Granger Causality

Definition

Granger causality is a predictive notion of causality: variable $y_3$ Granger-causes $y_1$ if past values of $y_3$ help forecast $y_1$ beyond what is forecastable from the past of $y_1$ alone (and all other variables in the system). In a vector autoregression (VAR), this reduces to a zero restriction on a block of lag coefficient matrices.

Key Ideas

Granger causality is not structural causation — it is a statement about incremental forecast content in linear projections.
In stationary systems, testing noncausality is a standard zero-restriction Wald test with asymptotic $\chi^2$ distribution.
In $I(1)$ / cointegrated systems the Wald test statistic no longer has a $\chi^2$ limit in general; the correct distribution depends on the cointegration structure (Toda-Phillips 1993).
Failure of standard chi-squared asymptotics is not rare or pathological — it is the generic case for cointegrated systems.

How It Works

Noncausality Restrictions in the VAR

Partition the $n$ -vector $y_t = (y_{1t}', y_{2t}', y_{3t}')'$ where $y_{1t}$ ( $n_1 \times 1$ ) is potentially caused and $y_{3t}$ ( $n_3 \times 1$ ) is the potential cause. The VAR( $k$ ) is $J(L)y_t = u_t$ . The noncausality null is:

$H_0: J_{13}(L) = 0 \quad \Leftrightarrow \quad J_{13,1} = \cdots = J_{13,k} = 0 \tag{1}$

where $J_{13,i}$ is the $(n_1 \times n_3)$ upper-right subblock of $J_i$ . The equivalent error correction model (ECM) formulation is:

$H_0^{\text{ECM}}: J_{13}^*(L) = 0 \quad \text{and} \quad \Gamma_1 \beta_3' = 0 \tag{2}$

where $\Pi = \alpha\beta'$ ( $\alpha$ = loading matrix, $\beta$ = cointegrating vectors), $\beta_3$ = last $n_3$ rows of $\beta$ , and $\Gamma_1$ = first $n_1$ rows of $\alpha$ . The long-run term $\Gamma_1\beta_3' = 0$ imposes a nonlinear restriction on the product of two matrices.

Stationary Case

When all variables are $I(0)$ , ordinary least squares (OLS) estimation of the levels VAR is consistent and asymptotically normal, and the standard Wald statistic $F_{WD}$ has the familiar $\chi^2_{n_1 n_3 k}$ limit under $H_0$ .

Levels VAR Tests with $I(1)$ Variables (Toda-Phillips Theorems 1–2)

When $y_t$ is $I(1)$ , OLS on the levels VAR (1) is still feasible but the Wald test asymptotics change:

Theorem 1 (Toda-Phillips 1993): If $y_t$ is cointegrated and rank $(\beta_3) = n_3$ then:

$F_{WD} \xrightarrow{d} \chi^2_{n_1 n_3 k} \tag{T1}$

The condition rank $(\beta_3) = n_3$ — "sufficient cointegration with respect to $y_3$ " — means the excluded variables' block of the cointegrating matrix has full row rank, so the long-run information about $y_3$ is fully present in the cointegrating space.

Theorem 2 (Toda-Phillips 1993): If $y_t$ is not cointegrated then:

$F_{WD} \xrightarrow{d} \text{nonstandard, nuisance-parameter-free distribution} \tag{T2}$

The limit involves stochastic integrals of Brownian motions (can be tabulated).

General case (cointegration exists but rank $(\beta_3) < n_3$ ): the limit distribution is a mixture of $\chi^2$ and nonstandard components and depends on nuisance parameters (correlations among innovations). No valid statistical test is available without estimating the nuisance parameters consistently — which cannot be done from OLS levels VAR because OLS estimators of cointegrating vectors suffer from simultaneous-equations bias.

Practical verdict: Granger causality tests in OLS levels VARs are not to be recommended in general when stochastic trends and possible cointegration are present.

