






































DEM_2015_129to156


© 2016 Nicolaus Copernicus University Press. All rights reserved.  
http://www.dem.umk.pl/dem 

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.2015.006  Vol. 15 (2015) 129−156 

Submitted October 22, 2015 ISSN (online) 2450-7067 
Accepted December 15, 2015 ISSN (print) 1234-3862 

Ewa Ratuszny*  

Risk Modeling of Commodities  
using CAViaR Models, the Encompassing Method  

and the Combined Forecasts  

A b s t r a c t. The aim of the research is to compare VaR methods/models for commodities. 
For risk measurement Conditional Autoregressive Value at Risk models (CAViaR), implied 
quantile model and encompassing method are used. The aim is to check whether simultaneous 
use of information both from historical time series and regarding markets' expectation can 
improve accuracy of forecasts. For this purpose four methods of combining forecasts are 
used: a simple average combining, an unrestricted linear combination, a weighted averaged 
combining and a weighted averaged combining using exponential weighting. In the case of 
the commodities neither the encompassing method nor the combining forecast method 
improve VaR forecasts. The method of choosing the most adequate model leads to simple 
CAViaR-SAV model as the source of most optimal measure of risk forecasts. The Kupiec 
test, the Christoffersen and the Dynamic Quantile test indicate the model as an adequate to 
forecast VaR for gold and oil for short positions at the 0.01 and the 0.05 significance level, 
and for a long position at the 0.05 significance level. 

K e y w o r d s: CAViaR, VaR, encompassing method, combined forecasts, commodities  

J E L Classification: C22, G17. 

Introduction  

 Value at Risk models should provide an adequate risk forecast both in 
stability period and in period with high volatility. Accurate assessment of the 
risk is required for capital management purpose, limit settings and position 

                                                 
* Correspondence to: Ewa Ratuszny, Warsaw School of Economics, al. Niepodległości 

162, 02-554 Warsaw, Poland, e-mail: ewa.ratuszny@doktorant.sgh.waw.pl 



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

130 

management. Nowadays there exist many risk measurement methods but 
none of the models surpasses the others. This paper extends research 
proposed by Jeon and Taylor (2013) for commodities. Moreover, we apply 
more complex way to compare VaR methods which helps to avoid 
overestimation and underestimation of risk.  
 Value at Risk is defined as the maximum potential loss in portfolio value 
over a given time period due to adverse market movements (i.e. 500 days), 
with a given significance level of α (Doman, Doman, 2009; Iwanicz-
Drozdowska, 2005). Engle and Manganelli (2004) have classified the 
existing VaR methods into three broad categories: parametric, 
semiparametric and nonparametric. Parametric approach includes 
RiskMetrics methodology and GARCH models (Piontek, 2000; Fiszeder, 
2009; Jajuga, 2011; Mazur, Pipień, 2012), but the weakness of those 
methods lies in possibility of incorrect specification both of variance model 
and the error distribution. An interesting parametric method, which becomes 
increasingly popular, is based on implied volatility. Implied volatility is the 
expectation of volatility implied by the option market (Chong, 2004; 
Christoffersen, Mazzotta, 2005; Giot, 2005).  
 The most common nonparametric approach is the historical simulation, 
used by about 73% of banks (Pérignon, Smith, 2010). The main advantage is 
that the historical VaR does not require an assumption about parametric form 
of the distribution of the risk factor returns. Nevertheless the VaR forecast 
might be inaccurate due to inadequate rolling window of risk factors 
(Boudoukh et al., 1998). A long data history will typically encompass 
several regimes with different behavior of market risk factors. Boudoukh et 
al. (1998), Mittnik and Paolella (2000) and Taylor (2008) propose to apply 
exponentially weighted approaches to VaR estimation to overcome those 
difficulties.  
 In our research we apply semiparametric approach based on Conditional 
Autoregressive Value at Risk models (CAViaR). Engle and Manganelli 
(2004) have proposed models that derive a time-varying VaR directly via 
autoregression. The models are estimated using robust method, i.e. a non-
linear quantile regression proposed by Koenker and Bassett (1978). The 
robust approach is widely applied in risk measurement, hedging and 
portfolio allocation (Taylor, 1999; Umantsev, Chernozhukov, 2001). This 
approach allows the shape of the conditional returns distributions to vary in 
time, and for the time-variation to differ for the different quantiles of 
distribution (Jeon, Taylor, 2013). The autoregressive structure is adequate in 
case of clustered time series. The previous researches by Ratuszny (2013), 



Risk Modeling of Commodities using CAViaR Models…  

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131

Ratuszny (2015) indicated that CAViaR models successfuly compete with 
other VaR methods. 
 Jeon and Taylor (2013) proposed to combine quantile forecasts – 
elaborated above 25 year ago by Granger (1989) and Granger, White and 
Kamstra (1989). In their approach the quantile forecasts, obtained from 
CAViaR models and from method based on implied volatility, are combined. 
They applied their method not to economic indicators but to risk 
measurement of position in equity indices such as S&P500 and DAX30. 
Moreover, they included in the CAViaR models an additional regressor: 
a quantiles predictor based on implied volatility (encompassing method). 
 The authors concluded that linear combining method and arithmetical 
method generate better forecast over the sample. The observation motivated 
us to apply their approach to commodities. The Polish researches on 
combined forecast performed by Grajek (2002), Greszta, Maciejewski 
(2005), Piłatowska (2009) have also indicated the predominance of 
combined forecasts over the method based on single approach. In our 
research we apply CAViaR models, the encompassing method and four 
combining forecast methods: Simple Average Combining, Unrestricted 
Linear Combination, Weighted Averaged Combining and Weighted 
Averaged Combining Optimized using Exponential Weighting. We try to 
verify the following hypothesis: 
 The encompassing method or combining forecast methods based on 
CAViaR models and implied quantile model improve accuracy of VaR for 
commodities. 
 The paper is organized as follows. Firstly, we review Value at Risk 
methodology based on CAViaR models, implied quantile, encompassing 
method and combining forecast methodology. The part contains also the 
performance criteria. The second part contains empirical applications of the 
models. The last part contains concluding remarks. 

1. Review of Value at Risk Methodology  

1.1. CAViaR Models 

 The Conditional Autoregressive Value at Risk model has been 
introduced by Engle and Manganelli (2004). The basic intuition is to model 
directly the evolution of the quantile over time, rather than the whole 
distribution of portfolio returns. The general form of CAViaR models is 
defined by (Engle, Manganelli, 2004; Doman, Doman, 2009):  



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

132 

,)|,,(),(=),(=),( 11
1=

0 −++− ++∑ tqppiti

p

i
tt lVaRfVaR Fβββαβββα α Kβx   (1) 

 where 1−tF  is the information set available at time 1−t , and

),(= 0 ′+qpββα Kβ  is the vector of parameters which is estimated using non-

linear regression quantile techniques. In most practical cases the above 
formulation is reduced to a first order model: 

 ,)),(,,(),(=),( 112110 βαββαβββα −−− ++ tttt VaRylVaRVaR  (2) 

where )|(= 1−− tttt rEry F , tr  is rate of return, )|( 1−ttrE F  is the expected 

value of rate of returns. The autoregressive term ),(11 βαβ −tVaR  ensures that 

the VaR changes smoothly over time. The role of )),(,,( 112 βαβ −− tt VaRyl  is 

the linking the level of explained variable )(αtVaR  to the level of y  at the 

moment 1−t . That is, it measures the impact of new information in y  on 
the level of VaR. The following CAViaR models are analysed in our 
research both for long position (l) and short position (s) (Engle, Manganelli, 
2004; Doman, Doman, 2009):  
1. Symmetric Absolute Value – SAV 

 ,),(=),( 12110 −− ++ t
l
t

l
t yVaRVaR ββαβββα  (3) 

 .),(=),( 12110 −− ++ t
s
t

s
t yVaRVaR ββαβββα  (4) 

Current VaR depends on the past value 1−tVaR  and absolute value of past 

rate of return. The model symmetrically responds to both negative and 
positive past returns. 
2. Asymmetric Slope – AS 

 ,0)<(0)(),(=),( 1131111 −−−−− +≥+ tttt
l
t

l
t yIyyIyVaRVaR βββαβα  (5) 

 ,0)<(0)(),(=),( 1131111 −−−−− +≥+ tttt
s
t

s
t yIyyIyVaRVaR βββαβα  (6) 

where )(⋅I  is the indicator function. Current VaR depends on its past value 

1−tVaR  and on positive and negative returns that are treated in different way. 

