1 VARs and IRFs: the basics
Often, impulse response functions (IRFs) are generated in the context of vectorial autoregressive (VAR) models. This section presents these models and show how they can be used to compute IRFs.
1.1 Definition of VARs (and SVARMA) models
Definition 1.1 ((S)VAR model) Let \(y_{t}\) denote a \(n \times1\) vector of (endogenous) random variables. Process \(y_{t}\) follows a \(p^{th}\)-order (S)VAR if, for all \(t\), we have \[\begin{eqnarray} \begin{array}{rllll} VAR:& y_t &=& c + \Phi_1 y_{t-1} + \dots + \Phi_p y_{t-p} + \varepsilon_t,\\ SVAR:& y_t &=& c + \Phi_1 y_{t-1} + \dots + \Phi_p y_{t-p} + B \eta_t, \end{array}\tag{1.1} \end{eqnarray}\] with \(\varepsilon_t = B\eta_t\), where \(\{\eta_{t}\}\) is a white noise sequence whose components are mutually and serially independent.
The first line of Eq. (1.1) corresponds to the reduced-form of the VAR model (structural form for the second line). While the structural shocks (the components of \(\eta_t\)) are mutually uncorrelated, this is not the case of the innovations, that are the components of \(\varepsilon_t\). However, in boths cases, vectors \(\eta_t\) and \(\varepsilon_t\) are serially correlated (through time).
As is the case for univariate models, VARs can be extended with MA terms in \(\eta_t\), giving rise to VARMA models:
Definition 1.2 ((S)VARMA model) Let \(y_{t}\) denote a \(n \times1\) vector of random variables. Process \(y_{t}\) follows a VARMA model of order (p,q) if, for all \(t\), we have \[\begin{eqnarray} \begin{array}{rllll} VARMA:& y_t &=& c + \Phi_1 y_{t-1} + \dots + \Phi_p y_{t-p} + \\ &&&\varepsilon_t + \Theta_1\varepsilon_{t-1} + \dots + \Theta_q \varepsilon_{t-q},\\ SVARMA:& y_t &=& c + \Phi_1 y_{t-1} + \dots + \Phi_p y_{t-p} + \\ &&& B_0 \eta_t+ B_1 \eta_{t-1} + \dots + B_q \eta_{t-q}, \end{array}\tag{1.2} \end{eqnarray}\] with \(\varepsilon_t = B_0\eta_t\), and \(B_j = \Theta_j B_0\), for \(j \ge 0\) (with \(\Theta_0=Id\)), where \(\{\eta_{t}\}\) is a white noise sequence whose components are are mutually and serially independent.
1.2 IRFs in SVARMA
One of the main objectives of macro-econometrics is to derive IRFs, that represent the dynamic effects of structural shocks (components of \(\eta_t\)) though the system of variables \(y_t\).
Formally, an IRF is a difference in conditional expectations: \[\begin{equation} \boxed{\Psi_{i,j,h} = \mathbb{E}(y_{i,t+h}|\eta_{j,t}=1) - \mathbb{E}(y_{i,t+h})}\tag{1.3} \end{equation}\] (effect on \(y_{i,t+h}\) of a one-unit shock on \(\eta_{j,t}\)).
IRFs closely relate to the Wold decomposition of \(y_t\). Indeed, if the dynamics of process \(y_t\) can be described as a VARMA model, and if \(y_t\) is covariance stationary (see Def. 11.1), then \(y_t\) admits the following infinite MA representation (or Wold decomposition): \[\begin{equation} y_t = \mu + \sum_{h=0}^\infty \Psi_{h} \eta_{t-h}.\tag{1.4} \end{equation}\] With these notations, we get \(\mathbb{E}(y_{i,t+h}|\eta_{j,t}=1) = \mu_i + \Psi_{i,j,h}\), where \(\Psi_{i,j,h}\) is the component \((i,j)\) of matrix \(\Psi_h\) and \(\mu_i\) is the \(i^{th}\) entry of vector \(\mu\). Since we also have \(\mathbb{E}(y_{i,t+h})=\mu_i\), we obtain Eq. (1.3).
Hence, estimating IRFs amounts to estimating the \(\Psi_{h}\)’s. In general, there exist three main approaches for that:
- Calibrate and solve a (purely structural) Dynamic Stochastic General Equilibrium (DSGE) model at the first order (linearization). The solution takes the form of Eq. (1.4).
