Estimating VAR!MGARCH Models in Multiple Steps - WebMeets.com

Estimating VAR-MGARCH Models in 

Multiple Steps 

M. Angeles Carnero and M. Hakan Eratalay y 

Dpt. Fundamentos del Análisis Económico 

Universidad de Alicante 

PRELIMINARY VERSION 

June 12, 2009 

Abstract 

This paper analyzes the performance of multiple steps estimators of Vector Autoregressive 

Multivariate Conditional Correlation GARCH models by means of Monte Carlo 

experiments. High dimensionality is one of the main problems in the estimation of the 

parameters of these models and therefore maximizing the likelihood function of the 

full model leads commonly to numerical problems. We show that if innovations are 

gaussian, estimating the parameters in multiple steps is a reasonable alternative to the 

full maximization of the likelihood. Our results also suggest that for the sample sizes 

usually encountered in …nancial econometrics, the di¤erences between the volatility 

estimates obtained with the more e¢ cient estimator and the multiple steps estimators 

are negligible. However, this does not seem to be the case if the distribution is 

a Student-t for some Conditional Correlation GARCH models. Then, multiple steps 

estimators lead to volatility estimates which could be far from the ones implied by the 

true parameters. The results obtained in this paper are used for estimating volatility 

interactions in 10 European stock markets. 

JEL Classi…cation: C32 

Keywords: Volatility Spillovers, Financial Markets. 

E-mail: acarnero@merlin.fae.ua.es 

y Corresponding Author. E-mail: eratalay@merlin.fae.ua.es 

1

1 Introduction 

Understanding how stock market returns and volatilities move over time has been of 

great concern to researchers into the time series literature. In addition, as the recent 

…nancial crisis has shown, it is also very important to realize that stock markets move 

together. There are many examples con…rming this incidence: a very big investment 

bank in US, Lehman Brothers, declares bankruptcy and this increases the losses in the 

stock market of some other country. Barack Obama’s $787 billion economic stimulus 

package has been passed by the US Congress and there were positive reactions to 

this in local and global stock markets. Therefore, trying to model stock markets in a 

univariate way ignoring this kind of interaction would be insu¢ cient. In this sense, Multivariate 

generalized autoregressive conditional heteroskedasticity (MGARCH) models 

have been popular tools in the literature to model volatility-covolatility of assets and 

markets; see, for example, Bauwens et al. (2006) and Silvennoinen et al. (2008) for a 

survey. 

The problem with many MGARCH models is that it is often di¢ cult to verify if the 

conditional variance-covariance matrix is positive de…nite. Engle et al. (1984) provide 

necessary conditions for the positive de…niteness of the variance-covariance matrix in 

a bivariate ARCH setting, but extensions of these results to more general models are 

very complicated. Moreover, imposing restrictions on the log-likelihood function during 

optimization to obtain such conditions is often di¢ cult. 

A model that could avoid these problems is the Constant Conditional Correlation 

GARCH (CCC-GARCH) model proposed by Bollerslev (1990). He shows that the 

gaussian maximum likelihood (ML) estimator of the correlation matrix is the sample 

correlation matrix which is always positive de…nite, leading to the result that the optimization 

will not fail as long as the conditional variances are positive. On the other 

hand, he notes that the correlation matrix can be concentrated out of the log-likelihood 

function, what simpli…es more the optimization problem. Due to this computational 

simplicity, the CCC-GARCH model has become very popular in the literature. However, 

the CCC-GARCH model has several limitations. One of them is that it fails to explain 

possible volatility interactions as pointed out by Ling and McAleer (2003). They 

favor the extension proposed by Jeantheau (1998), ECCC-GARCH, where volatility 

spillovers can be considered. Another limitation of the CCC-GARCH model is the 

constant correlation assumption; see Christodoulakis and Satchell (2002), Engle (2002) 

and Tse and Tsui (2002) who propose the Dynamic Conditional Correlation GARCH 

(DCC-GARCH) model where the correlation changes over time. 

The DCC-GARCH model is computationally more complicated than the CCC- 

GARCH since the correlation dynamics require the estimation of more parameters. 

Fortunately, the variance targeting approach, in which the intercept in the correlation 

equation is replaced by its sample counterpart, makes the estimation managable. 

2

Aielli (2006) points our severe problems with the variance targeting approach used 

when estimating the DCC-GARCH model in multiple steps. He suggests a correction 

to the DCC-GARCH model, which is referred to as Corrected DCC-GARCH (cDCC- 

GARCH) model. However, as also pointed out in Caporin and McAleer (2009), the 

cDCC-GARCH model does not allow for variance targeting. Therefore the model of 

Aielli (2006) su¤ers from the curse of dimensionality since as the number of time series 

considered increases, the number of correlation parameters to be estimated increases 

at a geometric rate. 

Pelletier (2006) introduces the Regime Switching Dynamic Correlation GARCH 

(RSDC-GARCH) model in which correlations between series are allowed to be constant 

over time but changing between di¤erent regimes and driven by an unobserved markov 

switching chain. This model is, somehow, in between the CCC-GARCH model and 

DCC-GARCH model, with the problem that the number of correlation parameters to 

be estimated increases rapidly with the number of series considered. 

When dealing with stock market returns, it is not unusual to …nd some dynamics in 

the conditional mean, that could be approximated by a VARMA model. One way to 

estimate the parameters of the full VARMA-MGARCH conditional correlation model 

would be solving the optimization problem of a full gaussian log-likelihood function; 

therefore obtaining the estimates for all the parameters in one step. If the true data 

generating process (DGP) is also gaussian, then it is known that ML estimators are 

consistent and asymptotically normal. In the case that the true DGP is not gaussian, 

then we would be using quasi-maximum likelihood (QML) estimators. Bollerslev and 

Wooldridge (1992) show that under quite general conditions, these QML estimators are 

consistent and asymptotically normal. Bollerslev (1990), Longin and Solnik (1995) and 

Nakatani and Teräsvirta (2008) are few of the papers using one-step estimation. Estimating 

all parameters in one step would be the best one could achieve, however when 

there are many equations and parameters involved, it is very heavy computationally, 

when feasible. As Longin and Solnik (1995) state, sometimes even the convergence in 

the optimization process is only possible for very speci…c starting values for the parameters. 

In addition, if some of the coe¢ cients are not strongly signi…cant, it is hard 

to achieve iterative convergence with a small sample. 

Under the normality assumption for the DGP, the parameters could also be estimated 

in two steps by estimating …rst the mean and variance parameters assuming no 

correlation and then, in a second step, estimating the correlation parameters given the 

estimates from the …rst step; see, for example, Engle (2002). However, as Engle and 

Sheppard (2001) suggest for the DCC-GARCH model, these two-step estimators will 

be consistent and asymptotically normal but not fully e¢ cient since they are limited 

information estimators. 

The three-steps estimation method is mentioned in Bauwens (2006) and consists 

of estimating the mean equation parameters in a …rst step, obtaining the variance 

3

equation parameters in a second step and …nally in the third and last step, given all 

other parameter estimates, estimating the correlation parameters. If the parameters of 

the …rst step are consistently estimated, then the estimators from the second step will 

be consistent and therefore the third step estimators will also be consistent. The second 

and third steps of the three-steps estimation procedure will be equivalent to the twosteps 

estimation method for a zero-mean series. Therefore the three-steps estimators 

are also asymptotically normal. Engle and Sheppard (2001) use such a three-steps 

procedure in the empirical part of their paper. 

We also consider estimating the models using the log-likelihood function based on 

Student-t distribution. If the true distribution of the errors is also a t-distribution, 

then the estimators are ML estimators, if not then they will be QML estimators. In 

one-step estimation, all the parameters of the model are estimated maximizing the loglikelihood 

function based on the multivariate t-distribution; see for example Harvey 

et al:(1992) and Fiorentini et al: (2003). Although it would not have any theoretical 

support, we also tried two-step and three-step estimation methods for this case. We 

…nd that some, although not all, of the parameter estimates obtained in multiple steps 

can be very far from their true values. 

For the MC studies where we obtained unbiased estimators, we checked the robustness 

of the results to the distribution assumption. For example, since the two-step and 

three-step estimators for CCC-GARCH model simulated and estimated with gaussian 

errors were unbiased, we checked the robustness of this result by simulating the model 

with t-distribution and estimating with gaussian distribution. Our results show that, 

the estimators that are unbiased under the correctly speci…ed error distribution are 

also unbiased when the true distribution was changed. 

