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DYNAMIC ECONOMETRIC MODELS 
Vol. 11 – Nicolaus Copernicus University – Toruń – 2011 

Magdalena Osińska 
Nicolaus Copernicus University in Toruń   

On the Interpretation of Causality in Granger’s Sense 

A b s t r a c t. The concept of causality formulated in 1969 by C.W.J. Granger is mostly popular in 
the econometric literature. The central assumption of the concept is the fact that the cause pre-
cedes the effect and can help in forecasting the effect. Years of application of Granger causality 
idea have resulted in many misunderstandings related with the interpretation of the empirical 
findings. The paper focuses on systematization of the definitions based on Granger concept and 
their proper interpretation.  

K e y w o r d s: Granger causality, systematic causality, informational causality, nonlinear cau-
sality.  

Introduction  

The concept of causality formulated in 1969 by Clive Granger, based on earlier 
paper by Wiener (1956), is mostly popular in the econometric literature. The 
central assumption of the concept is the fact that the cause precedes the effect 
and can help in forecasting it. It is further assumed that the cause includes 
unique information of the effect, which is not available in any other way. The 
potential causes are chosen from all information connected with the effect.  

 The general definition of Granger’s causality is formulated in the frame-
work of conditional probability distribution. Let )|( XYF  denote conditional 

distribution of Y given X, and t  represents all information in the universe in 
time t. It is said that Xt does not cause Yt if for all k > 0 the following relation is 
true: 

),\|()|( ttkttkt XYFYF     

where: tt X\  denotes all information in the universe except for those included 

in tX . Otherwise tX  cause tY  (Granger, Newbold, 1986). The above definition 
cannot be called operational for the reason of using the phrase ‘all information 



Magdalena Osińska 130 

in the universe’ that cannot be identified in practice. Since 1969 several opera-
tional definitions have been formulated. They were the subject of statistical 
verification. The word ‘causality’ present in all these definitions is the source of 
misunderstanding.  

 In the paper the attention is focused on the operational definitions of cau-
sality in Granger’s sense and their interpretation in the two contexts: philosoph-
ical as well as empirical.  

1. Systematic Granger Causality  

Modelling conditional mean of the endogenous variable via the econometric 
model means that we are looking for a systematic, repeatable relation which can 
be used, among others, in forecasting. We call Granger causality defined for 
linear representation of time series ‘the systematic Granger causality’ because it 
refers to such cases.  

 Operational definition of Granger causality, corresponding to systematic 
causality concept, is formulated for wide sense stationary time series and it is 
measured in terms of forecasting errors. It is assumed that autoregressive (or 
vector autoregressive) representation of the time series that constitutes ‘all the 
information in the universe’ can be used in forecasting. The direction of causali-
ty is the subject of testing. Granger causality tests were discussed widely in 
Osinska (2008) and will not be the subject of further analysis. The attention is 
concentrated at the definitions and their understanding. 

 We say that tX  Granger-cause tY  if: 

),\|()|( ''2'2
ttttt XYY    (1) 

where )|( '2
ttY   is the prediction error in the case where all information from 

the past are used and )\|( ''2
ttt XY   is the prediction error corresponding to 

the situation when Xt was excluded from the information set. Key function in 
the above definition is played by the information set t , because the results of 
testing for Granger causality strongly depend on it. Granger did not define ex-
plicitly t , leaving it to the researchers. If the variables included into the in-
formation set come directly from the economic theory then the definition, de-
spite of its limitations, is closer to the meaning of the word ‘causality’. If how-
ever, the information set is based on the data, we cannot expect finding out any 
unknown relations which will occur a new economic law in future. As Cart-
wright (2007) wrote ‘no cause in -  no cause out’. So we should not expect  
anything more than it comes out from the data. 

