Inference conventions

This page summarizes which reference distribution each method uses for its test statistics, p-values and confidence intervals, how to read the diagnostic statistics, and where the defaults differ from R, statsmodels and linearmodels. It matters when you compare output against another package.

Reference distributions by method

Method	Coefficient statistics (`test_stats`)	Overall test	Degrees of freedom
OLS	t	F (`f_statistic`)	`n - k`; `G - 1` with `cov_type="cluster"`
WLS	t	F	same as OLS
FE	t	F	`df_resid` (`n - n_entities - k` for one-way) for `classical`/`hc1`-`hc3`; `G - 1` with `cov_type="cluster"`; `t_periods - 1` with `cov_type="dk"`
RE	t	F	`df_resid` (`n - k`) for `classical`/`hc1`-`hc3`; `G - 1` with `cov_type="cluster"`; `t_periods - 1` with `cov_type="dk"`. The F statistic is the same Wald test as FE (it follows `cov_type`), so `f_df_denom` equals the degrees of freedom of the t tests
IV, `estimator="2sls"`	t	F (`wald_dist="f"`)	`df_resid`; `G - 1` with `cov_type="cluster"`
IV, `estimator="gmm"`	normal	chi-squared (`wald_dist="chi2"`)	none
Logit / Probit	normal	likelihood-ratio chi-squared (`lr_statistic`, `lr_df`)	none
Tobit	normal	Wald chi-squared (`wald_dist="chi2"`, `wald_df`)	none

Every results object reports the distribution actually used in stat_dist ("t" or "normal") and the t degrees of freedom in stat_df (None for the normal distribution). stat_df is the value used for the p-values and confidence intervals, so it can differ from df_resid (for example with cov_type="cluster" in OLS, WLS, FE, RE and 2SLS, or cov_type="dk" in FE and RE).
The degrees of freedom of every test statistic are exposed so you can recompute a p-value yourself: f_df_num / f_df_denom (OLS, WLS, FE, RE), wald_df_num / wald_df_denom (IV), wu_hausman_df_num / wu_hausman_df_denom, weak_instrument_f_df_num / weak_instrument_f_df_denom, overid_df (IV), lr_df (Logit, Probit), wald_df (Tobit), hausman_df (RE). A degrees-of-freedom field is None when its statistic is not available.
wald_dist (IV and Tobit) tells whether the overall Wald statistic is F or chi-squared, since the name alone does not.
marginal_effects() always uses the normal distribution.

Why the choice differs between methods

t / F (OLS, WLS, FE, RE, 2SLS). Under the classical assumptions the standardized coefficient follows a t distribution exactly in finite samples. The same t distribution is used for every cov_type (classical, HC0-HC3, cluster, HAC) so that switching the standard error type does not silently change the reference distribution.
Normal / chi-squared (GMM, Logit, Probit, Tobit). The theory behind GMM and maximum likelihood is purely asymptotic. There is no finite-sample degrees-of-freedom argument, so a t distribution would claim a justification that does not exist.

Differences from other packages

The defaults below were checked against R 4.5.3 (sandwich 3.1.3, lmtest 0.9.40, fixest 0.14.2, plm 2.6.7, ivreg 0.6.8, AER 1.2.17), statsmodels 0.15.0 and linearmodels 7.0.

R

R	Reference distribution	Difference from econometricsmodels
`lm` + `summary`	t, `n - k`	same as OLS
`lm` + `lmtest::coeftest` with `sandwich::vcovHC` / `vcovCL`	t, `n - k`	same as OLS for HC; with clustering, `cov_type="cluster"` uses `G - 1` degrees of freedom instead of `n - k`
`glm(family = binomial)` (logit / probit)	z	same as Logit / Probit
`glm(family = gaussian)`	t	not applicable (use OLS)
`fixest::feols`	t, `n - k`; with `cluster`, t with `G - 1`	OLS matches; FE's small-sample corrections (`cov_type="cluster"`/`"dk"`) and their degrees of freedom (`G - 1` / `t_periods - 1`) are aligned with fixest's `ssc()` defaults (`K.fixef="nonnested"` for cluster, `"full"` for `hc1`-`hc3`/`dk`)
`fixest::feglm`	z	same as Logit / Probit
`plm` (`within`, `random`)	p-values from the normal distribution	FE and RE always use t (never normal), so p-values and confidence intervals differ; RE's `cov_type="cluster"` small-sample correction matches `plm::vcovHC(type="sss")` (Stata/R-style `G/(G-1)·(n-1)/(n-K)`)
`ivreg::ivreg`	t	same as 2SLS
`AER::tobit`	z	same as Tobit

