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

Joanna Górka 
Nicolaus Copernicus University in Toruń 

The Sign RCA Models: Comparing Predictive  
Accuracy of VaR Measures† 

A b s t r a c t. Evaluating Value at Risk (VaR) methods of predictive accuracy in an objective and 
effective framework is important for both efficient capital allocation and loss prediction. From 
this reasons, finding an adequate method of estimating and backtesting is crucial for both the 
regulators and the risk managers’. The Sign RCA models may be useful to obtain the accurate 
forecasts of VaR. In this research one briefly describes the Sign RCA models, the Value at Risk 
and backtesting. We compare the predictive accuracy of alternative VaR forecasts obtained from 
different models. Empirical example is mainly related to the PBG Capital Group shares on the 
Warsaw Stock Exchange. 

K e y w o r d s: Family of Sign RCA Models, Value at Risk, backtesting, loss function.  

1. Introduction 
 Nowadays, accurate modelling of risk is very important in risk management. 
This is a result of the globalisation of financial market, the evolution of the 
derivative markets and the technological development. Value at Risk (VaR) has 
become the standard measure to quantify market risk1. This measure can be 
used by the financial institutions to assess their risks or by a regulatory 
committee to set margin requirements.  
 In the literature, many parametric VaR models and many forecasting 
accuracy assessments for VaR methods exist. The important representation of 
the parametric VaR models are the generalized autoregressive conditional 
heteroskedasticity models (GARCH) (Bollerslev, 1986; Engle, 1982). These 
models describe non-linear dynamics of financial time series. A different, 
alternative approach to the description of financial time series represent the 

                                                 
† This work was financed from the Polish science budget resources in the years 2008-2010 as 

the research project N N111 434034. 
1 It was introduced by JP Morgan in 1996.  



Joanna Górka 62

random coefficient autoregressive models (RCA) (which were proposed by 
Nicholls, Quinn, 1982). Thavaneswaran et al. proposed a number of expansions 
of the random coefficient autoregressive model order one. The new models, 
such as Sign RCA(1), RCAMA(1,1), Sign RCAMA(1,1), RCA(1)-GARCH(1,1) 
and Sign RCA(1)-GARCH(1,1) can be used to obtain Value-at-Risk measure.    
 The aim of this paper is to use the family of Sign RCA models to obtain the 
VaR forecasts and compare the results obtained from Sign RCA models with 
other selected VaR models. 

2. The Family of Sign RCA Models 
 Random coefficient autoregressive models (RCA) are straightforward 
generalization of the constant coefficient autoregressive models. A full 
description of this class of models including their properties, estimation 
methods and some applications can be found in Nicholls and Quinn (1982). 
 Thavaneswaran, Appadoo and Bector (2006) proposed a first order random 
coefficient autoregressive model with a first order moving average component, 
i.e. RCAMA(1,1). In another paper Thavaneswaran and Appadoo (2006) 
proposed to add the sign function to RCA(1) and RCAMA(1,1) models. 
 The last modification is based on assumption that residuals from the RCA 
model or the Sign RCA model can be described by a GARCH model. In this 
way, the RCA(1)-GARCH(1,1) model and Sign RCA(1)-GARCH(1,1) model 
were created. All these modifications influence the increase of variance and 
kurtosis of processes2.  
 In Table 1 equations of individual models from the family of Sign RCA 
models and their names are presented. 
 To ensure the existence of the I-VI models (Table 1) the following 
assumptions must be satisfied: 

⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛
⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
2

2

0
0

,
0
0

~
ε

δ

σ
σ

ε
δ iid

t

t , (1) 

122 <+ δσφ . (2) 

 The sign function, described by the following formula 

1 for 0,
0 for 0,
1 for 0,

t

t t

t

y
s y

y

>⎧
⎪= =⎨
⎪− <⎩

 (3) 

                                                 
2 Theoretical properties of  the family of Sign RCA models can be found in articles, i.e.: 

Appadoo, Thavaneswaran, Singh (2006), Aue  (2004), Górka, (2008), Thavaneswaran, Appadoo, 
Bector (2006), Thavaneswaran, Appadoo (2006), Thavaneswaran, Appadoo, Ghahramani, (2009), 
Thavaneswaran, Peiris, Appadoo (2008).  



The Sign RCA Models: Comparing Predictive Accuracy of VaR Measures 63

has the interpretation: if Φ>+ tδφ , the negative value of Φ  means that the 
negative (positive) observation values at time 1−t  correspond to a decrease 
(increase) of observation values at time t . In the case of stock returns it would 
suggest (for returns) that after a decrease of stock returns, the higher decrease of 
stock returns occurs than expected, and in the case of the increase of stock 
returns the lower increase in stock returns occurs than expected. 

Table 1.  The family of Sign RCA models (without conditions) 
Model Model equations No. 

RCA(1) ( ) tttt yy εδφ ++= −1  I 
Sign RCA(1) ( ) ttttt ysy εδφ +Φ++= −− 11  II 
RCAMA(1,1) ( ) 11 −− +++= ttttt yy θεεδφ  III 

Sign RCAMA(1,1) ( ) 111 −−− ++Φ++= tttttt ysy θεεδφ  IV 

RCA(1)-GARCH(1,1) 

( ) tttt yy εδφ ++= −1 , 

ttt zh=ε  
11

2
110 −− ++= ttt hh βεαα  

V 

Sign RCA(1)-GARCH(1,1) 

( ) ttttt ysy εδφ +Φ++= −− 11 , 

ttt zh=ε  
11

2
110 −− ++= ttt hh βεαα  

VI 

Note: ts – sign function is described by equation (3); φ , θ , Φ , iα , 1β  – model parameters.  

 Condition (2) is necessary and sufficient for the second-order stationarity of 
process described by equation I, however conditions (1)-(2) ensure strict 
stationarity of this process. If conditions (1)-(2) are satisfied, then processes 
described by equations II-IV are stationary in mean. 
 If residuals from the RCA model are described by a GARCH model, then 
the RCA(1)-GARCH(p,q) model described by equation V, where ( )2,0~ zt Nz σ , 

00 >α , 0≥iα  and 0≥jβ , is obtained. If the sign function is added to the 
RCA-GARCH model, then the process described by equation VI is obtained. 
The conditions ensuring the positive value of conditional variance of this 
process are the following: ( )2,0~ zt Nz σ , 00 >α , 0≥iα , 0≥jβ , 0α≤Φ . 

 Predictors of the conditional mean and conditional variance of Sign RCA 
models are presented in Table 2 and 3 respectively. 
  



