Chapter map

1. 1. Target
2. 2. Centre
3. 3. Scale
4. 4. Standardized values
5. 5. H0
6. 6. H1
7. 7. H2
8. 8. H3
9. 9. H4
10. 10. Moment average
11. 11. BM3-4
12. 12. Weight matrix
13. 13. HAC option
14. 14. Degrees of freedom
15. 15. Monte Carlo
16. 16. Claim

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

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1. Target

A small example makes the mechanism visible. Define the variable or residual whose marginal distribution matters.

A concrete situation

Use 1–5 for arithmetic.

What the mathematics does not license

Testing the wrong residual answers the wrong question. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Freeze model. 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.

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2. Centre

This is where modelling choices begin to affect evidence. Estimate and subtract the mean.

A concrete situation

Mean=3.

What the mathematics does not license

Estimated parameters affect moments. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record estimator. 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.

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3. Scale

The central mathematical idea is concrete. Divide by the declared standard deviation.

A concrete situation

Population-form s=√2.

What the mathematics does not license

Near-zero scale is unstable. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check spread. 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.

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4. Standardized values

Here the geometry becomes operational. List z values.

A concrete situation

−1.414,−.707,0,.707,1.414.

What the mathematics does not license

Rounding changes high powers. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep precision. 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.

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5. H0

This step deserves a careful pause. Set normalized H0(z)=1.

A concrete situation

This anchors recurrence.

What the mathematics does not license

A different normalization changes covariance. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Declare basis. 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.

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6. H1

The useful question is not only how to calculate, but what the calculation assumes. Set normalized H1(z)=z.

A concrete situation

Its sample mean is zero after centring.

What the mathematics does not license

It is consumed by parameter fitting. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Respect uncertainty. 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.

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7. H2

A small example makes the mechanism visible. Use (z²−1)/√2.

A concrete situation

It checks variance direction.

What the mathematics does not license

Estimated scale makes it special. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Follow theory. 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.

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8. H3

This is where modelling choices begin to affect evidence. Use (z³−3z)/√6.

A concrete situation

The five values cancel to zero.

What the mathematics does not license

Zero H3 does not imply normality. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Keep H4. 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.

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9. H4

The central mathematical idea is concrete. Use (z⁴−6z²+3)/√24.

A concrete situation

The values sum about −1.328.

What the mathematics does not license

High powers are influence-sensitive. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Show contributions. 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.

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10. Moment average

Here the geometry becomes operational. Divide each polynomial sum by √n.

A concrete situation

H4 scaled sum≈−.594.

What the mathematics does not license

Do not divide twice. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Audit units. 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.

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11. BM3-4

This step deserves a careful pause. Square and add selected scaled moments.

A concrete situation

Illustrative total≈.353.

What the mathematics does not license

This simple identity-weight example is not every GMM variant. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Name specification. 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.

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12. Weight matrix

The useful question is not only how to calculate, but what the calculation assumes. Estimate covariance of selected moments.

A concrete situation

Use identity only when justified.

What the mathematics does not license

Poor covariance estimates distort evidence. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Diagnose conditioning. 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.

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13. HAC option

A small example makes the mechanism visible. Estimate long-run covariance for dependent data.

A concrete situation

Choose kernel and bandwidth before results.

What the mathematics does not license

HAC does not cure nonstationarity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check series. 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.

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14. Degrees of freedom

This is where modelling choices begin to affect evidence. Match the number of valid restrictions after estimation.

A concrete situation

State selected orders.

What the mathematics does not license

Data-driven order choice changes null law. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Preselect. 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.

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15. Monte Carlo

The central mathematical idea is concrete. Simulate the fitted process and repeat estimation.

A concrete situation

Mirror residual construction.

What the mathematics does not license

IID simulation is wrong for dependent series. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Preserve dependence. 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.

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16. Claim

Here the geometry becomes operational. State moment-condition evidence.

A concrete situation

Selected Hermite directions disagree with normality.

What the mathematics does not license

Unselected alternatives may remain. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report scope. 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.

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A seven-day practice lab

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

Bontemps–Meddahi tests express normality as moment conditions. For standardized normal data, normalized Hermite polynomials have mean zero. The GMM statistic collects sample averages of selected Hermite terms, weights their departures by their covariance structure and asks whether the joint vector is too far from zero.

For 1,2,3,4,5 standardized with mean 3 and population-form scale sqrt(2), z is approximately -1.414,-.707,0,.707,1.414. The normalized H3=(z^3-3z)/sqrt(6) values cancel to zero. The normalized H4=(z^4-6z^2+3)/sqrt(24) values sum to about -1.328, giving an illustrative BM3-4 contribution near .353 after division by sqrt(n) and squaring.

The example shows why the chosen polynomial orders matter. H3 is a skewness direction; H4 is a kurtosis direction; higher orders target more complex departures. Adding many moments does not automatically improve a finite sample because estimation noise and multiple directions also grow.

The GMM framework also addresses estimated parameters and time dependence. Hermite conditions have useful robustness to parameter uncertainty, while a heteroskedasticity-and-autocorrelation-consistent weighting matrix can handle unspecified serial dependence. That does not rescue a badly chosen lag bandwidth or nonstationary series; those choices remain part of the model.

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Frequently asked questions

What is a Stein equation?

It is a moment identity that holds under the target normal distribution.

Why Hermite polynomials?

Their expectations are zero under standard normality and they provide interpretable orthogonal directions.

