# Core Statistical Modeling Capabilities

Linear models, generalized linear models, discrete choice models, time series analysis,
and statistical tests and diagnostics — what statsmodels offers in each area and which
class to use.

## Core Statistical Modeling Capabilities

### 1. Linear Regression Models

Comprehensive suite of linear models for continuous outcomes with various error structures.

**Available models:**
- **OLS**: Standard linear regression with i.i.d. errors
- **WLS**: Weighted least squares for heteroskedastic errors
- **GLS**: Generalized least squares for arbitrary covariance structure
- **GLSAR**: GLS with autoregressive errors for time series
- **Quantile Regression**: Conditional quantiles (robust to outliers)
- **Mixed Effects**: Hierarchical/multilevel models with random effects
- **Recursive/Rolling**: Time-varying parameter estimation

**Key features:**
- Comprehensive diagnostic tests
- Robust standard errors (HC, HAC, cluster-robust)
- Influence statistics (Cook's distance, leverage, DFFITS)
- Hypothesis testing (F-tests, Wald tests)
- Model comparison (AIC, BIC, likelihood ratio tests)
- Prediction with confidence and prediction intervals

**When to use:** Continuous outcome variable, want inference on coefficients, need diagnostics

**Reference:** See `references/linear_models.md` for detailed guidance on model selection, diagnostics, and best practices.

### 2. Generalized Linear Models (GLM)

Flexible framework extending linear models to non-normal distributions.

**Distribution families:**
- **Binomial**: Binary outcomes or proportions (logistic regression)
- **Poisson**: Count data
- **Negative Binomial**: Overdispersed counts
- **Gamma**: Positive continuous, right-skewed data
- **Inverse Gaussian**: Positive continuous with specific variance structure
- **Gaussian**: Equivalent to OLS
- **Tweedie**: Flexible family for semi-continuous data

**Link functions:**
- Logit, Probit, Log, Identity, Inverse, Sqrt, CLogLog, Power
- Choose based on interpretation needs and model fit

**Key features:**
- Maximum likelihood estimation via IRLS
- Deviance and Pearson residuals
- Goodness-of-fit statistics
- Pseudo R-squared measures
- Robust standard errors

**When to use:** Non-normal outcomes, need flexible variance and link specifications

**Reference:** See `references/glm.md` for family selection, link functions, interpretation, and diagnostics.

### 3. Discrete Choice Models

Models for categorical and count outcomes.

**Binary models:**
- **Logit**: Logistic regression (odds ratios)
- **Probit**: Probit regression (normal distribution)

**Multinomial models:**
- **MNLogit**: Unordered categories (3+ levels)
- **Conditional Logit**: Choice models with alternative-specific variables
- **Ordered Model**: Ordinal outcomes (ordered categories)

**Count models:**
- **Poisson**: Standard count model
- **Negative Binomial**: Overdispersed counts
- **Zero-Inflated**: Excess zeros (ZIP, ZINB)
- **Hurdle Models**: Two-stage models for zero-heavy data

**Key features:**
- Maximum likelihood estimation
- Marginal effects at means or average marginal effects
- Model comparison via AIC/BIC
- Predicted probabilities and classification
- Goodness-of-fit tests

**When to use:** Binary, categorical, or count outcomes

**Reference:** See `references/discrete_choice.md` for model selection, interpretation, and evaluation.

### 4. Time Series Analysis

Comprehensive time series modeling and forecasting capabilities.

**Univariate models:**
- **AutoReg (AR)**: Autoregressive models
- **ARIMA**: Autoregressive integrated moving average
- **SARIMAX**: Seasonal ARIMA with exogenous variables
- **Exponential Smoothing**: Simple, Holt, Holt-Winters
- **ETS**: Innovations state space models

**Multivariate models:**
- **VAR**: Vector autoregression
- **VARMAX**: VAR with MA and exogenous variables
- **Dynamic Factor Models**: Extract common factors
- **VECM**: Vector error correction models (cointegration)

**Advanced models:**
- **State Space**: Kalman filtering, custom specifications
- **Regime Switching**: Markov switching models
- **ARDL**: Autoregressive distributed lag

**Key features:**
- ACF/PACF analysis for model identification
- Stationarity tests (ADF, KPSS)
- Forecasting with prediction intervals
- Residual diagnostics (Ljung-Box, heteroskedasticity)
- Granger causality testing
- Impulse response functions (IRF)
- Forecast error variance decomposition (FEVD)

**When to use:** Time-ordered data, forecasting, understanding temporal dynamics

**Reference:** See `references/time_series.md` for model selection, diagnostics, and forecasting methods.

### 5. Statistical Tests and Diagnostics

Extensive testing and diagnostic capabilities for model validation.

**Residual diagnostics:**
- Autocorrelation tests (Ljung-Box, Durbin-Watson, Breusch-Godfrey)
- Heteroskedasticity tests (Breusch-Pagan, White, ARCH)
- Normality tests (Jarque-Bera, Omnibus, Anderson-Darling, Lilliefors)
- Specification tests (RESET, Harvey-Collier)

**Influence and outliers:**
- Leverage (hat values)
- Cook's distance
- DFFITS and DFBETAs
- Studentized residuals
- Influence plots

**Hypothesis testing:**
- t-tests (one-sample, two-sample, paired)
- Proportion tests
- Chi-square tests
- Non-parametric tests (Mann-Whitney, Wilcoxon, Kruskal-Wallis)
- ANOVA (one-way, two-way, repeated measures)

**Multiple comparisons:**
- Tukey's HSD
- Bonferroni correction
- False Discovery Rate (FDR)

**Effect sizes and power:**
- Cohen's d, eta-squared
- Power analysis for t-tests, proportions
- Sample size calculations

**Robust inference:**
- Heteroskedasticity-consistent SEs (HC0-HC3)
- HAC standard errors (Newey-West)
- Cluster-robust standard errors

**When to use:** Validating assumptions, detecting problems, ensuring robust inference

**Reference:** See `references/stats_diagnostics.md` for comprehensive testing and diagnostic procedures.