






































Microsoft Word - 00_tresc.docx


DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Joanna Górka 
Nicolaus Copernicus University in Toruń 

Application of the Family of Sign RCA Models  
for Obtaining the Selected Risk Measures† 

A b s t r a c t. Accurate modelling of risk is very important in finance. There are many alternative 
risk measures, however none of them is dominating. This paper proposes to use the family of Sign 
RCA models to obtain the Value-at-Risk (VaR) and Expected Shortfall (ES) measures. For mod-
els from the family of Sign RCA models and AR-GARCH model the one-step forecasts of VaR 
were calculated based on rolling estimates from the given model using different window sizes. To 
obtain the VaR and ES measures the filtered historical simulation was used in new version pro-
posed by Christoffersen. The results were verified using backtesting and the loss function.  

K e y w o r d s: Family of Sign RCA Models, risk measures, Value at Risk, Expected Shortfall.  

1. Introduction 
 Random coefficient autoregressive models (RCA) are the straightforward 
generalization of the constant coefficient autoregressive models. A full descrip-
tion of this class of models including their properties, estimation methods and 
some application was originally presented by Nicholls and Quinn (1982). In 
later years, these models have been not so popular like GARCH models (Boller-
slev, 1986; Engle, 1982) in general. GARCH models are easy to understand and 
estimate and they could describe a non-linear dynamics of financial time series. 
However, in the last decade one can see that RCA models gained more interest 
again. As a result some of RCA model were produced.   
 The aim of this paper is to apply the family of Sign RCA models to obtain 
the selected risk measures for daily and weekly data. Risk measures through 
different calculation method are obtained.   

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

the research project N N111 434034. 



Joanna Górka 40

2. The Family of Sign RCA Models 
 In the Table 1 equations of individual models from the family of Sign RCA 
models and their names were presented. 

Table 1. The family of Sign RCA models (without conditions) 
Model Model equations Equation 
RCA(1) ( ) tttt yy εδφ ++= −1 I 
Sign RCA(1) ( ) ttttt ysy εδφ +Φ++= −− 11 II 
RCA(1)-MA(1) ( ) 11 −− +++= ttttt yy θεεδφ III 
Sign RCA(1)-MA(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.  

 To ensure the existence of the I-VI models the following assumption 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 following formula 

⎪
⎩

⎪
⎨

⎧

<−
=
>

=
0dla1
0dla0
0dla1

t

t

t

t

y
y
y

s , (3) 

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 re-
turns the lower increase in stock returns occurs than expected. 
 Condition (2) is necessary and sufficient for the second-order stationarity of 
process described by equation I, however conditions (1)-(2) ensure the strict 



Application of the Family of Sign RCA Models … 41

stationarity of this process. If conditions (1)-(2) are satisfied, then processes 
described by equations II-IV are stationary in mean. Theoretical properties of 
processes described by equations I-VI, satisfying conditions (1)-(2) can be 
found in several articles (Appadoo, Thavaneswaran, Singh, 2006; Aue, 2004; 
Górka, 2008; Thavaneswaran, Appadoo, Bector, 2006; Thavaneswaran, Appa-
doo, 2006). 
 Residuals from the RCA model can be described by the GARCH model 
(Thavaneswaran, Peiris, Appadoo, 2008; Thavaneswaran, Appadoo, Ghahrama-
ni, 2009). 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. 

 When the sign function is added to the RCA-GARCH model, then the 
process described by equation VI is obtained (Thavaneswaran, Appadoo, Gha-
hramani, 2009). The conditions ensuring the positive value of conditional vari-
ance of this process are following: ( )2,0~ zt Nz σ , 00 >α , 0≥iα , 0≥jβ , 

0α≤Φ . 

3. The Selected Risk Measures 
 In this paper two tools for measuring market risk were used, i. e. Value-at-
Risk (VaR) and Expected Shortfall (ES). Value-at-Risk is the maximum loss 
over a target horizon such that there is a low, prespecified probability that the 
actual loss will be larger. Expected Shortfall is a coherent alternative to Value-
at-Risk (Acerbi, Tasche, 2002). It is the expected loss conditional on exceeding 
VaR. 
 One-step-ahead conditional forecasts of Value-at-Risk are calculated in two 
ways. Firstly, Value-at-Risk is calculated by formula:  

( ) ,111 ασμα zVaR tttt
l
t +++ +=  

where tt |1+μ , tt |1+σ  are one-step-ahead conditional forecasts of mean and volatil-
ity respectively. 
Secondly, the formula proposed by Christoffersen (2009) is used:  

( ) ,111
tu

tttt
l
t qVaR ασμα +++ +=  

where tuqα  denotes the percentile of the set of standardization historical shocks 

tu . 
Expected Shortfall for the one-step forecast can be calculated as: 
− average of values exceeding VaR (YY) (Yamai, Yoshiba, 2002), 
− weighted average of values exceeding VaR (Dowd) (Dowd, 2002), 



Joanna Górka 42

− filtered historical simulation (FHS), in the version proposed by Christof-
fersen (2009), i. e. 

