Piotr Fiszeder

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 7 – Nicolaus Copernicus University – Toruń – 2006 
 
 
 
 
 
Piotr Fiszeder 
Nicolaus Copernicus University in Toruń 
 
 
 
Consequences of Congruence for GARCH Modelling
1  
 
 
1. Introduction 
  
  In 1981 Granger formulated the idea of congruence, as similarity of main 
properties of the endogenous variable and main properties of exogenous vari-
ables. Zielinski (1984) introduced the idea of dynamic congruent modelling. 
Congruence of the model in the meaning of Zieliński means that harmonic 
structure of the endogenous process is congruent with the joint harmonic struc-
ture of explanatory processes and the error term, which is independent of ex-
planatory processes. The idea of dynamic congruent modelling is based on cap-
turing information about the internal structure of processes at the model specifi-
cation stage and building the congruent model on a basis of white noise error 
terms equation. Models which were built according to the procedure of dynamic 
congruent modelling are often better models concerning statistical properties, if 
only the internal structures of analysed processes are correctly discovered and 
specified in the model. The use of information about the internal structure of 
economic processes was also a basis for the idea of congruent modelling intro-
duced by Granger (1990). 
  The concept of congruence of conditional variances is not mentioned in 
those two ideas of modelling. In this article the idea of congruence is extended 
to conditional variances. In this paper congruence of the model in conditional 
variances means equality of conditional variance of endogenous variable with 
conditional variance of linear function of explanatory processes and error term. 
It is assumed that all analysed processes have constant and finite unconditional 
variances. A model is congruent in conditional variance if the unconditional 
                                                      
1 Financial support of the Polish Committee for Scientific Research for the project  
1 H02B 033 29 carried out in 2005–2007 is gratefully acknowledged.  © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder  144 
variance of the endogenous variable is equal to the unconditional variance of 
linear function of explanatory processes and the error term and there is congru-
ence of harmonic structure of the endogenous process squared with harmonic 
structure of the square of linear function of explanatory processes and the error 
term
2. 
The article is laid out in four sections. Section 2 introduces the models con-
gruent in variance for GARCH processes. Section 3 describes consequences of 
modelling which is not congruent in variance and introduces a new volatility 
measures. Section 4 concludes.  
 
 
2. Models Which Are Congruent in Conditional Variance 
 
 The  GARCH  process can be written as:  ) , ( q p
 
  )   , 0 ( ~     1 t t t h D − ψ ε , (1) 
  , (2)  ∑∑
==
+ + =
q
  i 
p
  j 
t -j j t-i i o t h β     ε α   α h
11
2
 
where  1 − t ψ  is the set of information available at time t–1 and  is prob-
ability distribution (usually the normal or Student–t distribution) with zero 
mean and variance . 
) , 0 (   t h D
t h
Let  yt ε ,  t xs ε    and  ) ,..., 2 , 1 ( k s = t ε be white noise GARCH processes (with zero 
mean, constant unconditional variance and no autocorrelation). The model: 
  
  , (3)  t
k
s
t x s yt s ε ε ρ ε + =∑
=1
 
where  0 ) ( = t x t s E ε ε , is congruent, when harmonic structures of the left and 
right hand side terms of the equation are identical. 
If  yt ε  is an error process in the model describing internal structure of dependent 
variable, and  t xs ε  are error processes in the models explaining internal struc-
tures of explanatory variables , then congruent model may be constructed in a 
traditional way using in equation (3) internal structure of the processes (see 
Talaga and Zieliński, 1986). Returns of financial processes are often stationary 
and usually they may be sufficiently described by autoregressive models with 
low order of autoregression. 
                                                      
2 The idea of congruence can be extended to higher conditional moments of a dis-
tribution, however in this case interpretation is not easy. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Consequences of Congruence for GARCH Modelling 
 
145
  Presented congruent models do not have to be congruent in conditional 
variance. First let us consider an example with one explanatory variable. Let  xt ε  
and  yt ε  be GARCH processes of orders:   and  . If  ) , ( 1 1 q p ) , ( 2 2 q p yt ε ,  xt ε  and 
t ε  are white noise processes, then model: 
  
  t xt yt ε ρε ε + =  (4) 
 
is congruent. 
Model (4) is congruent in conditional variance, if unconditional variance  yt ε  is 
equal to unconditional variance  t xt ε ρε + , and harmonic structures of the proc-
esses   and   are identical, that is if the model: 
2
yt ε
2 ) ( t xt ε ρε +
  
   (5) 
2 2 2 2 2 t t xt xt yt ε ε ρε ε ρ ε + + =
 
is congruent. 
The equality of unconditional variances of  t ε  and  t xt ε ρε +   is required in 
every congruent model of type (4). 
Assuming  , GARCH  model may be written as an ARMA 
(m,p) model for (where m = max {p,q}). It follows that   and   may be 
presented as: 
t t t h − =
2 ε ν ) , ( q p
2
t ε
2
xt ε
2
yt ε
 
