





































Microsoft Word - 13_Olbryś_J.docx


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

Joanna Olbryś 
Bialystok University of Technology 

ARCH Effect in Classical Market-Timing Models with 
Lagged Market Variable: the Case of Polish Market† 

A b s t r a c t. The main goal of this study is to present the regressions of the GARCH versions of 
classical market-timing models of Polish equity funds. We examine the models with lagged val-
ues of the market factor as an additional variable because of the Fisher’s effect1 in the case of the 
main Warsaw Stock Exchange indexes. The market-timing and selectivity abilities of fund man-
agers are evaluated for the period Jan 2003 – June 2011. Results on both the HAC and the 
GARCH estimates are qualitatively similar, and even better in the case of the simpler HAC meth-
od. For this reason, it is not necessary to estimate the GARCH versions of market-timing models 
in the case of Polish mutual funds, even despite the strong ARCH effects that exist in these  
models.  
K e y w o r d s: market-timing, non-trading, ARCH effect, GARCH model.  

Introduction 

Market-timing is one strategy by which portfolio managers might attempt to 
obtain returns in excess of those expected of an unmanaged portfolio. One of 
the benefits of market-timing is the production of a  positively skewed distribu-
tion of returns. Treynor and Mazuy (1966) produce a single factor model de-
rived from CAPM in which a quadratic term is added to reflect the market-
timing. The T-M coefficient measures the co-skewness with the benchmark 
portfolio. Henriksson and Merton (1981) start from a similar idea, but provide 
a different interpretation of market-timing ability. Adding a term in the CAPM 
model that contains a dummy variable based on the difference between market 
return and the risk-free rate, they permit managers to choose between two levels 
of market risk: an up-market and a down-market beta (Cogneau, Hübner, 2009). 
Some other researchers extend market-timing models to multifactor as well as 
                                                 

† The author acknowledges the financial support in 2009 – 2011 from the Polish Ministry of 
Science and Higher Education within the grant no. N N113 173237. 

1 Lawrence Fisher’s effect (1966) 



Joanna Olbryś 186 

to conditional versions. In relation to the Polish market, in Olbryś (2009) the 
usefulness of the conditional multifactor market-timing models for the invest-
ment managers’ performance evaluation on the Polish market has been exam-
ined. Ferson and Schadt (1996) use a collection of public information variables. 
In Poland, the suitable variables are: (1) the lagged monthly dividend yield of 
the WSE stock index, (2) the lagged monthly level of the 1M WIBOR, (3) the 
lagged monthly measure of the slope of the term structure (Olbryś, 2009, 
p. 522). The evidence on Polish market shows that the quality and usefulness of 
these models is rather low. As for the other multifactor models, in light of the 
empirical results for the Polish market, the influence of the Fama and French’s 
(1993) size (SMB) and book-to-market (HML) spread variables, and Carhart’s 
(1997) momentum (WML) factor on the Polish equity funds’ market seems to 
be rather controversial (Olbryś, 2010a b; 2012). It is worth stressing that the 
SMB, HML, and WML factors have a diverse explanatory power for the sample 
of Polish funds. Another important finding is that the investigated funds are not 
homogeneous regarding the influence of the size, book-to-market and momen-
tum factors, despite the fact that all of them are Polish equity open-end mutual 
funds (Olbryś, 2011a). For this reason, the SMB, HML, and WML factors have 
not been taken into account as explanatory variables in our models. 

According to the literature, the methods most widely applied in market-
timing models estimation are the two proposed by White (1980) or Newey-West 
(1987) (see e.g. Ferson, Schadt, 1996; Bollen, Busse, 2001; Romacho, Cortez, 
2006; Olbryś, 2010a, b). Some previous publications also describe applications 
of the GLS procedure with correction for heteroskedasticity (see e.g. Henriks-
son, Merton, 1981; Henriksson, 1984) or the Fama-MacBeth cross-sectional 
regression approach from 1973 (Carhart, 1997). Kao et al. (1998) employ an 
autoregressive conditional heteroskedastic (ARCH) model, but without testing 
the ARCH effects. Recent studies in multifactor market-timing models in the 
case of Polish equity funds by Olbryś (2010a) present possibilities and exam-
ples of applying the seemingly unrelated regression method (SUR) which was 
described by Zellner (1962). The author’s recent research provide evidence of 
pronounced ARCH effects (Engle, 1982) in the market-timing models of Polish 
equity open-end mutual funds. For this reason, the main goal of this study is to 
present the regression results of the new GARCH(p, q) models of these funds. 
We estimate the GARCH versions of classical market-timing models with 
lagged values of the market factor as an additional independent variable because 
of the pronounced Fisher’s effect in the case of the main Warsaw Stock Ex-
change indexes. The market-timing and selectivity abilities of fund managers 
are evaluated for the period January 2003 – June 2011. In comparison to robust 
Newey-West method results, our findings suggest that the GARCH(p, q) model 
is suitable but not necessary for such applications. To the best of author’s 
knowledge, no such investigation has been undertaken for the Polish market.    



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 187

The remainder of the paper is organized as follows. Section 1 specifies 
a methodological background and a brief literature review. First, we stress the 
validity of the non-trading problem and the Fisher’s effect in the case of market-
index returns. Next, we present classical market-timing models with lagged 
values of the market factor as additional explanatory variable. We also present 
a brief theoretical framework concerning the ARCH(q) and the GARCH(p,q) 
models. In the end of Section 1, we describe tests for the ARCH effect in an 
econometric model. In Section 2, we present the data and methodology in the 
case of Polish emerging market and discuss the results obtained. Section 3 re-
calls the main findings and presents the conclusions.  

1.  Methodological Background 

1.1. Non-trading Problem and the Fisher’s Effect 

 It is worth stressing, that the empirical market microstructure literature is 
an extensive one recently. High-frequency financial data are important in study-
ing a variety of issues related to the trading process and market microstructure 
(Tsay, 2010, p. 231). For some purposes, such aspects of the market microstruc-
ture as non-trading or bid-ask spread effects can be safely ignored. However, 
for other purposes, market microstructure is central (Campbell et al., 1997). 
In 1980 Cohen et al. point to various frictions in the trading process that can 
lead to a distinction between “true” and observed returns. They have focused on 
the fact that transaction prices differ from what they would otherwise be in 
a frictionless environment. It has been reported in the literature that some em-
pirical phenomena can be attributed to frictions in the trading process (see e.g. 
Fisher, 1966; Scholes and Williams, 1977; Dimson, 1979), also on the Polish 
capital market (see e.g. Doman, Doman, 2004; Doman, 2010; Brzeszczyński et 
al., 2011; Olbryś, 2011b). It is worthwhile to note that two common elements 
among most of the phenomena are evident, the “interval effect” and the impact 
of a security’s “thinness”. The evidence that daily market-index returns exhibit 
a pronounced positive first-order autocorrelation has been called the Fisher’s 
effect since Lawrence Fisher in 1966 hypothesized its probable cause. Fisher 
suggested it was caused by a non-trading of the component securities. The ob-
served correlation is higher in those indexes that give greater weight to the secu-
rities of smaller firms. To detect for the Fisher’s effect in the market-index re-
turns, partial autocorrelations functions (PACF) can be calculated. To calculate 
PACF, first it should be determined (based on the ADF test) that the analyzed 
index series are stationary. In the next step partial autocorrelations functions for 
individual stationary processes can be calculated and the significance of the 
first-order daily serial correlation coefficients 1  can be tested, using the Que-
nouille’s test (Kufel, 2009). The evaluation of first-order serial correlation is 
carried out by testing the null hypothesis:  