 Furthermore, the formula (1) turns our attention to the understanding of 
causal relations in terms of forecasting. This is not satisfactory from the philo-
sophical point of view but very practical. Forecasting ability is definitely one of 



On the Interpretation of Causality in Granger’s Sense 131

the desired characteristics of causal relation, and when we are able to identify 
the cause and its effect we expect that the cause (or rather set of causes) will 
result in the occurrence of a given effect. But causality in Granger’s sense can-
not be identified as the relation determining that the cause is able to induce the 
effect. It is obvious that in the case when economic theory states that there is the 
causal relation between two variables, say, the magnitude of money supply and 
the inflation rate, testing for causality in Granger’s sense will confirm that 
statement. In many cases however the theory says nothing and the researchers  
are looking for any relation confirmed by the data. A good example of such an 
investigation is testing for Granger causality between variables characterizing 
financial markets, for example indices of two or more stock exchanges. The 
question arises then what we should expect using Granger causality tests.  

 First of all let us turn the attention to the case when Granger causality can 
be thought as an idea of finding causal relations using structural econometric 
models. This was explained by LeRoy (2004). 

Assume simple structural econometric model of the form: 

,1121211112121 ttttt ybybyay    (2) 

,2122211211212 ttttt ybybyay    (3) 

where the explanatory variables (on the RHS of each equation) are not correlat-
ed with the residuals. The model is not identified and lagged values of variables 
are earlier than those observed in time t. The condition 012 a  defines condi-

tional causality of the form 121121 ,|  tttt yyyy  (LeRoy, 2004). This condition 
cannot be tested directly because observable implications for it do not exist.  

 On the other hand, Granger causality is defined for reduced form of the 
model: 

1 11 1 1 12 2 1 1t t t ty c y c y u ,     (4) 

,2122211212 tttt uycycy    (5) 

where:    211222121212 1/ aababc  . We say that ty2  does not Granger cause 

ty1 , if 012 c . This condition is neither necessary nor sufficient for conditional 

causality 121121 ,|  tttt yyyy . That is why Granger causality does not allow 

stating what is prior: ty1  or ty2 . Only if 012 b and 022 b , 012 c  is equal to 

012 a  and then Granger causality is equivalent to the condition 

121121 ,|  tttt yyyy . Such a case can occur very rarely in practice.  

 Respecting the explanation given above, one may ask whether the structural 
econometric model is able to state causal relation between variables. The an-
swer we often put is ‘no’. To say ‘yes’ we need some additional information of 
interventions (Hoover, 2001).  



Magdalena Osińska 132 

 Granger causality concept was the subject of criticism in econometric litera-
ture (Basmann, 1988; Zellner, 1988 and LeRoy, 2004).  On the other hand the 
explosion of causality tests based on Granger’s idea shows great support for this 
concept (see for example: Caporale, Pittis, Spagnolo, 2002; Geweke 1984; 
Hsiao, 1979; Pierce, Haugh, 1977)). 

 The main objection is that in Granger’s definition no stress is put to the role 
of the economic theory. Granger did not also take into account the definition of 
causality formulated by Feigl. Feigl understood causality as forecasting accord-
ing to a law or a set of laws that joins prediction with the theory (Basmann, 
1988; Zellner, 1979). Hoover (2001), in his classification of different concepts 
of causality in econometrics, includes causality in Granger’s sense to the pro-
cess approach, defined as the inference based on the data. This puts Granger in 
line with the continuators of Hume’s philosophy who rejected any metaphysics 
and indicated observation as the main source of scientific finding.   

 Important limits of Granger causality were indicated by Zellner (1979). He 
argues, among others, that minimum prediction error cannot be used as a crite-
rion of causal relation because the prediction error can be reduced using many 
techniques, not necessarily by including causes into the model.  

 Having defined the limits of Granger causality concept let’s take a glance at 
its advantages: 

1. The definition is operational and allows for testing such important as-
pects of causal relations as: time sequence of variables, asymmetry of 
relationships and forecasting ability in the sample and out of the sam-
ple.  

2. The definition resulted in many statistical tests, developed for different 
classes of data. 

3. The definition was extended for causality in conditional variance as 
well as causality in risk and nonlinear causality. This allows finding out 
many relationships of different nature present in the observable eco-
nomic reality.  

4. The information set included into the model is chosen by the researcher. 
This is a chance for combing of data analysis with a certain theory and  
developing an individual approach to a specific problem. 

