Working Paper 01-12 Departamento de Estadística y Econometría Statistics and Econometrics Series 08 Universidad Carlos III de Madrid January 2001 Calle Madrid, 126 28903 Getafe (Spain) Fax (34-91) 624-9849

PROPERTIES OF THE SAMPLE AUTOCORRELATIONS IN AUTOREGRESSIVE STOCHASTIC VOLATILITY MODELS

Ana Pérez and Esther Ruiz* Abstract __________________________________________________________________ Time series generated by Stochastic Volatility (SV) processes are uncorrelated although not independent. This has consequences on the properties of the sample autocorrelations. In this paper, we analyse the asymptotic and finite sample properties of the correlogram of series generated by SV processes. It is shown that the usual uncorrelatedness tests could be misleading. The properties of the correlogram of the log-squared series, often used as a diagnostic of conditional heteroscedasticity, are also analysed. It is proven that the more persistent and the larger the variance of volatility, the larger the negative bias of the sample autocorrelations of that series. ________________________________________________________________________ Keywords: Heteroscedastic time series; correlogram; non-linear transformations; asymptotic distribution. *Pérez, Department of Applied Economics (Statistics and Econometrics), Universidad de Valladolid, Spain; Ruiz, Departamento de Estadística y Econometría, Universidad Carlos III de Madrid. C/ Madrid, 126 28903 Madrid. Spain. Ph: 34-91-624.98.51, Fax: 34-91-624.98.49, e-mail: ortega@est-econ.uc3m.es. Financial support from projects SEC97-1379 (CICYT) and PB98-0026 (DGICYT) by the Spanish Government is acknowledged. This research was started while Ana Pérez was visiting the Department of Statistics and Econometrics (Universidad Carlos III de Madrid). She thanks the department for its hospitality. We are also grateful to A. Maravall, F. Marmol, M. Regúlez and S. Velilla for their helpful comments and suggestions. Any remaining errors are our own.

Properties of the Sample Autocorrelations in AutoRegressive

Stochastic Volatility Models

Ana Pérez

Department of Applied Economics (Statistics and Econometrics), Universidad de Valladolid, Spain,

Avda. Valle Esgueva 6, 47150 Valladolid, Spain; email: perezesp@eaee.uva.es.

Esther Ruiz

Department of Statistics and Econometrics, Universidad Carlos III de Madrid, Spain,

c/ Madrid 126, 28903 Getafe, Spain; email: ortega@est-econ.uc3m.es.

Abstract: Time series generated by Stochastic Volatility (SV) processes are uncorrelated

although not independent. This has consequences on the properties of the sample

autocorrelations. In this paper, we analyse the asymptotic and finite sample properties of

the correlogram of series generated by SV processes. It is shown that the usual

uncorrelatedness tests could be misleading. The properties of the correlogram of the log-

squared series, often used as a diagnostic of conditional heteroscedasticity, are also

analysed. It is proven that the more persistent and the larger the variance of volatility, the

larger the negative bias of the sample autocorrelations of that series.

Keywords: Heteroscedastic time series, correlogram, non-linear transformations,

asymptotic distribution.

Classification code: C22

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1. INTRODUCTION

High frequency financial time series are often characterised by being serially

uncorrelated although not independent. They exhibit time varying volatilities that imply

autocorrelated squared observations. The AutoRegressive Stochastic Volatility (ARSV)

model was proposed by Taylor (1986) to represent the dynamic evolution of volatility

which is specified as an unobservable process, σt, the logarithm of which is modelled as a

linear autoregressive process. Stochastic Volatility (SV) processes have the advantage of

being very flexible to represent the stochastic properties of real time series with

conditional heteroscedasticity; see Ghysels, Harvey and Renault (1996) for a detailed

review. Moreover, their dynamic properties are easily obtained from the properties of the

process generating the variance component. ARSV processes generate non-Gaussian and

uncorrelated series. However, they imply correlations in the squared observations with a

structure similar to an ARMA(1,1) process; Taylor (1986, p.75). Therefore, knowledge of

the properties of the correlogram is important for modelling temporal dependence in

volatility. In this paper, we investigate the properties of the sample autocorrelations of

time series generated by ARSV processes. We also consider the properties of the

correlogram of the log-squared observations, that is often used to test for the presence of

time varying volatility.

The sample autocorrelations of linear time series are known to be downward biased

estimators of their population counterparts. Fuller (1996, chap.6) proposes approximated

formulas to compute this bias in finite samples. Regarding the asymptotic distribution of

the sample autocorrelations, Anderson and Walker (1964) derive the main result for

linear models with independent and identically distributed (I.I.D.) disturbances. Hannan

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and Heyde (1972) and Anderson (1992) generalise this result to the case where the

disturbances are a martingale difference. Further results can be found in Anderson (1971,

chap.8) and Brockwell and Davies (1991, chap.7).

With respect to heteroscedastic time series, Milhoj (1985) studies the effect of

conditional heteroscedasticity on the asymptotic properties of the correlogram of series

generated by AutoRegressive Conditionally Heteroscedasticity (ARCH) processes. Based

upon these results, Diebold (1988) shows that the usual Barlett confidence bands and the

Box-Pierce test can be misleading for such processes. Bollerslev (1988) and He and

Terasvirta (1999) also show, by Monte Carlo experiments, that in GARCH (Generalized

ARCH) processes, the correlogram of the squared series exhibits severe downwards bias.

The behaviour of the correlogram of series generated by SV processes has not

received much attention in the literature. Taylor (1986, chap. 5) proposes an estimate of

the variance of the sample autocorrelations for ARSV processes and suggests that using

the standard methodology for those processes will make the usual uncorrelatedness tests

unreliable. The objective of this paper is to analyse the asymptotic and finite sample

properties of the sample autocorrelations of series generated by the ARSV process and

evaluate their consequences on the tests involving these autocorrelations. We will also

derive the properties of the correlogram of the log-squared observations.

The outline of the paper is as follows. In section 2, the population moments and

correlation structure of ARSV processes are presented. In section 3, we analyse the

asymptotic and finite sample properties of the correlogram of series generated by ARSV

processes and the properties of the usual tests for uncorrelatedness. We prove that the

asymptotic distribution is not the usual one and we propose a correction for volatility. We

also show, by means of Monte Carlo experiments, that this correction works

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appropriately in finite samples. In section 4, the asymptotic distributions of the sample

autocovariances and the sample autocorrelations of log-squared observations are

established. The asymptotic results are accompanied with the results of a simulation study

to assess their small sample performance. Section 5 includes the empirical analysis of a

series of daily returns of the Spanish Stock Market index IBEX-35. Finally, section 6

contains a few concluding remarks.

2. STATISTICAL PROPERTIES OF SV PROCESSES

SV processes represent the series of interest, yt, as the product of a sequence of I.I.D.

random variables, εt, and a non-observable stochastic process, σt, called volatility. In the

simplest set up, the logarithm of the volatility, ht=log( 2tσ ), is generated by an

autoregressive process of order one. The resultant SV process, called ARSV, is given by:

yt = σ∗ σt εt , σt = exp(ht/2), (1a)

ht = φ ht-1 + ηt (1b)

where σ* is a scale factor that removes the need of including a constant term in equation

(1b), φ is an unknown parameter that satisfies the condition |φ|<1, εt is an independent

Gaussian process with zero mean and unit variance, i.e., εt is NID(0,1), and ηt is

NID(0, 2ησ ), generated independently of εs for all t,s. Although the assumption of

Gaussianity for ηt can seem ad hoc at first sight, Andersen, Bollerslev, Diebold and

Labys (1999) show that the log-volatility distribution can be well approximated by a

Normal distribution. On the other hand, the assumption of Gaussianity for εt is not as

restrictive as it is in GARCH processes. In fact, the ARSV process with εt being Normal

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still allows simultaneously for excess kurtosis in the marginal distribution of yt and low

but persistent autocorrelations in the squares; see Carnero, Peña and Ruiz (2001).

The statistical properties of the basic ARSV process in (1) are given, for example, by

Harvey (1993, chap.8) who shows that yt is a martingale difference. Therefore, E(yt)=0

and Cov(yt, ys)=0 for all t≠s. As εt is always stationary, the condition |φ|<1 is enough for

yt to be stationary with finite variance. Moreover, using the properties of the logNormal

distribution, the second and fourth order moments of yt are obtained as follows:

Var(yt)= 2*σ E( 2

tε )E[exp(ht)]= 2*σ exp( 2

hσ /2) (2)

E( 4ty )= 4

*σ E( 4tε )E[exp(2ht)]=3 4

*σ exp(2 2hσ ) (3)

where 2hσ is the variance of ht, given by )1( 222

h φ−σ=σ η . Therefore, |φ|<1 is also the

condition for the fourth order moment of yt to exist and to be finite.

Although the series yt is uncorrelated, it is not an independent sequence. The

dynamics of the series appear in the squared observations. The autocovariance function of

2ty is given by:

)k(2Yγ = )y,y(Cov 2kt

2t + = )exp( 2

h4* σσ exp[γh(k)]-1, for k≥1, (4)

where γh(k) is the autocovariance function of the process ht given by:

γh(k)= 2hσ φk, for k≥1. (5)

Taylor (1986, pp. 74-75) notes that if 2hσ is small and/or φ is close to one, the

autocorrelation function of 2ty is very close to that of an ARMA(1,1) process.

