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Working Paper 8303
EXTENSION OF GRANGER CAUSALITY
IN MULTIVARIATE TIME SERIES MODELS
by Michael L. Bagshaw
Working papers of the Federal Reserve Bank of
Cleveland are preliminary materials,
circulated to stimulate discussion and
critical comment. The views stated herein
are the author's and not necessarily those of
the Federal Reserve Bank of Cleveland or of
the Board of Governors of the Federal Reserve
System.
August 1983
Federal Reserve Bank of Cleveland
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EXTENSION OF GRANGER CAUSALITY
IN MULTI.VARIATE TIME SERIES MODELS
Abstract
This paper presents an examp,le of a situation where Granger causality
does not exist but an extended definition of causality does.
The
extended definition of causality is discussed, along with methods to
determine its existence in multivariate time series models.
Key words:
Granger causality, multivariate time series.
I.
Introduction
The concept of Granger causality (Granger 1969) has become widely
used in discussing relationships among variables.
Some relevant
references to Granger causal ity are Sims (1972), Haugh (1972), Pierce
(1977), and Pierce and Haugh (1977).
Generally, Granger causal ity has
been discussed in terms of bivariate models.
This paper proposes an extension of Granger causality when more than
two variables are used in a multivariate time series model and it is
necessary to consider more than one-period-ahead forecasts.
Granger causality more appropriately may be called one-period-ahead
forecasting ability.
Variable z is said to "Granger cause" variable y,
with respect to a given information set that includes z and y, if
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-
2
-
forecasts of present y are more accurate when using past values of z than
when not doing so, all other available information being used in either
case. The measure of accliracy usually used in the definition o f Granger
causality is the mean square error of one-period-ahead forecasts.
This idea can be expressed as follows: Let
tion set (including at least zt and yt),
it
At
be the given informa-
=(AS:s < t),
u~(~~B)
be the minimum mean square error from forecasting y (one period ahead)
given the information set El, and At-zt be the set
it without z.
Then
z is said to Granger cause y if
Thus, Granger causality refers to only one-period-ahead forecasts.
When forecasting for more than one period ahead, it is necessary to
know whether Granger causal i ty would include a1 1 possible causal ity.
situations (in terms of forecasting ability).
In section I1 is an example
of a multivariate model for three variables (x, y, and z). This model
demonstrates that while z may not Granger cause y, the two-period-ahead
forecasts using z have a smaller forecast error than the forecasts not
using z. Thus, Granger causality does not encompass all situations where
one would conclude that some type of causality exists (in terms of
forecasting ability).
In section 11, we also introduce an extension of
Granger causality that includes the multiperiod, multivariate situation.
11. Extension of Granger Causality
When dealing with only two variables, Pierce (1975) proves that if
better L-period-ahead forecasts for any L > 2 are produced by the addition
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o f past z, then z must Granger cause y w i t h respect t o t h e s e t (zt, yt).
However, as t h e f o l l o w i n g example demonstrates, t h i s i s not necessarily
t r u e f o r systems w i t h more than two variables:
where t h e a,l
a2, and a3 are mutually orthogonal white noise processes
2
w i t h variances u,l
2
2
u2, u3, respectively.
In
t h i s model, At = (yt,
xt,
zt).
The minimum mean-square-error, one-period-ahead forecasts f o r t h i s
model are
where Gt
-
(1) i s the one-period-ahead f o r e c a s t o f w a t time t-1.
2
2
2
forecasts have mean square e r r o r s of u,l
These
u2, and u3, respectively.
(See
Tiao and Box (1981) f o r a general discussion o f how t o c a l c u l a t e
forecasts from these types o f models and t h e i r mean square errors. )
I f z i s n o t included i n the model, then t h e appropriate b i v a r i a t e
model can be derived from t h e model given i n equation (1) by matching t h e
variances, covariances, and cross covariances of y and x ' i m p l i e d by t h i s
model.