Although the two-step estimation takes less computational time compared to onestep 

estimation, it is still costly compared to three step estimation method, speci…cally 

when dealing with big samples, many equations and/or parameters. These three estimation 

methods will be explained later in detail in section 2 of the paper. 

In this paper, we consider the e¢ ciency loss that Engle (2002) and Engle and 

Sheppard (2001) talk about when estimating the correlation parameters separately 

from the mean and variance parameters. This type of e¢ ciency loss will be analyzed 

by comparing the one-step and two-step estimation methods. As we shall see, when 

the errors are assumed to be gaussian, the small sample behavior of one-step and twosteps 

estimators are very similar. On the other hand, when the estimation is based 

on t-distribution, there are cases where two-steps estimators deviate from one-step 

estimators. 

Additionally, we analyze the e¤ects of separating the estimation of mean and variance 

parameters in small samples. Bauwens (2006) points out that the conditional 

mean parameters may be estimated consistently in the …rst stage for example for a 

Vector Autoregressive Moving Average (VARMA) model, prior to estimating the con- 

4

ditional variance parameters. He notes that estimating all the parameters in one stage 

wouldn’t bring about gains in e¢ ciency in large samples, since the variance-covariance 

matrix is asymptotically block diagonal between the mean and variance parameters. In 

addition, since one-step estimation is computationally more di¢ cult, what is generally 

done is taking a very simple model for the conditional mean of the series or …tting 

the demeaned series, obtained after a mean regression, to estimate the variance parameters; 

see for example Engle and Sheppard (2001). We analyze the e¤ects of such 

an estimation procedure in small samples by comparing the two-step and three-step 

methods. 

In this paper we conduct Monte Carlo (MC) studies to …nd out how much e¢ ciency 

is lost in small samples when estimating the above-mentioned conditional correlation 

models if the estimation of the parameters are done in two or three steps instead of one 

step. Assuming gaussian errors, we …rst analyze the e¤ects of estimation in multiple 

steps of a CCC-GARCH model by considering di¤erent values for the parameters to 

simulate two time series. To see that the e¤ects are also robust for the number of the 

series or parameters considered, we consider also three time series and a study with 

a di¤erent mean structure. Later, we study the other conditional correlation models, 

ECCC, DCC, cDCC, RSDC-GARCH. As reported in the results of the MC studies, 

when the estimations are based on gaussian distribution the di¤erence between the onestep 

and multiple steps estimators in small samples is negligible. When we change the 

assumption on the distribution of errors to t-distribution, for some models we conclude 

that the di¤erences between the estimators could be relevant. 

Up to our knowledge, this is the …rst work comparing these multiple steps estimators 

and analyzing the e¢ ciency lost when employing them instead of one-step estimation 

procedure in small samples. Finally, we use the three-steps procedure to estimate the 

volatility spillovers in ten major European stock markets. 

The rest of the paper is structured as follows: in section 2 we present the models that 

are to our interest. Di¤erent estimators of the parameters of these models are discussed 

in section 3. In section 4 we introduce the basis of the MC experiments we conducted 

and discuss the results we obtained. In section 5, we use our results to estimate a 

VAR-ECCC-GARCH model with volatility spillovers using the data obtained from 10 

major stock markets in Europe. Finally in section 6 concludes the paper. 

2 Econometric Models 

For simplicity we consider a k-variate vector autoregressive (VAR) model of order one 

for the mean equation with the following notation: 

5

Y t = + Y t 1 + " t 

(1) 

where V ar(" t jY t 1 ; :::Y 1 ) = H t , Y t is a kx1 vector of returns, is a kx1 vector of 

constants, is a kxk matrix of autoregressive coe¢ cients and " t is a kx1 vector of error 

terms as follows: 

Y t = y 1t y 2t : : y kt 

0, = 

 

1 2 : : k 

0 

2 

3 

11 12 : : 1k 

21 : : 

= 

: : : 

6 

7 

, " t = " 1t " 2t : : " 0 

kt 

4 : : : 5 

k1 : : : kk 

The stationarity of this VAR(1) model is assured if all the values of z that solve 

the following equation are outside of the unit circle: jI k zj = 0 

The coe¢ cient matrix could be chosen as diagonal which wouldn’t allow for the 

return spillovers. In this case there will be 2k mean parameters to estimate. Otherwise 

this number will be k(k + 1): 

The conditional variance H t is modelled in the following way: 

" t = H 1=2 

t t , where t is a kx1 vector with E( t ) = 0 and V ar( t ) = I k 

H t = D t R t D t , where D t = diag(h 1=2 

1t ; h 1=2 

2t ; :::; h 1=2 

kt ) and R t is the kxk conditional 

correlation matrix such that: 

2 

3 

1 12t : : 1kt 

12t 1 : 

H t = diag(h 1=2 

1t ; h 1=2 

2t ; :::; h 1=2 

kt ) : : : 

diag(h 1=2 

1t ; h 1=2 

2t ; :::; h 1=2 

kt 

6 

7 

) 

4 : : : 5 

1kt : : : 1 

2 

p p 3 

h 1t 12t h1t h 2t : : 1kt h1t h p kt 

12t h1t h 2t h 2t : 

= 

: : : 

6 

7 

4 : : : 5 

p 

1kt h1t h kt : : : h kt 

which ensures that as long as the conditional correlation matrix is positive de…nite 

and conditional variances are positive, the conditional variance-covariance will be positive 

de…nite as well. For parsimony, the variances of the error terms of each series are 

modelled as GARCH(1,1): 

6

~ ht = W + A~" 2 t + G ~ h t 1 

(2) 

where ~ h t = (h 1t ; h 2t ; :::; h kt ) 0 and ~" 2 t = (" 2 1t; " 2 2t; :::; " 2 kt )0 are kx1 vector of conditional 

variances and squared errors respectively and W is a kx1 and A and G are kxk matrices 

of coe¢ cients. A and G matrices could be restricted to be diagonal, which wouldn’t 

allow for volatility spillovers, as in Bollerslev (1990) or Engle (2002). On the other hand, 

it could be non-diagonal as in Jeantheau (1998) or Ling and McAleer (2003) allowing 

for volatility spillovers. In the former case there will be 3k variance parameters to 

estimate, while in the latter this number will be k(2k + 1). 

If we denote by ! i = [W ] i , ij = [A] i;j and ij = [G] i;j , for the variances to be always 

positive, Jeantheau (1998) requires the following trivial set of su¢ cient conditions: 

! i > 0, ij > 0, ij > 0 for all i, j. 

Nakatani and Teräsvirta (2008) provide necessary and su¢ cient conditions where 

they allow for negative volatility spillovers. For an empirical work, see Conrad and 

Karanasos (2008). For the covariance stationarity, the roots of the following determinant 

has to be outside of unit circle is required: 

jI k (A + G)zj = 0 

which in the diagonal case implies that the persistence term " + " should be: 

ii + ii < 1 for all i. 

We consider four di¤erent models for the conditional correlations. The …rst and 

simplest is the Constant Conditional Correlation (CCC) GARCH model of Bollerslev 

(1990) where he restricts the correlations to be constant over time. He shows in his 

paper that, under this restriction, gaussian QML estimator of the correlation matrix, 

R, is equal to the sample correlation of the standardized residuals such that: 

[R] ij = b ij = 

P 

t v itv jt 

q 

( P t v2 it )(P t v2 jt ) (3) 

where v t = Dt 

1 " t are the standardized errors. Therefore the number of correlation 

parameters to be estimated is only k(k 1)=2. 

7

The second model we consider is the Dynamic Conditional Correlation (DCC) 

GARCH of Engle (2002) given by: 

R t = P t Q t P t 

P t = diag(Q 1=2 

t ) 

Q t = (1 1 2 ) Q + 1 v t 1 vt 0 1 + 2 Q t 1 , 

where Q t is generally referred to as non-standardized correlations and Q is the long 

run correlation matrix that could be replaced by the correlation matrix of the standardized 

errors which is a consistent estimator of Q. This latter process is called "variance 

targeting" and it makes the estimation easier, reducing the number of parameters to 

estimate from k(k 1)=2 + 2 to only 2. v t is the vector of standardized errors de…ned 

as before. 1 and 2 are non-negative scalars which satisfy 1 + 2 < 1, to have the 

correlation matrix, R t ; positive de…nite. As one can notice, these scalar coe¢ cients impose 

the same dynamics on di¤erent correlations. Hafner and Franses (2003) provide 

a more general de…nition of the model where they consider coe¢ cient matrices instead 

of scalar coe¢ cients allowing for di¤erent dynamics on di¤erent correlations, however 

this increases the number of parameters considerably. For simplicity, we will focus on 

the set up with the scalar coe¢ cients. 