3. Indirect GARCH for both short and long position:  

 [ ] .),(=),(
1/22

13
2

121 −− ++ ttt yVaRVaR ββαβββα  (7) 

Current VaR is described as GARCH process. The model is correctly 
specified for rate of returns from GARCH(1,1) model. 



Risk Modeling of Commodities using CAViaR Models…  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

133

4. Adaptive – AD 

( )[ ]{ },)],([exp1),(=),(
1

1111 αβαββαβα −−++
−

−−−
l
tt

l
t

l
t VaRyGVaRVaR   (8) 

( )[ ]{ },)],([exp1),(=),(
1

1111 αβαββαβα −−++
−

−−−
s
tt

s
t

l
t VaRyGVaRVaR   (9) 

 where G  is some positive finite number. If ∞→G , the second term 
converges to ])),( ([ 11 αβαβ −− −− tt VaRyI  for long position and to

])),(([ 11 αβαβ −≥ −− tt VaRyI  for short position, where )(⋅I  is the indicator 
function. In case of a VaR breach the VaR forecast should be increased, 
otherwise should be slightly decreased. The model aims to reduce the 
probability of sequences of VaR breaches and will also make unlikely that 
the VaR has never been reached. The disadvantage of this type of CAViaR 
models is lack of rate of return in explanatory variables set so that the 
information about extremal market movements is not effectivelly included in 
model (Doman, Doman, 2009).  
  Estimation of CAViaR models is performed on the basis of Koenker and 
Basset (1978) regression quantile methodology, which minimalises the 
regression quantile objective function of the following form for long (l) and 
short (s) position, respectively:  

 

,),()(1

),(min

),(>

),( 









−−+









+−

∑

∑

−

−

βαα

βαα

βα

βα
β

l
tt

l
tVaRtyt

l
tt

l
tVaRtyt

VaRy

VaRy

 (10)  

 

.),()(1

),(min

),(<

),(









−−









+−

∑

∑
≥

βαα

βαα

βα

βα
β

s
tt

s
tVaRtyt

s
tt

s
tVaRtyt

VaRy

VaRy

 (11) 

  
  



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

134 

1.2. Implied Volatility 

 Implied volatility reflects market’s expectations regarding future 
volatility. Implied volatility is the key variable in financial investment 
decision, risk management, derivative pricing, market making, market 
timing and portfolio selection. 
 In spite of huge volume of research, no consensus has been reached on 
usefulness of implied volatility as a predictor for future volatility in 
comparison with predictions from time series models. There are many 
empirical studies in which implied volatility overcomes the historical 
volatility (Szakmary et al. (2003) for futures on equity indices, interest rates, 
currencies, commodities and crude oil; Pong et al. (2004) for FX; Corredor 
and Santamaria (2004) for Ibex; Giot and Laurent (2007) for stock indices 
such as the S&P100 and S&P500). Noh and Kim (2006) conclude that both 
implied volatility and historical volatility using high-frequency returns can 
outperform each other in forecasting volatility. In their empirical test, 
historical volatility from high frequency returns performed better in the 
FTSE100 futures, which tend to be relatively close to normally distributed, 
while the result of implied volatility was better in the S&P500 futures, which 
displays excess skewness even with volatilities from high frequency returns. 
Implied volatility is also considered as useful variable for estimating quantile 
of the returns distribution (Giot, 2005; Chong, 2004). 
Jeon and Taylor (2013) applied implied volatility to VaR for equity indices 
S&P500 and DAX30. They construct an implied quantile (IQ ) estimator as 

the product of the implied volatility recorded in the previous period Implied
t 1−σ , 

and the empirical distribution quantile )(αEmpQ  of ( ty ) standardised by the 

implied volatility. The IQ  estimator for long and short position is expressed 
in the following form (Jeon, Taylor, 2013):  

 ,)(=)( 1
)()( Implied

t
lEmplIQ

t QVaR −σαα  (12) 

 .)(=)( 1
)()( Implied

t
sEmpsIQ

t QVaR −σαα  (13) 

The IQ approach captures the market’s expectation of future risk. Another 
advantage is that the method does not assume a particular distribution for the 
asset returns, and it involves no parameter estimation. Jeon and Taylor 
(2013) note that this simple approach to capture an ‘implied quantile’ 
assumes returns standardised with implied volatility are i.i.d.  
 



Risk Modeling of Commodities using CAViaR Models…  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

135

1.3. Encompassing Method 

 Jeon and Taylor (2013) propose to construct one model encompassing 
competitive forecast models. Such model should generate better forecast 
than every model separately. This approach is called encompassing (Chong, 
Hendry, 1986; Diebold, 1989; Grajek, 2002) or plug-in (Jeon, Taylor, 2013). 
Day and Lewis (1992) in their research show that the implied volatility and 
models based on historical volatility (EGARCH or GARCH models) does 
not reflect whole information about volatility. Blair et al. (2001) received 
completely different results. They shows that implied volatility VIX, is 
a significant explanatory variable for volatility forecast of S&P100. Claessen 
and Mittnik (2002) and Giot (2005) performed research over the sample. 
Claessen and Mittnik (2002) included the implied volatility VDAX to 
GARCH model, and show that implied volatility reflect market expectation 
about future volatility of DAX. Similarly Giot (2005) for Nasdaq and 
S&P500 shows that implied volatility VIC and VXN included in GARCH 
models improves a volatility forecast.  
 Jeon and Taylor (2013) analyzing the results of previous research of 
encompassing method, decided to check the possibility to receive a better 
estimate of VaR if information from historical time series and information 
regarding risk expected by market are combined. The rationale for their 
research was that if the implied volatility forecasts the future well, it should 
be useful in estimating future quantile of returns distribution. They include 
the implied quantile expressed by equation (12) or (13) to the CAViaR 
models as an explanatory variable. The impact of implied volatility on VaR 
can be determined by coefficient IQβ . 

 The following models are analysed in our research (Jeon, Taylor, 2013): 
1. Symmetric Absolute Value PlugIn: (SAV-PlugIn):  

 ,)(),(=),( )(
12110 αβββαβββα lIQ

tIQt
l
t

l
t VaRyVaRVaR +++ −−  (14) 

 .)(),(=),( )(
12110 αβββαβββα sIQ

tIQt
s
t

s
t VaRyVaRVaR +++ −−  (15) 

2. Asymmetric Slope PlugIn: (AS-PlugIn):  

 
,)(

0)<(0)(),(=),(

)(

1131111

αβ

βββαβα
lIQ

tIQ

tttt
l
t

l
t

VaR

yIyyIyVaRVaR

+

++≥+ −−−−−
 (16) 



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

136 

 
.)(

0)<(0)(),(=),(

)(

1131111

αβ

βββαβα
sIQ

tIQ

tttt
s
t

s
t

VaR

yIyyIyVaRVaR

+

++≥+ −−−−−
 (17) 

3. Indirect GARCH(1,1) PlugIn (IGARCH-PlugIn):  

 [ ] [ ]{ }2

1
2)(2

13

2

121 )(),(=),( αβββαβββα lIQ
tIQt

l
t

l
t VaRyVaRVaR +++ −−   (18) 

 [ ] [ ]{ } .)(),(=),( 2

1
2)(2

13

2

121 αβββαβββα sIQ
tIQt

s
t

s
t VaRyVaRVaR +++ −−  (19) 

4. Adaptive PlugIn (AD-PlugIn):  

 
( )[ ]{ } ,)()],([exp1

),(=),(

)(1

113

121

αβαβαβ

βαβββα

lIQ
tIQ

l
tt

l
t

ll
t

VaRVaRyG

VaRVaR

+−−+

++
−

−−

−
 (20) 

 
( )[ ]{ } .)()],([exp1

),(=),(

)(1

113

121

αβαβαβ

βαβββα

sIQ
tIQ

s
tt

s
t

s
t

VaRVaRyG

VaRVaR

+−−++

++
−

−−

−
 (21) 

Notations in the models are the same as in part 1.1. and 1.2. 1.3.  

