Understanding Multivariate Relationships Across Time: Introduction to the Theory of Vector Autoregression(VAR)

Vector Autoregression (VAR) is one of the latest developments in time series econometrics.  It’s ability to make more accurate forecast than univariate models and the insight that it provides into the dynamics of the macroeconomy have made it extremely popular among economic researchers.  Forecast can be made by considering the expected value of input variables which adds to the flexibility and realism to processes which develop with some uncertainty.  The theory of vector autoregression is an extension of the univariate autoregressive models.  A vector autoregression is a system of equations that describe the evolution and interdependencies of multiple time series. Each variable of the time series is affected by it’s own lags and the lags of the other time series variables in the system.

Primitive System

Key Assumptions:

  • The error terms in both equations are normally distributed with constant mean and variance
  • Error terms show no serial correlation in either equation
  • There is no correlation between the error term of equation y and z

Why can’t the primitive system be estimated directly?

The system described above is called a first-order vector autoregression because the largest lag is unity.  The is an indirect relationship between the error term in the y equation and the z equation which violates an assumption of the classical linear regression model that “regressors be uncorrelated with error terms.

Deriving the Reduced Form Equations from the Primitive Equations

In order to eliminate the violate which prohibits the estimation of the primitive system, it must be transformed into a system with that adheres to the assumptions in the classical linear regression model. This can be done by using matrix algebra and testing the assumptions of the derived system.

Move all the contemporaneous variables to the left hand side

Write the equation above in matrix notation and solve for y and z

The equation above can be simplified if we name the different matrices above by the following conventions. This simplification will only simplify the notation to aid in calculation and has doesn’t compromise any of the information in the system.

Using the matrix notation above the system and it’s solution can be written in the following way:

Move from Matrix Notation Back to Linear Equations

Closing Remarks

The primitive VAR system has now been transformed into the reduced form VAR by the matrix algebra manipulations above. In this system the error terms have zero mean, constant variance, the error terms are serially and mutually uncorrelated. The properties of the reduced for VAR enable theorist and economic practitioners to measure and understand economic relationships without having to worry about the violation of regression assumptions. In the following post the properties of the reduced form error terms will be be mathematically proven and further analysis will be provided to examine the interesting properties of VAR.