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DYNAMIC ECONOMETRIC MODELS 
Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 

Blanka Łęt 
Poznań University of Economics  

Dynamics of Multivariate Return Series of U.S.  
Automotive Stock Companies in Conditions of Crisis†  

A b s t r a c t. This article contains an analysis of dynamic interrelations between log-returns 
series of three automotive companies listed on the New York Stock Exchange: GM, F and DAI. 
We consider two periods: before and during crisis. We apply DiagBEKK model and we calculate 
dynamic conditional correlations. As a result of our research we found that in conditions of crisis 
there were strong connections between considered stock companies. 

K e y w o r d s: DiagBEKK model, dynamic conditional correlation.  

1. Introduction  
 General Motors, Ford and Chrysler, known as the Detroit’s Big Three, are 
the major companies of American automotive industry. They have in common 
not only strong worldwide position but also problems, among others, with high 
labor costs as a result of activities of United Auto Workers. This inevitably re-
sults in high prices of cars offered, which can be afforded by fewer and fewer 
potential customers. General Motors, Ford and Chrysler face such problems for 
several years. 
 Very high oil prices, soaring in the period from January 2007 to mid-2008, 
also resulted in fewer Big Three car sales, because of high fuel costs.  
 Credit crunch, due to the prevailing economic crisis, caused deeper and 
deeper problems of the Big Three. Dismal financial performance of these com-
panies inevitably resulted in a weakening listing on the New York Stock Ex-
change1. 
                                                 

† This work was financed from the Polish science budget resources in the years 2007-2010 as 
the research project NN 111 1256 33. 

1 Chrysler is not listed on NYSE. From 1998 to mid-May 2007, Chrysler were part of Daim-
lerChrysler AG, later the shares were held by an investment fund Cerberus. At the end of April 
2009, declared bankruptcy, Fiat hold 20% of his shares. 



Blanka Łęt 44

 This paper contains an analysis of daily log-returns series of three automo-
tive companies listed on the New York Stock Exchange: GM (General Motors), 
F (Ford Motor Company) and DAI (Daimler AG, for mid-May 2007 as Daim-
lerChrysler). The goal of this paper is to investigate and describe dependencies 
between them. The research is made for two periods separately: before and dur-
ing crisis. This allows us to observe the changes that have taken place.  

2. Methodology 

 Let ),...,,( ,,2,1 ′= tnttt rrrr  denote a multivariate time series of returns with the 
following decomposition 

,ttt yμr +=   (1)  

where: 
)|( 1−Φ= ttt E rμ  is conditional mean, 

1−Φ t  is the information set available at time 1−t . 

Conditional expected value )|( 1−Φ= ttt E rμ  can be modeled by VARMA mod-
els (Tsay, 2002): 

,)()( 0 tt BB aΘφμΦ +=   (2) 

where: 

0φ  is n-dimensional vector, 
p

p BBB ΦΦIΦ −−−= ...)( 1  and q
q BB ΘΘΙΘ −−−= ...)( 1  are two nn×  

matrix polynomials,  
qB  is back-shift operator: ,,, qtiti

qB −= aa   

{ }ta  is a sequence of serially uncorrelated random vectors with mean zero 
and covariance matrix .Σ  

A general multivariate GARCH model for ty is given by equation 

,2/1
ttt εHy =   (3)                                           

where: 

tε  is n-dimensional i.i.d. process with zero mean and identity covariance ma-
trix, 

2/1
tH  is a nn×  matrix satisfying  ( ) ,2/12/1

ttt HHH =
′

 

0y =Φ − )|( 1ttE  and .)|( 1 ttttE Hyy =Φ′ −  



Dynamics of Multivariate Return Series of U.S. Automotive Stock Companies…  45

Specific MGARCH (multivariate GARCH) model is described by parameteriza-
tion of positive definite covariance matrix tH . 

 The BEKK model have been proposed by Engle and Kroner (1995).  
The following equation defines BEKK(p,q,K) model: 

,
1 1 1 1

00 ∑∑ ∑∑
= = = =

−−− ′+′′+′=
K

k

q

i

K

k

p

i
ikitikikititikt GHGAyyACCH  (4)             

where: 

ikAC ,0  and ikG  are matrices of dimension nn×  but 0C  is upper triangular  
matrix, 

K term determines generality of the model. 
On this model positivity of matrix tH can be easily imposed. To reduce number 
of parameters one can impose a diagonal BEKK model (DiagBEKK) where 
matrices ikA  and ikG  are diagonal. Consequently, the generality of the model 
decreases. We can write equation for the simplest DiagBEKK model with 

1=K  for GARCH(1,1) model in following way: 
.1111111111100 GHGAyyACCΗ −−− ′+′′+′= tttt   (5) 

