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D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2018.005  Vol. 18 (2018) 81−97 

Submitted November 25, 2018  ISSN (online) 2450-7067 
Accepted December 17, 2018 ISSN (print) 1234-3862 

Dominik Krężołek* 

Testing Day of the Week Effect  
on Precious Metals Market 

A b s t r a c t. Market efficiency assumes that asset prices should be characterized by random-
ness and unpredictability, so that potential market participants are not able to generate above-
average profits. This means that there should be no seasonal phenomenon in time series, which 
clearly projects a certain pattern of behavior of financial assets. The paper is an attempt to verify 
some specific seasonal effect called “the day of the week” on the precious metals market using 
AR-GARCH(APARCH) models. The selection of this area is not accidental. Precious metals 
are an alternative to classic capital investments, especially in the case of financial and economic 
crises. In addition, the literature shows a gap in this area in terms of dynamics analysis on 
commodity markets, if compared to capital market. The results are not unambiguous and the 
effect of seasonality was observed for the volatility of gold returns in the entire period and in 
the period of upward trend (positive returns on Fridays). The returns of palladium suffer from 
seasonality during entire period and in the period of upward trend (positive returns on Fridays, 
negative on Mondays, Tuesdays and Thursdays). Moreover, it was observed that the AR-
APARCH models is more appropriate when taking into account the heavy-tail distributions of 
residuals. 

K e y w o r d s: APARCH model; day of the week; GARCH model; precious metals; time se-
ries. 

J E L Classification: C22; G01; G14. 

Introduction 
 In the theory of economics and finance the term “market efficiency” as-
sumes that at any time asset prices accurately reflect the full available 

                                                
* Correspondence to: Dominik Krężołek, Department of Demography and Economic Sta-

tistics, University of Economics in Katowice, 1 Maja 50 Street, 40-287 Katowice, Poland,  
e-mail: dominik.krezolek@ue.katowice.pl. 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

82 

information about them. The term "effective market" first appeared in 
Bachelier's paper (Bachelier, 1900), but it did not get much interest from the 
scientific community. In the second half of XX century, the American econo-
mist Eugene Fama discussed this problem and considered so-called efficient-
market hypothesis (Fama, 1970) in the context of weak, semi-strong and 
strong efficiency. According to his perception, week market efficiency as-
sumes that currently realized prices reflect the historical data and it is not pos-
sible to forecast future level of prices from historical data. The hypothesis of 
semi-strong efficiency proclaims that current asset prices reflect all public in-
formation available, including historical data, reports and any available eco-
nomic forecasts The last one – strong market efficiency assumes the existence 
of full public and non-public information about assets considered, and this 
information is available for everyone at any given time. This hypothesis is the 
most restrictive and means that all kind of analyzes carried out on financial 
market, including technical and fundamental analysis, would be unnecessary, 
as it wouldn’t be possible to obtain above-average profits based on full avail-
able information.  
 As one can see, the efficiency of the market doesn’t go hand in hand with 
the main goal of investment activity, which is multiplying assets. In this paper 
some selected calendar anomalies are considered. As “calendar anomalies” 
we define statistically significant differences in levels of prices/returns for 
a given financial assets depending on the adopted period.  

1. Calendar Effects on Financial Market – Literature Review 
 When introducing the problem of calendar anomalies, it is worth to define 
those which are most often considered in the analysis of financial market data. 
One can mention here, among others, such anomalies like effect of the month 
in the year, effect of the turn of the month, holidays effect, effect of the day of 
the week or even the effect of the hour on the day. Each of these anomalies 
undermines the efficiency of the market and is the topic of analyzes in scien-
tific communities around the world.  
 The effect of the month in the year reflects significant differences in the 
average values of returns of financial assets in particular months of the year. 
The most popular anomalies in this group are the effect of January, May or 
September. In the first case, it was observed that average level of returns in 
January is significantly higher than these observed in other months. In turn, 
May and September effects are associated with semi-low level of returns. 
 The turn of the month is related by the phenomenon when we can observe 
the average higher prices (returns) at the end of month if compared to their 
values in the first days of the following month. This situation may result from 
the allocation of funds obtained in the middle of the month in investment funds 



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DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