As Harvey, Ruiz and Shephard (1994) point out, the dynamic properties of the ARSV

process appear more clearly in the series log( 2ty ). Taking logarithms of squares in (1a)

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gives the following linear representation of the ARSV process:

xt = log( 2ty ) = µ + ht + ξt (6a)

ht = φ ht-1 + ηt (6b)

where µ=log( 2*σ )+E[log( 2

tε )] and ξt=log( 2tε )-E[log( 2

tε )] is a non-Gaussian, zero mean,

white noise process whose properties depend on the distribution of εt. Given that εt is

N(0,1), we have E[log( 2tε )]≅-1.27, 2

ξσ =Var(ξt)=π2/2≅4.93 and E( 4tξ )=3 4

ξσ +π4; see

Abramowitz and Stegun (1970, p. 943).

The series xt in equation (6a) is the sum of two independent processes, a stationary

AR(1) process, ht, plus a non-Gaussian white noise, ξt. This decomposition makes it

easier to obtain the variance, the autocovariance and the autocorrelation function (ACF)

of xt, namely:

γ(0) = Var(xt) = 22h ξσ+σ (7)

γ(k) = Cov(xt, xt+k) = Cov(ht + ξt , ht+k + ξt+k) = γh(k)= 2hσ φk, for k≥1 (8)

ρ(k)=22

h

h )k()0()k(

ξσ+σγ

=γγ =ρh(k)

22h

2h

ξσ+σσ

=φk22

h

2h

ξσ+σσ

, for k≥1 (9)

Expression (9) states that the ACF of xt is proportional to the ACF of ht. Therefore, the

shape of the latter is carried over to the former, except by the factor of proportionality

that is less than one.

From (6), it is possible to obtain the following alternative expression of xt:

(1-φL)(xt-µ) = ηt + ξt - φξt-1

From this representation, it is immediate to show that the ACF of (1-φL)(xt-µ) exhibits

the cut-off at lag one characteristic of the MA(1) process. Therefore, the reduced form of

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xt is an ARMA(1,1) process given by:

(1-φL)(xt-µ) = zt + θ zt-1 (10)

where φ

φ−φ+++φ++−=θ

24)1q()1q( 2222

with q= 22 / ξη σσ , (φ,θ) are constrained to be

in 0<φ<1, -1<θ<0 or -1<φ<0, 0<θ<1 and E( 2tz )=−(φ/θ) 2

ξσ . Nerlove, Grether and

Carvalho (1979, p.73) point out that as ξt is not Gaussian, the disturbance zt of model (10) is

uncorrelated but not independent. As we will see later, this will have important implications

for the asymptotic properties of the sample autocorrelations of the series xt.

The fourth-order moment of xt can also be obtained from equation (6a) as follows:

E(xt -µ)4 = E( ht + ξt )4 = )(E6)h(E 4t

22h

4t ξ+σσ+ ξ =3[γ(0)]2 + π4 (11)

where γ(0) is the variance of xt in (7). Therefore, the fourth-order moment of xt exists and

is finite if |φ|<1.

We now turn to the evaluation of the fourth-order cumulant of xt, denoted by κx(s,r,q).

Taking into account that xt is the sum of two mutually independent processes, ht and ξt, it

turns out that:

κx(s,r,q) = κh(s,r,q) + κξ(s,r,q) (12)

where κh(s,r,q) and κξ(s,r,q) are the fourth-order cumulants of ht and ξt, respectively; see

Brillinger (1981, p.19). Since ht is a stationary zero-mean Gaussian process, all their

fourth-order cumulants are zero (Anderson 1971, p. 444). Moreover, since ξt is an

independent sequence, all their fourth-order cumulants are also zero, except

κξ(0,0,0)=E( 4tξ )-3 4

ξσ =π4. Therefore, all the fourth-order cumulants of xt in (12) are zero,

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except κx(0,0,0) = π4.

Finally, the sixth order moment of xt can be derived as:

E(xt -µ)6 =E( 6th )+15 2

ξσ E( 4th )+15 2

hσ E( 4tξ )+E( 6

tξ ) (13)

where )h(E 6t =15 6

hσ . As εt is N(0,1), then )(E 6tξ =10µ )(E 4

tξ +15 2ξσ π4+10[ )(E 3

tξ ]2+8π6,

where µ=ψ(1/2)-ln(1/2) and )(E 3tξ =-14ζ(3), where ψ(·) is the Euler psi function and ζ(·)

is the Riemman zeta function; see Abramowitz and Stegun (1970, pp. 258, 807-810, 943).

Therefore, all the moments in (13) are finite, and so is the sixth order moment of xt.

3. TESTING FOR SERIAL CORRELATION IN ARSV SERIES

Consider a zero-mean I.I.D. time series, ytt=1,…,T, with constant variance. It can be

shown that the sample autocorrelations of yt given by

rY(k)= ∑∑=+=

− −−−T

1t

2t

T

1ktktt )yy()Yy)(yy( (14)

where y = ∑=

T

1tty

T1 , are asymptotically independent and normally distributed with zero

mean and variance 1/T; see, for example, Brockwell and Davies (1991, pp. 222-223).

This result leads to the usual tests for zero autocorrelation in yt, namely, the so-called

Barlett 95% confidence bands, ±1.96/ T , and the Box-Pierce statistic, Q(K), defined as:

Q(K)=T∑=

K

1j

2Y ])j(r[ (15)

Under the null, H0:ρY(1)=…=ρY(K)=0, Q(K) is asymptotically distributed as a χ2 with K

degrees of freedom.

In models for heteroscedastic time series, the variables yt are uncorrelated but

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dependent, and so the results above do not hold. In this section, we study the properties of

rY(k) and Q(K) in the basic ARSV process defined in (1).

3.1 Asymptotic Properties of the Correlogram of yt

In this subsection we show that the sample autocorrelations of yt in ARSV processes,

are asymptotically normally distributed with zero mean and variance greater than 1/T.

This could lead to errors in the usual tests for uncorrelatedness that are based on these

autocorrelations. Appropriate SV-corrected tests are then proposed.

Since the expected value of yt in ARSV processes is zero, we consider the following

sample autocorrelations, )k(r~Y = )k(c~Y / )0(c~Y , where )k(c~Y = ∑+=

−

T

1ktktt yy

T1 .

Proposition 1. Consider the stationary ARSV process in (1) with |φ|<1. The sample

autocorrelations )k(r~Y are asymptotically independent and normally distributed with zero

mean and variance given by

C(T,k) = T1

expγh(k) (16)

where γh(k) is the autocovariance function of ht given in (5).

The proof is in Appendix A. Notice that the only assumptions on εt needed in the

proof are the symmetry and finite fourth order moment of the distribution of εt. However,

as it is often the case that, in empirical applications it is assumed that εt is Gaussian, in

this paper we give the results for this particular case.

Proposition 1 confirms the result in Taylor (1986, pp.120-122), who proves that the

estimate of the variance of the sample autocorrelations that he proposes, converges to

expression (16) in series generated by ARSV processes.

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If the mean of yt is unknown and is estimated with its sample mean, the sample

autocorrelation rY(k), defined in (14), is used instead of )k(r~Y . Notice that as yt is a

stationary, uncorrelated zero-mean time series, it can easily be shown, from theorem 1 in

Hannan and Heyde (1972), that the sample mean, y , converges almost sure to the

population mean. So the asymptotic behaviour of rY(k) and )k(r~Y is the same.

Proposition 1 implies that the usual Barlett 95% confidence bands, ±1.96/ T , are

inappropriate for ARSV processes, and leads directly to the SV-corrected 95% bands

given by:

±1.96[C(T,k)]1/2 = ±T96.1 exp0.5γh(k)= ±

T96.1 exp0.5 φk

2

2

1 φ−ση (17)

It is clear that these bands are different for each lag. Moreover, if φ>0, the Barlett bands

are “too narrow” and can erroneously reject the hypothesis of uncorrelation. Notice also

that the SV-correction factor is bigger, the bigger φ and 2ησ and the smaller k.

Furthermore, note that:

T1)k(exp

T1lim)k,T(Clim h

kk=γ=

∞→∞→

since γh(k)→0 as k→∞, and therefore, for large values of k, the Barlett confidence bands

will be quite close to the SV-corrected bands. As an illustration, figure 1 displays the

Barlett and the SV-corrected 95% bands for ARSV processes with φ=0.98 and 2ησ =0.1,

0.05, 0.01. Clearly, the divergence is different at each lag and the largest divergence

occurs at the low-ordered autocorrelations and in the processes with higher variance

( 2ησ =0.1). However, the differences become progressively smaller when the lag order

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gets larger and/or when the value of 2ησ becomes smaller.

The size of the uncorrelatedness test that uses the Barlett 95% bands in ARSV series,

may be calculated as follows:

p(ERROR I)=0H

p (|rY(k)|≥1.96/ T ) ≅ 2 [1-Φ(1.96 exp-0.5γh(k))]

where Φ denotes the cumulative distribution function of a standard normal variable.