( ) ( ) ,11

1

1

1
11 ⎟

⎟

⎠

⎞

⎜
⎜

⎝

⎛
<⋅=

+

+

=
++ ∑

tt

l
t

t

T

t
tttt

VaRuu
T

ES
σ

α
α

σα  

where ( )∗1  denotes the indicator function returning a unit if the argument is 
true, and zero otherwise; α is tolerance level. 

 To check the accuracy of risk measures the backtesting of VaR and ES us-
ing the traditional VaR tests and the loss function was conducted. 
 The traditional VaR tests used to compare results are following: 
− proportion of failures test1 – LRPOF, 
− independence test – LRIND,  
− time between failures test – LRTBF, 

 The loss functions used to compare results are following:  
− regulatory loss function – RL 
− firm’s loss function – FL 
− loss function with the expected loss proposed by Angelidis, Degiannakis 

(2006). It can be described as MSEMAELFES +=  where
 

∑
=

=
N

t
tf

N
MAE

1
|1

1 , 

∑
=

=
N

t
tf

N
MSE

1
|2

1

 
while: 

⎪⎩

⎪
⎨
⎧

−≤+

−>
=

++

+

+ ,

,0

,1,1

,1
1|1

trttrt

trt
t VaRrESr

VaRr
f  

( )⎪⎩

⎪
⎨
⎧

−≤+

−>
=

++

+
+ .

,0

,1
2

,1

,1
1|2

trttrt

trt
t VaRrESr

VaRr
f  

4. Empirical Results 
 The data used in the empirical application are eight stock exchange indexes 
and thirty three share prices of the Polish firms’ from the Warsaw Stock Ex-
change. It gives forty one time series. The data were obtained from bossa.pl for 
the period from November 30, 1998 to November 4, 2008, what gives 2490 
daily percentage log returns and 493 weekly percentage log returns.  
 The calculations were carried out in the Gauss and Microsoft Excel. 

                                                 
1 Other name of this test is the Kupiec test. 



Application of the Family of Sign RCA Models … 43

 Firstly, for each returns series the descriptive statistics and some tests were 
calculated. All series have positive kurtosis (leptokurtic). Some of returns series 
are autocorrelated. 
 Secondly, parameters of six models from the Sign RCA family were esti-
mated using maximum likelihood (MLE). The number of models from the fami-
ly Sign RCA models with statistically significant parameters for the total sam-
ple is presented in the Table 2. 

Table 2. The number of models with significant parameters from the family of Sign 
RCA models for percentage log returns (2490 daily data and 493 weekly data)  

Model α = 5% α = 10% 
daily data weekly data daily data weekly data 

AR(1) 26 9 27 10 
RCA(1) 24 5 24 10 

Sign RCA(1) 1 2 3 2 
RCA(1)-MA(1) 15 21 19 22 

Sign RCA(1)-MA(1) 5 3 7 4 
RCA(1)-GARCH(1,1) 25 2 25 2 

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

It is seen that the models like AR(1), RCA(1), RCA(1)-MA(1) and RCA(1)-
GARCH(1,1) were found in about 50 percentage of cases. For smaller samples, 
similar results are obtained (see the Table 3). Models with sign function occur 
very seldom in empirical time series (see also the Table 3) both at the 5% and 
10% significance level, and also for different size of sample and different level 
of data aggregation (both daily and weekly data). 

Table 3. The number of models with significant parameters from the family of Sign 
RCA models for percentage log returns (1500 daily data and 300 weekly data)  

Model α = 5% α = 10% 
  daily data weekly data daily data weekly data 

AR(1) 23 12 25 13 
RCA(1) 22 10 24 11 

Sign RCA(1) 2 1 3 3 
RCA(1)-MA(1) 20 24 22 25 

Sign RCA(1)-MA(1) 2 7 3 10 
RCA(1)-GARCH(1,1) 18 - 20 - 

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

 For example, the RCA models for the selected indexes are presented in the 
Table 4. 



