Regression & Generalized Linear Models
Statios's model namespace covers 50+ regression and generalized linear estimation procedures, from OLS to latent-variable extensions. Methods range from standard parametric models (linear, logistic, count) to specialized estimators for endogenous/selected samples, censoring, and complex errors. All predictor-taking handlers auto-expand categorical predictors R-style (labeled indicator columns, named reference level, factors field) — no manual dummy coding required.
142 procedures · 3 with documented limitations. Every result is computed by the open Python engine and is exportable to APA tables and reports.
| Procedure | What it does | Status | Key reference |
|---|---|---|---|
| Linear Regression (OLS) | Standard ordinary least squares with diagnostics (R², F-test, VIF, Durbin-Watson, residual normality). Categorical predictors are auto-expanded R-style (labeled indicators, named reference level, factors field) — no manual dummy coding required. | ✅ Full | Akaike (1974); Schwarz (1978) |
| Linear Regression — Robust SEs (HC0–HC3) | OLS with heteroskedasticity-consistent standard errors; selectable HC0 (White), HC1 (Stata default), HC2, or HC3 (MacKinnon & White 1985). | ✅ Full | Akaike (1974); Schwarz (1978) |
| Linear Regression — Newey-West HAC SEs | OLS with Newey-West heteroskedasticity- and autocorrelation-consistent (HAC) standard errors (Bartlett kernel, set hac_lags); standard for time-series regressions. | ✅ Full | Newey & West (1987) |
| Errors-in-Variables Regression | Method-of-moments correction for classical measurement error (attenuation bias) using per-regressor reliability ratios; reports naive OLS vs corrected coefficients. | ✅ Full | Fuller (1987); Wooldridge (2010) |
| SIMEX (measurement-error correction) | Corrects attenuation bias from a regressor measured with known additive error variance by simulation-extrapolation: adds increasing extra noise, refits OLS at each lambda (averaging over B Monte-Carlo replicates), and extrapolates the coefficient quadratically back to lambda=-1 to recover the error-free estimate. | ✅ Full | Cook & Stefanski (1994, JASA); Carroll et al. (2006) |
| Frisch-Waugh-Lovell Partial Slope | Partials the controls out of both Y and the focal regressor and regresses the residuals through the origin; numerically equals the focal coefficient in the full OLS, with the partial correlation. | ✅ Full | Frisch & Waugh (1933) |
| Omitted-Variable Bias Decomposition | Decomposes omitted-variable bias for each included regressor into short-model vs long-model coefficients and the exact gamma*delta contribution of each omitted variable. | ✅ Full | Wooldridge (2010) |
| Log-Y Retransformation (Duan Smearing) | Fits a log-scale OLS and reports Duan's smearing factor S=mean(exp(residuals)) plus naive vs bias-corrected (smeared) retransformed predictions on the original scale. | ✅ Full | Duan (1983) |
| Potential Outcomes (ATE/ATT/ATU) | Reports the average treatment effect (ATE), the effect on the treated (ATT) and on the untreated (ATU) in Rubin's counterfactual notation by regression adjustment under unconfoundedness: an OLS fit of the outcome on treatment, covariates and their interactions predicts each unit's pair (Y(1), Y(0)), with delta-method or bootstrap standard errors. | ✅ Full | Rubin (1974); Imbens & Wooldridge (2009) |
| Selection Model (MNAR missingness) | Heckman (1979) two-step selection model for non-ignorable (MNAR) missingness: a probit models whether the outcome is observed, the outcome equation is corrected for selection on unobservables via the inverse Mills ratio, and rho (with its rho=0 ignorability test) quantifies the selection correlation, reported alongside naive complete-case OLS for comparison. | ✅ Full | Heckman (1979); Little & Rubin (2019) |
| Pattern-Mixture Model (MNAR) | Pattern-mixture model (Little 1993) for non-ignorable (MNAR) missing outcomes: partitions the sample by the outcome's missingness, fits the completers' regression, imputes the missing outcomes under a sensitivity restriction whose delta shifts the missing-group mean (delta=0 = MAR), and reports the pattern-averaged mean, fraction missing, and a sensitivity curve. | ✅ Full | Little (1993, JASA) |
| Continuous-Treatment Dose-Response (GPS) | Estimates the average dose-response function of a continuous treatment with the Hirano-Imbens generalized propensity score: models the treatment given covariates as Normal, regresses the outcome on a flexible function of the treatment and the GPS, and averages to trace E[Y(t)] over a grid of doses, returning the dose-response curve, the GPS-adjusted average marginal effect, and the naive unadjusted slope for comparison. | ✅ Full | Hirano & Imbens (2004) |
| Abadie Kappa-Weighting (compliers) | Uses Abadie's (2003) kappa weights kappa = 1 - D(1-Z)/(1-p(X)) - (1-D)Z/p(X) to estimate the complier share, the complier mean outcome, the mean covariates among compliers versus the full sample, and a kappa-weighted LATE in a binary-instrument / binary-treatment IV setting. | ✅ Full | Abadie (2003, J. Econometrics 113:231-263) |
| Marginal Treatment Effect (MTE) | Estimates the Heckman-Vytlacil marginal treatment effect via local IV / generalized Roy: fits the propensity score and a flexible control function in p, differentiates to trace the MTE over the unobserved-resistance dimension, integrates it with the appropriate weights to recover ATE/ATT/ATUT/LATE, and tests for essential heterogeneity. | ✅ Full | Heckman & Vytlacil (2005), Econometrica 73:669-738 |
| Synthetic DiD | Synthetic Difference-in-Differences (Arkhangelsky et al. 2021) combines synthetic-control unit weights with difference-in-differences time weights to estimate the ATT on a balanced treated/control panel, reporting the SDID effect, a placebo/jackknife standard error, the sparse non-negative unit weights and the pre-treatment fit. | ✅ Full | Arkhangelsky, Athey, Hirshberg, Imbens & Wager (2021, AER) |