Engle and Sheppard (2001) and later Engle, Sheppard, Shepherd (2007) point out 

that when using large systems of equations, i.e. k is large, estimating the correlation 

parameters of a DCC-GARCH model separately, as in two-steps or three-steps, brings 

about signi…cant biases in the estimates of the dynamic parameters 1 and 2 . Aielli 

(2006) shows that two-step or three-step estimation of DCC-GARCH models with variance 

targeting is inconsistent since the correlation matrix of the standardized residuals 

is not a consistent estimator of the long run correlation matrix. He suggests a corrected 

version of the DCC-GARCH model, in this paper referred to as cDCC-GARCH, which 

replaces the Q t equation of DCC-GARCH model: 

Q t = + 1 Q t 1(v t 1 v 0 t 1 R t 1 )Q t 1 + ( 1 + 2 )Q t 1 

where Q t = diag(Q 1=2 

t ) = inv(P t ). 

Although Aielli (2006) suggested this model to resolve the inconsistency problem 

in estimating the unconditional correlation matrix with sample correlation matrix, his 

model does not allow variance targeting. Therefore, his model su¤ers from the curse of 

dimensionality since the number of parameters to be estimated, including the intercept 

matrix, is k(k 1)=2 + 2 which is O(k 2 ). On the other hand, it is worth mentioning 

that Engle, Sheppard and Shepherd (2007) employs Aielli’s model and they use variance 

targeting, where they estimate the unconditional correlation matrix through a singlefactor 

model. 

8

The last model of conditional correlations we will consider is the Regime Switching 

Dynamic Correlations GARCH, (RSDC-GARCH) model of Pelletier (2006). In 

his model, the conditional correlations follow a switching regime driven by an unobserved 

Markov chain such that they are …xed in each regime but may change across 

regimes. For simplicity we assume a two-state Markov process such that R t , at any 

time t, could be equal to either R L or R H , which stand for low and high state correlation 

matrices respectively. The matrix of transition probabilities are given by 

= ff L;L ; H;L g; f L;H ; H;H gg, where i;j is the probability of passing from state 

j to state i. Given that j;j + i;j = 1, in this model the number of parameters to 

estimate is equal to 2k(k 1) + 2: 

Next section discusses how the parameters of these models are estimated. The 

estimation of RSDC-GARCH models will be discussed separately. 

3 Estimation Procedures 

For notation purposes, we denote all the mean parameters by = ( 0 ; vec() 0 ) 0 , all 

the variance parameters by = (W 0 ; vec(A 0 ); vec(G 0 )) 0 , and all the correlation parameters 

by which changes according to the model considered above. For example, 

for a CCC-GARCH model = vech(R), while for cDCC-GARCH it becomes: = 

(vech() 0 ; 1 ; 2 ) 0 : 1 

3.1 CCC, DCC and cDCC GARCH models 

3.1.1 One-step Estimation 

In the one-step estimation procedure, all parameters of the model, = ( 0 ; 0 ; 0 ) 0 , 

are estimated in one step simultaneously. If the DGP is gaussian (not gaussian), we 

assume that it is normal, then the estimation is a ML (QML) estimation and it is 

carried out by maximizing the multivariate gaussian log-likelihood function: 

L() = 

T k 

2 log(2) 1 

2 

P T 

t=1 (log jH tj + " 0 tH 1 

t " t ) 

Under the assumption that ijt = ijt it jt , this log-likelihood function boils down 

to: 

L() = 

T k log(2) P 

1 T 

2 2 t=1 log jD P 

1 

tR t D t j T 

2 t=1 "0 t(D t R t D t ) 1 " t 

= T k 

2 log(2) 1 

2 

P T 

t=1 log jR tj 

P T 

t=1 log jD tj 

1 

2 

P T 

t=1 v0 tRt 

1 v t 

Since all the parameters are estimated using the full log-likelihood function in one 

step, the estimators are consistent, asymptotically normal and if the DGP is also 

1 vec operator stacks the colums of a matrix while vech operator stacks the columns of the lower 

triangular part of a matrix. 

9

normal they are asymptotically e¢ cient. The problem with this estimation would be 

that in every iteration, the correlation matrix has to be inverted which slows down 

the optimization process. On the other hand, when the number of series, k, increases, 

the number of parameters to be estimated will increase rapidly, which might cause 

convergence problems speci…cally in small samples. 

If we would assume that the errors follow a t-distribution, then the multivariate 

t-loglikelihood function to be maximized would be as in Fiorentini et al. (2003): 

L(; ) = T log( ( k+1 

1 

)) T log( ( 

2 

P n 

 

T 1 

t=1 log jH k+1 

2 tj log(1 + 

2 

2 )) 

T k 

2 

1 2 

log( 

o 

 

1 2 v0 tR 1 

t v t ) 

 

) 

T k 

2 log() 

where is the inverse of the degrees of freedom as a measure of tail thickness such 

that it is always positive and strictly less than 0:5 for the existence of the second order 

moments. One concern on using a loglikelihood function based on t-distribution is 

that, as shown in Newey and Steigerwald (1997), although it will yield more e¢ cient 

estimators than using gaussian loglikelihood when the assumed distribution is correct, 

there is the danger that one may end up sacri…cing consistency when it is not. Since this 

inconsistency problem will not occur as long as both the true and assumed distributions 

are symmetric, it will not be a problem for our Monte Carlo simulations in section 4. 

3.1.2 Two-steps Estimation 

This is the procedure proposed in Engle (2002) or in Engle and Sheppard (2001) for a 

DCC-GARCH model. The idea is to separate the estimation of the correlation parameters 

from the mean and variance parameters. Assuming gaussian errors, in the …rst 

step all parameters except the correlation coe¢ cients, = ( 0 ; 0 ) 0 , are estimated by 

maximizing the gaussian log-likelihood function mentioned above replacing the correlation 

matrix R t with the identity matrix of size kxk; therefore maximizing the following 

quasi-log-likelihood function: 

QL() = 

T k 

2 log(2) P T 

t=1 log jD tj 

1 

2 

P T 

t=1 v0 tv t 

Notice that in the case where there are no volatility spillovers allowed, this …rst 

step estimation is equivalent to estimating k univariate models separately. See Engle 

and Sheppard (2001) for details. However, when there are volatility spillovers, this …rst 

step estimation must be done in a multivariate fashion. 

In the second step, given the estimates from the …rst step, ^, the correlation coef- 

…cients are estimated by: 

QL( j^) = 1 2 

P T 

t=1 

 

log jRt j ^v tR 0 t 

1 ^v t 

10

Where the ^v t is the standardized residuals obtained in the …rst step. As shown in 

Engle and Shepperd (2001), this two step estimation yields estimators which are consistent 

and asymptotically normal but not fully e¢ cient since they are limited information 

estimators. Their proof follows closely the discussion of the asymptotic properties of 

two-step GMM estimators in Newey and McFadden (1994). Bollerslev (1990) shows 

that when the correlations are constant over time, this second step estimators are equal 

to the sample correlation matrix of the standardized residuals: 

b ij = 

P 

t ^v it^v jt 

p 

( 

P 

t ^v2 it )(P t ^v2 jt ) 

Although itdoesn’t have theoretical support, we also consider a two-step estimation 

using log-likelihood function based on t-distribution. In the multivariate t-based loglikelihood 

function, we replace the correlation matrix R t with I k in the …rst step: 

QL(; ) = T log( ( k+1 

1 

)) T log( ( 

2 

P n 

 

T 

t=1 

log jD t j log(1 + 

k+1 

2 

2 )) 

1 2 

log( 

o 

T k 

2 

 

1 2 v0 tv t ) 

 

) 

T k 

2 log() 

Similar to the gaussian log-likelihood case, in the case where there are no volatility 

spillovers, we employ univariate estimation for each series while when there are 

volatility spillovers, we solve the multivariate problem. In the second step of the estimation 

based on t-distribution, we use the second step estimation with the gaussian 

loglikelihood. 

3.1.3 Three-steps Estimation 

This procedure is mentioned in Bauwens (2006) for the gaussian log-likelihood estimation. 