  , (6)  ∑∑ ∑
==
−
=
− − + − + + =
11 1
11
1
1
2
1
2
1 01
2
q
i
p
j
xt j xt j
p
j
j xt j i xt i xt ν ν β ε β ε α α ε
  , (7)  ∑∑ ∑
==
−
=
− − + − + + =
22 2
11
2
1
2
2
2
2 02
2
q
i
p
j
yt j yt j
p
j
j yt j i yt i yt ν ν β ε β ε α α ε
 
where  yt ν ,  xt ν  are white noise processes (if  yt ε  and  xt ε  have finite fourth order 
moments). Most financial processes have finite fourth order moments. The ex-
ceptions are mainly financial instruments from money market and less liquid 
instruments from other markets.  
The structure of   will depend on internal structure of   and   processes. 
Conditional variance of the error term 
2
t ε
2
yt ε
2
xt ε
t ε  in equation (4) may then be described 
using GARCH  model. The values of   and   will depend on the 
character of  the relationship between   and  and the properties of these 
processes (more about the properties of processes which are a sum of other 
autoregressive processes may be found in Kufel, Piłatowska and Zieliński, 
1996, Stawicki and Górka, 1996). If 
) , ( 3 3 q p 3 p 3 q
2
yt ε
2
xt ε
t t u A Y u B 1 1 1 1 ) ( ) ( ε =  and © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder  146 
t t u A Y u B 2 2 2 2 ) ( ) ( ε =  are independent ARMA processes of degree   and 
 respectively, then 
) , ( 1 1 q p
) , ( 2 2 q p
 
     (8)  t t t t t u A u B u A u B Y Y Y 1 1
1
1 2 2
1
2 1 2 ) ( ) ( ) ( ) ( ε ε
− − − = − =
 
and 
 
  . (9)  t t t u A u B u A u B Y u B u B 1 1 2 2 2 1 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ε ε − =
 
 
Thus   process is of ARMA  type, where   and 
. If some parameters are close to zero, then the process 
may be identified as a process with fewer lags. In the case of financial time 
series there is usually a great number of observations, thus   will almost al-
ways be an identifiable ARMA process. Similarly, when   and   are inde-
pendent then the variance 
t Y ) , ( q p 2 1 p p p + =
) , ( max 1 2 2 1 q p q p q =
t Y
2
yt ε
2
xt ε
t ε  in equation (4) may be presented as a 
ARMA  process. If equation (4) is a model for error processes in the mod-
els describing internal structure of dependent and explanatory variables, then 
only correct specification of model (4) and GARCH model for 
) , ( q p
t ε  will assure 
congruence in variance of model (4). In specification one should properly de-
scribe internal structure of dependent and explanatory variables (in order to 
receive white noise properties) and properly describe orders in GARCH model. 
  One reaches similar conclusions from a general model for k  explanatory 
variables (equation (3)), assuming  0 ) (
' = t x t x s s E ε ε . When explanatory vari-
ables are correlated then variance of  t ε  will also depend on covariances of par-
ticular explanatory variables.  
  It is worth to mention that there is no possibility to create a classical con-
gruent model between GARCH processes  yt ε  and  xt ε  on the basis of standard-
ised processes. Let  ,   and   be standardised white noise processes with 
constant conditional variances: 
yt z xt z t z
 
  yt yt yt h z / ε = ,  xt xt xt h z / ε = ,  t t t h z / ε = . (10) 
 
The model: 
 
  t xt yt z z z + = ρ  (11)   
 
is congruent. Substituting (10) in equation (11) and multiplying the equation by 
 one obtains:  yt h
 
 
t
yt
t xt
xt
yt
yt h
h
h
h
ε ε ρ ε + = . (12) © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Consequences of Congruence for GARCH Modelling 
 
147
 
 Substituting 
*
t
xt
yt
h
h
ρ ρ =  and 
*
t
t
yt
t h
h
ε ε = one acquires a congruent model: 
  
  , (13) 
* *
t xt t yt ε ε ρ ε + =
 
with stochastic parameter  . 
*
t ρ
Obviously there exists classical congruent model between processes:  yt ε  and 
1 −
yt xt
xt
h h
ε
 that is between  yt ε  and properly standardized process  xt ε . 
 
