The finite-sample size of the BDS test for

GARCH standardized residuals

Marcelo Fernandes

Graduate School of Economics, Fundacao Getulio Vargas

Praia de Botafogo, 190, 22253-900 Rio de Janeiro, Brazil

E-mail: mfernand@fgv.br

Pierre-Yves Preumont

CREST and Universite Libre de Bruxelles

50, Avenue F.D. Roosevelt, CP140, B-1050 Brussels, Belgium

E-mail: pypreumo@ulb.ac.be

Abstract. This paper uses a multivariate response surface methodology to

analyze the size bias of the BDS test when applied to standardized residu-

als of GARCH process. In particular, the distortion is set as a function of

the parameter vector, the embedding dimension and the sample size. The

results show that the asymptotic standard normal distribution is an unre-

liable approximation even in large samples. On the other hand, a simple

log-transformation of the squared standardized residuals of GARCH pro-

cesses seems to correct most of the size problems. The estimated response

surfaces can nonetheless provide not only a measure of the size distortion,

but also more adequate critical values for the BDS test in small samples.

JEL Classification: C15, C52.

Keywords: BDS test, GARCH process, response surface, size distortion.

Acknowledgements: We wish to thank the helpful comments from numer-

ous colleagues and seminar participants at the EC2 Conference “Simulation

Methods in Econometrics”(Florence, 1996), ENTER Jamboree (Tilburg,

1997), and PAI Workshop in Financial Modeling and Econometric Anal-

ysis (Louvain-la-Neuve, 1997). The usual disclaimer applies.

1

1 Introduction

The extensive literature on GARCH-type processes is clearly a consequence

of their success modeling financial time series. The GARCH class of models,

introduced by Engle (1982) and Bollerslev (1986), is designed to handle two

specific features of these series: volatility clustering and leptokurtosis. Fur-

thermore, it is possible to interpret GARCH models as discrete approxima-

tions of jump-diffusion processes (Nelson, 1990a; Drost and Werker, 1996).

On the other hand, some papers present evidence that GARCH models

are not able to fully explain all nonlinear dependence in financial data (e.g.

Hsieh, 1991; Peel and Speight, 1994; Abhyankar, Copeland and Wong, 1995).

A common procedure is to apply the BDS test described in Brock,

Dechert, Scheinkman and LeBaron (1996) to the standardized residuals

of GARCH models. The BDS test has good power against a wide class

of data generating processes departing from the property of independence

and identical distribution (iid). Moreover, the BDS test does not require

the existence of high-order moments as opposed to most alternative tests

that usually assume the existence of the fourth or even higher moments

(de Lima, 1997). Since financial data does not satisfy often these require-

ments, the robustness of the BDS test to failure of moment conditions is a

particularly desirable property.

However, pre-filtering the data using a GARCH-type process distorts the

asymptotic distribution of the BDS statistic in consequence of the nonzero

variance of the estimates (see Brooks and Heravi, 1999). There are some

solutions in the literature. Brock, Hsieh and LeBaron (1991) perform Monte

Carlo simulations to derive the distribution of the BDS test on standardized

residuals of a specific GARCH model. Hsieh (1993) uses the same pro-

cedure to determine proper critical values for the BDS test when applied

to standardized residuals of EGARCH and autoregressive volatility models.

Chappell, Padmore and Ellis (1996) and Fernandes (1998) bootstrap the

standardized residuals of conditional heteroskedastic models for exchange

2

rate series to illustrate the degree of size distortion. Finally, de Lima (1996)

proves the nuisance parameter free property of the BDS test for additive

models, which requires working with the logarithm of the squared GARCH

standardized residuals as in Brock and Potter (1992). Taking the squares

of the standardized residuals dooms the BDS test to have no power against

alternatives featuring asymmetry, such as the leverage effect singled out by

Black (1976).