statsmodels

With the default cov_type="nonrobust", OLS uses the t distribution. With any other cov_type (HC0-HC3, "cluster", "HAC") it switches to the normal distribution (use_t=False). econometricsmodels uses t for every cov_type, so pass use_t=True to statsmodels when comparing.
Logit and Probit use the normal distribution, as here.

linearmodels

IV (IV2SLS, IVGMM): fit(debiased=False) is the default, which reports normal and chi-squared statistics. econometricsmodels 2SLS always reports t and F (equivalent to debiased=True); GMM always reports normal and chi-squared (equivalent to debiased=False).
Panel (PanelOLS, RandomEffects): fit(debiased=True) is the default, which reports t and F, consistent with FE and RE here. Point estimates and the classical/hc1-hc3 standard errors still match linearmodels' PanelOLS/RandomEffects. The cluster and dk standard errors and their degrees of freedom no longer match linearmodels: FE/RE switched to fixest (R) / Stata-style small-sample corrections (G/(G-1)·(n-1)/(n-K)) instead of linearmodels' n/(n-extra_df-k), because that is what fixest, xtreg and reghdfe users expect. See docs/spec/fe-spec.md / docs/spec/re-spec.md for the exact formulas. The RE F statistic is the Wald quadratic form (the same as plm::pwaldtest(test = "F", vcov = ...) and linearmodels' f_statistic_robust), not linearmodels' f_statistic, which is a sum-of-squares version that ignores cov_type: the two agree only for the classical covariance on balanced panels, and linearmodels' value can even be negative on very unbalanced panels.

Weak instruments

weak_instrument_f_statistics returns, for each endogenous variable, the raw partial F statistic of the excluded instruments in the first stage. It always assumes homoskedasticity and does not depend on cov_type.

The result is not compared with the Stock-Yogo critical values, so no weak / strong verdict is reported. A commonly quoted rule of thumb is an F statistic above about 10; whether that is adequate for your application is left to you. Joint diagnostics for several endogenous variables (Cragg-Donald statistic) are not provided.

Overidentification tests

overid_statistic and overid_p_value (chi-squared with overid_df degrees of freedom, None when just-identified) are computed differently for the two estimators.

2SLS reports the Sargan test. It is always computed under homoskedasticity and does not depend on cov_type.
GMM reports Hansen's J test. It uses the weight matrix of the point estimate and therefore follows gmm_weight_type.

If you need an overidentification test that is robust to heteroskedasticity or clustering, use estimator="gmm" with a robust or cluster gmm_weight_type and read Hansen's J.

Other diagnostics

Hausman test (RE): hausman_statistic / hausman_p_value / hausman_df are the regression-based (auxiliary regression) version, equivalent to plm::phtest(method = "aux", effect = "individual"). The comparison is always against one-way FE. The Wald test on the auxiliary regression follows the cov_type of the RE fit (the default "cluster" gives the cluster-robust Hausman test; "classical" gives the classical version, which assumes RE is fully efficient under the null). The auxiliary regression uses the small-sample corrections of OLS (Stata/R-style, e.g. G/(G-1)·(n-1)/(n-k) for cluster), which now use the same style of correction as the RE standard errors themselves (both fixest/Stata-style since FE/RE moved off the linearmodels-style corrections). The statistic is non-negative by construction. If the robust covariance is structurally singular (e.g. the number of clusters does not exceed the 2k auxiliary slopes, or there are too few periods for "dk"), fit() raises an error instead of returning None.
Wu-Hausman test (IV): follows the cov_type passed to fit().