Joanna Olbryś 188 

  0 1: 0H    (1) 

If the estimate 1̂  satisfies the inequality  1

1.96
ˆ

T
  , then we have no reason to 

reject the null hypothesis (1).  

The non-trading effect induces potentially serious biases in the moments 
and co-moments of asset returns such as their means, variances, covariances, 
betas, and autocorrelation and cross-autocorrelation coefficients (Campbell et 
al., 1997, p. 84). For this reason, Busse (1999) proposed lagged values of the 
market factor as an additional independent variable in the regressions of market-
timing models using Dimson’s (1979) correction. We emphasize that the Polish 
market is an emerging market and it can be expected that non-trading problem 
should be more visible than in other, developed markets. 

1.2. Classical Market-Timing Models with Lagged Market Variable 

The classical parametric Treynor – Mazuy market-timing model with lagged 
values of the market factor as additional explanatory variable can be expressed 
as: 

2
, 1 , 2 , 1 , ,( ) ,P t P P M t P M t P M t P tr r r r                 (2) 

where tFR ,  is the one-period return on riskless securities, , , ,P t P t F tr R R   is the 

excess return on portfolio P in the period t, , , ,M t M t F tr R R   is the excess return 

on portfolio M in the period t, 1, tMr  is the lagged excess return on portfolio M 

in the period t, P  measures the selectivity skills of the manager (Jensen, 1968), 

P1  is the systematic risk measure of portfolio P to changes in the market fac-

tor returns, 2P  is the systematic risk measure of portfolio P to changes in the 

lagged market factor returns, P  measures the market-timing skills of the man-

ager of portfolio P (the T-M coefficient), and tP,  is a residual term, with the 

following standard CAPM conditions: ,( ) 0P tE   , , , 1( ) 0P t P tE     . 

In a way analogous to (2), the classical parametric Henriksson - Merton 
model with lagged values of the market factor as additional explanatory variable 
can be expressed as: 

, 1 , 2 , 1 , , ,P t P P M t P M t P M t P tr r r y                 (3) 

where tPr , , tMr , , 1, tMr , P , 1P , 2P , ,P t  are as in equation (2), P  

measures the market-timing skills of the manager of portfolio P (the H-M coef-
ficient), and , ,max{0, }M t M ty r  . 



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 189

If the portfolio manager has the ability to forecast security prices, the inter-
cept P  in equations (2)–(3) will be positive (Jensen, 1968). Indeed, it repre-
sents the average incremental rate of return on the portfolio per unit time which 
is due solely to the manager’s ability to forecast future security prices. In this 
way, ˆP  measures the contribution of security selection to portfolio perfor-
mance, which corresponds to testing the null hypothesis: 

0 : 0PH        (4) 

i.e., the manager does not have any microforecasting ability. The evaluation of 
market-timing skills is carried out by testing the null hypothesis: 

  0 : 0PH    (5) 

i.e., the manager does not possess any timing ability or does not on his forecast 
(Henriksson 1984). A negative value for the regression estimate ˆP  would im-

ply a negative value for market-timing. The size of the estimate ˆP  informs us 
about the manager’s market skills. 

1.3. The GARCH(p, q) Model 

The first model that provides a systematic framework for volatility model-
ing is the ARCH model of Engle (1982). Engle proposed the ARCH models to 
capture the serial correlation of volatility (Campbell et al., 1997, p. 482). Engle 
suggested the ARCH model as an alternative to the usual time-series process. 
More recent studies of financial markets suggest that the phenomenon is quite 
common (Greene, 2002). The basic idea of the ARCH models is that 1) the in-
novation t  of the regression is serially uncorrelated, but dependent, and 2) the 

dependence of t  can be described by a simple quadratic function of its lagged 
values (Tsay, 2010). The ARCH(q) regression model is obtained by assuming 
that the mean of random variable ty , which is drawn from the conditional den-

sity function 1( )t tf y y  , is given as tx b , a linear combination of lagged endog-

enous and exogenous variables included in the information set 1t  , with b  a 
vector of unknown parameters (Engle, 1982). Formally: 

  

1

2
1 2 0

1

0

~ ( , ),

,

( , , , , ) ,

0, 0, 1, , .

t t t t

t t t

q

t t t t q i t i
i

i

y N x h

y x

h h

i q




     

 



   


 

   

  



b

b

α



 (6) 



Joanna Olbryś 190 

where t  is the innovation in a linear regression with 2( )V   , q  is the order 

of the ARCH(q) process, α  is the vector of unknown parameters, th  is the vari-
ance function. 

The null hypothesis of white noise disturbances in (6) is: 

  0 1: 0qH      (7) 

The GARCH(p, q) model generalizes the ARCH(q) model of Engle (1982) 
and is proposed by Bollerslev (1986). The GARCH(p, q) is given by: 

  

1

1 2 1 2

2
0

1 1

0

~ (

0 0 1 0 0 1 0

t t t t

t t t

t t t t q t t t p

q p

i t i j t j
i j

i j

y N x ,h ),

y x ,

h h( , , , ,h ,h , ,h , , )

h ,

, ,i , ,q,q , , j , , p, p .




  

   

  



     

 
 

 

 

    

      

 

b

b

α β 

 

 (8)  

where t , q , α , th  are as in equation (6) and β  is a vector of unknown param-
eters. 