This paper investigates the source of these distortions in the BDS test

distribution using response surface regressions. In particular, we analyze

the effects of the GARCH parameters, the embedding dimension used to

compute the statistic and the sample size on the effective level of the BDS

test. The possibility of examining the response of the size bias with respect

to a great number of situations, avoiding specificity, motivates the response

surface analysis. For instance, it is interesting to verify whether the size

distortions depends upon the values of the GARCH parameters or upon a

function of them (e.g. persistence of volatility).

The results confirm that, in finite samples, the selection of the embedding

dimension for the BDS test plays a role in the size distortion, even though,

under the null hypothesis, the distribution of the test should be the same

regardless the dimension. One novel evidence consists in the significant

relationship between the persistence of volatility and the nuisance parameter

effect. Finally, the results for the logarithm of the squared standardized

residuals are encouraging, since it seems to correct, even in moderate sample

sizes, the distortions due to the presence of nuisance parameters.

The paper is organized as follows. Section 2 discusses in detail the BDS

test and its properties. Section 3 outlines the response surface methodology

for determining the size bias implied by the asymptotic distribution of the

BDS test in the presence of nuisance parameters. Section 4 compares the

results of the response surface estimation for the case of the standardized

residuals and the case of the transformed residuals. Section 5 illustrates

with an example how the results of the test can qualitatively change if

3

one considers more precise critical values, such as the ones provided by the

surface response. Section 5 offers some concluding remarks.

2 A closer look to the BDS test

The nonparametric test of Brock et al. (1996) is derived from the correlation

integral, which is a measure of spatial correlation of scattered points in the

m-dimensional space. In a time-series context, {xt} is embedded in the m-

space by forming m-histories xmt = (xt, xt−1, . . . , xt−m+1). The correlation

integral reads

C(δ,m) =

∫

u

∫

v

I(u, v, δ)dFm(u)dFm(v),

where the indicator kernel function I(·) is one when |u − v| < δ, zero oth-

erwise, and Fm(·) is the distribution function of xmt . Hence, it indicates the

concentration of the joint distribution of m-consecutive observations.

Brock et al. (1996) have shown that the generalized U-statistic

C(δ,m, T ) =2

(T −m)(T −m+ 1)

∑

t<s

I(xmt , xms , δ)

is a consistent estimator of C(δ,m) provided that {xt} is an absolutely

regular and strictly stationary stochastic process. The fact that C(δ,m, T )

is an U-statistic entails some interesting properties. For instance, under

certain conditions, U-statistics are minimum variance estimators in the class

of all unbiased estimators and converge rapidly to normality (Serfling, 1980).

If the process {xt} is iid, then Fm(xmt ) =∏m−1i=0 F1(xt−i), and C(δ,m) =

C(δ, 1)m almost surely. Brock et al. (1996) use this relation to construct the

following test statistic for detecting deviations from the iid property

BDS(δ,m, T ) =√T

C(δ,m, T )− C(δ, 1, T )m

σ(δ,m, T ),

where σ(δ,m, T ) is a nontrivial function of the correlation integral. Strong

consistency and asymptotically standard normality are proven using the

theory of U-statistics. Moreover, the BDS test has high power against a

vast class of linear, nonlinear and nonstationary models.

4

The asymptotic distribution of the BDS statistic is also invariant to the

estimation process of smooth filters under some modest conditions (de Lima,

1996). In particular, the process {xt} must be strong mixing (with mixing

coefficients satisfying the summability condition(∑∞

k=1

√α(k) <∞

)and

the filter must be an additive noise model (Tong, 1990) with parameters√T -consistently estimated. Although the GARCH(p, q) model

yt =√htεt, εt|It−1 ∼ iid(0, 1), ht = ω +

p∑

i=1

αiy2t−i +

q∑

j=1

βjht−j

is multiplicative, it is readily converted to an additive noise model. Brock

and Potter (1992) and de Lima (1996) indeed suggest to transform the stan-

dardized residuals as follows

ηt = log ε2t = log y2t − log ht.

Under the null hypothesis of correct specification, the standardized error εt is

iid, which implies that ηt is also iid. In addition, the parametric restrictions

normally imposed to achieve covariance stationarity and positivity of the

conditional variance suffice to guarantee the invariance property (de Lima,

1996), since the parameters can be√T -consistently estimated by pseudo-

maximum likelihood (Bollerslev and Wooldridge, 1992).