In the …rst step, assuming constant variance for each error term, i.e. 2 it = 2 i 8 

t, and assuming that the the correlation matrix R t is equal to the identity matrix for 

all t; we estimate parameters of the mean equation, , and the variance parameters 2 i 

by maximizing the following quasi-loglikelihood function: 

QL(; 2 i ) = 

T k 

2 log(2) P T 

t=1 log jDj 1 

2 

P T 

t=1 v0 tv t 

where D is a vector of conditional standard deviations: D = diag(h 1=2 

1 ; h 1=2 

2 ; :::; h 1=2 

k ). 

This …rst step estimation is equivalent to doing OLS estimation for the mean equations, 

equation by equation, given that under the assumptions made, the variance-covariance 

matrix is diagonal between the mean and variance parameters. Therefore the estimators 

obtained from this …rst step are consistent, disregarding that the errors are actually 

heteroskedastic. Later in the second step, the parameters of the variance equations, , 

are estimated, given the estimates of the parameters of the mean equation, b , imposing 

to the full loglikelihood function only the assumption that the correlation matrix R t is 

replaced with identity matrix. This gives the following quasi-loglikelihood function: 

11

QL(j b ) = 

T k log(2) P T 

2 t=1 log jD 1 

tj 

2 

P T 

t=1t ^"0 tDt 

2 ^" t 

where ^" t are the residuals obtained from the …rst step. After obtaining the estimators 

b and ^ from the …rst two steps, in the third step, the correlation parameters are 

estimated as in the two-step estimation: 

QL( j b ; ^) = 1 2 

P T 

t=1 

 

log jRt j ^v tR 0 t 

1 ^v t 

where ^v t are the standardized residuals obtained from the …rst two steps. Similarly, 

as in the two-step estimation, when the correlations are constant over time, this third 

step estimators could be calculated from the sample correlations of the standardized 

residuals: 

b ij = 

P 

t ^v it^v jt 

p 

( 

P 

t ^v2 it )(P t ^v2 jt ) 

In case of a log-likelihood function based on t-distribution, these three steps are 

performed in a similar manner. In the …rst step, mean parameters, , are estimated 

along with the inverse of the degrees of freedom assuming homoskedasticity, i.e. 2 it = 

2 i 8 t, by maximizing the …rst step loglikelihood function: 

QL(; ) = T log( ( k+1 

1 

)) T log( ( 

2 

P n 

 

T 

t=1 


k+1 

2 

2 )) 

T k 

2 

 

1 2 v0 tv t ) 

1 2 

log( 

o 

 

) 

T k 

2 log() 

The estimated variances, ^ 2 i , and degrees of freedom, ^, of the …rst step estimation 

are disregarded. In the second step, the variance parameters, , and the inverse of 

the degrees of freedom, , that is reported in the results are estimated given the mean 

parameter estimates, ^, maximizing the following log-likelihood function: 

QL(; j^) = T log( ( k+1 

1 

)) T log( ( 

2 

P n 

 

T 

t=1 


k+1 

2 

2 )) 

T k 

2 

 

1 2^"0 tD 2 

t ^" t ) 

1 2 

log( 

o 

 

) 

T k 

2 log() 

As in the two-step estimation procedure, we use the gaussian log-likelihood function 

to estimate the correlation parameters, in the third step of the three-step estimation 

with t-based log-likelihood function. Our results from the MC studies didn’t change 

when we chose a t-based log-likelihood function in the estimation of correlation parameters. 

3.2 Estimation with RSDC-GARCH Model 

The RSDC-GARCH model of Pelletier (2006) suggests correlations to be constant over 

time but di¤erent between regimes where these regimes are driven by an unobserved 

12

markov chain. In our paper, we will make use of his unrestricted two regime RSDC 

model. In this model, the correlation matrix at any time t could be either R L or R H . 

The matrix of transition probabilities is given by = ff L;L ; H;L g; f L;H ; H;H gg, 

where i;j is the probability of passing from state j to state i. If we denote by t 1 all 

the information up to t 1, then a one-step estimation, with f(:) being the likelihood 

function based on either Gaussian or Student-t distribution, would be obtained by 

maximizing the following log-likelihood function for estimating = ( 0 ; 0 ; 0 ) 0 , the 

mean, variance and correlation parameters respectively is: 

L() = P T 

t=1 log f(Y tj t 1 ) 

f(Y t j t 1 ) = f(Y t jS t = L; t 1 ) Pr(S t = Lj t 1 ) 

+f(Y t jS t = H; t 1 ) (1 Pr(S t = Lj t 1 )) 

where f(Y t jS t ; t 1 ) is the likelihood conditional on the state and past values while 

f(Y t j t 1 ) is the likelihood when the state is marginalized out. Pr(S t j t 1 ) denotes 

the probability of being in a certain state given the past information. This probability 

is derived using Hamilton …lter (Hamilton, 1994, Chapter 22). 

In our two state model, the law of motion of Pr(S t j t 1 ) will boil down to: 

Pr(S 

h t = Lj t 1 ) = (1 H;H ) + ( L;L + H;H 1) 

 

f(Y t 1 jS t 1 =L; t 2 )Pr(S t 1 =Lj t 2 ) 

f(Y t 1 jS t 1 =L; t 2 )Pr(S t 1 =Lj t 2 )+f(Y t 1 jS t 1 =H; t 2 )(1 Pr(S t 1 =Lj t 2 )) 

For the initial condition of this iterative process, we use the long run probabilities 

for each state which are calculated using the transition probabilities and that the sum 

of the long run probabilities should be equal to one. 

Since the number of correlation parameters to be estimated is 2k(k 1) + 2 for k 

time series, as Pelletier (2006) also mentions, one-step estimation is not practicable, 

especially when k is not small. Pelletier (2006) uses a data set consisting of four exchange 

rate series. After removing the mean of the data, he estimates the correlation 

parameters separately from estimation of the variance parameters. In our paper, this 

corresponds to a three-step estimation without paying attention to the mean parameters 

or then a two-step estimation for a zero mean series. The estimation of mean 

and variance parameters can be processed in a single step or in successive two steps 

as discussed in the case of other conditional correlation models. When estimating the 

correlation parameters in a separate step, the log-likelihood function will be maximized 

taking the mean and variance parameter estimates as known. On the other hand, if k 

is large, estimating the correlation parameters in a separate step could be even troublesome 

in terms of convergence. For this reason, Pelletier (2006) suggests using the 

EM algorithm of Dempster et al:(1977), noting that EM algorithm gives estimates that 

are very close to the numerical maximum. To obtain the exact numerical maximum, 

he suggests using a limited number of iterations of a Newton type algorithm. 

13 

i

3.3 Asymptotic Properties 

The one-step gaussian ML/QML estimators for the CCC, DCC and cDCC GARCH 

models are consistent and asymptotically normal such that: 

p n(^n 0 ) A N(0; A 1 

0 B 0 A 1 

0 ) 

where A 0 is the negative expectation of the Hessian matrix evaluated at the true 

parameter vector 0 and B 0 is the expectation of the outer product of the score vector 

evaluated at 0 obtained from the full-likelihood function mentioned in one-step 

estimation. 

The two-step gaussian QML estimators mentioned are consistent as well and under 

mild assumptions listed in Engle and Sheppard (2001) for a DCC-GARCH model, 

they are asymptotically normal such that A 0 is a block lower-triangular matrix and 

together with B 0 , are obtained from the likelihood functions used in each step. The 

part of A 0 that corresponds to the mean and variance parameters is block diagonal 

where each block comes from the second derivatives of the log-likelihood functions of 

each corresponding step. To obtain the part of A 0 corresponding to the correlation 

parameters, we use the second derivatives from the full-likelihood function. Since A 0 is 

block lower triangular, the asymptotic variance of the two-step estimators are weakly 

greater than the case of a block diagonal A 0 , which belongs to the one-step estimators. 

This loss of e¢ ciency is due to the estimation of the correlation parameters in a separate 

step. 

Three-step QML estimators are consistent and their asymptotic distribution is very 

similar to that of the two-step estimators. The part of A 0 that corresponds to the 

correlation parameters is obtained in the same manner, however the part for the mean 

and variance equations parameters come from the second derivatives of the likelihood 

functions of the corresponding …rst two steps, therefore for the parameters of each series 

forming a block diagonal matrix where the …rst block is for the mean and second block 

is for the variance parameters. In the three-step estimation procedure there is loss of 

e¢ ciency both because of separating the estimation of variance parameters from the 

mean parameters, as mentioned by Bauwens (2001) and of separating the estimation 

of correlation parameters from the mean and variance parameters as pointed out in 

Engle and Sheppard (2001). 