- Directly estimate the \(\Psi_{h}\) based on projection approaches (see Section 8).
- Approximate the infinite MA representation by estimating a parsimonious type of model, e.g. VAR(MA) models (see Section 1.4). Once a (Structural) VARMA representation is obtained, Eq. (1.4) is easily deduced using the following proposition:
Proposition 1.1 (IRF of an ARMA(p,q) process) If \(y_t\) follows the VARMA model described in Def. 1.2, then the matrices \(\Psi_h\) appearing in Eq. (1.4) can be computed recursively as follows:
- Set \(\Psi_{-1}=\dots=\Psi_{-p}=0\).
- For \(h \ge 0\), (recursively) apply: \[ \Psi_h = \Phi_1 \Psi_{h-1} + \dots + \Phi_p \Psi_{h-p} + \Theta_h B_0, \] with \(\Theta_0 = Id\) and \(\Theta_h = 0\) for \(h>q\).
Proof. This is obtained by applying the operator \(\frac{\partial}{\partial \eta_{t}}\) on both sides of Eq. (1.2).
Typically, consider the VAR(2) case. The first steps of the algorithm mentioned in the last bullet point are as follows: \[\begin{eqnarray*} y_t &=& \Phi_1 {\color{blue}y_{t-1}} + \Phi_2 y_{t-2} + B \eta_t \\ &=& \Phi_1 \color{blue}{(\Phi_1 y_{t-2} + \Phi_2 y_{t-3} + B \eta_{t-1})} + \Phi_2 y_{t-2} + B \eta_t \\ &=& B \eta_t + \Phi_1 B \eta_{t-1} + (\Phi_2 + \Phi_1^2) \color{red}{y_{t-2}} + \Phi_1\Phi_2 y_{t-3} \\ &=& B \eta_t + \Phi_1 B \eta_{t-1} + (\Phi_2 + \Phi_1^2) \color{red}{(\Phi_1 y_{t-3} + \Phi_2 y_{t-4} + B \eta_{t-2})} + \Phi_1\Phi_2 y_{t-3} \\ &=& \underbrace{B}_{=\Psi_0} \eta_t + \underbrace{\Phi_1 B}_{=\Psi_1} \eta_{t-1} + \underbrace{(\Phi_2 + \Phi_1^2)B}_{=\Psi_2} \eta_{t-2} + f(y_{t-3},y_{t-4}). \end{eqnarray*}\]
In particular, we have \(B = \Psi_0\). Matrix \(B\) indeed captures the contemporaneous impact of \(\eta_t\) on \(y_t\). That is why matrix \(B\) is sometimes called impulse matrix.
Example 1.1 (IRFs of an SVARMA model) Consider the following VARMA(1,1) model: \[\begin{eqnarray} \quad y_t &=& \underbrace{\left[\begin{array}{cc} 0.5 & 0.3 \\ -0.4 & 0.7 \end{array}\right]}_{\Phi_1} y_{t-1} + \underbrace{\left[\begin{array}{cc} 1 & 2 \\ -1 & 1 \end{array}\right]}_{B}\eta_t \\ && - \underbrace{\left[\begin{array}{cc} -0.4 & 0 \\ 1 & 0.5 \end{array}\right]}_{\Theta_1} \underbrace{\left[\begin{array}{cc} 1 & 2 \\ -1 & 1 \end{array}\right]}_{B}\eta_{t-1}.\tag{1.5} \end{eqnarray}\]
We can use function simul.VARMA of package IdSS to produce IRFs (using indic.IRF=1 in the list of arguments):
library(IdSS)
distri <- list(type=c("gaussian","gaussian"),df=c(4,4))
n <- length(distri$type) # dimension of y_t
nb.sim <- 30
eps <- simul.distri(distri,nb.sim)
Phi <- array(NaN,c(n,n,1))
Phi[,,1] <- matrix(c(.5,-.4,.3,.7),2,2)
p <- dim(Phi)[3]
Theta <- array(NaN,c(n,n,1))
Theta[,,1] <- matrix(c(-.4,1,0,.5),2,2)
q <- dim(Theta)[3]
Mu <- rep(0,n)
C <- matrix(c(1,-1,2,1),2,2)
Model <- list(
Mu = Mu,Phi = Phi,Theta = Theta,C = C,distri = distri)
Y0 <- rep(0,n)
eta0 <- c(1,0)
res.sim.1 <- simul.VARMA(Model,nb.sim,Y0,eta0,indic.IRF=1)
eta0 <- c(0,1)
res.sim.2 <- simul.VARMA(Model,nb.sim,Y0,eta0,indic.IRF=1)
par(plt=c(.15,.95,.25,.8))
par(mfrow=c(2,2))
for(i in 1:2){
if(i == 1){res.sim <- res.sim.1
}else{res.sim <- res.sim.2}
for(j in 1:2){
plot(res.sim$Y[j,],las=1,
type="l",lwd=3,xlab="",ylab="",
main=paste("Resp. of y",j,
" to a 1-unit increase in eta",i,sep=""))
abline(h=0,col="grey",lty=3)
}}
Figure 1.1: Impulse response functions (SVARMA(1,1) specified above).
