Vector autoregression

Vector autoregression (VAR) is an econometric model used to capture the evolution and the interdependencies between multiple time series, generalizing the univariate AR models. All the variables in a VAR are treated symmetrically by including for each variable an equation explaining its evolution based on its own lags and the lags of all the other variables in the model. Based on this feature, Christopher Sims advocates the use of VAR models as a theory-free method to estimate economic relationships, thus being an alternative to the "incredible identification restrictions" in structural models.

Definition
A VAR model describes the evolution of a set of n variables (called endogenous variables) measured over the same sample period (t = 1, ..., T) as a linear function of only their past evolution. The variables are collected in a n x 1 vector yt, which has as the ith element yi,t the time t observation of variable yi. For example, if the ith variable is GDP, then yi,t is the value of GDP at t.

A (reduced) p-th order VAR, denoted VAR(p), is


 * $$y_{t}=c + A_{1}y_{t-1} + A_{2}y_{t-2} + \cdots + A_{p}y_{t-p} + e_{t},$$

where c is a n x 1 vector of constants (intercept), Ai is a n x n matrix (for every i = 1, ..., p) and et is a n x 1 vector of error terms satisfying


 * 1) $$\mathrm{E}(e_{t}) = 0\,$$ — every error term has mean zero;
 * 2) $$\mathrm{E}(e_{t}e_{t}') = \Omega\,$$ — the contemporaneous covariance matrix of error terms is Ω (a n x n positive definite matrix);
 * 3) $$\mathrm{E}(e_{t}e_{t-k}') = 0\,$$ for any non-zero k — there is no correlation across time; in particular, no serial correlation in individual error terms.

The k-periods back observation yt-k is called the k-th lag of y. Thus, a p-th order VAR is also called a VAR with p lags.

Example
A VAR(1) in two variables can be written in matrix form (more compact notation) as


 * $$\begin{bmatrix}y_{1,t} \\ y_{2,t}\end{bmatrix} = \begin{bmatrix}c_{1} \\ c_{2}\end{bmatrix} + \begin{bmatrix}A_{1,1}&A_{1,2} \\ A_{2,1}&A_{2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\end{bmatrix} + \begin{bmatrix}e_{1,t} \\ e_{2,t}\end{bmatrix},$$

or, equivalently, as the following system of two equations


 * $$y_{1,t} = c_{1} + A_{1,1}y_{1,t-1} + A_{1,2}y_{2,t-1} + e_{1,t}\,$$
 * $$y_{2,t} = c_{2} + A_{2,1}y_{1,t-1} + A_{2,2}y_{2,t-1} + e_{2,t}.\,$$

Note that there is one equation for each variable in the model. Also note that the current (time t) observation of each variable depends on its own lags as well as on the lags of each other variable in the VAR.

Writing VAR(p) as VAR(1)
A VAR with p lags can always be equivalently rewritten as a VAR with only one lag by appropriately redefining the dependent variable. The transformation amounts to merely stacking the lags of the VAR(p) variable in the new VAR(1) dependent variable and appending identities to complete the number of equations.

For example, the VAR(2) model


 * $$y_{t}=c + A_{1}y_{t-1} + A_{2}y_{t-2} + e_{t}$$

can be recast as the VAR(1) model


 * $$\begin{bmatrix}y_{t} \\ y_{t-1}\end{bmatrix} = \begin{bmatrix}c \\ 0\end{bmatrix} + \begin{bmatrix}A_{1}&A_{2} \\ I&0\end{bmatrix}\begin{bmatrix}y_{t-1} \\ y_{t-2}\end{bmatrix} + \begin{bmatrix}e_{t} \\ 0\end{bmatrix},$$

where I is the identity matrix.

The equivalent VAR(1) form is more convenient for analytical derivations and allows more compact statements.

Structural VAR
A structural VAR with p lags is


 * $$B_{0}y_{t}=c_{0} + B_{1}y_{t-1} + B_{2}y_{t-2} + \cdots + B_{p}y_{t-p} + \epsilon_{t},$$

where c0 is a n x 1 vector of constants, Bi is a n x n matrix (for every i = 0, ..., p) and εt is a n x 1 vector of error terms. The main diagonal terms of the B0 matrix (the coefficients on the ith variable in the ith equation) are scaled to 1.

The error terms εt (structural shocks) satisfy the conditions (1) - (3) in the definition above, with the particularity that all the elements off the main diagonal of the covariance matrix $$\mathrm{E}(\epsilon_{t}\epsilon_{t}') = \Sigma$$ are zero. That is, the structural shocks are uncorrelated.