83 

(Fiszeder et al., 2013). In the case of the holidays effect, which s related to the 
effect of the month, increases of financial assets returns are observed before 
holidays, while declines in the period just after holidays. A similar relationship 
can be observed in the case of holiday periods, when no trading sessions are 
held. 
 If the anomalies during the week are observed, it means so-called the day 
of the week effect. It can be described as significant differences of the level of 
returns on financial assets depending on the day of the week (most often five-
day week is considered, however, there are some assets listed in the seven-day 
trading system (e.g. the returns of electricity prices (Ganczarek-Gamrot, 
2013)). 
 The phenomenon of volatility in daily returns during a week is the prob-
lem analyzed not only by practitioners but also by scientists. K. French (1980), 
while analyzing the U.S. market, observed high, statistically significant vola-
tility in stocks returns. In his research he showed that the returns realized on 
Friday were significantly higher comparing to the other days, while the returns 
realized on Monday – respectively lower, negative on average. Other research-
ers also came to similar conclusions, including Lakonishok and Smidt (1988). 
Jaffe and Westerfield analyzed the capital markets in Japan and Australia and 
they observed average negative, statistically significantly different returns on 
Tuesdays comparing with other days of the week (Jaffe et al., 1985). On the 
Polish capital market that kind of research was conducted, among others, by 
Szyszka (1999). In his research based on data from the Warsaw Stock Ex-
change, he observed positive returns on Mondays and negative on Tuesdays. 
Similar results were obtained by Landmesser (2006), Witkowska, Kompa 
(2007). Additionally, they pointed out positive returns on Fridays. A slightly 
different study was conducted by a Fiszeder and Kożuchowska (2013). They 
studied anomalies of indices WIG and WIG20. The obtained results indicating 
the turn of the month effect, no seasonal and mild disturbances during the 
week. 
 When analyzing the precious metals market, there is few papers in which 
the issue of seasonality in the context of anomalies would be discussed. Ma 
(1986) in his research showed positive returns on Fridays and negative Mon-
days for gold. More recent studies, taking into account the last 30 years, result 
in slightly different conclusions. Aksoy (2013) observed negative returns form 
gold on Mondays and Fridays on Istanbul Gold Exchange. Arora et al. (2013) 
discovered positive, statistically significant returns on Tuesdays and Fridays. 
In research of Kohli (2012) the positive returns on Fridays and Wednesdays 
for gold and positive returns on Wednesdays for silver were detected. 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

84 

 The use of appropriate statistical tools allows to create models that ade-
quately reflect such kind of seasonality. The results will allow to effectively 
forecast level of returns or prices in economic (financial) time series. 

2. Methodology and Statistical Tools 
 Based on the regressive approach, the model describing the seasonality of 
the returns, taking into account different days of the week, can be written in 
the following way: 

𝑟" = ∑ 𝜔&𝑑&"(
&)* + 𝜀"  (1) 

where 𝑟" is the return on the asset at the moment 𝑡, while 𝑑&" is the binary 
variable representing the 𝑘 − 𝑡ℎ day of the week. 
 To describe volatility on financial time series one of the most popular 
model is the one proposed by Engle (1982) – the ARCH model extended by 
Bollerslev (1986) into the GARCH model. One of the main assumptions of 
ARCH model is the time-independence of returns. The expected return and 
variance can be described by historical information and presented by formu-
las: 

𝜇" = 𝐸(𝑟"|𝐼"6*) (2) 
𝜎"9 = 𝜎9(𝑟"|𝐼"6*) (3) 

where 𝜇" and 𝜎"9 define conditional expected return and conditional variance 
in time 𝑡, whereas 𝐼"6* defines the information set available in time 𝑡 − 1. The 
GARCH(𝑚, 𝑠) model of Bollerslev can be described using formulas (Tsay, 
2005): 

𝑟" − 𝜇" = 𝑎" = 𝜎"𝜀" (4) 
𝜎"9 = 𝛼@ + ∑ 𝛼A𝑎"6A9B

A)* + ∑ 𝛽D𝜎"6D9E
D)*  (5) 

where 𝛼@ > 0. 𝛼A ≥ 0 for 𝑖 > 0. 𝛽D ≥ 0 and ∑ (𝛼A + 𝛽A)
BKL(B,E)
A)* < 1.  

 The family of GARCH models is comprehensively described in the liter-
ature where many interesting properties are highlighted, i.e. the ability of mod-
elling heavy-tailed distribution. The main disadvantage of GARCH models is 
that do not describe asymmetry observed in the and neither both leverage and 
long-memory effects. To solve this problems Ding et al. (1993) proposed new 
family of models describing these stylized facts observed in financial time 
series – the class of APARCH (Asymmetric Power ARCH) models. Mathe-
matical formula takes a form (Karanasos et al., 2006): 

𝜎"N = 𝛼@ + ∑ 𝛼A(|𝑎"6A| − 𝛾A𝑎"6A)N
P
A)* + ∑ 𝛽D𝜎"6DNQ

D)*  (6) 



Testing Day of the Week Effect on Precious Metals Market 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

85 

where −1 < 𝛾A < 1 and 𝛿 > 0. The parameter 𝛾 describes the leverage effect. 
A positive value of  𝛾 means that past negative shocks have a deeper impact 
on current conditional volatility than past positive shocks. A negative value 
means the opposite (negative value means that positive information has 
stronger impact than the negative information on the price volatility). Moreo-
ver, ARCH and GARCH models are special cases of APARCH (ARCH(q) for 
𝛿	 = 	2, 𝛾A 	= 	0. 𝛽A = 	0. GARCH (p, q) for 𝛿	 = 	2, 𝛾A = 	0). The estimation 
of unknown parameters of APARCH model is usually conducted using the 
MLE. 
 In this paper, to model volatility, we used the combination of two models: 
AR and GARCH (APARCH). The first one represents conditional expected 
return (with a part related to the seasonality effect) whereas the second one is 
related to conditional variance. Therefore, we can write the formulas for 
AR(m)-GARCH(p, q) and AR(m)-APARCH(p, q) models as below: 
- AR-GARCH: 