Joanna Górka 64

Table 2. Conditional mean predictors 

Models Conditional mean 

RCA(1), RCA(1)-GARCH(1,1) ( )11
P

t t tt ty E y F yϕ++ = =  
Sign RCA(1),  

Sign RCA(1)-GARCH(1,1) ( ) ( )11
P

t t t tt ty E y F s yϕ++ = = +Φ  

RCAMA(1,1) ( )11
P

t t t tt ty E y F yϕ θε++ = = +
 

Sign RCAMA(1,1) ( ) ( )11
P

t t t tt ty E y F s yϕ θε++ = = +Φ +
 

Table 3. Conditional variance predictors 

Models Conditional variance 
RCA(1), Sign RCA(1),  

RCAMA(1,1), Sign RCAMA(1,1) ( )2 2 2 2 2
11 t t tt t E u F yε δσ σ σ++ = = +  

RCA(1)-GARCH(1,1),  
Sign RCA(1)-GARCH(1,1) ( )2 2 2 2 2

11 ( )t t z t tt t E u F E h yδσ σ σ++ = = +  

3. Value-at-Risk 
 Value-at-Risk (VaR) is used as a tool for measuring market risk. It is 
defined as „the maximum potential loss that a portfolio can suffer within a fixed 
confidence level during a holding period”. 
 Formal definition of VaR is following (Artzner, Delbaen, Eber, Heath, 
1999): 

( ) ( ){ } ( ){ }VaR inf : inf : 1XX x F x x P X xα α α= ≥ = > ≤ − , (4) 

where ( )0,1α ∈  is a particular confidence level, XF – the cumulative density 
function. 
 Consider a time series of daily ex post returns ( ( )1100 ln lnt t tr P P−= − where 

tP  is the share price at time t) and corresponding time series of ex ante VaR 
forecasts ( VaRα ), the formula (4) takes the form: 

( )1 VaRtP r α α+ ≤ − = . (5) 

The negative sign arises from the convention of reporting VaR as a positive 
number. 
 One-step-ahead conditional forecasts of Value-at-Risk are calculated by the 
formula:  

( )1 1 1VaR ,l
t t t t t zαα μ σ+ + += +  (6)

 
 

where tt |1+μ , tt |1+σ  are one-step-ahead conditional forecasts of mean and 
volatility respectively. 



The Sign RCA Models: Comparing Predictive Accuracy of VaR Measures 65

3.1. Estimation Methods for VaR 

 This section briefly describes the alternative models that we use for 
estimating VaR forecasts in this paper.  
 The following models are used in the research to obtain VaR forecasts: 
− The historical simulation (HS)3. The VaR is estimated as the α-th quantile 

of the empirical distribution of returns. HS is based on the assumption that 
returns are iid time series of an unknown distribution.    

− The equally weighted moving average (EWMA) model, i.e.  

2 2
1

1

1 t

it t
i t k

r
k

σ +
= − +

= ∑ , (7) 

where k  –  size of window, 2
ir  – returns. The returns are assumed to be 

normally distributed. 
− The RiskMetrics (RM) model, i.e. 

( ) ( )2 2 2 2
1

1

1 1
t

t i
i t tt t

i t k

r rσ λ λ λσ λ−
+

= − +

= − = + −∑ , (8) 

where ( )0,1λ∈  is known as the decay factor, 2
tλσ  is the previous volatility 

forecast weighted by the decay factor, and ( ) 21 trλ−  is the latest squared 
returns weighted by ( )1 λ− . The VaR is estimated under the assumption 
that returns are normally distributed (as in the case of EWMA).   

− The AR(1)-GARCH(1,1) model, i.e. 

1t t tr rφ ε−= + , (9) 

where t t tzε σ= ,  )1,0(~ Nzt , 
2 2 2

1 1t t tσ ω αε βσ− −= + + . (10) 

In this case, returns series is assumed to be conditionally normally 
distributed. 

− Models from the family of Sign RCA models4. 
  

                                                 
3 HS is the oldest and still very popular estimator of the VaR.  
4 They were presented in previous section. 



Joanna Górka 66

3.2. Backtesting VaR Estimates 

 Backtesting is based on testing whether the VaR estimates are statistically 
accurate.  
 The ,,failure process” is defined as:  

( )1 VaR , 1, ...,l
t t tI r t T T N= < − = + + , (11) 

where ( )*1  denotes the indicator function returning a unit if the argument is 
true, and zero otherwise; T  is the size of the sample used to estimate 
parameters of the model; N  is the number of one-step-ahead VaR forecasts 
computed. The VaR forecasts are accurate if the { }tI series is iid with mean α , 

i.e. | 1t tE I α−⎡ ⎤ =⎣ ⎦ . To test the statistical accuracy we used the standard 
likelihood ratio tests: 

1. The proportion of failures test – LRpof (Kupiec, 1995) 5: 

[ ] [ ]0 1: . :t tH E I vs H E Iα α= ≠ , 

12ln
ˆ ˆ1

N n n

pofLR α α
α α

−⎡ ⎤−⎛ ⎞ ⎛ ⎞= − ⎢ ⎥⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
      ~ 2

1χ , (12) 

where n  is the number of failures VaR, α̂  is the MLE of α , i. e. n
N . 

2. The Christoffersen independence test – LRind (Christoffersen, 1998): 

0 01 11:H α α= , 

( )
( ) ( )

00 10 01 11

00 1001 11
01 01 11 11

1
2ln

1 1

T T T T

ind T TT T
LR

α α

α α α α

+ +−
= −

− −
      ~ 2

1χ , (13) 

where: 

0 1

ˆ ij
ij

i i

T
T T

α =
+

, 
11011000

1101

TTTT
TT

+++
+

=α , 

ijT  – number of i values followed by a j value in the tI  series ( ), 0,1i j = . 

3. The time between failures test – LRtbf (Haas, 2001) 6:  
1

1

12 ln
1

ivN

tbf
i i i

LR α α
α α

−

=

⎡ ⎤⎛ ⎞−⎢ ⎥= − ⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦
∑       ~ 2

Nχ , (14) 

                                                 
5 Similar, the LR test of unconditional coverage by Christoffersen (1998) was proposed. Other 

symbol of this test is the LRuc. 
6 Haas extended the Kupiec’s time until first failure test (TUFF test) by adding test for every 

exception (second and next).  



The Sign RCA Models: Comparing Predictive Accuracy of VaR Measures 67

where 
i

i v
1

=α , 1v  – time until first failure,  iv  – time between exception 

( )1−i  and  exception i  for 2, ...,i N= . 