On the other hand, the distribution of the BDS statistic is greatly af-

fected whenever one considers the standardized residuals. In general, the

results of Monte Carlo simulations point out a tendency to under-reject the

null of correct specification of the GARCH model. One solution relies on

resampling techniques so as to mitigate the size bias. Notwithstanding, it is

important to verify whether distortions arise solely in the estimation process

or they are also influenced by the parameters of the test, more specifically,

the embedding dimension. In the next section, we put forth a response

surface analysis to tackle these issues and to investigate the finite-sample

adequacy of the transformation proposed by the literature.

5

3 A response surface analysis

In the sequel, we describe the experimental design using Hendry’s (1984)

terminology. The data generating process is a GARCH(1,1) model with

normal conditional distribution, viz.

yt =√htεt, εt|It−1 ∼ N(0, 1), ht = ω + αy2

t−1 + βht−1.

The Monte Carlo design variables are the parameters θ, the embedding

dimension m and the sample size T , where

θ = (ω, α, β)′ ∈ Θ = {θ | ω > 0, α > 0, β > 0, α+ β < 1},

m ∈ M = [ m,m ] and T ∈ T =[T , T

]. An underline indicates the

smallest and an overline the largest value we consider for any given variable.

The parameter space thus is Θ ×M× T . The relationship of interest for

the BDS statistic is the correct null hypothesis {H0 : εt is iid}, so as to

address the size of the test. The aim of the simulation exercise is at in-

vestigating the deterioration of the asymptotic distribution of the BDS test

when applied to the standardized residuals. The nuisance parameter bias

presumably depends upon θ, m and T , implying that wemust pursue an

adequate approximation over Θ×M× T .

We hold two parameters constant across experiments: ω and δ. There

is no loss of generality setting ω to one for it is only a scale factor. By the

same token, the actual standard deviation of εt is assigned to the tuning

parameter δ so as to optimize the power and size of the test (Brock et

al., 1991). Therefore, δ = 1 when the test is applied to the standardized

residuals and δ = 2.22 when applied to the transformed residuals. The

values of the GARCH parameters cover a large range of processes:

(α, β) ∈ {α ∈ Θ∗, β ∈ Θ∗|α+ β < 1},

where Θ∗ = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8}. We also compute the BDS

statistic for several embedding dimensions: m ∈ {2, 3, 4, 5, 6, 7, 8}. Albeit

four sample sizes are investigated T ∈ {100, 250, 500, 1000}, the number of

6

replications N is the same for all experiments (N = 1, 000). Given the

selection of the key parameters and the covariance stationarity constraint

(α+ β < 1),1 we adopt a full factorial design, totaling 1,008 experiments.

Albeit we draw 5,000 standard normal variates to generate each GARCH

process, we use only the last T observations for estimation purposes to avoid

any spurious effect due to of the initial conditions. Further, we set h0 to the

unconditional variance, that is to say, h0 = ω/(1 − α − β). The estimation

of θ is by maximum likelihood. We the compute the BDS statistic for both

the standardized residuals εt and the transformed residuals ηt.

We gauge the size distortion for the lower and upper tail of the distribu-

tion, since one usually performs the BDS test considering both tails. More

precisely, we compute the ratio between the critical values CVα derived from

the Monte Carlo simulations and the α-quantiles CV ∞α of the asymptotic

standard normal distribution, where α ∈ {0.025, 0.050, 0.950, 0.975}. The

absence of size distortion would then imply Dα ≡ CVα/CV∞α equal to one

for every α.

As the exact functional form is unknown, we adopt the usual power-

series approximation for the response surface analysis. More precisely, we

start with three initial specifications for D = Q(θ, Tm) + υ, namely

Q1(θ, Tm) = g(α, β, 1/Tm)

Q2(θ, Tm) = g(α+ β, 1/Tm)

Q3(θ, Tm) = g(1/Tm)

where Tm stands for the effective sample size after adjusting by the embed-

ding dimension,2 υ is a white noise vector and g(·) is a polynomial of second

order. In contrast to the first approximation Q1 that sets the size distortion

as a function of the individual values of the parameters, Q2 assumes that

1 Actually, this restriction is sufficient, but not necessary, for the covariance-stationarity of the process (see Nelson, 1990b).