𝑟" − 𝜇 − ∑ 𝜑A𝑟"6AB
A)* − ∑ 𝜔&𝑑&"(

&)* = 𝑎" = 𝜎"𝜀" (7) 
𝜎"9 = 𝛼@ + ∑ 𝛼A𝑎"6A9P

A)* + ∑ 𝛽D𝜎"6D9Q
D)*  (8) 

- AR-APARCH: 
𝑟" − 𝜇 − ∑ 𝜑A𝑟"6AB

A)* − ∑ 𝜔&𝑑&"(
&)* = 𝑎" = 𝜎"𝜀" (9) 

𝜎"N = 𝛼@ + ∑ 𝛼A(|𝑎"6A| − 𝛾A𝑎"6A)N
P
A)* + ∑ 𝛽D𝜎"6DNQ

D)*  (10) 

where 𝜑A, 𝑖 = 1,… ,𝑚 stands form autoregressive model’s parameters and 𝑑&" 
is dummy variable representing the 𝑘 − 𝑡ℎ day of the week in the equation of 
conditional expected return. To avoid the effect of collinearity between inter-
cept and dummy variables representing 𝑘 − 𝑡ℎ day of the week in equations 
(7) and (9) while estimating unknown parameters, one of the dummy variables 
has to be omitted and its missing value should be assessed using appropriates 
identities (Fiszeder et al., 2013) 

From the literature we know that the correct financial time series analysis 
requires to examine simultaneously the conditional mean, conditional variance 
and standardised residuals described by the proper probability function.  The 
most popular distributions describing residuals 𝜀" are conditional standard nor-
mal, t-Student or Generalized Error Distribution (GED) (Piontek, 2002). This  
t-Student and GED distributions have gained popularity because of some spe-
cific features as high level of kurtosis or heavy tails (similar to distributions 
based on Extreme Value Theory). The conditional standard normal, t-Student 
and GED distributions for residuals can be described in terms of probability 
distribution functions as follow: 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

86 

𝑓X(𝜀", 𝜎"9; 𝜃) =
*

[\√9^
𝑒𝑥𝑝 b− c\d

9[\d
e (11) 

𝑓f"gh(𝜀", 𝜎"9; 𝜃) =
ijklmd n

opij
k
dnq^(r69)

j1 + c\d

(r69)[\d
n
klm
d  (12) 

𝑓stu(𝜀", 𝜎"9; 𝜃) = 26
klm
k

r

[\v
wxkymz
wx{kymz9

ydki(rym)

𝑒𝑥𝑝

⎩
⎪
⎨

⎪
⎧

−*
9 �
� c\

[\v
wxkymz
wx{kymz9

ydk

��

r

⎭
⎪
⎬

⎪
⎫

   (13) 
where {𝜀"} stands for the sequence of iid random variables, 𝜎"9 is the condi-
tional variance of the process, 𝜃 is the vector of estimated parameters, 𝑣 is the 
number of degrees of freedom, and Γ(𝑘) = ∫ 𝑥&6*𝑒6* 𝑑𝑥��

@  is the gamma 
function with parameter 𝑘. 
 If the model is fitted, the next step is to assess the goodness of this fitting. 
We can use the information criteria of: Akaike (AIC), Schwarz (BIC) and 
Hannan-Quinn (HQC). The values of these criteria are calculated using for-
mulas as below: 

𝐴𝐼𝐶 = −2𝑙𝑛[𝐿𝐿𝐹(𝜃�)] + 2𝑘 (14) 
𝐵𝐼𝐶 = −2𝑙𝑛[𝐿𝐿𝐹(𝜃�)] + 𝑘𝑙𝑛(𝑛) (15) 
𝐻𝑄𝐶 = −2𝑙𝑛[𝐿𝐿𝐹(𝜃�)] + 2𝑘𝑙𝑛[𝑙𝑛(𝑛)] (16) 

where 𝐿𝐿𝐹(𝜃�) is the log-likelihood function of the parameters vector 𝜃�, 𝑘 is 
the number of estimated parameters and 𝑛 is the number of observations. The 
final selection is based on the values of one the information criteria: the lower 
values, the better the model. 