1.4. Combining Method to VaR Forecast 

 If it is not clear which of two forecasts performs better, a combination 
can be the best option (Bates, Granger, 1969). Combining methods include 
information contained in each of individual forecast. According to 
Armstrong (2001) the combined forecasts should be applied if several 
different models can be combined to obtain better forecast, there is no 
certainty about the future state of the object forecast, and where large 
forecasting error involves a high cost. By combining forecasters should able 
to reduce inconsistency in estimates and to cancel out biases to some extent. 
 The work by Bates and Granger (1969) often is considered to be the 
seminal article on combining forecasts. They combined two separate sets of 
forecasts of airline passenger data to form a composite set of forecasts. They 
concluded that the composite set of forecasts can yield lower mean-square 
error than either of the original forecasts. Past errors of each of the original 
forecasts are used to determine the weights to attach to these two original 
forecasts in forming the combined forecasts. They also examined different 
methods of deriving these weights. Combined forecasts for economy 



Risk Modeling of Commodities using CAViaR Models…  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

137

indicators are subject of research of Crane and Crotty (1967), Zarnowitz 
(1967), Nelson (1972, 1984). 
 Despite the criticism of combined forecasts (e.g. Diebold (1989) showed 
that it is better to improve one of the single models, rather than relying on 
combining methods of forecast derived from models with incorrect 
specifications), this approach has become the subject of further research.  
 Combining forecasts for the variation is subject of research of Doidge 
and Wei (1998), Armendola and Storti (2008), Donaldson and Kamstra 
(2005).  
 There are very few studies about combining quantile forecasts. Granger 
(1989) and Granger et al. (1989) introduce the idea of using quantile 
regression to combine quantile forecasts. Taylor and Bunn (1998) assess the 
usefulness of different restrictions on the parameters of the quantile 
regression combination. Giacomini and Komunjer (2005) describe how 
encompassing tests can be performed for two quantile predictors using the 
quantile regression combining framework. They apply their proposal to VaR 
estimates of the S&P500 based on two time series volatility forecasting 
methods. 
 Jeon and Taylor (2013) in their research applied the following four 
combined mehods: Simple Average Combining (SimpAvg), Unrestricted 
Linear Combination (LinearComb), Weighted Averaged Combining 
(WtdAvg) and Weighted Averaged Combining Optimized using Exponential 
Weighting (WtdAvgExp).  

2. Simple Average Combining (SimpAvg) 

 The simplest and most widely used forecast combining method is to take 
the simple arithmetic mean of the individual forecasts. We consider the 
simple average of the quantile forecasts from the IQ method and one 
CAViaR model, as in expression (3–9) (Jeon and Taylor, 2013):  

 ,),(
2

1
)(

2

1
=)( )()( βααα lCAViaR

t
lIQ

t
l
t VaRQVaR +  (22) 

 .),(
2

1
)(

2

1
=)( )()( βααα sCAViaR

t
sIQ

t
s
t VaRQVaR +  (23) 

The method will be here denoted as SimpAvg according to Jeon and Taylor 
(2013) nomenclature. The aim of this approach is to determine the 
combination of forecasts with lower error variance than in case of individual 
forecasts. 



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

138 

3. Unrestricted Linear Combination (LinearComb) 

 A traditional approach to combining is to compute linear combinations 
of forecasts, called also regression method (Jeon and Taylor, 2013). The 
method will be dnoted as LinearCom according to Jeon and Taylor (2013) 
nomenclature. Forecast is formed on the basis of an IQ forecast and one of 
CAViaR models (Jeon and Taylor, 2013):  

 ,),()(=)( )(
3

)(
21 βαγαγγα lCAViaR

t
lIQ

t
l
t VaRQVaR ++  (24) 

 .),()(=)( )(
3

)(
21 βαγαγγα sCAViaR

t
sIQ

t
s
t VaRQVaR ++  (25) 

 The parameters 2γ  and 3γ  inform about the dynamics of forecasted 

variable. If the sum of the parameters 2γ  and 3γ is less than unity, the 
individual predictions are more volatile than the risk measure VaR. If the 
sum of the parameters is greater than one, then the individual forecasts are of 
less dynamic than VaR.  
 There are several difficulties with the combination method. The first is 
related to collinearity of individual forecasts. If the individual predictions are 
quite good, they would not differ significantly and this entails the 
phenomenon of collinearity. Consequently, the low-significance and high 
randomness of estimated weights are obtained. Another issue is the 
autocorrelation of the random component, caused by autocorrelation of 
dependent variable. In order to solve this problem Diebold (1988) proposed 
to estimate the ARCH model. The third issue is related with the inability to 
impose zero restrictions for correlation between the errors of individual 
forecasts, when examining the behavior of individual forecasts in the past. In 
addition, regression method requires a large data sets, which in case of time 
series is fullfilled. The advantage of this method is the lack of restrictions on 
the parameters and lack of assumptions about unbiasedness of individual 
forecasts. 

4. Weighted Averaged Combining (WtdAvg) 

 The Weighted Averaged Combining method is based on the relation 
between forecast error in the past. In this approach the unbiasedness of 
quantile forecast is assumed (Granger (1989)). Error variance of combined 
forecast will be equal or smaller than of the individual forecasts. The method 
in our research will be noted as WtdAvg according to Jeon and Taylor (2013) 
nomenclature. The resultant quantile forecast is of the form (26–27), without 



Risk Modeling of Commodities using CAViaR Models…  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

139

constant, where combining weights are constrained to be between zero and 
one. 

 ,),()(1)(=),( )()( βαωαωωα lCAViaR
t

lIQ
t

l
t VaRQVaR −+  (26) 

 .),()(1)(=),( )()( βαωαωωα sCAViaR
t

sIQ
t

s
t VaRQVaR −+  (27) 

Clemen (1986) advocates the use of the weighted average even if the 
forecasts are biased, arguing that gains in efficiency can be made at the cost 
of some bias. Bunn (1989) noted greater robustness of the method compared 
with regression method. Taylor and Bunn (1998) pointed out that the value 
of the weight indicates the relative explanatory powers of the two quantile 
predictors. 