Therefore, if φ>0, and consequently, γh(k)>0, as it is usual in real data, the probability of

type-I error is greater than 0.05. Table 1 shows the values of these probabilities for ARSV

processes with 2ησ =0.1, 0.05, 0.01 and φ=0.5, 0.9, 0.95, 0.98. In this table, it is

possible to observe that in ARSV processes with small values of φ, the error with the

uncorrected bands is negligible, but in those processes which are closer to the unit root,

implying more persistence of the volatility process, and in those processes with higher

variance, the error is quite dramatic. In these cases, the correction for volatility is

essential to avoid detecting spurious autocorrelations in yt.

In practice, the true parameter values are unknown and the variance of rY(k) in (16)

must be estimated from the data. As ht is unobservable, the estimate is constructed from

expression (A.4) in Appendix A, as

C*(T,k)=

γγ+

2Y

Y

])0(ˆ[)k(ˆ

1T1 2 (18)

where ∑=

=γT

1t

2tY y

T1)0(ˆ and )k(ˆ 2Yγ = ∑

+=− γ−γ−

T

1ktY

2ktY

2t ))0(ˆy))(0(ˆy(

T1 . From (18), the

following sample SV-corrected 95% bands are obtained:

±1.96[C*(T,k)]1/2 (19)

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Notice that once we use sample moments to construct the SV-corrected bands in (19),

they become the same as those proposed by Diebold (1988) for ARCH processes.

We now turn to the analysis of the Box-Pierce statistic defined in (15). This should

also be modified in order to keep the nominal size of the correspondent uncorrelatedness

test. From proposition 1, the following correction turns out:

Qc(K) = ∑=

K

1j

2Y

)j,T(C])j(r[ = T 2Y

K

1jh ])j(r[)j(exp∑

=

γ− . (20)

In practice, the asymptotic variance of rY(k) must be estimated from the data and the

following sample SV-corrected version of the statistic Q(K) is obtained:

)K(Q*c = 2Y

K

1j Y2

Y

2Y

K

1j

2Y )]j(r[

)j(ˆ)]0(ˆ[)]0(ˆ[T

)j,T(C)]j(r[

2* ∑∑

==

γ+γγ= . (21)

3.2 Finite Sample Properties of the Tests for Uncorrelatedness in yt

To analyse the finite sample behaviour of the proposed tests for uncorrelatedness in

yt, we generate series by ARSV processes with parameters φ=0.9,0.95,0.98,

2ησ =0.1,0.05,0.01, σ*=1 and εt~NID(0,1). Three sample sizes are considered, T=512,

T=1024 and T=4096. For each parameter set and sample size, 5000 independent

replicates are generated, and for each replicate, the sample autocorrelations of yt, up to

order 50, are calculated, along with the Box-Pierce statistic in (15) and the SV-corrected

Box-Pierce statistics in (20) and (21). All the simulations have been carried out using

GAUSS version 3.2 on a Pentium 166 MHz. The Gaussian noise, εt, and the AR(1)

process, ht, are generated with commands RNDNS and RECSERAR, respectively.

Table 2 displays the empirical size of the test for uncorrelatedness of order k in yt,

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for k=1,10,50, using the Barlett bands and the SV-corrected bands. The proportion of

rejections with the Barlett 95% bands is denoted as P, while Pc and *cP denote the

proportion of rejections with the theoretically SV-corrected and the sample SV-corrected

bands in (17) and (19), respectively. In this table, it is possible to observe that, the bigger

φ and/or 2ησ , the bigger the empirical size of the Barlett confidence bands. For instance,

when φ=0.98, 2ησ =0.05 and k=10, the empirical size of the test in a sample of size

T=512 is more than twice the nominal 5% and it becomes four times bigger when

T=4096. On the other hand, the empirical size of the sample SV-corrected bands, *cP , is

always quite close to the nominal 5%, even in the more persistent cases and with the

smaller sample size. Note also that in the nearly unit root case (φ=0.98), the empirical

size of the theoretical SV-corrected bands, Pc, is always below the nominal 5%, even with

sample size T=4096. This could be due to the fact that the process with φ=0.98 is so close

to being non-stationary that the convergence to the asymptotic distribution in Proposition

1 is very slow and huge sample sizes are needed to achieve reasonable results. In table 2

we also observe that the problem of detecting spurious autocorrelations with the

uncorrected Barlett bands becomes less serious in higher-ordered lags, as predicted by the

asymptotic theory. Finally, it is also worth noting that the values of P in table 2

approximately equal the probabilities of type-I error displayed in table 1, obtained using

the asymptotic distribution of rY(k).

As an illustration, figure 2 represents histograms of the empirical distribution of

T rY(k), rY(k)/[C(T,k)]1/2 and rY(k)/[C*(T,k)]1/2 together with the standard Normal

density, for k=1,10,50, in an ARSV process with φ=0.98, 2ησ =0.05 and T=512, 4096.

This figure shows up clearly that the empirical size of T rY(k) is larger than the nominal

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5%, especially in the low-order lags, while the asymptotic distribution seems to provide

an adequate approximation to the empirical distribution of rY(k)/[C*(T,k)]1/2. On the other

hand, it is also possible to observe that the theoretically SV-corrected distribution is not a

good approximation in small samples when φ=0.98. However, as T increases the

approximation becomes much better.

Table 3 shows, for the same ARSV processes considered in table 2, the empirical sizes

obtained with the Box-Pierce statistic in (15) and the two SV-corrected statistics in (20)

and (21) when the null hypothesis is H0:ρY(1)=…=ρY(K)=0 for K=10 and 50. These

values, denoted by Q, Qc and *cQ , respectively, are calculated as the proportion of

rejections (out of the 5000 replications) with respect to the 95% percentile of the 2Kχ

distribution. In this table, it is possible to see that the sample SV-corrected statistic does a

good job keeping the nominal size for all the parameter values and lags considered. The

largest deviations from nominal size with the Box-Pierce statistic, Q(K), are always

associated with the more persistent processes. Furthermore, in these cases, the SV-

corrected statistic Qc(K) gives worse results than *cQ (K), especially in the smaller

samples. As we said before, this could be due to the fact that the convergence to the

asymptotic distribution is very slow when φ is close to one.

As an illustration, figure 3 represents histograms of the empirical distribution of the

Box-Pierce statistic, Q(K), and the SV-corrected statistics Qc(K) and *cQ (K), together

with the 2Kχ density, for K=10,50, in an ARSV process with φ=0.98, 2

ησ =0.05 and

T=4096. In this figure, it is possible to observe that the 2Kχ distribution is completely

inappropriate for Q(K) and it is not a good approximation to the empirical distribution of

Qc(K) either. However, the 2Kχ distribution fits very well the empirical distribution of the

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sample SV-corrected statistic *cQ (K).

4. SAMPLE AUTOCORRELATIONS OF LOG-SQUARED OBSERVATIONS

As we mentioned before, one of the main tools to identify the presence of stochastic

volatility in a given time series is the correlogram of the log-squared observations. In this

section, the asymptotic distribution of the sample autocovariances and the sample

autocorrelations of the series xt=log( 2ty ) in the ARSV process are analysed. Though the

sample autocovariances themselves have a limited use in model identification, its

distribution is essential to derive the asymptotic properties of the sample autocorrelations.

Finally, the behaviour of the sample autocorrelations in finite samples is also assessed

through a Monte Carlo experiment.

Unless otherwise stated, the following definitions of the sample autocovariance and

the sample autocorrelation of xt will be used:

c(k) = ∑+=

− −−T

1ktktt )xx)(xx(

T1

r(k) = ∑∑=+=

− −−−=T

1t

2t

T

1ktktt )xx()xx)(xx()0(c)k(c

where x is the sample mean across all log-squared observations.

4.1 Asymptotic Distribution of the Sample Autocovariances of log( 2ty )

Proposition 2. In the stationary ARSV process defined in (6), the asymptotic

distribution of T c(0)-γ(0),c(1)-γ(1),…,c(k)-γ(k)’ is multivariate Normal with zero

mean and variance-covariance matrix V, whose elements, viji,j=0,1,…,k, are given by:

15

vij = ])jiu()u()jiu()u([u

hhhh∑∞

−∞=++γγ+−+γγ +2 2

ξσ [γh(i-j)+γh(i+j)]+ 4ξσ Ii=j+( 4

ξσ +π4)Ii=j=0

(22)

where γh(k) is the autocovariance function of the process ht, Ii=j is an indicator function

defined as Ii=j=1 if i=j, Ii=j=0 if i≠j, and Ii=j=0 is an indicator function defined similarly.

The proof is in Appendix B.

Corollary 1. In the stationary ARSV process defined in (6), the asymptotic marginal

distribution of T (c(k)-γ(k)) is Normal with mean zero and variance given by:

vkk = 2

24h

2

12)0(

φ−φσ+γ +

+φ−φ+σ+σφσ ξ k2

112

2

22h

2k22h .

This expression follows immediately from (22) considering i=j=k and replacing γh(u)

by its value in (5) and working out the summations.

4.2 Asymptotic Distribution of the Sample Autocorrelations of log( 2ty )

Proposition 3. In the stationary ARSV process defined in (6), the asymptotic

distribution of T r(1)-ρ(1),…,r(k)-ρ(k)’ is multivariate Normal with zero mean and

variance-covariance matrix W whose elements, wiji,j=1,…,k, are given by:

wij= [ ]∑∞

−∞=+ρρρ−+ρρρ−ρρρ+++ρρ+−+ρρ

u

2 )iu()u()j(2)ju()u()i(2)u()j()i(2)jiu()u()jiu()u( +π4

)0()j()i(

2γρρ

(23)

where γ(0) and ρ(k) are respectively, the variance and autocorrelation function of xt.