Joanna Górka 44

 On the basis of models from the family of Sign RCA models and AR-
GARCH model fitted to the different window size, i.e, N =250, 500 and 1500 
the one-step ahead forecasts of VaR and ES were made forecasting one-step-
ahead from the end of window till the next 500 observations which were hold 
out. It should be pointed out that the last observation of each sample (N =250, 
500 and 1500) is placed at the same point at time, hence these samples can be 
treated as overlapping. Each one-step ahead forecast was generated from esti-
mates of the given model2 using a sequence of rolling windows (with window 
size of 250, 500 and 1000 observations) which were moved 500 times by one 
observation on time axis.  

Table 4. The RCA model for the selected indexes  

 

WIG-BUDOW WIG-SPOZYW 

daily data weekly data daily data weekly data 
φ  0.131 0.133 0.119 0.186 
( )φS  0.024 0.053 0.026 0.059 

2
εσ  1.799 12.441 1.300 6.835 
2
δσ  0.199 0.135 0.291 0.357 

ln L -4480.33 -1348.43 -4146.82 -1244.88 
Q(3) 4.810 5.784 19.371* 2.728 

ARCH(3) 73.649* 11.911* 90.960* 24.364* 
AIC 8966.66 2702.86 8299.64 2495.76 
BIC 8984.12 2715.46 8317.10 2508.36 

Note: Q(3) – the value of the Ljung-Box Q-statistic up to 3 lags, ARCH(3) – the value of the Engle ARCH 
test statistics up to 3 lags, AIC – Akaike information criterion, BIC – Bayesian information criterion.  

 The backtesting of forecasts of VaR and ES measures were carried out (the 
example results were presented in Table 5–7).  
 The empirically determined probability for the proportion of failures test for 
different window size is presented in Table 5. Almost all forecasts of VaR are 
underestimated. Only for RCA-MA with FHS method the VaR forecasts are 
overestimated. It is seen that as the window size decreases the scale of underes-
timation decreases, too.   
  The results of traditional VaR tests and loss function for the VaR forecasts 
were carried out from rolling estimation of models using window size of 250 
observations for one of selected index (see Table 6). 

 

                                                 
2 Forecasts were carried out form models with statistically significant parameters obtained in 

the basic sample.  



Application of the Family of Sign RCA Models … 45

Table 5. The empirically determined probability for the proportion of failures test for 
different window size (WIG-spozyw) 

Model 
Empirically determined probability 

N = 1500 N = 500 N = 250 
AR-GARCH 6.8% 6.8% 6.6% 

AR-GARCH (FHS) 10.8% 10.8% 8.8% 
RCA 11,0% 9,0% 6.6% 

RCA-GARCH 6.8% 6.6% 6.6% 
RCA-GARCH (FHS) 10.8% 10,0% 8.8% 

RCA (FHS) 13.4% 10.6% 7.4% 
RCA-MA 11.8% 8.2% 6.4% 

RCA-MA (FHS) 3.8% 3.6% 2.2% 
Sign RCA 11.4% 9.2% 6.8% 

Sign RCA-GARCH 6.8% 6.8% 7.2% 
Sign RCA-GARCH (FHS) 10.8% 10,0% 8.4% 

Sign RCA (FHS) 13.8% 10.6% 7.4% 
Sign RCA-MA 12,0% 8.6% 6,0% 

Sign RCA-MA (FHS) 9.8% 6.8% 4.8% 
Note: N denotes the window size.  

Table 6. Results of traditional VaR tests for forecasts carried out from rolling estimation 
of models using window size of 250 observations for the WIG-spozyw index  

Model LRPOF LRIND  LRTBF RL FL 
AR-GARCH 2.459 0.897 36.501 171.509 1386.669 

AR-GARCH (FHS) 12.518*** 1.291 64.588** 205.050 1259.617 
RCA 2.459 0.897 49.856** 205.547 1281.037 

RCA-GARCH 2.459 0.897 31.090 190.525 1411.081 
RCA-GARCH (FHS) 12.518*** 1.291 57.746* 219.107 1289.683 

RCA (FHS) 5.317** 0.256 58.441** 215.699 1267.312 
RCA-MA 1.903 4.389** 43.849* 191.254 1287.360 

RCA-MA (FHS) 10.347*** 0.496 25.287*** 102.243 1927.830 
Sign RCA 3.081 1.065 51.368*** 207.307 1279.988 

Sign RCA-GARCH 4.511** 0.171 39.764 200.195 1389.469 
Sign RCA-GARCH (FHS) 10.194*** 0.101 53.739 224.806 1289.986 

Sign RCA (FHS) 5.317** 0.256 58.441 217.562 1272.351 
Sign RCA-MA 0.992 3.840* 37.125 193.384 1289.115 

Sign RCA-MA (FHS) 0.043 2.426 27.264 157.853 1412.718 
Sym. Hist 9.1102*** 0.134 56.766 235.584 1232.198 

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 func-
tion.  



