Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
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Six Applied Audit Case Studies · Chapter V-08

Root-n Simulation Error

Simulation-based dynamic estimators involve the sample size, the number of simulated alternatives or paths, and the number of draws per object. Root-n inference depends on their joint rates. A simulated first stage can add variance or bias to the score even when the population target remains unchanged.

Conceptual map

  1. V-08.01Condition ledgers
  2. V-08.02A small dynamic model
  3. V-08.03Competing sequences
  4. V-08.04Corrected limiting variance
  5. V-08.05Unresolved boundaries

1. Sampling size and simulation effort enter separately

Consistency of a simulated object does not determine its contribution at root-\(n\) scale. Bias and variance have different rate requirements.

Let \(X_{i}\) be iid with mean \(\theta\) and variance \(\sigma _{X}^{2}\). For each observation use \(m_{n}\) independent simulation draws \(\eta _{ir}\) with mean zero and variance \(\sigma _{\eta}^{2}\). Consider

\[\hat{\theta }_{n}=n^{-1}\sum _{i=1}^{n}[X_{i}+b/m_{n}+m_{n}^{-1}\sum _{r=1}^{m_{n}}\eta _{ir}].\](1)
Definition 1 · Root-\(n\) simulation negligibility

The simulation approximation is negligible for a target limit when its centered stochastic component and deterministic approximation bias both vanish after multiplication by \(\sqrt{n}\) in the estimator expansion.

2. Bias and variance have separate limits

Proposition 1 · Triangular-array decomposition

Assume the variables in (1) are mutually independent across observations, simulation draws are independent of data, and both components have uniformly bounded moments of some order 2+\(\delta\). If 1/\(m_{n}\to \tau\), then \(\sqrt{n}[\hat{\theta }_{n}-\theta-b/m_{n}]\) converges in distribution to \(N(0,\sigma _{X}^{2}+\tau \sigma _{\eta}^{2})\). If additionally \(\sqrt{n}/m_{n}\to \lambda<\infty\), the same result centered at \(\theta\) has mean \(\lambda b\).

Proof. Subtract \(b/m_{n}\) from each summand. The centered row variables are independent with variance \(\sigma _{X}^{2}+\sigma _{\eta}^{2}/m_{n}\). The moment bound gives a Lyapunov central limit theorem with the limiting variance stated above. Under the additional rate, restoring the deterministic term adds \(\sqrt{n b}/m_{n}\to \lambda b\). ∎

When \(m_{n}\to \infty\), \(\tau=0\), so independent centered simulation noise disappears from this sample average. Bias requires the stronger rate \(m_{n}/\sqrt{n}\to \infty\) to disappear when \(b\ne0\). Shared draws, nonlinear first stages, or simulation inside an optimizer can produce a different expansion.

3. Three simulation budgets at \(n=10\),000

Set \(b=1\) and both variances to one. With \(\sqrt{n}=100\), simulation budgets \(m=10\),100,1000 give root-\(n\) bias terms 10,1,0.1 and centered asymptotic variances 1.1,1.01,1.001. Increasing \(m\) reduces both quantities, with different powers.

For \(m_{n}=n^{\alpha}\), root-\(n\) bias is \(b n^{1/2-\alpha}\): it diverges for \(\alpha<1/2\), has a finite nonzero limit for \(\alpha=1/2\), and vanishes for \(\alpha>1/2\). Centered simulation variance vanishes for every \(\alpha>0\) in this independent-draw design.

4. Dependence and nonlinearity change the ledger

Failure case · Fixed simulation budget justified by consistency

With \(b\ne0\) and fixed \(m\), the estimator converges to \(\theta+b/m\). Root-\(n\) inference centered at \(\theta\) is invalid.

Common random numbers correlate simulation errors across observations and can prevent averaging. Antithetic or quasi-random draws change variance rates. Estimated policies create nonsmooth selection, and optimizer tolerance can add another deterministic sequence. If the number of alternatives, paths, and draws all grow, each rate belongs in the ledger. A single draw gives finite variance but may fail moment conditions under heavy tails. Random stopping can create bias. At zero simulation variance, only approximation bias remains; at zero bias, fixed \(m\) can still alter the limiting variance.

5. Implementation, exercises, and sources

Derive the estimator expansion before choosing simulation budgets. Separate deterministic approximation, centered simulation noise, data sampling, optimization residual, and discretization. Record whether draws are independent, shared, nested, antithetic, or stopped adaptively. Replicate across independent random streams and compare empirical scaling with the predicted powers.

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Exercises

  1. Reproduce the three bias and variance pairs in Section 3.
  2. Derive the fixed-\(m\) limit when \(b=0\).
  3. Show how a simulation error shared by every observation changes the variance of the sample mean.
Partial solutions

1. Use \(\sqrt{n b}/m\) and 1+1/\(m\). 2. The centered limit variance is 1+1/\(m\). 3. If the same error \(Z_{m}\) enters every summand, averaging over \(i\) leaves it unchanged; \(\sqrt{n Z_{m}}\) requires a joint \(n,m\) rate.

  1. Ariel Pakes and David Pollard (1989), “Simulation and the Asymptotics of Optimization Estimators,” Econometrica 57, 1027–1057.Simulation error in extremum estimators.
  2. Christian Gouriéroux and Alain Monfort, Simulation-Based Econometric Methods.Bias, variance, and simulation design.
  3. Patrick Billingsley, Probability and Measure, sections on triangular-array central limit theorems.Limit argument used in the proposition.

6. Audit checkpoint

Sample size, simulations per object, alternatives and paths, deterministic bias, centered variance, joint rates, independence, shared or nested draws, common random numbers, moment conditions, heavy tails, random stopping, policy-selection kinks, optimizer tolerance, discretization, seed streams, empirical rate plots, centering, and corrected limiting variance.

7. Scope boundary

The chapter proves a sample-average benchmark. Simulated likelihood, indirect inference, particle filters, and stochastic optimization require model-specific expansions.

Prerequisites