The proof is in Appendix C.

Let us observe the extra term that appears in expression (23) if compared to the usual

16

expression of the asymptotic covariances of r(k) given by Anderson and Walker (1964)

for linear models with I.I.D. disturbances. This extra term accounts for the non-

Gaussianity of the disturbance ξt in equation (6a). As we mentioned in section 2, this

would make the disturbance of the linear representation of xt in (10) to be non-

independent and therefore the conditions in Anderson and Walker (1964) does not hold.

Corollary 2. In the stationary ARSV process defined in (6), the sample

autocorrelation r(k) is asymptotically normally distributed with mean ρ(k) and variance:

−

+

φ−φ+σ+σ

γρ+σ

φ−φ

γρ+

+ ξ k2112

)0()k2(

1)0(])k(21[2

1T1~))k(r(Var

2

22h

24h2

2

2

2

γ

π+ρ+

+φ−φ+σ+σ

γρ− ξ )0(

2)k(k112

)0()k(4

2

42

2

22h

22

(24)

The proof of this corollary follows immediately from Proposition 3 noting that:

Var(r(k))~T

wkk = [ ]

+ρρρ−ρρ++ρρ+ρ∑∞

−∞=u

222 )ku()u()k(4)u()k(2)k2u()u()u(T1 +π4

γρ

)0()k(

2

2 (25)

and replacing in expression (25), ρ(k) by its value in (9) and working out the summations.

An alternative expression of the asymptotic variance of r(k), which will be more

useful in practice, can be obtained, after some little effort, from expression (25), as:

Var(r(k)) ~ −−ρρρ

−

ρ+ρ++ρ ∑∑

=

∞

=

k

0u1u

22 )uk()u()k(4)u(1])k(21[)k2(2T1

γρπ++ρρ+−ρρ++ρρρ− ∑∑∑

∞

=

−

=

∞

= 1u2

24

1k2

1u1u )0()k()k2u()u(2)uk2()u()uk()u()k(8 (26)

In practice, the true values of the ACF, ρ(k), are unknown and the variance of r(k) can be

17

estimated from the data replacing in (26) ρ(k) by the corresponding sample

autocorrelation, r(k), and truncating the infinite summations at a sufficiently large lag.

Expression (26) can be used to build up confidence bands to test for uncorrelation in

the series xt. Under the null hypothesis H0:ρ(k)=0 for any k>q, the confidence bands

obtained for the log-squared observations are the same as those obtained in linear models

with I.I.D. disturbances; see Brockwell and Davies (1991, pp. 223-224). Though this

result seems to be quite unexpected, that is not the case if we observe that the additional

term in (25), π4

)0()k(

2

2

γρ , vanishes when we compute Var(r(k)) under the null H0:ρ(k)=0 for

any k>q. Therefore, it turns out that, under the null, expression (25) becomes the classical

expression of Var(r(k)) in Anderson and Walker (1964).

Another important consequence of Proposition 3 is that the sample autocorrelations of

the series xt in ARSV processes are not asymptotically independent, because the non-

diagonal elements of their covariance matrix W=wij, defined in (23), are not zero.

Therefore, the asymptotic distribution of the Box-Pierce statistic to test for joint

uncorrelatedness in the series xt, would not be a χ2 distribution with K degrees of

freedom. In consequence, using the critical values of such distribution could result in a

probability of Type-I error different than the nominal level.

4.3 Finite Sample Properties of the Sample Autocorrelations of log( 2ty )

In this subsection we present a simulation study to assess the finite sample properties

of the sample autocorrelations of the series xt. The Monte Carlo design is the same as in

section 3.

Table 4 reports the sample bias and standard deviation of r(k), for k=1,10,50, together

18

with the values of the theoretical finite sample bias and the asymptotic standard deviation

derived from (24), for some selected cases that are representative of the overall results.

Table 4 also displays the values of the true autocorrelation function ρ(k) of the series xt

calculated with formula (9). The theoretical bias of r(k) for the ARSV process has been

derived in Pérez (2000), following the results in Fuller (1996, chap. 6), and is given by

the following expression:

E(r(k))-ρ(k)= −γ

πρ+

ρ−+ρ−−

ρ−++−ρ− ∑∑

−

=

−

= )0(T)k()j()

Tj1(21

T)k(1)j()

Tj1(42k

T)k(

2

41T

1j

1T

1j

2

)T(O)jk()j()kT()jk()j()jkT(2T2 2

k

0j

1kT

1j2

−

=

−−

=+

−ρρ−++ρρ−−− ∑∑ (27)

Replacing in this equation ρ(k) by its value in (9) and working out the summations, the

bias of r(k) can be written, as a function of the parameters of the model, as follows:

E(r(k))-ρ(k)=

= )T(O1)0(T

)k(4)0(T

)k(k1

21)0(T

)k(2)1k(12

)0(1

T1 2

2

2

2

4h

2

42hk

2h −+

φ−φ

γσρ

+γπρ

+

−

φ+φ+

γσρ

+

φ++

φ−φ

γσ

+−

(28)

The main conclusion emerging from the results in table 4 is that the bias of r(k) is

always negative, so the correlogram of xt always underestimates the corresponding true

ACF. However, the bias becomes negligible in large samples. Another important

conclusion is that both the bias and variance of r(k) increase with φ and 2ησ . Therefore,

the more persistent the volatility and the greater its variance, the worse the sample

autocorrelation estimates the true ACF. On the other hand, table 4 shows that the

asymptotic distribution provides an adequate approximation to the finite sample bias and

19

variance of r(k). However, in the more persistent cases (φ=0.98), the ARSV process is so

close to being non-stationary that the asymptotic approximation in proposition 5 is not

appropriate in small samples. In fact, the largest differences between the asymptotic and

the empirical values always arise in these cases. Finally, from table 4 it is also clear that

the usual value 1/ T underestimates the standard deviation of the sample

autocorrelations of xt, except on those models with 2ησ =0.01.

As an illustration of the previous results, figure 4 displays the mean correlogram across

replications, )k(r , for k=1,2,…,50, together with the corresponding values of the ACF of

the series xt, for ARSV processes with φ=0.98. From left to right and top to bottom, the

nine correlograms displayed in this figure correspond to T=512, T=1024 and T=4096,

and to 2ησ =0.1, 2

ησ =0.05 and 2ησ =0.01, respectively. This figure clearly illustrates that

there are important biases in the correlogram when the variance is high ( 2ησ =0.1) and T is

small (T=512). However, the ACF fits quite well the mean correlogram if 2ησ =0.01

and/or T=4096. This sample size is not unusual in financial time series and so in

empirical applications with real data, the bias of r(k) would be negligible. Note also that

the ARSV process with 2ησ =0.01 stands for a situation very close to an homocedastic

white noise, where the autocorrelations themselves are nearly zero and so is their bias.

To complete the analysis of the sample autocorrelations of the series xt, figure 5

displays the empirical distribution across the 5000 replications of T [r(k)-ρ(k)], [r(k)-

ρ(k)]/[V1(T,k)]1/2 and [r(k)-ρ(k)]/[V2(T,k)]1/2, for k=1,10,50, where V1(T,k) denotes the

asymptotic variance of r(k) in (24), calculated with the true parameter values, and V2(T,k)

denotes the estimated asymptotic variance, computed from expression (26) replacing ρ(k)

by its corresponding sample counterpart, r(k). The model considered is an ARSV process

20

with parameters φ=0.98, 2ησ =0.05 and sample sizes T=512 and T=4096. The standard

Normal density is also drawn over all the empirical distributions in order to check the

adequacy of the asymptotic theory developed in section 4.2.

Several important conclusions emerge from figure 5. First, we can see that in

ARSV processes, the usual asymptotic variance, 1/T, is completely inappropriate for

sample autocorrelations of the log-squared observations. Secondly, figure 5 confirms that

the bias of the sample autocorrelations of the series xt is quite important when T=512.

However, if the sample size increases, the asymptotic distribution of the standardized

sample autocorrelations using the asymptotic value of Var(r(k)) in (24), becomes a good

approximation to the empirical distribution in finite samples. On the other hand, the

standard Normal density does not seem to provide an adequate approximation to the

standardized distribution with the estimated asymptotic variance.

The approximation to the asymptotic distribution of r(k) in small samples can be

further improved by correcting its systematic bias. Figure 6 displays the bias-corrected

standardized distribution of r(k), [r(k)-ρ(k)-bias(r(k))]/[V1(T,k)]1/2, for k=1,10,50, in the

ARSV process with φ=0.98, 2ησ =0.05 and with sample sizes T=512 and T=4096. The

bias of r(k) has been calculated using formula (28). This figure shows up clearly that the

correction for bias is essential when the sample size is small and it is also recommended

for larger sample sizes, where the improvement is quite remarkable. In practice, the true

parameter values are unknown, and both the bias and the variance of r(k) must be

estimated from the data. In this case, the bias of r(k) should be obtained from expression

(27) replacing ρ(k) by its corresponding sample counterpart, r(k). However, we have

checked that the distribution of r(k) standardized with the estimated bias and variance

21

does not provide good results in finite samples. In these cases, the bootstrap methodology

proposed by Romano and Thombs (1996) would be a good alternative to approximate the

empirical distribution of r(k).