5.  Weighted Averaged Combining Optimized using Exponential 
Weighting (WtdAvgExp).  

  The method is similar to Weighted Averaged Combining but additionally 
the Exponential Weighting factor for the optimisation of the combining 
weight is applied. The factor gives greater weight to the more recent 
observations in the quantile regression optimisation (Taylor (2008)). In this 
way the nonstationarity problem of weights is solved. This is particularly 
important when the time series exhibits time-varing and cyclical volatility. 
Boudoukh et al. (1998) insist that such an approach is a reasonable 
compromise between statistical precision and adaptation to the latest 
information. Exponentially Weighted Quantile Regression (EWQR) method 
solves the following minimizing problem (Jeon, Taylor, 2013): 







−−+







+−

−

−

−

−≤

∑

∑

),()(1

),(min

),(>|

),( |

ωααλ

ωααλ

ωα

ωαω

l
tt

tT

l
tVaRtyt

l
tt

tT

l
tVaRtyt

VaRy

VaRy

 (28) 







−−+







+−

−

−

≥

∑

∑

),()(1

),(min

),(<|

),(|

ωααλ

ωααλ

ωα

ωαω

s
tt

tT

s
tVaRtyt

s
tt

tT

s
tVaRtyt

VaRy

VaRy

 (29) 



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

140 

where )(αl
tVaR  and )(αs

tVaR  are expressed in equations (26)–(27). A lower 

value of the decay parameterλ  implies faster exponential decay, and hence 
more weight is given to the recent observations and less historical 
information is captured. This method is noted as WtdAvgExp according to 
Jeon and Taylor (2013).  

5.1. Out-of-sample diagnostics 

 Regulators can apply backtest for evaluating the accuracy of the VaR 
models, but this method misclassifies forecasts from inaccurate models as 
acceptably accurate. In our research the out-of-sample diagnostic of VaR is 
performed on the basis tests and measures, i.e. backtests, tests based on 
Bernoulli trials model, the Dynamic Quantile test, regulatory loss, binary 
loss, firm’s loss.  
 The LR Test of Unconditional Coverage (Kupiec test) evaluates the 
model, taking into account both too much and too few exceedances. Its 
disadvantage, however, is that it does not take into account the distribution 
of exceedances in the sample. A well-functioning VaR model should be 
characterized by the absence of autocorrelation in the indicator function 
what can be done by performing the Dynamic Quantile test. The model is 
considered adequate if the number of exceedances corresponds to the 
assumptions and there is no autocorrelation. In case of no exceedances we 
consider that the model is inadequate, because of overestimation of VaR. 
Recall the construction and interpretation of these tests/measures. 
 Kupiec (1995), Christoffersen (1998), Rachev and Mittnik (2002) 
proposed the indicator variable (tξ ) for time t, made at time t–1, which is 

defined as:  

 ,
)(<1
)(0

=




−
−≥

+

+

α
αξ l

tnt

l
tntl

t VaRyfi
VaRyfi

 (30) 

 .
)(>1
)( 0

=




+

+

α
αξ s

tnt

s
tnts

t VaRyif
VaRyfi

 (31) 

 On basis of the variable we perform backtesting. In the backtest we check ex 
post if observed loss ( nty + ) breaches the forecast VaR at the time t. If the 

VaR is indicated on the significance level of α , an appriopriate model 
should also indicate fraction of exceedances of the realised loss at the level 
of α . If the fraction of breaches is much greater than assumed, it means that 



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141

the model underestimates the size of VaR. Lower number of exceedances 
means in turn that the model overestimates the Value at Risk (Doman, 
Doman, 2009). 

 Kupiec (1995) proposes to treat TTTTtpoz
t ′++ ,1,,=: Kξ , wherepoz

denotes instrument position, i.e. l  – long ors  – short, as sequence of 
Bernoulli trials of independent variables with the same probability of 
success:  

 ,
)(<1

1)(0
=




−
−−≥

+

+

αα
ααξ

ithwVaRyif
ithwVaRyfi

l
tnt

l
tntl

t  (30) 

 .
)(>1

1)( 0
=


 −

+

+

αα
ααξ

.ithwVaRyif
ithwVaRyif

s
tnt

s
tnts

t  (31) 

The null hipotesis of Kupiec test is: αα =:0H . 

The likelihood ratio test statistic ucLR  (ang. the LR Test of Unconditional 
Coverage) is given by an equation (Pipień, 2006):  

 ( ){ } ( ){ }[ ],1lnˆ1ln2= 11 SSTSST
ucLR αααα −+′−−′ −−−  (32) 

 where poz
t

TT

Tt
S ξ∑

′+
=

=  means the total number of exceedances and 

poz
t

TT

TtT
ξα ∑
′+

+′ =1

1
=ˆ  is the assessment of the likelihood of success. With a 

true null hypothesis, test statistic has asymptotic distribution 2
1χ . The null 

hypothesis is rejected if the statistical value is above a critical value, i.e.
84.3>ucLR .  

Christoffersen (1998) combines the above tests for unconditional coverage 
and independence. In effect, the null hypothesis of the unconditional 
coverage test will be tested against the alternative of the independence test. 
The statistics of the Joint Test of Coverage and Independence is expressed 
by the following equation (Christoffersen, 1998):  
 .= induccc LRLRLR +  (33) 
where:  

 ( ) ( ){ }[ −−− 11
11

10
11

01
01

00
01 ˆˆ1ˆˆ1ln2= TTTT

indLR ππππ  

 ( )( ) ( ){ }].ˆˆ1ln 1101
2

1000
2

TTTT ++−− ππ  (34) 



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

142 

where )/(=ˆ 10 iiijij TTT +π ; TTT )/(=ˆ 1101 +π for 0,1=, ij , ijT  – number of 

points at time { }Ttt ≤≤;2 for which the iI t =  follows jI t =1+ . 

The test statistic has asymptotic distribution 2
2χ . The null hypothesis is 

rejected if the statistical value is above a critical value, i.e. 99.5>ccLR .  

 The Dynamic Quantile test was constructed to check the absence of 

autocorrelation in sequence },1,,=:{ TTTTtpoz
t ′++ Kξ , where poz

tξ  is 
binary variable expressed by equations (30)–(31). Define for long position 

ααα −− ))(<(=)( tt
l
t VaRyIHit  and for short position: 

)())(>(=)( ααα −tt
s
t VaRyIHit , where )(⋅I  is indication function. The 

Dynamic Quantile test verifies two hypothesis simultaneously:  

• ,0=))((:01 αpoz
tHitEH  

• :02H  variable )(αpoz
tHit  is uncorrelated with the variables included 

into information set. 
Engle and Manganelli (2004) jointly verify the above hypothesis by the 
regression of the following form: 

 ,=)( t
pozHit ελα +X  (35) 

 where X  is the matrix ][= , jtxX  of dimension kT × , where in the first 

column are the ones, then p columns contains variables ptt HitHit −− K,1 , and 

k–p–1 remaining columns – an additional independent variables (including 

tVaR)cThe Dynamic Quantile test statistic is expressed by the equation:  

 ,
)(1

ˆˆ

αα
λλ

−
′XX'

 (38) 

 where ( ) YXXX ′′ −1=λ̂  is the OLS estimate of parameters λ . The test 

statistic is asymptotically distributed 2kχ .  
Detailed test results are available from the author on the request. In table 
(12)–(13) bolded value indicate models which are adequate under assumed 
criterion. 
 Lopez (1998) proposed the loss functions evaluation method not based 
on hypothesis testing framework, but rather on assigning to the VaR 
estimates a numerical score that reflects specific regulatory or firm’s 



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143

concern. In our research we take into account binary loss, regulatory loss and 
firm’s loss. 
Let loss function will be implied by the binomial method which takes value 
0 or 1 related with observed VaR breach:  

 




−
−≥

+

+
+ )(<1

)(0
=))(,(

α
αα l

tnt

l
tntl

tnt
l

t VaRyfor
VaRyfor

VaRyf , (39) 

 .
)(1
)(<0

=))(,(




≥+

+
+ α

αα s
tnt

s
tnts

tnt
s

t VaRyfor
VaRyfor

VaRyf  (36) 

Binary loss (BL) is described by the number of exceptions observed in 
period from nTt +=  to nTTt +′+=  (Lopez, 1998; Pipien, 2006):  

 poz
t

TT

Tt

poz fBL ∑
′+

=

= . (37) 

