choices begin to affect evidence. State sharp or fuzzy treatment rule.

A concrete situation

Use a teaching sharp jump.

What the mathematics does not license

Actual take-up may differ. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Audit treatment. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

3. Cutoff

The central mathematical idea is concrete. Record the policy threshold.

A concrete situation

Use c=0.

What the mathematics does not license

Other rules may change there too. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check institutions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

4. Local sample

Here the geometry becomes operational. Choose observations near the cutoff.

A concrete situation

Left outcomes 4,5,6; right 6,8,10.

What the mathematics does not license

Tiny samples are unstable. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use justified bandwidth. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Left CDF

This step deserves a careful pause. Order left outcomes.

A concrete situation

Values are 4,5,6.

What the mathematics does not license

Mass points need a declared inverse. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show counts. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

6. Right CDF

The useful question is not only how to calculate, but what the calculation assumes. Order right outcomes.

A concrete situation

Values are 6,8,10.

What the mathematics does not license

Composition can change across sides. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check covariates. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

7. Median effect

A small example makes the mechanism visible. Compare lower-inverse medians.

A concrete situation

8-5=3.

What the mathematics does not license

Raw difference is not yet a valid RD estimator. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Fit local distributions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

8. Lower quantile

This is where modelling choices begin to affect evidence. Compare .25 quantiles.

A concrete situation

6-4=2.

What the mathematics does not license

Extreme local quantiles need support. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report counts. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

9. Upper quantile

The central mathematical idea is concrete. Compare .75 quantiles.

A concrete situation

10-6=4.

What the mathematics does not license

Tail differences can be noisy. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use bands. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

10. Distribution regression

Here the geometry becomes operational. Estimate CDF values locally at thresholds y.

A concrete situation

Use local linear fits.

What the mathematics does not license

Separate CDF fits may violate monotonicity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Rearrange if needed. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

11. Inversion

This step deserves a careful pause. Invert estimated potential-outcome CDFs.

A concrete situation

Obtain Q1(tau)-Q0(tau).

What the mathematics does not license

Difference of quantiles is not individual rank movement. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

State estimand. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

12. Fuzzy first stage

The useful question is not only how to calculate, but what the calculation assumes. Estimate treatment probability jump.

A concrete situation

Require a visible discontinuity.

What the mathematics does not license

Weak jumps widen uncertainty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report strength. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

13. Manipulation

A small example makes the mechanism visible. Inspect running-variable density and sorting.

A concrete situation

Check bunching near zero.

What the mathematics does not license

A density test is not the only diagnostic. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use institutional evidence. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

14. Balance

This is where modelling choices begin to affect evidence. Compare predetermined covariates near cutoff.

A concrete situation

Look for smoothness.

What the mathematics does not license

Passing balance cannot prove all assumptions. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report limits. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

15. Sensitivity

The central mathematical idea is concrete. Vary bandwidth, kernel and polynomial order.

A concrete situation

Track quantile effects.

What the mathematics does not license

Choosing the best-looking window is bias. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show full grid. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

16. Claim

Here the geometry becomes operational. State local population, cutoff and assumptions.

A concrete situation

Teaching differences are 2,3,4.

What the mathematics does not license

Do not claim universal or individual effects. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Bound conclusion. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

A seven-day practice lab

Day 1: define the decision, observation unit, X variables and Y or reference condition.
Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
Day 3: build a transparent baseline and write what it assumes.
Day 4: fit the main method inside a training-only pipeline; record every parameter.
Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
Day 6: compare validation metrics with the practical workload created by errors.
Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

Which part of the model is a mathematical definition, and which part is a choice?
What small calculation would catch a sign, scale or dimension error?
What evidence comes from training, validation and the untouched test?
What benign condition could look like a fault or prediction failure?
How would the answer change for another response, asset, school project or time period?
What should a human review before acting on the score?