Finally, we carry out some Monte Carlo experiments to asses the finite sample

behaviour of the Box-Pierce statistic for the series xt. Our results show that the χ2

distribution with K degrees of freedom is not a good approximation to the sample

distribution of the Box-Pierce statistic even after correcting the sample autocorrelations

by their asymptotic variance. Notice that, as we mentioned before, the sample

autocorrelations of xt are highly correlated and, therefore, the dependency between the

estimated autocorrelations should be taken into account when computing the Box-Pierce

statistic. To illustrate the problem, Figure 7 displays several scatter plots with sample

autocorrelations of xt of order 1, 10 and 50, obtained in the 5000 replicates. These plots

show clearly that both sample autocorrelations are highly positively correlated.

The results in this section can explain why, in practice, it is inadequate to use the

sample autocorrelations of log-squared observations to identify the model to be used to

represent the dynamic evolution of volatility.

5. EMPIRICAL APPLICATION TO THE IBEX-35 INDEX

Figure 8(a) plots daily returns of the IBEX-35 index of the Madrid Stock Market

observed from 7/1/87 to 30/12/98 (2991 observations). This series, denoted by yt, is the

result of filtering the original series of returns to remove a small autocorrelation of order

one and the effect of two outliers, the mini-crash in Wall Street (13/10/89) and the

kidnapping of Gorbachev (19/8/91). The series yt moves randomly around a constant zero

mean while the volatility evolves over time, with periods of high volatility followed by

22

periods of low changes. Moreover, the kurtosis of the series is 8.321, indicating a fat tail

marginal distribution.

Table 5 displays sample autocorrelations of yt, for some selected lags. Following the

results in section 3, table 5 displays the SV-corrected standard deviation of rY(k), obtained

from expression (18), together with the usual value 1/ T for I.I.D. processes. These

values are denoted by [s.e.c] and (s.e.), respectively. We can see that some

autocorrelations that appear to be significant when compared with the 95% Barlett bands,

become not significant when tested against the corrected SV-bands, ±1.96[s.e.c]. This

feature shows up even more clearly in Figure 8(b), where the correlogram of yt up to lag

k=100 is displayed together with both confidence bands.

Finally, table 6 reports the Box-Pierce statistics for joint uncorrelation of yt up to order

K, for K=5,10,20,50,100, together with the corresponding SV-corrected Box-Pierce

statistic, denoted by Q(K) and Qc(K), respectively. Once more, we observe that Q(K) is

significant for K=10,20,50,100. However, when the statistic corrected for volatility,

Qc(K), is used, the hypothesis of uncorrelation in yt is no longer rejected. Therefore, it is

clear that the usual Barlett confidence bands and the Box-Pierce test can be misleading

for processes with time varying volatility.

Regarding the behaviour of the correlations of log-squared returns, table 5 displays the

asymptotic standard deviation of the sample autocorrelations of log( 2ty ) for ARSV

processes, together with the usual value 1/ T for I.I.D. processes. These values are

again denoted by [s.e.c] and (s.e.), respectively. The values of [s.e.c] are obtained from

expression (26), by replacing ρ(k) by its corresponding sample autocorrelation, r(k), and

truncating the infinite summations at a sufficiently large lag. Figure 8(c) displays the

23

correlogram of log( 2ty ) together with the 95% bands obtained recursively under the

assumption of conditional homocedasticity as explained in section 4.2. The evidence for

conditional heteroscedasticity is overwhelming: the first order autocorrelation of log( 2ty )

is 0.131 and the Box-Pierce statistic for up to tenth order serial correlation in this series is

Qlog(10)=609.275, which is highly significant. It is also possible to observe that the series

of log-squared returns have significant autocorrelations even for high lags, with a very

slow decay to zero; see Figure 8(c). Notice that the correlogram of log( 2ty ) may suggest the

presence of long-memory in volatility. However, taking into account that the standard

deviation of the sample autocorrelations is much bigger than the one used to obtain the

confidence bands in Figure 8(c), it is possible that this apparent long memory is spurious.

6. CONCLUSIONS

In this paper we analyse the asymptotic and finite sample properties of the

correlogram of series generated by stationary ARSV processes. First, we derive the

asymptotic distribution of the correlogram. In particular, we show that the asymptotic

variance of the sample autocorrelations is larger than 1/T. Therefore, the usual tests for

uncorrelatedness based on the Barlett bands or the Box-Pierce statistic could lead to

detect spurious autocorrelations in series generated by ARSV processes. This effect is

especially remarkable in the first lags and in the more persistent cases. Appropriate

corrections of both statistics are proposed and it is empirically shown that they provide

highly satisfactory results.

We also derive the asymptotic properties of both the sample autocovariances and

sample autocorrelations of the series log( 2ty ). It is shown that the correlogram of this

series is always downward biased, and the bias increases with the variance and the

24

persistence of the volatility. It is also proven that the variance of the sample

autocorrelations of log( 2ty ) is greater than 1/T. These results can explain why, in practice,

it is inappropriate to use the usual confidence bands for the sample autocorrelations of

log-squared observations to identify the model adequate to represent the dynamic

evolution of the volatility.

The empirical application to the IBEX-35 stock market index illustrates how the usual

Barlett bounds and the Box-Pierce statistic could detect spurious autocorrelations in

heteroscedastic time series, both in the levels and in the log-squared series.

This paper focused on the basic SV process, where the logarithm of the volatility is an

AR(1) process. The results do not extend directly to more general cases. We have some

work in progress on the properties of sample autocorrelations of series generated by SV

processes with long memory, where the logarithm of the volatility is a fractionally

integrated process.

Furthermore, it is possible to use the results on the asymptotic distribution of the

sample autocorrelations of log-squared observations to propose a modification of the

Box-Pierce statistic that takes into account the correlation between these autocorrelations.

We left this correction for further research.

ACKNOWLEDGEMENTS

Financial support from projects SEC97-1379 (CICYT) and PB98-0026

(DGICYT) by the Spanish Government is acknowledged. This research was started while

Ana Pérez was visiting the Department of Statistics and Econometrics (Universidad

Carlos III de Madrid). She thanks the department for its hospitality. We are also grateful

25

to A. Maravall, F. Marmol, M. Regúlez and S. Velilla for their helpful comments and

suggestions. Any remaining errors are our own.

APPENDIX A: PROOF OF PROPOSITION 1

Consider the variable:

)k(c~T Y = ∑+=

−

T

1ktktt yy

T1 = ∑

+=

T

1kttU

T1

where Ut=ytyt-k, with k>0. In the stationary ARSV process, it is easily shown that Ut is a

martingale difference sequence and therefore, Ut has zero mean and is uncorrelated.

Moreover, by the symmetry of εt, E( mm

11

nt

nt y...y )=0 with ti≠tj, if at least one of the

exponents ni is odd (Milhoj 1985). The existence of the fourth moment of yt is ensured

because of the existence of the fourth moment of εt. Therefore:

Cov(Ut,Ut+r)=E(UtUt+r)=

=γ+γ=≠=

−

−++−

0rif)k(])0([)yy(E0rif0)yyyy(E

2Y2

Y2

kt2t

krtrtktt (A.1)

where γY(0) is the variance of yt and )k(2Yγ is the autocovariance function of 2ty . The

corollary 6.1.1.2 in Fuller (1996) yields the ergodicity of Ut. Under these conditions, the

central limit theorem for martingales in Billingsley (1961) applies and consequently:

∑+=

T

1kttU

T1 →d N(0, E( 2

tU ))

where →d denotes convergence in law. Substituting in this expression E( 2tU ) by its

value in (A.1) provides the asymptotic distribution of )k(c~T Y , namely,

)k(c~T Y = ∑+=

T

1kttU

T1 →d N(0, )k(])0([ 2Y

2Y γ+γ )

26

The covariances between the sample autocovariances are calculated by

Cov( )i(c~Y , )j(c~Y )=

= [ ]∑ ∑+= +=

−−−− −−−−++T

1it

T

1jsyjstsitjsitst2

)jts,ts,i()y,y(Cov)y,y(Cov)y,y(Cov)y,y(CovT1 κ

where κy(s,r,q) is the mixed fourth-order cumulant of yt. When i≠j, i>0, j>0, the mixed

cumulants κy(-i,s-t,s-t-j) are zero because all odd moments of yt are zero by the symmetry

of εt (Milhoj 1985). Moreover, as the series yt is serially uncorrelated, it turns out that

Cov( )i(c~Y , )j(c~Y )=0 when i≠j, i>0, j>0.

The asymptotic normality of ])k(c~),...,1(c~[T YY follows by considering arbitrary

linear combinations. Therefore, we have that:

])k(c~),...,1(c~[T YY ’ →d N

γ+γ

γ+γ

)k(])0([0

0)1(])0([,

0

0

2

2

Y2

Y

Y2

Y

L

MOM

L

M (A.2)

Finally, from corollary 6.1.1.1 in Fuller (1996) and since the autocovariance function

of 2ty defined in (4) converges to zero, it is straightforward to prove that:

)0(c~Y = ∑=

T

1t

2ty

T1 →p γY(0) (A.3)

where →p denotes convergence in probability. The conjunction of (A.2) and (A.3)

implies that the sample autocorrelations )k(r~Y = )k(c~Y / )0(c~Y are asymptotically

independent and jointly normally distributed with a marginal distribution given by

)k(r~T Y =)0(c~

)k(c~TY

Y →d N(0,2

Y

Y

])0([)k(1 2

γγ+ )

27

Therefore, the asymptotic variance of )k(r~Y can be approximated in finite samples by

Var( )k(r~Y ) =

γγ

+2

Y

Y

])0([)k(

1T1 2 (A.4)

Now, substituting in (A.4) γY(0) and )k(2Yγ by their values in (2) and (4), respectively,

the required expression of Var( )k(r~Y ) in (16) is obtained.