The smaller is the number of exceedances, the better rating for the models 
will be assigned. This criterion favors models which overestimate the VaR 
and assign low score for the models that generate liberal VaR forecasts. In 
the assessment of VaR forecasts an important issue is to take into account 
the size of the losses that are associated with exceptions of VaR by 
observation nty + . Function (41) includes only the fact of exceptions, and 

does not take into account the size of the losses arising from an extremal 
market movement. 
 The second loss function proposed by Lopez (1999) contains both the 
magnitude and the number of exceptions: 





−++
−≥

++

+
+ )(<))((1

)(0
=))(,( 2 αα

αα l
tnt

l
tnt

l
tntl

tnt
l

t VaRyforVaRy
VaRyfor

VaRyf , (38) 

.
)())((1

)(<0
=))(,( 2




≥−+ ++

+
+ αα

αα s
tnt

s
tnt

s
tnts

tnt
s

t VaRyforVaRy
VaRyfor

VaRyf  (39) 

Regulatory loss (RL) is expressed as follows (Sarma et al., 2003): 

 .=
=

poz
t

TT

Tt

poz fRL ∑
′+

 (40) 

Thus, as before, a score of one is imposed when an exception occurs, but 
also, an additional term based on its magnitude is included. The numerical 
score increases with magnitude of the exception and can provide additional 



Ewa Ratuszny  

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144 

information on how the underlying VaR model forecasts the lower tail of the 
inderlying distribution.  
 Sarma et al. (2003) pointed out that in financial institutions exists 
conflict between profit maximization and the duty of protection against 
market risk. The duty is related with Basel III which imposes an obligation 
to maintain the capital requirements to cover potential losses. Sarma et al. 
(2003) propose to incorporate in the loss function the additional costs arising 
from the capital adequacy. The loss function is: 





−++
−≥

++

+
+ )(<))((1

)()(
=))(,( 2 αα

ααα l
tnt

l
tnt

l
tnt

l
tl

tnt
l

t VaRyforVaRy
VaRyforcVaR

VaRyf , (45) 

.
)())((1

)(<)(
=))(,( 2




≥−+ ++

+
+ αα

ααα s
tnt

s
tnt

s
tnt

s
ts

tnt
s

t VaRyforVaRy
VaRyforcVaR

VaRyf  (46) 

where the parameter 0>c  specifies the opportunity cost associated with 
non-use of the capital which the institution must hold in order to hedge 
against the risk predicted by VaR (Sarma et al.,2003; Pipień, 2006). In our 

research we assume 1=c . The cumulated value of pozf  is expressed as 
follows (Pipień, 2006):  

 .=
=

poz
t

TT

Tt

poz fFL ∑
′+

 (47) 

The function is called firm’s loss (FL) and enables to compare the VaR 
forecasts generated by different models in scope of market risk hedging. The 
model that generates too conservative VaR predictions will – unlike to (44) – 
be penalized by the (47) due to inefficient maintenance of excess capital in 
order to hedge against market risks.  

6. Empirical Study 

6.1. Descriptive statistics 

 We analysed close price (tP ) from August 1st, 2008 to October 10th, 

2014 in case of gold (1593 observation), and from May 10th, 2008 to October 
10th, 2014 (1851 observation) for oil. As an implied volatility we used 
CBOE Gold Volatility Index and CBOE Crude Oil Volatility Index 
( )Implied

tσ . The indices measure the market's expectation of volatility implicit 
in the prices of options. The indices are leading barometers of investor 



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145

sentiment and market volatility relating to listed options on an instrument 
with different strike prices, at the money (ATM) and out of the money 
(OTM), which are then averaged to provide hypothetical price of ATM 
options with a maturity of one month (22 days business). Daily volatility is 
calculated using scaling rule: .252= 1252

ImpliedImplied σσ  

Table  1.  Descriptive statistics 

Instrument Median Mean 
Std 

Minimum Maximum Skewness Kurtosis 
deviation 

Gold 0.0005 0.0002 0.0127 –0.0888 0.1044 –0.2497 10.0226 
Oil 0.0008 0.0002 0.0239 –0.1274 0.1503 0.0379 9.0206 

Table  2.  The Lomnicki-Jarque-Bera test results 

Instrument Statistics p-value 

Gold 24.5934 0.2174 
Oil 83.5494 0.0000 

Table  3. The Ljung-Box test results 

Lags 10 15 20 

Instrument Statistics p-value Statistics p-value Statistics p-value 

Gold 13.7835 0.1831 22.9158 0.0859 24.5934 0.2174 
Oil 46.6134 0.0000 74.5049 0.0000 83.5494 0.0000 

Table  4.  The Engle test results 

Lags 10 15 20 

Instrument Statistics p-value Statistics p-value Statistics p-value 

Gold 89.3389 0.0000 105.6773 0.0000 148.0272 0.0000 
Oil 413.6957 0.0000 488.4326 0.0000 522.1252 0.0000 

Table  5.  The McLeod-Li test results 

Lags 10 15 20 

Instrument Statistics p-value Statistics p-value Statistics p-value 

Gold 150.8873 0.0000 207.7756 0.0000 303.2220 0.0000 
Oil 1 300.5209 0.0000 1 908.8351 0.0000 2 553.6536 0.0000 

 The time series of the quotations, prices and rates of return were checked 
for the presence of the following features: fatter tails than in the normal 
distribution (identified on the basis of the quantile-quantile plots, histograms 
and the Lomnicki-Jarque-Bera test); stationarity; autocorrelation of the rates 
of returns (checked with the Ljung-Box test); skewness, kurtosis of rates of 
return. The rate of returns have high degrees of kurtosis, a negative skewness 
is evident in case of gold. The oil time series is characterised by positive 
skewness. The Lomnicki-Jarque-Bera test rejects normality at the 5%-level 



Ewa Ratuszny  

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146 

in case of oil. The standard deviation of the rate of returns is the highest in 
the case of oil. The Q test Ljung-Box in case of oil indicates autocorrelation. 
The Engle and McLeod-Li test confirms the existence of a strong and 
permanent nonlinear dependence.  

Table  6. Gold. Estimated parameters of CAViaR and CAViaR-PlugIn models 

Model Parametr 

 CAViaR   PlugIn  

 Long position   Short position   Long position   Short position  

 0.01   0.05   0.01   0.05   0.01   0.05   0.01   0.05  

SAV 

1β  
 0.00041   0.00008   0.00037   0.00025   –0.00223  –0.00414  –0.00156  –0.00271 

 (0.0004)   (0.0002)   (0.0004)   (0.0002)   (0.0073)   (0.0050)   (0.0043)   (0.0027)  

2β  
 0.93857   0.94362   0.95419   0.94818   –0.35921  –0.28538  0.53545   0.33501  

 (0.0222)   (0.0244)   (0.0277)   (0.0163)   (0.5209)   (1.1716)   (0.4215)   (0.4223)  

3β  
 0.18587   0.11934   0.10555   0.07401   0.24365   0.08086   –0.27554  –0.14731 

 (0.0698)   (0.0537)   (0.0612)   (0.0189)   (0.2447)   (0.1983)   (0.1014)   (0.0757)  

IQβ  
                 1.47378   1.55136   0.64168   0.95140  

                 (0.6732)   (1.4107)   (0.3212)   (0.5991)  

AS 

1β  
 0.00068   0.00005   0.00032   0.00024   –0.00194  –0.00365  –0.00995  –0.00163 

 (0.0007)   (0.0002)   (0.0003)   (0.0002)   (0.0063)   (0.0021)   (0.0099)   (0.0021)  

2β  
 0.94322   0.94893   0.96706   0.94679   –0.14498  0.34562   0.07883   0.40237  

 (0.0373)   (0.0216)   (0.0141)   (0.0179)   (0.6099)   (0.2039)   (0.3584)   (0.2845)  

3β  
 0.25113   0.13757   0.10653   0.07312   –0.32960  –0.24975  –0.21780  –0.10120 