1.3 Covariance-stationary VARMA models
Let’s come back to the infinite MA case (Eq. (1.4)): \[ y_t = \mu + \sum_{h=0}^\infty \Psi_{h} \eta_{t-h}. \] For \(y_t\) to be covariance-stationary (and ergodic for the mean), it has to be the case that \[\begin{equation} \sum_{i=0}^\infty \|\Psi_i\| < \infty,\tag{1.6} \end{equation}\] where \(\|A\|\) denotes a norm of the matrix \(A\) (e.g. \(\|A\|=\sqrt{tr(AA')}\)). This notably implies that if \(y_t\) is stationary (and ergodic for the mean), then \(\|\Psi_h\|\rightarrow 0\) when \(h\) gets large.
What should be satisfied by \(\Phi_k\)’s and \(\Theta_k\)’s for a VARMA-based process (Eq. (1.2)) to be stationary? The conditions will be similar to that we have in the univariate case. Let us introduce the following notations: \[\begin{eqnarray} y_t &=& c + \underbrace{\Phi_1 y_{t-1} + \dots +\Phi_p y_{t-p}}_{\color{blue}{\mbox{AR component}}} + \tag{1.7}\\ &&\underbrace{B \eta_t - \Theta_1 B \eta_{t-1} - \dots - \Theta_q B \eta_{t-q}}_{\color{red}{\mbox{MA component}}} \nonumber\\ &\Leftrightarrow& \underbrace{ \color{blue}{(I - \Phi_1 L - \dots - \Phi_p L^p)}}_{= \color{blue}{\Phi(L)}}y_t = c + \underbrace{ \color{red}{(I + \Theta_1 L + \ldots + \Theta_q L^q)}}_{=\color{red}{\Theta(L)}} B \eta_{t}. \nonumber \end{eqnarray}\]
Process \(y_t\) is stationary iff the roots of \(\det(\Phi(z))=0\) are strictly outside the unit circle or, equivalently, iff the eigenvalues of \[\begin{equation} \Phi = \left[\begin{array}{cccc} \Phi_{1} & \Phi_{2} & \cdots & \Phi_{p}\\ I & 0 & \cdots & 0\\ 0 & \ddots & 0 & 0\\ 0 & 0 & I & 0\end{array}\right]\tag{1.8} \end{equation}\] lie strictly within the unit circle. Hence, as is the case for univariate processes, the covariance-stationarity of a VARMA model depends only on the specification of its AR part.
Let’s derive the first two unconditional moments of a (covariance-stationary) VARMA process.
Eq. (1.7) gives \(\mathbb{E}(\Phi(L)y_t)=c\), therefore \(\Phi(1)\mathbb{E}(y_t)=c\), or \[ \mathbb{E}(y_t) = (I - \Phi_1 - \dots - \Phi_p)^{-1}c. \] The autocovariances of \(y_t\) can be deduced from the infinite MA representation (Eq. (1.4)). We have: \[ \gamma_j \equiv \mathbb{C}ov(y_t,y_{t-j}) = \sum_{i=j}^\infty \Psi_i \Psi_{i-j}'. \] (This infinite sum exists as soon as Eq. (1.6) is satisfied.)
