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.Time-series analysis

Michael L. Bagshaw

English

Research Papers in Economics

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 
http://clevelandfed.org/research/workpaper/index.cfm
Best available copyEXTENSION 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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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 of  Granger 
causality is the mean square error of one-period-ahead forecasts. 
This idea can be expressed as follows:  Let At be the given informa- 
tion set  (including  at least zt  and yt),  it  =(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  i  ty. 
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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Best available copyof past z,  then  z  must  Granger  cause y  with respect to  the set (zt,  yt). 
However,  as  the following example  demonstrates,  this is  not necessarily 
true for systems  with more  than two variables: 
where  the al,  a2,  and  a3  are mutually orthogonal white noise processes 
222  with variances ul,  u2,  u3,  respectively.  In this model,  At  =  (yt,  xt,  zt). 
The minimum mean-square-error,  one-period-ahead  forecasts for this 
model  are 
where  Gt - (1) is the one-period-ahead  forecast of w  at time t-1.  These 
2  2  2  forecasts have  mean  square  errors of  ul,  u2,  and  u3,  respectively.  (See 
Tiao and  Box  (1981)  for a general discussion of how  to  calculate 
forecasts from these types of models  and  their mean  square errors. ) 
If z  is  not included in  the model,  then the appropriate bivariate 
model  can be  derived from the model  given in  equation  (1) by matching the 
variances,  covariances,  and  cross covariances of  y  and  x'implied by this 
model.  This reduced model  is  given by 
2 
where  o2I2 =  4:3<  +  u2. 
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Best available copyThe minimum mean-square-error,  one-period-ahead  forecasts  from this 
model  are 
with mean  square errors of  U:  and  6E30:  +  u;,  respectively. 
We  thus have  shown  that z  does  not Granger  cause y  (the mean-square- 
forecast error is  U:  in  both cases),  but that z  does  Granger  cause  x 
+  a). Similarly,  we  can  show  that x  Granger  causes y  but not z.  ('2  <  '2303 
Also,  y  does  not Granger  cause  x  or z.  The  causality chain is  thus an 
example  of  indirect causality (Tjostheim 1981)  between  z and y: 
where -  means  Granger  causes. 
When  we  examine  the two-period-ahead forecasts,  the result is 
different.  The  two-period-ahead  forecasts from  model 1 are 
2  with mean  square  errors of  0:  +.4:2~g,  U:  +  4:3~$,  and  u3,  respectively. 
(See  Tiao and  Box  (1981)  for  a  general  discussion of  how  to calculate 
mu1 t  iple-period forecasts and  their mean  square errors. ) 
For model  3,  the two-period-ahead  forecasts are given by 
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2  with mean square forecast errors of a:  +  dl2(a;  + 
a2) and  0;  +  $:3a:,  respectively.  '23  3 
Thus, the mean square forecast error for y two periods ahead is  less 
2  2  in the moiel including z  (a  +  d12a2) than it  is  in  the model not  including 
2  22  222 
z  (al  +  d12a2 +  $12423a3).  This illustrates the principle that 
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~(~(L)  I A) 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 
for some  j >  1.  In this model, y is related to z with .a j-period lag, 
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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Best available copy111.  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  );  W  is the vector of n-1 variables excluding v;  and  TI  (n-1 x 1) 
and  E (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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Best available copycorresponding 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-length  chain 
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 
+
  alt 
xt=4  23 z  t-k 
+
  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  + j) period causes y 
independently of k.  Thus, the value of L depends not only on the length 
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Best available copy-8- 
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 i  ty 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  1  ity. 
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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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Best available copyReferences 
Granger, C.W.J.  I1Investigating  Causal 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, 
VO~.  62 (1972),  pp.  540-52. 
http://clevelandfed.org/research/workpaper/index.cfm
Best available copyTiao, 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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Extension of Granger causality in multivariate time series models

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Extension of Granger causality in multivariate time series models

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