In the GARCH(p, q) model, q  refers to the number of lags of t  and p  re-

fers to the number of lags of th  to include in the model of the regression vari-

ance (Adkins, 2010). For 0p   the process reduces to the ARCH(q) process, 

and for 0p q  , t  is simple white noise. The null hypothesis of white noise 
disturbances in (8) is: 

  0 1 1: 0; 0q pH            (9)  

In the ARCH(q) process the conditional variance is specified as a linear 
function of past sample variances only, whereas the GARCH(p, q) process al-
lows the lagged conditional variances to enter as well (Bollerslev 1986). A wide 
range of GARCH models have now appeared in the econometric literature  
(e.g. Engle, 2000; Fiszeder, 2009). The parameters of GARCH(p, q) models are 
almost invariably estimated via Maximum Likelihood (ML) or Quasi-Maximum 
Likelihood (QML: see Bollerslev, Wooldridge, 1992) methods, which bring up 
the subject of a suitable choice for the conditional distribution of t . Several 
likelihood functions are commonly used in ARCH (GARCH) estimation, de-
pending on the distributional assumption of t  (Tsay, 2010).    

1.4. Testing for ARCH Effect in an Econometric Model 

Before estimating the GARCH(p, q) model it might be useful to test for the 
ARCH effect. The simplest approach is to examine the squares of the least 



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 191

squares residuals. The autocorrelations of the squares of the residuals provide 
evidence about ARCH effects (Greene, 2002). Two tests are available. The first 
test is to apply the Ljung-Box statistics ( )Q q  (Ljung, Box, 1978). The null hy-

pothesis is that the first q  lags of ACF of the squares of the least squares resid-

uals series are zero. In practice, the choice of q  may affect the performance of 

the ( )Q q  statistic. Simulation studies suggest that the choice of ln( )q T , 
where T is the number of time periods, provides better power performance 
(Tsay, 2010, p. 33). The second test for conditional heteroskedasticity is the 
Lagrange multiplier (LM) test of Engle (1982). Lee (1991) found that the LM 
test of white noise disturbances against GARCH(p, q) disturbances in a linear 
regression model is equivalent to that against ARCH(q) disturbances. This im-
plies that under the null hypothesis of white noise, the GARCH(p, q) effect and 
the ARCH(q) effect are locally equivalent alternatives. Hence we can proceed 
by testing the ARCH(q) effect against the GARCH(p, q) effect (Lee, 1991, pp. 
269–270).   

An LM test of ARCH(q) against the hypothesis of no ARCH effects can be 
carried out by computing 2 2

q T R    in the regression of 2
te  on a constant and 

q  lagged values. Under the null hypothesis of no ARCH effects, the statistic 

has a limiting chi-squared distribution with q  degrees of freedom. Values larg-
er than the critical table value give evidence of the presence of ARCH (or 
GARCH) effects (Greene, 2002, p. 244).  

2.  Empirical Results 

2.1. The Fisher’s Effect on the Warsaw Stock Exchange 

The Fisher’s effect on the Warsaw Stock Exchange has been detected by 
Olbryś (2011b). The empirical results show a pronounced Fisher’s effect in the 
case of the WIG, mWIG40 and sWIG80 series. We observe the most clear ef-
fect for the sWIG80 series. We have no reason to reject the null hypothesis (1) 
only in the case of the WIG20 series. As mentioned above, this evidence is con-
sistent with most of the literature on friction in the trading process because the 
observed correlation is higher in those indexes that give greater weight to the 
securities of smaller firms. For the Fisher’s effect reason, we can use Dimson’s 
(1979) correction and include lagged values of the market factor (i.e. the main 
index of WSE companies – WIG) as an additional independent variable in the 
regressions of market-timing models of Polish equity open-end mutual funds to 
accommodate infrequent trading (Olbryś, 2011a).  

2.2. Data 

The period investigated is January 2, 2003 – June 30, 2011 (T=2137 obser-
vations). To detect for the ARCH effect in market-timing models of Polish 



Joanna Olbryś 192 

funds in subsamples of various length, the entire sample has been divided into 
eight subsamples: P1, P2, P3, P4, P5, P6, P7, P8 (Table 1). 

Table 1.  Subsamples in the period from Jan 2, 2003 to June 30, 2011 

Subsample  T 
P1 Jan 2, 2003–June 30, 2011 2137 
P2 Jan 2, 2004–June 30, 2011 1886 
P3  Jan 3, 2005–June 30, 2011 1631 
P4  Jan 2, 2006–June 30, 2011 1380 
P5  Jan 2, 2007–June 30, 2011 1129 
P6  Jan 2, 2008–June 30, 2011 880 
P7  Jan 5, 2009–June 30, 2011 629 
P8  Jan 4, 2010–June 30, 2011 377 

Note: T is the number of data points. 

Table 2. Equity open-end mutual funds in Poland by the end of 2002 

 Equity fund (current name) Short Name Year of 
creation 

1 Arka BZ WBK FIO Subfundusz Arka Akcji Arka 1998 
2 Aviva Investors FIO Subfundusz Aviva Investors Polskich Akcji Aviva 2002 
3 BPH FIO Parasolowy BPH Subfundusz Akcji  BPH 1999 
4 ING Parasol FIO ING Subfundusz Akcji ING 1998 
5 Investor Top 25 Małych Spółek FIO Investor 25 2002 
6 Investor Akcji Dużych Spółek FIO Investor ADS 1998 
7 Investor Akcji FIO Investor 1998 
8 Legg Mason Akcji FIO Legg Mason 1999 
9 Millennium FIO Subfundusz Akcji Millennium 2002 

10 Novo FIO Subfundusz Novo Akcji Novo 1998 
11 Pioneer FIO Subfundusz Pioneer Akcji Polskich Pioneer 1995 
12 PKO Akcji – FIO PKO 1998 
13 PZU FIO Parasolowy Subfundusz PZU Akcji Krakowiak PZU 1999 
14 Skarbiec FIO Subfundusz Akcji Skarbiec – Akcji Skarbiec 1998 
15 UniFundusze FIO Subfundusz UniKorona Akcje UniKorona 1997 

Note: The source of this Table is the Polish Financial Supervision Authority http://www.knf.gov.pl (Sept 8, 
2011).  

We have examined the performance of 15 selected equity open-end Polish 
mutual funds which were created up to the end of 2002. Our dataset includes 
returns on all the equity funds in existence in Poland from 2002 to 2011, there-
fore our results are free of survivorship bias (Table 2). We have studied daily 
logarithmic excess returns from Jan 2003 to June 2011. Daily returns on the 
main index of WSE companies are used as the returns on the market portfolio. 
The daily average of returns on 52-week Treasury bills are used as the returns 
on riskless assets. All calculations were done using Gretl 1.9.5. 

2.3. ARCH Effect in Market-Timing Models with Lagged Market Variable 

Volatility clustering, which is a common cause of heteroskedasticity, is 
more likely to be present in financial models built using higher-frequency data, 
such as daily data (Brzeszczyński et al., 2011). To detect for the ARCH effect in 



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 193

market-timing models of Polish equity open-end mutual fund portfolios in the 
period investigated Jan 2, 2003 – June 30, 2011, the LM (Lagrange Multiplier) 
and the LB (Ljung-Box) tests have been applied. The empirical results present-
ed in Table 3 show strong ARCH effect in the case of all of the funds. The null 
hypothesis (7) is rejected in these cases. Because we are using daily logarithmic 
excess returns on fund portfolios, the LM test at the lag 5q  has been applied. 