3. Empirical Analysis 
 This paper contains an analysis of daily log-returns series of three automo-
tive companies listed on the New York Stock Exchange: General Motors 
(NYSE:GM), Ford Motor Company (NYSE:F) and Daimler AG (NYSE: DAI). 
There are some indications that the crisis in the automobile industry connected 
with the current economic crisis began in mid-2008. In our analysis, a start date 
for crisis in the automotive industry is set on 1 July 2008. An investigation is 
made for two periods separately: before crisis (January 3, 2007 to June 30, 
2008) and during crisis (July 1, 2008 to May 5, 2009). We use following desig-
nations for time-series: GM_1, F_1, DAI_1 and GM_2, F_2, DAI_2  
respectively.  
 Figure 1 presents the plots of daily log-returns series for investigated com-
panies. One can see that in mid-2008 dynamics of examined series has changed. 
There is a significant increase in volatility. This is most strongly marked in the 
period from mid-September to the end of December. This was the period of 
greatest uncertainty in financial markets: September 15 – the collapse of Leh-
man Brothers, which caused a crash on the American Stock Exchange, early 
October – events in Iceland. In addition, due to rapid increase in oil prices  
(by mid-2008) and credit crunch, car manufacturers have noted the next drop in 
sales. General Motors, along with Chrysler, has started efforts to receive aid 
from the American governments because of the threat of bankruptcy.  
These events coincide in time with periods of increased volatility in the ex-



Blanka Łęt 46

amined series. December 19, 2008, U.S. president has approved financial help 
for General Motors and Chrysler. From that moment, we can observe some 
sedation, which lasts only until mid-February 2009. This corresponds to the 
moment when General Motors and Chrysler have asked the government for 
more financial support (February 18). 

 
Figure 1. GM, F and DAI – daily log-returns. Period 3.01.2007 – 5.05.2009 

 We calculate the most important descriptive statistics of the return series for 
first and second period separately. The results are contained in Table 1. 

Table 1. Descriptive statistics of the return series 
 Min Mean Max Std. deviation Skewness Kurtosis 

GM_1 -0.114 -0.003 0.096 0.029 0.096 1.164 
GM_2 -0.373 -0.009 0.301 0.097 -0.281 1.749 
F_1 -0.113 -0.001 0.111 0.026 0.012 2.167 
F_2 -0.288 -0.001 0.259 0.072 0.088 3.120 

DAI_1 -0.076 0.000 0.079 0.020 -0.032 1.417 
DAI_2 -0.165 -0.002 0.199 0.053 0.050 1.219 

 Clear differences between log-returns series of the investigated companies 
have been noted based on chart already. These observations are confirmed by 
calculated descriptive statistics. In all cases during the crisis, there is stronger 
volatility and extreme values – both positive and negative – are higher. 
The changes of skewness vary for each series. We observe that log-return series 
of GM had positive skew in the first period and during the crisis it changed to 
negative. Negative returns of General Motors occurred more often because of 
the many problems of this company and necessity of government help. Return 
series of Daimler AG behaved conversely. Ford was characterized by positive 
skewness and during the crisis asymmetry was stronger.  More frequent positive 



Dynamics of Multivariate Return Series of U.S. Automotive Stock Companies…  47

returns for Ford probably resulted from a better perception of the manufacturer 
by investors – company did not ask for government aid despite the prevailing 
situation. In all cases series were leptokurtic. 

Table 2.  Parameter estimates for the fitted DiagBEKK model for two periods: before 
(first period) and during crisis (second period). DF is the degree of freedom for 
Student’s t error distribution 

 First period Second period 
 Estimate p-value Estimate p-value 

01φ  -0.0019 0.1685 - - 

02φ  -0.0005 0.6900 - - 

03φ  0.0004 0.6621 - - 

11,0C  0.0143 0.0000 0.0428 0.0126 

12,0C  0.0207 0.0000 0.0171 0.0007 

13,0C  0.0033 0.0000 0.0169 0.0000 

22,0C  0.0040 0.2478 0.0183 0.0130 

23,0C  0.0043 0.0118 0.0041 0.4872 

33,0C  0.0000 0.7009 0.0000 0.1715 

11,11G  0.8511 0.0000 0.8154 0.0000 

22,11G  0.5137 0.0000 0.9081 0.0000 

33,11G  0.9541 0.0000 0.9200 0.0000 

11,11A  0.1685 0.0626 0.3839 0.0260 

22,11A  -0.2554 0.0054 0.2056 0.0703 

33,11A  -0.1260 0.0209 -0.2426 0.0000 
DF - - 5.3920 0.0000 

 We applied DiagBEKK model to characterize the dynamics of multivariate 
time series and changing dependencies between examined companies before 
and during the crisis. The results are presented in Table 2. We fit DiagBEKK 
model with Normal distribution in the first period and Student’s t in the second. 
 Figure 2 presents the plots of dynamic conditional correlations from model 
fitted for the first period. In all cases there were only positive conditional corre-
lations between examined series. 
 The strongest dependency was between General Motors and Ford. 
The mean of conditional correlations between them equals 0.67, the lowest val-
ue equals 0.12, the highest: 0.78. The mean of conditional correlations between 
GM and DAI equals 0.67, the lowest value equals 0.12, the highest: 0.78. Con-
ditional correlations between stock returns of Ford and Daimler were at the 



Blanka Łęt 48

lowest level with mean 0.35, minimum 0.11, maximum 0.48 and characterized 
by strongest dynamics. 

