10 Vector autoregression model

• Vector autoregression (VAR) is a system of equations assuming that all variables are endogenously related (there is no exogenous variable)

• It is a multivariate dynamic model with number of equations equal to the number of endogenous variables $k$

• Reduced form of VAR($p$) model is commonly used in applications, which means that all RHS variables are lagged with $p$ time lags ($t-1$, $t-2$, $\dots$, $t-p$)

• System of equations with two endogenous variables ($k=2$) and only one time lag $p=1$ is called bivariate VAR($1$) model

$$\begin{equation} \begin{aligned} y_t&=\beta_{1,0}+\beta_{1,1}y_{t-1}+\beta_{1,2}x_{t-1}+u_{1,t} \\ x_t&=\beta_{2,0}+\beta_{2,1}y_{t-1}+\beta_{2,2}x_{t-1}+u_{2,t} \end{aligned} \tag{10.1} \end{equation}$$

• Using matrix notation bivariate VAR($1$) is

$$\begin{equation} \underbrace{\begin{bmatrix}y_t \\ x_t \end{bmatrix}}_{z_t}=\underbrace{\begin{bmatrix} \beta_{1,0} \\ \beta_{2,0} \end{bmatrix}}_{a_0}+ \underbrace{\begin{bmatrix} \beta_{1,1} & \beta_{1,2} \\ \beta_{2,1} & \beta_{2,2} \end{bmatrix}}_{A_1} \underbrace{\begin{bmatrix}y_{t-1} \\ x_{t-1} \end{bmatrix}}_{z_{t-1}}+\underbrace{\begin{bmatrix} u_{1,t} \\ u_{2,t} \end{bmatrix}}_{u_t} \tag{10.2} \end{equation}$$

• For simplicity, a matrix equation is written as

$$\begin{equation} z_t=a_0+A_1 z_{t-1}+u_t ~~~~~~~~~u_t\sim WN(0,~\Sigma) \tag{10.3} \end{equation}$$

• In matrix equation (10.3) $z_t$ is a vector of two time-series, $a_0$ is a two dimensional vector of constant terms, $A_1$ is matrix of coefficients $k \times k$ with respect to two dimensional vector of lagged time-series $z_{t-1}$, $u_t$ is two dimensional vector of error terms, while $\Sigma$ is error terms covariance matrix $k \times k$

• Note that every equation in the system (10.1) is a special case of ARDL($1,1$) without contemporaneous or instantaneous terms (non-lagged time-series on the RHS are omitted)

• To include contemporaneous terms, we would need to impose certain restrictions in the structure, such as using structural VAR (SVAR)

• Main advantage of reduced VAR($p$) model is OLS estimation for each equation separately

• Number of estimated parameters in reduced VAR($p$) model is $k+pk^2$

• Additionaly, $k(k+1)/2$ terms should be estimated in the covariance matrix $\Sigma$ (e.g. two variances on the diagonal and one covariance off-diagonal)

• It is assumed that error terms from both equations $u_{1,t}$ and $u_{2,t}$ are independent and follow white noise with zero mean and constant variances

• This indicates that $\Sigma$ is diagonal matrix (equal diagonal elements and off-diagonal elements are all zero)

• Matrices $A_1$ and $\Sigma$ are most important in this type of multivariate time-series analysis

• Matrix $\Sigma$ can be estimated using residuals from both equations

$$\begin{equation} \hat{\Sigma}=\frac{1}{T-1} \begin{bmatrix}\sum_{t=1}^{T}\hat{u}^2_{1,t}~~~~~~~~ \sum_{t=1}^{T} \hat{u}_{1,t} \hat{u}_{2,t} \\ \sum_{t=1}^{T} \hat{u}_{1,t} \hat{u}_{2,t}~~~~~~~~ \sum_{t=1}^{T}\hat{u}^2_{2,t} \end{bmatrix} \tag{10.4} \end{equation}$$

• Consistent and efficient OLS estimates require that all time-series are stationary

• Stationarity of every time-series can be checked by Augmented Dickey Fuller test - ADF

• Estimated parameters of VAR($1$) model are not interpreted in a traditional way

• Results of VAR($1$) are interpreted in context of:

1. Granger causality
2. Impulse response function
3. Cointegration