T h i s reduced model i s given by
where o 2 I 2 = 4:3<
+
2
u2.
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The minimum mean-square-error, one-period-ahead forecasts from t h i s
model are
w i t h mean square e r r o r s of
U
:
and 6E30:
+
u
,;
respectively.
We thus have shown t h a t z does n o t Granger cause y ( t h e mean-squareforecast e r r o r i s
('2
< '2303
+ a ) .
U
:
i n both cases), b u t t h a t z does Granger cause x
S i m i l a r l y , we can show t h a t x Granger causes y but n o t z.
Also, y does n o t Granger cause x o r z.
The c a u s a l i t y chain i s thus an
example of i n d i r e c t c a u s a l i t y (Tjostheim 1981) between
where
-
z and
y:
means Granger causes.
When we examine t h e two-period-ahead forecasts, t h e r e s u l t i s
different.
The two-period-ahead f o r e c a s t s from model 1 are
w i t h mean square e r r o r s of
0
:
+.4:2~g,
U:
+
4:3~$,
2
and u3, r e s p e c t i v e l y .
(See Tiao and Box (1981) f o r a general discussion of how t o c a l c u l a t e
mu1t iple- period f o r e c a s t s and t h e i r mean square e r r o r s . )
For model 3, t h e two-period-ahead f o r e c a s t s are given by
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-5-
2
+
with mean square forecast errors of :a + dl2(a;
2
respectively.
'23 a3) and 0; + ,:a3$
Thus, the mean square forecast error for y two periods ahead is less
2 2 than it is in the model not including
in the moiel including z ( a + d12a2)
2
2 2
2 2 2
This illustrates the principle that
z (al + d12a2
+
$12423a3).
even if a variable z does not Granger cause another variable y, z may be
useful in forecasting y more than one period ahead.
This motivates the following extension of Granger causality.
A
variable z is said to cause another variable y, with respect to the set
(6)
U ~ ( ~ ( ILA)
)
4 U ~ ( ~ ( L IA
)-)
for any L > 0,
where y ( ~ )is the L-period-ahead forecast of y at time t-1.
We call this
type of causality L-period causality, where L is the smallest value so
that inequality 6 is true. Thus, Granger causality is one-period
causal i ty.
The concept of L-period.causality is not the same as the idea that z
is related to y with an L-period lag.
Consider the model
In this model, y is related to z with .a j-period lag,
for some j > 1.
but z Granger causes y; the value of j is immaterial.
This also
illustrates the idea that Granger causality does not necessarily involve
only one-period lags.
.
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111. Determining Multiperiod Causality
Given a multivarjate time series model., we wish to determine what
patterns of causality are represented by the model. One method of doing
this (as demonstrated in the example in section 11) is to compare the
mean square forecast errors from the reduced models resulting from
deleting one and only one variable with those of the full model f.or
different forecast lengths. One advantage in ,doing this is to learn how
much the mean square errors change. For example, we saw in section I1
2 2 2
that the two-period-ahead mean square error for y was reduced by 412623a
when z was included in the given model.
.
.
Consider the following model for n variables:
where K (n-1 x n-1), J (n-1 x l ) , M (1 x n-1), and N (1 x 1) are
polynomial matrixes in B (where B is the backshift operator, i .e., Bvt
=
v t-1 );
and
E
W
is the vector of n-1 variables excluding v; and
TI
(n-1 x 1)
(1 x 1) are the corresponding error terms. If the variable v is
omitted from this model, the resulting model is given by
(See Quenouil le 1957, p. 43. ) The autoregressive operators wi 11 be given
by the right-hand side of equation (7).
The moving average operators
will have to be dete,rmined by combining the two sources of error n and
.
Once the submodels are determined, the mean square forecast errors
for the submodels for different forecast lengths can be compared with the
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corresponding quantities of the full model to determine L-period
causality. This method also could be used to determine the effect of
Granger causality on the mean square error for one-period-ahead
forecasts.
