Short run and long run causality in time series: inference∗

Jean-Marie Dufour,† Denis Pelletier‡ and Éric Renault§

Université de Montréal

September 2003

∗ This work was supported by the Canada Research Chair Program (Chair in Econometrics, Université de Mon-tréal), the Alexander-von-Humboldt Foundation (Germany), the Institut de Finance mathématique de Montréal (IFM2),the Canadian Network of Centres of Excellence [program onMathematics of Information Technology and Complex Sys-tems(MITACS)], the Canada Council for the Arts (Killam Fellowship), the Natural Sciences and Engineering ResearchCouncil of Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds FCAR (Governmentof Québec). The authors thank Jörg Breitung, Helmut Lütkepohl, Peter Schotman, participants at the EC2 Meeting onCausality and Exogeneity in Louvain-la-Neuve (December 2001), two anonymous referees, and the Editors Luc Bauwens,Peter Boswijk and Jean-Pierre Urbain for several useful comments. This paper is a revised version of Dufour and Renault(1995).

† Canada Research Chair Holder (Econometrics). Centre interuniversitaire de recherche en analyse des organisa-tions (CIRANO), Centre interuniversitaire de recherche en économie quantitative (CIREQ), and Département de scienceséconomiques, Université de Montréal. Mailing address: Département de sciences économiques, Université de Montréal,C.P. 6128 succursale Centre-ville, Montréal, Québec, Canada H3C 3J7. TEL: 1 514 343 2400; FAX: 1 514 343 5831;e-mail: jean.marie.dufour@umontreal.ca. Web page: http://www.fas.umontreal.ca/SCECO/Dufour .

‡ CIRANO, CIREQ, Université de Montréal (Département de sciences économiques), and North Carolina StateUniversity. Mailing address: Department of Economics, North Carolina State University, Campus Box 8110, Raleigh,NC, USA 27695-8110. E-mail: denis_pelletier@ncsu.edu .

§CIRANO, CIREQ, and Université de Montréal (Département de sciences économiques). Mailing address: Départe-ment de sciences économiques, Université de Montréal, C.P. 6128 succursale Centre-ville, Montréal, Québec, CanadaH3C 3J7. TEL: 1 514 343 2398; FAX: 1 514 343 5831; e-mail: eric.renault@umontreal.ca .

ABSTRACT

We propose methods for testing hypothesis of non-causality at various horizons, as defined in Du-four and Renault (1998,Econometrica). We study in detail the case of VAR models and we proposelinear methods based on running vector autoregressions at different horizons. While the hypothesesconsidered are nonlinear, the proposed methods only require linear regression techniques as well asstandard Gaussian asymptotic distributional theory. Bootstrap procedures are also considered. Forthe case of integrated processes, we propose extended regression methods that avoid nonstandardasymptotics. The methods are applied to a VAR model of the U.S. economy.

Key words: time series; Granger causality; indirect causality; multiple horizon causality; autore-gression; autoregressive model; vector autoregression; VAR; stationary process; nonstationary pro-cess; integrated process; unit root; extended autoregression; bootstrap; Monte Carlo; macroeco-nomics; money; interest rates; output; inflation.

JEL classification numbers: C1; C12; C15; C32; C51; C53; E3; E4; E52.

i

RÉSUMÉ

Nous proposons des méthodes pour tester des hypothèses de non-causalité à différents horizons, telque défini dans Dufour et Renault (1998,Econometrica). Nous étudions le cas des modèles VARen détail et nous proposons des méthodes linéaires basées sur l’estimation d’autorégressions vecto-rielles à différents horizons. Même si les hypothèses considérées sont non linéaires, les méthodesproposées ne requièrent que des techniques de régression linéaire de même que la théorie distribu-tionnelle asymptotique gaussienne habituelle. Dans le cas des processus intégrés, nous proposonsdes méthodes de régression étendue qui ne requièrent pas de théorie asymptotique non standard.L’application du bootstrap est aussi considérée. Les méthodes sont appliquées à un modèle VAR del’économie américaine.

Mots clés : séries chronologiques; causalité; causalité indirecte; causalité à différents horizons;autorégression; modèle autorégressif; autorégression vectorielle; VAR; processus stationnaire; pro-cessus non stationnaire; processus intégré; racine unitaire; autorégression étendue; bootstrap; MonteCarlo; macroéconomie; monnaie; taux d’intérêt; production; inflation.

Classification JEL: C1; C12; C15; C32; C51; C53; E3; E4; E52.

ii

Contents

List of Definitions, Propositions and Theorems iv

1. Introduction 1

2. Multiple horizon autoregressions 2

3. Estimation of (p, h) autoregressions 7

4. Causality tests based on stationary(p, h)-autoregressions 9

5. Causality tests based on nonstationary(p, h)-autoregressions 12

6. Empirical illustration 15

7. Conclusion 22

iii

List of Tables

1 Rejection frequencies using the asymptotic distribution and the simulated proce-dure when the true DGP is an i.i.d. Gaussian sequence. . . . . . . . . . . . . . 15

2 Causality tests and simulated p-values for series in first differences (of logarithm)for the horizons 1 to 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Causality tests and simulated p-values for series in first differences (of logarithm)for the horizons 13 to 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Summary of causality relations at various horizons for series in first difference. . 215 Causality tests and simulated p-values for extended autoregressions at the horizons

1 to 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Causality tests and simulated p-values for extended autoregressions at the horizons

13 to 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Summary of causality relations at various horizons for series in first difference with

extended autoregressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

List of Definitions, Propositions and Theorems

3.1 Proposition : Asymptotic normality of LS in a(p, h) stationary VAR. . . . . . . . . 94.1 Proposition : Asymptotic distribution of test criterion for non-causality at horizonh

in a stationary VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

iv

1. Introduction

The concept of causality introduced by Wiener (1956) and Granger (1969) is now a basic notion forstudying dynamic relationships between time series. The literature on this topic is considerable; see,for example, the reviews of Pierce and Haugh (1977), Newbold (1982), Geweke (1984), Lütkepohl(1991) and Gouriéroux and Monfort (1997, Chapter 10). The original definition of Granger (1969),which is used or adapted by most authors on this topic, refers to the predictability of a variableX(t),wheret is an integer, from its own past, the one of another variableY (t) and possibly a vectorZ(t)of auxiliary variables,one period ahead: more precisely, we say thatY causesX in the sense ofGranger if the observation ofY up to timet (Y (τ) : τ ≤ t) can help one to predictX(t + 1)when the corresponding observations onX andZ are available(X(τ), Z(τ) : τ ≤ t); a moreformal definition will be given below.

Recently, however, Lütkepohl (1993) and Dufour and Renault (1998) have noted that, for multi-variate models where a vector of auxiliary variablesZ is used in addition to the variables of interestX andY, it is possible thatY does not causeX in this sense, but can still help to predictX severalperiods ahead; on this issue, see also Sims (1980), Renault, Sekkat and Szafarz (1998) and Giles(2002). For example, the valuesY (τ) up to timet may help to predictX(t + 2), even though theyare useless to predictX(t+1). This is due to the fact thatY may help to predictZ one period ahead,which in turn has an effect onX at a subsequent period. It is clear that studying such indirect effectscan have a great interest for analyzing the relationships between time series. In particular, one candistinguish in this way properties of “short-run (non-)causality” and “long-run (non-)causality”.

In this paper, we study the problem of testing non-causality at various horizons as defined inDufour and Renault (1998) for finite-order vector autoregressive (VAR) models. In such models,the non-causality restriction at horizon one takes the form of relatively simple zero restrictions onthe coefficients of the VAR [see Boudjellaba, Dufour and Roy (1992) and Dufour and Renault(1998)]. However non-causality restrictions at higher horizons (greater than or equal to 2) aregenerally nonlinear, taking the form of zero restrictions on multilinear forms in the coefficientsof the VAR. When applying standard test statistics such as Wald-type test criteria, such forms caneasily lead to asymptotically singular covariance matrices, so that standard asymptotic theory wouldnot apply to such statistics. Further, calculation of the relevant covariance matrices _ which involvethe derivatives of potentially large numbers of restrictions _ can become quite awkward.