| Changes-in-Changes (Athey-Imbens) | Nonlinear generalization of difference-in-differences (Athey & Imbens 2006) that maps the treated-pre outcome distribution through the control group's pre-to-post distributional change to identify the entire counterfactual treated distribution, reporting the ATT plus quantile treatment effects (0.25/0.5/0.75) that reveal heterogeneous effects ordinary DiD would miss. | ✅ Full | Athey & Imbens (2006, Econometrica 74:431-497) |
| Doubly-Robust DiD (Sant'Anna-Zhao) | Estimates the ATT in a 2x2 difference-in-differences design by combining an outcome-regression DiD with an inverse-propensity-weighted DiD into the doubly-robust estimator of Sant'Anna & Zhao (2020), which stays consistent if either the propensity-score model or the untreated-outcome model is correct, reports influence-function standard errors with the IPW and OR components and the simple DiD for contrast, and reduces exactly to the canonical mean 2x2 DiD when no covariates are supplied. | ✅ Full | Sant'Anna & Zhao (2020, J. Econometrics 219:101-122) |
| Honest DiD (parallel-trends sensitivity) | Rambachan & Roth (2023) sensitivity analysis for difference-in-differences: estimates the event-study coefficients, then bounds the post-treatment effect under a relative-magnitudes restriction (Mbar) on how far post-treatment trend violations can exceed observed pre-trends, reporting robust confidence sets across Mbar and the breakdown Mbar at which significance is lost. | ✅ Full | Rambachan & Roth (2023, Rev. Econ. Stud.) |
| Sun-Abraham Event Study | Interaction-weighted (IW) event-study estimator that saturates the model with cohort-by-relative-period interactions and aggregates them by cohort shares, recovering the true dynamic treatment path (and clean pre-trends) under staggered adoption with heterogeneous effects where conventional dynamic TWFE is contaminated. | ✅ Full | Sun & Abraham (2021, J. Econometrics 225:175-199) |
| Callaway-Sant'Anna Staggered DiD | Callaway & Sant'Anna (2021) staggered difference-in-differences estimating group-time average treatment effects ATT(g,t) via clean 2x2 DiD against not-yet-treated/never-treated comparison units, aggregated to an overall ATT, an event-study (dynamic) profile, and per-cohort effects, with a naive two-way FE benchmark to expose heterogeneous-timing bias. | ✅ Full | Callaway & Sant'Anna (2021, J. Econometrics); Goodman-Bacon (2021) |
| Causal Forest (heterogeneous effects) | Honest generalized random forest (Wager-Athey) that estimates the conditional average treatment effect tau(x) via split-sample honest trees with R-learner local centering, returning the ATE with a forest-based SE, the distribution of individual CATEs, and the leading covariate driving effect heterogeneity. | ✅ Full | Wager & Athey (2018), JASA; Athey, Tibshirani & Wager (2019), Annals of Statistics |
| Double / Debiased ML | Estimates a debiased treatment effect in the partially-linear model Y = theta*D + g(X) + e by cross-fitting gradient-boosted nuisance regressions for E[Y|X] and E[D|X] and solving the orthogonal (Neyman) moment on the residuals, reporting theta with robust SE/CI alongside the biased naive single-regression estimate for contrast. | ✅ Full | Chernozhukov et al. (2018) |
| McCrary Density Test (RD manipulation) | McCrary (2008) density-discontinuity test that fits separate triangular-kernel local-linear density estimates on each side of an RD cutoff and tests whether the log-density of the running variable jumps at the threshold, detecting sorting/manipulation that would invalidate the RD design. | ✅ Full | McCrary (2008, J. Econometrics) |
| Propensity-Score Subclassification | Estimates the ATE and ATT of a binary treatment by fitting a propensity-score logit, splitting units into quantile strata of the score, and combining the within-stratum treated-minus-control mean differences weighted by stratum size (ATE) or treated share (ATT), with per-stratum effects and a stratified standard error. | ✅ Full | Rosenbaum & Rubin (1983, Biometrika); Rosenbaum & Rubin (1984, JASA) |
| Mahalanobis Matching (ATT) | Nearest-neighbour matching estimator of the Average Treatment effect on the Treated that matches each treated unit to its closest control(s) in Mahalanobis covariate distance (pooled-control inverse covariance), imputes the counterfactual from the matched controls, and reports the ATT with an approximate Abadie-Imbens standard error and covariate balance. | ✅ Full | Rubin (1980); Abadie & Imbens (2006) |
| Box-Cox Transformed Regression | Estimates the Box-Cox power-transform lambda maximizing the concentrated profile log-likelihood, then fits OLS on the transformed (strictly-positive) response. | ✅ Full | Box & Cox (1964) |
| Stochastic Frontier Analysis | Maximum-likelihood Aigner-Lovell-Schmidt stochastic frontier (half-normal inefficiency) for production or cost functions, returning frontier coefficients, sigma_v/sigma_u, lambda, gamma, mean technical efficiency E[exp(-u)] and per-observation efficiency scores. | ✅ Full | Aigner, Lovell & Schmidt (1977); Meeusen & van den Broeck (1977); Jondrow et al. (1982); Battese & Coelli (1988) |
| Finite-Mixture Regression (latent class) | Fits a finite mixture of K linear regression regimes by EM, recovering each latent class's own coefficient vector, error variance, and mixing proportion together with the log-likelihood, BIC, and posterior class-assignment summary. | ✅ Full | Quandt & Ramsey (1978); McLachlan & Peel (2000) |
| Dirichlet-Process Mixture Density | Bayesian nonparametric density estimation that fits an infinite (Dirichlet-process) Gaussian mixture via truncated stick-breaking, automatically inferring the effective number of mixture components and returning their means, variances, weights, and the estimated density curve. | ✅ Full | Escobar & West (1995); Ferguson (1973); sklearn |