On the other hand, the LB test at the lag 8)2137ln( q  has been used (Tsay, 

2010). The p -values of all statistics are very close to zero. 

Table 3. The ARCH effect in market-timing models (2) and (3) of Polish equity mutual 
funds in the entire sample P1 (period from Jan 2, 2003 to June 30, 2011) 

 Equity fund  
(short name) 

T-M model  H-M model 
LM p-value LB p-value  LM p-value LB p-value 

1 Arka 326.8 110-68 157.6 410-30  346.9 710-73 159.5 210-30 

2 Aviva 257.1 110-53 299.9 410-60  258.3 910-54 306.7 110-61 
3 BPH 424.6 110-89 434.2 910-89  427.0 410-90 436.5 210-89 
4 ING 443.9 110-93 442.4 110-90  445.2 510-94 444.5 510-91 
5 Investor 25 404.4 310-85 145.1 210-27  390.4 310-82 142.1 810-27 
6 Investor ADS 524.4 410-111 474.7 110-97  531.8 110-112 475.1 110-97 
7 Investor 460.3 210-97 498.0 110-102  459.4 410-97 497.2 210-102 
8 Legg Mason 402.1 110-84 333.1 310-67  408.6 410-86 334.2 210-67 
9 Millennium 437.6 210-92 371.2 210-75  439.8 710-93 374.7 410-76 

10 Novo 622.2 310-132 489.3 110-100  609.5 110-129 485.7 810-100 
11 Pioneer 423.7 210-89 372.1 110-75  426.6 510-90 374.4 510-76 
12 PKO 485.9 810-103 379.1 510-77  477.1 710-101 379.0 510-77 
13 PZU 402.0 110-84 387.0 110-78  404.9 210-85 391.8 110-79 
14 Skarbiec 384.4 610-81 427.0 310-87  385.5 410-81 426.7 310-87 
15 UniKorona 371.4 410-78 519.3 510-107  376.8 210-79 519.7 410-107 

Note: The table is based on the entire sample P1; T-M (2) is the classical Treynor-Mazuy model with the 
lagged excess return on market portfolio M as additional factor; H-M (3) is the classical Henriksson-Merton 
model with the lagged excess return on market portfolio M as additional factor; LM is the Engle (1982) 
statistic at the lag equal to five, which should be distributed as chi-squared; LB is the Ljung-Box (1978) 
statistic at the lag equal to eight, which should be distributed as chi-squared. 

Tables 4a–4b present further analysis, including more details about empiri-
cal results of testing the ARCH effect in the T-M and the H-M market-timing 
models. The ARCH effect has been tested in the case of all funds and in all 
subsamples. 

 Several results in Tables 4a–4b are worth special notice. The ARCH effect 
disappears as the interval is shortened and only in the case of 5 out of 15 funds  
(i.e. Arka, Investor 25, Novo, Skarbiec and UniKorona) it persists in all samples 
P1–P8. Furthermore, if the ARCH effects are not present in the model, simple 
OLS regression is quite sufficient (Brzeszczyński et al., 2011). 



Table 4a. Result summary of the ARCH effect in the T-M and H-M models; subsamples 
P1–P4 

Fu
nd

. 
N

o.
 P1 P2 P3 P4 

T-M H-M T-M H-M T-M H-M T-M H-M 
LM LB LM LB LM LB LM LB LM LB LM LB LM LB LM LB 

1 + + + + + + + + + + + + + + + + 
2 + + + + + + + + + + + + + + + + 
3 + + + + + + + + + - + - + - + - 
4 + + + + + + + + + + + + + + + + 
5 + + + + + + + + + + + + + + + + 
6 + + + + + + + + + + + + + + + + 
7 + + + + + + + + + + + + + + + + 
8 + + + + + + + + + - + - + - + - 
9 + + + + + + + + + - + - + - + - 

10 + + + + + + + + + + + + + + + + 
11 + + + + + + + + + - + - + - + - 
12 + + + + + + + + + + + + + + + + 
13 + + + + + + + + + - + - + - + - 
14 + + + + + + + + + + + + + + + + 
15 + + + + + + + + + + + + + + + + 

Table 4b. Result summary of the ARCH effect in the T-M and H-M models; subsamples 
P5–P8 

Fu
nd

. 
N

o.
 P5 P6 P7 P8 

T-M H-M T-M H-M T-M H-M T-M H-M 
LM LB LM LB LM LB LM LB LM LB LM LB LM LB LM LB 

1 + + + + + + + + + + + + + + + + 
2 + + + + - + - + - + - + + + + + 
3 + - + - + - + - - - - - + - + - 
4 + + + + + - + - + - + - - + - + 
5 + + + + + + + + + + + + + + + + 
6 + + + + + + + + + + + + - - - - 
7 + + + + + + + + + + + + + - + - 
8 + - + - + - + - + + + + - - - - 
9 + - + - + - + - + - + - + - + - 

10 + + + + + + + + + + + + + + + + 
11 + - + - + - + - + - + - - - - - 
12 + + + + + + + + + - + - - - - - 
13 + - + - + - + - + + + + + + + + 
14 + + + + + + + + + + + + + + + + 
15 + + + + + + + + + + + + + + + + 

Note: Table 4a is based on the samples P1– P4 and Table 4b is based on the samples P5– P8 (Table 1); T-M 
(2) is the classical Treynor-Mazuy model with the lagged excess return on market portfolio M as additional 
factor; H-M (3) is the classical Henriksson-Merton model with the lagged excess return on market portfolio M 
as additional factor; LM is the Engle (1982) statistic at the lag q, which should be distributed as chi-squared; 
LB is the Ljung-Box (1978) statistic at the lag q, which should be distributed as chi-squared; + denotes that 
statistic value is larger than the critical table value of chi-squared and gives evidence of the presence of 
ARCH effect; –  denotes that statistic value is smaller than the critical table value of chi-squared. 