ECM-Based Tests (Toda-Phillips Theorem 3)

Johansen's maximum likelihood (ML) estimator of $(\alpha, \beta)$ is consistent and asymptotically mixed normal. This makes it possible to check and test the rank conditions. The Wald statistic $F_{ML}$ based on ML estimates of the ECM decomposes as:

$F_{ML} = F_{ML,(1)} + F_{ML,(2)} \tag{3}$

where $F_{ML,(1)}$ corresponds to short-run dynamics restrictions (asymptotically $\chi^2_{n_1 n_3(k-1) + n_1 g}$ , $g = \text{rank}(\beta_3)$ ) and $F_{ML,(2)}$ corresponds to long-run cointegrating restrictions (asymptotically $\chi^2_{n_1(n_3-g)}$ ), so $F_{ML,(1)} + F_{ML,(2)} \sim \chi^2_{n_1 n_3 k}$ .

Theorem 3 (Toda-Phillips 1993): If $y_t$ is cointegrated and either rank $(\beta_3) = n_3$ or rank $(\Gamma_1) = n_1$ then:

$F_{ML} \xrightarrow{d} \chi^2_{n_1 n_3 k} \tag{T3}$

When both $\beta_3$ and $\Gamma_1$ are rank-deficient, the limit is a nonlinear function of chi-squared variates (Example 2 in the paper: the statistic converges to $\chi_a^2 \chi_b^2 / (\chi_a^2 + \chi_b^2)$ , which is more concentrated near zero than $\chi^2_1$ ).

Sequential Testing Procedure

Because rank conditions can be tested from ML estimates:

Test cointegrating rank $r$ via Johansen trace or max-eigenvalue statistic.
Test rank of $\beta_3$ (last $n_3$ rows of $\hat\beta$ ) — e.g., via likelihood ratio (LR) test of rank restriction.
If rank $(\beta_3) = n_3$ or rank $(\Gamma_1) = n_1$ , apply $F_{ML}$ with $\chi^2_{n_1 n_3 k}$ critical values.
Otherwise, test the rank conditions empirically and proceed accordingly.

Simulation evidence (Toda-Phillips 1991b) indicates the sequential procedure works well for $n = 3$ – $4$ and $T > 100$ observations.

Block Exogeneity Tests (Sims 1980)

A natural application of Granger causality in macroeconomics is block exogeneity testing: asking whether an entire group of variables Granger-causes another group, rather than testing one variable at a time. Sims (1980) used this to evaluate the rational expectations market-clearing hypothesis, which predicts that real variables $(Y, U, W, P)$ should not be Granger-caused by money $M$ or import prices $PM$ .

The test null is that all cross-block lag coefficients from money into the real-sector equations are jointly zero — a multi-equation, multi-lag version of the Granger non-causality restriction (1). This is implemented as a Wald statistic on the appropriate subblocks of the VAR lag matrices.

Empirical results (Sims 1980, Table V):

Hypothesis	U.S. $\chi^2$ (df)	$p$ -value	Germany $\chi^2$ (df)	$p$ -value
Real sector exogenous to $M$	64.63 (24)	$< 0.001$	52.10 (32)	$\approx 0.01$
Real + money jointly exogenous	42.54 (36)	0.21	—	—

The real-sector exogeneity hypothesis is rejected in both countries, contradicting the pure market-clearing rational expectations model.
The U.S. joint hypothesis that both real and money sectors are exogenous (i.e., only $P$ and $PM$ are driven by external factors) is not rejected — suggesting a "two-block" structure where nominal variables drive real ones in the U.S. context.

This framework for using Granger causality tests to evaluate structural hypotheses about which blocks of variables are informationally prior to others became a standard tool in applied macroeconometrics, subsequently refined by Toda-Phillips (1993) to account for the non-standard asymptotics that arise when variables are integrated.

Higher-Dimensional Systems and the Dolado-Lütkepohl Condition (Lütkepohl 1999)

In a $K > 2$ variable VAR, testing $\alpha_{21,i} = 0$ for $i = 1, \ldots, p$ (direct zero restrictions on the coefficients of $y_{1t}$ on $y_{2t}$ ) is not equivalent to Granger noncausality. Even when these coefficients are all zero, $y_{1t}$ may still Granger-cause $y_{2t}$ through an indirect channel: $y_{1t}$ affects $y_{3t}$ , which in turn affects $y_{2t}$ . True Granger noncausality in higher-dimensional systems requires the absence of such indirect links at all horizons, which translates into nonlinear restrictions on the full VAR coefficient matrices — not just a single block (Dufour and Renault 1998).

