Identification tests¶

Download the script: identification.py

What the tests ask. GMM estimates are only as good as the instruments’ power to predict the endogenous regressors. These are the regressors that are not their own instruments: here, the lagged dependent variable and the predetermined x. dpd-stream reports the tests of Windmeijer (2025) that Stata’s underid reports with kp sw. They are:

robust to the fit’s clustering, by panel unless vcov= says otherwise;
computed in the weighted regressions, with weights;
computed with the principal components as the instruments, with PCA;
net of the exogenous regressors (those that are their own instruments, the constant among them), and of any absorbed effect.

The algebra is hdfe-stream’s core.ivtests, which feols_stream’s 2SLS uses too.

Test	Asks	Rejection means
Kleibergen–Paap rk LM	is the instruments’ first-stage coefficient matrix short of full rank?	the model is identified
Kleibergen–Paap rk Wald F	(weak-identification statistic, with ivreg2’s small-sample factor)	small values mean weak instruments
Sanderson–Windmeijer, one per endogenous regressor	is this regressor identified given the others?	that regressor is identified

No Stock–Yogo critical values. They are derived for homoskedastic errors and few instruments, so they do not apply here.

for rho in (0.3, 0.8, 0.98):
    df = simulate_dynamic(3_000, 7, rho=(rho,), seed=7)
    r = dpd_stream("y ~ l(y, 1) + x + w | year | gmm(y, 2:4, collapse=True) "
                   "+ gmm(x, 1:3, collapse=True) + iv(w)", df, panel=("id", "year"))
    r.idtests

rho = 0.3: KP rk LM p = 0.0000, KP Wald F = 311.4; Sanderson-Windmeijer l(y, 1): p = 0.000, x: p = 0.000
rho = 0.8: KP rk LM p = 0.0000, KP Wald F = 74.8; Sanderson-Windmeijer l(y, 1): p = 0.000, x: p = 0.000
rho = 0.98: KP rk LM p = 0.0095, KP Wald F = 2.5; Sanderson-Windmeijer l(y, 1): p = 0.010, x: p = 0.009

As ρ approaches 1, lagged levels stop predicting differences, and the statistics show it.

One endogenous regressor: effective F and Anderson–Rubin¶

With one endogenous regressor (here x is strictly exogenous, so only y’s lag is), r.idtests also has:

effective_f: Montiel Olea and Pflueger’s (2013) effective F, robust to heteroskedasticity and clustering, with the critical value that bounds 2SLS’s worst-case bias at 10% of OLS’s (their simplified test);
anderson_rubin: the Anderson–Rubin (1949) test, and its 95% confidence set for the coefficient. Both stay valid however weak the instruments are.

Every fit also reports each endogenous regressor’s first-stage partial R² and Shea’s partial R² (partial_r2), and the Anderson–Rubin test that the endogenous regressors’ coefficients are all zero.

df = simulate_dynamic(3_000, 7, rho=(rho,), x_type="exogenous", seed=7)
r = dpd_stream("y ~ l(y, 1) + x + w | year | gmm(y, 2:4, collapse=True) + iv(x + w)",
               df, panel=("id", "year"))
r.idtests["anderson_rubin"]["confidence_set"]

rho = 0.3: rho^ = 0.294, Wald 95% [0.268, 0.321], Anderson-Rubin 95% [0.259, 0.330]; effective F = 575.1 (critical value 18.1)
rho = 0.8: rho^ = 0.782, Wald 95% [0.723, 0.842], Anderson-Rubin 95% [0.704, 0.862]; effective F = 56.2 (critical value 18.6)
rho = 0.98: rho^ = 0.914, Wald 95% [0.684, 1.144], Anderson-Rubin 95% [-inf, -0.689] U [0.169, 2.782] U [6.003, inf]; effective F = 3.8 (critical value 19.4)

With strong instruments the two intervals agree. At ρ = 0.98 the effective F is far below its critical value. The Wald interval still looks tidy, but the Anderson–Rubin set is unbounded: the data cannot pin ρ down.

No endogeneity test here. In the transformed equation, the lagged dependent variable is endogenous by construction. Whether another regressor is exogenous is what the difference-in-Hansen test of its own IV-style group checks (put it in iv() and read that group’s test).

These are not Stata’s underid numbers

The statistics follow ivreg2: Kleibergen and Paap’s rk by the SVD algebra (closed form, the same as Anderson’s LM under homoskedasticity) and Sanderson and Windmeijer’s by 2SLS. After xtabond2, Stata’s underid computes something else:

Kleibergen–Paap: ranktest’s nosvd form, a J statistic at the LIML estimates. (ranktest has no SVD form that takes xtabond2’s H matrix, which is why underid uses this one.)
Sanderson–Windmeijer: a two-step GMM that keeps the exogenous regressors among the moments, with a first step weighted by \(Z'HZ\).

The tests are asymptotically equivalent under the null, but not equal in finite samples. For a simulated panel the Kleibergen–Paap LM is 33.6 here and 33.0 from underid; in two other fits the two differ by 7% and 10%. Neither form is known to be consistently more conservative. Do not compare these numbers digit for digit with underid’s output.

Cost. The tests cost a pass, and the Kleibergen–Paap covariance grows with the square of (excluded instruments × endogenous regressors). By default (idtests="auto") they are computed when that product is at most 500. idtests=True forces them, and collapse=True makes them cheap.