testing for jumps in the stochastic volatility models

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 79 (2009) 2597–2608

Testing for jumps in the stochastic volatility models

Masahito Kobayashi ∗Faculty of Economics, Yokohama National University, Yokohama 240-8501, Japan

Available online 24 December 2008

Abstract

This paper proposes the Lagrange multiplier (LM) test, or the score test, for jumps in the stochastic volatility (SV) model inthe cases where the innovation term follows the normal and Student t-distributions. The tested null hypothesis is that the jumpdensity has zero variance, which is expressed by Dirac’s delta function. It is shown that the unknown jump probability, which isan unidentified parameter under the null hypothesis, is cancelled out in the LM test statistic, and hence this test is free from theestimation problem of unidentified parameters, which is known as the Davies problem [R.B. Davies, Hypothesis testing when anuisance parameter is present only under the alternative, Biometrika 64 (1977) 247–254]. Monte Carlo experiments show that thenull distribution of the LM test statistic can be approximated by the normal distribution with sufficient accuracy.© 2008 IMACS. Published by Elsevier B.V. All rights reserved.

JEL classification: C12; C22

Keywords: Davies Problem; Dirac’s delta function; Jump process; Lagrange multiplier test; Stochastic volatility process

1. Introduction

The presence of jumps is an important topic in time series analysis, because the excess skewness and kurtosis thatcannot be explained by the stochastic volatility (SV) and ARCH-type models are often attributed to jumps in levelsor volatility. In spite of the growing literature on their estimation, less attention has been paid to standard parametrictests for jumps. Khalaf et al. [20] demonstrated, in the context of the GARCH model, that parametric testing forjumps is a difficult problem because the standard test statistics for jumps include the jump probability, which is anunidentified nuisance parameter and cannot be estimated consistently under the null hypothesis. Then the asymptoticnull distribution of the standard test statistics, such as the Wald and likelihood ratio test statistics, is not easily tractableand this difficulty is called as the Davies problem [9].

In this paper the jump is defined as an occasional and unexpected increase of volatility of levels in the discrete timeSV model. This approach is different form that of Barndorff-Nielsen and Shephard [2], where the jump is defined ina continuous time framework. Our specification of jumps is similar to the models considered by Chib et al. [6] andEraker et al. [13].

The maximum likelihood (ML) estimate used in constructing the test statistic is obtained by Kitagawa’s nonlinearfiltering algorithm [21] for the SV model, where the likelihood is evaluated by a one-dimenstional numerical integration.This method was applied by Watanabe [28] and Friedman and Harris [14] to the estimation of the SV model. The useof state space representation of the SV model is essential in our paper, since we use the formula of Hamilton [15].

∗ Tel.: +81 45 339 3544; fax: +81 45 339 3518.E-mail address: [email protected].

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2598 M. Kobayashi / Mathematics and Computers in Simulation 79 (2009) 2597–2608

He showed that the LM test statistic of the Markov switching model, a special case of the state-space model, can beexpressed as an expected value with respect to smoothed density. See McAleer [24] and Asai et al. [1] for extensivesurvey of the estimation and modelling of the stochastic volatility models.

In this paper the null hypothesis is defined as the zero variance of the jump distribution and the jump variance is anuisance parameter unidentified under the null, because the jump probability cannot be identified when the jump sizeis zero. It is shown that the nuisance parameter unidentified under the null is cancelled out in the score test statistic andhence the test is free from the estimation problem of the unidentified parameter.

The score test is derived by regarding the jump distribution density with zero variance as Dirac’s delta function.The use of Dirac’s delta function can be regarded as a formalization of the method proposed by Cox [7] and Dean [10]in deriving the score test statistic for the presence of overdispersion in a count data analysis and random effects.

It is also possible to consider the null hypothesis that the jump probability is zero, regarding the jump variance as anuisance parameter unidentified under the null. However, the nuisance parameter cannot be cancelled out in the scoretest statistic so that this approach is not pursued here.

We show that the actual distribution of the test statistic is approximated with sufficient accuracy by the normaldistribution by Monte Carlo experiments. An empirical example is also reported.

2. Estimation of the SV model with normal errors

We first define the SV model with normal errors as

yt = σtut, (2.1)

θt+1 = α+ βθt + σvt+1, σ2t = exp(θt), |β| < 1, (2.2)

where the serially independent random vector (ut, vt+1)′ is normally distributed with mean zero, unit variance, andzero correlation coefficient. The first Eq. (2.1) is the measurement equation and the second Eq. (2.2) is the transitionequation in the framework of the state space modelling. The conditional densities of the observed variable yt and thelog volatility θt are written as

f (yt|θt) = 1√2πσ2

t

exp

(− y2

t

2σ2t

), (2.3)

f (θt+1|θt) = 1√2πσ2

exp

(− (θt+1 − α− βθt)2

2σ2

). (2.4)

We here assume that

θ1 ∼ N

(α

1 − β,

σ2

1 − β2

).