2 In theory, the embedding dimension does not play any role other than reducingthe effective sample size available to compute the correlation integral. We consider twomeasures of adjusted sample size: T −m + 1 and T/m. The former corresponds to thetotal number of observations available to compute the BDS statistic, whereas the latter isclose to the number of nonoverlapping m-histories.

7

the persistence of volatility implied by the GARCH parameters summarizes

all the information in the parameters. Finally, Q3 suggests that the level

distortion is solely due to finite sample sizes.

The estimation is by SUR to take into account the correlation among

the residuals of each equation and it is performed in a training subsam-

ple of 800 (randomly chosen) experiments. We left out the remaining 208

experiments to assess more carefully the validity and precision of the dif-

ferent approximations through out-of-sample analysis. The model selection

procedure therefore contains two stages: in-sample and out-of-sample. In

the training set, for each starting functional form (Q1, Q2 and Q3), we

adopt a general-to-specific approach to find the more adequate approxima-

tion. We consecutively delete the less significant parameter of the system

until all coefficients are statistically different from zero at the 5.0% level of

significance. We then compare the in- and out-of-sample performance of the

resulting systems of each starting specification.

4 Modeling the size distortion

In this section, we report the outcome of the model selection procedure and

discuss the estimation results of the response surface systems, comparing

the best representations for the cases of the standardized and transformed

residuals. In particular, we show some evidence that the nuisance parameter

effect on the distribution of the BDS test increases with the persistence of

volatility. On the other hand, for the case of the transformed residuals, the

size bias depends only upon the effective sample size and the test statistic

seems to converge in distribution to a standard normal.

Before discussing in detail the selection and estimation of the response

surfaces, it is important to observe some features of the BDS test distribu-

tion for both standardized and transformed residuals. Figure 1 exhibits the

estimated distributions of the test for both cases to highlight their differ-

ences in relatively large sample. Figure 2 plots the nonparametric estimation

8

for the distribution of the test when applied to the standardized (Figure 2a)

and transformed residuals (Figure 2b), stressing the effects of sample size.3

As the sample size increases, the distributions in the Figure 2b becomes

closer to the normal distribution, which is clearly not the case in Figure 2a.

Indeed, despite the high skewness, the distribution of the BDS statistic in

Figure 2a is more similar to the standard normal when T = 100. This pe-

culiarity is probably a consequence of the interaction of two opposite forces:

the finite sample bias and the nuisance parameter effect.

Table 1 reports the average effective size of the BDS test over the dif-

ferent sets of GARCH parameters, according to the sample size and the

embedding dimension. In small samples (100 observations), the lack of pre-

cision to estimate both the GARCH parameters and the correlation integral

causes over-rejection of the correct null hypothesis on both standardized

and transformed residuals. As the sample size increases, the BDS test on

the standardized residuals becomes more and more conservative, while the

size distortion correction provided by the transformation on the residuals

improves.

Tables 2 and 3 document the in- and out-of-sample performance of each

specification, respectively. We assess the out-of-sample performance using

two different metrics, namely the Euclidean norm and the norm of the max-

imum, to gauge how large are, on average, the residual vectors of the multi-

variate response surfaces. For the standardized residuals, using the number

of nonoverlapping m-histories to gauge the effective sample size dominates

both in- and out-of-sample irrespective of the specification. The approxima-

tion based on the persistence of volatility, Q∗2(θ, Tm), provides a marginally

superior out-of-sample fit. The fitted values are

D0.975 = 0.51 + 13.1m/T + 51.6 (m/T )2 + 7.6 (α+ β)m/T

−117.7 (α+ β)(m/T )2 + 81.9 [(α+ β)m/T ]2

D0.950 = 0.49 + 12.4m/T + 7.5 (α+ β)m/T

3 All nonparametric density estimations are performed using a quartic kernel and across-validation procedure to select the bandwidth.

