Vector Autoregressions and Granger Causality

00 · Symbol Glossary

$Y_t$Y sub t — the vector series

A $K\times 1$ random vector at date $t$ : several yields, returns, or macro series stacked together. A VAR is one equation per component.

$A_i$A sub i — lag-i coefficient matrix

Each $A_i$ is $K\times K$ . The $(j,k)$ entry of $A_i$ is the coefficient on the $i$ -th lag of series $k$ in the equation for series $j$ .

$\Sigma$Sigma — residual covariance

$\Sigma=\mathbb{E}[\varepsilon_t\varepsilon_t^\top]$ , the contemporaneous covariance of the reduced-form innovations. Off-diagonals are unidentified structural mixing until an extra assumption is imposed.

$\Psi_h$Psi sub h — VMA coefficients

$Y_t=\mu+\sum_{h=0}^{\infty}\Psi_h\varepsilon_{t-h}$ when the VAR is stable, with $\Psi_0=I$ . Reduced-form impulse responses live here — not yet orthogonal shocks.

01 · From AR to VAR

A univariate ARMA cannot answer “do lagged yields help forecast inflation after lags of inflation are already in the model?” A vector autoregression is the system analogue of an AR.

Definition — VAR$(p)$

A $K$ -dimensional VAR $(p)$ is

Y_t = c + A_1 Y_{t-1} + \cdots + A_p Y_{t-p} + \varepsilon_t, \qquad \varepsilon_t\sim\mathrm{WN}(0,\Sigma)

Each of the $K$ equations is an OLS regression of one component on $p$ lags of all $K$ series (and a constant). Under standard conditions OLS is consistent; with Gaussian $\varepsilon_t$ it is Gaussian MLE.

Stack the lags into a $Kp\times 1$ vector. The VAR becomes a giant VAR $(1)$ whose companion matrix has $A_1,\ldots,A_p$ on the first block-row — the same trick as writing a scalar AR $(p)$ as an AR $(1)$ in lag space. Stability: every eigenvalue of that companion lies inside the unit circle, exactly as a scalar AR root must lie outside the unit circle. If any eigenvalue has modulus $\geq 1$ , impulse responses do not die out and the VMA $(\infty)$ representation below fails to exist.

When the VAR is stable it has a VMA $(\infty)$ representation

Y_t = \mu + \sum_{h=0}^{\infty} \Psi_h \varepsilon_{t-h}, \qquad \Psi_0 = I

The matrices $\Psi_h$ are the building blocks of impulse responses. A VAR $(p)$ has $K+K^2 p$ mean parameters plus a symmetric $\Sigma$ ( $K(K+1)/2$ free entries): degrees of freedom vanish quickly in $K$ and $p$ .

02 · Granger Causality

Definition — Granger Causality

In a given information set, $X$ Granger-causes $Y$ if lags of $X$ help predict $Y$ beyond lags of $Y$ itself (and any other included variables). It is a statement about predictive content, not about “ $X$ is the structural cause of $Y$ .”

In a bivariate VAR $(1)$ with $Y_t=(y_t,x_t)^\top$ , “ $x$ does not Granger-cause $y$ ” means every entry in the first row of $A_1$ that multiplies a lag of $x$ is zero — for example $a_{12}=0$ in the $y$ equation. You are testing whether past $x$ adds forecast power for $y$ after past $y$ is already included, not whether $x$ structurally drives $y$ today. Test with a Wald / $F$ statistic (or a likelihood ratio). The test is only as good as the information set: omitted variables can create or hide Granger links.

❌ Reading a Granger $p$-value as a structural cause

Oil prices “Granger-causing” equity returns in a two-variable VAR does not identify an oil-supply shock.

Why it breaks: Granger is a forecasting restriction inside a reduced form. Contemporaneous mixing in $\Sigma$ is unidentified without extra assumptions. Policy conclusions need a structural identification scheme on top.

Consequence: report Granger tests as predictive evidence, and say so. Do not write “ $X$ causes $Y$ ” from a $p$ -value alone.