For example, a two variable structural VAR(1) is:


 * $$\begin{bmatrix}1&B_{0;1,2} \\ B_{0;2,1}&1\end{bmatrix}\begin{bmatrix}y_{1,t} \\ y_{2,t}\end{bmatrix} = \begin{bmatrix}c_{0;1} \\ c_{0;2}\end{bmatrix} + \begin{bmatrix}B_{1;1,1}&B_{1;1,2} \\ B_{1;2,1}&B_{1;2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\end{bmatrix} + \begin{bmatrix}\epsilon_{1,t} \\ \epsilon_{2,t}\end{bmatrix},$$

where


 * $$\Sigma = \mathrm{E}(\epsilon_{t}\epsilon_{t}') = \begin{bmatrix}\sigma_{1}&0 \\ 0&\sigma_{2}\end{bmatrix};$$

that is, the variances of the structural shocks are denoted $$\mathrm{var}(\epsilon_{i}) = \sigma_{i}^2$$ (i = 1, 2) and the covariance is $$\mathrm{cov}(\epsilon_{1},\epsilon_{2}) = 0$$.

Writing the first equation explicitly and passing y2,t to the right hand side one obtains


 * $$y_{1,t} = c_{0;1} - B_{0;1,2}y_{2,t} + B_{1;1,1}y_{1,t-1} + B_{1;1,2}y_{2,t-2} + \epsilon_{1,t}\,$$

Note that y2,t can have a contemporaneous effect on y1,t if B0;1,2 is not zero. This is different from the case when B0 is the identity matrix (all off-diagonal elements are zero - the case in the initial definition), when y2,t can impact directly y1,t+1 and subsequent future values, but not y1,t.

Because of the parameter identification problem, ordinary least squares estimation of the structural VAR would yield inconsistent parameter estimates. This problem can be overcome by rewriting the VAR in reduced form.

From an economic point of view it is considered that, if the joint dynamics of a set of variables can be represented by a VAR model, then the structural form is a depiction of the underlying, "structural", economic relationships. Two features of the structural form make it the preferred candidate to represent the underlying relations:


 * 1) Error terms are not correlated. The structural, economic shocks which drive the dynamics of the economic variables are assumed to be independent, which implies zero correlation between error terms as a desired property. For instance, there is no reason why an oil price shock (as an example of a supply shock) should be related to a shift in consumers' preferences towards a style of clothing (as an example of a demand shock).


 * 1) Variables can have a contemporaneous impact on other variables. This is a desirable feature especially when using low frequency data. For example, an indirect tax rate increase would not affect tax revenues the day the decision is announced, but one could find an effect in that quarter's data.

Reduced VAR
By premultiplying the structural VAR with the inverse of B0


 * $$y_{t} = B_{0}^{-1}c_{0} + B_{0}^{-1}B_{1}y_{t-1} + B_{0}^{-1}B_{2}y_{t-2} + \cdots + B_{0}^{-1}B_{p}y_{t-p} + B_{0}^{-1}\epsilon_{t},$$

and denoting


 * $$B_{0}^{-1}c_{0} = c,$$  $$B_{0}^{-1}B_{i} = A_{i}$$ for any $$i = 1, \cdots, p\,$$ and $$B_{0}^{-1}\epsilon_{t} = e_{t}$$

one obtains the p-th order reduced VAR


 * $$y_{t}=c + A_{1}y_{t-1} + A_{2}y_{t-2} + \cdots + A_{p}y_{t-p} + e_{t}$$

Note that in the reduced form all right hand side variables are predetermined at time t. As there are no time t endogenous variables on the right hand side, no variable has a direct contemporaneous effect on other variables in the model.

However, the error terms in the reduced VAR are composites of the structural shocks et = B0-1εt. Thus, the occurrence of one structural shock εi,t can potentially lead to the occurrence of shocks in all error terms ej,t, thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR


 * $$\Omega = \mathrm{E}(e_{t}e_{t}') = \mathrm{E}(B_{0}^{-1}\epsilon_{t}\epsilon_{t}'(B_{0}^{-1})') = B_{0}^{-1}\Sigma(B_{0}^{-1})'\,$$

can have non-zero off-diagonal elements, thus allowing non-zero correlation between error terms.

Estimation
Ordinary least squares estimation of each equation in the reduced VAR is both consistent and asymptotically efficient.