| Finite-Mixture Count (latent class) | Fits a finite mixture of K latent-class Poisson regressions by EM, recovering each regime's own log-link coefficient vector and mixing proportion together with the log-likelihood, BIC/AIC, and posterior class-assignment summary to capture unobserved heterogeneity and overdispersion in count data. | ✅ Full | Wedel et al. (1993); Cameron & Trivedi (2013) |
| Simultaneous Quantile Regression | Fits linear quantile regressions at several quantiles at once and uses a joint paired (xy) bootstrap covariance to report per-quantile coefficient tables and to test whether slopes change across the conditional distribution (e.g. tau=0.25 vs 0.75). | ✅ Full | Koenker & Bassett (1978); Koenker (2005) |
| Quantile Regression Forest | Estimates full conditional quantiles (Meinshausen 2006) with a random forest by reading the weighted empirical response distribution in each leaf, revealing heteroskedastic conditional intervals that widen with predictors plus an out-of-bag interval-coverage check and variable importance. | ✅ Full | Meinshausen (2006, JMLR) |
| Additive Quantile Regression (quantile GAM) | Fits a tau-quantile regression in which each predictor enters through a smooth cubic B-spline term, minimizing the asymmetric check-loss via linear programming to recover nonlinear conditional quantile curves. | ✅ Full | Koenker (2011); Fasiolo et al. (2021, JASA) |
| Composite Quantile Regression | Estimates a single shared slope vector by pooling the check-loss across an equally-spaced grid of quantiles (with quantile-specific intercepts), giving slope estimates that match OLS under Gaussian errors and beat it in efficiency under heavy-tailed errors, with bootstrap standard errors. | ✅ Full | Zou & Yuan (2008, Ann. Statist.) |
| Expectile Regression | Newey-Powell asymmetric-least-squares expectile regression, fit by iteratively reweighted least squares to estimate any conditional tau-expectile (tau=0.5 reproduces OLS exactly), with heteroskedasticity-consistent sandwich standard errors and an asymmetry interpretation. | ✅ Full | Newey & Powell (1987); Aigner, Amemiya & Poirier (1976) |
| Quantile Count Regression (jittering) | Estimates conditional quantiles of an overdispersed count response by the Machado-Santos Silva (2005) jittering method -- uniform noise is added to the integer counts, quantile regression is run on the log-transformed scale, and coefficients are averaged over many jitter draws to give stable, monotone-in-tau multiplicative quantile effects. | ✅ Full | Machado & Santos Silva (2005, JASA) |
| Generated-Regressor Inference (Murphy-Topel) | Two-step estimation that regresses a stage-2 outcome on the fitted values from a first-stage OLS, reporting both the naive stage-2 standard errors and the Murphy-Topel/Pagan-corrected standard errors that propagate the first-stage estimation uncertainty into the generated regressor. | ✅ Full | Pagan (1984, IER); Murphy & Topel (1985, JBES) |
| Heteroskedastic Probit | Harvey (1976) binary probit in which the latent-error standard deviation is modelled as sigma_i = exp(z_i'gamma), estimated by BFGS maximum likelihood, returning separate mean (beta) and variance (gamma) coefficient tables plus a likelihood-ratio test of homoskedasticity (H0: gamma=0). | ✅ Full | Harvey (1976, Econometrica); Greene (2018) |
| Kernel Regression (Nadaraya-Watson) | Nonparametric Nadaraya-Watson local-constant kernel smoother estimating the conditional mean of one outcome on one predictor (Gaussian or Epanechnikov kernel, Silverman's rule-of-thumb or user bandwidth), returning fitted values, a 100-point smooth curve, an R-squared-like fit measure, and the bandwidth used. | ✅ Full | Nadaraya (1964); Watson (1964) |
| Kernel Conditional Density | Estimates the full conditional density f(y|x) with a double Gaussian kernel (Nadaraya-Watson weighting in x combined with a kernel in y), returning conditional 0.1/0.5/0.9 quantile curves and the conditional mean as functions of x so heteroskedastic spread, skew and multimodality are visible where a mean regression cannot show them. | ✅ Full | Rosenblatt (1969); Hyndman, Bashtannyk & Grunwald (1996) |
| Single-Index Model (Ichimura) | Semiparametric single-index regression y = g(x'beta) + e estimated by Ichimura's semiparametric least squares, recovering the index direction beta (identified up to scale, unit-norm normalized) by minimizing the leave-one-out kernel-regression residual sum of squares while leaving the link g(.) fully nonparametric, and reporting standardized and raw-scale index coefficients, the data-driven bandwidth, a leave-one-out pseudo-R-squared, and the estimated link curve. | ✅ Full | Ichimura (1993), J. Econometrics 58:71-120; Klein & Spady (1993), Econometrica 61:387-421. |
| Maximum Score (Manski) | Manski's semiparametric maximum-score estimator for binary choice, recovering the scale-normalized coefficient direction by maximizing correctly-predicted signs of x'beta under only a conditional-median error restriction, staying consistent where logit/probit fail under heteroskedasticity or non-normal errors. | ✅ Full | Manski (1975, 1985) |
| Translog Production/Cost Function | Estimates a second-order flexible (transcendental logarithmic) production or cost function by OLS on log-inputs, their squares and pairwise cross-products, reporting output elasticities at the sample mean, returns-to-scale, and an F-test of the Cobb-Douglas (no-second-order-terms) restriction. | ✅ Full | Christensen, Jorgenson & Lau (1973) |