03 · Impulse Responses Need Identification

The reduced-form residual $\varepsilon_t$ is a mixture of contemporaneous shocks. An orthogonal impulse response needs a factorization $\Sigma=PP^\top$ (or a more general $B_0$ in SVAR language) so that $u_t=P^{-1}\varepsilon_t$ has identity covariance.

The usual default is the Cholesky factor: $P$ lower triangular. That imposes a recursive contemporaneous ordering — the first variable does not react within the period to later shocks; the last reacts to everything. The ordering is an economic assumption, not a statistical free lunch. Reordering can change the IRF shapes.

Example — Rate and output, two orderings

Let $Y_t=(\text{policy rate},\ \text{output})^\top$ . Cholesky with rate first means the rate does not respond to output within the month, while output can respond to the rate. Swap the order and you have a different within-period story — and different IRFs. Always report the ordering.

Variance decompositions apportion forecast-error variance of each series to each orthogonal shock. They inherit the same identification caveats. The $(i,j)$ entry of $\Psi_h$ itself is not a structural IRF: it is the reduced-form response to a reduced-form residual.

❌ Treating default Cholesky IRFs as a structural policy shock

Software’s default orthogonal IRFs come from a Cholesky ordering you may never have chosen consciously.

Why it breaks: reordering variables permutes which series can react within the period. The same reduced-form VAR can imply opposite “within-month” stories.

Consequence: always report the ordering, treat IRFs as one identification scheme among many, and do not label them “the effect of a policy shock” without extra structure (sign restrictions, instruments, or narrative timing).

04 · Lag Length and What We Skip

Choose $p$ by AIC/BIC on the system (criteria that penalize $K^2 p$ free lag coefficients). Too short a lag length leaves residual autocorrelation and biased Granger tests. Too long eats degrees of freedom.

What this chapter does not do

Structural identification beyond Cholesky (sign restrictions, external instruments / proxy SVAR, narrative shocks). Bayesian VARs with Minnesota priors — the practical fix when $K$ is large relative to $T$ . Both belong in a macro-econometrics sequel, not this core pass.

05 · Exercises

EXERCISE 7.1

Write the two equations in scalars; count intercepts, the four entries of $A_1$ , and the three free entries of symmetric $\Sigma$ .

$y_t=c_1+a_{11}y_{t-1}+a_{12}x_{t-1}+\varepsilon_{1t}$ and $x_t=c_2+a_{21}y_{t-1}+a_{22}x_{t-1}+\varepsilon_{2t}$ . Four slope coefficients, two intercepts, and three free entries in $\Sigma$ . Total: $9$ .

Write the two equations of a bivariate VAR $(1)$ in scalar form. How many free parameters are in $(A_1,c,\Sigma)$ ?

EXERCISE 7.2

Look at the $y$ equation only.

The restriction $a_{12}=0$ : lags of $x$ do not enter the $y$ equation. For a VAR $(p)$ every lag of $x$ in the $y$ equation is zero.

In that VAR $(1)$ , what coefficient restriction is “ $x$ does not Granger-cause $y$ ”?

EXERCISE 7.3

$\Psi_0=I$ maps $\varepsilon$ onto $Y$ one-for-one at horizon $0$ . Orthogonal IRFs need a factor of $\Sigma$ .

False. $\Psi_h$ is the reduced-form moving-average response to a reduced-form residual. Orthogonal / structural IRFs require a factorization of $\Sigma$ (Cholesky or otherwise).

True or false: the $(i,j)$ entry of $\Psi_h$ is already an identified structural impulse response of series $i$ to a shock in series $j$ .

06 · Chapter Summary

Object	Role
VAR $(p)$	each series on $p$ lags of the whole vector
Stability	companion eigenvalues inside the unit circle
Granger	extra predictive content of $X$ for $Y$ , not structural cause
$\Psi_h$	reduced-form MA coefficients
Cholesky IRF	convenient orthogonalization; the ordering is identification

Next: Chapter 08 — Cointegration and Error Correction, when the series share a common stochastic trend and a levels VAR is the wrong reduced form.