Consequently, we propose simple tests for non-causality restrictions at various horizons [asdefined in Dufour and Renault (1998)] which can be implemented only through linear regressionmethods and do not involve the use of artificial simulations [e.g., as in Lütkepohl and Burda (1997)].This will be done, in particular, by considering multiple horizon vector autoregressions [called(p, h)-autoregressions] where the parameters of interest can be estimated by linear methods. Re-strictions of non-causality at different horizons may then be tested through simple Wald-type (orFisher-type) criteria after taking into account the fact that such autoregressions involve autocorre-lated errors [following simple moving average processes] which are orthogonal to the regressors.The correction for the presence of autocorrelation in the errors may then be performed by using anautocorrelation consistent [or heteroskedasticity-autocorrelation-consistent (HAC)] covariance ma-trix estimator. Further, we distinguish between the case where the VAR process considered is stable

1

(i.e., the roots of the determinant of the associated AR polynomial are all outside the unit circle)and the one where the process may be integrated of an unknown order (although not explosive). Inthe first case, the test statistics follow standard chi-square distributions while, in the second case,they may follow nonstandard asymptotic distributions involving nuisance parameters, as already ob-served by several authors for the case of causality tests at horizon one [see Sims, Stock and Watson(1990), Toda and Phillips (1993, 1994), Toda and Yamamoto (1995), Dolado and Lütkepohl (1996)and Yamada and Toda (1998)]. To meet the objective of producing simple procedures that can beimplemented by least squares methods, we propose to deal with such problems by using an exten-sion to the case of multiple horizon autoregressions of the lag extension technique suggested byChoi (1993) for inference on univariate autoregressive models and by Toda and Yamamoto (1995)and Dolado and Lütkepohl (1996) for inference on standard VAR models. This extension will al-low us to use standard asymptotic theory in order to test non-causality at different horizons withoutmaking assumption on the presence of unit roots and cointegrating relations. Finally, to alleviatethe problems of finite-sample unreliability of asymptotic approximations in VAR models (on bothstationary and nonstationary series), we propose the use of bootstrap methods to implement theproposed test statistics.

In section 2, we describe the model considered and introduce the notion of autoregression athorizonh [or (p, h)-autoregression] which will be the basis of our method. In section 3, we studythe estimation of(p, h)-autoregressions and the asymptotic distribution of the relevant estimatorsfor stable VAR processes. In section 4, we study the testing of non-causality at various horizons forstationary processes, while in section 5, we consider the case of processes that may be integrated.In section 6, we illustrate the procedures on a monthly VAR model of the U.S. economy involv-ing a monetary variable (nonborrowed reserves), an interest rate (federal funds rate), prices (GDPdeflator) and real GDP, over the period 1965-1996. We conclude in section 7.

2. Multiple horizon autoregressions

In this section, we develop the notion of “autoregression at horizonh” and the relevant notations.Consider aVAR(p) process of the form:

W (t) = µ(t) +p∑

k=1

πkW (t− k) + a (t) , t = 1, . . . , T, (2.1)

whereW (t) =(w1t, w2t, . . . , wmt

)′is anm× 1 random vector,µ(t) is a deterministic trend, and

E[a (s) a (t)′

]= Ω, if s = t ,= 0, if s 6= t ,

(2.2)

det(Ω) 6= 0 . (2.3)

The most common specification forµ(t) consists in assuming thatµ(t) is a constant vector,i.e.

µ(t) = µ , (2.4)

2

although other deterministic trends could also be considered.TheVAR(p) in equation (2.1) is an autoregression at horizon 1. We can then also write for the

observation at timet + h:

W (t + h) = µ(h)(t) +p∑

k=1

π(h)k W (t + 1− k) +

h−1∑

j=0

ψja (t + h− j) , t = 0, . . . , T − h ,

whereψ0 = Im andh < T . The appropriate formulas for the coefficientsπ(h)k , µ(h)(t) andψj are

given in Dufour and Renault (1998), namely:

π(h+1)k = πk+h +

h∑

l=1

πh−l+1π(l)k = π

(h)k+1 + π

(h)1 πk , π

(0)1 = Im , π

(1)k = πk , (2.5)

µ(h)(t) =h−1∑

k=0

π(k)1 µ(t + h− k) , ψh = π

(h)1 , ∀h ≥ 0 . (2.6)

Theψh matrices are the impulse response coefficients of the process, which can also be obtainedfrom the formal series:

ψ(z) = π(z)−1 = Im +∞∑

k=1

ψkzk , π(z) = Im −

∞∑

k=1

πkzk . (2.7)

Equivalently, the above equation forW (t + h) can be written in the following way:

W (t + h)′ = µ(h)(t)′+

p∑

k=1

W (t + 1− k)′ π(h)′k + u(h) (t + h)′

= µ(h)(t)′+ W (t, p)′ π(h) + u(h) (t + h)′ , t = 0, . . . , T − h , (2.8)

whereW (t, p)′ =[W (t)′ , W (t− 1)′ , . . . , W (t− p + 1)′

], π(h) =

[π

(h)1 , . . . , π

(h)p

]′and

u(h) (t + h)′ =[u

(h)1 (t + h) , . . . , u(h)

m (t + h)]

=h−1∑

j=0

a (t + h− j)′ ψ′j .

It is straightforward to see thatu(h) (t + h) has a non-singular covariance matrix.We call (2.8) an “autoregression of orderp at horizonh” or a “(p, h)-autoregression”. In the

sequel, we will assume that the deterministic part of each autoregression is a linear function of afinite-dimensional parameter vector,i.e.

µ(h)(t) = γ(h)D(h)(t) (2.9)

whereγ(h) is am×n coefficient vector andD(h)(t) is an×1 vector of deterministic regressors. If

3

µ(t) is a constant vector,i.e. µ(t) = µ, thenµ(h)(t) is simply a constant vector (which may dependonh):

µ(h)(t) = µh . (2.10)

To derive inference procedures, it will be convenient to consider a number of alternative formu-lations of(p, h)-autoregression autoregressions.a) Matrix (p, h)-autoregression_ First, we can put (2.8) in matrix form, which yields:

wh (h) = W p (h) Π(h) + Uh (h) , h = 1, . . . , H , (2.11)

wherewh (k) andUh (k) are(T − k + 1)×m matrices andW p (k) is a(T − k + 1)× (n + mp)matrix defined as

wh (k) =

W (0 + h)′

W (1 + h)′...

W (T − k + h)′

= [w1 (h, k) , . . . , wm (h, k)] , (2.12)

W p (k) =

Wp (0)′

Wp (1)′...

Wp (T − k)′

, Wp (t) =

[D(h)(t)

′

W (t, p)

], (2.13)

Π(h) =[

γ(h)′

π(h)

]= [β1 (h) , β2 (h) , . . . , βm (h)] , (2.14)

Uh (k) =

u(h) (0 + h)′

u(h) (1 + h)′...

u(h) (T − k + h)′

= [u1 (h, k) , . . . , um (h, k)] , (2.15)

ui (h, k) =[u

(h)i (0 + h) , u

(h)i (1 + h) , . . . , u

(h)i (T − k + h)

]′. (2.16)

We shall call the formulation (2.11) a “(p, h)-autoregression in matrix form”.b) Rectangular stacked(p, H)-autoregression_ To get the same regressor matrix on the right-hand side of (2.11), we can also consider:

wh (H) = W p (H) Π(h) + Uh (H) , h = 1, . . . , H. (2.17)

This, however, involves losing observations. Using (2.17), we can also stack theH systems aboveas follows:

wH = W p (H) ΠH + UH (2.18)

wherewH andUH are(T −H + 1)×(mH) matrices andWp (H) is an(mp)×(mH) matrix such

4

that

wH = [w1 (H) , w2 (H) , . . . , wH (H)] ,

ΠH =[Π(1), Π(2), . . . , Π(H)

],

UH = [U1 (H) , U2 (H) , . . . , UH (H)] .

Since the elements ofUH are linear transformations of the random vectorsa (t) , t = 1, . . . , T,which containTm random variables, it is clear that the vectorvec (Um) will have a singular covari-ance matrix when

Tm < (T −H) mH = TmH −mH2,

which will be the case whenH ≥ 2 andTm > H.

c) Vec-stacked(p, H)-autoregression_ We can also write equation (2.8) as

w (t + h) =[Im ⊗Wp (t)′

]Π(h) + u(h) (t + h)

= W p (t)′ Π(h) + u(h) (t + h) , t = 0, . . . , T − h, (2.19)

where

Π(h) = vec(Π(h)

)=

β(h)1

β(h)2...

β(h)m

,

W p (t)′ =

Wp (t)′ 0 · · · 00 Wp (t)′ · · · 0...

......

...0 0 · · · Wp (t)′

,

which yields the linear modelwh = ZhΠ(h) + uh (2.20)

where

wh =

W (0 + h)W (1 + h)

...W (T )

= vec

[wh (h)′

],

Zh =

W p (0)′

W p (1)′...

W p (T − h)′

=

Im ⊗Wp (0)′

Im ⊗Wp (1)′...

Im ⊗Wp (T − h)′

,

5

uh =

u(h) (0 + h)u(h) (1 + h)

...u(h) (T )

= vec

[Uh (h)′

].

It is also possible to stack together the models (2.20) forh = 1, . . . , H :

w (H) = Z (H) ΠH + u (H) (2.21)

where

w (H) =

w1

w2...

wH

, u (H) =

u1

u2...

uH

, Z (H) =

Z1 0 · · · 00 Z2 · · · 0...

......

0 0 · · · ZH

.

d) Individual (p, H)-autoregressions_ Consider finally a single dependent variable

Wi (t + h) = Wp (t)′ βi (h) + u(h)i (t + h) , t = 0, . . . , T −H, (2.22)

for 1 ≤ h ≤ H, where1 ≤ i ≤ m. We can also write:

Wi (t + H) =[IH ⊗Wp (t)′

]βi (H) + ui (t + H) , t = 0, . . . , T −H , (2.23)

where

Wi (t + H) =

Wi (t + 1)Wi (t + 2)

...Wi (t + H)

, ui (t + H) =

ui (t + 1)ui (t + 2)

...ui (t + H)

, βi (H) =

βi (1)βi (2)

...βi (H)

,

which yields the linear modelWi (H) = ZH βi (H) + uH (2.24)

where

Wi (H) =

Wi (0 + H)Wi (1 + H)

...Wi (T )

, βi (H) =

βi (1)βi (2)

...βi (H)

,

6

ZH =

IH ⊗Wp (0)′

IH ⊗Wp (1)′...