| Empirical Likelihood (mean) | Owen's nonparametric empirical-likelihood inference for a population mean or mean vector, returning the -2 log EL-ratio statistic with its chi-square p-value plus a distribution-free EL confidence interval that requires no normality assumption. | ✅ Full | Owen (1988, Biometrika); Owen (2001), Empirical Likelihood |
| Copula Fitting (bivariate dependence) | Fits Gaussian, Clayton, Gumbel and Frank copulas to two variables' rank pseudo-observations by maximum likelihood, reporting each family's parameter, log-likelihood and AIC, the AIC-selected copula, and Kendall's tau and Spearman's rho. | ✅ Full | Nelsen (2006); Joe (1997) |
| Gaussian-Process Regression | Nonparametric Bayesian regression with an RBF + white-noise kernel that learns its length-scale and noise level by maximizing the log-marginal-likelihood, returning a smooth posterior mean plus per-point predictive uncertainty. | ✅ Full | Rasmussen & Williams (2006) |
| BART (Bayesian Additive Regression Trees) | Flexible nonparametric regression that models the conditional mean as a regularized sum of trees fit by Bayesian backfitting MCMC, returning posterior-mean fitted values, predictive uncertainty, variable-inclusion importance, and in-sample R-squared. | ✅ Full | Chipman, George & McCulloch (2010), Ann. Appl. Stat. 4:266-298 |
| Gaussian-Process Classification | Fits a Gaussian-process classifier (RBF kernel, Laplace approximation) to a binary 0/1 outcome, reporting in-sample accuracy/log-loss/AUC, learned kernel hyperparameters, and a predicted-probability summary to capture smooth nonlinear class boundaries a linear logit would miss. | ✅ Full | Rasmussen & Williams (2006), ch. 3 |
| k-Nearest-Neighbours Regression | Nonparametric regression that predicts each outcome as the average of its k nearest neighbours in standardized (Euclidean) predictor space, reporting an in-sample fit, a leave-one-out cross-validated RMSE and CV R-squared, and the chosen k. | ✅ Full | Altman (1992), The American Statistician 46:175-185. |
| Spline Regression (B-splines) | Fits a flexible nonlinear curve by building a B-spline basis of one predictor (patsy bs() with given degrees of freedom and polynomial degree) and running OLS of the response on that basis, returning the fitted curve on a 100-point grid, the data-quantile knot locations, the basis dimension, R-squared and residual MSE. | ✅ Full | de Boor (1978) |
| Generalized Additive Model (splines) | Fits y = intercept + sum_j f_j(x_j) + e with each term a penalized cubic B-spline (statsmodels GLMGam, Gaussian family), selecting the smoothing penalty by GCV and reporting per-term effective degrees of freedom, deviance, pseudo-R^2, and a fitted-value summary. | ✅ Full | Hastie & Tibshirani (1990); Wood (2017) |
| GAM (Poisson / Logit) | Generalized additive model for count (Poisson/log) or binary (Binomial/logit) outcomes that fits each predictor as a penalized cubic B-spline smooth via penalized IRLS with GCV/UBRE smoothing selection, reporting per-smooth effective degrees of freedom, deviance explained, fitted curves with confidence bands, and approximate significance. | ✅ Full | Hastie & Tibshirani (1990); Wood (2017) |
| Local-Linear Regression | Nonparametric degree-1 local-polynomial kernel smoother that fits a kernel-weighted least-squares line at each point and takes its intercept as the conditional-mean estimate (Gaussian or Epanechnikov kernel, Silverman rule-of-thumb or user bandwidth), correcting the boundary bias of Nadaraya-Watson and returning fitted values, a 100-point smooth curve, effective degrees of freedom, and an R-squared-like fit measure. | ✅ Full | Fan (1992); Fan & Gijbels (1996) |
| Local Polynomial Regression | Nonparametric kernel-weighted local polynomial smoother of arbitrary degree that generalizes Nadaraya-Watson (degree 0) and local-linear (degree 1) fits, returning the fitted curve, its estimated first derivative, and a smoother effective-df fit measure, with higher degrees reducing boundary bias. | ✅ Full | Fan & Gijbels (1996) |
| Isotonic Regression (PAVA) | Nonparametric monotone least-squares fit of a response on a single predictor via the pool-adjacent-violators algorithm, returning the fitted step function (level sets), R^2, and the imposed direction. | ✅ Full | Barlow et al. (1972); Robertson, Wright & Dykstra (1988) |
| Wavelet Regression (denoising) | Nonparametric denoising of a response treated as a signal ordered by a predictor via discrete wavelet transform with soft-thresholding (VisuShrink universal or hybrid SureShrink), returning the denoised fitted curve, robust MAD noise estimate, retained-coefficient sparsity, and reconstruction diagnostics. | ✅ Full | Donoho & Johnstone (1994, Biometrika) |
| Wavelet Coherence | Computes squared wavelet coherence R^2 in [0,1] between two time series across time and scale via a hand-rolled FFT-based Morlet continuous wavelet transform, smoothed cross-wavelet and power spectra (Torrence & Compo 1998; Grinsted et al. 2004), summarizing the dominant common period, coherence by short/medium/long scale bands, and the cone-of-influence-masked global coherence. | ✅ Full | Torrence & Compo (1998); Grinsted, Moore & Jevrejeva (2004) |
| Partially Linear (Robinson) | Robinson (1988) sqrt(n)-consistent semiparametric partially-linear regression y = x'beta + g(z) + e, estimating the linear slopes beta by the double-residual method (Nadaraya-Watson partialling-out of y and each x on z, then residual-on-residual OLS) while leaving g(z) fully nonparametric. | ✅ Full | Robinson (1988, Econometrica 56:931-954) |
| Overdispersion Test (Cameron-Trivedi) | Cameron-Trivedi (1990) regression-based test of the Poisson equidispersion assumption: fits a Poisson GLM, then runs auxiliary through-origin OLS of the studentised squared-residual statistic on g(mu) for g(mu)=mu (NB1) and g(mu)=mu^2 (NB2), reporting per-variant alpha-hat, a one-sided t-test of H0 alpha=0, and an overdispersion verdict. | ✅ Full | Cameron & Trivedi (1990, J. Econometrics 46:347-364) |
