Numerical Illustration & Monte Carlo (ectheory-data-analysis)

When to trigger

• Your theorem is proved and you need simulations showing it bites in finite samples
• Reviewers will ask whether the asymptotic approximation is accurate at realistic n
• You include an illustrative empirical application and want it to serve the theory, not the reverse
• The simulation design feels arbitrary and you need principled choices

Role of "data analysis" at a theory journal

ET is theorem-proof first; numerical work is evidence that the asymptotics are useful, not the contribution itself. Two distinct, optional components:

1. Monte Carlo — the standard companion to a limit result. Its job is to show that finite-sample size/power/bias/coverage track the theory, and to map where the approximation breaks down.
2. Empirical illustration — an optional applied example showing the method on real data. It illustrates; it does not carry the paper. Keep it proportionate.

Designing a credible Monte Carlo for ET

• DGP coverage. Span the assumptions: include cases near the boundary (weak identification, near-unit-root, growing dimension, heavy tails, dependence) where the theory is most stretched.
• What you report. For an estimator: bias, RMSE, and the gap between empirical and nominal coverage. For a test: empirical size under the null and power under local/fixed alternatives.
• Comparisons. Benchmark against the natural existing method, so the simulation shows what your theory buys.
• Sample sizes. A grid of n that reveals the convergence rate visually, not a single n.
• Honesty. Show where the asymptotic approximation is poor; a candid breakdown region strengthens credibility more than uniformly green tables.

Reproducible computation

• Fix and report random seeds; report the number of Monte Carlo replications and all n.
• Specify the DGP precisely enough to regenerate every table/figure.
• Keep simulation code clean and runnable; long simulation evidence can go to the online Supplementary Material (already-reviewed, separate labeled file, not copyedited).

Checklist

• Monte Carlo DGPs span the assumptions, including the boundary cases
• Reported metrics match the claim (coverage/size/power/bias/RMSE as appropriate)
• A grid of n reveals the rate; convergence visible
• Benchmarked against the natural existing method
• Breakdown region of the approximation shown honestly
• Seeds, replication count, and DGP fully specified
• Empirical illustration (if any) kept proportionate to its illustrative role

Anti-patterns

• A single favorable n and DGP chosen to flatter the method
• Simulations that never probe the boundary where the theory is delicate
• An empirical "application" that overshadows the theorem
• Unreported seeds / replication counts (non-reproducible)
• Reporting size but not power for a test (or vice versa)

What an ET referee checks in the Monte Carlo first

At a theorem-proof venue the referee treats simulations as a stress test of whether the limit approximation is useful, not as the result. The first checks:

A Monte Carlo that never visits the regime where the proof's delicate step lives is desk-reject-adjacent.

Worked vignette and the simulation fixes

For a refinement that reduces the error in rejection probability of a t-test from order n^(-1/2) to n^(-1) under local-to-unity asymptotics, report the design:

# Monte Carlo skeleton for the refinement illustration
seed   = 20260610               # fixed and reported
reps   = 50000                  # per cell
n_grid = [50,100,200,400,800]
c_grid = [0,-5,-10,-20]         # local-to-unity drift c, root rho = 1 + c/n
# size under H0 (first-order vs refined); power under local alt theta0 + h/sqrt(n)
# show ERP=|size-0.05| decays faster for the refined test; flag breakdown at large |c|

The fixes: "rate without distribution theory" → upstream (route ectheory-identification-strategy), since a Monte Carlo cannot supply a missing limiting law; "no finite-sample evidence" → add the boundary-spanning design; "simulations avoid the hard regime" → extend the c-grid into the regime where the proof's delicate step operates. The ET structure is theorem → proof → simulation; confirm Supplement conventions against the author guidelines.

Output format

【Components】Monte Carlo / empirical illustration / both
【DGP coverage】boundary cases included? [Y/N]
【Metrics】size / power / coverage / bias / RMSE
【n grid】reveals rate? [Y/N]
【Benchmark】existing method compared? [Y/N]
【Reproducibility】seeds + reps + DGP specified? [Y/N]
【Next step】ectheory-tables-figures