IH ⊗Wp (T −H)′

, uH =

ui (0 + H)ui (1 + H)

...ui (T )

.

In the sequel, we shall focus on prediction equations for individual variables and the matrix(p, h)-autoregressive form of the system in (2.11).

3. Estimation of (p, h) autoregressions

Let us now consider each autoregression of orderp at horizonh as given by (2.11):

wh (h) = W p (h) Π(h) + Uh (h) , h = 1, . . . , H . (3.1)

We can estimate (3.1) by ordinary least squares (OLS), which yields the estimator:

Π(h) =[W p (h)′W p (h)

]−1W p (h)′wh (h) = Π(h) +

[W p (h)′W p (h)

]−1W p (h)′ Uh (h) ,

hence √T [Π(h) −Π(h) ] =

[1T

W p (h)′W p (h)]−1 1√

TW p (h)′ Uh (h)

where

1T

W p (h)′W p (h) =1T

T−h∑

t=0

Wp (t) Wp (t)′ ,1√T

W p (h)′ Uh (h) =1√T

T−h∑

t=0

Wp (t) u(h) (t + h)′ .

Suppose now that

1T

T−h∑

t=0

Wp (t)Wp (t)′ p−→T→∞

Γp with det(Γp) 6= 0 . (3.2)

In particular, this will be the case if the processW (t) is second-order stationary, strictly indeter-ministic and regular, in which case

E[Wp (t) Wp (t)′

]= Γp , ∀t. (3.3)

Cases where the process does not satisfy these conditions are covered in section 5. Further, since

u(h) (t + h) = a (t + h) +h−1∑

k=1

ψka (t + h− k)

7

(where, by convention, any sum of the form∑h−1

k=1 with h < 2 is zero), we have:

E[Wp (t) u(h) (t + h)′

]= 0 , for h = 1, 2, ... ,

V

vec[Wp (t) u(h) (t + h)′

]= ∆p (h) .

If the processW (t) is strictly stationary with i.i.d. innovationsa(t) and finite fourth moments, wecan write:

E[Wp (s)u(h)i (s + h) u

(h)j (t + h) Wp (t)′] = Γij(p, h, t− s) = Γij(p, h, s− t) (3.4)

where1 ≤ i ≤ m, 1 ≤ j ≤ m, with

Γij(p, h, 0) = E[Wp (t) u

(h)i (t + h) u

(h)j (t + h) Wp (t)′

]

= σij (h) E[Wp (t) Wp (t)′

]= σij (h) Γp , (3.5)

Γij(p, h, t− s) = 0 , if |t− s| ≥ h . (3.6)

In this case,1

∆p (h) = [σij (h) Γp]i, j=1, ... , m = Σ(h)⊗ Γp (3.7)

whereΣ(h) is nonsingular, and thus∆p (h) is also nonsingular. The nonsingularity ofΣ(h) followsfrom the identity

u(h) (t + h) =[ψh−1, ψh−2, . . . , ψ1, Im

] [a (t + 1)′ , a (t + 2)′ , . . . , a (t + h)′

]′.

Under usual regularity conditions,

1√T

T−h∑

t=0

vec[Wp (t) u(h) (t + h)′ ] L−→T→∞

N[0, ∆p (h)

](3.8)

where∆p (h) is a nonsingular covariance matrix which involves the variance and the autocovari-ances ofWp (t)u(h) (t + h)′ [and possibly other parameters, if the processW (t) is not linear].Then,

√T vec

[Π(h) −Π(h)

]=

Im ⊗

[1T

W p (h)′W p (h)]−1

vec

[1√T

W p (h)′ Uh (h)]

=

Im ⊗

[1T

W p (h)′W p (h)]−1

1√T

T−h∑

t=0

vec[Wp (t) u(h) (t + h)′

]

1Note that (3.5) holds under the assumption of martingale difference sequence ona (t) . But to get (3.6) and allow theuse of simpler central limit theorems, we maintain the stronger assumption that the innovationsa(t) are i.i.d. accordingto some distribution with finite fourth moments (not necessarily Gaussian).

8

L−→T→∞

N[0,

(Im ⊗ Γ−1

p

)∆p (h)

(Im ⊗ Γ−1

p

) ]. (3.9)

For convenience, we shall summarize the above observations in the following proposition.

Proposition 3.1 ASYMPTOTIC NORMALITY OF LS IN A (p, h) STATIONARY VAR. Underthe assumptions(2.1), (3.2), and (3.8), the asymptotic distribution of

√T vec

[Π(h) − Π(h)

]is

N [0, Σ(Π(h))] , whereΣ(Π(h)) =(Im ⊗ Γ−1

p

)∆p (h)

(Im ⊗ Γ−1

p

).

4. Causality tests based on stationary(p, h)-autoregressions

Consider thei-th equation(1 ≤ i ≤ m) in system (2.11):

wi (h) = W p (h) βi (h) + ui (h) , 1 ≤ i ≤ m , (4.1)

wherewi (h) = wi (h, h) andui (h) = ui (h, h) , wherewi (h, h) andui (h, h) are defined in(2.12) and (2.15). We wish to test:

H0(h) : Rβi (h) = r (4.2)

whereR is a q × (n + mp) matrix of rankq. In particular, if we wish to test the hypothesis thatwjt does not causewit at horizonh [i.e., using the notation of Dufour and Renault (1998),wj 9

h

wi | I(j), whereI(j)(t) is the Hilbert space generated by the basic information setI(t) and thevariableswkτ , ω < τ ≤ t, k 6= j, ω being an appropriate starting time(ω ≤ −p + 1)], therestriction would take the form:

H(h)j9i : π

(h)ijk = 0 , k = 1, . . . , p , (4.3)

where π(h)k =

[π

(h)ijk

]i, j=1, ... , m

, k = 1, . . . , p . In other words, the null hypothesis takes

the form of a set of zero restrictions on the coefficients ofβi (h) as defined in (2.14).The matrix of restrictionsR in this case takes the formR = R(j), where R(j) ≡[δ1( j), δ2( j), . . . , δp( j)]′ is a p × (n + mp) matrix, δk( j) is a (n + pm) × 1 vector whoseelements are all equal to zero except for a unit value at positionn + (k − 1)m + j, i.e.δk( j) = [δ (1, n + (k − 1)m + j) , . . . , δ (n + pm, n + (k − 1)m + j)]′ , k = 1, . . . , p, withδ(i, j) = 1 if i = j, andδ(i, j) = 0 if i 6= j. Note also that the conjunction of the hypothesis

H(h)j9i, h = 1, . . . , (m − 2)p + 1, is sufficient to obtain noncausality at all horizons [see Dufour

and Renault (1998, section 4)]. Non-causality up to horizonH is the conjunction of the hypothesisH

(h)j9i, h = 1, . . . , H.

We have:βi (h) = βi (h) +

[W p (h)′W p (h)

]−1W p (h)′ ui (h) ,

9

hence

√T

[βi (h)− βi (h)

]=

[1T

W p (h)′W p (h)]−1 1√

T

T−h∑

t=0

Wp (t) u(h)i (t + h) .

Under standard regularity conditions [see White (1999, chap. 5-6)],

√T

[βi (h)− βi (h)

] L−→T→∞

N[0, V

(βi

)]

with det[V

(βi

)] 6= 0, whereV(βi

)can be consistently estimated:

VT

(βi

) p−→T→∞

V(βi

).

More explicit forms forVT

(βi

)will be discussed below. Note also that

Γp = plimT→∞

1T

W p (h)′W p (h) , det (Γp) 6= 0 .

Let

Vip (T ) = Var

[1√T

W p (h)′ ui (h)]

=1T

Var

[T−h∑

t=0

Wp (t)u(h)i (t + h)

]

=1T

T−h∑

t=0

E[Wp (t) u

(h)i (t + h)u

(h)i (t + h) Wp (t)′

]

+h−1∑

τ=1

T−h∑

t=τ+1

[E[Wp (t) u

(h)i (t + h)u

(h)i (t− τ + h) Wp (t− τ)′

]

+E[Wp (t− τ) u

(h)i (t− τ + h) u

(h)i (t + h) Wp (t)′

]].

Let us assume thatVip (T ) −→

T→∞Vip , det (Vip) 6= 0 , (4.4)

whereVip can be estimated by a computable consistent estimatorVip (T ) :

Vip (T )p−→

T→∞Vip . (4.5)

Then, √T

[βi (h)− βi (h)

]p−→

T→∞N

[0, Γ−1

p VipΓ−1p

]

10

so thatV(βi

)= Γ−1

p VipΓ−1p . Further, in this case,

VT

(βi

)= Γ−1

p Vip (T ) Γ−1p

p−→T→∞

V(βi

),

Γp =1T

T−h∑

t=0

Wp (t) Wp (t)′ =1T

W p (h)′W p (h)p−→

T→∞Γp .