| Conway-Maxwell-Poisson Regression | COM-Poisson GLM (log link on lambda) with a jointly-estimated dispersion parameter nu that flexibly handles under- (nu>1) and over-dispersed (nu<1) counts, fit by maximum likelihood with a stable truncated normalizing constant and a Wald test of nu=1 against the Poisson. | ✅ Full | Shmueli et al. (2005); Sellers & Shmueli (2010) |
| Endogenous Count (control function) | Poisson count regression with an endogenous regressor corrected by two-stage residual inclusion (control function): a first-stage OLS residual on instruments is added to the count model, yielding the endogeneity-corrected effect, a control-function residual coefficient that doubles as an endogeneity test, and bootstrap standard errors. | ✅ Full | Mullahy (1997); Terza, Basu & Rathouz (2008) |
| System OLS (multi-equation) | Estimates a system of linear equations equation-by-equation by OLS (intercept added), reporting per-equation coefficient tables (coef/se/t/p/CI), R2 and n, plus the cross-equation residual covariance Sigma (symmetric, PSD) and residual correlation on the common listwise-complete sample - the SUR/FGLS baseline. | ✅ Full | Zellner (1962); Greene (2018) |
| Joint Confidence Region | Fits OLS and reports the joint F test that a chosen set of coefficients is simultaneously zero, the exact confidence-ellipse geometry (centre, covariance submatrix, eigen-axes and half-axis lengths) for exactly two coefficients, and Bonferroni simultaneous intervals, contrasting the tilted exact region with the conservative axis-aligned box. | ✅ Full | Greene (2018); Scheffe (1959) |
| General GLS (specified Omega) | Feasible generalized least squares (Aitken) under a user-specified error covariance: AR(1) Toeplitz errors, groupwise heteroskedasticity (one variance per group level), or power heteroskedasticity (var proportional to |fitted|^delta), reporting the estimated structure parameter and an efficient coefficient table. | ✅ Full | Aitken (1936); Greene (2018) |
| Complementary Log-Log Regression | Binary regression with the asymmetric complementary log-log link P(y=1)=1-exp(-exp(x'beta)) for rare-event data and grouped/discrete-time proportional-hazards (survival) models, returning a coef/se/z/p/CI table with log-likelihood and AIC/BIC. | ✅ Full | McCullagh & Nelder (1989) |
| GEE (Population-Averaged) | Generalized estimating equations fit a population-averaged (marginal) regression for clustered or longitudinal data, modelling within-cluster dependence through a working correlation (independence / exchangeable / AR-1) while reporting robust sandwich standard errors valid even under correlation misspecification. | ✅ Full | Liang & Zeger (1986); Zeger & Liang (1986) |
| LIML (Limited-Information ML) | Limited-information maximum likelihood IV estimator that is less biased than 2SLS under weak or many instruments, reporting the k-class kappa eigenvalue, robust coefficient table with confidence intervals, first-stage instrument-strength diagnostics, and an overidentification test. | ✅ Full | Anderson & Rubin (1949); Davidson & MacKinnon (2004) |
| Two-Sample IV (TS2SLS) | Two-sample two-stage least squares: estimates a single endogenous regressor's structural coefficient by fitting the first stage in one half of the data and carrying the fitted endogenous value into the reduced-form/structural equation in the other half (Inoue-Solon TS2SLS; Angrist-Krueger TSIV design), reporting the coefficient with a robust SE, first-stage strength, and sample sizes. | ✅ Full | Angrist & Krueger (1992, JASA); Inoue & Solon (2010, ReStat) |
| Semiparametric IV (sieve) | Newey-Powell / Ai-Chen series (sieve) two-stage instrumental-variables estimation of an unknown dose-response g(endog) approximated by a polynomial sieve and instrumented by a sieve basis of the instruments, returning the estimated g(.) curve on a grid, the average derivative, HC1-robust standard errors, first-stage relevance, and a naive sieve-OLS comparison to expose endogeneity bias. | ✅ Full | Newey & Powell (2003); Ai & Chen (2003) |
| Lewbel Heteroskedasticity IV | Estimates a regression with an endogenous regressor when no external instruments exist, by constructing Lewbel (2012) generated instruments from heteroskedasticity in the first-stage residual and running heteroskedasticity-robust 2SLS, reporting the first-stage F and an OLS-vs-Lewbel comparison. | ✅ Full | Lewbel (2012, JBES) |
| Generalized Empirical Likelihood (GEL) | Information-theoretic estimation of a linear IV / moment-condition model via the GEL saddle-point (empirical likelihood, exponential tilting, or continuous-updating), returning coefficients with sandwich SEs, an LR-type overidentification test, and nonnegative implied probabilities; reproduces 2SLS exactly when just-identified. | ✅ Full | Smith (1997); Newey & Smith (2004) |
| Empirical-Likelihood Regression | Estimates linear-regression coefficients on the OLS estimating-equation moments E[x(y - x'beta)]=0 and tests/intervals them via Owen's empirical likelihood, giving distribution-free, generally-asymmetric EL confidence intervals plus chi-square-calibrated EL-ratio p-values whose point estimates coincide with OLS. | ✅ Full | Owen (1991); Qin & Lawless (1994) |
| Continuously-Updated GMM (CUE) | Continuously-updated GMM for a linear IV model that re-estimates the optimal weight matrix as a function of the parameters inside the objective and minimises jointly, returning CUE coefficients with robust standard errors, z/p-values, the Hansen J overidentification test, and (when just-identified) an exact 2SLS match. | ✅ Full | Hansen, Heaton & Yaron (1996, JBES) |