5. It is easy to be used in practice.  

 Granger’s definition and its modifications, as well as the tests based on that 
concept should be used with care and interpreted only in the domain they are 
specified for. We cannot interpret any relation confirmed by Granger causality 
test as ‘causal’ in the broad philosophical sense. Analyzing the data without any 
theoretical background one cannot avoid spurious causality or symptomatic 
causality resulting from the presence of the third variable not included into the 
information set. If, however a certain theory is tested using Granger causality 
tests it can be stated that it is accepted or rejected by the data in the sense of 



On the Interpretation of Causality in Granger’s Sense 133

asymmetry and forecasting ability. The question whether it ensures repeatability 
in future cannot be answer directly because it depends on the economic policy 
in a given country, which – in normal conditions - changes from one economic 
regime to another.  

 It should be stated that attempts at formulating operational definition of 
causality are not very popular because it is very difficult to do (Cartwright, 
2007). For that reason Granger’s definition, although far from excellence, is so 
much important. There are some further implications that can be applied jointly 
with forecasting ability like the analysis of interventions (Hoover, 2001), thick 
causality (Cartwright, 2007) or modular concept (Pearl, 2000) but none of these 
concepts can be thought of as universal.  

 At first Granger causality concept was formulated for stationary time series 
possessing autoregressive representations. Further, it was developed for cointe-
grated time series (Granger, 1981; Toda, Yamamoto, 1995) and, what is very 
important, Granger causality considered in the sample and out of the sample 
that allows determining whether Granger cause in the sample is still able to help 
predicting the effect out of the sample (Ashey, Granger, Schmalensee, 1980; 
Chao, Corradi, Swanson, 2001). All the mentioned concepts refer to the direct 
causality in Granger’s sense such as Xt  Yt+1. In the recent years Dufour and 
Renault (1998) turned their attention to the indirect causality concept, based on 
the definition given by Hsiao (1979). It is not the subject of further analysis here 
because Hsiao definition of causality can be decomposed into Granger under-
standing of causality in the terms of forecasting.  

2. Informational Granger Causality 

In the previous part the attention was concentrated on causality in mean based 
on linear systems. However Granger causality can be also considered in terms 
of causality in conditional variance, and the related concept of causality in risk. 
We will call it ‘the informational Granger causality’ because it is related mainly 
with financial markets where relations between different stocks, portfolios or 
derivatives are usually the subject of influence of the information affecting 
both: the cause and the effect.  

 Cheung and Ng (1996), basing on Granger’s definition have proposed the 
following formulation for causality in variance. Let tX  and tY  denote stochas-

tic processes which are covariance stationary and  0; ,   jYX jtjtt  is 

a set of all information from the past, available at time t and tt X\  is the cor-

responding information set excluding tX .  



Magdalena Osińska 134 

 It can be said that tX  Granger does not cause tY  in variance, if: 

      ttYttttYt YEXYE  |\| 22  , (6) 

where: 
tY  is a conditional mean of tY , assuming the information set tt X\ .  

 Further development, made by Hong (2001), corresponding directly with 
the above definition concerns causality in risk. Granger causality in risk is de-
fined as follows. Let  tt YY 21 ,  is a bivariate not necessarily stationary stochastic 

time series. Let   1 tlltit IAA  l = 1,2 be the Value at Risk (VaR) at level

 1;0  for Ylt predicted using the information set    )2()1(1 ,   tltltl YYI  

available at time 1t  . ltA  satisfies     1,| tlltlt IAYP . In the case of 

Granger non-causality the null hypothesis takes the form: 

    1111,1110 ||:   tttttt IAYPIAYPH   almost surely (7) 

where  1 1 1 2 1t ( t ) ( t )I I ,I ,     

with the alternative 

    1111,1111 ||:   tttttt IAYPIAYPH . (8) 

 Comparing the above definition with the original one we may state that it 
concentrates only on the violations of value at risk computed for a given portfo-
lio represented by Y1t. So we interpret it as if information about the second port-
folio represented by Y2t could help change the probability of breaking the VaR. 
The definition captures the general characteristics of Granger causality concept.  
It can be extended for other risk measures, belonging to the class of coherent 
risk measures (Artzner, Delbaen, Eber, Heath, 1998). 