One disadvantage of this method is the number of submodels that must
be determined. In general, if there are n variables in the model, then
we must determine n-1 submodels, each of which has n-1 variables. The
determination of these submodels is difficult when there are more than
three variables in the model.
We now hypothesize an additional method of determining L-period
causality. The method is presented without proof, but it is intuitively
appealing. The hypothesis is that y is L-period caused by z for some L
if there exists a chain of Granger causali3es between z and y. If each
Granger causality in the system involves only one lag, then we
hypothesize that L is equal to the length of the ~inimum-lengthchain
between z and y. However, if some of the lags involved in the Granger
causality chain are longer than one, then L may be larger than this
minimum, depending on the position of the longer lags. For example,
consider the following model:
(8)
Yt = d12 't-j
x t = 423 zt-k
+
+
alt
a2t
zt = a3t.
In section 11, we saw that if j = k
=
1, then z two period causes y.
In general, it can be shown that for this model z (1
independently of k.
+
j) period causes y
Thus, the value of L depends not only on the length
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of the Granger causal chain but also on the lags involved and their
location in the chain.
In terms of forecasting ability, L-period causality occurs when each
variable in the chain is better forecast in the Granger sense using the
previous variable. The L-period forecast is thus a better forecast
because each variable in the chain is better forecast. That is, the
L-period forecast of y depends on the forecast of x, which depends on the
forecast of w,
... which depends on lagged z.
In the example of equation
(l), y is two-period caused by z, because the two-period-ahead forecast
of y depends on the one-period-ahead forecast of x, which depends on
lagged z.
The advantage of this method over the previous method is its ease of
use. Determination of Granger causal ity is fairly easy (Tjostheim
1981).
Once the Granger causalities are ascertained, it is trivial to
determine the chains of causalities. However, this method does not
provide an indication of how much the L-period-ahead mean square error is
reduced. Thus, we may have L-period causality with no practical
significance. This would probably be true when L becomes large.
IV. Summary
Because Granger causality is determined only in terms of one-period
forecasts, and because it is often necessary to forecast more than one
period ahead, an extension of Granger causality is necessary in
multivariate models.. We have presented an example that illustrates this
idea and a proposal for an extension of Granger causality that addresses
this problem. This extension involves L-period forecasting abi 1ity.
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That is, the ability to forecast a variable for L periods ahead is
improved by using another variable versus not using it in the same sense
of Granger causality for one period ahead.
We have provided methods for
determining L-period causality when a model is known.
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References
Granger, C.W.J.
I1InvestigatingCausal Relations by Econometric Models
and Cross-Spectral Methods," Econometrica, vol. 37 (1969), pp. 424-38.
Haugh, L.D. "The Identification of Time Series Interrelationships with
Special Reference to Dynamic Regression," Ph.D. Thesis, Department of
Statistics, University of Wisconsin, Madison, 1972.
Pierce, D.A. "Forecasting in Dynamic Models with Stochastic Regressors,'
Journal of Econometrics, vol. 3 (1975), pp. 349-74.
. "Relationships--and the Lack Thereof--Between Economic Time
Series, with Special Reference to Money and Interest Rates," Journal
of the American Statistical Association, vol. 72 (1977), pp. 11-22.
Pierce, D.A., and L.D. Haugh. "Causality in Temporal Systems:
Characterizations and a Survey," Journal of Econometrics, vol. 5
Quenouille, M.H. The Analysis of Multiple Time Series, London:
E. Griffin and Co., 1957.
Sims, C.A. "Money, ,Income, and Causality," American Economic Review,
V O ~ .62
(1972), pp. 540-52.
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Tiao, G.C.,
and G.E.P.
Box. "Modeling Multiple Time Series with
Applications," Journal of the American Statistical Association, vol.
76, no. 376 (1981), pp. 802-16.
Tjostheim, D. "Granger-Causal i ty in Mu1 tiple Time Series, " Journal of
Econometrics, vol. 17 (1981), pp. 157-76.
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