We can thus state the following proposition.

Proposition 4.1 ASYMPTOTIC DISTRIBUTION OF TEST CRITERION FOR NON-CAUSALITY AT

HORIZON h IN A STATIONARY VAR. Suppose the assumptions of Proposition3.1hold jointly with(4.4)− (4.5). Then, under any hypothesis of the formH0(h) in (4.2), the asymptotic distribution of

W[H0(h)] = T[Rβi (h)− r

]′[RVT

(βi

)R′]−1[

Rβi (h)− r]

(4.6)

is χ2 (q). In particular, under the hypothesisH(h)j9i of non-causality at horizonh from wjt to

wit

(wj 9

hwi | I(j)

), the asymptotic distribution of the corresponding statisticW[H0(h)] is χ2 (p) .

The problem now consists in estimatingVip. Let ui (h) =[u

(h)i (t + h) : t = 0, . . . , T − h

]′

be the vector of OLS residuals from the regression (4.1),g(h)i (t + h) = Wp (t) u

(h)i (t + h) , and

set

R(h)i (τ) =

1T − h

T−h∑t=τ

g(h)i (t + h) g

(h)i (t + h− τ)′ , τ = 0, 1, 2, . . . .

If the innovations are i.i.d. or, more generally, if (3.6) holds, a natural estimator ofVip, which wouldtake into account the fact that the prediction errorsu(h) (t + h) follow an MA(h− 1) process, isgiven by:

V(W )ip (T ) = R

(h)i (0) +

h−1∑

τ=1

[R

(h)i (τ) + R

(h)i (τ)′

].

Under regularity conditions studied by White (1999, Section 6.3),

V(W )ip (T )− Vip

p−→T→∞

0.

A problem withV(W )ip (T ) is that it is not necessarily positive-definite.

An alternative estimator which is automatically positive-semidefinite is the one suggested byDoan and Litterman (1983), Gallant (1987) and Newey and West (1987):

V(NW )ip (T ) = R

(h)i (0) +

m(T )−1∑

τ=1

κ (τ , m (T ))[R

(h)i (τ) + R

(h)i (τ)′

], (4.7)

11

whereκ (τ , m) = 1− [τ/ (m + 1)] , limT→∞

m (T ) = ∞, and limT→∞

[m (T ) /T 1/4

]= 0 . Under the

regularity conditions given by Newey and West (1987),

V(NW )ip (T )− Vip −→

T→∞0 .

Other estimators that could be used here includes various heteroskedasticity-autocorrelation-consistent (HAC) estimators; see Andrews (1991), Andrews and Monahan (1992), Cribari-Neto,Ferrari and Cordeiro (2000), Cushing and McGarvey (1999), Den Haan and Levin (1997), Hansen(1992), Newey and McFadden (1994), Wooldridge (1989).

The cost of having a simple procedure that sidestep all the nonlinearities associated with thenon-causality hypothesis is a loss of efficiency. There are two places where we are not using allinformation. The constraints on theπ(h)

k ’s are giving information on theψj ’s and we are not usingit. We are also estimating the VAR by OLS and correcting the variance-covariance matrix insteadof doing a GLS-type estimation. These two sources of inefficiencies could potentially be overcomebut it would lead to less user-friendly procedures.

The asymptotic distribution provided by Proposition4.1, may not be very reliable in finite sam-ples, especially if we consider a VAR system with a large number of variables and/or lags. Due toautocorrelation, a larger horizon may also affect the size an power of the test. So an alternative tousing the asymptotic distribution chi-square ofW[H0(h)], consists in using Monte Carlo test tech-niques [see Dufour (2002)] or bootstrap methods [see, for example, Paparoditis (1996), Paparoditisand Streitberg (1991), Kilian (1998a, 1998b)]. In view of the fact that the asymptotic distribution ofW[H0(h)] is nuisance-parameter-free, such methods yield asymptotically valid tests when appliedto W[H0(h)] and typically provide a much better control of test level in finite samples. It is alsopossible that using better estimates would improve size control, although this is not clear, for impor-tant size distortions can occur in multivariate regressions even when unbiased efficient estimatorsare available [see, for example, Dufour and Khalaf (2002)].

5. Causality tests based on nonstationary(p, h)-autoregressions

In this section, we study how the tests described in the previous section can be adjusted in order toallow for non-stationary possibly integrated processes. In particular, let us assume that

W (t) = µ(t) + η(t) , (5.1)

µ(t) = δ0 + δ1t + · · ·+ δqtq , η (t) =

p∑

k=1

πkη (t− k) + a (t) , (5.2)

t = 1, . . . , T, whereδ0, δ1, . . . , δq arem × 1 fixed vectors, and the processη (t) is at mostI(d)whered is an integer greater than or equal to zero. Typical values ford are0, 1 or 2. Note that theseassumptions allow for the presence (or the absence) of cointegration relationships.

12

Under the above assumptions, we can also write:

W (t) = γ0 + γ1t + · · ·+ γqtq +

p∑

k=1

πkW (t− k) + a (t) , t = 1, . . . , T, (5.3)

whereγ0, γ1, . . . , γq arem× 1 fixed vectors (which depend onδ0, δ1, . . . , δq, andπ1, . . . , πp);see Toda and Yamamoto (1995). Under the specification (5.3), we have:

W (t + h) = µ(h)(t) +p∑

k=1

π(h)k W (t + 1− k) + u(h) (t + h) , t = 0, . . . , T − h. (5.4)

whereµ(h)(t) = γ(h)0 + γ

(h)1 t + · · ·+ γ

(h)q tq andγ

(h)0 , γ

(h)1 , . . . , γ

(h)q arem× 1 fixed vectors. For

h = 1, this equation is identical with (5.3). Forh ≥ 2, the errorsu(h) (t + h) follow a MA(h− 1)process as opposed to being i.i.d. . For any integerj, we have:

W (t + h) = µ(h)(t) +p∑

k=1k 6=j

π(h)k

[W (t + 1− k)−W (t + 1− j)

]

+

(p∑

k=1

π(h)k

)W (t + 1− j) + u(h) (t + h) , (5.5)

W (t + h)−W (t + 1− j) = µ(h)(t) +p∑

k=1k 6=j

π(h)k

[W (t + 1− k)−W (t + 1− j)

]

−(Im −

p∑

k=1

π(h)k

)W (t + 1− j) + u(h) (t + h) , (5.6)

for t = 0, . . . , T − h . The two latter expressions can be viewed as extensions to(p, h)-autoregressions of the representations used by Dolado and Lütkepohl (1996, pp. 372-373) forVAR(p) processes. Further, on takingj = p + 1 in (5.6), we see that

W (t + h)−W (t− p) = µ(h)(t) +p∑

k=1

A(h)k ∆W (t + 1− k)

+B(h)p W (t− p) + u(h) (t + h) (5.7)

where∆W (t) = W (t) −W (t − 1), A(h)k =

∑kj=1 π

(h)k ,andB

(h)k = A

(h)k − Im . Equation (5.7)

may be interpreted as an error-correction form at the horizonh,with baseW (t− p).

13

Let us now consider the extended autoregression

W (t + h) = µ(h)(t) +p∑

k=1

π(h)k W (t + 1− k) +

p+d∑

k=p+1

π(h)k W (t + 1− k) + u(h) (t + h) , (5.8)

t = d, . . . , T−h. Under model (5.3), the actual values of the coefficient matricesπ(h)p+1, . . . , π

(h)p+d

are equal to zero(π(h)p+1 = · · · = π

(h)p+d = 0), but we shall estimate the(p, h)-autoregressions

without imposing any restriction onπ(h)p+1, . . . , π

(h)p+d.

Now, suppose the processη (t) is eitherI(0) or I(1), and we taked = 1 in (5.8). Then, onreplacingp by p + 1 and settingj = p in the representation (5.6), we see that

W (t + h)−W (t− p− 1) = µ(h)(t) +p∑

k=1

π(h)k

[W (t + 1− k)−W (t− p− 1)

]

−B(h)p+1 W (t− p− 1) + u(h) (t + h) , (5.9)

whereB(h)p+1 =

(Im −

∑p+1k=1 π

(h)k

). In the latter equation,π(h)

1 , . . . , π(h)p all affect trend-stationary

variables (in an equation where a trend is included along with the other coefficients). Using argu-ments similar to those of Sims et al. (1990), Park and Phillips (1989) and Dolado and Lütkepohl(1996), it follows that the estimates ofπ

(h)1 , . . . , π

(h)p based on estimating (5.9) by ordinary least

squares (without restrictingB(h)p+1 ) _ or, equivalently, those obtained from (5.8) without restricting

π(h)p+1 _ are asymptotically normal with the same asymptotic covariance matrix as the one obtained

for a stationary process of the type studied in section 4.2 Consequently, the asymptotic distributionof the statisticW[H(h)

j9i] for testing the null hypothesisH(h)j9i of non-causality at horizonh from wj

to wi

(wj 9

hwi | I(j)

), based on estimating(5.8), is χ2(p). When computingH(h)

j9i as defined in

(4.3), it is important that only the coefficients ofπ(h)1 , . . . , π

(h)p are restricted (but notπ(h)

p+1).If the processη (t) is integrated up to orderd, whered ≥ 0, we can proceed similarly and addd

extra lags to the VAR process studied. Again, the null hypothesis is tested by considering the restric-tions entailed onπ(h)

1 , . . . , π(h)p . Further, in view of the fact the test statistics are asymptotically

pivotal under the null hypothesis, it is straightforward to apply bootstrap methods to such statistics.Note finally that the precision of the VAR estimates in such augmented regressions may eventuallybe improved with respect to the OLS estimates considered here by applying bias corrections suchas those proposed by Kurozumi and Yamamoto (2000)]. Adapting and applying such corrections to(p, h)-autoregressions would go beyond the scope of the present paper.