 (0.1068)   (0.0492)   (0.0474)   (0.0333)   (0.3141)   (0.1194)   (0.1142)   (0.0753)  

4β  
 –0.03450  –0.08393  –0.02525  –0.08316  –0.23637  –0.15212  0.55140   0.22534  

 (0.0785)   (0.0491)   (0.0316)   (0.0208)   (0.3469)   (0.1318)   (0.3182)   (0.1333)  

IQβ  
                 1.25475   0.87372   1.52731   0.80735  

                 (0.6264)   (0.3152)   (0.8703)   (0.3249)  

Indirect 
GARCH 

1β  
 0.00002   0.00000   0.00001   0.00000   0.00004   0.00015   0.00023   0.00003  

 (0.0004)   (0.0002)   (0.0004)   (0.0001)   (0.0036)   (0.0011)   (0.0006)   (0.0002)  

2β  
 0.91453   0.95049   0.95819   0.95292   0.32477   0.37988   0.03281   0.40679  

 (0.4147)   (0.4265)   (0.7352)   (0.3599)   (4.6846)   (2.7296)   (0.6178)   (0.8225)  

3β  
 0.47486   0.15482   0.16744   0.07178   0.71078   0.03784   0.22551   0.04628  

 (0.5825)   (0.6343)   (1.0813)   (0.1408)   (7.8410)   (2.0967)   (3.0804)   (0.5717)  

IQβ  
                 1.50097   1.98863   1.59270   0.79010  

                 (9.4800)   (6.6522)   (1.6841)   (0.9189)  

AD 

1β  
 –0.00003  –0.00001  –0.00001  –0.00001  –0.05703  –0.04389  0.05029   0.00480  

 (0.0000)   (0.0000)   (0.0000)   (0.0000)   (0.0546)   (0.1450)   (0.0470)   (0.0470)  

2β  
                 0.01650   0.56209   –0.34653  0.42903  

                 (0.2477)   (1.0987)   (0.3483)   (0.9800)  

3β  
                 0.11134   0.08988   –0.12186  –0.01272 

                 (0.0232)   (0.0000)   (0.0018)   (0.0000)  

IQβ  
                 1.36557   0.84537   1.53685   0.61808  

                 (0.0546)   (0.1540)   (0.0677)   (0.1400)  
Note: 0.01; 0.05 – α-significance level of VaR; standard errors in bracktes; bolded values indicate 
significant parameters according to t statistics. 
 



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6.2. Empirical research 

  We perform estimation of CAViaR models (equations (3)–(9)), implied 
quantile model (equations (12)–(13)), encompassing method (equations 
(14)–(21)) and combining method of forecast (equations (22)–(27)). We use 
1093 periods for gold and 1350 periods for oil to estimate parameters (in-
sample) and 500 periods for post-sample evaluation of day-ahead quantile 
estimates using rolling window. For both instruments we indicate rate of 
returns tr  and and an average returns in the sample (µ ). We perform 

estimation for residuals µ−tt ry = .  
 We estimate the parameters using, as Engle and Manganelli (2004), 
Doman, Doman (2009), Jeon and Taylor (2013), the Differential Evolution 
algorithm in C++ and Matlab. The algorithm was presented by Price and 
Storn (1997).  
 Parameters of estimated models are contained in tables (6) and (7). For 
gold in the SAV models for long and short positions at the 0.05 level of 
probability and in the case of the model Indirect GARCH both long and 
short positions explanatory variable as empirical quantile turns out to be 
significant. For oil we observe a different situation. Only in the case of 
models SAV and the AS for a short position at the 0.05 significance level 
and for the AD model for long and short positions at the 0.01 significance 
level attached explanatory variable of empirical quantile turned out to be 
irrelevant. 
 To optimize the parameters of regression methods and the variance-
covariance method we expressed the quantile regression minimization as 
a linear programme and applied the Nelder-Mead Simplex algorithm. The 
estimated parameter for the combined forecasts are contained in the  
Table (8).    
In case of gold the forecast based on implied volatility receives a higher 
weight than the predictions from the CAViaR models. In case of the oil the 
implied quantile receives significantly higher weight when forecast is 
combined on the basis of CAViaR-AD model and implied quantile for long 
and short positions at the 0.01 significance level, and for a long position at 
the 0.05 significance level. But for a short position at the 0.05 significance 
level the implied quantile receives a significantly higher weight in the 
combination of implied quantile with the forecast based on SAV or Indirect 
GARCH model.  
  



Ewa Ratuszny  

DYNAMIC ECONOMETRIC MODELS 15 (2015) 129–156 

148 

Table  7.  Oil. Estimated parameters of CAViaR and CAViaR-PlugIn models 

Model 

  
Parametr 

  

 CAViaR   PlugIn  

 Long position   Short position   Long position   Short position  

 0.01   0.05   0.01   0.05   0.01   0.05   0.01   0.05  

*SAV 

  1β     0.00251   0.00011   0.00278   0.00006   –0.00062  –0.00196  –0.00074  –0.00288 
 (0.0007)   (0.0002)   (0.0015)   (0.0002)   (0.0032)   (0.0029)   (0.0037)   (0.0013) 

  2β     0.87120   0.91757   0.76330   0.90003   0.68751   0.78916   0.49449   0.79478  
 (0.0211)   (0.0370)   (0.0683)   (0.0265)   (0.1239)   (0.1580)   (0.2313)   (0.0520) 

  3β     0.27242   0.17969   0.55563   0.20181   0.24017   0.14208   0.51237   0.22536  
 (0.0466)   (0.0882)   (0.2154)   (0.0561)   (0.0486)   (0.1666)   (0.2829)   (0.0638) 

  IQβ                     0.26605   0.20524   0.33959   0.18180  
                 (0.1846)   (0.1679)   (0.2533)   (0.0843) 

AS 

  1β      0.00303   0.00010   0.00218   0.00048   –0.00396  –0.00053  0.00225   –0.00089 
 (0.0008)   (0.0002)   (0.0008)   (0.0002)   (0.0084)   (0.0011)   (0.0013)   (0.0009) 

  2β     0.84521   0.93895   0.86735   0.91141   0.60910   0.91166   0.86953   0.85358  
 (0.0358)   (0.0301)   (0.0375)   (0.0279)   (0.2047)   (0.0795)   (0.0387)   (0.0638) 

  3β     0.25223   0.09902   0.48895   0.24716   0.22894   0.07790   0.48150   0.26358  
 (0.0585)   (0.0823   (0.1723)   (0.0589)   (0.0530)   (0.1848)   (0.1674)   (0.1156) 

  4β     –0.44066  –0.16163  –0.01570  –0.06286  –0.38298  –0.16073  –0.02043  –0.05617 
 (0.1968)   (0.0653)   (0.0790)   (0.0564)   (0.2921)   (0.0923)   (0.0785)   (0.0831) 

  IQβ                     0.40265   0.05087   –0.00280  0.09615  
                 (0.3612)   (0.1275)   (0.0396)   (0.0533) 

Indirect 
GARCH 

  1β     0.00009   0.00002   0.00027   0.00001   0.00010   0.00001   0.00020   0.00014  
 (0.0008)   (0.0004)   (0.0005)   (0.0003)   (0.0007)   (0.0008)   (0.0009)   (0.0010) 

  2β     0.92011   0.89439   0.57487   0.88718   0.76380   0.85947   0.33369   0.69506  
 (0.2345)   (0.4421)   (0.1830)   (0.3737)   (0.5136)   (0.5754)   (1.6465)   (0.9587) 

  3β     0.33190   0.30253   2.23002   0.26633   0.36438   0.30606   2.15737   0.27593  
 (0.0955)   (0.2939)   (1.6959)   (0.2245)   (0.0785)   (0.5608)   (1.8255)   (0.5857) 

  IQβ                     0.16353   0.05555   0.41244   0.33313  
                 (0.4457)   (0.4795)   (1.6413)   (1.0470) 

AD 

  1β    –0.00006  –0.00001  –0.00009  –0.00001  –0.01450  –0.01260  –0.00243  0.01063  
 (0.0000)   (0.0000)   (0.0000)   (0.0000)   (0.0327)   (0.0000)   (0.0587)   (0.0000) 

  2β                     0.46859   0.87643   0.13146   0.79313  
                 (0.1660)   (0.0000)   (0.2360)   (0.0000) 

 3β                     0.01776   0.02359   –0.01067  –0.02933 
                 (0.0000)   (0.0000)   (0.0000)   (0.0000) 

  IQβ                     0.72811   0.24512   0.99871   0.20844  
                 (0.0636)   (0.0000)   (0.0955)   (0.0000) 

Note: 0.01; 0.05 – α-significance level of VaR; standard errors in bracktes; bolded values indicate signifi-
cant parameters according to t statistics. 