| Optimal-Instrument GMM | Efficient GMM for a linear IV model that builds Chamberlain (1987) optimal instruments A(z)=E[x|z]/Var(e|z) from a flexible projection of the regressors on the instruments plus a fitted conditional-variance model, reporting beta with efficient-GMM standard errors, first-stage fit, Hansen's J overidentification test, and an efficiency comparison versus plain 2SLS. | ✅ Full | Chamberlain (1987); Newey (1990); Hansen (1982) |
| Nonlinear GMM (exponential moments) | Two-step efficient GMM for the exponential / multiplicative-error model y = exp(x'beta)eta with instruments, using Mullahy's multiplicative moment E[z(y*exp(-x'beta)-1)]=0 to identify possibly-endogenous regressors (a nonlinear analogue of 2SLS) and reporting the coefficient table plus Hansen's J overidentification test. | ✅ Full | Hansen (1982); Mullahy (1997) |
| Method of Simulated Moments | Method of Simulated Moments / indirect inference for a univariate AR(1)-plus-measurement-noise model whose likelihood is intractable: parameters (phi, sigma_eta, sigma_e) are chosen so simulated auxiliary statistics (variance and first two autocovariances) match the data's, minimizing a scale-free weighted distance with common random numbers and reporting estimates, sandwich SEs with the (1+1/S) simulation-noise inflation, the objective value, and the moment-match table. | ✅ Full | McFadden (1989); Pakes & Pollard (1989); Gourieroux, Monfort & Renault (1993) |
| Minimum Distance (CMD) | Classical minimum-distance (minimum-chi-square) estimation that recovers a structural IV slope from the OLS reduced-form coefficients by minimizing (pi_hat - h(theta))' W (pi_hat - h(theta)) with the efficient weight W = inv(Var(pi_hat)), reporting theta_hat, CMD standard errors, the overidentification chi-square, and (in the just-identified case) exact agreement with 2SLS. | ✅ Full | Rothenberg (1973); Chamberlain (1982); Wooldridge (2010) |
| IV Quantile Regression (Chernozhukov-Hansen) | Estimates the structural effect of an endogenous regressor on a chosen conditional quantile of the outcome using one or more instruments, via the Chernozhukov-Hansen inverse-quantile-regression grid search that drives the instrument coefficient to zero, correcting the endogeneity bias that contaminates ordinary quantile regression. | ✅ Full | Chernozhukov & Hansen (2005, Econometrica; 2006, J. Econometrics) |
| Oaxaca-Blinder Decomposition | Decomposes the mean outcome gap between two groups from separate within-group OLS fits into the threefold endowments/coefficients/interaction components and the twofold pooled-reference (Neumark) explained vs unexplained split. | ✅ Full | Oaxaca (1973); Blinder (1973); Jann (2008) |
| Prais-Winsten AR(1) Regression | Iterative AR(1) FGLS that quasi-differences the data and retains a scaled first observation, correcting OLS for first-order serial correlation; reports rho and before/after Durbin-Watson. | ✅ Full | Prais & Winsten (1954) |
| Linear Probability Model | OLS of a binary 0/1 outcome with HC1 robust SEs; coefficients are marginal effects on P(y=1), and the share of out-of-[0,1] fitted values is flagged. | ✅ Full | Wooldridge (2010) |
| Generalized Linear Model (family x link) | Unified GLM: choose the exponential family (gaussian, poisson, binomial, gamma, inverse-Gaussian) and link (identity, log, logit, probit, inverse, sqrt). Reports coefficients, deviance, AIC/BIC. | ✅ Full | Nelder & Wedderburn (1972) |
| Linear Combination of Coefficients (lincom) | Tests an arbitrary linear combination g'β of OLS coefficients with its standard error, t, p-value, and 95% CI (Stata-style lincom). | ✅ Full | Wooldridge (2010) |
| Delta-Method (nonlinear combination) | Delta-method standard error for a sum / difference / ratio / product of two OLS coefficients, with t, p-value, and 95% CI. | ✅ Full | Wooldridge (2010) |
| Restriction Test (F / Wald / LR) | General joint-exclusion test: F, Wald, and likelihood-ratio statistics for dropping a subset of regressors (unrestricted vs restricted OLS). | ✅ Full | Wooldridge (2010) |
| Binary Logistic Regression | Binary outcome (0/1) modeled as probability; reports odds ratios, McFadden R², classification accuracy. | ✅ Full | Nelder & Wedderburn (1972); McFadden (1974) |
| Probit Regression | Binary choice via normal CDF link; includes marginal effects and classification accuracy. | ✅ Full | Seabold & Perktold (2010) |
| Multivariate Probit | Jointly models 3+ correlated binary outcomes by maximum simulated likelihood with the GHK simulator, returning per-equation probit coefficients plus the full cross-equation error correlation matrix. | ✅ Full | Cappellari & Jenkins (2003); Train (2009) |
| Bivariate Probit | Jointly estimates two binary-outcome probit equations with correlated normal errors by maximizing the bivariate-normal log-likelihood, reporting both coefficient tables, the error correlation rho with a Wald rho=0 test that detects whether the outcomes can be modeled as two independent probits, the log-likelihood and n. | ✅ Full | Ashford & Sowden (1970); Greene (2018) |
| Ordered Logit | Ordinal outcome; reports coefficients, threshold parameters, and McFadden pseudo-R². | ✅ Full | Seabold & Perktold (2010) |
| Generalized Ordered Logit | Fits an ordinal-outcome model that relaxes the proportional-odds (parallel-lines) assumption by letting each cumulative split have its own ML-estimated coefficient vector, reporting per-threshold coefficient tables, a Brant-style Wald test (with an LR cross-check) of parallel lines, log-likelihood, AIC/BIC, and an interpretation. | ✅ Full | Williams (2006); Fu (1998) |
| Sequential / Continuation-Ratio Logit | Models an ordinal outcome as a sequence of binary continuation-ratio logits -- one per ordered transition, fit on the at-risk subset {Y>=r} -- yielding stage-specific coefficients for outcomes whose categories are stages literally passed through in order. | ✅ Full | Fienberg (1980); Tutz (1991) |