The asymptotic properties of one-step t-based ML estimators can be found in 

Fiorentini et al: (2003). It is well known that if the true distribution of the errors 

is also a Student-t, ML estimators based on t-errors are more e¢ cient than QML estimators 

based on gaussian errors. However, as shown by Newey and Steigerwald (1997), 

care should be taken when computing ML estimations based on t-distribution since if 

the true distribution of the errors is not Student-t, one might end up sacri…cing consistency. 

In our MC studies, as Newey and Steigerwald (1997) requires for consistency 

14

of the t-based estimators, the data is either generated from a symmetric gaussian or a 

symmetric t-distribution. 

Finally, the asymptotic properties of the one-step and multiple steps estimators of 

the RSDC-GARCH model are similar and can be found in Pelletier (2006). 

4 Monte Carlo Experiments 

We …rst analyze the …nite sample performance of multiple steps estimators in di¤erent 

settings using CCC-GARCH model with gaussian errors. Although we could focus on 

the performance of each parameter estimator, it is not really practical when we consider 

di¤erent models, sample sizes, estimators and parameters. In this sense, later when 

we consider CCC, DCC, cDCC and RSDC GARCH models with gaussian or Student-t 

errors, we rather look at the in-sample predictions of volatilities and correlations to 

compare di¤erent estimators. For RSDC GARCH models the correlations are driven 

by an unobservable markov chain and therefore the correlation parameter estimators 

will be analyzed instead of correlation predictions. Finally, for the models that the 

multiple steps estimation yields estimators which are very close to one-step estimators, 

we will do a robustness check with respect to the distribution of the errors. 

We consider a replication size of 1000 and in each replication a data set of k series 

is generated according to the relevant model and distribution. We consider two time 

series in all the experiments except one in the …rst part where we check the robustness 

of the results when there are three time series. Then we estimate the parameters of 

the model using one-step, two-step and three-step estimators assuming either gaussian 

or t-distribution for the errors. All simulations are performed by MATLAB computer 

language. We consider three di¤erent sample sizes 500, 1000 and 5000 for the …rst 

part. However, having realized that the 5000 sample size is already too large, in the 

remaining parts we consider the sample sizes 200, 500 and 1000. 

In the …rst part only, to compare di¤erent estimators, the true parameter values 

and the Monte Carlo average of the parameter estimates together with their corresponding 

standard errors from each procedure are reported. In addition, for each 

parameter, average deviations of two-steps from one-step estimators and of three-steps 

from two-steps estimators are reported with their corresponding Monte Carlo standard 

deviations. In all the parts, kernel density estimates of di¤erent estimators of each 

parameter are also plotted to compare the performance of multiple steps estimators for 

each sample size. Also these density estimates allow us to observe graphically the bias 

and standard deviations of the di¤erent parameter estimators. 

15

eal 

values 

Onestep Est Twostep Est Threestep Est 

Deviation of 2stepest from 

1stepest 

Deviation of 3stepest from 

2stepest 

s500 s1000 s5000 s500 s1000 s5000 s500 s1000 s5000 s500 s1000 s5000 s500 s1000 s5000 

μ1 0.2 0.20677 0.20410 0.20113 0.20695 0.20413 0.20117 0.20759 0.20382 0.20119 0.00018 0.00003 0.00005 0.00064 0.00031 0.00001 

(0.04973) (0.03642) (0.01631) (0.05029) (0.03692) (0.01628) (0.05318) (0.03901) (0.01749) (0.00660) (0.00433) (0.00186) (0.01854) (0.01387) (0.00593) 

β1 0.8 0.79324 0.79636 0.79928 0.79313 0.79645 0.79927 0.79200 0.79621 0.79909 0.00011 0.00009 0.00001 0.00113 0.00024 0.00017 

(0.02826) (0.02006) (0.00873) (0.02888) (0.02046) (0.00874) (0.03048) (0.02183) (0.00955) (0.00583) (0.00398) (0.00173) (0.01051) (0.00811) (0.00360) 

μ2 0.4 0.40322 0.40339 0.40022 0.40311 0.40327 0.40006 0.40302 0.40389 0.40012 0.00010 0.00012 0.00016 0.00009 0.00062 0.00005 

(0.06017) (0.04303) (0.01740) (0.06091) (0.04353) (0.01750) (0.06215) (0.04435) (0.01806) (0.00837) (0.00549) (0.00236) (0.01446) (0.01109) (0.00507) 

β2 0.6 0.59620 0.59763 0.59980 0.59607 0.59774 0.59994 0.59615 0.59731 0.60008 0.00014 0.00011 0.00014 0.00009 0.00043 0.00013 

(0.03807) (0.02621) (0.01135) (0.03893) (0.02683) (0.01150) (0.03943) (0.02743) (0.01190) (0.00762) (0.00516) (0.00222) (0.00966) (0.00745) (0.00348) 

ω1 0.1 0.17979 0.12374 0.10339 0.18172 0.12290 0.10334 0.18273 0.12374 0.10335 0.00192 0.00084 0.00005 0.00102 0.00084 0.00001 

(0.17909) (0.07936) (0.01929) (0.18270) (0.07195) (0.01947) (0.18413) (0.07482) (0.01948) (0.11269) (0.05054) (0.00322) (0.11759) (0.05370) (0.00025) 

α1 0.1 0.10835 0.10306 0.09988 0.10853 0.10298 0.09990 0.10643 0.10231 0.09977 0.00018 0.00008 0.00002 0.00209 0.00068 0.00013 

(0.04394) (0.03022) (0.01234) (0.04429) (0.03042) (0.01253) (0.04313) (0.03021) (0.01251) (0.00918) (0.00594) (0.00187) (0.01070) (0.00413) (0.00019) 

γ1 0.8 0.70641 0.77192 0.79637 0.70502 0.77285 0.79640 0.70455 0.77244 0.79652 0.00139 0.00093 0.00003 0.00047 0.00041 0.00012 

(0.20318) (0.09603) (0.02695) (0.20553) (0.08880) (0.02727) (0.20786) (0.09253) (0.02727) (0.12339) (0.05506) (0.00431) (0.13076) (0.05995) (0.00039) 

ω2 0.1 0.26975 0.12014 0.05309 0.27334 0.13184 0.05318 0.29011 0.14590 0.05400 0.00359 0.01170 0.00009 0.01677 0.01407 0.00082 

(0.30773) (0.17689) (0.01474) (0.31067) (0.19837) (0.01492) (0.33851) (0.23086) (0.03143) (0.21035) (0.13564) (0.00215) (0.21710) (0.15677) (0.02819) 

α2 0.1 0.06083 0.05417 0.05014 0.06092 0.05379 0.05011 0.06058 0.05388 0.05005 0.00009 0.00038 0.00003 0.00034 0.00010 0.00006 

(0.03588) (0.02312) (0.00869) (0.03653) (0.02386) (0.00874) (0.03540) (0.02336) (0.00884) (0.01187) (0.00796) (0.00121) (0.00865) (0.00532) (0.00125) 

γ2 0.9 0.65975 0.82210 0.89662 0.65620 0.81012 0.89656 0.63727 0.79581 0.89577 0.00355 0.01198 0.00006 0.01893 0.01431 0.00080 

(0.32238) (0.19192) (0.02074) (0.32471) (0.21186) (0.02095) (0.35500) (0.24291) (0.03530) (0.21535) (0.14075) (0.00297) (0.22300) (0.15847) (0.02950) 

ρ 0.2 0.19921 0.20071 0.19999 0.19751 0.19985 0.19983 0.19760 0.19981 0.19982 0.00170 0.00086 0.00016 0.00010 0.00004 0.00000 

(0.04359) (0.03147) (0.01379) (0.04325) (0.03134) (0.01378) (0.04322) (0.03134) (0.01378) (0.00155) (0.00084) (0.00011) (0.00124) (0.00072) (0.00012) 

Table 1: Means and standard deviations of different estimators in Basic Setting. Bold: Statistically significant at 5 %. 