What do H3 and H4 target?

H3 is related to skewness and H4 to kurtosis.

What does GMM add?

It weights the joint moment departures by their covariance structure.

How can time dependence be handled?

A suitable heteroskedasticity-and-autocorrelation-consistent covariance estimator can be used.

Should many high-order moments always be added?

No. More moments add directions but can make finite-sample estimation noisy.

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Primary source and fact-check note

Christian Bontemps and Nour Meddahi, “Testing Normality: A GMM Approach,” Journal of Econometrics 124(1), 149–186 (2005), DOI 10.1016/j.jeconom.2004.02.014. 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

• D’Agostino–Pearson K² test
• BHEP multivariate normality test
• How Mathematics Works

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 the target observations or residuals, independence assumptions and inclusion rules.

Transformation audit

Reproduce ordering, centring, scaling, plotting positions or polynomial recurrence.

Statistic audit

Show every rank score, moment coordinate, component and coefficient.

Calibration audit

Match sample size, estimated parameters, dependence and rejection direction.

Graph audit

Preserve Q–Q plots, contour maps, component plots and influence displays.

Magnitude audit

Explain practical consequences in original units, not only p-values.

Sensitivity audit

Use only prespecified formula, resolution and influence checks.

Final claim

Keep the conclusion inside the measured population and fitted model.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Target

Begin with this mechanism: Define the variable or residual whose marginal distribution matters. Recreate the concrete case: Use 1–5 for arithmetic. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Testing the wrong residual answers the wrong question. Complete the transfer step: Freeze model. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 2: Centre

Begin with this mechanism: Estimate and subtract the mean. Recreate the concrete case: Mean=3. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Estimated parameters affect moments. Complete the transfer step: Record estimator. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 3: Scale

Begin with this mechanism: Divide by the declared standard deviation. Recreate the concrete case: Population-form s=√2. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Near-zero scale is unstable. Complete the transfer step: Check spread. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 4: Standardized values

Begin with this mechanism: List z values. Recreate the concrete case: −1.414,−.707,0,.707,1.414. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Rounding changes high powers. Complete the transfer step: Keep precision. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 5: H0

Begin with this mechanism: Set normalized H0(z)=1. Recreate the concrete case: This anchors recurrence. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: A different normalization changes covariance. Complete the transfer step: Declare basis. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 6: H1

Begin with this mechanism: Set normalized H1(z)=z. Recreate the concrete case: Its sample mean is zero after centring. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: It is consumed by parameter fitting. Complete the transfer step: Respect uncertainty. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 7: H2

Begin with this mechanism: Use (z²−1)/√2. Recreate the concrete case: It checks variance direction. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Estimated scale makes it special. Complete the transfer step: Follow theory. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 8: H3

Begin with this mechanism: Use (z³−3z)/√6. Recreate the concrete case: The five values cancel to zero. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Zero H3 does not imply normality. Complete the transfer step: Keep H4. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 9: H4

Begin with this mechanism: Use (z⁴−6z²+3)/√24. Recreate the concrete case: The values sum about −1.328. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: High powers are influence-sensitive. Complete the transfer step: Show contributions. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 10: Moment average

Begin with this mechanism: Divide each polynomial sum by √n. Recreate the concrete case: H4 scaled sum≈−.594. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Do not divide twice. Complete the transfer step: Audit units. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 11: BM3-4

Begin with this mechanism: Square and add selected scaled moments. Recreate the concrete case: Illustrative total≈.353. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: This simple identity-weight example is not every GMM variant. Complete the transfer step: Name specification. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 12: Weight matrix

Begin with this mechanism: Estimate covariance of selected moments. Recreate the concrete case: Use identity only when justified. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Poor covariance estimates distort evidence. Complete the transfer step: Diagnose conditioning. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 13: HAC option

Begin with this mechanism: Estimate long-run covariance for dependent data. Recreate the concrete case: Choose kernel and bandwidth before results. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: HAC does not cure nonstationarity. Complete the transfer step: Check series. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 14: Degrees of freedom

Begin with this mechanism: Match the number of valid restrictions after estimation. Recreate the concrete case: State selected orders. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Data-driven order choice changes null law. Complete the transfer step: Preselect. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 15: Monte Carlo

Begin with this mechanism: Simulate the fitted process and repeat estimation. Recreate the concrete case: Mirror residual construction. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: IID simulation is wrong for dependent series. Complete the transfer step: Preserve dependence. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.

Workshop 16: Claim

Begin with this mechanism: State moment-condition evidence. Recreate the concrete case: Selected Hermite directions disagree with normality. Write standardized observations, Hermite values, moment sums, covariance entries, HAC weights, quadratic forms, degrees of freedom and calibrated probabilities with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it independently. Challenge this limitation: Unselected alternatives may remain. Complete the transfer step: Report scope. Change exactly one observation, plotting position, quantile convention, moment divisor, polynomial order, transformation coefficient, contour, covariance entry, lag bandwidth, rejection direction or Monte Carlo seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep design, transformation, statistic, calibration, magnitude, interpretation and validation on separate notebook lines. Explain the result twice: first to a student using the visible ordered values or component table, then to a parent using the real decision and original units. Add a counterexample in which the same statistic arises from rounding, censoring, mixture structure, dependence, skewness or tail contamination. State what extra graph, design record or held-out observation would distinguish the explanations. Finish with one supported sentence, one unsupported sentence and one useful next measurement.