A method-specific audit before deployment

Quantile treatment effects in a regression discontinuity design ask how an outcome distribution changes locally where treatment assignment jumps at a threshold. Frandsen, Frölich and Melly estimate local potential-outcome distributions with local linear distribution regression and invert them to obtain quantile treatment effects. Their framework covers fuzzy designs and treats sharp assignment as a special case under the required continuity and identification assumptions.

For a teaching sharp threshold at zero, suppose nearby left outcomes are 4,5,6 and nearby right outcomes are 6,8,10. Lower-inverse empirical medians are 5 and 8, giving a local median difference of 3. The .25 quantiles are 4 and 6, a difference of 2; the .75 quantiles are 6 and 10, a difference of 4. These raw local differences illustrate distributional heterogeneity, but they are not a valid estimator without bandwidth, smoothness, assignment and uncertainty checks.

Regression discontinuity identifies a local effect at the cutoff, not a universal programme effect. Manipulation of the running variable, other rules changing at the same threshold, weak treatment take-up jumps and sparse observations undermine the design. Quantile differences compare potential-outcome distributions; they do not track how the same individual moves from one percentile to another without stronger rank assumptions.

Report the running variable, cutoff, assignment rule, sharp or fuzzy design, treatment jump, exclusion restrictions, bandwidth, kernel, polynomial order, density and balance checks, distribution-regression grid, inversion rule and robust uncertainty. Show observations around the cutoff, sensitivity to plausible bandwidths and support for every quantile. Keep the original programme application separate from any new education or policy claim.

Frequently asked questions

What does the design exploit?

A discontinuous change in treatment probability at a known running-variable cutoff.

Why estimate local CDFs?

They identify distributional changes that can be inverted into quantile effects under the assumptions.

What is a fuzzy design?

Treatment probability jumps at the cutoff without changing from zero to one.

Is the effect global?

No. It applies locally to the population and threshold represented by the design.

Does a quantile effect track one person?

No. It compares potential-outcome distributions unless stronger rank assumptions are added.

What must be reported?

Rule, cutoff, first stage, bandwidth, kernel, CDF grid, diagnostics, inversion and uncertainty.

Primary source and fact-check note

Brigham R. Frandsen, Markus Frölich and Blaise Melly, “Quantile Treatment Effects in the Regression Discontinuity Design,” Journal of Econometrics 168(2), 382–395 (2012), DOI 10.1016/j.jeconom.2012.02.004. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Design audit

Freeze time-series vintages, sampling frequency, margins, pseudo-observations, panel dimensions, missing-data rules, running variable, cutoff and treatment assignment before estimation.

Objective audit

Write the cointegrating check loss, copula conditional-CDF plug-in, low-rank quantile-factor objective or local distribution-regression discontinuity estimand explicitly.

Tuning audit

Record tau grids, lag augmentation, long-run covariance bandwidth, copula family, vine order, factor count, starts, RD bandwidth, kernel, polynomial order and CDF grid.

Computation audit

Check residual integration, correction terms, response weights, copula likelihood, alternating-factor convergence, local CDF monotonicity, first-stage jumps and numerical tolerance.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Support audit

Inspect time breaks, marginal and joint tails, concentrated copula weights, sparse panel cells, quantile-specific factor support and observations immediately around the cutoff.

Uncertainty audit

Propagate long-run dependence, marginal estimation, copula fitting, factor selection, nonconvex optimisation, local bandwidth and CDF inversion uncertainty through the complete workflow.

Validation audit

Use future periods, held-out units, blocked panels or bandwidth-stable cutoff windows for quantile hits, weighted-CDF calibration, reconstruction loss and distributional effect sensitivity.