3. Empirical Analysis on Precious Metals Market 
 The precious metal market is part of the commodity market – an alterna-
tive area for classical financial investments. Its popularity has increased due 
to the opportunities that offers to investors. The main reason for investing 
funds on the metals market is to hedge against financial crises. The empirical 
research shows a negative correlation of precious metals prices and stock ex-
change indices (Draper et al., 2010). Precious metals have a wide range of 
applications. They are used in many industries (e.g. automotive, space indus-
try etc.), medicine, biology, jewelry, cosmetology and even in gastronomy.  
It is difficult to synthetically indicate the exact use of precious metals, but it 
is worth mentioning about their psychological aspects: precious metals are 



Testing Day of the Week Effect on Precious Metals Market 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

87 

usually associated with wealth and economic stability, especially during crisis. 
Therefore, the analysis of that kind of assets is justified. 
 The hypothesis says that precious metals market is characterized by cal-
endar anomalies (in this case: the day of the week effect). The daily log-returns 
for gold, silver, platinum and palladium from the London Metal Exchange are 
considered. The period of analysis is January 2000 – December 2016, finally 
divided into two sub-periods: a period of upward trend (January 2000 – Au-
gust 2011) and a downward trend respectively (September 2011 – December 
2016). A comparison of the volatility of gold and silver prices with the vola-
tility of global stock indices seems worth commenting. For example, the 
S&P500 and NASDAQ COMPOSITE quotes together with gold and silver 
are presented in the same time period (Fig. 1).  

 
Figure 1.  Quotations of gold (top-left), silver (top-right), S&P500 (bottom-left) and 

NASDAQ COMPOSITE (bottom-right) – all period 

 It is clearly visible that the upward trend after the crisis on the financial 
markets (2007–2009) coincides with a significant downward trend observed 
for gold/silver. In addition, when considering the period of the crisis, it is 
worth mentioning that then were recorded the highest gold prices comparing 
to the entire period. The driving force behind this is the withdrawal of funds 
from the capital market and allocation in an alternative way (e.g. on metals 
market) in order to hedge against price drops. 
 The main goal of the analysis is the volatility of returns observed within 
precious metals. Based on the data from the LME, daily log-returns were 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

88 

estimated. The time series of the analyzed data during entire period are pre-
sented in figure 2. 

 
Figure 2.  Log-returns of gold (top-left), silver (top-right), platinum (bottom-left) and 

palladium (bottom-right) – all period 

Table 1.  Average levels of returns for days of the week 
ALL ENTIRE PERIOD 

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY 
GOLD 0.00006 –0.00011 –0.00005 0.00018 0.00151 

SILVER 0.00021 –0.00006 0.00026 –0.00049 0.00134 
PALLADIUM –0.00071 0.00015 –0.00013 –0.00028 0.00147 
PLATINUM 0.00016 –0.00029 0.00009 0.00015 0.00074 
PERIOD 1 PERIOD OF UPWARD TREND 

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY 
GOLD 0.00038 0.00016 0.00045 0.00021 0.00193 

SILVER 0.00064 0.00014 0.00111 –0.00042 0.00202 
PALLADIUM –0.00044 0.00023 0.00027 –0.00068 0.00155 
PLATINUM 0.00082 –0.00034 0.00049 0.00031 0.00118 
PERIOD 2 PERIOD OF DOWNWARD TREND 

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY 
GOLD –0.00064 –0.00071 –0.00114 0.00012 0.00060 

SILVER –0.00073 –0.00051 –0.00163 –0.00062 –0.00014 
PALLADIUM –0.00130 –0.00004 –0.00101 0.00059 0.00129 
PLATINUM –0.00126 –0.00019 –0.00080 –0.00019 –0.00023 

 
 The highest volatility was observed for SILVER and palladium return 
rates. Data clusters and grouping of variances are clearly visible. In addition, 
all figures show a significant increase in the level of volatility corresponding 
to the beginning of the upward trend on global economy. In table 1 the average 



Testing Day of the Week Effect on Precious Metals Market 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

89 

levels of returns for every day of the week for entire period and all subperiods 
are shown. 
 The results show that during entire period the average positive returns on 
Fridays were observed, regardless of the type of metal. Gold realized negative 
returns on Tuesdays and Wednesdays, silver on Tuesdays and Thursdays, pal-
ladium on Mondays, Wednesdays and Thursdays, while platinum only on 
Tuesdays. Positive returns were observed for other days. During the upward 
trend, the average returns on Fridays were much higher comparing to the other 
days of the week. All metals generated on average positive returns and the 
differences determined by the day of the week are clearly visible. Similar con-
clusions can be drawn in the case of a downward trend, with the average neg-
ative returns observed on all days. Nevertheless, we can still see the diversity 
due to the day of the week. In the case of gold and silver, the lowest returns 
were observed on Wednesdays, while for platinum and palladium on Mon-
days. Summarizing this results we can suppose that the day of the week is 
observed on metals market. 

