What is Vector Autoregression?

Vector autoregression (VAR) is a statistical method that captures the relationship between multiple quantities as they change over time. Autoregression is a statistical technique used in time-series analysis that assumes that the current value of a time series is a function of its past values. Autoregressive models use similar mathematical techniques to determine the probabilistic correlation between elements in a sequence.

VAR models are a special category of autoregressive models that allow you to estimate multiple (multivariate) outcomes from the same time series. For example, a VAR model might be used to predict future trends in economic indicators like inflation and interest rates. It would do this by analyzing other market indicators like fuel prices and average household income over a period of time.

Vector autoregression (VAR) is one approach to time-series forecasting for multivariate data. We give more use cases below.

Forecasting

A forecast is a prediction made by studying historical data and past patterns. Forecasting is the primary use case of VAR. For example, you can use VAR to assess the interdependencies of exchange rates, inflation, gross domestic production (GDP), and other factors when forecasting economic growth.

Impulse response analysis

Shocks, or sudden changes in one variable, impact the results of a statistical model. Mathematicians use vector autoregressive models to analyze how a shock in a system affects correlated outcomes. For example, you can use VAR to assess the effect of a sudden earnings decline on a person’s savings and expenditures.

Forecast error variance decomposition

Shocks may have varying effects on a statistical model’s results. FEVD allows mathematicians to attribute the impact to their respective variables proportionately. For example, long lead times might be more influential in manufacturing delays than machine inefficiencies.

Granger causality tests

When designing a multivariable model, mathematicians use the vector autoregressive method to determine if a variable significantly influences another variable. We call this the Granger causality test. For example, you can use VAR to find out if customer sentiment is a helpful indicator in predicting sales growth.

Time-series data is categorized as univariate and multivariate. For example, the total electricity consumption for a single household is a univariate time series over a period of time. When multiple univariate time series are stacked on each other, it’s called a multivariate time series. For example, the total electricity consumption of 10 different (but correlated) households in a single neighborhood makes up a multivariate time-series dataset.

Univariate autoregressive models predict the probabilistic outcome based on a single variable. For example, they use a person’s past earnings to predict their savings in upcoming months. In contrast, vector autoregression models allow the statistical model to consider several interdependent variables. For example, you can use expenditure, salary, month, and other indicators to predict future earnings.

Variable relationship

Vector autoregression allows data analysts to perform time-series analysis on multiple endogenous variables. Endogenous variables are statistical variables influenced by multiple factors. Vector autoregression analyzes endogenous variables and their past values.

Variables in the vector autoregressive model share a bi-directional relationship, which differs from univariate time-series models. In a VAR model, the predictor variable not only influences the forecast outcome but can also be influenced by the modeling results. For example, you can use a VAR model to predict revenue growth based on marketing budget. At the same time, you can also use the model to determine if an increase in revenue influences budget allocation.

Equation

Consider the mathematical equation below.

y_t = c₁ + φ₁y_t-1 + φ₂y_t-2 + φ_py_t-p + ε_t

This equation represents the VAR model as a linear function where:

• y represents a set of endogenous variables of interest.

• c is a constant value that acts as a baseline.

• φ is a coefficient that influences past values of y.

• e is the covariance matrix consisting of error values.

• p is the number of lags to be calculated.

When applied, a vector autoregressive model makes predictions based on relevant variables and their lag values. The model analyzes endogenous values up to a historical time period given by the p-value to forecast possible values for a dependent variable. A two-variable VAR will have two linear equations, each predicting a different variable.

Several variants of vector autoregression models are derived from the multivariate time-series analysis approach.

Standard vector autoregressive modeling

Standard VAR is the simplest statistical model for multivariate time-series analysis. Also known as reduced form VAR models, they use the past values of other variables and those of its key predictor to forecast their future outcomes. When modeling with standard VAR, you can create several equations to model the future outcomes of all the variables. Consider the following variables:

• Inflation rate.

• Unemployment rate.

• Interest rate.

With a standard VAR system, you can create:

• A first equation that predicts the future inflation rate by analyzing all three variables.

• A second equation that predicts the future unemployment rate by analyzing all three variables.

• A third equation that predicts interest rate fluctuation based on all variables.

Assuming all variables are correlated, all three equations will share a common error term, e. The error term denotes common shock events contributing to sudden changes in all variables.

Recursive vector autoregressive modeling

Recursive VAR works similarly to reduced form VAR, except that equations don’t share a common correlated error value. Moreover, each equation in a recursive VAR model applies the result of the past equation in its prediction. For example:

• The model predicts the inflation rate with all variables.

• The model predicts the unemployment rate with all variables plus the current inflation rate.

• The model predicts the interest rate with all variables plus the current inflation and unemployment rate.

Structural vector autoregressive modeling

Structural VAR allows analysts to perform complex policy analysis on multiple time-lagged variables. Besides analyzing lag values, analysts can set assumptions and use the model to prove if specific variables are correlated in one way or another. For example, you can use structural VAR to determine if national policy implementation positively impacts the unemployment rate.

Bayesian vector autoregression modeling

Bayesian VAR modeling shares a similar linear equation with the standard VAR, except that it includes coefficient priors. A coefficient prior is a probabilistic expression that indicates the range of distribution of specific parameters. With Bayesian VAR, you might improve forecast accuracy when analyzing a small dataset that contains too many parameters.

Long-memory vector autoregressive modeling

Long-memory vector autoregressive (LMVAR) modeling expands the time range of standard VAR models. It’s suitable for studying the impact of long histories. With LMVAR, data analysts can analyze multivariate components that extend further back in time and uncover correlations that are not apparent from a shorter interval. For example, LMVAR is helpful in analyzing financial data that spans decades.

We share how VAR compares with other autoregressive models below.

VAR vs. ARMA

An autoregressive moving average (ARMA) model predicts future outcomes based on lagged values and errors generated by past results. ARMA is helpful in predicting time-series variables such as weather, stock prices, and sales. Like ARMA, VAR is a statistical model that compares lagged values and errors to make predictions. However, ARMA is a univariate model, whereas VAR can analyze the influence of multiple variables in the model.

VAR vs. ARIMA

Autoregressive integrated moving average (ARIMA) is an expansion of ARMA with the addition of an integration component. Integration allows ARIMA to prevent peaks, outliers, or seasonal trends from significantly impacting its prediction. For example, you can use ARIMA to predict sales while taking seasonal holidays into account. ARIMA is also univariate. On the other hand, VAR is more suitable for causal analysis by discovering how different sales factors influence each other.

VAR vs. ARIMAX

ARIMAX is similar to ARIMA, but the model allows the inclusion of exogenous variables when predicting future outcomes. Exogenous variables are factors that may influence the forecast but are not directly part of the time-series variables. Conversely, VAR doesn’t use exogenous variables for calculation but compares multiple lagged values of endogenous variables.

AWS provides fully managed services that make building, deploying, and scaling vector time-series forecasting applications easier. From identifying causal relationships to predicting time-based outcomes, you can create diverse VAR applications affordably, accurately, and efficiently with AWS services.

Organizations use Amazon Forecast to forecast business outcomes without creating VAR models. With Amazon Forecast, you can analyze business metrics with forecasting models powered by world-class machine-learning technologies.

Amazon SageMaker provides data scientists and business analysts with machine learning and generative AI models for implementing custom VAR applications. You can create customized analytics workflows in a few clicks with SageMaker’s no-code interface.

Get started with VAR modeling on AWS by creating a free account today.