 



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 195

2.4. The GARCH Versions of Market-Timing Models of Polish Equity 
Mutual Funds 

Hamilton (2008) stresses that even if the researcher’s primary interest is in es-
timating the conditional mean, having a correct description of the conditional 
variance can still be quite important. By incorporating the observed features of 
the heteroskedasticity into the estimation of the conditional mean, substantially 
more efficient estimates of the conditional mean can be obtained. The most 
popular White or Newey–West corrections may not fully correct for the influ-
ence problems introduced by ARCH. The testing results from the Polish equity  

Table 5a. The HAC estimates of the T-M market-timing models of Polish equity mutual 
funds in the entire period from Jan 2, 2003 to June 30, 2011 

 Equity fund P̂  
1P̂  2P  P̂  2R  

1 Arka 410-4 ** (110-4) 0.71*** (0.03) 0.14*** (0.03) -1.86** (0.83) 0.66 
2 Aviva 310-4 ** (110-4) 0.75*** (0.03) 0.15*** (0.02) -1.45** (0.69) 0.73 
3 BPH 210-5 (110-4) 0.72*** (0.02) 0.12*** (0.02) -0.56  (0.48) 0.76 
4 ING -210-6 (110-4) 0.76*** (0.02) 0.13*** (0.02) -0.48  (0.43) 0.74 
5 Investor 25 210-4 (210-4) 0.39*** (0.02) 0.31*** (0.03) -1.76** (0.85) 0.39 
6 Investor ADS -110-4 (210-4) 0.63*** (0.03) 0.38*** (0.03) -0.55  (1.26) 0.55 
7 Investor -410-5 (110-4) 0.54*** (0.03) 0.37*** (0.03) -0.75  (0.86) 0.55 
8 Legg Mason 210-4 *(910-5) 0.69*** (0.02) 0.12*** (0.02) -0.68  (0.51) 0.74 
9 Millennium -410-6 (110-4) 0.69*** (0.02) 0.13*** (0.02) -0.76  (0.57) 0.72 

10 Novo 810-5 (210-4) 0.48*** (0.03) 0.45*** (0.02) -1.20  (1.14) 0.55 
11 Pioneer -910-5 (110-4) 0.80*** (0.03) 0.16*** (0.02) -1.13  (0.81) 0.74 
12 PKO 610-5 (210-4) 0.55*** (0.03) 0.28*** (0.03) -1.66  (1.19) 0.56 
13 PZU 310-5 (110-4) 0.71*** (0.02) 0.12*** (0.02) -0.98** (0.49) 0.73 
14 Skarbiec -410-6 (110-4) 0.46*** (0.03) 0.38*** (0.03) 0.15  (0.73) 0.51 
15 UniKorona 110-4 (110-4) 0.48*** (0.03) 0.45***(0.03) -0.54 (0.89) 0.55 

Note: The table is based on the entire sample P1; T-M (2) is the classical Treynor-Mazuy model with the 
lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors 
are in parentheses next to the coefficient estimates; the values of the adjusted determination coefficient are in 
the last column; * significant at the 10 per cent level; ** significant at the 5 per cent level; *** significant at 
the 1 per cent level. 

mutual funds dataset show pronounced ARCH effect in market-timing models 
(Tables 3–4). For this reason, the estimation of the market-timing models as the 
GARCH(p, q) models is well-founded. Although the ARCH(q) model (6) is 
simple, it often requires many parameters to adequately describe the volatility 
process. The modeling procedure of the ARCH(q) model can also be used to 
build a GARCH(p, q) model (8). However, specifying the order of 
a GARCH(p, q) model is not easy. Only the lower order GARCH models are 
used in most applications, i.e. GARCH(1,1), GARCH(1,2), GARCH(2,1), and 
GARCH(2,2) models (Tsay, 2010). According to the literature, GARCH(p, q) 
models are usually compared and selected by the information criterion of 
Akaike (AIC) and the information criterion of Schwartz (SC). Lower values of 



Joanna Olbryś 196 

the AIC and SC indexes indicate the preferred model, that is, the one with the 
fewest parameters that still provides an adequate fit to the data2. 

As an example, we present the comparison of the estimation results of mar-
ket-timing models T-M and H-M of Polish equity mutual funds in the entire 
period from Jan 2, 2003 to June 30, 2011. We use the Newey-West robust esti-
mates (HAC) as well as the robust quasi-maximum likelihood estimates (QML) 
of the parameters of the suitable GARCH(p, q) version of the market-timing 
model. Tables 5a–5b provide details on the robust HAC estimates of the T-M 
and H-M market-timing models, respectively. 

Table 5b. The HAC estimates of the H-M market-timing models of Polish equity mutual 
funds in the entire period from Jan 2, 2003 to June 30, 2011 

 Equity fund P̂  
1P̂  2P  P̂  2R  

1 Arka 710-4 ***(210-4) 0.65*** (0.04) 0.14*** (0.03) -0.14** (0.06) 0.66 
2 Aviva 610-4 **(210-4) 0.70*** (0.04) 0.15*** (0.02) -0.11** (0.05) 0.73 
3 BPH 110-4 (110-4) 0.70*** (0.03) 0.12*** (0.02) -0.04  (0.04) 0.76 
4 ING 110-4 (110-4) 0.74*** (0.03) 0.13*** (0.02) -0.04  (0.04) 0.74 
5 Investor 25 410-4 (310-4) 0.33*** (0.03) 0.31*** (0.03) -0.13* (0.06) 0.39 
6 Investor ADS -210-4 (310-4) 0.63*** (0.04) 0.38*** (0.03) -0.01  (0.08) 0.55 
7 Investor 210-5 (310-4) 0.52***(0.03) 0.37*** (0.03) -0.04  (0.06) 0.55 
8 Legg Mason 310-4 *(110-4) 0.67***(0.03) 0.12*** (0.02) -0.05  (0.04) 0.74 
9 Millennium 210-4 (210-4) 0.65*** (0.03) 0.13*** (0.02) -0.07  (0.04) 0.72 
10 Novo 110-4 (310-4) 0.45*** (0.03) 0.45*** (0.03) -0.06  (0.07) 0.55 
11 Pioneer 910-5 (210-4) 0.76*** (0.04) 0.16*** (0.02) -0.08  (0.06) 0.74 
12 PKO 310-4 (310-4) 0.50*** (0.04) 0.28*** (0.03) -0.12  (0.07) 0.56 
13 PZU 210-4 (210-4) 0.67*** (0.03) 0.12*** (0.02) -0.08* (0.04) 0.73 
14 Skarbiec -810-5 (310-4) 0.47*** (0.03) 0.38*** (0.03) 0.02  (0.06) 0.51 
15 UniKorona 110-4 (310-4) 0.47*** (0.03) 0.45*** (0.03) -0.02  (0.06) 0.55 

Note: The table is based on the entire sample P1; H-M (3) is the classical Henriksson-Merton model with the 
lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors 
are in parentheses next to the coefficient estimates; the values of the adjusted determination coefficient are in 
the last column; * significant at the 10 per cent level; ** significant at the 5 per cent level; *** significant at 
the 1 per cent level.  