9

D0.050 = 0.52 + 36.3m/T − 299.5 (m/T )2 − 56.3 (α+ β)m/T

+801.1 (α+ β) (m/T )2 + 40.3 (α+ β)2 m/T

− 612.0 [(α+ β)m/T ]2

D0.025 = 0.51 + 36.0m/T − 317.2 (m/T )2 − 57.4 (α+ β)m/T

+855.7 (α+ β) (m/T )2 + 41.7 (α+ β)2 m/T

− 654.5 [(α+ β)m/T ]2.

The most striking feature is that the intercepts are remarkably distant from

one, implying that the standard normal is not a good approximation for the

test distribution even asymptotically. Indeed, the asymptotic critical values

corrected by the nuisance parameter effect are about half of the noncorrected

ones provided by the standard normal distribution.

The shape of the response surfaces indicates that the size distortion is

strongly dependent on the persistence of volatility only in small samples

(see Figure 3). In the upper tail, the size bias linearly increases with the

persistence of volatility (Figure 3a,b). In the lower tail, the size distortion

is a nonlinear function of the degree of persistence and effective sample size

(Figure 3c,d). Nevertheless, irrespective to the tail, the convergence towards

the adjusted asymptotic critical values is rather fast.

For the case of the transformed residuals, there is no clear dominant mea-

sure of effective sample size. Although the out-of-sample performance con-

tinues to be superior when using the number of nonoverlapping m-histories

to proxy the effective sample size, the bias in the lower tail is better explained

using the alternative measure. Hence, we combine both measures to specify

an alternative response surface system, Q4(θ, Tm), where the lower tail bias

depends upon the total number of available observations (T −m + 1) and

the upper tail bias is a function of the number of nonoverlapping m-histories

(T/m).

This last system specification clearly outperforms the others both in-

10

sample and out-of-sample, and is given by

D0.975 = 0.99 + 10.34m/T − 39.37 (m/T )2

D0.950 = 0.99 + 5.88m/T

D0.050 = 1.00 + 14.26/(T −m+ 1)

D0.025 = 0.96 + 12.59/(T −m+ 1).

Although all intercepts are in the vicinity of one, which corresponds to

the asymptotic absence of size distortion, only the constant of the third

equation is not statistically different from one (see Table 4). We conjecture

that this inconsistency with de Lima’s (1996) analytical results is an artifact

attributable to the small number of replications. The Monte Carlo estimator

of a critical value depends upon the accuracy of the tail estimate, hence is

quite natural to expect some imprecision in the results.

In summary, the size properties of the BDS test for the standardized

residuals are poor even in large samples. The transformation on the resid-

uals proposed by Brock and Potter (1992) and de Lima (1996) corrects this

problem in a satisfying way. Although the restriction implied by the ab-

sence of size distortion does not hold, Figure 4 shows that the critical values

converge rapidly toward the vicinity of their asymptotic values. The size dis-

tortion of the asymptotic test is indeed marginal even for moderate sample

sizes. Finally, the estimated response surfaces provide not only a measure of

the size distortion, but also help determining more accurate critical values

in small samples.

5 Example

To illustrate how the size distortions of the BDS test may be misleading

when applied to standardized GARCH residuals, we revisit the empirical

exercise performed by Serletis and Dormaar (1996). They aim is at testing

the standardized residuals of GARCH(1,1) processes for serial dependence.

They utilize the critical values simulated by Hsieh (1991) to set the 5% two-

tailed rejection region of the test as a function of the embedding dimension.

11

These critical values are however not appropriate for they assume a particu-

lar EGARCH filtering that differs substantially from the GARCH processes

Serletis and Dormaar (1996) estimate. We therefore reexamine their results

through the eyes of the critical values given by the response surface, which

accounts not only for the embedding dimension but also for the different

sample sizes and parameter values.

