Varmax

A VARMAX model (Vector Autoregressive Moving Average with Exogenous Regressors) is a multivariate time-series model extending two classical ideas:

1. VAR($p$): Models interactions among multiple endogenous series using autoregressive lags.
2. VMA($q$): Captures correlations in multivariate residuals using moving-average terms.
3. X: Allows the inclusion of exogenous predictors.

Formally, a $k$-dimensional VARMAX($p$, $q$) for endogenous vector $y_t$ and exogenous predictors $x_t$ can be written as:

$y_t=c+{\sum }_{i=1}^p{A}_i{y}_{t-i}+{\sum }_{j=1}^q{M}_j{\epsilon }_{t-j}+B{x}_t+{\epsilon }_t$

Where:

• $y_t$: vector of endogenous variables
• $A_i$: autoregressive coefficient matrices
• $M_j$: moving-average coefficient matrices
• $B$: matrix of coefficients for exogenous variables
• ${\epsilon }_t$: error terms (possibly correlated across series)

Statsmodels implements this within the statespace framework, giving access to Kalman filtering, smoothing, and forecast intervals.

Multivariate Analysis

How VARMAX is used

1. Multivariate forecasting

Useful when several time-dependent variables influence each other. Examples:

• Forecasting $[\text{demand},\text{price}]$ together
• Forecasting multiple KPIs: $[\text{revenue},\text{users},\text{conversion}]$

Because the model estimates cross-lag impacts, you obtain richer dynamics than running separate univariate models.

2. Understanding cross-relationships

You can quantify how a change in one variable affects others over future periods. Common analyses include:

• Impulse-response functions
• Forecast error variance decomposition

3. Including external drivers

Exogenous regressors allow structured drivers to influence all series.

Examples:

• $x_t$: temperature affecting energy demand and cost
• $x_t$: marketing spend affecting multiple product KPIs

4. Scenario analysis

Once fitted, you can simulate future paths under different exogenous scenarios.

Example usage in Statsmodels

import pandas as pd
from statsmodels.tsa.statespace.varmax import VARMAX
 
# Assume df contains:
#   'revenue', 'customers', plus exogenous column 'marketing_spend'
endog = df[['revenue', 'customers']]
exog = df[['marketing_spend']]
 
model = VARMAX(endog, exog=exog, order=(2, 1))
fit = model.fit(disp=False)
 
# Forecast 12 periods ahead
forecast = fit.get_forecast(steps=12, exog=future_exog)
forecast_df = forecast.predicted_mean

When to consider VARMAX

Choose VARMAX when:

• Your variables interact across time.
• Residuals show multivariate autocorrelation.
• You need forecasts that account for cross-effects.
• You need to incorporate exogenous drivers.

Avoid it when:

• You want a purely univariate model (consider ARIMA/ETS).
• You have very short time series (many parameters).
• You need nonlinear or regime-switching dynamics (consider deep learning or structural models).