0135 — Instrumental-variable and panel estimators have no .NET incumbent, and both could be written at parity

Status: accepted · Date: 2026-09-15 · Applies: 0074, 0075, 0095, 0110, 0129, 0130, 0134

Context

Issue #338 named instrumental variables and panel data as the estimator families an econometrician reaches for after least squares, with no reading behind either. #776 exists to read them before anything is claimed, and it could conclude written, delegated or left for a caller, per family.

Neither family is a covariance option on the existing fits. Each is a new estimator with its own table. Their reference is also not statsmodels:

• IV: statsmodels carries IV2SLS only under statsmodels.sandbox.
• Panel: statsmodels has no panel estimator at all.

So the reference had to be read first. 0130 and 0134 each found a reference that does not reproduce itself at the corpus tolerance, so reproducibility was the first thing to measure.

The reference

linearmodels 7.0, released 2025-10-21, installed into .venv-oracles for this reading and not pinned in tools/requirements*.txt, which waits for a lot that writes a corpus.

Licence, read from the wheel as 0075 requires:

• License-Expression: NCSA in METADATA;
• licenses/LICENSE.md, "Copyright (c) 2017 Kevin Sheppard", granting use, copying, modification and redistribution with the notice kept.

It is permissive, and reading it to build a corpus is what every other oracle here does.

The .NET side

nuget.org, 2026-09-15:

query	hits	anything that estimates either family
instrumental variables, two stage least squares, econometrics	0	—
2sls	6	none: SLS log sinks for Aliyun and an Epicor client
panel data	53	none: UI panels, dashboards, data-feed clients
fixed effects	4	none: UI effects and a rendering sample

Surfaces read with tools/survey.cs (0110), commercial libraries included as 0129 did. The pattern was (Instrument|TwoStage|2Sls|2SLS|Tsls|TSLS|IV2|Liml|LIML|Gmm|GMM|MethodOfMoments|Panel|FixedEffect|RandomEffect|WithinEstimat|BetweenEstimat|FirstDifferen|Endogen|Hausman|Sargan|Durbin.?Wu):

package	exported types	members read	matches
Accord.Statistics 3.8.0	561	8,105	none
MathNet.Numerics 5.0.0	336	6,938	none
Meta.Numerics 4.2.0	177	2,374	none
NumFlat 1.3.4	123	1,181	Clustering.ToGmm, ToDiagonalGmm — Gaussian mixtures, not the generalized method of moments
Numerics.NET 10.7.0 (commercial)	905	17,727	none
Dew.Stats.Core 6.3.10 (commercial)	51	967	none
ILNumerics.Toolboxes.Statistics 7.4.62 (commercial)	513	3,844	none

The commercial libraries are read and not run: nothing is timed or fitted under a trial licence.

What reproduces

Every estimator below was fitted by linearmodels 7.0 and computed a second time directly in numpy on the same seeded data. For IV the data was 500 rows, one exogenous regressor, one endogenous regressor and two instruments. For panel it was 60 entities over 8 periods, balanced and with a quarter of the rows dropped.

estimator	compared against	largest relative gap
IV2SLS, unadjusted	(XᵀP_Z X)⁻¹ XᵀP_Z y, s² = eᵀe/n	2.4e-16 (estimates), 1.5e-16 (errors)
IV2SLS, debiased=True	s² = eᵀe/(n − k); and statsmodels.sandbox IV2SLS	3.1e-16; 3.7e-16
IV2SLS, cov_type="robust"	the HC0 sandwich on the projected regressors	3.9e-16
IVLIML	κ as the smallest eigenvalue of (YᵀM_Z Y)⁻¹ YᵀM_W Y, then k-class	8.8e-16 (κ), 8.5e-16 (estimates)
IVGMM, two-step	the two-step estimator with the heteroskedastic weight	6.6e-16
IVGMM, iterated	itself at tol=1e-6 against tol=1e-12 (3 and 6 iterations)	6.6e-6
PanelOLS, entity effects	least squares on the within-transformed rows	3.5e-16
PanelOLS, entity and time effects	least squares with a dummy per entity and per period, balanced / unbalanced	2.2e-15 / 3.9e-15
BetweenOLS	least squares on the entity means	1.3e-15
FirstDifferenceOLS	least squares on the within-entity differences	1.2e-15
RandomEffects	itself, fitted twice	0.0

Every estimator but iterated GMM is a closed form, and the reference reproduces it to the last digits. Iterated GMM stops on a tolerance and moves with it, which is 0134's finding again at a smaller scale.

The errors follow linearmodels' conventions, not statsmodels'. On the unbalanced panel:

• Unadjusted fixed-effects errors are least squares on the within rows scaled by √((n − k)/(n − k − N)), 1.0996 here, because the absorbed effects count against the degrees of freedom.
• Entity-clustered errors are 0.99306 of what statsmodels' cluster correction gives on those same rows.

The sandwich #775 added computes the filling either way; the small-sample factor is the part that differs, and a corpus has to pin it rather than assume it.

Decision

1. Both families could be written at linearmodels parity, and neither has an incumbent to delegate to. Across seven .NET libraries, three of them commercial, no exported member estimates an instrumental-variable or panel model. The reference reproduces each closed-form estimator at 1e-15. The 1e-9 corpora every package here is held to are therefore available, as they were not for 0130's mixed models or 0134's ARIMA.
2. Instrumental variables: IV2SLS, IVLIML and two-step IVGMM, with the table the reference prints beside the estimates.
   • What the reference returns: first_stage reports R², partial R², Shea's R² and the first-stage F per endogenous regressor. The results also carry the Wu–Hausman, Durbin, Sargan, Basmann, Anderson–Rubin and three Wooldridge tests.
   • Which of those ship is the spec's to decide; none was measured here.
   • Iterated GMM is left out: its answer moves with its tolerance at 1e-6.
3. Panel data: PanelOLS with entity and time effects, BetweenOLS, FirstDifferenceOLS and RandomEffects, with the reference's unadjusted, robust and clustered covariances and its own small-sample factors, two of which are measured above.
   • PanelOLS(...).fit(use_lsmr=True), an iterative solver, is out, for the same reason as iterated GMM.
   • The default path matched the dummy-variable solve at 3.9e-15 on the unbalanced panel.
4. Neither is written by this record. Whether each earns a lot is a question for its issue, proposed rather than opened under 0095's rule. The spec of each lot:
   • names the package (Lodestar.Stats.Regression holds the least-squares table both would reuse);
   • pins linearmodels 7.0 in tools/requirements.txt;
   • benchmarks against linearmodels through the cross-language harness, since there is no .NET row to time against.
5. docs/migration/statsmodels.md gains the two rows, and docs/equivalence.md a section recording what was measured.

Rejected

• statsmodels.sandbox.regression.gmm.IV2SLS as the reference. It agrees with linearmodels' debiased 2SLS to 3.1e-16, but it lives in the sandbox, carries none of the diagnostics, and there is no panel counterpart for it to share conventions with.
• Iterated GMM at a looser tolerance. A tolerance chosen to make a corpus pass is chosen by the implementation (0134).
• Opening both lots now. The reading answers whether they could be written; issue #776's rule is that each family that earns a lot leaves as its own issue, and issues are proposed first.

Consequences

• #776 closes on this record.
• The migration page names both families as unwritten and writable, pointing here.
• An IV lot or a panel lot, if Cyril wants either, starts from the reproducibility table above and the covariance conventions it measured.