regularized-var 0.1.1

Regularized Vector Autoregression (VAR) with ridge shrinkage, Minnesota prior, metrics, and walk-forward validation.

regularized-var

Vector Autoregression (VAR) with l2 regularization, optional Minnesota prior, error metrics, and walk-forward validation.

Install

pip install regularized-var

Quickstart

import pandas as pd
from regularized_var.var import VAR, MinnesotaVAR
from regularized_var.metrics import mse
from regularized_var.model_selection import WalkForward, WalkForwardValidator

# Data
df = pd.DataFrame(...)

# VAR
model = VAR(n_lags=2, alpha=0.1).fit(df)
print(model.predict(steps=5))

# MinnesotaVAR
mvar = MinnesotaVAR(n_lags=3, alpha_own=1.0, alpha_cross=2.0).fit(df)
print(mvar.predict(steps=3))

# Walk-forward validation
splitter = WalkForward(train_size=1000, min_train_size=500, horizon=5)
wf = WalkForwardValidator(VAR, {"n_lags": 2, "alpha": 0.1}, splitter, metric=mse)
wf.run(df)
print(wf.overall_score_)

Mathematical Formulation

Ridge-regularized VAR

The baseline VAR class solves a ridge-penalized least squares problem:

where Y are the responses, Z is the lagged design matrix, and the intercept (if present) is excluded from the penalty.

This is equivalent to multivariate ridge regression (scikit-learn’s Ridge) applied equation-by-equation, but solved in closed form for all equations at once.

MinnesotaVAR as Weighted Ridge

The Minnesota prior (see Litterman, 1986: "Forecasting with Bayesian Vector Autoregressions — Five Years of Experience") is a Bayesian shrinkage approach that imposes beliefs about how coefficients in a VAR should behave:

• Own-lag coefficients are shrunk toward 0 (for stationary/differenced data: growth rates, returns, etc.)
• Cross-lag coefficients (other variables) are shrunk more heavily toward 0
• Higher lags are shrunk more strongly than shorter lags

In practice, this leads to a ridge-style penalty on the coefficient matrix:

with weights

• i: target equation index
• j: predictor variable index
• ℓ: lag index

This weighted ℓ₂ penalty reproduces a structure similar to the Minnesota prior, where:

• own lags are shrunk mildly,
• cross-variable lags are shrunk more heavily,
• higher-order lags are shrunk progressively more.

Thus, MinnesotaVAR can be seen as a structured ridge regression equivalent of the Minnesota Bayesian prior.