APPENDIX B: PROOF OF PROPOSITION 2

Consider the vector time series Wt=(ht,ξt)’, where ht and ξt are defined as in the linear

representation of the ARSV process in (6). The variable Wt can be considered as a linear

process such as Wt=∑∞

=−υ

0jjt)j(B , where:

Wt=

ξ t

th, υt=

ξη

t

t , B(j)=

β

β)j(0

0)j(22

11

The elements β11(j), j=0,1,2,…, are the coefficients of the Wold representation of the

process ht in (6b), that is, β11(j)=φj, for every j. The coefficients β22(j) are all equal to

zero, except β22(0)=1. Moreover, since ηt and ξt are uncorrelated themselves and

independent of each other, the first and second order moments of υt are as follows:

E(υt)=

=

ξη

00

)(E)(E

t

t , E(υtυs’)=

σ

σδ=

ξξηξ

ξηηη

ξ

η

2

2

stst

stst

0

0)s,t(

)(E)(E

)(E)(E

where δ(t,s) is the Kronecker delta function, δ(t,s)=1 if t=s, δ(t,s)=0 if t≠s.

Let us define the k-th sample autocovariance and cross-covariance of ht and ξt as:

28

c11(k) = ∑+=

− −−T

1ktktt )hh)(hh(

T1 , c22(k)= ∑

+=− ξ−ξξ−ξ

T

1ktktt ))((

T1 (B.1)

c12(k)= ∑+=

− ξ−ξ−T

1ktktt ))(hh(

T1 , c21(k)= ∑

+=− −ξ−ξ

T

1ktktt )hh)((

T1 (B.2)

respectively, where ∑=

=T

1tth

T1h and ∑

=ξ=ξ

T

1ttT

1 . Similarly, let γ11(k), γ22(k), γ12(k), γ21(k)

be their population counterparts, that is,

γ11(k)=Cov(ht,ht+k)=γh(k), γ22(k)=Cov(ξt,ξt+k)= 2ξσ Ik=0 (B.3)

γ12(k)=Cov(ht,ξt+k)=0, γ21(k)=Cov(ξt,ht+k)=0 (B.4)

where Ik=0 is the indicator function.

In the stationary ARSV process, the spectra of ξt and ht are both square-integrable.

Therefore, the Central Limit Theorem for multivariate linear processes in Hannan (1976)

applies to the series Wt=(ht,ξt)’. This theorem implies that the 4(k+1)x1 vector:

Z= Tc11(0)-γ11(0), c12(0), c21(0), c22(0)-γ22(0),…, c11(k)-γ11(k), c12(k), c21(k), c22(k)’

with k>0, is asymptotically normally distributed with zero mean and a covariance

structure given by the following elements:

∞→Tlim T Cov(cab(i)-γab(i),ccd(j)-γcd(j)) =

= [ ]∑∞

−∞=

++γγ+−+γγu

bcadbdac )jiu()u()jiu()u( + ======

otherwise00ji,2dcbaif2222κ

(B.5)

where i=0,1,…,k; j=0,1,…,k; a=1,2; b=1,2; c=1,2; d=1,2; and κ2222=E( 4tξ )–3[E( 2

tξ )]2=π4.

Putting the values of γ11(k), γ22(k), γ12(k) and γ21(k) in (B.3) and (B.4) back into equation

(B.5), it is seen that the limiting covariances of Z are only different from zero when

a=b=c=d=1, a=c=1,b=d=2, a=c=2,b=d=1, a=d=1,b=c=2, a=d=2,b=c=1 and

29

a=b=c=d=2. Therefore, the 4(k+1)x4(k+1) asymptotic variance-covariance matrix of Z

can be written as a partitioned matrix as follows:

∑∑∑∑=

kkk1k0

k111k0

k00100

AAA

AAAAAA

L

MOMM

L

L

where the 4x4 matrices Aij, for i,j=0,1,…,k, are given by:

Aij =

π+σ+σ−γσ+γσ+γσ−γσ

++γγ+−+γγ

==ξ=ξ

ξξ

ξξ

∞

−∞=∑

0ji44

ji4

h2

h2

h2

h2

uhhhh

I)(I0000)ji()ji(00)ji()ji(0

000])jiu()u()jiu()u([

The asymptotic distribution of the sample autocovariances of xt follows readily from

the distribution of the vector Z. Noting that the series xt in (6a) is the sum of ht and ξt, the

k-th sample autocovariance of xt, say c(k), can be written as:

c(k) = c11(k) + c12(k) + c21(k) + c22(k), for k≥0 (B.6)

where c11(k), c12(k), c21(k), c22(k) are defined in (B.1) and (B.2). From (7) and (8), the

population autocovariance function of xt, γ(k), can be decomposed as:

γ(k) = γh(k) + 2ξσ Ik=0 = γ11(k) + γ22(k) (B.7)

From (B.6) and (B.7) it is easily shown that the (k+1)x1 vector,

U= T c(0)-γ(0), c(1)-γ(1),…, c(k)-γ(k)’

is a linear transformation of the vector Z of the form U=ΩZ, where Ω is the following

(k+1)x4(k+1) matrix,

30

=Ω

111100000000

000011110000000000001111

L

MOMM

L

L

As Z is asymptotically normally distributed with zero mean and variance-covariance

matrix ∑∑∑∑, the variable U=ΩZ converges in law to a multivariate Normal distribution with

mean zero and variance-covariance matrix V=Ω∑∑∑∑Ω’. Working out this product of

matrices, it is seen that V is a (k+1)x(k+1) matrix whose elements vij are given in (22).

APPENDIX C: PROOF OF PROPOSITION 3

Let c(0) and c(k) be the variance and the k-th sample autocovariance of the series xt,

and γ(0) and γ(k) their population counterparts. From proposition 2,

T c(0)-γ(0),c(1)-γ(1),…,c(k)-γ(k)’ →d N(0,V)

where the elements of the matrix V are given in (22). Let g(.) be the real vector-valued

function from ℜk+1 into ℜk defined as g([x0,x1,…,xk]’)=(x1/x0,…,xk/x0)’, for x0≠0, so that:

Tr(1)-ρ(1),…,r(k)-ρ(k)’= Tg([c(0),c(1),…,c(k)]’)-g([γ(0),γ(1),…,γ(k)]’)

Then, from the result on the convergence of differentiable functions of asymptotically

Normal vectors (Serfling 1980, p. 122),

Tr(1)-ρ(1),…,r(k)-ρ(k)’ →d N(0, DVD’) (C.1)

where D is the matrix of first partial derivatives of g(.) evaluated at [γ(0),γ(1),…,γ(k)]’,

D=

ρ−

ρ−ρ−

γ100)k(

010)2(001)1(

)0(1

L

MOMMM

L

L

31

Denote the variance-covariance matrix in (C.1) by W=DVD’ and its (i,j)-th element by

wij. Then, it can be readily shown that wij has the following expression:

wij = )0(

12γ

[ ρ(i)ρ(j)v00 - ρ(i)v0j - ρ(j)v0i + vij ] (C.2)

for i,j=1,2,…,k. Finally, substituting in (C.2) v00, v0j, v0i and vij by their values in (22), the

required expression of wij in (23) is obtained.

REFERENCES

Abramowitz, M., and Stegun, N. C. (1970), Handbook of Mathematical Functions, New York: Dover Publications Inc.

Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labys, P. (1999), “The Distribution of Exchange Rate Volatility,” Wharton Financial Institutions Center Working Paper 99-08 and NBER Working Paper 6961.

Anderson, T. W. (1971), The Statistical Analysis of Time Series, New York: Wiley.

(1992), “The Asymptotic Distribution of Autocorrelation Coefficients,” in The Art of Statistical Science: A tribute to G. S. Watson, ed. K. V. Madia, New York: Wiley, pp. 9-25.

Anderson, T. W., and Walker, A. M. (1964), “On the Asymptotic Distribution of the Autocorrelations of a Sample From a Linear Stochastic Process,” Annals of Mathematical Statistics, 35, 1296-1303.

Billingsley, P. (1961), “The Lindeberg-Lévy Theorem for Martingales,” Proceedings of the American Statistical Society, 12, 788-792.

Bollerslev, T. (1988), “On the Correlation Structure for the Generalized Autoregressive Conditional Heteroskedastic Process,” Journal of Time Series Analysis, 9, 121-131.

Brillinger, D. R. (1981), Time Series: Data Analysis and Theory, New York: Mc-Graw Hill.

Brockwell, P. J., and Davis, R. A. (1991), Time Series: Theory and Methods (2nd ed.), Berlin: Springer Verlag.