4.1 Parameter Estimators of the CCC-GARCH Model 

In this …rst part, we consider di¤erent settings where we change either some parameter 

values or model structure or the number of time series. In all these settings we simulate 

data with gaussian errors and estimate the parameters with a gaussian log-likelihood 

function. We start our Monte Carlo experiments with a bivariate model given by 

equations (1) to (3) with a diagonal matrix . We …x the unconditional mean and 

variance equal to 1. The mean and variance persistence is set to be di¤erent one from 

each other but quite high. Therefore in this bivariate model, which we will refer to as 

basic setting later, we have 11 parameters to estimate. The true parameter values for 

this basic setting, as well as Monte Carlo means and standard deviations of one-step 

and multiple steps estimators are given in Table 1. To check the robustness of the 

results, in the other settings we change only one assumption or property of this basic 

setting keeping others …xed. 

In the second setting, we set the unconditional variance of one series approximately 

six times the other by changing the true parameter values in the second variance 

equation to: w 2 = 1, 2 = 0:15, 2 = 0:7. In the third setting, we set the unconditional 

mean of one series six times the other by changing the true mean parameters for the 

…rst series to: 1 = 0:3, 1 = 0:8. In the fourth setting we increase the power of a shock 

that …rst return series receives on its conditional variance by changing the coe¢ cients 

to: 1 = 0:35, 1 = 0:55. We allow for return interactions in the …fth setting by 

chosing a non diagonal matrix with the following true parameter values: 1 = 0:1, 

11 = 0:8; 12 = 0:1, 2 = 0:3, 21 = 0:1; 22 = 0:6. Finally, in the sixth setting, we 

consider a trivariate model where we add another equation to the basic setting with 

the following mean and variance parameters: 3 = 0:3, 3 = 0:7, w 3 = 0:05, 2 = 0:15, 

2 = 0:8. The true values for the correlation parameters are chosen in this sixth setting 

16

as: 12 = 0:1, 13 = 0:2, 23 = 0:3. To save space, Table 1 contains the results for only 

the basic setting. Although not included in this paper, similar tables for the results of 

other settings are available from the authors upon request. 

The results obtained are similar in all these settings. First thing to note is that 

the Monte Carlo bias and standard error of all estimators considered are decreasing 

as the sample size increases, as shown in Table 1 for the basic setting. The results 

of coe¢ cient tests in every setting suggest that signi…cant biases in the estimators for 

small samples seem to disappear when we move to larger samples, as expected when 

estimators are consistent. 

In some cases, multiple steps estimators seem to perform better than the one-step 

estimator; see, for example Table 1, where the average of the three-step estimates of 1 

and 2 are closer to the true parameter values compared to one-step estimates. Therefore, 

we can not conclude that in …nite samples, multiple steps estimators over/under 

estimate the parameters in a systematic manner. 

Like for the basic setting, in all these settings we analyzed the deviation of twosteps 

estimators from one-step estimators for the loss of e¢ ciency due to estimating the 

correlation separately and also the deviation of three-steps estimators from two-step 

estimators for the loss of e¢ ciency due to estimating mean and variance parameters 

separately. For all the settings, the results suggest that the two-steps estimators follow 

the one-step estimators very closely such that even in small samples the deviation is 

very small. Similarly, the three-steps estimators are very close to two-steps estimators. 

We can also con…rm these results visually from Figure 1 for the basic setting where we 

plot kernel density estimates of the one-step and multiple steps estimators for sample 

size 500. Even for this small sample size the modes of the estimated densities of one-step 

and multiple steps estimators are very close to the true value of the parameter. There is 

very little, if any, di¤erence between the density estimates for di¤erent estimators. As 

the sample size increases, these densities become more and more concentrated around 

true parameter values. We don’t include the …gures for higher sample sizes in this …rst 

part to save space. 

17

CCCnCCCn 

Sample size: 500 

15 

10 

5 

0 

15 

10 

5 

0 

15 

10 

5 

0 

15 

10 

5 

0 

15 

10 

5 

0 

µ 1 

1step 

2step 

3step 

true value 

15 

10 

5 

β 1 

0 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

β 2 

15 

10 

5 

µ 2 

0 0.2 0.4 0.6 0.8 1 

0 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

0 0.2 0.4 0.6 0.8 1 

γ 1 

15 

15 

10 

10 

5 

5 

0 

0 

0 0.2 0.4 0.6 0.8 1 

ω 1 α 1 γ 2 

15 

15 

10 

10 

5 

5 

0 

0 

0 0.2 0.4 0.6 0.8 1 

ω 2 α 2 ρ 

15 

15 

10 

10 

5 

5 

0 

0 

0 0.2 0.4 0.6 0.8 1 

α +γ 1 1 α +γ 2 2 Figure 1: Kernel density estimates of di¤erent estimators in Basic Setting 

18

CCC 

GARCH 

DCC 

GARCH 

cDCC 

GARCH 

ECCC 

GARCH 

RSDC 

GARCH 

μ1 0.20 0.20 0.20 0.20 0.20 

β11 0.80 0.80 0.80 0.80 0.80 

μ2 0.40 0.40 0.40 0.40 0.40 

β22 0.60 0.60 0.60 0.60 0.60 

ω1 0.10 0.10 0.10 0.10 0.10 

α11 0.10 0.10 0.10 0.10 0.10 

α12 0.10 

γ11 0.80 0.80 0.80 0.80 0.80 

γ12 0.80 

ω2 0.05 0.05 0.05 0.05 0.05 

α21 0.05 

α22 0.05 0.05 0.05 0.05 0.05 

γ21 0.90 

γ22 0.90 0.90 0.90 0.90 0.90 

ρ 0.20 0.20 

ρL 0.20 

ρH 0.80 

λ 0.20 0.20 

δ1 0.04 0.04 

δ2 0.94 0.94 

πL 0.80 

πH 0.90 

Table 2. The true parameter values used to simulate the 

data used in the MC experiments in part 2. "λ" is the 

offdiagonal parameter of the intercept in DCC and cDCC 

GARCH models. 

4.2 Predicted Volatilities and Correlations 

Focusing on the di¤erences of one-step and multiple steps estimators for each parameter 

for all the models is not practical. Therefore instead, in this part and the next one, we 

will focus on the in-sample predictions of volatilities and correlations. Since correlation 

parameter estimators depend on the standardized residuals which in turn depends on 

mean parameter estimators through the residuals and variance parameter estimators 

through the variances used for standardization, the in-sample predictions of volatilities 

and correlations could be used to see the e¤ects of estimating in multiple steps. 

As we could obtain the in-sample predictions of volatilities and correlations using 

di¤erent estimators, we could also derive the volatilities and correlations implied by 

the true parameter values. For a given sample size T , if we call the volatility predictors 

^h j i;t as the volatilities of series i over time t obtained from estimator j (one-step, twosteps 

or three-steps estimator) and if we denote h true 

i;t as the true volatilities, then the 

estimator for the deviation in the volatilities of series i will be: 

^h j i;t = P 1 T 

1000 t=1 

n^hj i;t 

h true 

i;t 

o 

Similarly, estimators for the deviations in correlations over time between series i 

and j, p ij;t , can be calculated as: 

^p j ij;t = 1 

1000 

P T 

t=1 

^p 

j 

ij;t 

p true 

ij;t 

When the conditional volatility and conditional correlation predictions were calculated 

the starting values were taken as the unconditional volatilities and unconditional 

correlations. 

19

Finally we plot the kernel density estimates of these deviation estimators to compare 

the performances of one-step, two-steps and three-steps estimators. 

In this part we consider a range of GARCH models. For the CCC-GARCH model of 

Bollerslev (1990), we take the parameters and structure of the basic setting of the …rst 

part. We also consider DCC-GARCH model of Engle (2002), cDCC-GARCH model 

of Aielli (2006), ECCC-GARCH of Jeantheau (1998) and RSDC-GARCH model of 

Pelletier (2006). The true values of the mean, variance and correlation parameters 

considered are given in Table 2. On the other hand, when simulating the data and 

estimating the parameters we assume either Gaussian or Student-t distribution. When 

using the Student-t distribution, we chose the degrees of freedom equal to …ve, which 

translates to fairly fat tails for the distribution of the errors. 

For the models where the one-step and multiple steps estimators yield similar 

volatility and correlation predictions, we do a robustness check for the distribution 

in the third part. Checking the robustness of the results to simulating with one model 

and estimating with another model is outside of the focus of this paper and a subject 

for future research. 

4.2.1 With Gaussian Errors 

We …rst assume that the true distribution of the errors is normal and we therefore 

estimate using a gaussian loglikelihood function. As mentioned before, the one-step 

and multiple steps estimators in this case are consistent asymptotically normal however 

the standard errors will be weakly greater for the latter estimators. See Engle and 

Sheppard (2001) for details. 