Table 2.  AR-GARCH and AR-APARCH models – gold – entire period 
Parameter AR(1)-GARCH(1,1)-N p-value AR(1)-GARCH(1,1)-S p-value AR(1)-GARCH(1,1)-GED p-value 

𝜑@ 0.00158 0.000*** 0.00132 0.000*** 0.00124 0.001*** 
𝜑* –0.00111 0.950 –0.02239 0.097* –0.02077 0.192 
𝜔��� –0.00164 0.001*** –0.00109 0.005*** –0.00106 0.035** 
𝜔�g� –0.00159 0.002*** –0.00129 0.002*** –0.00113 0.034** 
𝜔��h –0.00169 0.001*** –0.00112 0.007*** –0.00100 0.042** 
𝜔��g� –0.00145 0.007*** –0.00121 0.005*** –0.00118 0.017** 
𝜔��A 0.00158 0.000*** 0.00132 0.000*** 0.00124 0.000*** 
𝛼@ 0.00000 0.031** 0.00000 0.001*** 0.00000 0.003*** 
𝛼* 0.03952 0.002*** 0.04504 0.000*** 0.04256 0.000*** 
𝛽* 0.94563 0.000*** 0.94754 0.000*** 0.94636 0.000*** 
𝜈 – – 4.71052 0.000*** 1.18248 0.000*** 

Parameter AR(1)-APARCH(1,1)-N p-value AR(1)-APARCH(1,1)-S p-value AR(1)-APARCH(1,1)-GED p-value 
𝜑@ 0.00164 0.000*** 0.00138 0.000*** 0.00130 0.000*** 
𝜑* –0.00220 0.357 –0.02688 0.057* –0.02415 0.000*** 
𝜔��� –0.00164 0.001*** –0.00112 0.004*** –0.00111 0.001*** 
𝜔�g� –0.00161 0.002*** –0.00131 0.001*** –0.00117 0.000*** 
𝜔��h –0.00170 0.001*** –0.00118 0.003*** –0.00104 0.002*** 
𝜔��g� –0.00151 0.005*** –0.00127 0.002*** –0.00125 0.000*** 
𝜔��A 0.00164 0.000*** 0.00138 0.000*** 0.00130 0.000*** 
𝛼@ 0.00000 0.030** 0.00000 0.001*** 0.00000 0.001*** 
𝛼* 0.04547 0.000*** 0.05035 0.000*** 0.04965 0.000*** 
𝛽* –0.07825 0.503 –0.28993 0.004*** –0.19508 0.090** 
𝛾* 0.94625 0.000*** 0.95404 0.000*** 0.95079 0.000*** 
𝛿 1.66345 0.000*** 1.24743 0.000*** 1.36498 0.000*** 
𝜈 – – 4.74212 0.000*** 1.18170 0.000*** 

 
 In the next stage of the analysis, the effect of the day of the week was 
verified using the models presented in theoretical part of this paper. The 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

90 

seasonality effect was included in the conditional mean equation. At the be-
ginning, the assumption of normality of returns was examined. The results 
showed that this hypothesis has to be rejected. In the next step, based on the 
appropriate statistical tests (the information criteria, Jung-Box autocorrelation 
test and the ARCH effect test), the following models were finally selected for 
given lags: AR(1)-GARCH(1,1) and AR(1)-APARCH(1,1). It was also as-
sumed that the rest of the model follow normal (N), t-Student (S) and GED 
distributions. The stability of model parameters was assessed using the Chow 
and Nyblom tests. In tables 2–4, as example, the results obtained for testing 
the day of the week effect for gold (entire period), palladium (a period of up-
ward trend) and platinum (a period of downward trend) are presented. 

Table 3.  AR-GARCH and AR-APARCH models – palladium – period of upward 
trend 

Parameter AR(1)-GARCH(1,1)-N p-value AR(1)-GARCH(1,1)-S p-value AR(1)-GARCH(1,1)-GED p-value 
𝜑@ 0.00151 0.037** 0.00118 0.044** 0.04351 0.000*** 
𝜑* 0.11368 0.000*** 0.08499 0.000*** 0.00041 0.000*** 
𝜔��� –0.00228 0.028** –0.00173 0.038** –0.00053 0.000*** 
𝜔�g� –0.00166 0.102 –0.00103 0.210 –0.00041 0.000*** 
𝜔��h –0.00128 0.196 –0.00040 0.636 –0.00012 0.905 
𝜔��g� –0.00157 0.139 –0.00105 0.223 –0.00041 0.000*** 
𝜔��A 0.00151 0.037** 0.00118 0.044** 0.00041 0.551 
𝛼@ 0.00002 0.006*** 0.00001 0.013** 0.00001 0.009*** 
𝛼* 0.13384 0.000*** 0.17310 0.000*** 0.15462 0.000*** 
𝛽* 0.84171 0.000*** 0.82803 0.000*** 0.83330 0.000*** 
𝜈 – – 4.20127 0.000*** 1.08528 0.000*** 