32

Carnero, M. A., Peña, D., and Ruiz, E. (2001), “Is Stochastic Volatility More Flexible Than GARCH?,” Working Paper 01-08(05), Statistics and Econometrics Series, Universidad Carlos III de Madrid, Spain.

Diebold, F. X. (1988), Empirical Modelling of Exchange Rate Dynamics, New York: Springer Verlag.

Fuller, W. A. (1996), Introduction to Statistical Time Series (2nd ed.), New York: Wiley.

Ghysels, E., Harvey, A. C., and Renault, E. (1996), “Stochastic Volatility,” in Handbook of Statistics (Vol. 14), eds. G. S. Maddala and C. R. Rao, Amsterdam: North Holland, pp. 119-191.

Hannan, E. J. (1976), “The Asymptotic Distribution of Serial Covariances,” Annals of Statistics, 4, 396-399.

Hannan, E. J., and Heyde, C. C. (1972), “On Limit Theorems for Quadratic Functions of Discrete Time Series,” Annals of Mathematical Statistics, 43, 2058-2066.

Harvey, A. C. (1993), Time Series Models (2nd ed.), New York: Harvester Wheatsheaf.

Harvey, A. C., Ruiz, E., and Shephard, N. G. (1994), “Multivariate Stochastic Variance Models,” Review of Economic Studies, 61, 247-264.

He, C., and Terasvirta, T. (1999), “Properties of Moments of a Family of GARCH Processes,” Journal of Econometrics, 92, 173-192.

Milhoj, A. (1985), “The Moment Structure of ARCH Processes,” Scandinavian Journal of Statistics, 12, 281-292.

Nerlove, M., Grether, D. M., and Carvalho, J. L. (1979), Analysis of Economic Time Series: A Synthesis, New York: Academic Press.

Pérez, A. (2000), “Estimación e Identificación de Modelos de Volatilidad Estocástica con Memoria Larga,” unpublished Ph.D. dissertation, University of Valladolid, Spain, Dept. of Statistics and Econometrics.

Romano, J. P., and Thombs, L. A. (1996), “Inference for Autocorrelations Under Weak Assumptions,” Journal of the American Statistical Association, 91, 590-600.

Serfling, R. J. (1980), Approximation Theorems of Mathematical Statistics, New York: Wiley.

Taylor, S. (1986), Modelling Financial Time Series, New York: Wiley.

33

Table 1. Asymptotic size of Bartlett 95% confidence bands to test H0:ρY(k)=0.

2ησ Lag φ=0.5 φ=0.9 φ=0.95 φ=0.98

0.1 k=1 k=10 k=50

0.058 0.050 0.050

0.122 0.074 0.050

0.229 0.149 0.060

0.570 0.485 0.216

0.05 k=1 k=10 k=50

0.054 0.050 0.050

0.082 0.061 0.050

0.125 0.093 0.055

0.291 0.242 0.119

0.01 k=1 k=10 k=50

0.051 0.050 0.050

0.056 0.052 0.050

0.062 0.057 0.051

0.083 0.077 0.061

Table 2. Empirical size of tests for H0:ρY(k)=0 based on Bartlett bands and on SV-corrected bands.

T=512 T=1024 T=4096 2ησ Lag Bands φ=0.9 φ=0.95 φ=0.98 φ=0.9 φ=0.95 φ=0.98 φ=0.9 φ=0.95 φ=0.98

0.1 k=1 P Pc

*cP

0.103 0.041 0.044

0.165 0.030 0.045

0.268 0.002 0.045

0.1230.0500.051

0.200 0.041 0.050

0.338 0.006 0.051

0.124 0.052 0.051

0.222 0.050 0.054

0.430 0.022 0.047

k=10 P Pc

*cP

0.064 0.041 0.045

0.110 0.029 0.046

0.197 0.001 0.046

0.0760.0490.052

0.132 0.037 0.052

0.251 0.005 0.045

0.076 0.051 0.050

0.144 0.050 0.049

0.347 0.018 0.047

k=50 P Pc

*cP

0.035 0.035 0.036

0.038 0.033 0.035

0.058 0.005 0.036

0.0410.0410.042

0.044 0.038 0.043

0.080 0.008 0.043

0.045 0.045 0.044

0.051 0.043 0.046

0.137 0.023 0.044

0.05 k=1 P Pc

*cP

0.071 0.043 0.045

0.102 0.039 0.046

0.167 0.016 0.048

0.0840.0520.051

0.122 0.050 0.053

0.208 0.027 0.052

0.087 0.052 0.051

0.124 0.055 0.054

0.257 0.041 0.052

k=10 P Pc

*cP

0.054 0.042 0.045

0.079 0.038 0.048

0.137 0.015 0.047

0.0600.0500.050

0.091 0.044 0.049

0.164 0.021 0.046

0.062 0.051 0.051

0.092 0.048 0.049

0.207 0.036 0.048

k=50 P Pc

*cP

0.038 0.038 0.038

0.038 0.036 0.039

0.053 0.016 0.037

0.0410.0410.040

0.042 0.038 0.041

0.069 0.022 0.042

0.046 0.046 0.048

0.051 0.047 0.047

0.098 0.039 0.045

0.01 k=1 P Pc

*cP

0.051 0.046 0.047

0.054 0.045 0.047

0.070 0.041 0.047

0.0590.0530.051

0.064 0.054 0.052

0.086 0.051 0.054

0.055 0.050 0.050

0.067 0.053 0.054

0.083 0.057 0.054

k=10 P Pc

*cP

0.044 0.043 0.045

0.053 0.043 0.044

0.063 0.040 0.045

0.0500.0480.048

0.056 0.049 0.050

0.070 0.043 0.047

0.052 0.049 0.050

0.056 0.048 0.049

0.076 0.047 0.049

k=50 P Pc

*cP

0.040 0.040 0.039

0.040 0.040 0.039

0.043 0.034 0.040

0.0410.0410.041

0.041 0.040 0.039

0.049 0.035 0.040

0.049 0.049 0.050

0.049 0.049 0.049

0.056 0.047 0.046

NOTE: P, Pc and *cP denote the empirical sizes based on the usual Bartlett bands, on the theoretically SV-corrected bands in (17) and on the sample SV-corrected bands in (19), respectively. The calculations are based on 5000 replications. The nominal size is 5%.

34

Table 3. Empirical size of Box-Pierce test and SV-corrected Box-Pierce tests for H0:ρY(1)=…=ρY(K)=0.

T=512 T=1024 T=4096 2ησ Lag Bands φ=0.9 φ=0.95 φ=0.98 φ=0.9 φ=0.95 φ=0.98 φ=0.9 φ=0.95 φ=0.98

0.1 K=10 Q Qc

*cQ

0.1670.0480.046

0.379 0.032 0.048

0.661 0.001 0.050

0.191 0.045 0.049

0.440 0.045 0.050

0.798 0.006 0.053

0.202 0.057 0.056

0.541 0.058 0.050

0.930 0.031 0.054

K=50 Q Qc

*cQ

0.0970.0420.043

0.277 0.034 0.051

0.653 0.000 0.059

0.118 0.047 0.045

0.309 0.053 0.049

0.855 0.005 0.062

0.131 0.048 0.044

0.553 0.062 0.049

0.985 0.048 0.060

0.05 K=10 Q Qc

*cQ

0.1030.0490.048

0.202 0.043 0.051

0.437 0.014 0.052

0.107 0.046 0.048

0.228 0.045 0.049

0.536 0.027 0.051

0.108 0.056 0.055

0.257 0.057 0.053

0.690 0.049 0.054

K=50 Q Qc

*cQ

0.0630.0400.041

0.152 0.043 0.050

0.452 0.012 0.058

0.073 0.044 0.045

0.197 0.047 0.048

0.647 0.035 0.063

0.079 0.048 0.046

0.245 0.051 0.049

0.873 0.069 0.058

0.01 K=10 Q Qc

*cQ

0.0530.0460.049

0.066 0.048 0.049

0.109 0.039 0.051

0.057 0.048 0.049

0.069 0.046 0.049

0.129 0.043 0.049

0.063 0.055 0.054

0.081 0.057 0.058

0.140 0.053 0.057

K=50 Q Qc

*cQ

0.0470.0400.041

0.058 0.040 0.043

0.123 0.037 0.050

0.052 0.048 0.045

0.064 0.046 0.046

0.154 0.046 0.049

0.051 0.045 0.046

0.070 0.046 0.047

0.199 0.049 0.051

NOTE: Q, Qc and *cQ denote the empirical sizes based on the usual Box-Pierce statistic in (15), the SV-corrected Box-Pierce statistic in (20) and the sample SV-corrected Box-Pierce statistic in (21), respectively. The calculations are based on 5000 replications. The nominal size is 5%.

35

Table 4. Bias and standard deviation (in parenthesis) of sample autocorrelations of xt=log( 2

ty ), together with autocorrelation function (ACF) values, for k=1,10,50.