The robust QML estimates of the parameters of the suitable GARCH(p, q) 
version of market-timing models are presented in Tables 6a–6b, respectively.  
It is worth stressing that some restrictions for the parameters in the 
GARCH(p, q) models (8) can be relaxed. For example, it is not necessary for 
the 2  parameter in the conditional variance equation in the GARCH(1,2) 
model to be nonnegative (Fiszeder, 2009). Note that in the case of all funds, 
both for the T-M model and for the H-M model the same variant of the 
GARCH(p, q) model has been chosen (Tables 6a–6b). 

                                                 
2 When the values of the information criterions AIC or SC for different variants of the 

GARCH(p, q) models are almost equal, the statistical significance of the parameters in the condi-
tional mean and conditional variance equations of the GARCH(p, q) model has been analyzed to 
choose the appropriate model. 



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 197

In summary, the results in Tables 5a–5b and 6a–6b clearly show that despite 
the strong ARCH effect in all models built based on the sample P1, the simpler 
robust HAC method is quite sufficient. Therefore, in our opinion, the 
GARCH(p, q) model is suitable but not necessary for such applications.  

Table 6a. The GARCH(p, q) versions of the T-M market-timing models of Polish equity 
mutual funds in the entire period from Jan 2, 2003 to June 30, 2011 

Fu
nd

. 
N

o.
 T-M model – conditional mean equation 

(p, q) 
conditional variance equation 

P̂  
1P̂  2P  P̂  0̂  1  2̂  

1̂  2̂  

1 
310-4 

(110-4) 
0.80 

(0.01) 
0.04 

(0.01) 
-1.60 
(0.52) 

(1,1) 
210-7 

(110-7) 
0.09 

(0.02) 
- 

0.90 
(0.02) 

- 

2 
310-4 

(710-5) 
0.86 

(0.01) 
0.01 

(0.005) 
-1.95 
(0.33) 

(1,2) 
210-7 

(610-8) 
0.47 

(0.12) 
-0.30 
(0.12) 

0.85 
(0.03) 

- 

3 
-710-5 

(510-5) 
0.84 

(0.005) 
0.005 

(0.004) 
0.06 

(0.18) 
(1,1) 

710-8 

(210-8) 
0.08 

(0.01) 
- 

0.91 
(0.01) 

- 

4 
-110-5 
(510-5) 

0.89 
(0.005) 

0.003 
(0.005) 

0.06 
(0.21) 

(1,1) 
710-8 

(210-8) 
0.07 

(0.01) 
- 

0.91 
(0.01) 

- 

5 
710-5 

(210-4) 
0.34 

(0.02) 
0.25 

(0.02) 
-1.06 
(1.07) 

(2,2) 
410-6 

(110-6) 
0.17 

(0.04) 
0.20 

(0.04) 
-0.16 
(0.05) 

0.74 
(0.05) 

6 
-210-4 
(910-5) 

0.94 
(0.03) 

0.07 
(0.02) 

0.64 
(0.69) 

(1,2) 
410-8 

(310-8) 
0.28 

(0.04) 
-0.21 
(0.04) 

0.93 
(0.01) 

- 

7 
-110-4 
(710-5) 

0.79 
(0.01) 

0.08 
(0.01) 

-0.28 
(0.40) 

(1,2) 
510-8 

(310-8) 
0.35 

(0.04) 
-0.26 
(0.04) 

0.91 
(0.01) 

- 

8 
910-5 

(610-5) 
0.83 

(0.007) 
0.02 

(0.005) 
-0.009 
(0.29) 

(1,2) 
510-8 

(210-8) 
0.14 

(0.03) 
-0.07 
(0.03) 

0.93 
(0.01) 

- 

9 
-210-4 
(610-5) 

0.81 
(0.008) 

0.009 
(0.005) 

-210-4 
(0.31) 

(2,1) 
810-8 

(410-8) 
0.13 

(0.03) 
- 

0.33 
(0.18) 

0.54 
(0.18) 

10 
210-5 

(110-4) 
0.14 

(0.08) 
0.64 

(0.04) 
-0.89 
(0.79) 

(1,2) 
910-8 

(610-8) 
0.40 

(0.07) 
-0.30 
(0.07) 

0.91 
(0.02) 

- 

11 
-210-4 
(610-5) 

0.89 
(0.006) 

0.02 
(0.005) 

-0.38 
(0.28) 

(2,1) 
910-8 

(410-8) 
0.13 

(0.02) 
- 

0.39 
(0.10) 

0.47 
(0.09) 

12 
-410-5 

(910-5) 
0.73 

(0.02) 
0.06 

(0.01) 
-0.34 
(0.52) 

(1,2) 
610-8 

(310-8) 
0.27 

(0.04) 
-0.17 
(0.03) 

0.91 
(0.01) 

- 

13 
-110-4 

(510-5) 
0.84 

(0.005) 
0.01 

(0.004) 
-0.17 
(0.18) 

(1,1) 
610-8 

(210-8) 
0.10 

(0.02) 
- 

0.90 
(0.02) 

- 

14 
210-5 

(110-4) 
0.42 

(0.05) 
0.41 

(0.05) 
0.35 

(0.88) 
(1,2) 

910-7 

(410-7) 
0.24 

(0.04) 
-0.16 
(0.04) 

0.91 
(0.02) 

- 

15 
-610-5 

(110-4) 
0.40 

(0.04) 
0.53 

(0.03) 
0.80 

(1.08) 
(1,2) 

810-7 

(510-7) 
0.29 

(0.03) 
-0.20 
(0.04) 

0.90 
(0.03) 

- 

Note: The table is based on the entire sample P1; T-M (2) is the classical Treynor-Mazuy model with the 
lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors 
are in parentheses below the coefficient estimates; the variance-covariance matrix of the estimated parameters 
is based on the QML algorithm; the distribution for the innovations is supposed to be normal. 

