Vector autoregression

Statistical model to calculate the value of multiple quantities as they change over time

Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single variable (univariate) autoregressive model by allowing for multivariate time series. VAR models are often used in economics and the natural sciences. Like the autoregressive model, each variable has an equation modelling its evolution over time. This equation includes the variable's lagged (past) values, the lagged values of the other variables in the model, and an error term. VAR models do not require as much knowledge about the forces influencing a variable as do structural models with simultaneous equations. The only prior knowledge required is a list of variables which can be hypothesized to affect each other over time.

A VAR model describes the evolution of a set of k variables, called endogenous variables, over time. Each period of time is numbered, t = 1, , T. The variables are collected in a vector, yt, which is of length k. (Equivalently, this vector might be described as a (k × 1) matrix.) The vector is modelled as a linear function of its previous value. The vector's components are referred to as yi,t, meaning the observation at time t of the i th variable. For example, if the first variable in the model measures the price of wheat over time, then y1,1998 would indicate the price of wheat in the year 1998. VAR models are characterized by their order, which refers to the number of earlier time periods the model will use. Continuing the above example, a 5th order VAR would model each year's wheat price as a linear combination of the last five years of wheat prices. A lag is the value of a variable in a previous time period. So in general a pth order VAR refers to a VAR model which includes lags for the last p time periods. A pth order VAR is denoted "VAR(p)" and sometimes called "a VAR with p lags". A pth order VAR model is written as

The variables of the form yt−i indicate that variable's value i time periods earlier and are called the "ith lag" of yt. The variable c is a k vector of constants serving as the intercept of the model. Ai is a time invariant (k × k) matrix and et is a k vector of error terms. The error terms must satisfy three conditions:
Every error term has a mean of zero. The contemporaneous covariance matrix of error terms is a k × k positive semidefinite matrix denoted Ω.
for any non zero k. There is no correlation across time. In particular, there is no serial correlation in individual error terms. The process of choosing the maximum lag p in the VAR model requires special attention because inference is dependent on correctness of the selected lag order.

Vector autoregression models often involve the estimation of many parameters. For example, with seven variables and four lags, each matrix of coefficients for a given lag length is 7 by 7, and the vector of constants has 7 elements, so a total of 49×4 + 7 = 203 parameters are estimated, substantially lowering the degrees of freedom of the regression (the number of data points minus the number of parameters to be estimated). This can hurt the accuracy of the parameter estimates and hence of the forecasts given by the model.

Properties of the VAR model are usually summarized using structural analysis using Granger causality, impulse responses, and forecast error variance decompositions.

An estimated VAR model can be used for forecasting, and the quality of the forecasts can be judged, in ways that are completely analogous to the methods used in univariate autoregressive modelling.

Christopher Sims has advocated VAR models, criticizing the claims and performance of earlier modeling in macroeconomic econometrics. VAR models are also increasingly used in health research for automatic analyses of diary data or sensor data.