 As it was mentioned above, the concept of Granger causality was often crit-
icized because it depreciates the philosophical nature of causal relation. On the 
other hand it is widely known and popular in econometric literature. In fact 
Granger’s definition is related with predictability of one variable using previous 
values of another one. Such an approach takes into consideration only one of 
many characteristics of causal relation, however in practice it is often the unique 
possibility of measuring interdependencies between variables. It is particularly 
important when causality in conditional variance is considered. The number of 
factors that cause the volatility of  financial returns is enormous. Furthermore, 
they change in time and occur only in some periods such as they cannot be ob-
served systematically. Their nature is also very much complicated, starting from 
fundamental causes coming from company itself, through causes located in the 
macroeconomic surroundings, ending at those of social and psychological na-
ture. However the results they cause are very important, observable and spread 
all over the world. Very similar situation takes place in the case of Granger 
causality in risk, where specified risk measures are applied. The causes, which 



On the Interpretation of Causality in Granger’s Sense 135

evoke the failure of the risk measures are rarely of systematic nature. So if such 
a raise in risk occurs at one market it is very likely to be moved to another one. 
It is due to the risk-transferring procedure realized by many market participants 
including banks. Avoiding the risk by closing positions and moving financial 
capital from one market to another are the main characteristics of contemporary 
markets. It changes the liquidity preference in the markets that cannot be avoid-
ed without the intervention. Such a situation is called the contagion phenome-
non (Allen, Gale, 2000).  

3. Nonlinear Granger Causality 

Nonlinear Granger causality is analyzed separately since it may consider other 
aspects of Granger causality that the ones discussed above. Baek and Brock 
(1992) formulated the general definition of Granger causality for nonlinear case.  
It is expressed in terms of the correlation integral that is a measure of local spa-
tial correlation of time series which belong to a specified space. Formally, for 
a multivariate random vector W , the associated correlation integral ( )WC   is 
the probability of finding two independent realizations of the vector at a dis-
tance smaller (or equal) than , i.e. 

         1 2 1 2 1 2 2 1W WW W WC P I s s f s f s ds ds ,          
(9) 

where: 1 2W ,W  are independent realizations of W , the integrals are taken over 

the sample of W ,  is the supremum norm and I() is the indicator function, 

which is equal to one if its argument is true, and is zero otherwise.  

 Denote the m-length lead vector (m-history) of tY  by 1 1( , ,..., )m
t t t t mY Y Y Y    

and the p-length and q-length lag vectors of tX  and tY , respectively by 

1 1( , ,..., )p
t p t p t p tX X X X      and 1 1( , ,..., )q

t q t q t q tY Y Y Y     . 

Baek and Brock proposed the following definition of Granger nonlinear  
causality:  

tX  does not nonlinearly Granger cause tY  if 

 

 

m m p p q q
t s t p s p t q s q

m m q q
t s t q s q

P Y Y | X X , Y Y

P Y Y | Y Y .

  

 

   

 

      

    
 

(10) 

where  is the supremum-norm distance. 

The definition says that, given  , p  lags of tX  does not incrementally help 

predict next period’s value of tY , given q lags of tY . It seems to be clear why 

the event  ‘ p
ptX   is close to p

psX  ’ may help incrementally predict tY  close to 



Magdalena Osińska 136 

sY , in case when ),( q
qt

p
ptt YXfY   for some deterministic and continuous 

function f. While ),( q
qt

p
ptt YXgY  , where g  is a stochastic function, the defi-

nition is motivated by a hope that at least part of the deterministic relation is 

present, especially when the conditional variance of tY , given q
qt

p
pt YX  ,  is 

smaller than the unconditional variance of tY . It is worth noting that the defini-

tion is based on the assumptions concerning the choice of  , the lags number 
p  and q  as well as the forecasting horizon m . 

Note that from the definition of conditional probability 

   
  .