Parameter AR(1)-APARCH(1,1)-N p-value AR(1)-APARCH(1,1)-S p-value AR(1)-APARCH(1,1)-GED p-value 
𝜑@ 0.00156 0.056* 0.00122 0.162 0.00044 0.000*** 
𝜑* 0.11159 0.000*** 0.08166 0.000*** 0.04126 0.000*** 
𝜔��� –0.00222 0.048** –0.00173 0.112 –0.00055 0.000*** 
𝜔�g� –0.00159 0.144 –0.00101 0.332 –0.00044 0.000*** 
𝜔��h –0.00124 0.263 –0.00038 0.742 –0.00014 0.000*** 
𝜔��g� –0.00154 0.183 –0.00100 0.406 –0.00044 0.000*** 
𝜔��A 0.00156 0.045** 0.00122 0.042** 0.00044 0.000*** 
𝛼@ 0.00002 0.008*** 0.00002 0.015** 0.00002 0.010** 
𝛼* 0.14094 0.000*** 0.17941 0.000*** 0.16495 0.000*** 
𝛽* –0.05296 0.409 –0.06803 0.224 –0.06944 0.150 
𝛾* 0.84864 0.000*** 0.84015 0.000*** 0.84345 0.000*** 
𝛿 1.67146 0.000*** 1.35168 0.000*** 1.38839 0.000*** 
𝜈 – – 4.15511 0.000*** 1.08185 0.000*** 

 
 Discussing the results presented in the tables above, it was indicated that 
for the analyzed metals, the statistically significant effects of the week were 
mainly observed during the entire period or during the upward trend period. 
In the case of a downward trend, these results can’t be unambiguously con-
firmed. Similar results were obtained for other metals in the examined sub-
periods. The values of 𝛾* for each model are positive which means that 



Testing Day of the Week Effect on Precious Metals Market 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

91 

negative information has stronger impact than the positive information on the 
returns volatility of analyzed metals. The figures below present synthetic re-
sults obtained for all metals tested for every day of the week. The sign (+) 
indicates the positive return, while the sign (–) – negative respectively. Shaded 
cells indicate statistically significant returns. 

Table 4.  AR-GARCH and AR-APARCH models – platinum – period of downward 
trend 

Parameter AR(1)-GARCH(1,1)-N p-value AR(1)-GARCH(1,1)-S p-value AR(1)-GARCH(1,1)-GED p-value 
𝜑@ –0.00022 0.761 –0.00030 0.663 –0.00030 0.873 
𝜑* 0.07461 0.021** 0.06262 0.038 0.06995 0.023** 
𝜔��� –0.00110 0.282 –0.00078 0.431 –0.00074 0.670 
𝜔�g� –0.00016 0.869 –0.00001 0.993 0.00008 0.969 
𝜔��h –0.00024 0.809 –0.00028 0.774 –0.00024 0.935 
𝜔��g� –0.00014 0.893 –0.00010 0.919 –0.00016 0.946 
𝜔��A –0.00022 0.761 –0.00030 0.663 –0.00030 0.682 
𝛼@ 0.00000 0.164 0.00000 0.142 0.00000 0.140 
𝛼* 0.03465 0.003*** 0.03374 0.001*** 0.03441 0.001*** 
𝛽* 0.95579 0.000*** 0.95879 0.000*** 0.95664 0.000*** 
𝜈 – – 13.18740 0.001*** 1.67895 0.000*** 

Parameter AR(1)-APARCH(1,1)-N p-value AR(1)-APARCH(1,1)-S p-value AR(1)-APARCH(1,1)-GED p-value 
𝜑@ –0.00044 0.489 –0.00042 0.491 –0.00045 0.276 
𝜑* 0.06781 0.018** 0.05963 0.051* 0.06531 0.040** 
𝜔��� –0.00095 0.340 –0.00069 0.420 –0.00065 0.434 
𝜔�g� 0.00001 0.989 0.00007 0.941 0.00017 0.500 
𝜔��h 0.00004 0.970 –0.00010 0.920 –0.00002 0.985 
𝜔��g� 0.00002 0.988 0.00000 0.997 –0.00003 0.971 
𝜔��A –0.00044 0.561 –0.00042 0.535 –0.00045 0.522 
𝛼@ 0.00000 0.256 0.00000 0.275 0.00000 0.262 
𝛼* 0.02989 0.018** 0.03046 0.003*** 0.03036 0.013** 
𝛽* 0.41536 0.391 0.28854 0.636 0.37204 0.489 
𝛾* 0.96999 0.000*** 0.96847 0.000*** 0.96935 0.000*** 

𝛿 1.25298 0.166 1.50355 0.434 1.30223 0.280 
𝜈 – – 14.20030 0.002*** 1.70171 0.000*** 
 
 The results confirm the existence of day of the week effect, mainly in the 
case of gold and palladium in the whole period and in the period of upward 
trend. Statistically significant positive returns were observed on Fridays, while 
during the other days the returns were negative. Different results were ob-
tained for the period of downward trend. There were statistically significant 
negative returns on Tuesdays and Wednesdays. Taking into account the mod-
els used, the statistical significance of the day of the week effect was primarily 
observed for models with the distribution of residuals described by the t-Stu-
dent or GED distribution. It may be the result of clustering in variance and 
existence of outliers. The selection of final model depends on the values of 
information criteria given by the formulas (14)–(16). The results are below. 