 If, in above tests the null hypothesis is not rejected, then a particular model 
gives accurate forecasts of VaR. However, if more than one model is deemed 
adequate, we cannot conclude which of VaR model should be selected.   
 Lopez (1998) suggested measuring the accuracy of VaR forecasts on the 
basis of distance between observed returns and forecasted VaR values. This 
approach does not give any formal statistical selection of model adequacy but it 
allows to rank the models.  

 Let 
1

N

t
t

f f
=

= ∑ means a total loss function. A model which minimizes the 

total loss function is preferred over the other models. In the literature, different 
loss functions were proposed (see Lopez, 1998, 1999; Blanco and Ihle, 1998; 
Sarma, Thomas and Shah, 2003, Caporin, 2003; Angelidis, Benos and 
Degiannakis, 2004). In this paper, the loss functions used to compare the 
accurate VaR forecasts are as follows: 
− The regulatory loss function – RL (Lopez, 1999)7: 

( )
1 ,

21
1 , 1 ,

0 VaR ,

1 VaR VaR .

t r t

t
t r t t r t

r
f

r r

+

+

+ +

> −⎧⎪= ⎨
+ + ≤ −⎪⎩

 (15) 

− The firm’s loss function – FL (Sarma, Thomas, Shah, 2003): 

( )
, 1 ,

21
1 , 1 ,

VaR VaR ,

1 VaR VaR .

r t t r t

t
t r t t r t

c r
f

r r

+

+

+ +

> −⎧⎪= ⎨
+ + ≤ −⎪⎩

 (16) 

where c is a measure of cost of capital opportunity. 
 Sarma, Thomas and Shah (2003) proposed testing for superiority of a model 
vis-á-vis another in terms of the loss function. They suggested a two-stage VaR 
model selection procedure. The first stage consists in testing the statistical 
accuracy for the competing VaR models. In the second stage of the VaR model 
selection procedure, the firm’s loss function is used to evaluate statistically VaR 
models8.  
  

                                                 
7 This name comes from Sarma, Thomas and Shah (2003) who explain that (16) is able to 

express the regulatory concerns in model evaluation. However, no score is attached in case if 
exception does not occur. 

8 Only that VaR model for which the average number of failures was equal to the expected 
and these failures are independently distributed is included in the second stage.   



Joanna Górka 68

 Consider two VaR models, i and j. The hypotheses are:  

0 1: 0 . : 0H vs Hθ θ= < , 

where θ  is the median of the distribution of , ,t i t j tz f f= − , where ,i tf  and ,j tf  
are the values of loss function generated by model i and model j respectively. 
Negative values of tz  indicate a superiority of model i over j. 

 The testing procedure is as follows: 
1. Define an indicator variable ( )0t tzψ = ≥1  and the number of non-negative 

tz ’s,  as 
1

T N

ij t
t T

S ψ
+

= +

= ∑ .  

2. Calculate the statistics as: 

( )
0.5

~ 0,1
0.25

ij
ij

S N
STS N

N
−

= asymptotically, (17) 

ijSTS  is based on assuming that the tz  is iid9.  

 Alternatively, we can compare competing VaR models using the predictive 
quantile loss function (see Giacomini and Komunjer, 2005; Bao et al., 2006). 
The expected loss function is given by: 

( ) ( )
1

1 1 VaR VaR
N

i i i i
i

Q r r
Nα α

=

⎡ ⎤= − < − +⎣ ⎦∑ . (18) 

The selected model is the VaR model which has the minimum of Qα . 

4. Empirical application 
 The data used in the empirical application are daily prices of twenty Polish 
firms’ shares from the WIG20 portfolio on the Warsaw Stock Exchange (WSE). 
The data were obtained from bossa.pl for the period from 23-rd September 2005 
to 18-th February 2009, which yields 852 observations. However, one of shares 
was excluded because it was not quoted on 23 September 2005. To analyze 
daily percentage log returns of each share were used. 
 This empirical study was composed of two parts. The first part (Analysis I) 
was carried out with regard to all of twenty shares from WSE. The research 
procedure was the following: 

1. For the first 500 observations of each returns series the descriptive 
statistics and some tests were calculated. Next, returns series with 

                                                 
9 For details on the sign test see Diebold and Mariano (1995). 



The Sign RCA Models: Comparing Predictive Accuracy of VaR Measures 69

autocorrelation and kurtosis bigger than for normal distribution were 
chosen10. 

2. Parameters of six models from the family of Sign RCA were estimated for 
the first 500 observations of time series selected in step one. Next, only 
models with statistically significant parameters were used.    

3. The estimation of parameters for models selected in step 2 was performed 
for rolling window of 100, 150, 200, 250, 300, 400, 500 observations. In 
the same way the estimation of AR(1)-GARCH(1,1) models was obtained.  

4. For all models from step 3 and for the historical simulation (HS), the 
equally weighted moving average (EWMA) model, the RiskMetrics (RM) 
models (with 0.95λ =  and 0.99λ = ) VaR measures were calculated11. 
One-step-ahead forecasts of VaR (that is 751, 701, 651, 601, 551, 451, 351 
forecasts, respectively) were calculated on the basis of these models. 

5. The traditional VaR tests and loss functions for each model and window 
were calculated. 

6. The obtained results in above step were compared. 
 In the second part (Analysis II) only the PBG shares (PBG Capital Group) 
was chosen. All presented models of VaR for the last 250 observations were 
calculated12. For obtained VaR forecasts the two-stage VaR model selection 
procedure was applied. 
 All model parameters (Analysis I and II) were estimated using maximum 
likelihood (MLE) with the BFGS algorithm. Calculations were carried out in 
the Gauss program. 

4.1. Results of the Analysis I 

 Selected results of the descriptive statistics and some tests are given in 
Table 4. All series have a mean between -0.052 and 0.561, kurtosis bigger than 
for normal distribution. The standard deviations are different, ranging from 
1.955 for PGNIG to 5.354 for BIOTON. The skewness and kurtosis differ 
among all series. Only 8 of 19 returns series have autocorrelation. The LBI test 
rejects the null hypothesis of non random coefficient to four stock returns. 

                                                 
10 This method of the elimination of initially selected companies can impact on the results. 

It would be worth to check out which results might be obtained for the whole set of companies. 
However, such analysis was omitted in this paper.  

11 The returns series were assumed either to be normally distributed or conditionally normally 
distributed, respectively.   

12 The set of 250 observations corresponds to roughly one year of trading days and according 
to the Basel II Accord requirement the minimum of 250 VaR forecasts should be used to the 
backtesting approach. Therefore, one-step-ahead forecasts of VaR at the same period (250 
observations) were calculated. Parameters were estimated for rolling windows of 125, 250, 375 
observations each. The returns series were assumed either to be conditionally normally distributed 
or normally distributed respectively.   