T=512 T=1024 T=4096 Parameters Lag ACF Theoretical MonteCarlo Theoretical MonteCarlo Theoretical MonteCarlo φ=0.9, 2

ησ =0.1 k=1 0.087 -0.005 (0.050)

-0.006 (0.049)

-0.003 (0.035)

-0.002 (0.035)

-0.001 (0.018)

-0.001 (0.017)

k=10 0.034 -0.006 (0.047)

-0.007 (0.046)

-0.003 (0.033)

-0.003 (0.033)

-0.001 (0.017)

-0.001 (0.017)

k=50 0.001 -0.005 (0.046)

-0.005 (0.043)

-0.003 (0.032)

-0.003 (0.032)

-0.001 (0.016)

-0.001 (0.016)

φ=0.95, 2ησ =0.1 k=1 0.164 -0.014

(0.062) -0.015 (0.060)

-0.007 (0.044)

-0.007 (0.043)

-0.002 (0.022)

-0.002 (0.022)

k=10 0.103 -0.017 (0.060)

-0.018 (0.058)

-0.008 (0.042)

-0.008 (0.041)

-0.002 (0.021)

-0.002 (0.021)

k=50 0.013 -0.016 (0.055)

-0.015 (0.048)

-0.008 (0.039)

-0.008 (0.037)

-0.002 (0.019)

-0.002 (0.019)

φ=0.98, 2ησ =0.1 k=1 0.332 -0.060

(0.110) -0.055 (0.095)

-0.030 (0.078)

-0.028 (0.071)

-0.007 (0.039)

-0.008 (0.038)

k=10 0.277 -0.069 (0.115)

-0.064 (0.097)

-0.034 (0.081)

-0.033 (0.073)

-0.009 (0.041)

-0.009 (0.040)

k=50 0.123 -0.084 (0.117)

-0.073 (0.080)

-0.042 (0.083)

-0.040 (0.069)

-0.011 (0.041)

-0.011 (0.039)

φ=0.98, 2ησ =0.05 k=1 0.200 -0.039

(0.086) -0.037 (0.075)

-0.019 (0.061)

-0.018 (0.056)

-0.005 (0.031)

-0.006 (0.030)

k=10 0.166 -0.044 (0.087)

-0.041 (0.074)

-0.022 (0.061)

-0.021 (0.056)

-0.005 (0.031)

-0.006 (0.030)

k=50 0.074 -0.050 (0.082)

-0.045 (0.060)

-0.025 (0.058)

-0.024 (0.050)

-0.006 (0.029)

-0.007 (0.028)

φ=0.98, 2ησ =0.01 k=1 0.048 -0.011

(0.050) -0.011 (0.049)

-0.006 (0.036)

-0.005 (0.035)

-0.001 (0.018)

-0.001 (0.017)

k=10 0.040 -0.012 (0.050)

-0.012 (0.047)

-0.006 (0.035)

-0.006 (0.034)

-0.001 (0.018)

-0.001 (0.017)

k=50 0.018 -0.013 (0.048)

-0.013 (0.043)

-0.007 (0.034)

-0.007 (0.033)

-0.002 (0.017)

-0.002 (0.017)

NOTE: For each lag and each sample size, the theoretical bias and standard deviation (in parenthesis), calculated from

(27) and (24), respectively, are displayed together with their Monte Carlo counterparts, based on 5000 replications.

36

Table 5. Sample autocorrelations of returns and log-squared returns of IBEX-35.

Lag Series k=1 k=2 K=3 k=4 k=5 k=10 k=20 k=50 k=100 yt (s.e.) [s.e.c]

0.002 (0.018) [0.029]

0.011 (0.018) [0.034]

-0.010 (0.018) [0.028]

0.033 (0.018) [0.033]

-0.007 (0.018) [0.030]

0.060ab (0.018) [0.029]

-0.049a (0.018) [0.025]

-0.018 (0.018) [0.022]

-0.011 (0.018)[0.021]

Log( 2ty )

(s.e.) [s.e.c]

0.131ab (0.018) [0.052]

0.183ab (0.018) [0.049]

0.172ab

(0.018) [0.049]

0.107ab

(0.018) [0.053]

0.142ab

(0.018) [0.051]

0.129ab

(0.018) [0.051]

0.118ab (0.018) [0.051]

0.091a (0.018) [0.051]

0.077a (0.018)[0.045]

NOTE: (s.e.) denote the asymptotic standard error of the sample autocorrelations in I.I.D. series, 1/ T . The values [s.e.c] denote the SV-corrected standard error calculated from equations (18) and (26), for the series

yt and log( 2ty ), respectively.

a Significant values with respect to Barlett 95% bounds ±1.96/ T . b Significant values with respect to the corresponding SV-corrected 95% bounds ±1.96[s.e.c].

Table 6. Box-Pierce statistic of returns and log-squared returns of IBEX-35.

Lag

Statistic K=5 K=10 K=20 K=50 K=100 Q(K) 4.024

(0.403)

20.199*

(0.017)

45.559*

(0.001)

82.703*

(0.002)

133.573*

(0.012)

Qc(K) 1.276

(0.866)

7.691

(0.566)

20.023

(0.393)

42.765

(0.723)

86.053

(0.820)

Qlog(K) 335.677*

(0.000)

609.275*

(0.000)

1043.247*

(0.000)

1936.319*

(0.000)

2830.897*

(0.000)

NOTE: Q(K) is the Box-Pierce statistic to test for joint uncorrelation in the series of returns; Qc(K) is the SV-corrected Box-Pierce statistic defined in (21); Qlog(K) is the Box-Pierce statistic to test for joint uncorrelation in the series of log-squared returns. The values in parenthesis are the p-values of the corresponding χ2 distribution.

* Significant values at 5% level.

37

Figure 1. Barlett 95% confidence bands (solid line) and SV-corrected 95% bands for ARSV processes with φ=0.98 and 2

ησ = 0.1 (dashed line), 2ησ =0.05 (short-dashed line) and 2

ησ =0.01 (dotted line).

38

Figure 2. Empirical distributions (uncorrected, theoretically SV-corrected and empirically SV-corrected) of the sample autocorrelations of yt for k=1, 10, 50, in an ARSV model with φ=0.98, 2

ησ =0.05.

(a) T=512

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(1)/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(1)/sqrt(C(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(1)/sqrt(C*(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(10)/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(10)/sqrt(C(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(10)/sqrt(C*(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(50)/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(50)/sqrt(C(T,50))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(50)/sqrt(C*(T,50))

(b) T=4096

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(1)/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(1)/sqrt(C(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(1)/sqrt(C*(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(10)/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(10)/sqrt(C(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(10)/sqrt(C*(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(50)/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(50)/sqrt(C(T,50))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

Histogram r(50)/sqrt(C*(T,50))

39

Figure 3. Empirical distributions of Box-Pierce statistic, Q(K), and the two SV-corrected statistics, Qc(K) and )K(Q*

c , for K= 10, 50, in an ARSV model with φ=0.98, 2ησ =0.05, T=4096.

0 20 40 60 80 100 120 140

0.00.0

50.1

0 Histogram Q(10)

0 10 20 30 40 50 60

0.00.0

50.1

0 Histogram Qc(10)

0 10 20 30 40 50 60

0.00.0

50.1

0 Histogram Qc*(10)

0 50 100 150 200 250 300

0.00.0

10.0

20.0

30.0

4

Histogram Q(50)

0 20 40 60 80 100 120 140

0.00.0

10.0

20.0

30.0

4

Histogram Qc(50)

0 20 40 60 80 100 120 140

0.00.0

10.0

20.0

30.0

4

Histogram Qc*(50)

40

Figure 4. Mean correlogram (vertical bars) and true autocorrelation function (solid line) of log( 2ty ) in ARSV

processes with φ=0.98, 2ησ =0.1, 0.05, 0.01 and T=512, 1024, 4096.

41

Figure 5. Standardised distributions (uncorrected, theoretically SV-corrected and empirically SV-corrected) of the sample autocorrelations of xt =log( 2

ty ) for k=1, 10, 50, in an ARSV model with φ=0.98, 2ησ =0.05.

(a) T=512

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)]/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)]/sqrt(V1(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)]/sqrt(V2(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)]/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)]/sqrt(V1(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)]/sqrt(V2(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)]/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)]/sqrt(V1(T,50))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)]/sqrt(V2(T,50))

(b) T=4096

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)]/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)]/sqrt(V1(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)]/sqrt(V2(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)]/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)]/sqrt(V1(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)]/sqrt(V2(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)]/sqrt(1/T)

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)]/sqrt(V1(T,50))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)]/sqrt(V2(T,50))

42

Figure 6. Standardised distribution of the sample autocorrelations of xt =log( 2ty ) theoretically SV-corrected

for bias and variance, for k=1, 10, 50, in an ARSV model with φ=0.98, 2ησ =0.05 and T=512,4096.

(a) T=512 (b) T=4096

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)-bias(r(1))]/sqrt(V1(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(1)-rho(1)-bias(r(1))]/sqrt(V1(T,1))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)-bias(r(10))]/sqrt(V1(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(10)-rho(10)-bias(r(10))]/sqrt(V1(T,10))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)-bias(r(50))]/sqrt(V1(T,50))

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

[r(50)-rho(50)-bias(r(50))]/sqrt(V1(T,50))

43

Figure 7. Scatter plot of sample autocorrelation coefficients, r(i)-ρ(i) versus r(j)-ρ(j), of the series xt=log( 2

ty ) of an ARSV model with φ=0.98, 2ησ =0.05 and sample sizes T=512 (top) and T=4096 (bottom).

44

Figure 8. Daily returns of IBEX-35 from 7/1/87 to 30/12/98 together with the correlograms of returns and log-squared returns.