# Time series models: VAR model definition

## September 30, 2015

The Vector AutoRegression (VAR) family of models has been widely used for modelling and forecasting since the early 1980s. A VAR model is a conceptually simple system of multivariate models where each variable is explained by its own past values and the past values of all other variables in the system.

# VAR model definition

The Vector AutoRegression (VAR) family of models has been widely used for modelling and forecasting since the work of Sims (1980)1. A VAR model is a system of multivariate models in which each variable is explained by its own past values and the past values of all other variables in the system. For example, with two variables $$x$$ and $$y$$, and two lagged values of each variables, the system is given by

$x_t=\alpha_0+\alpha_1 x_{t-1}+\alpha_2 x_{t-2} + \alpha_3y_{t-1} + \alpha_4y_{t-2} + \epsilon_{1t}$ $y_t=\beta_0+\beta_1 x_{t-1}+\beta_2 x_{t-2} + \beta_3y_{t-1} + \beta_4y_{t-2} + \epsilon_{2t}$

where $$\epsilon_1$$ and $$\epsilon_2$$ are generally correlated random errors. There are ten unknown parameters in the above system. More generally, the system of $$k$$ variables with $$l$$ legs on each variable plus a constant in each equation will have $$(k^2l)+k$$ parameters to estimate. As the right-hand side variables are the same in each equation, the ordinary least squares (OLS) can be used to estimate the system. Using matrix notation, the general VAR model can be written

$\mathbf{Y}_t=\mathbf{A}\mathbf{Y}_{t-1}+ \boldsymbol{\epsilon}_t$

where $$\mathbf{A}$$ is a matrix of polynomials in the lag operator, $$\mathbf{Y}$$ is the vector of variables, and $$\boldsymbol{\epsilon}$$ is a vector of random errors. Further details including the statistical and econometric research preceding the development of VAR models, problems of parameter estimation, cointegration and forecasting are available from Holden (1995).

Holden, K. (1995). Vector autoregression modeling and forecasting. Journal of Forecasting, 14(3), 159–166. http://doi.org/10.1002/for.3980140302

Sims, C. (1980). Macroeconomics and reality. Econometrica, 48(1), 1–48. http://doi.org/10.2307/1912017

1. In October 2011, Christopher Sims of Princeton University shared the Nobel Prize for economics with Thomas Sargent of New York University “for their empirical research on cause and effect in the macroeconomy”. Read more…