Conditional means and autocovariances can also be deduced from Eq. (1.4). For \(0 \le h\) and \(0 \le h_1 \le h_2\): \[\begin{eqnarray*} \mathbb{E}_t(y_{t+h}) &=& \mu + \sum_{k=0}^\infty \Psi_{k+h} \eta_{t-k} \\ \mathbb{C}ov_t(y_{t+1+h_1},y_{t+1+h_2}) &=& \sum_{k=0}^{h_1} \Psi_{k}\Psi_{k+h_2-h_1}'. \end{eqnarray*}\]
The previous formula implies in particular that the forecasting error \(y_{t+h} - \mathbb{E}_t(y_{t+h})\) has a variance equal to: \[ \mathbb{V}ar_t(y_{t+1+h}) = \sum_{k=0}^{h} \Psi_{k}\Psi_{k}'. \] Because the \(\eta_t\) are mutually and serially independent (and therefore uncorrelated), we have: \[ \mathbb{V}ar(\Psi_k \eta_{t-k}) = \mathbb{V}ar\left(\sum_{i=1}^n \psi_{k,i} \eta_{i,t-k}\right) = \sum_{i=1}^n \psi_{k,i}\psi_{k,i}', \] where \(\psi_{k,i}\) denotes the \(i^{th}\) column of \(\Psi_k\). This suggests the following decomposition of the variance of the forecast error (called variance decomposition): \[ \mathbb{V}ar_t(y_{t+1+h}) = \sum_{i=1}^n \underbrace{\sum_{k=0}^{h}\psi_{k,i}\psi_{k,i}'.}_{\mbox{Contribution of $\eta_{i,t}$}} \]
Let us now turn to the estimation of VAR models. Note that if there is an MA component (i.e., if we consider a VARMA model), then OLS regressions yield biased estimates (even for asymptotically large samples). Assume for instance that \(y_t\) follows a VARMA(1,1) model: \[ y_{i,t} = \phi_i y_{t-1} + \varepsilon_{i,t}, \] where \(\phi_i\) is the \(i^{th}\) row of \(\Phi_1\), and where \(\varepsilon_{i,t}\) is a linear combination of \(\eta_t\) and \(\eta_{t-1}\). Since \(y_{t-1}\) (the regressor) is correlated to \(\eta_{t-1}\), it is also correlated to \(\varepsilon_{i,t}\). The OLS regression of \(y_{i,t}\) on \(y_{t-1}\) yields a biased estimator of \(\phi_i\) (see Figure 1.2). Hence, SVARMA models cannot be consistently estimated by simple OLS regressions (contrary to VAR models, as we will see in the next section); instrumental-variable approaches can be employed to estimate SVARMA models (using past values of \(y_t\) as instruments, see, e.g., Gouriéroux, Monfort, and Renne (2020)).
N <- 1000 # number of replications
T <- 100 # sample length
phi <- .8 # autoregressive parameter
sigma <- 1
par(mfrow=c(1,2))
for(theta in c(0,-0.4)){
all.y <- matrix(0,1,N)
y <- all.y
eta_1 <- rnorm(N)
for(t in 1:(T+1)){
eta <- rnorm(N)
y <- phi * y + sigma * eta + theta * sigma * eta_1
all.y <- rbind(all.y,y)
eta_1 <- eta
}
all.y_1 <- all.y[1:T,]
all.y <- all.y[2:(T+1),]
XX_1 <- 1/apply(all.y_1 * all.y_1,2,sum)
XY <- apply(all.y_1 * all.y,2,sum)
phi.est.OLS <- XX_1 * XY
plot(density(phi.est.OLS),xlab="OLS estimate of phi",ylab="",
main=paste("theta = ",theta,sep=""))
abline(v=phi,col="red",lwd=2)}
Figure 1.2: Illustration of the bias obtained when estimating the auto-regressive parameters of an ARMA process by (standard) OLS.
1.4 VAR estimation
This section discusses the estimation of VAR models. Eq. (1.1) can be written: \[ y_{t}=c+\Phi(L)y_{t-1}+\varepsilon_{t}, \] with \(\Phi(L) = \Phi_1 + \Phi_2 L + \dots + \Phi_p L^{p-1}\).
