Verification Against FLM2#

This page documents how deep-inference validates against the original implementation by Farrell, Liang, and Misra.

Original Implementation#

Repository: maxhfarrell/FLM2

The FLM2 repository contains replication code for:

  • Farrell, Liang, Misra (2021): “Deep Neural Networks for Estimation and Inference” Econometrica

  • Farrell, Liang, Misra (2025): “Deep Learning for Individual Heterogeneity” Working Paper

Implementation Comparison#

Component

FLM2 (R)

deep-inference (Python)

Framework

R + PyTorch via reticulate

PyTorch native

Cross-fitting folds

K=50

K=50

Lambda regularization

λ=1e-8

λ=1e-8

Optimizer

Adam, lr=0.01

Adam, lr=0.01

Architecture

2 layers, 10-20 units

2-3 layers, 32-64 units

Epochs

5000

100-500

Influence Function Formula#

Both implementations compute:

\[\psi_i = \beta(X_i) - \Lambda^{-1} \nabla \ell_i \cdot e_\beta\]

Where:

  • \(\Lambda = \mathbb{E}[W_i \tilde{T}_i \otimes \tilde{T}_i]\) (Hessian)

  • \(\nabla \ell_i = -r_i \cdot [1, \tilde{T}_i]\) (score)

  • \(e_\beta = [0, 1]\) (targeting vector)

Validation Results#

Monte Carlo Study (M=100, N=20,000, K=50)#

Metric

FLM Target

deep-inference

Status

Coverage

93-97%

95%

PASS

SE Ratio

0.9-1.2

1.08

PASS

Bias

~0

-0.001

PASS

Parameter Recovery#

Metric

Value

Corr(β)

0.953 ± 0.004

Corr(α)

0.830 ± 0.005

RMSE(β)

0.105 ± 0.004

Diagnostics#

Check

Result

Min eigenvalue(Λ)

1.81 (stable)

Regularization rate

0.0%

Naive coverage

8% (confirms IF needed)

Key Alignment Points#

  1. Cross-fitting structure: 50-fold splitting ensures each observation’s inference uses a model trained without it

  2. Hessian regularization: Ridge penalty λ=1e-8 prevents numerical instability in Λ⁻¹

  3. Theorem 1 compliance: Influence function formula matches the asymptotic expansion in FLM (2021)

  4. Coverage validation: 95% coverage confirms valid confidence intervals

Alternative Implementations#

Other implementations of the FLM framework:

References#

  • Farrell, M.H., Liang, T., Misra, S. (2021). “Deep Neural Networks for Estimation and Inference.” Econometrica, 89(1), 181-213.

  • Farrell, M.H., Liang, T., Misra, S. (2025). “Deep Learning for Individual Heterogeneity: An Automatic Inference Framework.” Working Paper.