Joanna Górka 70

 Next, the 7 different models were estimated for 8 returns series.  Further, 
only models with statistically significant parameters were chosen. In this way 
models like RCA and Sign RCA were chosen. 
 To present backtesting results for VaR forecasts of the PBG shares was 
chosen because for that share the autoregressive parameter in the RCA models 
for all returns series has been the biggest. It is very important because we can  
expect the Sign RCA model to be better than other models.     
 The traditional VaR tests and loss functions for the PBG for all models are 
presented in Table 5 and the 5% at significance level. One can see that the 
accuracy test rejects the null hypothesis for windows size of 500, 400 
observations for HS, EWMA model, AR(1)-GARCH(1,1) model, RCA model 
and Sign RCA model. For example, for window size 250 the regulatory loss 
function is the smallest for RM ( 0.95λ = ). Next position in this ranking have 
AR(1)-GARCH(1,1) model, EWMA model, RCA model, Sign RCA model and 
the last position has RM ( 0.99λ = ). The HS method is not taken into 
consideration because the accuracy test rejects the null hypothesis for windows 
size of 250 observations. On the other hand, the firm’s loss function is the 
smallest for RM ( 0.99λ = ) and the next positions in ranking have Sign RCA 
model, RCA model, EWMA model, AR(1)-GARCH(1,1) model and RM  
( 0.95λ = ).  
 The differences between values of the firm’s loss function are small for 
estimated models. To compare these results, the tests for superiority of a model 
vis-á-vis another were used only for models included into the second stage at 
Sarma, Thomas and Shah procedure. The results are presented in Table 6. For 
the window size 300 we can see that the Sign RCA model is significantly better 
than other models, i.e. the null hypothesis is rejected in the test of superiority 
between the Sign RCA model and the other models presented in subsection 3.1.  
However, as the size of windows decreases the RM model ( 0.99λ = ) 
outperforms the Sign RCA model. RCA and Sign RCA models are statistically 
the same for the window size 100. In cases when results with HS are compared 
one can see that HS is almost everywhere significantly better than others. 
 The Table 7 includes the results of the VaR tests and the loss function at the 
2,5% significance level which are similar to the results obtained at the 5% 
significance level. Only for HS with the window size 250 and for RCA model 
with the window size of 300 observations, some differences can be noticed, i. e. 
In the case of HS the accuracy at the 2.5% is better than at the 5% significance 
level (except RCA model). For the loss function conclusions are the same with 
one exception, i. e. the HS has the last rank for regulatory loss function and the 
first rank for firm’s loss function. 
 At the 1% significance level we obtained more differences (see Table 8). 
Firstly, Risk Metrics models are accurate only for windows size 500 and 500, 
400 for 0.95λ = , 0.99λ = , respectively. The RCA, Sign RCA and EWMA 



The Sign RCA Models: Comparing Predictive Accuracy of VaR Measures 71

models are accurate for small windows (size 200, 150, 100). The regulatory loss 
function is the smallest for HS. The firm’s loss function has the lowest values 
for Sign RCA models for the window size 200. Very strange results were 
obtained for HS and therefore we are not able to find any rules for accuracy and 
value of the regulatory loss function.  

4.2. Results of the Analysis II 

 Firstly, we calculated the 250 one-step-ahead forecasts of VaR of the PBG 
share using all models of VaR (presented in 3.1)13. The VaR forecasts were 
received from different models estimated for the different window sizes, i.e,  
T =125, 250 and 375.  
 Secondly, the competing VaR models were testing for statistical accuracy. 
For the established period of forecasting, only Sign RCAMA(1,1) models (for  
T = 375 and all significance level, for T = 250 and α = 2.5%, 1%), Sign 
RCA(1)-GARCH(1,1) models (for α = 1%  and rolling window size T = 375, 
125) and Risk Metrics models (for λ= 0,99 and α = 1%  and T = 125) did not 
fulfill the conditions used at first stage of Sarma, Thomas and Shah procedure 
(the null hypothesis was rejected at least for one test, see (12)-(14)). For other 
models, the firm’s loss function (see the Table 9), the STS test and the 
predictive quantile loss function (see the Table 10) were calculated. Lower 
values of the firm’s loss function for VaR forecasts were received from 
RCAMA(1,1), RCA(1) and Sign RCAMA(1,1) (if it was included at second 
stage) models (with the exception of the HS for α = 5% and T = 375, 250 and 
with the exception of the RM(λ= 0,99) for T = 125 and α = 5%, 2.5%).  The test 
for superiority of a model vis-á-vis another indicates that: 

1. At the 5% significance level, for different rolling window sizes, each of 
models having first rank is superior over other models. 

2. At the 2.5% significance level, for rolling windows size of 125 
observations, the RM (λ= 0,99) is superior over other models. The 
RCAMA(1,1) model is better than almost all other models (with the 
exception of HS method and RCA(1) model for T = 375 and with the 
exception of the RCA(1)-GARCH(1,1) model for T = 250, for which the 
predictive ability is equal). 

3. For the α = 1%, for different rolling window sizes, each of models having 
first rank is superior over other models (with the exception of 
RCAMA(1,1) and RCA(1) models for T = 375 that have equal predictive 
ability). 

 Other conclusions are formulated based on the predictive quantile loss 
function (Table 10), which yields different position in the ranking. For VaR 
forecasts of the PBG share, for established forecasting period, the choice of the 
                                                 

13 One-step ahead forecasts on the period 19.02.2008-18.02.2009 were computed. 



Joanna Górka 72

best model from the competing models depends on the significance level and 
rolling window sizes. For the Sign RCA models the rolling window size of 125 
observations seemed too small. This conclusion is similar to one from 
Analysis I. 

5. Conclusions 
 Evaluating forecasts based solely on one criterion yield the limited 
information regarding the accuracy method. Thus, in the literature is commonly 
accepted that results of each evaluation criterion are presented separately and 
then best performing method is selected. However, it can be noticed that the 
different evaluation criteria give the different choice of the best estimation 
method of VaR. Therefore, it is difficult to make general remarks, nevertheless 
the empirical results showed that: 

1. None of the presented methods gave a satisfactory VaR estimates. 
2. The results showed no domination of either forecasting methods of VaR.      
3. Bigger sample did not lead to the better results.  
4. It seems that the family of Sign RCA models should be used for the sample 

size of 150 to 300 observations.  
5. In terms of the firm’s loss function the Sign RCA model was significantly 

better than the AR-GARCH model, RM ( 0.95λ = ) model and EWMA 
model. The Sign RCA model was not worse than the standard RCA model. 