 
3. Congruent Models in Conditional Variance – Propositions of 
New Volatility Measures 
 
  Volatility is an important quantity in many financial analyses, eg. deriva-
tives pricing, capital asset pricing, analysis of the flow of information between 
markets and financial instruments. Volatility of conditional variance of regres-
sion model error term lowers the efficiency of parameter estimators obtained 
with least squares method, and covariance matrix of estimators   is 
not valid if the regression involves lagged dependent variable. Standard testing 
for parameters statistical significance may in this case lead to wrong results. 
The alternative, more suitable procedures include the use of other estimation 
methods which are robust to changes of variance or describing the changes of 
error term variance directly in the model for example with GARCH or SV 
specifications. If the model which serves as a tool for creation of volatility 
measure is not congruent in conditional variance, then the variance is usually 
underestimated. Thus the congruence in variance of a model is important in 
every analysis using volatility forecasts. 
1 2 ) ' (
− X X σ
 Let   describe AR –GARCH  process:  t y ) (r ) , ( q p
 
  , (14)  t
r
i
i t i t y y ε φ φ ∑
=
− + + =
1
0
  )   , 0 ( ~     1 t t t h N − ψ ε , (15) 
  . (16)  ∑∑
==
+ + =
q
  i 
p
  j 
t -j j t-i i o t h β     ε α   α h
11
2
 
The estimates of   variability obtained on a basis of the conditional variance 
of  
t y
t ε  will be underestimated, because they do not account for changes of the 
autoregressive process. Variability of   is obviously greater than variability of  t y© Copyright by The Nicolaus Copernicus University Scientific Publishing House
Piotr Fiszeder  148 
error term  t ε . In empirical applications it may appear that inclusion of variabil-
ity resulting from autoregressive part of the model is important (for example in 
applications of Monte Carlo methods in pricing of derivatives or calculating the 
Value at Risk). One can use conditions of model congruence in conditional 
variance and apply a new volatility measure: 
  
 
∑
=
−
= r
i
i
t
t
h
hs
1
2 1 φ
 ,  (17) 
 
Proposed volatility measure is not a „pure” conditional variance, because it is 
rescaled up by a autoregressive variability. 
  Similarly the estimates of variability  yt ε  based on conditional variance  t ε  
in equation (3) will be underestimated, because changes of explanatory vari-
ables  t xs ε  are not accounted for. Let  0 ) , (
' = t x t x s s E ε ε . Proposed volatility 
measure for  yt ε  may be written as:  
 
  , (18)  ∑
=
+ =
k
s
t x s yt h h hs
s
1
2 ρ
 
where   )  and   are conditional variances of explanatory vari-
ables and the error term respectively. 
s x h ,..., 2 , 1 ( k s = t h
  If model (3) is not congruent in conditional variance, that is if   does not 
account for internal structure of   and   (
2
t ε
2
yt ε
2
st x ε k s ,..., 2 , 1 = ), then the estimates 
of variability generated by this model will be incorrect (underestimated or over-
estimated).  
 
 
4. Conclusions  
 
  In the paper the concept of model congruence has been extended to condi-
tional variances. The implications of neglecting the information about the inter-
nal structure of dependent variable or explanatory variables have been indi-
cated. New measures of variability have been proposed. Empirical application 
of proposed measures for financial analyses, in particular for volatility forecast-
ing is left for future research.    
 
References 
 
Granger, C. W. J. (1981), Some Properties of Time Series Data and their Use in 
Econometric Model Specification, Journal of Econometrics, 16, 121–130. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Consequences of Congruence for GARCH Modelling 
 
149
Granger, C. W. J. (1990), Where Are the Controversies In Econometric Methodology?, 
in: C. W. J Granger (ed.), Modelling Economic Series, Clarendonpress, Oxford. 
Kufel, T., Piłatowska, M., Zieliński, Z., (1996), Symulacyjna analiza poznawczych 
własności dynamicznych modeli zgodnych (Simulation Analysis of Cognitive 
Properties of Dynamic Congruent Models), Przestrzenno–czasowe modelowanie 
i prognozowanie zjawisk gospodarczych (Cross–Section–Time Modelling and 
Forecasting of Economic Events), AE, Kraków.  
Stawicki, J., Górka, J. (1996), An ARMA Representation for a Sum of Autoregressive 
Processes, Dynamic Econometric Models, vol. 2, UMK, Toruń. 
Talaga, L., Zieliński, Z. (1986), Analiza spektralna w modelowaniu ekonometrycznym 
(Spectral Analysis in Econometric Modelling), PWN, Warszawa. 
Zieliński, Z. (1984), Zmienność w czasie strukturalnych parametrów modelu ekonome-
trycznego (Time–Varying Structural Parameters of Econometric Model), Prze-
gląd Statystyczny (Statistical Survey), 31, 1/2, 135–148. 
 

An ARMA Representation for a Sum of Autoregressive Processes, Dynamic Econometric Models,

Analiza spektralna w modelowaniu ekonometrycznym (Spectral Analysis in Econometric Modelling),

Some Properties of Time Series Data and their Use in Econometric Model Specification,

Zmienność w czasie strukturalnych parametrów modelu ekonometrycznego (Time–Varying Structural Parameters of Econometric Model),

Conformable Models for GARCH Processes

http://www.dem.umk.pl/dem/archiwa/v7/14_Fiszeder_congruent.pdf

Conformable Models for GARCH Processes

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