  



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

92 

Table 5.  Day of the week effect – statistically significant results – entire period 
SILVER AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – + – – + 

FRIDAY + + + + + + 
GOLD AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – – – 

FRIDAY + + + + + + 
PALLADIUM AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – – – 

FRIDAY + + + + + + 
PLATINUM AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 
MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – + – + + 
THURSDAY – – – – – – 

FRIDAY + + + + + + 

Table 6.  Day of the week effect – statistically significant results – period of upward 
trend 

SILVER AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 
MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – – – 

FRIDAY + + + + + + 
GOLD AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – – – 

FRIDAY + + + + + + 
PALLADIUM AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – – – 

FRIDAY + + + + + + 
PLATINUM AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 
MONDAY – – + – + + 
TUESDAY – – – – – – 

WEDNESDAY – + + – + + 
THURSDAY – – – – + + 

FRIDAY + + + + + + 



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DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

93 

Table 7.  Day of the week effect – statistically significant results – period of down-
ward trend 

SILVER AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 
MONDAY – – + – – + 
TUESDAY – – – – – – 

WEDNESDAY – – – + – – 
THURSDAY – + + – + + 

FRIDAY – – – + – – 
GOLD AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – – – 

FRIDAY + + + + + + 
PALLADIUM AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 

MONDAY – – – – – – 
TUESDAY – – – – – – 

WEDNESDAY – – – – – – 
THURSDAY – – – – + + 

FRIDAY + + + + + + 
PLATINUM AR-GARCH-N AR-GARCH-S AR-GARCH-GED AR-APARCH-N AR-APARCH-S AR-APARCH-GED 
MONDAY – – – – – – 
TUESDAY – – + + + + 

WEDNESDAY – – – + – – 
THURSDAY – – – + + – 

FRIDAY – – – – – – 
 
 Considering the entire period we noticed that for gold and silver, the best 
fitted model was the AR-APARCH model with conditional t-Student distribu-
tion of residuals. For palladium and platinum information criteria give differ-
ent assessments, however, these are still models with residuals described by 
conditional t-Student distribution. Similar results were obtained for gold and 
silver during the upward trend. In the case of palladium, the AR-GARCH 
model with the GED distribution for residuals was proposed, while for plati-
num the AR-APARCH model with t-Student distribution. The last subperiod 
represents the downward trend. The results suggest different models: for sil-
ver, AR-GARCH model with GED distribution for residuals, AR-GARCH 
model for gold with residuals described by t-Student distribution, AR-
APARCH model for palladium with t-Student distribution for residuals, and 
finally AR-GARCH model for platinum with t-Student distribution for resid-
uals as well. We didn’t observe any model with conditional normal distribu-
tion for residuals. In figure 3 the AR(1)-APARCH(1, 1)-S model for gold and 
silver is presented (entire period). 

  



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

94 

Table 8. Information criteria for estimated models 

METAL MODEL ENTIRE PERIOD PERIOD OF UPWARD  
TREND 

PERIOD OF DOWNWARD  
TREND 

AIC BIC HQC AIC BIC HQC AIC BIC HQC 

SILVER 

AR-GARCH-N –23385 –23327 –23365 –16079 –16025 –16060 –7312 –7265 –7294 
AR-GARCH-S –23843 –23779 –23820 –16320 –16260 –16299 –7535 –7483 –7515 

AR-GARCH-GED –23829 –23765 –23807 –16301 –16240 –16279 –7542 –7490 –7522 
AR-APARCH-N –23400 –23330 –23375 –16119 –16053 –16095 –7310 –7253 –7289 
AR-APARCH-S –23873 –23797 –23846 –16352 –16280 –16326 –7531 –7468 –7508 

AR-APARCH-GED –23850 –23774 –23823 –16330 –16258 –16304 –7539 –7476 –7516 

GOLD 

AR-GARCH-N –27574 –27517 –27554 –18875 –18821 –18856 –8694 –8647 –8677 
AR-GARCH-S –27987 –27923 –27964 –19115 –19055 –19094 –8873 –8820 –8853 

AR-GARCH-GED –27975 –27911 –27952 –19112 –19052 –19091 –8864 –8812 –8844 
AR-APARCH-N –27575 –27505 –27550 –18908 –18842 –18884 –8707 –8649 –8685 
AR-APARCH-S –28005 –27928 –27977 –19139 –19067 –19113 –8876 –8813 –8852 

AR-APARCH-GED –27983 –27907 –27956 –19134 –19062 –19108 –8866 –8803 –8842 

PALLADIUM 

AR-GARCH-N –22449 –22391 –22429 –14987 –14933 –14968 –7468 –7421 –7450 
AR-GARCH-S –22755 –22692 –22733 –15275 –15215 –15253 –7491 –7439 –7472 