Dolado-Lütkepohl (1996) sufficient condition. A simpler path to valid Wald tests is available: if the null hypothesis does not restrict all elements of at least one complete coefficient matrix $A_i$ ( $i = 1, \ldots, p$ ), then the Wald statistic for that null has its standard $\chi^2_{q}$ limiting distribution regardless of whether $y_t$ contains unit roots or cointegration. In particular:

Causality restrictions $H_0 : \alpha_{21,1} = \cdots = \alpha_{21,p} = 0$ restrict every $A_i$ (one element from each), so standard asymptotics do not apply in general.
Fix: augment the model by one extra (redundant) lag, estimating a VAR $(p+1)$ under the assumption that the true data-generating process (DGP) is VAR $(p)$ . The matrix $A_{p+1}$ is unrestricted under $H_0$ (it is zero under the DGP but not constrained by the null), so the Dolado-Lütkepohl condition is satisfied and the Wald test on $A_1, \ldots, A_p$ is valid with $\chi^2$ critical values.
This procedure requires neither knowledge of the cointegrating rank nor of the integration order — it is a "safe" default for applied causality testing.
Trade-off: the extra lag adds $K^2$ redundant parameters, reducing power relative to correctly specified ECM-based tests.

Finite-Sample Size Failure and Maximized Monte Carlo Tests (Dufour-Jouini 2006)

Even the Dolado-Lütkepohl and Toda-Yamamoto lag-augmentation procedures can fail badly in finite samples. Dufour and Jouini (2006) document that standard asymptotic LR tests have empirical rejection rates as high as 97% at a nominal 5% level (6-variable VAR, $p=5$ lags, $T=50$ , $\rho=0.99$ ). Parametric bootstrap (Local Monte Carlo, LMC) tests reduce but do not eliminate overrejection (up to 59%). Lag-augmented Wald tests with bootstrap critical values also over-reject severely.

Three simulation procedures (Dufour 2006):

Procedure	Description	Level guarantee
MC test	Simulate $N$ i.i.d. replications of statistic under null (no nuisance params); reject when $\hat{p}_N(S_0) \leq \alpha$	Provably exact if $\alpha(N+1) \in \mathbb{Z}$
LMC test	Plug restricted MLE into simulation → parametric bootstrap $p$ -value	Asymptotically valid under regularity conditions; not exact
MMC test	Maximize simulated $p$ -value over nuisance parameter space: $\hat{p}_{\mathrm{MMC}} = \sup\{\hat{p}_N(S_0 \mid \theta) : \theta \in \Theta_0\}$ ; reject when $\hat{p}_{\mathrm{MMC}} \leq \alpha$	Provably exact; protects against bootstrap failures

Key identity. For the basic MC test, if $\alpha(N+1)$ is an integer, $P[\hat{p}_N(S_0) \leq \alpha] = \alpha$ under $H_0$ — the simulated p-value is exactly uniform, yielding exact level $\alpha$ without any asymptotic argument.

MMC for VAR Granger causality. The LR statistic $\text{LRG} = T \ln(|\hat{\Sigma}_0|/|\hat{\Sigma}|)$ can be simulated once $\theta = \mathrm{vec}[m, \Phi, R]$ is specified. The MMC algorithm: (1) compute LRG on observed data; (2) generate N i.i.d. Gaussian pseudo-errors; (3) compute pseudo-statistics as functions of $\theta$ ; (4) maximize $\hat{p}_N(\text{LRG}^{(0)} | \theta)$ over $\theta \in \Theta_0$ via simulated annealing. Starting from the restricted MLE: if the bootstrap p-value exceeds $\alpha$ , the MMC also exceeds $\alpha$ — so bootstrap provides a cheap filter before MMC.