2For related results, see also Choi (1993), Toda and Yamamoto (1995), Yamamoto (1996), Yamada and Toda (1998),and Kurozumi and Yamamoto (2000).

14

Table 1. Rejection frequencies using the asymptotic distribution and the bootstrap procedurea) i.i.d. Gaussian sequence

h = 1 2 3 4 5 6 7 8 9 10 11 12

Asymptotic5% level 27.0 27.8 32.4 36.1 35.7 42.6 47.9 48.5 51.0 55.7 59.7 63.610% level 37.4 39.4 42.2 46.5 47.8 52.0 58.1 59.3 60.3 66.3 69.2 72.5

Bootstrap5% level 5.5 5.7 4.7 6.5 4.0 5.1 5.5 3.9 4.7 6.1 5.2 3.810% level 10.0 9.1 10.1 10.9 9.6 10.6 10.2 9.4 9.5 10.9 10.3 8.9

b) VAR(16) without causality up to horizonh

Asymptotic5% level 24.1 27.9 35.8 37.5 55.9 44.3 52.3 55.9 54.1 60.1 62.6 72.010% level 35.5 38.3 46.6 47.2 65.1 55.0 64.7 64.6 64.8 69.8 72.0 79.0

Bootstrap5% level 6.0 5.1 3.8 6.1 4.6 4.7 4.4 4.5 4.3 6.3 4.9 5.810% level 9.8 8.8 8.7 10.4 10.3 9.9 8.7 7.4 10.3 11.1 9.3 9.7

6. Empirical illustration

In this section, we present an application of these causality tests at various horizons to macroeco-nomic time series. The data set considered is the one used by Bernanke and Mihov (1998) in orderto study United States monetary policy. The data set considered consists of monthly observationson nonborrowed reserves (NBR, also denotedw1), the federal funds rate(r, w2), the GDP deflator(P, w3) and real GDP(GDP, w4). The monthly data on GDP and GDP deflator were constructedby state space methods from quarterly observations [see Bernanke and Mihov (1998) for more de-tails]. The sample goes from January 1965 to December 1996 for a total of 384 observations. Inwhat follows, all the variables were first transformed by a logarithmic transformation.

Before performing the causality tests, we must specify the order of the VAR model. First, inorder to get apparently stationary time series, all variables were transformed by taking first differ-ences of their logarithms. In particular, for the federal funds rate, this helped to mitigate the effectsof a possible break in the series in the years 1979-1981.3 Starting with 30 lags, we then tested thehypothesis ofK lags versusK + 1 lags using the LR test presented in Tiao and Box (1981). Thisled to a VAR(16) model. Tests of a VAR(16) against a VAR(K) for K = 17, . . . , 30 also failed toreject the VAR(16) specification, and the AIC information criterion [see McQuarrie and Tsai (1998,chapter 5)] is minimized as well by this choice. Calculations were performed using the Ox program(version 3.00) working on Linux [see Doornik (1999)].

3Bernanke and Mihov (1998) performed tests for arbitrary break points, as in Andrews (1993), and did not findsignificant evidence of a break point. They considered a VAR(13) with two additional variables (total bank reserves andDow-Jones index of spot commodity prices and they normalize both reserves by a 36-month moving average of totalreserves.)

15

Vector autoregressions of orderp at horizonh were estimated as described in section 4 andthe matrixV (NW )

ip , required to obtain covariance matrices, were computed using formula (4.7) withm(T )−1 = h−1.4 On looking at the values of the test statistics and their correspondingp-values atvarious horizons it quickly becomes evident that theχ2(q) asymptotic approximation of the statisticW in equation (4.6) is very poor. As a simple Monte Carlo experiment, we replaced the data bya 383 × 4 matrix of random draw from anN(0, 1), ran the same tests and looked at the rejectionfrequencies over 1000 replications using the asymptotic critical value. The results are in Table 1a.We see important size distortions even for the tests at horizon 1 where there is no moving averagepart.

We next illustrate that the quality of the asymptotic approximation is even worse when we moveaway from an i.i.d. Gaussian setup to a more realistic case. We now take as the DGP the VAR(16)estimated with our data in first difference but we impose that some coefficients are zero such that thefederal funds rate does not causeGDP up to horizonh and then we test ther 9

hGDP hypothesis.

The constraints of non-causality fromj to i up to horizonh that we impose are:

πijl = 0 for 1 ≤ l ≤ p , (6.1)

πikl = 0 for 1 ≤ l ≤ h, 1 ≤ k ≤ m. (6.2)

Rejection frequencies for this case are given in Table 1b.In light of these results we computed thep-values by doing a parametric bootstrap,i.e. doing

an asymptotic Monte Carlo test based on a consistent point estimate [see Dufour (2002)]. Theprocedure to test the hypothesiswj 9

hwi | I(j) is the following.

1. An unrestricted VAR(p) model is fitted for the horizon one, yielding the estimatesΠ(1) andΩ for Π(1) andΩ .

2. An unrestricted(p, h)-autoregression is fitted by least squares, yielding the estimateΠ(h) ofΠ(h).

3. The test statisticW for testing noncausality at the horizonh from wj to wi

[H

(h)j9i : wj 9

h

wi | I(j)

]is computed. We denote byW(h)

j9i(0) the test statistic based on the actual data.

4. N simulated samples from (2.8) are drawn by Monte Carlo methods, usingΠ(h) = Π(h) andΩ = Ω [and the hypothesis thata(t) is Gaussian]. We impose the constraints of non-causality,

π(h)ijk = 0, k = 1, . . . , p. Estimates of the impulse response coefficients are obtained from

Π(1) through the relations described in equation (2.5). We denote byW(h)j9i(n) the test statis-

tic for H(h)j9i based on then-th simulated sample(1 ≤ n ≤ N).

4The covariance estimator used here is relatively simple and exploits the truncation property (3.6). In view of the vastliterature on HAC estimators [see Den Haan and Levin (1997) and Cushing and McGarvey (1999)], several alternativeestimators forVip could have been considered (possibly allowing for alternative assumptions on the innovation distribu-tion). It would certainly be of interest to compare the performances of alternative covariance estimators, but this wouldlead to a lengthy study, beyond the scope of the present paper.

16

5. The simulatedp-valuepN [W(h)j9i(0)] is obtained, where

pN [x] =

1 +

N∑

n=1

I[W(h)j9i(n)− x]

/(N + 1) ,

I[z] = 1 if z ≥ 0 andI[z] = 0 if z < 0.

6. The null hypothesisH(h)j9i is rejected at levelα if pN [W(0)

j9i(h)] ≤ α.

From looking at the results in Table 1, we see that we get a much better size control by usingthis bootstrap procedure. The rejection frequencies over 1000 replications (withN = 999) are veryclose to the nominal size. Although the coefficientsψj ’s are functions of theπi’s we do not constrain

them in the bootstrap procedure because there is no direct mapping fromπ(h)k to πk andψj . This

certainly produces a power loss but the procedure remains valid because theψj ’s are computed withtheπk, which are consistent estimates of the trueπk both under the null and alternative hypothesis.To illustrate that our procedure has power for detecting departure from the null hypothesis of non-causality at a given horizon we ran the following Monte Carlo experiment. We again took a VAR(16)fitted on our data in first differences and we imposed the constraints (6.1) - (6.2) so that there wasno causality fromr to GDP up to horizon 12 (DGP under the null hypothesis). Next the value ofone coefficient previously set to zero was changed to induce causality fromr to GDP at horizons4 and higher:π3(1, 3) = θ . As θ increases from zero to one the strength of the causality fromrto GDP is higher. Under this setup, we could compute the power of our simulated test procedureto reject the null hypothesis of non-causality at a given horizon. In Figure 1, the power curves areplotted as a function ofθ for the various horizons. The level of the tests was controlled through thebootstrap procedure. In this experiment we took againN = 999 and we did 1000 simulations. Asexpected, the power curves are flat at around 5% for horizons one to three since the null is true forthese horizons. For horizons four and up we get the expected result that power goes up asθ movesfrom zero to one, and the power curves gets flatter as we increase the horizon.

Now that we have shown that our procedure does have power we present causality tests athorizon one to 24 for every pair of variables in tables 2 and 3. For every horizon we have twelvecausality tests and we group them by pairs. When we say that a given variable cause or does notcause another, it should be understood that we mean the growth rate of the variables. Thep-valuesare computed by takingN = 999. Table 4 summarize the results by presenting the significantresults at the 5% and 10% level.

