Vector Autoregression (VAR)

Vector Autoregression (VAR) models multiple time series jointly using their past values. It captures interdependencies between variables by modeling each variable as a function of its own lagged values and the lagged values of all other variables in the system.

When to Use Vector Autoregression

VAR is best suited for:

• Multivariate time series forecasting (multiple target variables)
• When variables influence each other (e.g., price and volume, temperature and electricity)
• Economic and financial systems with interdependencies
• Forecasting multiple metrics that move together
• Analyzing impulse responses and causal relationships
• When you need to forecast multiple correlated series simultaneously

Strengths

• Captures interdependencies between multiple time series
• No need to specify which variables are "dependent" vs "independent"
• Provides forecasts for all variables simultaneously
• Each variable's forecast uses information from other variables
• Well-established econometric framework
• Can analyze Granger causality and impulse responses
• Symmetric treatment of all variables

Weaknesses

• Requires multiple target variables (not suitable for univariate forecasting)
• Large number of parameters scales with number of variables and lags (variables × lags × variables)
• Needs substantial historical data for stable estimates
• All variables must be stationary or integrated of the same order
• Does not handle seasonal patterns directly
• Cannot incorporate exogenous predictors easily
• Interpretability decreases with many variables
• Computationally intensive for high-dimensional systems

Parameters

Common Time Series Parameters

All time series models share these parameters:

• Timestamp Column (required): Column containing dates/times for time series indexing
• Target Columns (required): List of target variables for multivariate forecasting
  • Important: VAR requires multiple target columns (minimum 2)
  • Example: ['sales', 'inventory', 'price']
• Frequency (optional): Time spacing (D, H, W, M). Auto-inferred if not specified
• Forecast Steps (required, default=1): How many periods to predict ahead
• Aggregation (default='mean'): Method for duplicate timestamps (mean, sum, first, last, min, max)
• Fill Method (default='interpolate'): How to handle missing timestamps (interpolate, ffill, bfill)

VAR-Specific Parameters

Differencing Order

• Type: Integer
• Default: 0
• Min: 0
• Max: 2
• Description: Number of times each target series is differenced to achieve stationarity
  • 0: Use raw values (series is already stationary)
  • 1: First differences (removes linear trends)
  • 2: Second differences (removes quadratic trends)
• Guidance: Most economic time series need d=1. Check stationarity with statistical tests

Lag Order

• Type: Integer
• Default: null (auto-selected)
• Min: 1
• Description: Exact number of lagged time steps to include for each variable
  • If null, automatically selected using information criteria (AIC/BIC)
  • Higher lag order captures longer memory but increases parameters
• Example: lag_order=3 means each variable uses its past 3 values and past 3 values of all other variables

Max Lags

• Type: Integer
• Default: 10
• Min: 1
• Description: Maximum lag order considered when automatically selecting lag_order
• Visible When: lag_order is null
• Guidance:
  • For daily data: try 7-14
  • For monthly data: try 12-24
  • For quarterly data: try 4-8

Configuration Tips

Preparing Your Data for VAR

1. Ensure Stationarity: VAR requires stationary data

   • Check each variable for trends and unit roots
   • Set difference_order=1 if variables have trends
   • Consider logarithmic transformation for exponential growth

2. Select Related Variables: Include variables that likely influence each other

   • Example: [price, demand, inventory]
   • Avoid unrelated variables that add noise

3. Sufficient History: VAR needs enough data for stable estimation

   • Minimum: (number of variables × lag order × 10) observations
   • Example: 3 variables, lag=5 → need at least 150 observations

Choosing Lag Order

Automatic Selection (Recommended):

• Leave lag_order=null
• Set maxlags based on your frequency (7-14 for daily, 12 for monthly)
• The model will use information criteria to find optimal lag length

Manual Selection:

• Start with lag_order = number of periods in the shortest cycle
  • Daily data with weekly patterns: lag_order=7
  • Monthly data with yearly patterns: lag_order=12
• Increase if cross-validation shows improvement
• Decrease if model is overfitting or too slow

Handling Non-Stationarity

If your variables have trends:

1. Set difference_order=1 to difference all series
2. Forecasts will be in differenced space, then inverted back to original scale
3. Alternatively, use SARIMAX or Prophet for series with complex trends

Model Validation

• Use time series cross-validation to assess forecast accuracy
• Check that residuals are uncorrelated (white noise)
• Compare forecast accuracy across all target variables
• Consider multivariate metrics like trace RMSE

Common Issues and Solutions

Issue: Model Doesn't Converge or Produces Unstable Forecasts

Solution:

• Check that all variables are stationary (set difference_order=1)
• Reduce lag_order or maxlags
• Ensure sufficient data (at least 10× the number of parameters)
• Remove variables with very different scales (normalize if needed)

Issue: Too Many Parameters

Solution:

• Reduce the number of target variables (focus on most important)
• Decrease lag_order or maxlags
• Use a simpler model (ARIMA for univariate, or separate models per variable)
• Consider dimensionality reduction techniques before VAR

Issue: Poor Forecast for Some Variables

Solution:

• Check if those variables are truly interdependent with others
• Consider separate models for weakly related variables
• Ensure those variables don't have different stationarity properties
• Verify data quality for poorly forecasted variables

Issue: Need Seasonal Patterns

Solution:

• VAR doesn't handle seasonality directly
• Apply seasonal differencing before VAR (e.g., difference at lag=12 for monthly)
• Use seasonal decomposition, then apply VAR to deseasonalized data
• Consider SARIMA or Prophet for strongly seasonal data

Issue: Want to Include Exogenous Variables

Solution:

• VAR doesn't easily handle exogenous predictors
• Use SARIMAX (extends ARIMA with exogenous variables)
• Or use multivariate models like VECM if variables are cointegrated
• Alternatively, include exogenous variables as additional targets if they're available at forecast time

Issue: Forecasts Revert to Mean Too Quickly

Solution:

• This is expected for stationary VAR (forecasts converge to mean)
• Ensure difference_order is appropriate
• For persistent dynamics, consider VECM if variables are cointegrated
• Increase lag_order to capture longer-term dependencies

Issue: High Computational Cost

Solution:

• Reduce number of target variables
• Decrease maxlags to reduce search space
• Specify lag_order manually to skip automatic selection
• Use daily aggregation instead of hourly if frequency is high

Example Use Cases

• Retail forecasting: Forecast [sales, inventory, returns] simultaneously to capture their interactions
• Energy markets: Model [electricity_price, demand, temperature] jointly
• Financial markets: Predict [stock_price, volume, volatility] considering their interdependencies
• Macroeconomic modeling: Forecast [GDP, inflation, unemployment] in an integrated framework
• Supply chain: Model [orders, shipments, inventory_levels] to understand system dynamics
• Marketing: Forecast [impressions, clicks, conversions] to capture funnel effects

Technical Notes

Granger Causality

VAR models can test if one variable "Granger-causes" another (if past values of X help predict Y beyond Y's own history).

Impulse Response Analysis

VAR allows analyzing how a shock to one variable propagates through the system over time.

Model Order

Total parameters = (number of variables)² × lag_order + intercepts

• 3 variables, lag=5: 3² × 5 + 3 = 48 parameters
• Ensure you have at least 10× data points for stable estimation