6. One should treat every share individually and use different methods and 
models for obtaining a good forecast of VaR. 

7. The historical simulation gave better results (in terms of accuracy) at the 
1% significance level than for other significance levels. It seems that the 
minimum window size should be 250 observations but smaller than 500 
observations. 

8. The RCAMA(1,1) model can be competitive to other VaR measures from 
the firm’s loss function  point of view. 

9. The Sign RCA models with GARCH errors did not give better forecasts of 
VaR for the PBG share. 

References  
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Modele Sign RCA: Porównanie trafności prognoz VaR 

Z a r y s  t r e ś c i. Obiektywna i skuteczna ocena trafności prognozowania wartości narażonej na 
ryzyko (Value at Risk  – VaR) jest bardzo ważna zarówno dla efektywnego zarządzania kapitałem 
jak i do prognozowania strat. Z tego powodu znalezienie odpowiednich metod estymacji 
i weryfikacji VaR jest kluczowe zarówno dla instytucji nadzorujących jak i dla menadżerów. 
Modele Sign RCA mogą być użyteczne do otrzymywania trafnych prognoz VAR. W artykule, 
pokrótce przedstawione są modele Sign RCA, wartość narażona na ryzyko i weryfikacja prognoz 
VaR. Porównana jest trafność prognoz VaR otrzymanym z różnych alternatywnych modeli. 
Przykład empiryczny skoncentrowany jest głównie na cenach akcji spółki PBG notowanej na 
Giełdzie Papierów Wartościowych w Warszawie.  

S ł o w a  k l u c z o w e: Modele klasy Sign RCA, Value at Risk, testowanie wsteczne, funkcja 
strat.  



Table 4. Results of the descriptive statistics, Box-Ljung tests and locally best invariant 
test 

Company Mean Std. Dev. Skewness Kurtosis B-L (1) B-L (2) LBI 
AGORA -0,052 2,451 -0,204 4,853 8,925*** 9,029*** 1,672 

ASSECOPOL 0,169 2,493 -0,582 13,015 6,953*** 8,357*** 2,848** 
BIOTON -0,036 5,354 -8,286 138,621 1,673 2,111 -0,028 

BRE 0,256 1,972 0,263 4,055 3,915** 4,025 2,378** 
BZWBK 0,175 2,442 -0,135 3,472 1,478 2,738 1,034 

CERSANIT 0,247 2,361 0,567 6,312 0,156 1,887 1,639 
GETIN 0,231 2,646 0,523 11,370 0,008 0,837 0,954 
GTC 0,255 2,737 0,461 5,383 1,510 8,046*** 0,461 

KGHM 0,212 3,011 -0,591 5,303 0,001 4,766 1,156 
LOTOS 0,036 2,174 -0,329 4,835 1,596 2,249 -0,078 

PBG 0,363 2,094 0,095 5,344 3,466* 3,468 1,909 
PEKAO 0,071 2,160 0,219 3,616 0,005 0,044 0,273 
PGNIG 0,068 1,955 0,192 4,413 0,284 4,870* 2,929** 

PKNORLEN -0,021 2,170 -0,069 3,853 0,017 3,680 0,508 
PKOBP 0,117 2,055 0,324 3,912 3,625* 3,647 0,002 

POLIMEXMS 0,366 2,420 -0,172 6,835 2,402 3,945 1,449 
POLNORD 0,561 5,290 -1,387 28,269 2,085 2,489 -0,047 

TPSA -0,022 1,978 -0,161 3,775 0,310 1,757 1,109 
TVN 0,145 2,242 -0,083 3,716 3,004* 3,250 3,218** 

Note: *, **, *** indicate rejection of H0 at the 10% ,5% and 1% significant level, respectively.  
B-L (1) – estimates of the Box-Ljung test statistics of order 1. B-L (2) – estimates of the Box-Ljung test 
statistics of order 2.  LBI – estimates of the locally best invariant test statistics. 

Table 5. Results of the VaR tests (95% VaR for PBG) and the loss function 
Model α̂  LRpof LRind LRtbf RL FL 

SH             
500 10,54% 17,451*** 1,370 41,329 205,55 1366,59 
400 9,09% 12,929*** 1,139 33,525 240,15 1722,11 
300 7,62% 6,923*** 0,603 34,955 279,97 2106,98 
250 7,15% 5,213** 0,494 34,272 281,66 2246,19 
200 5,84% 0,914 0,026 32,805 261,26 2462,33 
150 5,56% 0,453 0,016 40,548 263,62 2668,63 
100 4,79% 0,068 0,394 35,815 224,13 2913,60 

EWMA      500 9,12% 10,179*** 1,967 32,075 186,89 1427,20 
400 7,98% 7,207*** 0,350 28,098 208,58 1818,34 
300 6,35% 1,961 0,027 26,143 236,04 2235,86 
250 5,82% 0,817 0,723 28,900 229,42 2428,40 
200 5,07% 0,007 0,348 31,243 223,04 2628,37 
150 4,42% 0,512 0,121 32,300 222,40 2856,85 
100 4,26% 0,907 0,117 35,007 215,48 3079,17 

 
 
 



 

Table 5. Continued 
Model α̂  LRpof LRind LRtbf RL FL 

RM (λ= 0,95)    500 7,12% 2,959* 3,850* 25,133 117,94 1629,97 
400 6,43% 1,788 0,544 25,727 148,73 2036,69 
300 5,99% 1,070 0,657 27,728 206,95 2394,60 
250 5,49% 0,296 0,481 29,504 206,95 2546,11 
200 5,07% 0,007 0,348 33,443 206,96 2686,53 
150 4,85% 0,033 0,326 36,837 210,83 2856,07 
100 4,66% 0,186 0,310 37,665 213,93 3052,66 

RM (λ= 0,99)    500 6,55% 1,630 3,238* 24,787 137,97 1576,26 
400 5,99% 0,872 0,306 25,381 172,62 1948,88 
300 6,17% 1,484 0,796 28,060 230,84 2272,02 
250 6,16% 1,581 0,041 28,334 240,02 2397,50 
200 6,14% 1,678 0,104 35,573 255,05 2502,31 
150 6,56% 3,293 0,000 46,305 287,75 2571,93 
100 7,19% 6,720*** 0,253 55,251 346,13 2559,78 