Simulation results summary:

Method	Max rejection rate ( $T=50$ , $k=4$ , $p=5$ )	Valid?
Asymptotic LR	97% at 5% nominal	No
LMC / parametric bootstrap	59% at 5% nominal	No
Lag-augmented Wald + bootstrap	Severely over-rejects in many configs	No
MMC	Exactly 5% throughout	Yes

Empirical application (Bernanke-Mihov 1998 data). 4-variable quarterly VAR (NBR=M, FFR=r, gross domestic product (GDP)=y, GDP deflator=P), 1965Q1–1996Q4, VAR(4). MMC Granger causality results:

At 5%: $M \to r \to y$
At 10%: $M \to r \to y \leftrightarrow P$

Monetarist interpretation: money Granger-causes interest rates, which in turn cause income. The feedback between $y$ and $P$ at 10% is consistent with standard aggregate supply-aggregate demand (AS-AD) dynamics.

Sims-Stock-Watson (1990): Causality Tests in Trivariate VARs

Sims, Stock, and Watson (1990) (Section 6) analyze a trivariate system $(y_{1t}, y_{2t}, y_{3t})'$ under four cases defined by the presence of cointegration and a deterministic time trend. Their Proposition 6.2 gives precise conditions for whether the Granger causality F-test for $y_3 \not\to y_1$ has a standard $\chi^2$ distribution.

The key structural distinction is whether the excluded variable's long-run coefficient $A_{12}(1)$ loads on the stationary equilibrium error $\zeta_{1t}$ (cointegrated cases) or on the I(1) stochastic trend component (non-cointegrated cases):

Case	Cointegration	$A_{12}(1)$ loads on	F-test distribution
1	No	I(1) trend	Nonstandard
2	No + deterministic trend	I(1) trend	Nonstandard
3	Yes	stationary $\zeta_{1t}$	$\chi^2_p$ (standard)
4	Yes + deterministic trend	stationary $\zeta_{1t}$	$\chi^2_p$ (standard)

The Sims-Stock-Watson (SSW) result identifies the structural reason why cointegration restores standard asymptotics for causality tests: when variables are cointegrated, the long-run coefficient loads entirely on the stationary equilibrium error, placing the restriction within the "mean-zero stationary canonical regressors" — the condition for $\chi^2$ validity in their Theorem 2. The lag-length F-test (last lag matrix equal to zero) has standard $\chi^2$ distribution in all four cases, since it restricts only short-run dynamics.

This is a more primitive result than Toda-Phillips (1993), which provides the exact operational rank conditions on $\beta_3$ and $\Gamma_1$ . SSW establishes the mechanism via the canonical decomposition; Toda-Phillips refines it into rank-testable conditions usable in practice.

Why It Matters

Standard Granger causality tests in levels VARs are among the most frequently misapplied procedures in applied macroeconometrics — the generic failure of $\chi^2$ asymptotics when variables are $I(1)$ invalidates routine inference.
Toda-Phillips provides the precise conditions under which tests are valid, motivating the Johansen ECM framework as the correct testing environment.
The rank conditions on $\beta_3$ and $\Gamma_1$ have economic content: rank $(\beta_3) = n_3$ means every potential causal variable appears in some cointegrating relation.
Block exogeneity tests (Sims 1980) translate structural economic hypotheses — market clearing, monetary neutrality — into testable Granger non-causality restrictions without requiring a full structural identification.
Even "safe" procedures (Dolado-Lütkepohl, Toda-Yamamoto, bootstrap) can have 97% actual rejection rates at 5% nominal — the size problem is not a small-sample curiosity but persists at T=300 in higher-dimensional VARs (Dufour-Jouini 2006). MMC is the only known procedure with exact finite-sample level control.

Open Questions

The sequential procedure pretests for rank, introducing size distortions in small samples.
A simpler alternative is the Toda-Yamamoto (1995) lag-augmentation approach: estimate a VAR( $k + d_{\max}$ ) in levels and test zero restrictions only on the first $k$ lag matrices via a standard Wald test — the extra lags absorb the integrated regressors and standard $\chi^2$ asymptotics apply regardless of integration order. This avoids the rank pretesting but loses some power.
Both procedures assume known lag length $k$ ; lag selection with integrated variables is itself non-trivial.