Consequently: \[ y_{t}\mid y_{t-1},y_{t-2},\ldots,y_{-p+1}\sim \mathcal{N}(c+\Phi_{1}y_{t-1}+\ldots\Phi_{p}y_{t-p},\Omega). \]
Using Hamilton (1994)’s notations, denote with \(\Pi\) the matrix \(\left[\begin{array}{ccccc} c & \Phi_{1} & \Phi_{2} & \ldots & \Phi_{p}\end{array}\right]'\) and with \(x_{t}\) the vector \(\left[\begin{array}{ccccc} 1 & y'_{t-1} & y'_{t-2} & \ldots & y'_{t-p}\end{array}\right]'\), we have: \[\begin{equation} y_{t}= \Pi'x_{t} + \varepsilon_{t}. \tag{1.9} \end{equation}\] The previous representation is convenient to discuss the estimation of the VAR model, as parameters are gathered in two matrices only: \(\Pi\) and \(\Omega\).
Let us start with the case where the shocks are Gaussian.
Proposition 1.2 (MLE of a Gaussian VAR) If \(y_t\) follows a VAR(p) (see Definition 1.1), and if \(\varepsilon_t \sim \,i.i.d.\,\mathcal{N}(0,\Omega)\), then the ML estimate of \(\Pi\), denoted by \(\hat{\Pi}\) (see Eq. (1.9)), is given by \[\begin{equation} \hat{\Pi}=\left[\sum_{t=1}^{T}x_{t}x'_{t}\right]^{-1}\left[\sum_{t=1}^{T}y_{t}'x_{t}\right]= (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y},\tag{1.10} \end{equation}\] where \(\mathbf{X}\) is the \(T \times (1+np)\) matrix whose \(t^{th}\) row is \(x_t\) and where \(\mathbf{y}\) is the \(T \times n\) matrix whose \(t^{th}\) row is \(y_{t}'\).
That is, the \(i^{th}\) column of \(\hat{\Pi}\) (\(b_i\), say) is the OLS estimate of \(\beta_i\), where: \[\begin{equation} y_{i,t} = \beta_i'x_t + \varepsilon_{i,t},\tag{1.11} \end{equation}\] (i.e., \(\beta_i' = [c_i,\phi_{i,1}',\dots,\phi_{i,p}']'\)).
The ML estimate of \(\Omega\), denoted by \(\hat{\Omega}\), coincides with the sample covariance matrix of the \(n\) series of the OLS residuals in Eq. (1.11), i.e.: \[\begin{equation} \hat{\Omega} = \frac{1}{T} \sum_{i=1}^T \hat{\varepsilon}_t\hat{\varepsilon}_t',\quad\mbox{with } \hat{\varepsilon}_t= y_t - \hat{\Pi}'x_t. \end{equation}\]
The asymptotic distributions of these estimators are the ones resulting from standard OLS formula.
Proof. See Appendix 11.2.
As stated by Proposition 1.3, when the shocks are not Gaussian, then the OLS regressions still provide consistent estimates of the model parameters. However, since \(x_t\) correlates to \(\varepsilon_s\) for \(s<t\), the OLS estimator \(\mathbf{b}_i\) of \(\boldsymbol\beta_i\) is biased in small sample. (That is also the case for the ML estimator.) Indeed, denoting by \(\boldsymbol\varepsilon_i\) the \(T \times 1\) vector of \(\varepsilon_{i,t}\)’s, and using the notations of \(b_i\) and \(\beta_i\) introduced in Proposition 1.2, we have: \[\begin{equation} \mathbf{b}_i = \beta_i + (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\varepsilon_i.\tag{1.12} \end{equation}\] We have non-zero correlation between \(x_t\) and \(\varepsilon_{i,s}\) for \(s<t\) and, therefore, \(\mathbb{E}[(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\boldsymbol\varepsilon_i] \ne 0\).
