9.1 SPECIFICATION OF VECTOR AUTOREGRESSIONS

9.1 SPECIFICATION OF VECTOR AUTOREGRESSIONS

Vector autoregressions generalize the simple univariate autoregressions to the multivariate case where ${\gamma }_t$ becomes an (n × 1) vector and so extend the information set to $Z_t=\{y_{it},y_{it-1},...{\}}_{i=1}^n$ . As with univariate data, nearly all the work on forecasting with VARs assumes MSE loss and uses VARs to approximate the conditional mean of the outcome variable of interest. Many of the properties of VARs are simple multivariate generalizations of the results for the univariate models covered in chapter 7. For example, just as an autoregression summarizes the first and second moments of a univariate time series, we can think of the VAR as a summary of the first and second moments of the vector ${\gamma }_t$ . The Wold representation theorem also extends to the multivariate case and hence VARs can be used to approximate covariance stationary multivariate processes. Indeed, some papers extend VARs to include moving average terms so as to better approximate the underlying process by means of a parsimonious representation.1

The pth order VAR for an $(n\times 1)$ vector ${\gamma }_t$ takes the following form:

$$y_t=c+\sum _{i=1}^P{A}_i{y}_{t-i}+u_t,\text{\hspace{1em}(9.1)}$$

where $A_i$ is an $(n\times n)$ matrix of autoregressive coefficients for $i=1,\dots ,p$ , and $E\left[u_t{u}_t^{\prime }\right]=\Sigma$ . The $(n\times 1)$ vector of intercepts c can easily be extended to contain a set of deterministic variables such as seasonals or trends.

Model (9.1) is a system of equations with the same regressors appearing in each equation. The innovations of $u_t,u_{it}$ , can be correlated across individual equations with general covariance matrix -, but the lag length, $P,$ is typically chosen to be large enough so $u_t$ is serially uncorrelated for all variables in the $\mathrm{VAR}$

Equation (9.1) is convenient shorthand but buries a lot of notation that is needed to examine these models. Letting $a_{1ij}$ be the coefficient capturing the effect of the j th lag of ${\gamma }_{1t}$ on ${\gamma }_{it},$ we can write

$$y_{it}=c_i+\sum _{j=1}^P{a}_{1ij}{y}_{1t-j}+\cdot \cdot \cdot +\sum _{j=1}^P{a}_{nij}{y}_{nt-j}+u_{it},i=1,\dots ,n.\text{\hspace{1em}(9.2)}$$

Here the first $\mathbf{p}$ terms show how ${\gamma }_{it}$ is affected by $y_{1t-1},\dots ,y_{1t-p}$ , while the last $\mathbf{p}$ terms show how ${\gamma }_{it}$ is affected by lags of $y_n$

Defining $z_t=(1,y_{t-1}^{\prime },\dots ,y_{t-p}^{\prime }{)}^{\prime }$ and $\beta =(c,A_1,\dots ,A_p{)}^{\prime }$ , (9.1) can conveniently be written as a set of stacked equations for the $(T\times n)$ matrices $Y=$ $(y_1,\dots ,y_T{)}^{\prime }$ and $U=(u_1,\dots ,u_T{)}^{\prime }$ and a $(T\times (np+1))$ matrix $Z=(z_1,\dots ,z_T{)}^{\prime }$

$$\begin{array}{r}Y=Z\beta +U,\end{array}\text{\hspace{1em}(9.3)}$$

where $\beta$ is $(np+1)\times n.$

VARs initially became a popular forecasting tool because of their relative simplicity in terms of which choices need to be made by the forecaster. When estimating a VAR by classical methods, only two choices need to be made to construct forecasts: which variables to include (selection of $(y_1,\dots ,y_n))$ , and how many lags of the variables to include (selection of $p)$ . However, the risk of overparameterization of VARs is high given that the inclusion of just a few variables with a few lags requires a large number of parameters to be estimated, often from relatively short data spans. For example, the model in (9.1) has $n(np+1)$ mean parameters, $\beta ,$ plus another $n(n+1)/2$ covariance parameters, -. To deal with parameter estimation error, very quickly forecasters turned to methods such as Granger causality tests which can help remove regressors that do not appear to be useful and hence reduce the size of the model. Bayesian procedures which reduce parameter estimation error by shrinking the parameter estimates towards some target value are also popular. The move towards Bayesian methods has, however, complicated the formulation of the models as they require choices on priors and algorithms for estimation. A large and sophisticated literature makes use of improvements in computational algorithms to address such issues.

Example 9.1.1 (Modeling the term structure of interest rates). Affine term structure models make extensive use of vector autoregressive dynamics to derive the complete term structure of interest rates under no-arbitrage restrictions. To see how this works, let $P_t^n$ be the price at time t of a zero-coupon bond that expires in period $t+n,$ so that n is the bond’s maturity. To account for risk, let $M_{t+1}$ be the stochastic discount factor, or so-called pricing kernel, used to price nominal payoffs at time t + 1. Under no-arbitrage, the current price of an n-period bond that pays one dollar at time $t+n$ is

$$P_t^n=E_t\left[\prod _{j=1}^n{M}_{t+j}\right].\text{\hspace{1em}(9.4)}$$

To derive pricing restrictions from this equation, suppose that the pricing kernel takes the form

$$M_{t+1}=\exp \left(-r_t-\frac{1}{2}{\lambda }_t^{\prime }{\lambda }_t-{\lambda }_t^{\prime }{\epsilon }_{t+1}\right),{\epsilon }_{t+1}\sim i.i.d.\mathrm{N}(0,I),\text{\hspace{1em}(9.5)}$$

where ${\lambda }_t={\lambda }_0+{\lambda }_1^{\prime }{X}_t$ and $r_t$ is the short rate. Thus, the coefficients, ${\lambda }_t,$ are assumed to be affine functions of a vector of state variables, $X_t.$ . Assuming that the short interest rate, ${\mathbf{r}}_t,$ is also an affine function of $X_t,$ i.e., $r_t={\delta }_0+{\delta }_1^{\prime }{X}_t,$ , the entire term structure will be driven by the vector of state variables $X_t$ . Suppose these follow a VAR(1) process,

$$X_{t+1}=\mu +A{X}_t+\Sigma {\epsilon }_{t+1},\text{\hspace{1em}(9.6)}$$

and so are affected by the same shocks as the pricing kernel in (9.5). Conjecturing a solution of the form

$$P_t^n=\exp (a_n+b_n^{\prime }{X}_t),$$

it can easily be verified from (9.4)–(9.6) that $a_n$ and $b_n$ must satisfy the recursive relations

$$\begin{array}{rl}& a_{n+1}=-{\delta }_0+a_n+b_n^{\prime }(\mu -\Sigma {\lambda }_0)+\frac{1}{2}{b}_n^{\prime }\Sigma {\Sigma }^{\prime }{b}_n,\\ & b_{n+1}=(A-\Sigma {\lambda }_1{)}^{\prime }{b}_n-{\delta }_1,\end{array}\text{\hspace{1em}(9.7)}$$

with $a_1=-{\delta }_0,b_1=-{\delta }_1$ as initial conditions. Given a set of estimates for the VAR coefficients μ, A, - in (9.6) and for the remaining parameters ${\delta }_0,{\lambda }_0,{\lambda }_1,{\delta }_1$ , we can compute the entire term structure of zero-coupon bond prices from (9.7); see Wright (2011) for further discussion of how this model can be estimated.