| Ordered Probit | Ordinal outcome via normal CDF; threshold parameters and pseudo-R² included. | ✅ Full | Seabold & Perktold (2010) |
| Multinomial Logit | Multi-category outcome; reports coefficients and relative-risk ratios per category relative to base. | ✅ Full | Seabold & Perktold (2010) |
| Choice-Based Sampling (WESML) | Corrects a conditional/multinomial logit for endogenous (choice-based) stratified sampling using the Manski-Lerman WESML estimator, reweighting each observation by population-share / sample-share of its chosen alternative to recover consistent population coefficients with robust sandwich standard errors, alongside a side-by-side comparison to the biased unweighted fit. | ✅ Full | Manski & Lerman (1977) |
| Multinomial Probit (GHK) | Estimates a random-utility multinomial probit on long-format choice data by maximum simulated likelihood with the Geweke-Hajivassiliou-Keane (GHK) recursive simulator, recovering attribute coefficients and the identified correlated-error covariance while relaxing the IIA restriction of the conditional logit. | ✅ Full | Geweke (1991); Hajivassiliou & McFadden (1998); Keane (1994) |
| Nested Logit | Two-level nested logit for discrete choice estimated by FIML, relaxing IIA within nests via per-nest dissimilarity (lambda) parameters with a 0<lambda<=1 RUM-consistency check and a likelihood-ratio test against the conditional logit. | ✅ Full | McFadden (1978); Train (2009) |
| Mixed (Random-Parameters) Logit | Random-parameters (mixed) logit estimated by simulated maximum likelihood on long-format discrete-choice data, drawing Normal mixing coefficients via Halton draws to recover preference means and the std devs that quantify unobserved heterogeneity. | ✅ Full | Train (2009); Revelt & Train (1998) |
| Maximum Simulated Likelihood | Fits a random-coefficient (mixed) binary-choice probit by Maximum Simulated Likelihood, approximating the intractable integral over unobserved slope heterogeneity with shifted-Halton low-discrepancy draws and reporting the mean coefficients, the heterogeneity standard deviation, the simulated log-likelihood, the number of draws, and a likelihood-ratio comparison to the pooled (no-heterogeneity) probit. | ✅ Full | Train (2009); Gourieroux & Monfort (1996) |
| Conditional Logit (McFadden) | Estimates McFadden's choice-specific conditional logit on long-format discrete-choice data, recovering how alternative-varying attributes drive the probability that each alternative is chosen within a choice occasion, with a full coefficient table, log-likelihood, and McFadden pseudo-R2. | ✅ Full | McFadden (1974) |
| Poisson Regression | Count data (non-negative integer); reports incidence-rate ratios (IRRs) and dispersion diagnostics; optional exposure/offset term for rate (frequency) models. | ✅ Full | Seabold & Perktold (2010) |
| Negative Binomial (NB1 / NB2) | Overdispersed count data; MLE-estimated dispersion (alpha). Selectable NB2 (quadratic variance, default) or NB1 (linear variance). Includes likelihood-ratio test of Poisson vs NB. | ✅ Full | Hilbe (2011); Cameron & Trivedi (2013) |
| Zero-Inflated Poisson (ZIP) | Count data with excess zeros via mixing; separate count and zero-inflation equations with Vuong test. | ✅ Full | Seabold & Perktold (2010) |
| Zero-Inflated Negative Binomial (ZINB) | Overdispersed count data with excess zeros; count + inflation + alpha parameter estimated jointly. | ✅ Full | Seabold & Perktold (2010) |
| Hurdle Model | Two-part: logit for zero vs nonzero + truncated Poisson for positive counts. | ✅ Full | Seabold & Perktold (2010) |
| Tobit Regression (Censored) | Outcome censored at known bound (lower/upper); MLE estimation with sigma parameter. | ✅ Full | Seabold & Perktold (2010) |
| Censored LAD (Powell CLAD) | Powell's (1984) Censored Least Absolute Deviations estimator for left-censored (Tobit-type) outcomes, fit by an iterated median-regression linear-programming scheme with bootstrap standard errors, consistent under conditional-median restrictions and robust to heteroskedastic or heavy-tailed errors where Tobit is biased. | ✅ Full | Powell (1984, J. Econometrics) |
| Truncated Regression | Sample truncated (observations above threshold retained); MLE with sigma. | ✅ Full | Seabold & Perktold (2010) |
| Heckman Selection Model (Two-Step) | Endogenous sample selection via probit + IMR augmentation; reports rho correlation coefficient. | ✅ Full | Seabold & Perktold (2010) |
| Endogenous Switching (Roy model) | Estimates a two-regime Roy switching regression with regime-specific outcome equations and a probit selection equation, correcting each regime for selection on unobservables via its own inverse Mills ratio and reporting the rho_0/rho_1 error-correlation terms. | ✅ Full | Roy (1951); Lee (1978); Maddala (1983) |
| Beta Regression | Outcome in (0,1); precision parameter (phi) and mean-model coefficients estimated jointly. | ✅ Full | Seabold & Perktold (2010) |
| Fractional-response quasi-MLE (Papke-Wooldridge) | Outcome in [0,1]; Papke-Wooldridge (1996) Bernoulli quasi-MLE — GLM with binomial family + logit link; includes marginal effects. | ✅ Full | Papke & Wooldridge (1996); Nelder & Wedderburn (1972) |
| Quantile Regression | Conditional quantile (median or specified tau); pseudo-R² and quantile-specific inference. | ✅ Full | Seabold & Perktold (2010) |
| Ridge Regression (L2) | L2 regularized OLS on standardized predictors; alpha controls shrinkage intensity. | ✅ Full | Seabold & Perktold (2010) |
| LASSO Regression (L1) | L1 regularized OLS; automatic feature selection via coefficient sparsity. Standardized fit. | ✅ Full | Seabold & Perktold (2010) |
| Elastic Net (L1+L2) | Combined L1/L2 penalty on standardized predictors; l1_ratio balances penalties. | ✅ Full | Seabold & Perktold (2010) |