However, when \(y_t\) is covariance stationary, then \(\frac{1}{n}\mathbf{X}'\mathbf{X}\) converges to a positive definite matrix \(\mathbf{Q}\), and \(\frac{1}{n}X'\boldsymbol\varepsilon_i\) converges to 0. Hence \(\mathbf{b}_i \overset{p}{\rightarrow} \beta_i\). More precisely:
Proposition 1.3 (Asymptotic distribution of the OLS estimate of VAR coefficients (for one variable)) If \(y_t\) follows a VAR model, as defined in Definition 1.1, we have: \[ \sqrt{T}(\mathbf{b}_i-\beta_i) = \underbrace{\left[\frac{1}{T}\sum_{t=p}^T x_t x_t' \right]^{-1}}_{\overset{p}{\rightarrow} \mathbf{Q}^{-1}} \underbrace{\sqrt{T} \left[\frac{1}{T}\sum_{t=1}^T x_t\varepsilon_{i,t} \right]}_{\overset{d}{\rightarrow} \mathcal{N}(0,\sigma_i^2\mathbf{Q})}, \] where \(\sigma_i = \mathbb{V}ar(\varepsilon_{i,t})\) and where \(\mathbf{Q} = \mbox{plim }\frac{1}{T}\sum_{t=p}^T x_t x_t'\) is given by: \[\begin{equation} \mathbf{Q} = \left[ \begin{array}{ccccc} 1 & \mu' &\mu' & \dots & \mu' \\ \mu & \gamma_0 + \mu\mu' & \gamma_1 + \mu\mu' & \dots & \gamma_{p-1} + \mu\mu'\\ \mu & \gamma_1 + \mu\mu' & \gamma_0 + \mu\mu' & \dots & \gamma_{p-2} + \mu\mu'\\ \vdots &\vdots &\vdots &\dots &\vdots \\ \mu & \gamma_{p-1} + \mu\mu' & \gamma_{p-2} + \mu\mu' & \dots & \gamma_{0} + \mu\mu' \end{array} \right],\tag{1.13} \end{equation}\] where \(\gamma_j\) is the unconditional autocovariance matrix of order \(j\) of \(y_t\), i.e., \(\gamma_i = \mathbb{C}ov(y_{t},y_{t-j})\).
Proof. See Appendix 11.2.
The following proposition extends the previous proposition and includes covariances between different \(\beta_i\)’s as well as the asymptotic distribution of the ML estimates of \(\Omega\).
Proposition 1.4 (Asymptotic distribution of the OLS estimates) If \(y_t\) follows a VAR model, as defined in Definition 1.1, we have: \[\begin{equation} \sqrt{T}\left[ \begin{array}{c} vec(\hat\Pi - \Pi)\\ vec(\hat\Omega - \Omega) \end{array} \right] \sim \mathcal{N}\left(0, \left[ \begin{array}{cc} \Omega \otimes \mathbf{Q}^{-1} & 0\\ 0 & \Sigma_{22} \end{array} \right]\right),\tag{1.14} \end{equation}\] where the component of \(\Sigma_{22}\) corresponding to the covariance between \(\hat\sigma_{i,j}\) and \(\hat\sigma_{k,l}\) (for \(i,j,l,m \in \{1,\dots,n\}^4\)) is equal to \(\sigma_{i,l}\sigma_{j,m}+\sigma_{i,m}\sigma_{j,l}\).
Proof. See Hamilton (1994), Appendix of Chapter 11.
In practice, to use the previous proposition (for instance to implement Monte-Carlo simulations, see Section 3.1), \(\Omega\) is replaced with \(\hat{\Omega}\), \(\mathbf{Q}\) is replaced with \(\hat{\mathbf{Q}} = \frac{1}{T}\sum_{t=p}^T x_t x_t'\) and \(\Sigma\) with the matrix whose components are of the form \(\hat\sigma_{i,l}\hat\sigma_{j,m}+\hat\sigma_{i,m}\hat\sigma_{j,l}\), where the \(\hat\sigma_{i,l}\)’s are the components of \(\hat\Omega\).