 
Figure 2. Dynamic conditional correlations. Period 3.01.2007 – 30.06.2008 

 It can be concluded that companies belonging to the U.S. producers of the 
Detroit Three (GM, F) are much more strongly linked to each other. In the pe-
riod before the crisis, linkages with companies of the Big Three (GM, F) were 
not high despite the fact that Daimler was connected to May 2007 with Chrys-
ler. 
 Figure 3 presents the plots of dynamic conditional correlations during the 
crisis. Once again, the strongest dependency was between General Motors and 
Ford Motor Company. The mean of conditional correlations between them 
equals 0.62 (small decrease, compared with the last period), the lowest value 
equals 0.43, the highest: 0.8. Strength of connections between GM, F and DAI 
increased in condition of crisis. The mean of conditional correlations between 
General Motors and Daimler equals 0.44, the lowest value equals -0.33, the 
highest: 0.82. Conditional correlations between Ford and Daimler were as fol-
lows: mean 0.47 (increase), minimum -0.24, maximum 0.76.  
 The most interesting behavior occur in October, 2008. Strength of depen-
dencies between General Motors and Ford increased until October 14, to the 
value 0.8. Then conditional correlations decreased rapidly to the value 0.44 
(October, 31). Quite the contrary in those days was behaviour of conditional 
correlations of General Motors – Daimler and Ford – Daimler. Interestingly, 
they changed the sign from positive to negative, which remained so far one 
week starting October, 10  and it reached a value close to -0.3. Perhaps it was 
a reaction to earlier events in Iceland, which shook the financial markets. 
A similar phenomenon, but with less power changes also occurred in late No-
vember and December. On November 28, 2008 conditional correlations of Gen-



Dynamics of Multivariate Return Series of U.S. Automotive Stock Companies…  49

eral Motors - Ford increased to a value of 0.77, while the conditional correla-
tions of General Motors - Daimler and Ford - Daimler fell almost to zero. This 
was perhaps the result of the events associated with the efforts of General Mo-
tors for government aid because of the threat of bankruptcy. Since the beginning 
of 2009 dynamics of the analyzed dependencies reduced. 

 
Figure 3. Dynamic conditional correlations. Period 1.07.2008 – 5.05.2009 

    We conclude that the strongest linkages were between General Motors and 
Ford Motor Company. It turned out that the estimated strength of linkages be-
tween them was high all the time, but decreased slightly during the crisis. This 
is an interesting phenomenon, because one would expect a completely different 
perception by investors of the two companies during the crisis, because of the 
much larger GM's financial problems that led to the threat of bankruptcy. The 
strength and dynamics of linkages between companies in the U.S. automotive 
market and Daimler rose during the crisis. Numerous problems faced by car 
manufacturers, have contributed to increase the relationship between GM, F and 
DAI. In time of crisis this may be an indication for investors, who should take 
into account the presence of strong dependencies between the companies be-
longing to the same industry.  

References  
Bauwens, L., Laurent, S., Rombouts, J. (2006), Multivariate GARCH Models: a Survey, Journal 

of Applied Econometrics, 21, 79–109. 
Doornik, J. A. (2007), Ox 5 – An Object-Oriented Matrix Programming Using Ox, Timberlake 

Consultants, London. 
Engle, R., Kroner, K. F. (1995), Multivariate Simultaneous Generalized ARCH, Econometric 

Theory , 11, 122–150. 



Blanka Łęt 50

Laurent, S. (2007), G@RCH 5, Estimating and Forecasting ARCH Models, Timberlake Consul-
tants Press, London. 

Leonhardt, D. (2008), $73 an Hour: Adding it Up, The New York Times, www.nytimes.com/ 
/2008/12/10/business/worldbusiness/10iht-10leonhardt.18542483.html (10.12.2008). 

McCullagh, D. (2008), Big Three Bailout? Not So Fast, CBS News, www.cbsnews.com/stories/ 
/2008/11/12/politics/otherpeoplesmoney/main4595068.shtml (12.11.2008). 

Osińska, M. (2006), Ekonometria finansowa (Financial Econometrics), PWE, Warszawa. 
Tsay, R. S. (2002), Analysis of Financial Time Series, John Wiley&Sons, New York. 

Dynamika wielowymiarowych szeregów czasowych notowań spółek 
amerykańskiego rynku motoryzacyjnego w warunkach kryzysu 

Z a r y s  t r e ś c i. W artykule przeprowadzono analizę dynamiki powiązań pomiędzy szeregami 
zwrotów logarytmicznych trzech spółek rynku motoryzacyjnego notowanych na nowojorskiej 
giełdzie: GM, F i  DAI. Badanie przeprowadzone zostało dla dwóch okresów: przed i w czasie 
kryzysu. Dopasowano model DiagBEKK, uzyskując oszacowania dynamicznych korelacji wa-
runkowych. Wyniki badania wskazują na występowanie w czasie kryzysu silnych powiązań 
pomiędzy badanymi spółkami.  

S ł o w a  k l u c z o w e: model DiagBEKK, dynamiczna korelacja warunkowa.  