AR(1)-GARCH(1,1)    500 8,26% 6,628** 5,247** 25,450 168,05 1497,28 
400 7,54% 5,331** 1,411 29,023 185,14 1884,31 
300 6,72% 3,095* 0,117 31,296 233,15 2267,60 
250 5,66% 0,525 0,596 27,342 219,27 2460,78 
200 5,38% 0,190 0,550 31,366 214,92 2629,52 
150 5,14% 0,027 0,514 36,663 215,14 2868,68 
100 4,26% 0,907 0,117 36,932 221,00 3138,36 

RCA     500 8,83% 8,924*** 1,693 26,278 187,36 1442,89 
400 7,76% 6,238** 1,630 28,136 204,92 1828,93 
300 6,53% 2,498 0,065 27,313 237,79 2223,04 
250 5,82% 0,817 0,723 26,199 230,09 2410,01 
200 5,07% 0,007 0,348 25,277 221,17 2594,08 
150 4,99% 0,000 0,415 37,995 227,54 2812,48 
100 4,39% 0,604 0,171 34,631 220,41 3025,38 

Sign RCA      500 8,83% 8,924*** 1,693 26,278 186,61 1438,09 
400 7,54% 5,331** 5,564** 27,559 204,61 1838,01 
300 6,53% 2,498 0,065 27,312 239,14 2219,78 
250 5,82% 0,817 0,723 26,199 230,74 2404,58 
200 5,07% 0,007 0,348 25,277 221,40 2586,80 
150 4,99% 0,000 0,038 39,013 227,35 2801,30 
100 4,79% 0,068 0,864 37,365 228,79 3003,25 

Note: *, **, *** indicate rejection of H0 at the 10% ,5% and 1% significant level, respectively, LRpof – the 
values of the proportion of failures test statistics, LRind – the values of the independence test statistics,  
LRtbf  – the values of the time between failures test statistics, RL – regulatory loss function, FL –  firm’s loss 
function. 



 

Table 6. The test for superiority of a model vis-á-vis another 
Sample:  300       

↓  better → Sign RCA RCA AR-GARCH RM(0.99) RM(0.95) EWMA HS 
Sign RCA x -7,455* -8,052* -10,863* -9,330* -12,397*  RCA 7,455 x -3,962* -10,182* -9,159* -3,451*  AR-GARCH 8,052 3,962 x -2,343* -9,245* 0,724  RM(0.99) 10,863 10,182 2,343 x -8,989* 8,904  RM(0.95) 9,330 9,159 9,245 8,989 x 8,563  EWMA 12,397 3,451 -0,724 -8,904* -8,563* x  HS       x 
Sample: 250       

↓  better → Sign RCA RCA AR-GARCH RM(0.99) RM(0.95) EWMA HS 
Sign RCA x -5,262* -4,691* 1,020 -8,199* -6,323  RCA 5,262 x -4,691* 1,999 -7,954* -5,099  AR-GARCH 4,691 4,691 x 4,854 -7,954* -0,612  RM(0.99) -1,020 -1,999 -4,854* x -9,994* -6,159  RM(0.95) 8,199 7,954 7,954 9,994 x 6,078  EWMA 6,323 5,099 0,612 6,159 -6,078* x  HS       x 
Sample:  200       

↓  better → Sign RCA RCA AR-GARCH RM(0.99) RM(0.95) EWMA HS 
Sign RCA x -5,369* -5,056* 11,640 -4,194* -8,662* 12,895 

RCA 5,369 x -3,253* 12,581 -3,880* -5,683* 13,992 
AR-GARCH 5,056 3,253 x 14,619 -1,842 -2,234* 13,522 

RM(0.99) -11,640* -12,581* -14,619* x -10,308* -16,971* 4,586 
RM(0.95) 4,194 3,880 1,842 10,308 x 1,999 11,092 
EWMA 8,662 5,683 2,234 16,971 -1,999 x 14,619 

HS -12,895* -13,992* -13,522* -4,586* -11,092* -14,619* x 
Sample:  150       

↓  better → Sign RCA RCA AR-GARCH RM(0.99) RM(0.95) EWMA HS 
Sign RCA x -0,567 -2,984* 19,905 -2,002* -8,120* 10,462 

RCA 0,567 x -3,059* 21,113 -1,775 -7,063* 12,502 
AR-GARCH 2,984 3,059 x 20,887 2,379 -2,757* 13,257 

RM(0.99) -19,905* -21,113* -20,887* x -14,315* -24,135* -6,761* 
RM(0.95) 2,002 1,775 -2,379* 14,315 x -0,944 10,613 
EWMA 8,120 7,063 2,757 24,135 0,944 x 15,448 

HS -10,462* -12,502* -13,257* 6,761 -10,613* -15,448* x 
Sample:  100       

↓  better → Sign RCA RCA AR-GARCH RM(0.99) RM(0.95) EWMA HS 
Sign RCA x -1,715 -5,218*  -2,153* -9,159* 4,634 

RCA 1,861 x -5,729*  -1,861 -6,386* 6,313 
AR-GARCH 5,218 5,729 x  4,415 2,007 9,086 

RM(0.99)    x    RM(0.95) 2,153 1,861 -4,415*  x 0,401 6,240 
EWMA 9,159 6,386 -2,007*  -0,401 x 9,597 

HS -4,634* -6,313* -9,086*  -6,240* -9,597* x 
Note: * indicate rejection of H0 at the 10% and 5% significant level. 



 

Table 7. Results of the VaR tests (97.5% VaR for PBG) and the loss functions 
Model α̂  LRpof LRind LRtbf RL FL 

SH 
500 5,98% 12,642*** 2,683 35,191** 123,33 1628,86 
400 5,10% 9,659*** 0,031 30,168 135,69 2074,56 
300 4,54% 7,587*** 0,019 27,865 171,95 2467,40 
250 3,66% 2,912* 0,047 17,641 155,85 2764,23 
200 3,69% 3,289* 0,015 24,976 154,39 2950,27 
150 3,14% 1,086 0,130 20,965 158,94 3208,25 
100 3,06% 0,911 0,117 23,765 157,49 3400,58 

EWMA 
500 4,84% 6,234** 1,736 28,034** 114,91 1661,45 
400 4,88% 8,226*** 0,006 29,047 130,49 2111,82 
300 3,81% 3,358* 0,049 25,024 151,85 2597,32 
250 3,33% 1,533 0,156 20,585 145,23 2833,12 
200 3,07% 0,816 0,217 21,926 144,02 3070,24 
150 2,85% 0,343 0,281 23,676 144,58 3332,11 
100 2,53% 0,003 0,455 25,225 138,45 3610,65 