Figure 2 shows for the CCC-GARCH model, the kernel density estimates of "devinh1", 

"devinh2" and "devincorr", the deviation estimators for the volatilities of the …rst and 

second series and the correlations, which are obtained from one-step and multiple steps 

estimators using sample sizes 200, 500 and 1000. "CCCn CCCn" is the name given 

to the …gure to represent that the data is simulated and estimated with a CCC-GARCH 

model with normal errors. The …gure suggests that the deviation estimators are distributed 

with a mode very close to zero and almost symmetric around zero for even 

the smallest sample size. Also, as the sample size grows, the distributions of these 

deviation estimators become degenerate around zero, suggesting that the volatility 

and correlation predictors are consistent estimators of the ones implied by the true 

parameters. 

Similar results were obtained for the DCC-GARCH, cDCC-GARCH, ECCC-GARCH 

and RSDC-GARCH models and are available upon request. 

20

CCCnCCCn 

10 

8 

6 

Sample size: 200 Sample size: 500 Sample size: 1000 

devinh1 

1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

10 


10 


10 


8 

8 

8 

6 

6 

6 

4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 

devincorr. 

0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 

devincorr 

0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

Figure 2: Kernel density estimates of the estimators of deviations of in-sample volatility 

and correlation predictors obtained from one-step and multiple steps Gaussian QML 

estimators of the parameters of CCC-GARCH model for di¤erent sample sizes. 

21

CCCtCCCt 

10 

8 

6 



1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

10 


10 


10 


8 

8 

8 

6 

6 

6 

4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 


and correlation predictors obtained from one-step and multiple steps Student-t QML 

estimators of the parameters of CCC-GARCH model for di¤erent sample sizes. 

4.2.2 With Student-t Errors 

In this second part, we assume that errors follow a t-distribution and we formulate our 

loglikelihood functions using Student-t distribution (univariate or multivariate depending 

on the method). We consider multiple steps estimators as explained in Section 3. 

Although not supported by theory, multiple steps estimation surprisingly yielded very 

close estimates to the ones obtained from one-step estimation for some models. 

Figure 3 plots kernel density estimates of the deviation estimators for the CCC- 

GARCH model. Since the errors both in the simulation and estimation process follow a 

Student-t distribution, the name of the …gure is "CCCt CCCt". As can be seen from 

the …gure, the deviation estimators, obtained from the di¤erence of the true volatilities 

and correlations and the ones obtained from one-step and multiple steps estimators, 

follow each other very closely. However, compared to the normal distribution case, 

the tails of the distributions are thicker. As the sample size grows, these density 

estimates of the deviation estimators become condensed around zero. Given that the 

data is distributed from a symmetric Student-t distribution, one-step estimation which 

is obtained from a t-based loglikelihood yields consistent estimators. In addition, the 

multiple steps volatility and correlation predictors follow the one-step ones very closely. 

For the DCC-GARCH similar results were obtained and are available from the authors 

22

ECCCtECCCt 

10 

8 

6 



1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

10 


10 


10 


8 

8 

8 

6 

6 

6 

4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 



estimators of the parameters of ECCC-GARCH model for di¤erent sample sizes. 

upon request. 

In the case of ECCC-GARCH, we obtained similar results, as shown in Figure 4, 

at least when the sample size was large. However, for the sample size 200, the density 

estimate for the deviation estimator between the three-steps volatility predictor and 

the volatility implied by the true parameters came out to be ‡at. This could be due 

to the extreme shocks generated by the t-distribution. 

On the other hand, the …gure for the cDCC-GARCH model, Figure 5, shows that 

the correlation predictor obtained using the three-step estimators is biased although 

the volatility predictors are not. When we looked in detail, we realized that the bias 

mostly comes from the intercept estimator in the correlation equation. As we can see 

from the …gure, the two-steps predictors follow one-step predictors very closely for both 

volatilities and correlations. Therefore although one-step predictors are theoretically 

feasible but two-steps predictors are not, we can conclude here that, practically by 

separating the estimation of correlation parameters from the estimation of mean and 

variance parameters, we don’t lose much from the e¢ ciency of the estimators. 

Similarly, Figure 6 shows the kernel density estimates for the volatility predictors 

and for the correlation parameter estimators of the RSDC-GARCH model. Although 

the density estimates for di¤erent correlation parameter estimators do not follow each 

23

cDCCtcDCCt 10 

8 

6 



1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

10 


10 


10 


8 

8 

8 

6 

6 

6 

4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 



estimators of the parameters of cDCC-GARCH model for di¤erent sample sizes. 

other very closely, the density estimates for the volatility predictors do. On the other 

hand, the densities for the two steps estimators and three-steps estimators are relatively 

close to each other. In addition, although the one-step estimator is consistent in this 

model and setting, we don’t see that the mode of the density estimates for the one-step 

estimator is very close to the true value of the parameter. Therefore we can conclude 

that the sample size could be too small for the results to emerge, but the existing 

results suggest that two-steps estimation could be still usable. 

Therefore in this section, we conclude that when the true distribution of the errors is 

Student-t and the estimations are performed using t-loglikelihoods, for the CCC, DCC, 

ECCC-GARCH models the two-steps or three-steps estimation is feasible although this 

is not supported by the theory. For the cDCC and RSDC-GARCH model, according to 

our …ndings, we don’t suggest using a three-steps estimation, but a two-steps estimation 

would yield very close estimates to the ones obtained by one-step estimation. 

4.3 Robustness of the Estimators to Distribution 

In this section, we changed the true distribution of the errors or the distribution assumption 

used to form the loglikelihood in order to check the robustness of the estimators. 

Whenever we generated a data using Student-t errors, we took the degrees of 

24

RSDCtRSDCt 

10 

8 

6 



1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


10 

8 

6 


10 

8 

6 


10 

8 

6 


4 

4 

4 

4 

4 

4 

2 

2 

2 

2 

2 

2 

0 

1 0 1 

0 

1 0 1 

0 

1 0 1 

0 

1 0 1 

0 

1 0 1 

0 

1 0 1 

corrst1 

corrst2 

corrst1 

corrst2 

corrst1 

corrst2 

15 

15 

15 

15 

15 

15 

10 

10 

10 

10 

10 

10 

5 

5 

5 

5 

5 

5 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

pcorrst1 

pcorrst2 

pcorrst1 

pcorrst2 

pcorrst1 

pcorrst2 

15 

15 

15 

15 

15 

15 

10 

10 

10 

10 

10 

10 

5 

5 

5 

5 

5 

5 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 

0 

0 0.5 1 



estimators of the parameters of RSDC-GARCH model for di¤erent sample sizes. 

freedom as 5. We considered only the models and distributions listed in Table 3, since 

from the previous part, we know that the three-step estimators based on t-loglikelihood 

don’t perform well when estimation cDCC-GARCG and RSDC-GARCH models. 

Figure 7 shows the kernel density estimates for di¤erent estimators for the CCC- 

GARCH model when the data is generated using a Student-t distribution and estimated 

assuming gaussian errors. Compared to the case when the true distribution and assumed 

distribution are both normal, in this …gure we see that the tails of the estimated 

densities are thicker. On the other hand, when we compare the results with the case 

when the true distribution and assumed distribution are both Student-t, we see that 

the …gures are very similar although in the latter case there are relatively more di¤erences 

between one-step and multiple steps estimators. Finally, in this part as well we 

con…rm that the results hold that the multiple-steps estimators follow one-step estimators 

very closely and the density estimates of the deviation estimators collapse to zero 

as the sample size grows. Similar …gures are obtained for the other models in the case 

of simulating the data with Student-t and estimating using gaussian loglikelihood. 

When we simulate the data with gaussian errors and estimate the model with 

Student-t loglikelihood, we see that the …gures look very similar to the case when the 

true distribution and the assumed distribution are both normal. Figure 8 shows the 

results for the CCC-GARCH model in this case. This similarity makes sense since 

25

Model Simulation Estimation 

CCCGARCH Studentt Normal 

CCCGARCH Normal Studentt 

DCCGARCH Studentt Normal 

DCCGARCH Normal Studentt 

cDCCGARCH Studentt Normal 

ECCCGARCH Studentt Normal 

ECCCGARCH Normal Studentt 

RSDCGARCH Studentt Normal 

Table 3: The models and distributions 

considered in the robustness analysis 

in Section 4.3. 