Final claim

Translate the result without turning cointegration, copula dependence, a latent factor or a local discontinuity effect into a universal causal guarantee.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Running variable

Begin with this mechanism: Define score and units. Recreate the concrete case: Centre cutoff at zero. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Data-driven recentering can manipulate design. Complete the transfer step: Freeze rule. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: Assignment

Begin with this mechanism: State sharp or fuzzy treatment rule. Recreate the concrete case: Use a teaching sharp jump. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Actual take-up may differ. Complete the transfer step: Audit treatment. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: Cutoff

Begin with this mechanism: Record the policy threshold. Recreate the concrete case: Use c=0. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Other rules may change there too. Complete the transfer step: Check institutions. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: Local sample

Begin with this mechanism: Choose observations near the cutoff. Recreate the concrete case: Left outcomes 4,5,6; right 6,8,10. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Tiny samples are unstable. Complete the transfer step: Use justified bandwidth. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Left CDF

Begin with this mechanism: Order left outcomes. Recreate the concrete case: Values are 4,5,6. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Mass points need a declared inverse. Complete the transfer step: Show counts. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: Right CDF

Begin with this mechanism: Order right outcomes. Recreate the concrete case: Values are 6,8,10. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Composition can change across sides. Complete the transfer step: Check covariates. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: Median effect

Begin with this mechanism: Compare lower-inverse medians. Recreate the concrete case: 8-5=3. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Raw difference is not yet a valid RD estimator. Complete the transfer step: Fit local distributions. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: Lower quantile

Begin with this mechanism: Compare .25 quantiles. Recreate the concrete case: 6-4=2. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Extreme local quantiles need support. Complete the transfer step: Report counts. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Upper quantile

Begin with this mechanism: Compare .75 quantiles. Recreate the concrete case: 10-6=4. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Tail differences can be noisy. Complete the transfer step: Use bands. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Distribution regression

Begin with this mechanism: Estimate CDF values locally at thresholds y. Recreate the concrete case: Use local linear fits. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Separate CDF fits may violate monotonicity. Complete the transfer step: Rearrange if needed. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: Inversion

Begin with this mechanism: Invert estimated potential-outcome CDFs. Recreate the concrete case: Obtain Q1(tau)-Q0(tau). Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Difference of quantiles is not individual rank movement. Complete the transfer step: State estimand. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Fuzzy first stage

Begin with this mechanism: Estimate treatment probability jump. Recreate the concrete case: Require a visible discontinuity. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Weak jumps widen uncertainty. Complete the transfer step: Report strength. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Manipulation

Begin with this mechanism: Inspect running-variable density and sorting. Recreate the concrete case: Check bunching near zero. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A density test is not the only diagnostic. Complete the transfer step: Use institutional evidence. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Balance

Begin with this mechanism: Compare predetermined covariates near cutoff. Recreate the concrete case: Look for smoothness. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Passing balance cannot prove all assumptions. Complete the transfer step: Report limits. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Sensitivity

Begin with this mechanism: Vary bandwidth, kernel and polynomial order. Recreate the concrete case: Track quantile effects. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Choosing the best-looking window is bias. Complete the transfer step: Show full grid. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State local population, cutoff and assumptions. Recreate the concrete case: Teaching differences are 2,3,4. Write running-variable units, cutoff distance, treatment indicators, outcomes, bandwidths, kernel weights, CDF thresholds, quantile levels, first-stage jumps, local effects and uncertainty bands with units, signs, identifiers and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not claim universal or individual effects. Complete the transfer step: Bound conclusion. Change exactly one observation, series transformation, long-run slope, residual, tau level, lag, correction bandwidth, marginal rank, copula family, query weight, response value, factor count, loading, factor path, missing panel cell, running-variable value, cutoff, bandwidth, kernel weight, treatment jump, CDF threshold or validation block. Explain which mathematical object changes, which estimator is selected, which conclusion survives and which claim must weaken. Keep design, objective, tuning, computation, support, uncertainty and interpretation on separate notebook lines. Explain the result once to a student with the complete table and once to a parent with the real decision and original units. Add a counterexample involving spurious trends, structural breaks, misspecified margins, tail-dependence failure, concentrated weights, factor rotation, local optima, cutoff manipulation, weak first stages or simultaneous policy changes. State what extra measurement would separate the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.