Serletis and Dormaar (1996) investigate 13 spot-month future prices,

namely Australian dollar, British pound, Canadian dollar, crude oil, copper,

Deutsche mark, gold, heating oil, unleaded gas, Japanese yen, platinum,

Swiss frank, and silver. Table 5 displays the figures one must plug into the

response surface to compute the critical values. It is striking how the critical

values differ from case to case, in contrast to the asymptotic and Hsieh’s

(1991) critical values. Table 6 evinces that the outcome of the BDS test

may change depending on the chosen set of critical values. In particular, our

critical values reject more often the null hypothesis as seen in the results for

the Canadian dollar, Deutsche mark, Japanese yen, platinum, Swiss frank,

and silver.

6 Conclusion

The BDS test is well known for the high power against a wide class of alter-

natives. However, the asymptotic standard normal distribution of the BDS

statistic does not hold when the test is applied to GARCH standardized

residuals due to nuisance parameter effects. de Lima (1996) demonstrate

that a log-transformation of the squared standardized residuals suffices to

meet the conditions for the nuisance parameter free property. This trans-

formation hurts however the power of BDS test against alternative of asym-

metric nature.

To avoid the specificity of Monte Carlo simulations, we employ a re-

sponse surface methodology to pinpoint the sources of the size distortions.

In particular, our multivariate response surface examines the influence of

12

the values of the GARCH parameters and the embedding dimension on the

finite-sample properties of the BDS test. It permits moreover revisiting

the empirical results found in the literature. We show, for instance, that

some results of Serletis and Dormaar (1996) are actually an artifact of the

inadequate critical values they consider.

Our results help understanding the behavior of the BDS test statistic

in other instances than GARCH filtering. Indeed, applications of the BDS

test to the standardized residuals of the autoregressive conditional duration

(ACD) models recently proposed by Engle and Russell (1998) also suffer for

nuisance parameter effects. In view that the ACD processes are very similar

to the GARCH filtering, we expect that the persistence of the duration

process will play a major role, as well.