 Weights received on the basis of Weighted Averaged Combining method 
are included in a table (9). For gold, we observe that the forecast based on 
implied volatility receives a higher weight than the forecast from CAViaR 
models, and in the case of oil implied quantile receives a higher weight only 
in combining forecast of the implied quantile and CAViaR-AD. 



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149

Table  8. Estimated parameters of linear combination method 

Model 
 
 

Long position Short position 

0.01 0.05 0.01 0.05 

1γ  2γ  3γ  1γ  2γ  3γ  1γ  2γ  3γ  1γ  2γ  3γ  

Gold 

SAV –0.002 0.826 0.265 –0.005 1.153 0.104 –0.009 2.121 –0.584 –0.001 1.624 –0.403 

AS –0.001 1.091 –0.025 –0.005 1.187 0.090 –0.011 1.571 0.020 –0.001 1.598 –0.381 

Indirect GARCH –0.001 1.120 –0.057 –0.005 1.057 0.178 –0.008 2.253 –0.746 0.000 1.622 –0.468 

AD 0.028 1.215 –0.956 0.005 1.294 –0.460 0.027 1.637 –1.287 0.005 1.188 –0.378 

Oil 

SAV –0.010 0.650 0.572 –0.008 0.581 0.639 –0.011 0.593 0.601 –0.013 0.907 0.454 

AS –0.006 0.561 0.575 –0.006 0.296 0.863 –0.008 0.198 0.950 –0.007 0.447 0.758 

Indirect GARCH –0.014 0.617 0.686 –0.007 0.201 0.969 –0.010 0.492 0.704 –0.012 0.943 0.417 

AD –0.016 1.338 0.035 –0.015 1.314 0.162 –0.029 1.235 0.236 –0.123 1.476 3.111 

Note: 0.01; 0.05 – α-significance level of VaR; bolded values indicate models with higher value of 
parameter for forecasts derived on the basis of implied quantile model 
 

 For determining the coefficient λ  the EWQR method is applied. EWQR 
estimation was carried out on the in-sample data with the last 500 
observations excluded and considering a grid values for λ  between 0.97 and 
1 with a step size of 0.001. The smallest value of the function (28)–(29) was 
the criterion to determine the optimum values for λ  for the analysed 
significance levels of VaR (α ). The discount factor λ  for long and short 
positions and considered significance levels are shown in the table (10). The 
lower values for long positions mean that the 0.01 and the 0.05 quantile 
change more dynamically over time than the 0.01 and the 0.05 quantile in 
the case of a short position. 

Table  9.  Estimated weights of Weighted Averaged Combining method 

Instrument    Model   Long position  Short position  

 α  0.01   0.05   0.01   0.05  

Gold 

 SAV   0.5996   0.7290   0.8339   0.9066  
 AS   0.7771   0.6058   0.7016  0.9067  

 Indirect GARCH  0.5198   0.7028   0.7731  0.9281  
 AD  0.9519   0.9696   0.9760  0.9762  

Oil 

 SAV   0.3193   0.3928   0.4083   0.2192  
 AS   0.0822   0.2564   0.0439   0.0966 

 Indirect GARCH  0.3288   0.0832   0.3052  0.2480  
 AD   0.9770   0.9280   0.9071  0.9295  

Note: 0.01; 0.05 – α-significance level of VaR; bolded values indicate models with higher weight ω  
assigned to forecasts from implied quantile model 
  



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Table 10. Estimated Exponential Weight λ  

Instrument    Model   Long position  Short position  

α-significance level of VaR  0.01   0.05   0.01   0.05  

Gold 

 SAV   0.996   0.996   1.000  1.000  
 AS   1.000   0.982   1.000   1.000  

 Indirect GARCH  1.000   0.991   0.993   1.000  
 AD  0.991   1.000   1.000   0.995  

Oil 

 SAV   0.970   0.998   1.000  0.999  
 AS   1.000   0.998   1.000  1.000  

 Indirect GARCH  1.000   0.998   1.000  0.998  
 AD   0.993   0.981   0.997  1.000  

The estimated values of weights for Weighted Averaged Combining 
Optimized using Exponential Weighting is included in the table (11). We see 
the opposite situation than in the case of Weighted Averaged Combining 
without discounting factor. In the considered combinations, the forecasts 
from the CAViaR models receive a significantly higher weight ( 93.0> ).  

Table 11.  Estimated weights of Weighted Averaged Combining Optimized using 
Exponential Weighting  

Instrument    Model   Long position  Short position  

α-significance level of VaR 0.01  0.05   0.01   0.05  

Gold 

 SAV   0.008   0.026   0.009  0.036  
 AS   0.007   0.028   0.009   0.036  

 Indirect GARCH  0.008   0.028   0.009  0.037  
 AD  0.007   0.023   0.009   0.036  

Oil 

 SAV   0.016   0.060   0.019  0.062  
 AS   0.016   0.058   0.016   0.058  

 Indirect GARCH  0.015   0.063   0.018   0.063  
 AD   0.015   0.062   0.025   0.066  

 Results of measures are presented in tables (12) for gold and (13) for oil.  

Conclusions  

 In the present study the CAViaR models, the encompassing method and 
the combined forecasts methods are applied to determine the risk measure 
VaR. Since none of the forecasts is dominant and there is no universally 
accepted ranking of the various methods, we decided to check if the 
encompassing method or the forecast combination methods may reduce the 
risk of a large forecast error compared to individual forecast. 
  



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151

Table 12. Gold. Loss functions for post sample 

Model/metoda 
 

BL FL RL 

long short long short long short 
position position position position position position 

α 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 

IQ 15 40 8 32 87.22 153.25 23.33 56.22 76.02 146.41 13.73 50.34 

SAV 

CAViaR 7 35 2 16 63.80 133.08 17.27 35.02 49.26 124.91 4.60 27.46 
PlugIn 13 40 10 35 66.86 144.30 25.66 58.53 54.80 137.31 16.25 52.83 

LinearComb 10 43 12 29 75.95 160.81 28.79 52.09 63.61 154.35 20.01 46.00 
SimpAvg 10 35 3 20 73.65 139.98 17.96 41.08 60.78 132.44 6.79 34.31 
WtdAvg 10 38 5 32 75.23 146.53 20.08 56.27 62.67 139.32 9.93 50.40 

WtdAvgExp 7 35 2 16 63.90 133.41 17.26 35.14 49.39 125.27 4.62 27.63 

AS 

CAViaR 8 35 2 16 79.72 142.30 17.04 35.28 65.61 134.49 4.03 27.79 
PlugIn 15 52 10 31 69.21 159.32 25.85 54.38 57.75 152.94 16.70 48.60 

