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GMM Benchmarks

This page presents detailed performance benchmarks for PanelBox's GMM estimators: one-step, two-step, CUE-GMM (Continuous Updated Estimator), and Bias-Corrected GMM (Hahn-Kuersteiner).

Benchmark Environment

CPU: Intel i7-10700K (8 cores, 3.8 GHz) | RAM: 32 GB DDR4 | Python: 3.12 | NumPy: 2.0 (MKL)

Each benchmark averaged over 10 runs. Data generated with fixed seed for reproducibility.

One-Step vs Two-Step vs CUE-GMM

Estimation Time Comparison

N T Instruments One-Step Two-Step CUE-GMM CUE/Two-Step Ratio
100 10 45 0.2s 0.4s 1.5s 3.8x
500 10 45 0.5s 1.0s 4.0s 4.0x
1000 10 45 1.2s 2.5s 10s 4.0x
5000 10 45 5.1s 9.8s 48s 4.9x

CUE-GMM: Instrument Count Scaling

CUE-GMM scales quadratically with the number of moment conditions due to repeated weighting matrix inversion (O(m^3) per iteration).

N Moments CUE-GMM Time Two-Step Time Ratio
500 10 2.1s 0.5s 4.2x
500 20 3.8s 0.7s 5.4x
500 50 12.4s 1.2s 10.3x

CUE-GMM Performance Thresholds

  • N < 2000, moments < 30: CUE-GMM completes in reasonable time (< 30s)
  • N > 5000 or moments > 50: Consider two-step GMM instead
  • CUE provides efficiency gains over two-step but at significant computational cost

CUE-GMM Profiling Breakdown

Function % of Runtime
scipy.optimize.minimize 45%
_compute_objective 28%
np.linalg.inv (weighting matrix) 15%
_compute_moments 8%
_compute_jacobian 4%

Bias-Corrected GMM

Bias-Corrected GMM (Hahn-Kuersteiner, 2002) adds a bias correction term that is most beneficial when T is small (T < 15). The overhead comes from computing derivative matrices.

N T Standard GMM BC-GMM Overhead
50 10 0.3s 0.5s +67%
100 20 0.6s 1.1s +83%
200 50 2.1s 4.3s +105%
500 50 5.8s 12.4s +114%

BC-GMM Profiling Breakdown

Function % of Runtime
_compute_bias_term 35%
scipy.optimize.minimize 30%
_numerical_derivative 20%
_compute_moments 10%
np.linalg.inv 5%

When to Use BC-GMM

  • Use when T < 15 (small time dimension where bias is significant)
  • Skip when T > 30 (bias is negligible, standard GMM is fine)
  • N < 1000 for reasonable runtime

Collapse Effect

Using collapse=True reduces the instrument matrix from O(T^2) to O(T) instruments, providing significant speedups for large T panels.

Instrument Count by T

T Standard Instruments Collapsed Instruments Reduction
5 10 4 60%
10 45 9 80%
20 190 19 90%
50 1225 49 96%

Collapse Performance Impact

N T Standard (instruments) Time Collapsed (instruments) Time Speedup
500 10 45 1.0s 9 0.4s 2.5x
500 20 190 3.2s 19 0.6s 5.3x
500 50 1225 28s 49 1.8s 15.6x

Collapse Trade-off

Collapsing instruments improves speed and reduces instrument proliferation but may sacrifice some efficiency. For large T, the speed gain is substantial and the efficiency loss is usually small.

Windmeijer Correction

The Windmeijer (2005) finite-sample correction for two-step standard errors adds minimal overhead:

N T Two-Step (no correction) Two-Step (Windmeijer) Overhead
100 10 0.38s 0.42s +10%
500 10 0.95s 1.05s +11%
1000 10 2.40s 2.65s +10%

The Windmeijer correction overhead is consistently ~10%, making it always worthwhile for two-step estimation.

Memory Usage

The GMM instrument matrix is the primary memory consumer, with dimensions L x (N*T) where L = number of instruments.

N T Instruments (L) Instrument Matrix Total Memory
100 10 45 ~3.4 MB ~10 MB
500 10 45 ~17 MB ~50 MB
1000 10 45 ~34 MB ~100 MB
500 20 190 ~145 MB ~200 MB
500 50 1225 ~4.7 GB ~5 GB

Memory with Large T

Standard (non-collapsed) instruments for T=50 require ~5 GB. Always use collapse=True for T > 20.

Scaling Summary

Dimension One-Step/Two-Step CUE-GMM BC-GMM
N (panels) O(N) O(N) O(N)
T (time periods) O(T) O(T) O(T)
m (moments) O(m^2) O(m^3) O(m^2)
Iterations 1-2 10-50 1-2

Recommendations

When to Use Each Estimator

Best for initial exploration and large datasets.

  • Fastest option
  • Consistent but not efficient
  • Use when N < L (fewer groups than instruments)
  • Good for quick model specification checks

Default choice for most applications.

  • 2x slower than one-step
  • Efficient, with Windmeijer-corrected SEs
  • Use for publication-quality results with N > L

For maximum efficiency when computational cost is acceptable.

  • 4-10x slower than two-step
  • Robust to instrument proliferation
  • Best for N < 2000, moments < 30
  • Not available in R's plm package

For small-T panels where bias is a concern.

  • ~70-115% overhead over standard GMM
  • Most beneficial for T < 15
  • Can combine with two-step or CUE

Quick Decision Guide

Is T < 15 and bias is a concern?
  → Use BC-GMM

Is N > 5000 or moments > 50?
  → Use Two-Step GMM (CUE too slow)

Do you need maximum efficiency?
  → Use CUE-GMM (if N < 2000)

Is this exploratory analysis?
  → Use One-Step GMM

Default:
  → Two-Step GMM with Windmeijer correction

Performance Optimization Checklist

  1. Use collapse=True for T > 10 (reduces instruments by 80-96%)
  2. Enable Windmeijer correction (only 10% overhead)
  3. Limit lag depth (gmm(y, 2:4) instead of gmm(y, 2:.))
  4. Use one-step when N < L (avoids rank-deficient weight matrix)
  5. Profile with %timeit to identify bottlenecks in your specific setup

Comparison with Other Software

Software N=1000, T=10, 20 moments Language
PanelBox (two-step) 2.5s Python
R plm::pgmm 3.8s R
Stata xtabond2 2.1s Stata (C backend)

PanelBox is competitive with R and approximately 1.2x slower than Stata's highly optimized C implementation. See Comparison with R/Stata for detailed cross-platform benchmarks.

References

  • Hansen, L. P., Heaton, J., & Yaron, A. (1996). "Finite-sample properties of some alternative GMM estimators." Journal of Business & Economic Statistics.
  • Hahn, J., & Kuersteiner, G. (2002). "Asymptotically unbiased inference for a dynamic panel model with fixed effects when both N and T are large."
  • Newey, W. K., & Smith, R. J. (2004). "Higher order properties of GMM and generalized empirical likelihood estimators."
  • Windmeijer, F. (2005). "A finite sample correction for the variance of linear efficient two-step GMM estimators." Journal of Econometrics.
  • Roodman, D. (2009). "How to do xtabond2: An introduction to difference and system GMM in Stata." The Stata Journal.

See Also