The simplicity of the VAR framework and the tractability of its MLE open the way to convenient econometric testing. Let’s illustrate this with the likelihood ratio test (see Def. 11.2). The maximum value achieved by the MLE is \[ \log\mathcal{L}(Y_{T};\hat{\Pi},\hat{\Omega}) = -\frac{Tn}{2}\log(2\pi)+\frac{T}{2}\log\left|\hat{\Omega}^{-1}\right| -\frac{1}{2}\sum_{t=1}^{T}\left[\hat{\varepsilon}_{t}'\hat{\Omega}^{-1}\hat{\varepsilon}_{t}\right]. \] The last term is: \[\begin{eqnarray*} \sum_{t=1}^{T}\hat{\varepsilon}_{t}'\hat{\Omega}^{-1}\hat{\varepsilon}_{t} &=& \mbox{Tr}\left[\sum_{t=1}^{T}\hat{\varepsilon}_{t}'\hat{\Omega}^{-1}\hat{\varepsilon}_{t}\right] = \mbox{Tr}\left[\sum_{t=1}^{T}\hat{\Omega}^{-1}\hat{\varepsilon}_{t}\hat{\varepsilon}_{t}'\right]\\ &=&\mbox{Tr}\left[\hat{\Omega}^{-1}\sum_{t=1}^{T}\hat{\varepsilon}_{t}\hat{\varepsilon}_{t}'\right] = \mbox{Tr}\left[\hat{\Omega}^{-1}\left(T\hat{\Omega}\right)\right]=Tn. \end{eqnarray*}\] Therefore, the optimized log-likelihood is simply obtained by: \[\begin{equation} \log\mathcal{L}(Y_{T};\hat{\Pi},\hat{\Omega})=-(Tn/2)\log(2\pi)+(T/2)\log\left|\hat{\Omega}^{-1}\right|-Tn/2.\tag{1.15} \end{equation}\]
Assume that we want to test the null hypothesis that a set of variables follows a VAR(\(p_{0}\)) against the alternative specification of \(p_{1}\) (\(>p_{0}\)). Let us denote by \(\hat{L}_{0}\) and \(\hat{L}_{1}\) the maximum log-likelihoods obtained with \(p_{0}\) and \(p_{1}\) lags, respectively. Under the null hypothesis (\(H_0\): \(p=p_0\)), we have: \[\begin{eqnarray*} 2\left(\hat{L}_{1}-\hat{L}_{0}\right)&=&T\left(\log\left|\hat{\Omega}_{1}^{-1}\right|-\log\left|\hat{\Omega}_{0}^{-1}\right|\right) \sim \chi^2(n^{2}(p_{1}-p_{0})). \end{eqnarray*}\]
What precedes can be used to help determine the appropriate number of lags to use in the specification. In a VAR, using too many lags consumes numerous degrees of freedom: with \(p\) lags, each of the \(n\) equations in the VAR contains \(n\times p\) coefficients plus the intercept term. Adding lags improve in-sample fit, but is likely to result in over-parameterization and affect the out-of-sample prediction performance.
To select appropriate lag length, selection criteria are often used. In the context of VAR models, using Eq. (1.15) (Gaussian case), we have for instance: \[\begin{eqnarray*} AIC & = & cst + \log\left|\hat{\Omega}\right|+\frac{2}{T}N\\ BIC & = & cst + \log\left|\hat{\Omega}\right|+\frac{\log T}{T}N, \end{eqnarray*}\] where \(N=p \times n^{2}\).
## Warning: package 'vars' was built under R version 4.3.2## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 3 3 2 3
##
## $criteria
## 1 2 3 4 5 6
## AIC(n) -0.3394120 -0.4835525 -0.5328327 -0.5210835 -0.5141079 -0.49112812
## HQ(n) -0.3017869 -0.4208439 -0.4450407 -0.4082080 -0.3761491 -0.32808581
## SC(n) -0.2462608 -0.3283005 -0.3154798 -0.2416298 -0.1725534 -0.08747275
## FPE(n) 0.7121914 0.6165990 0.5869659 0.5939325 0.5981364 0.61210908
estimated.var <- VAR(data,p=3)
#print(estimated.var$varresult)
Phi <- Acoef(estimated.var)
PHI <- make.PHI(Phi) # autoregressive matrix of companion form.
print(abs(eigen(PHI)$values)) # check stationarity## [1] 0.9114892 0.9114892 0.6319554 0.4759403 0.4759403 0.32469951.5 Block exogeneity and Granger causality
1.5.1 Block exogeneity
Let’s decompose \(y_t\) into two subvectors \(y^{(1)}_{t}\) (\(n_1 \times 1\)) and \(y^{(2)}_{t}\) (\(n_2 \times 1\)), with \(y_t' = [{y^{(1)}_{t}}',{y^{(2)}_{t}}']\) (and therefore \(n=n_1 +n_2\)), such that: \[ \left[ \begin{array}{c} y^{(1)}_{t}\\ y^{(2)}_{t} \end{array} \right] = \left[ \begin{array}{cc} \Phi^{(1,1)} & \Phi^{(1,2)}\\ \Phi^{(2,1)} & \Phi^{(2,2)} \end{array} \right] \left[ \begin{array}{c} y^{(1)}_{t-1}\\ y^{(2)}_{t-1} \end{array} \right] + \varepsilon_t. \] Using, e.g., a likelihood ratio test (see Def. 11.2), one can easily test for block exogeneity of \(y_t^{(2)}\) (say). The null assumption can be expressed as \(\Phi^{(2,1)}=0\).