LinearComb 14 45 6 29 83.31 163.03 21.51 51.85 71.70 156.58 11.41 45.72 
SimpAvg 11 38 3 20 82.60 147.74 17.77 41.23 69.94 140.42 6.43 34.50 
WtdAvg 12 38 4 32 83.83 148.39 18.84 56.27 71.96 141.16 8.17 50.40 

WtdAvgExp 8 35 2 16 79.72 142.41 17.04 35.40 65.62 134.63 4.05 27.95 

Indirect 
GARCH 

CAViaR 9 31 2 14 56.45 118.36 17.44 32.92 41.77 109.45 4.74 25.15 
PlugIn 3 4 2 12 36.75 61.98 17.73 29.92 18.47 46.66 2.80 21.95 

LinearComb 15 41 14 33 86.04 155.96 32.44 57.16 74.58 149.32 24.03 51.25 
SimpAvg 10 34 3 20 68.26 133.03 18.05 40.74 55.28 125.13 6.87 33.89 
WtdAvg 11 36 5 32 69.72 140.40 20.06 56.49 56.84 132.93 9.73 50.65 

WtdAvgExp 9 31 2 15 56.60 118.96 17.43 34.02 41.96 110.10 4.76 26.31 

AD 

CAViaR 10 25 4 32 81.23 123.45 19.68 61.71 68.48 114.45 5.98 55.48 
PlugIn 15 51 8 32 68.35 153.60 24.49 55.07 57.01 147.15 14.59 49.10 

LinearComb 5 40 4 15 62.64 149.55 19.06 34.03 47.34 142.51 7.67 26.68 
SimpAvg 12 28 4 30 83.04 132.13 18.93 56.13 71.06 124.17 7.24 50.06 
WtdAvg 14 42 7 32 86.03 159.76 22.29 56.27 74.75 153.26 12.57 50.38 

WtdAvgExp 10 26 4 32 81.22 124.64 19.66 61.38 68.48 115.70 6.00 55.16 

Note: 0.01; 0.05 – α-significance level of VaR; bolded value indicates models/methods that both tests 
based on Bernoulli trials model and dynamic quantile test indicate as adequate, underlined value is the 
lowest value among the models/methods for the analysed position and at the significance level. 

  The subject of the study were two assets: gold and oil. For crude oil, in 
the encompassing method we observe significant contribution of the implied 
volatility to the VaR. For gold, the implied quantile has been assigned 
a higher weight in the method of Linear Combination and Weighted 
Averaged Combining method. 
 Conclusions made by Jeon and Taylor (2013), who analyzed the 
CAViaR models, the encompassing method and the combination methods 
for the capital market indices differ from the one presented in this study for 
the commodities. Jeon and Taylor (2013) using backtests and the Dynamic 
Quantile test pointed out the forecast determined by combining methods as 
more adequate than coming from CAViaR models or the implied quantile 
model. In their view, the encompassing method derives more accurate 



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forecasts than the individual CAViaR models or the implied quantile model 
individually, but considering only the criterion of the smallest fraction of 
exceedances Hit (Table (5) in Jeon and Taylor (2013)). Among the methods 
of combining forecasts Jeon and Taylor (2013) pointed out that the linear 
combination method and the arithmetic mean method generate the most 
adequate VaR. 

Table 13. Oil. Loss functions for post sample 

Method/model 

BL FL RL 

long short long short long short 

position position position position position position 

α 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 0.01 0.05 

IQ 8 24 4 28 53.53 91.59 19.90 49.10 38.92 81.89 5.19 39.78 

SAV 

CAViaR 6 24 2 30 27.18 75.88 18.94 50.55 7.74 65.51 2.02 41.36 

PlugIn 9 51 10 65 44.49 132.40 25.07 112.84 29.46 124.69 11.22 106.70 

LinearComb 8 42 10 59 40.44 115.63 24.97 105.85 24.87 107.44 11.76 99.59 

SimpAvg 7 25 0 27 35.47 81.99 15.88 46.29 18.46 72.00 0.00 37.01 

WtdAvg 7 24 0 27 31.72 79.47 16.08 46.76 13.85 69.38 0.00 37.50 

WtdAvgExp 6 23 2 29 27.27 75.25 18.90 49.29 7.91 64.92 2.02 40.08 

AS 

CAViaR 5 23 1 20 26.86 77.04 18.58 35.75 6.62 66.64 1.10 25.07 

PlugIn 11 38 1 34 55.47 108.30 18.73 60.07 41.76 99.63 1.07 51.86 

LinearComb 7 35 4 27 36.01 101.16 19.97 48.09 19.04 92.17 4.60 39.09 

SimpAvg 6 25 0 25 34.13 83.56 16.15 41.35 16.70 73.55 0.00 31.40 

WtdAvg 5 24 1 22 27.17 79.86 18.43 37.66 7.39 69.66 1.08 27.14 

WtdAvgExp 5 23 1 21 26.90 77.38 18.52 36.69 6.75 67.01 1.09 26.10 

Indirect 
GARCH 

CAViaR 3 18 0 23 25.45 58.71 18.65 39.65 4.52 47.11 0.00 29.57 

PlugIn 6 20 0 0 33.06 63.24 18.08 16.03 15.50 52.08 0.00 0.00 

LinearComb 6 26 5 53 37.19 78.81 20.19 92.12 20.71 68.92 5.10 85.37 

SimpAvg 6 18 0 25 34.84 69.12 16.72 42.12 17.11 58.41 0.00 32.44 

WtdAvg 5 18 0 26 30.59 60.08 17.47 42.59 11.78 48.63 0.00 32.75 

WtdAvgExp 3 19 0 24 25.52 60.71 18.58 40.57 4.68 49.24 0.00 30.57 

AD 

CAViaR 0 5 0 5 29.80 34.08 19.28 20.99 0.00 16.00 0.00 5.50 

PlugIn 12 45 13 90 68.43 142.19 29.38 161.95 54.95 134.43 16.61 156.82 

LinearComb 15 44 59 205 77.90 132.14 114.91 470.52 65.61 124.19 107.58 468.66 

SimpAvg 0 8 0 11 34.45 49.03 18.96 25.55 0.00 35.05 0.00 13.10 

WtdAvg 10 34 4 55 59.29 113.07 19.77 101.50 45.54 104.52 5.42 94.84 

WtdAvgExp 0 5 0 5 30.73 35.10 19.21 20.69 0.00 17.51 0.00 5.57 

Note: 0.01; 0.05 – α-significance level of VaR; bolded value indicates models/methods that both tests 
based on Bernoulli trials model and dynamic quantile test indicate as adequate, underlined value is the 
lowest value among the models/methods for the analysed position and at the significance level. 

 In our research, the Kupiec test is used in conjunction with the Dynamic 
Quantile test and the Christoffersen test, so the models and methods that 
underestimate or overestimate VaR are rejected. We show that the criteria 
based on the Kupiec, the Christoffersen and the Dynamic Quantile test 



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together more precisely classify the VaR forecasting method than the one 
used by Jeon and Taylor (2013). Proposals of evaluation VaR models 
formulated in this study may protect the institution against errors arising 
from the application of methods pointed out by Jeon and Taylor (2013), and 
inefficiently maintaining too high level of capital. 
  In the case of the commodities neither the encompassing method nor the 
combining forecast method improve VaR forecasts. The method of choosing 
the most adequate model presented in this paper leads to the CAViaR-SAV 
model selection as the source of most optimal measure of risk forecasts. The 
Kupiec test, the Christoffersen and the Dynamic Quantile test indicate the 
model as an adequate to forecast VaR for gold and oil for short positions at 
the 0.01 and the 0.05 significance level, and for a long position at the 0.05 
significance level. The model generates also the lowest value of loss 
functions.  
 In our study, none of the models and none of the methods have predicted 
adequately the VaR for a long position at the 0.01 significance level, leading 
to the conclusion that the choice of model should also depend on the 
financial institution's investment strategy and portfolio structure. 

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