The first thing to notice is that we have significant causality results at short horizons for somepairs of variables while we have it at longer horizons for other pairs. This is an interesting illustrationof the concept of causality at horizonh of Dufour and Renault (1998).

The instrument of the central bank, the nonborrowed reserves, cause the federal funds rate athorizon one, the prices at horizon 1, 2, 3 and 9 (10% level). It does not cause the other two variablesat any horizon and except the GDP at horizon 12 and 16 (10% level) nothing is causing it. We seethat the impact of variations in the nonborrowed reserves is over a very short term. Another variable,the GDP, is also causing the federal funds rates over short horizons (one to five months).

17

0.0 0.5 1.0

50

100horizon 1

0.0 0.5 1.0

50

100horizon 2

0.0 0.5 1.0

50

100horizon 3

0.0 0.5 1.0

50

100horizon 4

0.0 0.5 1.0

50

100horizon 5

0.0 0.5 1.0

50

100horizon 6

0.0 0.5 1.0

50

100horizon 7

0.0 0.5 1.0

50

100horizon 8

0.0 0.5 1.0

50

100horizon 9

0.0 0.5 1.0

50

100horizon 10

0.0 0.5 1.0

50

100horizon 11

0.0 0.5 1.0

50

100horizon 12

Figure 1. Power of the test at the 5% level for given horizons.The abscissa (x axis) represents the values ofθ.

18

Tabl

e2.

Cau

salit

yte

sts

and

sim

ulat

edp-va

lues

for

serie

sin

first

diffe

renc

es(o

flog

arith

m)

for

the

horiz

ons

1to

12.

p-v

alue

sar

ere

port

edin

pare

nthe

sis.

h1

23

45

67

89

1011

12N

BR

9r

38.5

205

26.3

851

24.3

672

22.5

684

24.2

294

27.0

748

21.4

347

17.2

164

21.8

217

18.4

775

18.5

379

19.6

482

(0.0

41)

(0.2

40)

(0.3

27)

(0.3

95)

(0.3

79)

(0.3

13)

(0.5

50)

(0.7

99)

(0.6

03)

(0.7

80)

(0.8

24)

(0.7

89)

r9

NB

R22

.539

020

.862

117

.235

717

.422

217

.894

418

.646

225

.205

940

.989

641

.688

238

.165

641

.220

336

.127

8(0

.386

)(0

.467

)(0

.706

)(0

.738

)(0

.734

)(0

.735

)(0

.495

)(0

.115

)(0

.142

)(0

.214

)(0

.206

)(0

.346

)N

BR

9P

50.5

547

45.2

498

49.1

408

33.8

545

33.8

943

35.3

923

39.1

767

40.0

218

36.4

089

39.0

916

28.4

933

26.5

439

(0.0

04)

(0.0

14)

(0.0

09)

(0.1

21)

(0.1

62)

(0.1

57)

(0.0

99)

(0.1

41)

(0.2

34)

(0.2

02)

(0.4

85)

(0.6

09)

P9

NB

R17

.069

618

.950

416

.688

021

.238

128

.726

420

.035

918

.229

123

.670

423

.341

927

.842

730

.917

136

.515

9(0

.689

)(0

.601

)(0

.741

)(0

.550

)(0

.307

)(0

.664

)(0

.771

)(0

.596

)(0

.649

)(0

.492

)(0

.462

)(0

.357

)N

BR

9G

DP

27.8

029

25.0

122

25.5

123

23.6

799

15.0

040

17.5

748

16.6

781

19.6

643

29.9

020

34.0

930

32.3

917

34.8

183

(0.1

84)

(0.3

02)

(0.2

94)

(0.3

93)

(0.8

30)

(0.7

57)

(0.7

90)

(0.7

46)

(0.3

68)

(0.3

00)

(0.3

70)

(0.3

57)

GD

P9

NB

R17

.633

820

.656

824

.533

418

.322

018

.312

332

.874

633

.997

939

.670

140

.735

626

.342

440

.226

253

.348

2(0

.644

)(0

.520

)(0

.348

)(0

.670

)(0

.682

)(0

.211

)(0

.242

)(0

.145

)(0

.161

)(0

.551

)(0

.216

)(0

.092

)r

9P

32.2

481

32.9

207

32.0

362

25.0

124

25.2

441

25.8

110

29.2

553

29.6

021

36.0

771

43.2

116

30.2

530

20.8

982

(0.1

04)

(0.1

08)

(0.1

38)

(0.3

42)

(0.3

83)

(0.3

94)

(0.3

28)

(0.3

51)

(0.2

41)

(0.1

38)

(0.4

22)

(0.7

63)

P9

r22

.438

516

.445

514

.307

314

.293

214

.014

816

.713

811

.559

916

.573

113

.069

714

.575

915

.031

724

.407

7(0

.413

)(0

.670

)(0

.790

)(0

.826

)(0

.844

)(0

.774

)(0

.951

)(0

.809

)(0

.936

)(0

.909

)(0

.899

)(0

.659

)r

9G

DP

26.3

362

28.6

645

35.9

768

38.8

680

37.6

788

39.8

145

64.1

500

80.2

396

89.6

998

101.

4143

105.

9138

110.

3551

(0.2

62)

(0.1

59)

(0.0

73)

(0.0

59)

(0.0

82)

(0.0

79)

(0.0

06)

(0.0

02)

(0.0

02)

(0.0

02)

(0.0

01)

(0.0

03)

GD

P9

r42

.532

743

.007

849

.836

641

.108

241

.794

536

.096

529

.547

225

.789

827

.980

739

.636

639

.047

740

.154

5(0

.016

)(0

.017

)(0

.005

)(0

.033

)(0

.044

)(0

.109

)(0

.263

)(0

.429

)(0

.366

)(0

.148

)(0

.178

)(0

.205

)P

9G

DP

20.6

903

24.1

099

27.4

106

23.3

585

22.9

095

18.5

543

20.8

172

23.6

942

30.5

340

28.8

286

24.9

477

25.7

552

(0.4

95)

(0.3

22)

(0.2

33)

(0.4

23)

(0.4

97)

(0.7

13)

(0.6

39)

(0.5

55)

(0.3

75)

(0.4

14)

(0.6

12)

(0.6

29)

GD

P9

P24

.336

824

.492

524

.612

522

.816

026

.790

040

.082

536

.485

549

.616

146

.257

436

.319

726

.852

024

.011

3(0

.329

)(0

.365

)(0

.380

)(0

.470

)(0

.348

)(0

.081

)(0

.157

)(0

.058

)(0

.072

)(0

.262

)(0

.540

)(0

.666

)

19

Tabl

e3.

Cau

salit

yte

sts

and

sim

ulat

edp-va

lues

for

serie

sin

first

diffe

renc

es(o

flog

arith

m)

for

the

horiz

ons

13to

24

h13

1415

1617

1819

2021

2223

24N

BR

9r

20.7

605

21.1

869

18.6

062

17.6

750

22.2

838

34.0

098

28.7

769

33.7

855

66.1

538

38.7

272

28.8

194

34.8

015

(0.7

83)

(0.7

71)

(0.8

52)

(0.9

05)

(0.8

11)

(0.5

32)

(0.6

96)

(0.6

32)

(0.1

43)

(0.5

58)

(0.8

03)

(0.7

08)

r9

NB

R42

.292

443

.819

632

.811

328

.177

531

.232

631

.758

725

.694

234

.668

037

.747

051

.419

633

.289

727

.336

2(0

.245

)(0

.254

)(0

.533

)(0

.707

)(0

.625

)(0

.651

)(0

.832

)(0

.682

)(0

.625

)(0

.406

)(0

.755

)(0

.869

)N

BR

9P

38.0

451

39.0

293

43.2

101

37.9

049

22.4

428

20.0

337

38.8

919

66.1

541

55.0

460

64.8

568

54.8

929

42.2

509

(0.3

36)

(0.3

51)

(0.3

15)

(0.4

13)

(0.8

35)

(0.8

83)

(0.4

88)

(0.1

49)

(0.2

55)

(0.1

80)

(0.3

32)

(0.5

62)

P9

NB

R29

.545

729

.454

327

.242

528

.754

731

.485

144

.125

436

.031

947

.160

850

.274

345

.807

854

.196

156

.786

5(0

.565

)(0

.667

)(0

.688

)(0

.703

)(0

.627

)(0

.400

)(0

.581

)(0

.427

)(0

.379

)(0

.486

)(0

.401

)(0

.399

)N

BR

9G

DP

30.3

906

25.2

694

29.7

274

23.3

347

16.7

784

18.7

922

26.8

942

30.2

550

39.0

652

43.6

661

56.9

724

67.7

764

(0.4

93)

(0.7

16)

(0.6

03)

(0.8

14)

(0.9

42)

(0.9

01)

(0.7

52)

(0.7

45)

(0.5

45)

(0.5

05)

(0.3

14)

(0.2

51)

GD

P9

NB

R44

.226

441

.579

559

.330

165

.764

753

.357

950

.464

551

.516

343

.113

240

.705

136

.396

038

.716

240

.846

6(0

.219

)(0

.292

)(0

.106

)(0

.086

)(0

.225

)(0

.282

)(0

.311

)(0

.466

)(0

.513

)(0

.659

)(0

.665

)(0

.644

)r

9P

27.3

588

37.7

170

37.2

499

37.1

005

27.3

052

35.4

128

39.2

469

53.2

675

57.1

530

54.6

753

69.6

377

59.2

184

(0.5

88)

(0.3

41)

(0.3

81)

(0.4

43)

(0.7

00)

(0.5

34)

(0.4

77)

(0.2

66)

(0.2

29)

(0.3

14)

(0.1

64)

(0.2

93)

P9

r22

.372

026

.358

828

.493

023

.015

226

.421

136

.363

234

.392

217

.426

916

.965

231

.933

628

.837

730

.001

3(0

.750

)(0

.608

)(0

.633

)(0

.794

)(0

.747

)(0

.512

)(0

.536

)(0

.956

)(0

.958

)(0

.715

)(0

.801

)(0

.795

)r

9G

DP

123.