 

 



Joanna Olbryś 198 

Table 6b. The GARCH(p, q) versions of the H-M market-timing models of Polish equi-
ty mutual funds in the entire period from Jan 2, 2003 to June 30, 2011 

Fu
nd

. 
N

o.
 H-M model – conditional mean equation 

(p, q) 
conditional variance equation 

P̂  
1P̂  2P  P̂  0̂  1  2̂  

1̂  2̂  

1 510-4 
(110-4) 

0.75 
(0.02) 

0.04 
(0.01) 

-0.10 
(0.03) 

(1,1) 
210-7 

(110-7) 
0.09 

(0.02) 
- 

0.90 
(0.02) 

- 

2 810-4 
(710-5) 

0.75 
(0.03) 

0.02 
(0.005) 

-0.18 
(0.03) 

(1,2) 
210-7 

(610-8) 
0.52 

(0.17) 
-0.37 
(0.16) 

0.86 
(0.03) 

- 

3 -610-5 

(710-5) 
0.83 

(0.008) 
0.005 

(0.004) 
-410-4 
(0.01) 

(1,1) 
710-8 

(210-8) 
0.08 

(0.01) 
- 

0.91 
(0.01) 

- 

4 -410-5 
(710-5) 

0.89 
(0.009) 

0.002 
(0.005) 

-0.01 
(0.01) 

(1,1) 
710-8 

(210-8) 
0.08 

(0.01) 
- 

0.91 
(0.01) 

- 

5 210-4 

(210-4) 
0.33 

(0.03) 
0.25 

(0.02) 
-0.06 
(0.06) 

(2,2) 
310-8 

(110-8) 
0.20 

(0.04) 
-0.20 
(0.04) 

1.70 
(0.06) 

-0.70 
(0.05) 

6 -310-4 
(110-4) 

0.97 
(0.03) 

0.07 
(0.02) 

0.05 
(0.04) 

(1,2) 
410-8 

(310-8) 
0.29 

(0.04) 
-0.21 
(0.04) 

0.93 
(0.01) 

- 

7 -110-4 
(110-4) 

0.78 
(0.02) 

0.08 
(0.01) 

-0.02 
(0.02) 

(1,2) 
510-8 

(310-8) 
0.35 

(0.04) 
-0.25 
(0.04) 

0.91 
(0.01) 

- 

8 910-5 

(910-5) 
0.83 

(0.01) 
0.02 

(0.005) 
-810-4 
(0.02) 

(1,2) 
610-8 

(210-8) 
0.14 

(0.03) 
-0.07 
(0.03) 

0.93 
(0.01) 

- 

9 -710-5 
(910-5) 

0.81 
(0.01) 

0.009 
(0.005) 

-0.02 
(0.02) 

(2,1) 
810-8 

(410-8) 
0.13 

(0.03) 
- 

0.32 
(0.18) 

0.54 
(0.18) 

10 -210-4 
(110-4) 

0.14 
(0.06) 

0.65 
(0.05) 

0.02 
(0.04) 

(1,2) 
910-8 

(510-8) 
0.39 

(0.07) 
-0.29 
(0.08) 

0.91 
(0.02) 

- 

11 -210-4 
(810-5) 

0.88 
(0.01) 

0.02 
(0.005) 

-0.02 
(0.02) 

(2,1) 
910-8 

(410-8) 
0.13 

(0.02) 
- 

0.39 
(0.10) 

0.46 
(0.09) 

12 510-5 

(110-4) 
0.71 

(0.03) 
0.06 

(0.01) 
-0.03 
(0.03) 

(1,2) 
610-8 

(310-8) 
0.27 

(0.03) 
-0.18 
(0.03) 

0.91 
(0.01) 

- 

13 -210-5 

(810-5) 
0.83 

(0.008) 
0.01 

(0.004) 
-0.02 
(0.01) 

(1,1) 
610-8 

(310-8) 
0.10 

(0.02) 
- 

0.90 
(0.02) 

- 

14 -110-4 

(210-4) 
0.44 

(0.05) 
0.41 

(0.05) 
0.04 

(0.05) 
(1,2) 

910-7 

(410-7) 
0.24 

(0.04) 
-0.16 
(0.04) 

0.91 
(0.02) 

- 

15 -210-4 

(210-4) 
0.43 

(0.04) 
0.53 

(0.03) 
0.07 

(0.05) 
(1,2) 

910-7 

(510-7) 
0.29 

(0.03) 
-0.20 
(0.04) 

0.90 
(0.03) 

- 

Note: The table is based on the entire sample P1; H-M (3) is the classical Henriksson-Merton model with the 
lagged excess return on market portfolio M as additional factor; the heteroskedastic consistent standard errors 
are in parentheses below the coefficient estimates; the variance-covariance matrix of the estimated parameters 
is based on the QML algorithm; the distribution for the innovations is supposed to be normal. 

Conclusions 

Our research provides evidence of pronounced ARCH effects in the classical 
market-timing models of Polish open-end mutual funds. We detect for the 
ARCH effects in the entire period from Jan 2, 2003 to June 30, 2011, as well as 
for the 7 subperiods. For comparison, we estimate the market-timing models 
using two methods. Results on both the HAC and the GARCH estimates are 
qualitatively similar, and even better in the case of the simpler HAC method. 
For this reason, it is not necessary to estimate the GARCH versions of market-



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 199

timing models in the case of Polish mutual funds, even despite the strong 
ARCH effects that exist in these models. As for the interpretation of the esti-
mated coefficients, our empirical results can be summarized as follows:  

1. There is no evidence that equity fund managers are successful in selecti-
vity ( ˆP ). 

2. The levels of systematic risks are significantly positive ( 1̂P ).  

3. The regressions including the lagged values of the market factor as an addi-
tional explanatory variable are well-founded ( 2P ).  

4. The empirical results show no statistical evidence that Polish equity fund 
managers  have outguessed the market in the entire period Jan 2, 2003–
June 30, 2011 ( ˆP ).  

Probably the point is that mutual fund performance is affected by its operating 
style and purpose. If the purpose of the fund is to follow the market, its perfor-
mance will be close to the market and should show no superior performance. 
Therefore, a possible direction for further investigation would be the perfor-
mance evaluation including the operating style and purpose of the funds as an-
other factor (Wermers, 2000).  

References  

Adkins, L.C. (2010), Using Gretl for Principles of Econometrics, 3rd Edition, Version 1.313. 
Bollen, N.P.B.,  Busse, J.A. (2001), On the Timing Ability of Mutual Fund Managers, The Jour-

nal of Finance, 56(3), 1075–1094. 
Bollerslev, T. (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal of 

Econometrics, 31, 307–327. 
Bollerslev, T., Wooldridge, J.M. (1992), Quasi-Maximum Likelihood Estimation and Inference in 

Dynamic Models with Time-Varying Covariances, Econometric Reviews, 11, 143–179. 
Brzeszczyński, J., Gajdka, J., Schabek, T. (2011), The Role of Stock Size and Trading Intensity in 

the Magnitude of the „Interval Effect” in Beta Estimation: Empirical Evidence from the 
Polish Capital Market, Emerging Markets Finance & Trade, 47(1), 28–49. 

Busse, J.A. (1999), Volatility Timing in Mutual Funds: Evidence from Daily Returns, The Review 
of Financial Studies, 12, 1009–1041. 

Campbell, J.Y., Lo A.W., MacKinlay, A.O. (1997), The Econometric of Financial Markets, 
Princeton University Press, New Jersey. 

Carhart, M.M. (1997), On Persistence in Mutual Fund Performance, The Journal of Finance, 52, 
57–82. 