RM (λ = 0,95) 
500 3,13% 0,536 0,714 12,178 60,26 1932,02 
400 3,10% 0,628 0,582 10,837 80,65 2404,45 
300 3,27% 1,214 0,256 13,397 129,14 2810,04 
250 3,00% 0,569 0,337 12,575 129,14 2990,57 
200 2,76% 0,181 0,419 13,916 129,15 3157,89 
150 2,85% 0,343 0,281 21,614 132,36 3352,30 
100 2,80% 0,261 0,255 21,282 134,38 3584,68 

RM (λ = 0,99) 
500 3,99% 2,710 1,167 23,185* 80,16 1836,63 
400 3,77% 2,586 0,186 21,844 104,95 2268,16 
300 3,63% 2,537 0,099 21,407 148,64 2644,88 
250 3,33% 1,533 0,156 20,585 153,45 2799,71 
200 3,23% 1,291 0,143 25,544 164,68 2925,99 
150 3,71% 3,668* 0,001 31,614 189,66 3000,01 
100 4,93% 14,208*** 0,018 54,397** 240,05 2947,31 

AR(1)-GARCH(1,1) 
500 5,41% 9,215*** 2,182 26,878 103,79 1732,20 
400 4,21% 4,517*** 1,676 22,917 109,42 2199,89 
300 3,63% 2,537 0,099 21,407 148,91 2650,81 
250 3,33% 1,533 0,156 20,585 139,01 2877,29 
200 3,07% 0,816 0,217 21,926 136,56 3081,41 
150 3,14% 1,086 0,130 27,158 134,82 3359,94 
100 3,06% 0,911 0,117 22,612 142,15 3662,46 

 
 
 
 



 

Table 7. Continued 
Model α̂  LRpof LRind LRtbf RL FL 

RCA 
500 4,84% 6,234** 1,736 24,499* 115,61 1674,19 
400 4,66% 6,888*** 2,057 29,051 127,91 2125,85 
300 3,99% 4,277** 0,017 25,205 154,07 2583,65 
250 3,33% 1,533 0,156 20,585 146,76 2814,50 
200 3,07% 0,816 0,217 21,926 142,87 3032,33 
150 2,85% 0,343 0,281 23,676 144,83 3292,28 
100 2,80% 0,261 0,255 24,688 141,09 3540,01 

Sign RCA 
500 5,13% 7,666*** 1,953 26,390 115,69 1665,44 
400 3,99% 3,494* 1,500 23,079 126,41 2142,75 
300 3,81% 3,358* 0,049 21,490 154,02 2582,48 
250 3,33% 1,533 0,156 20,585 147,33 2807,82 
200 3,23% 1,291 0,143 24,923 143,88 3020,79 
150 3,00% 0,665 1,299 27,904 145,29 3276,84 
100 2,93% 0,539 0,179 27,440 145,11 3517,04 

Note: *, **, *** indicate rejection of H0 at the 10% ,5% and 1% significant level, respectively, LRpof – the 
values of the proportion of failures test statistics, LRind – the values of the independence test statistics,  
LRtbf  – the values of the time between failures test statistics, RL – regulatory loss function, FL –  firm’s loss 
function. 

Table 8. Results of the VaR tests (99% VaR for PBG) and the loss functions 
Model α̂  LRpof LRind LRtbf RL FL 

SH 
500 2,56% 6,056** 0,475 17,577** 42,00 2163,96 
400 1,77% 2,218 0,290 13,426* 56,23 2775,29 
300 1,27% 0,375 0,180 4,422 77,63 3452,54 
250 2,16% 6,162** 0,576 17,518 91,70 3464,14 
200 1,08% 0,036 0,152 1,888 67,93 4456,03 
150 2,00% 5,459** 0,571 24,181** 105,19 4229,94 
100 0,93% 0,036 0,132 3,400 60,19 5611,84 

EWMA 
500 3,13% 10,313*** 0,714 23,320** 66,52 1933,39 
400 2,66% 8,633*** 0,658 25,503** 74,00 2476,85 
300 2,00% 4,285** 0,449 13,755 90,20 3046,29 
250 2,00% 4,676** 1,436 11,420 86,18 3320,07 
200 1,54% 1,624 0,313 7,374 86,15 3617,66 
150 1,14% 0,135 0,185 3,069 84,75 3936,79 
100 1,07% 0,032 0,173 3,535 81,86 4268,12 

 
 
 

 

 



 

Table 8. Continued 
Model α̂  LRpof LRind LRtbf RL FL 

RM (λ = 0,95) 
500 1,99% 2,719* 0,286 9,927 26,51 2271,53 
400 2,22% 5,014** 1,581 12,666 39,05 2820,60 
300 2,36% 7,441*** 1,053 14,764 77,73 3286,66 
250 2,16% 6,162** 1,181 12,415 77,73 3500,93 
200 2,00% 5,067** 1,303 12,230 77,73 3699,53 
150 2,00% 5,459** 1,184 14,802 78,92 3935,00 
100 1,86% 4,516** 1,288 12,587 79,58 4213,58 

RM (λ = 0,99) 
500 1,71% 1,472 0,209 5,627 38,83 2171,52 
400 1,55% 1,188 0,221 7,810 54,51 2679,38 
300 2,00% 4,285** 0,449 14,208 89,53 3103,32 
250 2,00% 4,676** 0,490 12,402 93,77 3280,42 
200 2,15% 6,547** 1,074 15,053 104,02 3421,39 
150 2,85% 16,200*** 0,281 39,564 124,04 3485,92 
100 3,06% 20,831*** 0,117 48,907 157,58 3431,58 

AR(1)-GARCH(1,1) 
500 2,28% 4,259** 0,374 11,647 55,22 2048,00 
400 2,22% 5,014** 0,455 15,475 58,34 2590,51 
300 1,81% 2,977* 0,370 11,299 89,91 3118,01 
250 1,83% 3,360* 0,411 9,867 81,74 3386,59 
200 1,69% 2,592 0,379 7,917 81,97 3635,11 
150 1,71% 2,958 0,419 14,533 79,46 3965,85 
100 1,33% 0,755 0,270 9,869 79,37 4329,54 

RCA 
500 3,13% 10,313*** 0,714 23,320** 67,06 1947,01 
400 2,88% 10,707*** 0,774 25,957** 73,93 2487,60 
300 2,18% 5,778** 0,535 18,055 92,98 3031,22 
250 2,00% 4,676** 0,490 13,800 88,61 3302,35 
200 1,69% 2,592 0,379 11,307 86,68 3571,25 
150 1,43% 1,138 0,290 9,825 86,71 3881,87 
100 1,20% 0,281 0,219 6,357 82,19 4187,14 

Sign RCA 
500 3,13% 10,313*** 0,714 23,320** 65,98 1940,98 
400 2,88% 10,707*** 0,774 25,957** 75,11 2494,86 
300 2,00% 4,285** 0,449 13,755 92,42 3029,72 
250 2,00% 4,676** 0,490 13,800 88,94 3293,79 
200 1,69% 2,592 0,379 11,307 86,81 3560,67 
150 1,43% 1,138 0,290 11,433 85,97 3867,81 
100 1,33% 0,755 0,270 11,177 84,15 4156,81 

Note: *, **, *** indicate rejection of H0 at the 10% ,5% and 1% significant level, respectively, LRpof – the 
values of the proportion of failures test statistics, LRind – the values of the independence test statistics,  
LRtbf  – the values of the time between failures test statistics, RL – regulatory loss function, FL –  firm’s loss 
function. 