,
|















 q
qs

q
qt

q
qs

q
qt

m
s

m
tq

qs
q

qt
m

s
m

t
YYP

YYYYP
YYYYP (11) 

Moreover, since the supremum norm implies that  

   ,,   





qm
qs

qm
qt

q
qs

q
qt

m
s

m
t YYPYYYYP

 
(12) 

the identity 

   
 

m q m q
t q s qm m q q

t s t q s q q q
t q s q

P Y Y
P Y Y , Y Y ,

P Y Y


 



 
 

 

 

 
    

 
 

(13) 

is satisfied. By analogy 

 
 
 

m m p p q q
t s t p s p t q s q

m q m q p p
t q s q t p s p

q q p p
t q s q t p s p

P Y Y | X X , Y Y

P Y Y , X X
,

P Y Y , X X

  

 

 

   

 
   

   

      

   


     

(14) 

thus the null hypothesis of Granger nonlinear noncausality given by (10) may 
be expressed as follows: 

 
 

 
 





,4

,3

,,2

,,1

qC

qmC

pqC

pqmC 



, (15)  

where: 

   ,,,,1   






p

ps
p

pt
qm

qs
qm

qt XXYYPpqmC  

   ,,,,2   
p

ps
p

pt
q

qs
q

qt XXYYPpqC  

     





qm
qs

qm
qt YYPqmC ,3  and 



On the Interpretation of Causality in Granger’s Sense 137

   .,4   
q

qs
q

qt YYPqC  

 In practice, nonlinear causality is tested above the linear one, so first of all 
the linear relation should be excluded, for example by estimating a bivariate 
stationary VAR model. The tests are usually carried out on the residuals of the 
linear model. The same indication can be applied for causality in variance. If the 
source of nonlinearity is known and can be modeled, for example by GARCH 
models, it should also be filtered out to show whether other possible source of 
nonlinear Granger causality can be found.  

Conclusions 

In the paper we discussed the concept of Granger causality and its main devel-
opments present in the econometric literature. It should be emphasized that the 
concept refers to predictability as the one of the characteristics of causal rela-
tion. That is why it is too narrow to satisfy all the important attributes of philo-
sophical nature of causal relations. On the other hand it is very practical and 
popular in economic as well as econometric applications. The user of Granger 
causality concept and the tests based on this background should bear in mind 
their limitations and not to expect finding out any unlikely relations between 
economic variables. From Granger definition it comes out only such character-
istics of causal relation like: sequence in time, asymmetry of the cause and the 
effect and forecasting ability. However it can be analyzed in very many aspects: 
in the conditional mean and in the conditional variance, in linear and in nonlin-
ear framework, in the short and in the long run as well as in the sample and out 
of the sample,. This variety of applications gives a possibility of wide empirical 
analyses which should be put in a theoretical framework if we want to consider 
them as causal in a broader philosophical sense.  

 In the paper the types of Granger causality were classified. We discriminat-
ed: systematic Granger causality, informational Granger causality as well as 
nonlinear Granger causality. The first one refers to linear causality in condition-
al mean, the second one corresponds to two related concepts of causality: i.e. 
causality in variance as well as causality in risk and the last one defines the 
relation which can happen over the two mentioned earlier. 

 Although Granger causality has many disadvantages from the philosophical 
point of view its usefulness can be compared with the Vector Autoregression 
model defined by C. Sims in 1980 who won the Nobel prize in 2011. The VAR 
model was thought of as atheoretical remedy for disadvantages of traditional, 
structural econometric models. In fact it became structural, among others, be-
cause of testing for Granger causality. That example shows that even a very 
narrow tool can be used in solving much more complicated and sophisticated 
theoretical  problems. In the same sense proper use of Granger causality can be 



Magdalena Osińska 138 

a method of verification of the economic theory and possibly the source of  
some new empirical facts. 

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O interpretacji przyczynowości w sensie Grangera 

Z a r y s  t r e ś c i. Koncepcja przyczynowości sformułowana w 1969 roku przez C.W.J. Grange-
ra jest najbardziej popularna w literaturze ekonometrycznej. Centralnym założeniem tej koncepcji 
jest fakt, że przyczyna poprzedza skutek i jest pomocna w prognozowaniu skutku w przyszłości. 
Lata stosowania koncepcji przyczynowości w sensie Grangera zaowocowały wieloma nieporo-
zumieniami związanymi z interpretacją wyników empirycznych. Artykuł dotyczy systematyzacji 
definicji przyczynowości w  sensie Grangera i ich właściwej interpretacji.  

S ł o w a  k l u c z o w e: przyczynowość w sensie Grangera, przyczynowość systematyczna, 
przyczynowość informacyjna, przyczynowość w zależnościach nieliniowych.  

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 

 

 