AR-GARCH-GED –22753 –22689 –22731 –15294 –15234 –15273 –7485 –7433 –7466 
AR-APARCH-N –22447 –22376 –22422 –14988 –14922 –14964 –7484 –7427 –7463 
AR-APARCH-S –22758 –22682 –22731 –15281 –15209 –15255 –7508 –7445 –7484 

AR-APARCH-GED –22754 –22677 –22727 –15299 –15226 –15273 –7492 –7439 –7476 

PLATINUM 

AR-GARCH-N –25876 –25819 –25856 –17649 –17595 –17629 –8246 –8199 –8228 
AR-GARCH-S –26065 –26001 –26042 –17833 –17772 –17811 –8258 –8205 –8238 

AR-GARCH-GED –26046 –25982 –26024 –17822 –17762 –17801 –8254 –8202 –8235 
AR-APARCH-N –25889 –25819 –25864 –17671 –17604 –17647 –8247 –8190 –8226 
AR-APARCH-S –26075 –25998 –26048 –17848 –17776 –17822 –8256 –8194 –8233 

AR-APARCH-GED –26056 –25979 –26029 –17838 –17766 –17812 –8254 –8191 –8231 
 

 
Figure 3.  AR(1)-APARCH(1,1)-S model for gold (left) and silver (right) – entire pe-

riod 

 In conclusion, it can be said that the models describing conditional mean 
and conditional variance should be described by heavy-tails distributions for 
residuals. It is also reasonable to use models that take into account the asym-
metry observed in a data. 



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DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

95 

Conclusions 
 The paper attempts to describe calendar anomalies in the case of precious 
metals market. That kind of anomalies deny the problem of market efficiency 
and their detection allows to obtain additional information about volatility of 
financial assets. The research area of precious metals is part of commodity 
market (metals market), an alternative to the capital one. Diversification of the 
structure of the financial portfolio with components from different markets 
allows to protect against unpredictable events that may affect the broadly un-
derstood economic situation. 
 The entire research period was divided into two sub-periods: period of 
upward trend and period of downward trend. An attempt was to verify whether 
general economic trends affect the occurrence of the day of the week effect. 
As a time series models the class of AR-APARCH models with conditional 
residual distributions (normal, t-Student and GED) were proposed. The sea-
sonality effect was included in the conditional mean equation. The selection 
of models was made on the basis of appropriate diagnostic tests. The results 
show that in the entire period the returns on Fridays (positive) were signifi-
cantly different from these obtained for the other days of the week. Similar 
results were observed during the period of upward trend. Negative results for 
individual days of the week were observed during the period of downward 
trend.  
 As we can see, the results are not unambiguous. In summary, the day of 
the week effect was observed for the volatility of gold returns in the entire 
period and in the period of upward trend (positive returns on Fridays, negative 
for the other days). The day of the week effect was also observed for the vol-
atility of the palladium returns in the entire period and in the period of upward 
trend (positive returns on Fridays, negative on Mondays, Tuesdays and Thurs-
days). Moreover, it was pointed out that the AR-APARCH models should be 
used when taking into account the heavy-tail distributions for describing 
model residuals (mainly the Student's t-distribution). 

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Testowanie efektu dnia tygodnia na rynku metali szlachetnych 

Z a r y s  t r e ś c i. Efektywność rynku zakłada, że ceny aktywów powinny cechować się loso-
wością i nieprzewidywalnością tak, aby potencjalni uczestnicy rynku nie byli wstanie genero-
wać ponadprzeciętnych zysków. Oznacza to, że w szeregach czasowych nie powinno występo-
wać zjawisko sezonowości, które jednoznacznie wyznacza pewien wzorzec zachowań 



Testing Day of the Week Effect on Precious Metals Market 

DYNAMIC ECONOMETRIC MODELS 18 (2018) 81–97 

97 

aktywów finansowych. W referacie podjęto próbę weryfikacji efektu dnia tygodnia na rynku 
metali szlachetnych. Wybór obszaru badawczego nie jest akcydentalny. Metale szlachetne sta-
nowią alternatywę dla klasycznych inwestycji kapitałowych, zwłaszcza w przypadku kryzysów 
finansowych i gospodarczych. Ponadto literatura przedmiotu wykazuje lukę w obszarze analiz 
dynamiki na rynkach towarowych w porównaniu z aktywami rynku kapitałowego. Wyniki nie 
są jednoznaczne. Efekt dnia tygodnia zaobserwowano przede wszystkim dla stóp zwrotu złota 
i palladu (cały okres i okres wzrostu) oraz sporadycznie dla stóp zwrotu srebra (okres trendu 
spadkowego). Badanie wykazało, że w kontekście kryteriów informacyjnych należy stosować 
modele AR-APARCH z gruboogonowymi rozkładami prawdopodobieństwa dla reszt 

S ł o w a  k l u c z o w e: efekt dnia tygodnia; metale szlachetne; model APARCH; model 
GARCH; szeregi czasowe.  