| Elastic Net (cross-validated) | Elastic-net penalized linear regression that standardizes predictors and uses K-fold cross-validation to pick the penalty strength lambda at minimum CV-MSE and at the 1-SE rule, reporting the CV-MSE curve, original-scale selected coefficients, the number selected, and R^2. | ✅ Full | Zou & Hastie (2005, JRSS-B); Friedman, Hastie & Tibshirani (2010, J. Stat. Softw. / glmnet) |
| Robust Regression (Huber M-estimator) | M-estimator resistant to outliers; asymptotic-normal Wald inference (z-stat); no t-distribution df. | ✅ Full | Seabold & Perktold (2010) |
| Stepwise Selection | Greedy forward/backward/both; adds/removes by AIC or BIC until no improvement. | ✅ Full | Seabold & Perktold (2010) |
| Weighted Least Squares (WLS) | User-supplied or estimated weights (FGLS); corrects heteroscedasticity. | ✅ Full | Seabold & Perktold (2010) |
| Feasible Generalized Least Squares (FGLS) | Cochrane-Orcutt iterative GLSAR; corrects serial correlation in errors. | ✅ Full | Seabold & Perktold (2010) |
| Nonlinear Least Squares (NLS) | User-provided formula and starting values; scipy.optimize curve_fit minimizes residual sum of squares. | ✅ Full | Seabold & Perktold (2010) |
| Seemingly Unrelated Regressions (SUR) | Multiple equations fitted jointly accounting for cross-equation correlation; Breusch-Pagan test included. | ✅ Full | Seabold & Perktold (2010) |
| Three-Stage Least Squares (3SLS) | System of equations with endogenous right-hand-side variables; IV + SUR combined. | ✅ Full | Seabold & Perktold (2010) |
| FIML (simultaneous system) | Full-Information Maximum Likelihood joint estimation of a two-equation linear simultaneous-equations system, maximizing the system Gaussian log-likelihood concentrated over the residual covariance (with the ln|det(B)| Jacobian) to recover structural coefficients with FIML standard errors, the residual covariance/correlation, the system log-likelihood, and a side-by-side comparison against equation-by-equation 2SLS. | ✅ Full | Koopmans, Rubin & Leipnik (1950); Hausman (1983); Greene (2018) |
| Generalized Method of Moments (GMM) | IV-GMM with moment conditions; J-test of overidentifying restrictions. | ✅ Full | Seabold & Perktold (2010) |
| Panel Fixed Effects | Within-entity estimator; removes time-invariant heterogeneity. Clustered SEs on entity. Optional two-way (entity + time) effects. | ✅ Full | Seabold & Perktold (2010) |
| Panel Random Effects | GLS random-effects estimator; assumes random effect uncorrelated with predictors. | ✅ Full | Seabold & Perktold (2010) |
| Linear Mixed Model (Random Intercepts/Slopes) | REML by default; fixed-effect Wald-z inference. AIC/BIC reported only under ML. | ✅ Full | Laird & Ware (1982); Harville (1977) |
| Intraclass Correlation (ICC) | ICC(1) from one-way random-intercept model; variance-partition coefficient for clustering. | ✅ Full | Shrout & Fleiss (1979); McGraw & Wong (1996) |
| Difference-in-Differences (DiD) | Treatment×Post interaction; cluster-robust or HC1 SEs depending on clustering. | ✅ Full | Card & Krueger (1994); Bertrand et al. (2004) |
| Regression Discontinuity (Sharp RDD) | Local-linear at cutoff with triangular or uniform kernel. Ad-hoc or user-supplied bandwidth. | ⚠️ Limited | Seabold & Perktold (2010) |
| RD Robust (CCT bias-corrected) | Sharp regression-discontinuity treatment effect via local-polynomial (triangular-kernel) fits on each side of the cutoff within an MSE-optimal bandwidth, reporting the conventional point estimate, the Calonico-Cattaneo-Titiunik (2014) bias-corrected estimate, conventional and robust standard errors, confidence intervals, and effective sample sizes. | ✅ Full | Calonico, Cattaneo & Titiunik (2014); Imbens & Kalyanaraman (2012) |
| Propensity Score Matching (PSM) | Logit/probit propensity score; 1:1/1:k matching with replacement or caliper. | ✅ Full | Rosenbaum & Rubin (1983) |
| Extended Regression Model — Linear (ERM-Linear) | Combines endogeneity (IV-2SLS), sample selection (Heckman), treatment effects in one model. | ✅ Full | Seabold & Perktold (2010) |
| Extended Regression Model — Probit (ERM-Probit) | Endogeneity (control function) + selection (bivariate-probit IMR) for binary outcome. | ⚠️ Limited | Seabold & Perktold (2010) |
| Extended Regression Model — Ordered (ERM-Ordered) | Ordered probit with sample selection correction via IMR augmentation. | ⚠️ Limited | Seabold & Perktold (2010) |
| Extended Regression Model — Interval (ERM-Interval) | Outcome observed as bounds [lower, upper]; handles left/right/interval censoring via MLE. | ✅ Full | Seabold & Perktold (2010) |
| ERM Diagnostics | Specification tests for endogeneity, selection bias, and treatment effect significance within ERM. | ✅ Full | Seabold & Perktold (2010) |
| LOESS Regression | Local polynomial regression; non-parametric smoothing via weighted least squares. | ✅ Full | Seabold & Perktold (2010) |
| Gamma GLM (severity) | Gamma generalized linear model with log link for strictly-positive responses (e.g. claim severity/cost); supports an exposure/offset term. Dispersion via Pearson, matching R glm(family=Gamma(link="log")). | ✅ Full | Nelder & Wedderburn (1972) |
| Tweedie GLM (pure premium) | Tweedie compound Poisson-Gamma GLM (log link) for non-negative data with a mass at zero — single-model pure-premium pricing; configurable variance power p in (1,2). | ✅ Full | Nelder & Wedderburn (1972); Jorgensen (1987) |
Notes & limitations
- Regression Discontinuity (Sharp RDD) — Bandwidth selection is ad-hoc (1.5*SD rule), not data-driven MSE-optimal.
- Extended Regression Model — Probit (ERM-Probit) — IMR lambda SE not corrected for generated-regressor problem (Murphy-Topel correction not implemented).
- Extended Regression Model — Ordered (ERM-Ordered) — Lambda SE uncorrected; no p-values reported due to generated-regressor issue.