Multivariate Forecasting

Vectrix supports multivariate time series forecasting through Vector AutoRegression (VAR) and Vector Error Correction Models (VECM).

VAR Model

VAR models capture the linear interdependencies between multiple time series. Each variable is modeled as a linear combination of its own lagged values and the lagged values of all other variables.

from vectrix.engine.var import VARModel
import numpy as np

# Y shape: (T, k) — T time steps, k variables
Y = np.column_stack([sales, advertising, inventory])

model = VARModel(maxLag=5, criterion="aic")
model.fit(Y)

predictions, lower, upper = model.predict(steps=12)
# predictions shape: (12, 3) — 12 steps, 3 variables

Automatic Lag Selection

The model tests all lag orders from 1 to maxLag and selects the best one based on information criteria

model = VARModel(maxLag=8, criterion="aic")   # Akaike IC (default)
model = VARModel(maxLag=8, criterion="bic")   # Bayesian IC (sparser)

model.fit(Y)
print(f"Selected lag order: {model.order}")

Granger Causality

Test whether one variable helps predict another

model = VARModel()
result = model.grangerCausality(Y, cause=0, effect=1, maxLag=5)

print(f"F-statistic: {result['fStat']:.3f}")
print(f"p-value: {result['pValue']:.4f}")

if result['pValue'] < 0.05:
    print("Variable 0 Granger-causes Variable 1")

VECM Model

Vector Error Correction Models are used when variables are cointegrated — they share a long-run equilibrium relationship despite being individually non-stationary.

from vectrix.engine.var import VECMModel

# Cointegrated series example: prices of related assets
prices = np.column_stack([stock_a, stock_b])

model = VECMModel(maxLag=4)
model.fit(prices)

predictions, lower, upper = model.predict(steps=10)

When to Use VECM vs VAR

Cointegration Rank

VECM automatically estimates the cointegration rank (number of independent cointegrating relationships)

model = VECMModel(maxLag=4, rank=None)  # auto-detect
model = VECMModel(maxLag=4, rank=1)     # force rank=1

model.fit(Y)
print(f"Cointegration rank: {model._rank}")

Complete Example

import numpy as np
from vectrix.engine.var import VARModel

rng = np.random.default_rng(42)
T = 200

# Generate two related series
Y = np.zeros((T, 2))
for t in range(1, T):
    Y[t, 0] = 0.5 * Y[t-1, 0] + 0.2 * Y[t-1, 1] + rng.normal(0, 0.1)
    Y[t, 1] = 0.3 * Y[t-1, 0] + 0.4 * Y[t-1, 1] + rng.normal(0, 0.1)

# Fit and predict
model = VARModel(maxLag=5, criterion="bic")
model.fit(Y)

print(f"Selected order: {model.order}")

pred, lo, hi = model.predict(steps=10)
print(f"Predictions shape: {pred.shape}")  # (10, 2)

# Granger causality
gc = model.grangerCausality(Y, cause=0, effect=1, maxLag=3)
print(f"Does series 0 cause series 1? p={gc['pValue']:.4f}")