8280

80.3

638

95.0

727

83.2

773

76.6

169

84.1

068

76.8

979

81.3

505

65.5

025

71.3

334

63.1

942

64.8

020

(0.0

01)

(0.0

18)

(0.0

05)

(0.0

18)

(0.0

33)

(0.0

40)

(0.0

55)

(0.0

73)

(0.1

61)

(0.1

53)

(0.2

41)

(0.2

86)

GD

P9

r35

.708

633

.020

040

.171

330

.322

720

.558

219

.114

417

.738

622

.641

038

.817

538

.711

038

.998

725

.057

7(0

.319

)(0

.420

)(0

.273

)(0

.545

)(0

.838

)(0

.898

)(0

.930

)(0

.856

)(0

.515

)(0

.565

)(0

.549

)(0

.860

)P

9G

DP

9.42

909.

9870

12.1

148

15.1

682

14.2

883

21.2

701

29.0

528

47.5

841

66.5

988

59.2

137

71.5

165

67.1

851

(0.9

88)

(0.9

94)

(0.9

74)

(0.9

47)

(0.9

70)

(0.8

68)

(0.7

08)

(0.3

63)

(0.1

62)

(0.2

56)

(0.1

65)

(0.2

31)

GD

P9

P29

.764

237

.009

533

.467

642

.219

031

.157

352

.375

751

.956

740

.379

031

.659

854

.068

465

.828

453

.082

3(0

.521

)(0

.351

)(0

.470

)(0

.328

)(0

.605

)(0

.238

)(0

.226

)(0

.459

)(0

.675

)(0

.244

)(0

.215

)(0

.342

)

20

Table 4. Summary of causality relations at various horizons for series in first difference

h 1 2 3 4 5 6 7 8 9 10 11 12NBR 9 r ??

r 9 NBR

NBR 9 P ?? ?? ?? ?P 9 NBR

NBR 9 GDPGDP 9 NBR ?

r 9 PP 9 r

r 9 GDP ? ? ? ? ?? ?? ?? ?? ?? ??GDP 9 r ?? ?? ?? ?? ??

P 9 GDPGDP 9 P ? ? ?

h 13 14 15 16 17 18 19 20 21 22 23 24NBR 9 r

r 9 NBR

NBR 9 PP 9 NBR

NBR 9 GDPGDP 9 NBR ?

r 9 PP 9 r

r 9 GDP ?? ?? ?? ?? ?? ?? ? ?GDP 9 r

P 9 GDPGDP 9 P

Note _ The symbols? and?? indicate rejection of the non-causality hypothesis at the 10% and 5%levels respectively.

An interesting result is the causality from the federal funds rate to the GDP. Over the first fewmonths the funds rate does not cause GDP, but from horizon 3 (up to 20) we do find significantcausality. This result can easily be explained by, e.g. the theory of investment. Notice that wehave the following indirect causality. Nonborrowed reserves do not cause GDP directly over anyhorizon, but they cause the federal funds rate which in turn causes GDP. Concerning the observationthat there are very few causality results for long horizons, this may reflect the fact that, for stationaryprocesses, the coefficients of prediction formulas converge to zero as the forecast horizon increases.

Using the results of Proposition 4.5 in Dufour and Renault (1998), we know that for this examplethe highest horizon that we have to consider is 33 since we have a VAR(16) with four time series.Causality tests for the horizons 25 through 33 were also computed but are not reported. Somep-values smaller or equal to 10% are scattered over horizons 30 to 33 but no discernible patternemerges.

We next consider extended autoregressions to illustrate the results of section 5. To cover thepossibility that the first difference of the logarithm of the four series may not be stationary, we ranextended autoregressions on the series analyzed. Since we used a VAR(16) with non-zero mean for

21

the first difference of the series a VAR(17),i.e. d = 1, with a non-zero mean was fitted. The MonteCarlo samples withN = 999 are drawn in the same way as before except that the constraints on theVAR parameters at horizonh is π

(h)jik = 0 for k = 1, . . . , p and notk = 1, . . . , p + d.

Results of the extended autoregressions are presented in Table 5 (horizons 1 to 12) and 6 (hori-zons 13 to 24). Table 7 summarize these results by presenting the significant results at the 5% and10% level. These results are very similar to the previous ones over all the horizons and variableevery pairs. A few causality tests are not significant anymore (GDP 9 r at horizon 5,r 9 GDPat horizons 5 and 6) and some causality relations are now significant (r 9 P at horizon one) but webroadly have the same causality patterns.

7. Conclusion

In this paper, we have proposed a simple linear approach to the problem of testing non-causalityhypotheses at various horizons in finite-order vector autoregressive models. The methods describedallow for both stationary (or trend-stationary) processes and possibly integrated processes (whichmay involve unspecified cointegrating relationships), as long as an upper bound is set on the orderof integration. Further, we have shown that these can be easily implemented in the context of afour-variable macroeconomic model of the U.S. economy.

Several issues and extensions of interest warrant further study. The methods we have proposedwere, on purpose, designed to be relatively simple to implement. This may, of course, involve ef-ficiency losses and leave room for improvement. For example, it seems quite plausible that moreefficient tests may be obtained by testing directly the nonlinear causality conditions described inDufour and Renault (1998) from the parameter estimates of the VAR model. However, such proce-dures will involve difficult distributional problems and may not be as user-friendly as the proceduresdescribed here. Similarly, in nonstationary time series, information about integration order and thecointegrating relationships may yield more powerful procedures, although at the cost of complexity.These issues are the topics of on-going research.

Another limitation comes from the fact we consider VAR models with a known finite order. Weshould however note that the asymptotic distributional results established in this paper continue tohold as long as the orderp of the model is selected according to a consistent order selection rule [seeDufour, Ghysels and Hall (1994), Pötscher (1991)]. So this is not an important restriction. Otherproblems of interest would consist in deriving similar tests applicable in the context of VARMAor VARIMA models, as well as more general infinite-order vector autoregressive models, usingfinite-order VAR approximations based on data-dependent truncation rules [such as those used byLütkepohl and Poskitt (1996) and Lütkepohl and Saikkonen (1997)]. These problems are also thetopics of on-going research.

22

Tabl

e5.

Cau

salit

yte

sts

and

sim

ulat

edp-va

lues

for

exte

nded

auto

regr

essi

ons

atth

eho

rizon

s1

to12

h1

23

45

67

89

1011

12N

BR

9r

37.4

523

24.0

148

24.9

365

22.9

316

25.4

508

26.6

763

19.5

377

19.3

805

21.9

278

19.8

541

21.2

947

19.5

569

(0.0

51)

(0.3

37)

(0.3

09)

(0.4

10)

(0.3

33)

(0.3

33)

(0.6

60)

(0.6

88)

(0.6

09)

(0.7

16)

(0.7

10)

(0.7

95)

r9

NB

R25

.918

517

.797

716

.618

517

.382

019

.642

520

.203

239

.849

644

.017

243

.992

839

.421

736

.780

235

.088

3(0

.281

)(0

.650

)(0

.747

)(0

.759

)(0

.673

)(0

.676

)(0

.129

)(0

.075

)(0

.130

)(0

.207

)(0

.286

)(0

.375

)N

BR

9P

50.8

648

44.8

472

56.5

028

33.8

112

36.1

026

42.1

714

38.7

204

38.8

076

35.3

353

33.8

049

28.6

205

26.9

863

(0.0

04)

(0.0

15)

(0.0

07)

(0.1

52)

(0.1

24)

(0.0

61)

(0.1

52)

(0.1

61)

(0.2

59)

(0.3

16)

(0.4

98)

(0.6

09)

P9

NB

R20

.749

117

.277

216

.389

126

.161

029

.732

918

.758

322

.782

123

.174

327

.007

624

.376

331

.186

938

.334

9(0

.506

)(0

.704

)(0

.764

)(0

.339

)(0

.276

)(0

.730

)(0

.605

)(0

.629

)(0

.508

)(0

.655

)(0

.484

)(0

.336

)N

BR

9G

DP

27.2

102

23.8

072

25.9

561

24.3

384

14.4

893

17.8

928

15.9

036

17.9

592

30.7

472

33.5

333

33.7

687

36.3

343

(0.2

24)