13

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15

Table 1

The average empirical size of the BDS test

standardized transformedsample dimension

5.0% 10% 5% 10%

T = 100 m = 2 0.08 0.13 0.10 0.17

m = 4 0.09 0.16 0.09 0.17

m = 6 0.14 0.22 0.10 0.17

m = 8 0.23 0.32 0.11 0.18

T = 250 m = 2 0.01 0.02 0.06 0.13

m = 4 0.01 0.03 0.06 0.12

m = 6 0.03 0.06 0.07 0.12

m = 8 0.06 0.11 0.07 0.13

T = 500 m = 2 0.00 0.01 0.06 0.12

m = 4 0.00 0.01 0.06 0.11

m = 6 0.01 0.02 0.06 0.11

m = 8 0.01 0.02 0.06 0.12

T = 1000 m = 2 0.00 0.01 0.05 0.11

m = 4 0.00 0.00 0.05 0.11

m = 6 0.00 0.01 0.04 0.10

m = 8 0.00 0.01 0.05 0.10

16

Table 2

In-sample performance of the response surface

D0.025 D0.050 D0.950 D0.975approximation

R2 DW R2 DW R2 DW R2 DW

standardized residuals

Q∗1(θ, T −m+ 1) 0.85 2.15 0.83 2.14 0.78 2.12 0.79 2.10

Q∗1(θ, T/m) 0.92 1.91 0.93 1.93 0.96 1.95 0.96 1.91

Q∗2(θ, T −m+ 1) 0.84 2.14 0.83 2.13 0.78 2.12 0.78 2.10

Q∗2(θ, T/m) 0.92 1.84 0.92 1.88 0.95 1.95 0.95 1.92

Q∗3(θ, T −m+ 1) 0.84 2.13 0.82 2.13 0.77 2.13 0.77 2.11

Q∗3(θ, T/m) 0.91 1.82 0.91 1.87 0.94 2.01 0.94 1.99

transformed residuals

Q∗1(θ, T −m+ 1) 0.57 1.98 0.66 2.07 0.67 2.07 0.64 2.08

Q∗1(θ, T/m) 0.24 2.02 0.32 2.09 0.90 2.03 0.88 1.92

Q∗2(θ, T −m+ 1) 0.56 1.98 0.65 2.05 0.67 2.06 0.64 2.08

Q∗2(θ, T/m) 0.24 2.01 0.32 2.09 0.90 2.04 0.88 1.96

Q∗3(θ, T −m+ 1) 0.55 2.00 0.64 2.09 0.67 2.06 0.64 2.08

Q∗3(θ, T/m) 0.23 2.03 0.31 2.10 0.89 2.04 0.87 2.00

Q∗4(θ, T/m) 0.55 2.01 0.64 2.09 0.89 2.02 0.87 1.99

R2 is the coefficient of determination adjusted by the degrees of freedom, whereas

DW denotes the Durbin-Watson statistic for the residuals of the response surface.

17

Table 3

Out-of-sample performance of the response surface

standardized transformedapproximation

aen amn aen amn

Q∗1(θ, T −m+ 1) 0.2409 0.1605 0.1132 0.0881

Q∗1(θ, T/m) 0.1499 0.1025 0.0878 0.0640

Q∗2(θ, T −m+ 1) 0.2379 0.1569 0.1133 0.0881

Q∗2(θ, T/m) 0.1458 0.1008 0.0877 0.0640

Q∗3(θ, T −m+ 1) 0.2470 0.1644 0.1152 0.0894

Q∗3(θ, T/m) 0.1635 0.1137 0.0875 0.0642

Q∗4(θ, T/m) 0.0312 0.0229

AEN denotes the average Euclidean norm, whereas AMN

corresponds to the average norm of the maximum.

18

Table 4

Estimation results for the multivariate response surfaces

approximation D0.025 D0.050 D0.950 D0.975

standardized residuals

constant 0.51 (0.006) 0.52 (0.006) 0.51 (0.005) 0.49 (0.004)

m/T 36.0 (2.004) 36.27 (1.978) 13.06 (0.628) 12.35 (0.349)

m2/T 2 -317 (32.27) -299 (31.91) 51.60 (10.57)

(α+ β)m/T -57.4 (6.756) -56.5 (6.650) 7.61 (0.843) 7.51 (0.486)

(α+ β)m2/T 2 855.7 (111.2) 801.1 (109.7) -118 (27.50)

(α+ β)2m/T 2 41.67 (5.449) 40.31 (5.360)

(α+ β)2m2/T 2 -654 (90.30) -612 (89.06) 81.87 (21.00)

transformed residuals

constant 0.96 (0.002) 1.00 (0.002) 0.99 (0.004) 0.99 (0.002)

1/(T −m+ 1) 12.6 (0.384) 14.3 (0.365)

m/T 10.3 (0.249) 5.88 (0.081)

m/T 2 -39.4 (3.120)

Robust standard errors appear in parentheses.

19

Table 5

Response surface critical values at the 5% significance level

nonrejection regiondata sample size α β

m = 2 m = 3 m = 4 m = 5

Australian dollar 330 0.011 0.956 [-1.22, 1.24] [-1.33, 1.37] [-1.43, 1.49] [-1.53, 1.61]

British pound 955 0.123 0.831 [-1.08, 1.08] [-1.12, 1.13] [-1.15, 1.17] [-1.19, 1.21]

Canadian dollar 854 0.161 0.615 [-1.08, 1.09] [-1.11, 1.13] [-1.15, 1.17] [-1.19, 1.22]

crude oil 530 0.226 0.802 [-1.15, 1.15] [-1.23, 1.23] [-1.30, 1.31] [-1.37, 1.39]

copper 905 0.093 0.877 [-1.08, 1.09] [-1.12, 1.13] [-1.17, 1.18] [-1.21, 1.22]

Deutsche mark 955 0.153 0.793 [-1.08, 1.08] [-1.11, 1.12] [-1.15, 1.17] [-1.19, 1.21]

gold 961 0.198 0.790 [-1.08, 1.08] [-1.12, 1.13] [-1.16, 1.17] [-1.20, 1.21]

heating oil 734 0.312 0.666 [-1.10, 1.11] [-1.15, 1.16] [-1.20, 1.22] [-1.25, 1.28]

unleaded gas 439 0.213 0.700 [-1.16, 1.18] [-1.24, 1.27] [-1.31, 1.36] [-1.39, 1.45]