Cogneau, P., Hübner, G. (2009), The 101 Ways to Measure Portfolio Performance, Working 
Paper, http://ssrn.com/abstract=1326076 (Sept 8, 2011). 

Cohen, K.J., Hawawini, G.A., Maier, S.F., Schwartz, R.A., Whitcomb, D.K. (1980), Implications 
of Microstructure Theory for Empirical Research on Stock Price Behaviour, The Journal 
of Finance, 35, 249–257. 

Dimson, E. (1979), Risk Measurement when Shares are Subject to Infrequent Trading, Journal of 
Financial Economics, 7, 197–226.  

Doman, M., Doman, R. (2004), Ekonometryczne modelowanie dynamiki polskiego rynku finan-
sowego, Wydawnictwo Akademii Ekonomicznej w Poznaniu, Poznań. 

Doman, M. (2010), Liquidity and Market Microstructure Noise: Evidence from the Pekao Data, 
Dynamic Econometric Models, 10, 5–14.  



Joanna Olbryś 200 

Engle, R.F. (1982), Autoregressive Conditional Heteroscedasticity with Estimates of the Variance 
of United Kingdom Inflations, Econometrica, 50, 987–1007. 

Engle, R.F. (ed.) (2000), ARCH. Selected Readings, Oxford University Press.  
Fama, E.F., French, K.R. (1993), Common Risk Factors in the Returns on Stocks and Bonds, 

Journal of Financial Economics, 33, 3–56. 
Fisher, L. (1966), Some New Stock Market Indexes, Journal of Business, 39, 191–225. 
Fiszeder, P. (2009), Modele klasy GARCH w empirycznych badaniach finansowych (The Class of 

GARCH Models in Empirical Finance Research), Wydawnictwo Naukowe Uniwersytetu 
Mikołaja Kopernika, Toruń. 

Ferson, W.E., Schadt R.W. (1996), Measuring Fund Strategy and Performance in Changing Eco-
nomic Conditions, The Journal of Finance, 51, 425–461. 

Greene, W.H. (2002), Econometric Analysis, Fifth Edition, Prentice Hall, New Jersey.  
Hamilton, J.D. (2008), Macroeconomics and ARCH, Working Paper 14151, NBER Working 

Paper Series, Cambridge. 
Henriksson, R., Merton, R. (1981), On Market Timing and Investment Performance. II. Statistical 

Procedures for Evaluating Forecasting Skills, Journal of Business, 54, 513–533. 
Henriksson, R. (1984), Market Timing and Mutual Fund Performance: an Empirical Investigation, 

Journal of Business, 57, 73–96. 
Jensen, M. (1968), The Performance of Mutual Funds in the Period 1945-1964, The Journal of 

Finance, 23, 389–416. 
Kao, G.W., Cheng, L.T., Chan, K.C. (1998), International Mutual Fund Selectivity and Market 

Timing During Up and Down Market Conditions, The Financial Review, 33, 127–144. 
Kufel, T. (2009), Ekonometria. Rozwiązywanie problemów z wykorzystaniem programu Gretl, 

PWN, Warszawa. 
Lee, J. H. H. (1991), A Lagrange Multiplier Test for GARCH Models, Economics Letters, 37, 

265–271. 
Ljung, G., Box, G.E.P. (1978), On a Measure of Lack of Fit in Time Series Models, Biometrika, 

66, 67–72. 
Olbryś, J. (2012), Wieloczynnikowe hybrydowe modele market-timing polskich funduszy inwe-

stycyjnych (Multifactor Hybrid Market-Timing Models of Polish Mutual Funds), Studia 
Ekonomiczne – Zeszyty Naukowe, Uniwersytet Ekonomiczny w Katowicach, accepted for 
publication. 

Olbryś, J. (2011a), Book-to-Market, Size and Momentum Factors in Market-Timing Models: the 
Case of Polish Emerging Market, Research in Economics and Business: Central and East-
ern Europe, Tallinn School of Economics and Business Administration of Tallinn Univer-
sity of Technology, 3(2), accepted for publication. 

Olbryś, J. (2011b), The Intertemporal Cross Price Behavior and the “Fisher Effect” on the War-
saw Stock Exchange, Ekonometria 31. Theory and Applications of Quantitative Methods. 
Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu, accepted for publication. 

Olbryś,  J. (2010a), Orthogonalized Factors in Market-Timing Models of Polish Equity Funds,  
Quantitative Methods in Economics, WULS Press, 11(1), 128–138. 

Olbryś, J. (2010b), Three-Factor Market-Timing Models with Fama and French’s Spread Varia-
bles, Operations Research and Decisions, 2/2010, 91–106. 

Olbryś, J. (2009), Conditional Market-Timing Models for Mutual Fund Performance Evaluation,  
Prace i Materiały Wydziału Zarządzania Uniwersytetu Gdańskiego, 4/2, 519–532. 

Romacho, J.C., Cortez, M.C. (2006), Timing and Selectivity in Portuguese Mutual Fund Perfor-
mance, Research in International Business and Finance, 20, 348–368. 

Scholes, M., Williams, J. (1977), Estimating Betas from Nonsynchronous Data, Journal of Finan-
cial Economics, 5, 309–327. 

Treynor, J., Mazuy, K. (1966), Can Mutual Funds Outguess the Market?, Harvard Business Re-
view, 44, 131–136. 

Tsay, R.S. (2010), Analysis of Financial Time Series, John Wiley, New York. 



ARCH Effect in Classical Market-Timing Models with Lagged Market Variable… 201

Wermers, R. (2000), Mutual Fund Performance: an Empirical Decomposition into Stock-Picking 
Talent, Style, Transaction Costs, and Expenses. The Journal of Finance, 55, 1655–1703. 

Efekt ARCH w klasycznych modelach market-timing z opóźnioną 
zmienną rynkową: przypadek rynku polskiego 

Z a r y s  t r e ś c i. W artykule przedstawiono badania dokumentujące występowanie efektu 
ARCH w klasycznych modelach market-timing z opóźnioną zmienną rynkową w przypadku 
polskich funduszy akcji, w okresie styczeń 2003-czerwiec 2011. Dokonano estymacji wersji 
GARCH odpowiednich modeli oraz porównano jakość modeli GARCH i modeli uzyskanych 
metodą HAC. Wyniki wskazują, że modele GARCH są odpowiednie, ale metoda HAC jest wy-
starczająca, pomimo występowania efektu ARCH. Podano również interpretacje parametrów 
otrzymanych modeli w badanej grupie funduszy.     

S ł o w a  k l u c z o w e: model market-timing, niesynchroniczne transakcje, efekt ARCH, model 
GARCH. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 