Joanna Górka 46

 For AR-GARCH, RCA-GARCH and Sign RCA-MA (with FHS method) 
models the null hypothesis for the traditional VaR tests is not rejected, what 
means that the proportion of failures is equal to the given tolerance level 5%, 
failures are serially independent and the time between failures is independent   
(see the Table 6). Slightly worse results are obtained for models: RCA (with 
and without FHS method), RCA-MA, Sign RCA and Sign RCA-MA.  
 In that case the regulatory loss function is the smallest for the Sign RCA-
MA (with FHS method) model. Hence this model is preferred by the regulatory 
institution. However, the difference between this value of the regulatory loss 
function and their values calculated from other models are small, so these mod-
els are comparable. The firm’s loss function takes the smallest value for AR-
GARCH (with FHS method), so this model is preferred from the firm point of 
view, but  other values of the firm’s loss function are not much higher than for 
the best model (AR-GARCH, with FHS method).  

Table 7. Values of loss function with the expected loss (LFES) for different window sizes 
for the WIG-spozyw index 

Model LFES 
N = 1500 N = 500 N = 250 

AR-GARCH (Dowd) 0,238 0,238 0,252 
AR-GARCH (FHS) 0,451 0,451 0,385 
AR-GARCH (YY) 0,235 0,235 0,266 

RCA GARCH (Dowd) 0,243 0,233 0,296 
RCA GARCH (FHS) 0,439 0,403 0,391 
RCA GARCH (YY) 0,241 0,248 0,317 

RCA (Dowd) 0,415 0,363 0,313 
RCA (FHS) 0,567 0,440 0,389 
RCA (YY) 0,408 0,381 0,332 

RCA-MA (Dowd) 0,401 0,353 0,301 
RCA-MA (FHS) 0,283 0,232 0,231 
RCA-MA (YY) 0,399 0,371 0,319 

Sign RCA GARCH (Dowd) 0,242 0,263 0,295 
Sign RCA GARCH (FHS) 0,451 0,421 0,428 
Sign RCA GARCH (YY) 0,240 0,277 0,316 

Sign RCA (Dowd) 0,422 0,368 0,311 
Sign RCA (FHS) 0,580 0,445 0,385 
Sign RCA (YY) 0,418 0,387 0,331 

Sign RCA-MA (Dowd) 0,411 0,370 0,306 
Sign RCA-MA (FHS) 0,661 0,486 0,383 
Sign RCA-MA (YY) 0,411 0,393 0,322 

Note: N denotes the window size. 



Application of the Family of Sign RCA Models … 47

 Next, the values of loss function with the expected loss (LFES) for different 
window size of rolling estimation for example for one of indexes presented in 
Table 7.  
For the window size of 500 and 250 observations the smallest value of the loss 
function with the expected loss is obtained for the RCA-MA model (with FHS 
method), but for the window size of 1500 observations the best result is ob-
tained from AR-GARCH model (with FHS method). It is seen that for the mod-
els without GARCH residuals the value of the LFES decreases as the window 
size decreases. Models with GARCH errors prefer bigger windows. The com-
parison of results of the LFES for models with and without the sign function 
shows that introducing the sign function into the model causes the increase of 
the loss function with the expected loss. 

5. Summary 
 In this paper, the family of Sign RCA models to obtain the selected risk 
measures was presented. Empirical results showed that: 
− Models with sign function occur seldom. It means that percentage log re-

turns do not represent the asymmetric reaction to good or bad news coming 
from the market. 

− Accuracy of the VaR measures for models with sign function and without 
GARCH errors depends on the size of window and almost all of them are 
underestimated (except the RCA-MA model with FHS method).  

− Accuracy of the VaR forecasts for models with GARCH errors (without us-
ing FHS method) does not depend on the size of window and all of them are 
underestimated.  

− Filtered historical simulation (FHS) (Christoffersen version) is sensitive to 
the size of window, i.e. for smaller window the empirically determined 
probability is closer to the nominal significance level for all models from 
the family of Sign RCA models. 

− Using the Sign RCA-MA model with FHS method the empirical and no-
minal significance level are almost the same.  