Japanese yen 865 0.025 0.957 [-1.09, 1.09] [-1.13, 1.14] [-1.17, 1.19] [-1.22, 1.23]

platinum 1,072 0.095 0.881 [-1.07, 1.07] [-1.11, 1.11] [-1.14, 1.15] [-1.18, 1.19]

Swiss frank 955 0.070 0.927 [-1.08, 1.08] [-1.12, 1.13] [-1.16, 1.17] [-1.20, 1.21]

silver 1,140 0.121 0.865 [-1.07, 1.07] [-1.10, 1.11] [-1.13, 1.14] [-1.17, 1.18]

20

Table 6

Results of the BDS test according to the critical values

data m = 2 m = 3 m = 4 m = 5

Australian dollar

British pound

Canadian dollar rs rs rs rs

crude oil

copper rs/h/n rs/h/n rs/h rs/h

Deutsche mark rs rs

gold

heating oil

unleaded gas

Japanese yen rs rs/h/n rs/h/n rs/h/n

platinum rs rs rs rs

Swiss frank rs rs

silver rs rs

RS denotes rejection at the 5% level of significance using the crit-

ical values derived from the response surface, whereas H and N

correspond to rejections at the 5% level of significance based on

the critical values given by Hsieh (1991) and by the asymptotic

standard normal distribution, respectively.

21

-5 -4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

standardized residuals

N (0,1)

Figure 1 – The distribution of the BDS test(m = 4, α = 0.1, β = 0.8, T = 1000)

transformedresiduals

-5 -4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

T = 1000

T = 500

N (0,1)

T = 100

N (0,1)

T = 1000

T = 500

T = 100

(A) STANDARDIZED RESIDUALS

(B) TRANSFORMED RESIDUALS

Figure 2 – The distribution of the BDS test in finite samples(m = 4, α = 0.1, β = 0.8)

0100

200300

400500

0

0.5

10.4

0.6

0.8

1

1.2

1.4

persistence of volatility

0.51

effective sample size

distortionsize

Figure 3a – Size distortion of the BDS test at 5% level of significance (upper tail, standardized residuals)

0100

200300

400500

00.2

0.40.6

0.81

0.4

0.6

0.8

1

1.2

1.4

persistence of volatility

0.49

effective sample size

distortionsize

Figure 3b – Size distortion of the BDS test at 10% level of significance(upper tail, standardized residuals)

0100

200300

400500

00.2

0.40.6

0.81

-0.5

0

0.5

1

1.5

0.52

Figure 3c – Size distortion of the BDS test at 10% level of significance(lower tail, standardized residuals)

sizedistortion

persistence of volatility effective sample size

0100

200300

400500

00.2

0.40.6

0.81

-0.5

0

0.5

1

1.5

0.51

distortionsize

persistence of volatility effective sample size

Figure 3d – Size distortion of the BDS test at 5% level of significance(lower tail, standardized residuals)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.95

1

1.05

1.1

1.15

1.2

effective sample size (T/m)

Figure 4a – Size distortion of the BDS test at 5% level of significance(upper tail, transformed residuals)

sizedistortion

0 100 200 300 400 5001

1.05

1.1

1.15

1.2

1.25

effective sample size (T/m)

Figure 4b – Size distortion of the BDS test at 10% level of significance(upper tail, transformed residuals)

sizedistortion

0 100 200 300 400 500 600 700 800 900 10001

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

sizedistortion

effective sample size (T--m+1)

Figure 4c – Size distortion of the BDS test at 10% level of significance(lower tail, transformed residuals)

0 100 200 300 400 500 600 700 800 900 1000

1

1.05

1.1

effective sample size (T-m+1)

Figure 4d – Size distortion of the BDS test at 5% level of significance(lower tail, transformed residuals)

distortionsize