  



 

Table 9. Results of the firm’s loss function 

Model T = 375 T = 250 T = 125 
FL rank FL rank FL rank 

 α = 5% 
Sym, Hist, 1075,956 1 1141,764 2 1263,7779 10 

EWMA 1109,013 5 1192,913 9 1257,8114 8 
RM (λ= 0,95) 1229,387 9 1251,9575 10 1263,0397 9 
RM (λ= 0,99) 1189,228 8 1191,2823 8 1101,7305 1 

AR(1)-GARCH(1,1) 1154,158 7 1169,4068 6 1207,2458 4 
RCA 1101,740 3 1153,1165 4 1212,1169 6 

Sign RCA 1103,836 4 1172,3837 7 1239,823 7 
RCAMA 1098,295 2 1149,6521 3 1204,0552 3 

Sign RCAMA  -  - 1102,4462 1 1180,7179 2 
RCA GARCH 1132,959 6 1158,4801 5 1209,4749 5 

Sign RCA GARCH 1296,929 10 1339,8815 11 1404,0839 11 
α = 2,5% 

Sym, Hist, 1284,699 5 1364,1283 5 1521,6798 10 
EWMA 1282,948 4 1377,3036 7 1454,5161 8 

RM (λ= 0,95) 1447,782 9 1473,5854 9 1481,442 9 
RM (λ= 0,99) 1379,883 8 1378,9041 8 1283,0513 1 

AR(1)-GARCH(1,1) 1334,391 7 1366,5462 6 1426,3841 6 
RCA 1272,254 2 1341,8047 2 1406,9397 5 

Sign RCA 1274,395 3 1358,9357 4 1433,1749 7 
RCAMA 1270,903 1 1337,7536 1 1397,165 3 

Sign RCAMA - -   - - 1361,558 2 
RCA GARCH 1305,188 6 1349,0442 3 1405,9143 4 

Sign RCA GARCH 1549,821 10 1577,6011 10 1630,4145 11 
α = 1% 

Sym, Hist, 1754,0863 9 1754,7449 9 2234,0469 9 
EWMA 1506,0564 4 1623,2342 7 1723,7757 7 

RM (λ= 0,95) 1697,2481 8 1719,5178 8 1727,7362 8 
RM (λ= 0,99) 1625,3806 7 1619,3369 6 - - 

AR(1)-GARCH(1,1) 1569,7932 6 1611,8139 5 1688,9632 5 
RCA 1494,4265 2 1573,8864 2 1663,8987 3 

Sign RCA 1496,858 3 1593,0414 4 1698,2509 6 
RCAMA 1492,2845 1 1568,9729 1 1650,9211 2 

Sign RCAMA - - - - 1600,3206 1 
RCA GARCH 1535,3552 5 1589,0235 3 1664,72 4 

Sign RCA GARCH - - 1849,3472 10 - - 
Note: T denotes the rolling window size, FL– the firm’s loss function. 

  



 

Table 10. Results of the the predictive quantile loss function 

Model 
T = 375  T = 250  T = 125  

Qα  rank Qα  rank Qα  rank 

 α = 5% 
Sym. Hist 0,3183 7 0,3159 7 0,3202 7 

EWMA 0,3169 5 0,3141 5 0,3181 5 
RM (λ= 0,95) 0,3191 8 0,3182 9 0,3189 6 
RM (λ= 0,99) 0,3163 4 0,3153 6 0,3213 9 

AR(1)-GARCH(1,1) 0,3215 9 0,3167 8 0,3122 1 
RCA 0,3156 1 0,3131 3 0,3173 3 

Sign RCA 0,3161 3 0,3122 1 0,3169 2 
RCAMA 0,3174 6 0,3130 2 0,3206 8 

Sign RCAMA - - 0,3363 11 0,3253 10 
RCA GARCH 0,3158 2 0,3137 4 0,3179 4 

Sign RCA GARCH 0,3421 10 0,3283 10 0,3657 11 
α = 2,5% 

Sym. Hist 0,1945 5 0,1908 2 0,1938 6 
EWMA 0,1943 4 0,1910 3 0,1914 2 

RM (λ= 0,95) 0,1916 1 0,1914 5 0,1915 3 
RM (λ= 0,99) 0,1923 2 0,1931 9 0,1995 9 

AR(1)-GARCH(1,1) 0,1985 9 0,1913 4 0,1845 1 
RCA 0,1947 6 0,1925 6 0,1956 7 

Sign RCA 0,1943 3 0,1896 1 0,1927 4 
RCAMA 0,1957 8 0,1931 8 0,1982 8 

Sign RCAMA - - - - 0,2016 10 
RCA GARCH 0,1953 7 0,1926 7 0,1928 5 

Sign RCA GARCH 0,2108 10 0,2048 10 0,2358 11 
α = 1% 

Sym. Hist 0,1015 9 0,0992 9 0,0957 6 
EWMA 0,0978 3 0,0947 1 0,0941 4 

RM (λ= 0,95) 0,0973 2 0,0962 5 0,0958 7 
RM (λ= 0,99) 0,0957 1 0,0960 3 - - 

AR(1)-GARCH(1,1) 0,1000 7 0,0961 4 0,0909 1 
RCA 0,0997 6 0,0963 6 0,0950 5 

Sign RCA 0,0988 5 0,0950 2 0,0934 2 
RCAMA 0,1004 8 0,0967 7 0,0969 8 

Sign RCAMA - - - - 0,1022 9 
RCA GARCH 0,0980 4 0,0967 8 0,0938 3 

Sign RCA GARCH - - 0,1143 10 - - 
Note: T denotes the rolling window size, Qα – the predictive quantile loss function. 

  



 

 