(0.3

46)

(0.3

17)

(0.4

02)

(0.8

37)

(0.7

32)

(0.8

21)

(0.8

17)

(0.3

68)

(0.3

33)

(0.3

55)

(0.3

48)

GD

P9

NB

R16

.132

219

.447

120

.679

819

.103

529

.122

936

.105

337

.119

440

.357

843

.783

537

.733

748

.300

452

.244

2(0

.746

)(0

.571

)(0

.537

)(0

.658

)(0

.269

)(0

.166

)(0

.167

)(0

.163

)(0

.138

)(0

.247

)(0

.125

)(0

.123

)r

9P

32.5

147

31.2

909

25.6

717

24.4

449

21.7

640

25.4

747

27.6

313

31.4

581

43.2

611

38.2

020

30.6

394

19.9

687

(0.1

00)

(0.1

28)

(0.3

52)

(0.3

90)

(0.5

68)

(0.3

97)

(0.3

53)

(0.3

11)

(0.1

11)

(0.2

12)

(0.4

20)

(0.8

12)

P9

r22

.737

415

.945

315

.200

115

.193

316

.233

415

.747

213

.419

615

.250

614

.432

415

.858

921

.363

721

.994

9(0

.415

)(0

.704

)(0

.762

)(0

.790

)(0

.768

)(0

.830

)(0

.905

)(0

.859

)(0

.909

)(0

.887

)(0

.749

)(0

.731

)r

9G

DP

27.0

435

29.5

913

37.5

271

35.7

130

34.9

901

35.1

715

79.9

402

92.6

009

94.9

068

107.

6638

108.

0581

138.

0570

(0.2

44)

(0.1

58)

(0.0

61)

(0.0

94)

(0.1

17)

(0.1

64)

(0.0

02)

(0.0

01)

(0.0

04)

(0.0

01)

(0.0

03)

(0.0

01)

GD

P9

r41

.847

541

.944

944

.759

738

.035

830

.477

628

.184

030

.586

727

.074

526

.587

639

.950

239

.861

833

.485

5(0

.032

)(0

.019

)(0

.019

)(0

.060

)(0

.209

)(0

.286

)(0

.250

)(0

.369

)(0

.431

)(0

.157

)(0

.181

)(0

.347

)P

9G

DP

23.7

424

26.7

148

24.6

605

24.6

507

23.3

233

19.8

483

20.3

581

28.9

318

29.0

384

27.2

509

26.3

318

22.6

991

(0.3

68)

(0.2

37)

(0.3

42)

(0.3

73)

(0.4

79)

(0.6

68)

(0.6

66)

(0.3

76)

(0.4

27)

(0.5

05)

(0.5

58)

(0.7

40)

GD

P9

P25

.126

424

.194

125

.568

319

.212

737

.598

438

.031

837

.625

445

.221

945

.245

836

.991

223

.168

722

.993

4(0

.278

)(0

.358

)(0

.350

)(0

.639

)(0

.108

)(0

.110

)(0

.153

)(0

.088

)(0

.092

)(0

.237

)(0

.647

)(0

.698

)

23

Tabl

e6.

Cau

salit

yte

sts

and

sim

ulat

edp-va

lues

for

exte

nded

auto

regr

essi

ons

atth

eho

rizon

s13

to24

h13

1415

1617

1819

2021

2223

24N

BR

9r

19.5

845

30.2

847

17.4

893

24.9

745

21.2

814

27.4

800

36.9

567

27.8

970

59.3

731

34.9

153

27.3

241

31.9

143

(0.8

14)

(0.5

13)

(0.9

01)

(0.7

50)

(0.8

58)

(0.7

06)

(0.5

37)

(0.7

70)

(0.2

36)

(0.6

40)

(0.8

32)

(0.7

71)

r9

NB

R41

.736

047

.850

130

.961

329

.180

935

.765

422

.996

629

.086

929

.139

638

.067

552

.551

229

.613

325

.487

2(0

.282

)(0

.212

)(0

.592

)(0

.692

)(0

.540

)(0

.847

)(0

.757

)(0

.766

)(0

.665

)(0

.401

)(0

.838

)(0

.917

)N

BR

9P

36.7

787

37.7

357

33.4

273

35.3

241

20.6

330

23.6

911

44.5

735

55.5

420

53.7

340

68.0

270

52.3

332

47.5

614

(0.3

59)

(0.3

66)

(0.5

12)

(0.4

55)

(0.8

60)

(0.8

25)

(0.3

79)

(0.2

49)

(0.2

90)

(0.1

65)

(0.3

63)

(0.4

81)

P9

NB

R29

.504

939

.207

618

.083

130

.648

639

.551

734

.836

339

.960

843

.656

340

.671

344

.025

461

.691

462

.934

6(0

.582

)(0

.401

)(0

.920

)(0

.671

)(0

.441

)(0

.606

)(0

.535

)(0

.471

)(0

.551

)(0

.553

)(0

.296

)(0

.286

)N

BR

9G

DP

30.9

525

22.0

737

30.1

165

22.7

429

17.2

546

22.2

686

28.6

752

31.8

817

39.5

031

53.6

466

58.8

413

79.0

569

(0.5

01)

(0.8

22)

(0.5

99)

(0.7

93)

(0.9

37)

(0.8

63)

(0.7

49)

(0.7

17)

(0.5

91)

(0.3

67)

(0.3

27)

(0.1

53)

GD

P9

NB

R46

.453

838

.042

464

.226

960

.879

257

.579

857

.223

742

.085

143

.168

343

.437

941

.714

139

.829

241

.712

3(0

.186

)(0

.347

)(0

.071

)(0

.125

)(0

.169

)(0

.205

)(0

.453

)(0

.491

)(0

.501

)(0

.568

)(0

.631

)(0

.632

)r

9P

30.7

883

37.3

585

30.4

331

35.8

788

26.9

960

39.2

961

44.7

334

46.6

740

56.9

680

53.1

830

67.3

818

61.2

522

(0.5

03)

(0.3

64)

(0.5

66)

(0.4

48)

(0.7

12)

(0.4

29)

(0.3

58)

(0.3

52)

(0.2

23)

(0.3

10)

(0.1

82)

(0.2

41)

P9

r25

.308

533

.102

728

.736

233

.575

833

.899

139

.303

129

.422

814

.309

517

.858

829

.957

225

.034

729

.190

3(0

.654

)(0

.450

)(0

.624

)(0

.546

)(0

.525

)(0

.451

)(0

.707

)(0

.984

)(0

.957

)(0

.764

)(0

.883

)(0

.825

)r

9G

DP

109.

5052

84.9

556

88.4

433

80.9

836

79.9

549

75.1

199

94.8

986

65.5

929

67.2

913

71.6

331

75.5

123

67.4

315

(0.0

02)

(0.0

11)

(0.0

21)

(0.0

16)

(0.0

31)

(0.0

73)

(0.0

26)

(0.1

47)

(0.1

36)

(0.1

64)

(0.1

51)

(0.2

54)

GD

P9

r34

.818

541

.421

838

.276

128

.532

622

.611

616

.799

220

.809

731

.876

938

.708

334

.466

335

.627

920

.500

7(0

.340

)(0

.264

)(0

.318

)(0

.606

)(0

.809

)(0

.923

)(0

.866

)(0

.644

)(0

.519

)(0

.649

)(0

.637

)(0

.949

)P

9G

DP

8.80

398.

9511

17.0

182

9.76

0816

.477

219

.694

246

.324

041

.742

964

.592

860

.287

556

.031

572

.282

3(0

.995

)(0

.995

)(0

.933

)(0

.995

)(0

.923

)(0

.916

)(0

.349

)(0

.484

)(0

.207

)(0

.250

)(0

.332

)(0

.215

)G

DP

9P

23.8

773

39.5

163

34.3

832

34.5

734

36.8

350

54.8

088

54.8

048

36.4

102

29.4

543

58.0

095

58.1

341

59.5

283

(0.7

09)

(0.3

31)

(0.4

68)

(0.4

88)

(0.4

72)

(0.2

12)

(0.2

18)

(0.5

57)

(0.7

39)

(0.2

54)

(0.2

65)

(0.2

86)

24

Table 7. Summary of causality relations at various horizons for series in first difference withextended autoregressions

h 1 2 3 4 5 6 7 8 9 10 11 12NBR 9 r ?

r 9 NBR ?

NBR 9 P ?? ?? ?? ?P 9 NBR

NBR 9 GDPGDP 9 NBR

r 9 P ?P 9 r

r 9 GDP ? ? ?? ?? ?? ?? ?? ??GDP 9 r ?? ?? ?? ?

P 9 GDPGDP 9 P ? ?

h 13 14 15 16 17 18 19 20 21 22 23 24NBR 9 r

r 9 NBR

NBR 9 PP 9 NBR

NBR 9 GDPGDP 9 NBR ?

r 9 PP 9 r

r 9 GDP ?? ?? ?? ?? ?? ? ??GDP 9 r

P 9 GDPGDP 9 P

Note _ The symbols? and?? indicate rejection of the non-causality hypothesis at the 10% and 5%levels respectively.

25

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28