Student-t distribution has another parameter, namely the degrees of freedom, such 

that this distribution could approximate Gaussian distribution when this parameter 

is su¢ ciently large. In fact, when we were doing our MC experiments for this last 

case, we obtained very large estimates for the degrees of freedom. Similar …gures were 

obtained and are available on request from the authors. 

5 Empirical Application 

This section illustrates how multiple step estimators can be used to estimate volatility 

interactions in several European stock markets. The question we would like to answer is 

the following: are there “ volatility leading" and/or “volatility lagging" stock markets in 

Europe. In other words, among European stock markets, is there a particular market, 

say M, whose volatility is not a¤ected by the previous volatility of the others markets 

while the volatility of several markets depend on lagged volatility of market M 

With this question in mind, we use data containing daily returns on 10 European 

stock markets: AEX (Amsterdam), ATX (Vienna), BEL-20 (Brussels), CAC-40 

(Paris), DAX (Frankfurt), FTSE-100 (London), IGBM (Madrid), MIBTEL (Milan), 

OMXS (Stockolm) and SMI (Switzerland) 2 observed from January 8, 2002 

to April 

p 

30, 2009. Descriptive statistics of the returns series, computed as 100 log t 

p t 1 

, of 

sample size 1770, are given in Table 2. 

Evidence against normality is provided by di¤erent normality tests. However, we 

can still estimate consistently the parameters of the ECCC-GARCH model using the 

results in Ling and McAleer (2003). Also, based on the results provided by the Monte 

Carlo experiments, using multiple steps estimators seems to be reasonable. Therefore, 

we have estimated the ECCC-GARCH model given in equations (1), (2) and (3) with 

the following results: 

2 Series have been obtained in the webpage http://…nance.yahoo.com. 

26


CCCtCCCn 

10 

8 

6 


1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

0 

1 0.5 0 0.5 1 

0 

1 0.5 0 0.5 1 

10 


10 


10 


8 

8 

8 

6 

6 

6 

4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 


and correlation predictors obtained from one-step and multiple steps Gaussian QML 

estimators of the parameters of CCC-GARCH model for di¤erent sample sizes when 

the data is simulated with Student-t errors. 

Table 1: Descriptive statistics of return series 

Mean SD Skewness Kurtosis 

AEX 0:04 1:77 0:04 9:02 

ATX 0:03 1:62 0:56 13:4 

BEL-20 0:02 1:45 0:05 9:12 

CAC-40 0:02 1:65 0:13 8:99 

DAX 0:01 1:74 0:15 7:80 

FTSE-100 0:01 1:41 0:03 10:3 

IGBM 0:01 1:42 0:09 10:1 

MIBTEL 0:02 1:33 0:03 10:7 

OMXS 0:01 1:53 0:21 8:11 

SMI 0:01 1:39 0:14 8:79 

27

CCCnCCCt 

10 

8 

6 



1s 

2s 

3s 

10 

8 

6 


10 

8 

6 


4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

4 

2 

0 

1 0.5 0 0.5 1 

10 


10 


10 


8 

8 

8 

6 

6 

6 

4 

4 

4 

2 

2 

2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 

0 

1 0.5 0 0.5 1 

120 

100 

80 

60 

40 

20 


0 

0.2 0.1 0 0.1 0.2 



estimators of the parameters of CCC-GARCH model for di¤erent sample sizes when 

the data is simulated with Gaussian errors. 

28

= 0:04 0:03 0:02 0:02 0:01 0:01 0:01 0:02 0:01 0:01 0 

b = diag 0:03 0:03 0:06 0:06 0:05 0:08 0:04 0:01 0:00 0:01 

cW = 0:00 0:03 0:02 0:01 0:02 0:00 0:03 0:06 0:16 0:06 0 

2 

bA = 

6 

4 

0:14 

0:06 0:05 

0:05 0:11 

0:05 

0:02 0:07 

0:05 

0:06 

0:03 

3 

7 

5 

2 

bG = 

6 

4 

0:81 

0:85 

0:79 

0:45 

0:48 

0:81 0:65 

0:39 

0:51 

0:41 

0:91 

3 

7 

5 

2 

bR = 

6 

4 

1 

0:60 1 

0:84 0:57 1 

0:93 0:60 0:84 1 

0:88 0:59 0:79 0:91 1 

0:84 0:60 0:77 0:86 0:80 1 

0:83 0:59 0:77 0:87 0:84 0:79 1 

0:86 0:60 0:79 0:89 0:86 0:81 0:84 1 

0:80 0:59 0:73 0:82 0:80 0:76 0:76 0:78 1 

0:83 0:57 0:76 0:85 0:80 0:80 0:78 0:79 0:75 1 

3 

7 

5 

The standard errors of the parameter estimators are calculated analytically following 

Hafner and Herwartz (2008). The VAR-ECCC-GARCH model could help us to 

29

60 

50 

40 

30 

AEX 

ATX 

BEL20 

CAC40 

DAX 

FTSE100 

IGBM 

MIBTEL 

OMXS 

SMI 

20 

10 

0 

01/08/02 08/30/05 04/30/09 

Figure 9: Estimated conditional volatilities of di¤erent time series over the period 

considered. 

answer our question given that in this model interactions among volatilities of di¤erent 

markets are allowed through parameters in matrices A and G. The estimated parameters 

suggest that, if there is a leading market in Europe, the …rst candidate would be 

the stock market index BEL-20 in Brussels followed by the FTSE-100 in London. When 

estimating the volatility of the 10 European stock markets, the signi…cant coe¢ cients 

reported in b A and b G show that previous day returns squared of the BEL-20 index a¤ect 

in a signi…cant way the volatility of AEX in Amsterdam and FTSE-100. Moreover, 

previous day volatility of the BEL-20 index a¤ect in a signi…cant way the volatility of 

all the other markets with the exception of AEX, ATX and DAX. On the other hand, 

the volatility of the BEL-20 index depends on its previous day volatility and squared 

returns together with previous day squared returns of the FTSE-100 index. 

6 Conclusions 

In this paper we have performed several Monte Carlo experiments to study how much 

e¢ ciency is lost in small samples when estimating Vector Autoregressive Multivariate 

Conditional Correlation GARCH models in multiple steps. The estimation of the correlation 

parameters could be separated from the estimation of the mean and variance 

parameters, which we refer to as two-steps estimation. Also, the mean parameters 

could be estimated in a …rst step, before doing this two-step estimation and we refer to 

this procedure as three-steps estimation. Although one-step estimators are preferable 

30

ecause of their theoretical properties, they are not always feasible and therefore, estimating 

the parameters of a model in multiple steps could be a reasonable alternative. 

To see how they perform in practice, we start by running several MC experiments using 

a CCC-GARCH model with di¤erent model structures, parameters values or number of 

series. Our results show that, when the distribution of the errors is gaussian, multiple 

steps estimators have a very good performance even in small samples. 

When other models are considered, namely ECCC and DCC-GARCH models, we 

compute kernel density estimates of the in-sample volatility and correlation predictions 

obtained from one-step and multiple steps estimators, in order to compare the 

performance of such estimators when …tting volatilities and correlations. If errors follow 

a gaussian (Student-t) distribution and estimation is based on the true gaussian 

(Student-t) distribution, we …nd that the volatility and correlation predictions obtained 

with di¤erent estimators are very similar. However, in the case of cDCC and RSDC- 

GARCH models with Student-t errors, we …nd that two-steps estimators outperform 

three-steps estimators which seem to be biased even when the sample size increases. 

Finally, our results also show that the performance of multiple step estimators in 

the models considered is robust to the error distribution. In other words, if the true 

error distribution is Student-t but estimation is based on the gaussian distribution, 

kernel density estimates of the in-sample volatility and correlation predictors obtained 

from one-step and multiple steps estimators are quite similar. Analogously, if the true 

error distribution is gaussian but estimation is based on the Student-t distribution, the 

density estimates obtained are very similar and in this case, also similar to the ones 

obtained when the true and assumed distributions are both gaussian. This is due to 

the fact that the Student-t distribution can approximate the gaussian distribution for 

large values of the degrees of freedom parameter. 

Acknowledgements 

Financial support from projects ECO2008-05721/ECON by the Spanish Government 

and IVIE9-09I by the Instituto Valenciano de Investigaciones Económicas is gratefully 

acknowledged. 

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34

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