1.5.2 Granger Causality
Granger (1969) developed a method to explore causal relationships among variables. The approach consists in determining whether the past values of \(y_{1,t}\) can help explain the current \(y_{2,t}\) (beyond the information already included in the past values of \(y_{2,t}\)).
Formally, let us denote three information sets: \[\begin{eqnarray*} \mathcal{I}_{1,t} & = & \left\{ y_{1,t},y_{1,t-1},\ldots\right\} \\ \mathcal{I}_{2,t} & = & \left\{ y_{2,t},y_{2,t-1},\ldots\right\} \\ \mathcal{I}_{t} & = & \left\{ y_{1,t},y_{1,t-1},\ldots y_{2,t},y_{2,t-1},\ldots\right\}. \end{eqnarray*}\] We say that \(y_{1,t}\) Granger-causes \(y_{2,t}\) if \[ \mathbb{E}\left[y_{2,t}\mid \mathcal{I}_{2,t-1}\right]\neq \mathbb{E}\left[y_{2,t}\mid \mathcal{I}_{t-1}\right]. \]
To get the intuition behind the testing procedure, consider the following bivariate VAR(\(p\)) process: \[\begin{eqnarray*} y_{1,t} & = & c_1+\Sigma_{i=1}^{p}\Phi_i^{(11)}y_{1,t-i}+\Sigma_{i=1}^{p}\Phi_i^{(12)}y_{2,t-i}+\varepsilon_{1,t}\\ y_{2,t} & = & c_2+\Sigma_{i=1}^{p}\Phi_i^{(21)}y_{1,t-i}+\Sigma_{i=1}^{p}\Phi_i^{(22)}y_{2,t-i}+\varepsilon_{2,t}, \end{eqnarray*}\] where \(\Phi_k^{(ij)}\) denotes the element \((i,j)\) of \(\Phi_k\). Then, \(y_{1,t}\) is said not to Granger-cause \(y_{2,t}\) if \[ \Phi_1^{(21)}=\Phi_2^{(21)}=\ldots=\Phi_p^{(21)}=0. \] The null and alternative hypotheses therefore are: \[ \begin{cases} H_{0}: & \Phi_1^{(21)}=\Phi_2^{(21)}=\ldots=\Phi_p^{(21)}=0\\ H_{1}: & \Phi_1^{(21)}\neq0\mbox{ or }\Phi_2^{(21)}\neq0\mbox{ or}\ldots\Phi_p^{(21)}\neq0.\end{cases} \] Loosely speaking, we reject \(H_{0}\) if some of the coefficients on the lagged \(y_{1,t}\)’s are statistically significant. Formally, this can be tested using the \(F\)-test or asymptotic chi-square test. The \(F\)-statistic is \[ F=\frac{(RSS-USS)/p}{USS/(T-2p-1)}, \] where RSS is the Restricted sum of squared residuals and USS is the Unrestricted sum of squared residuals. Under \(H_{0}\), the \(F\)-statistic is distributed as \(\mathcal{F}(p,T-2p-1)\) (See Table 11.4).1
According to the following lines of code, the output gap Granger-causes inflation, but the reverse is not true:
grangertest(US3var[,c("y.gdp.gap","infl")],order=3)## Granger causality test
##
## Model 1: infl ~ Lags(infl, 1:3) + Lags(y.gdp.gap, 1:3)
## Model 2: infl ~ Lags(infl, 1:3)
## Res.Df Df F Pr(>F)
## 1 214
## 2 217 -3 3.9761 0.008745 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
grangertest(US3var[,c("infl","y.gdp.gap")],order=3)## Granger causality test
##
## Model 1: y.gdp.gap ~ Lags(y.gdp.gap, 1:3) + Lags(infl, 1:3)
## Model 2: y.gdp.gap ~ Lags(y.gdp.gap, 1:3)
## Res.Df Df F Pr(>F)
## 1 214
## 2 217 -3 1.5451 0.2038