− For RCA-MA models with FHS method the forecasts of VaR are overesti-
mated at the 5% significance level.  

− The smallest values of the regulatory loss function were obtained for the 
VaR forecasts from RCA-MA model (FHS method). 

− The smallest values of the firm’s loss function were obtained for the VaR 
forecasts from the Sign RCA, RCA, AR-GARCH models (all models with 
FHS method). 

− Loss function with expected loss (LFES) takes the smallest values for  
AR-GARCH models (Dowd and YY method) for each window size and 



Joanna Górka 48

from the RCA-MA models (FHS method) for windows of 500 and 250 ob-
servations. 

− Filtered historical simulation (Christoffersen version) generates bigger value 
of LFES than other analyzed method (except the RCA-MA model with FHS 
method). 

To sum up, some models from the family of Sign RCA models can generate 
useful results of the VaR and ES measures only in some cases.  

References  
Acerbi, C., Tasche, D. (2002), Expected Shortfall: A Natural Coherent Alternative to Value at 

Risk, Economic Notes, 31, 379–388. 
Angelidis, T., Degiannakis, S. (2006), Backtesting VaR Models: An Expected Shortfall Ap-

proach, Working Paper Series, http://econpapers.repec.org/paper/crtwpaper/0701.htm 
(2.09.2009). 

Appadoo, S., Thavaneswaran, A., Singh J. (2006), RCA Models with Correlated Errors Applied 
Mathematics Letters, 19, 824–829. 

Aue, A. (2004), Strong Approximation for RCA(1) Time Series with Applications, Statistics & 
Probability Letters, 68, 369–382. 

Bollerslev, T. (1986), Generalized Autoregressive Conditional Heteroscedasticity, Journal of 
Econometrics, 31, 307–327. 

Christoffersen, P.  F. (2009), Value-at-Risk Models, in Andersen, T. G., Davis, R. A., Kreiss,  
J.-P., Mikosch, T.  (ed.), Handbook of Financial Time Series, Springer Verlag. 

Dowd, K. (2002), Measuring Market Risk, John Wiley & Sons, ltd. 
Engle, R. F. (1982), Autoregressive Conditional Heteroscedasticity with Estimates of the Vari-

ance of United Kingdom Inflation, Econometrica, 50, 987–1006. 
Górka, J. (2008), Description the kurtosis of distributions by selected models with sing function, 

Dynamic Econometric Models, vol.8, Toruń  
Nicholls, D., Quinn, B. (1982), Random Coefficient Autoregressive Models: An Introduction, 

Springer, New York. 
Thavaneswaran, A., Appadoo, S. (2006), Properties of a New Family of Volatility Sing Models, 

Computers and Mathematics with Applications, 52, 809–818. 
Thavaneswaran, A., Appadoo, S., Bector, C. (2006), Recent Developments in Volatility Modeling 

and Application, Journal of Applied Mathematics and Decision Sciences, 1–23. 
Thavaneswaran, A., Peiris, S., Appadoo, S. (2008), Random Coefficient Volatility Models, Statis-

tics & Probability Letters, 78, 582–593. 
Thavaneswaran, S., Appadoo, S., i Ghahramani, M. (2009), RCA models with GARCH innova-

tions, Applied Mathematics Letters, 22, 110–114. 
Yamai, Y., Yoshiba, T. (2002), Comparative Analyses of Expected Shortfall and Value-at-Risk: 

Their Estimation Error, Decomposition and Optimization, Monetary and Economic Stud-
ies, 20(1), 87–121. 

Zastosowanie modeli klasy Sign RCA  
do wyznaczenia wybranych miar zagrożenia 

Z a r y s  t r e ś c i. W finansach bardzo ważne jest aby dokładnie ocenić ryzyko. Istnieje wiele 
metod szacowania ryzyka jednak żadna z istniejących już metod nie jest najlepsza. W niniejszym 
artykule, do wyznaczenie takich miar ryzyka jak Value at Risk (VaR) i Expected Shortfall (ES) 
zastosowano modele klasy Sign RCA Obliczone zostały jednookresowe prognozy VaR oraz ES 



Application of the Family of Sign RCA Models … 49

dla ostatnich 500 obserwacji z wykorzystaniem modeli oszacowanych w oknach na podstawie 
prób wielkości 250, 500 i 1500 obserwacji. Otrzymane wyniki zweryfikowano wykorzystując 
testowanie wsteczne oraz funkcje strat.  

S ł o w a  k l u c z o w e: Modele klasy Sign RCA Models, miary ryzyka, Value at Risk, Expected 
Shortfall. 



 