First, given the density of θt−1 conditional on the information up to t − 1, namely f (θt−1|y1, . . . , yt−1), the densityof the predicted state variable θt is expressed as the integral

f (θt|y1, . . . , yt−1) =∫ ∞

−∞f (θt|θt−1)f (θt−1|y1, . . . , yt−1)dθt−1, (2.5)

since it is implicitly assumed that the transition Eq. (2.2) given θt−1 is independent of y1, . . . , yt−1, namely f (θt|θt−1) =f (θt|θt−1, y1, . . . , yt−1). The integration interval (−∞,∞) will be suppressed hereafter where there is no fear ofambiguity. Next, we can update the conditional density of θt by obtaining the new observation yt as

f (θt|y1, . . . , yt) = f (yt|θt)f (θt|y1, . . . , yt−1)

f (yt|y1, . . . , yt−1), (2.6)

where

f (yt|y1, . . . , yt−1) =∫f (yt|θt)f (θt|y1, . . . , yt−1)dθt, (2.7)

M. Kobayashi / Mathematics and Computers in Simulation 79 (2009) 2597–2608 2599

since it is assumed implicitly that the measurement Eq. (2.1) is independent of the past observations y1, . . . , yt−1,namely f (yt|θt) = f (yt|θt, y1, . . . , yt−1). The process of obtaining the conditional density of the state variable θtgiven y1, . . . , yt is called “filtering” in the state-space framework, and we have the conditional likelihood of yt inthe denominator of (2.6) as a byproduct of filtering. We can obtain the conditional likelihood f (y1), f (y2|y1),. . .,f (yT |y1, . . . , yT−1) using (2.6) and (2.7) recursively by evaluating the integrals in (2.5) and (2.7) by means of numericalintegration such as the trapezoidal rule or Monte Carlo integration, and hence we obtain the unconditional densityf (y1, . . . , yT ) and the maximum likelihood estimator.

In obtaining the score statistic the conditional density of the state variable θt given y1, . . . , yT , namelyf (θt|y1, . . . , yT ), is required, as will be shown in the next section. This process is referred to by “smoothing”. We seethat

f (θt−1, θt|y1, . . . , yT ) = f (θt|y1, . . . , yT )f (θt−1|θt, y1, . . . , yt−1)

= f (θt|y1, . . . , yT )f (θt|θt−1)f (θt−1|y1, . . . , yt−1)

f (θt|y1, . . . , yt−1), (2.8)

from the Bayes theorem and

f (θt−1|θt, y1, . . . , yt−1) = f (θt−1|θt, y1, . . . , yT ). (2.9)

The Eq. (2.9) is intuitive, since it is evident from (2.1) and (2.2) that, given θt , the future observations yt, . . . , yThave no additional information with respect to θt−1. The smoothed density

f (θt−1|y1, . . . , yT ) =∫f (θt−1, θt|y1, . . . , yT )dθt (2.10)

is derived by integrating out θt in (2.8). Then we can obtain the smoothed density at t − 1 using the smoothed densityat t, the transition density f (θt|θt−1), the filtered density f (θt−1|y1, . . . , yt−1) in (2.6) and the predicted densityf (θt|y1, . . . , yt−1) in (2.5).

In our paper the results of the Monte Carlo experiments and empirical analysis are reported using the trapezoidalrule in evaluating the integrals in (2.6), (2.7), and (2.10) numerically. However, the algorithm of numerical integrationin filtering and smoothing is inessential, since they can be evaluated by Monte Carlo integration proposed by Durbinand Koopman [11].

3. LM test for jumps in SV with normal errors

We here assume that a jump in levels occurs with probability p and that the jump size has normal distribution withmean zero and variance λ and that the occurrence and size of a jump are mutually independent. The process can beexpressed as

yt = σtut + et, (3.1)

θt+1 = α+ βθt + σvt+1, σ2t = exp(θt), (3.2)

and the distribution of the jump variable et is a mixture of a normal distribution and 0 with weights p and 1 − p, namely

et ∼{N(0, λ) with probabilityp,

0 with probability 1 − p,(3.3)

where 0 denotes a degenerate distribution with all probability mass at 0. It is also assumed that the jump variable et isdistributed independently of vt and ut , and et itself is a serially independent series. The occurrence of a jump and itssize are also assumed to be independent. Then the density of the jump variable can be expressed as

fλ(et) = pφλ(et) + (1 − p)δ(et), (3.4)

where

φλ(et) = 1√2πλ

exp

(− e2

t

2λ

), (3.5)


and δ(·) is Dirac’s delta function, which is a degenerate density function with all probability mass at 0. It can be easilyshown by integration by parts that∫

k(x)δ(x) dx = k(0), (3.6)

∫k(x)

d

dxδ(x) dx = − d

dxk(0), (3.7)

∫k(x)

d2

dx2 δ(x) dx = d2

dx2 k(0) (3.8)

for an arbitrary regular function k(x). The normal density function with infinitely small variance, such as φλ(et) in (3.3)when λ is infinitely small, and the t-density with infinitely small scale parameter are examples of Dirac’s delta function.We will use the formulas (3.7) and (3.8) in deriving the test statistic oager. For the usage of Dirac’s delta function inother contexts, see Kobayashi [22] and Kobayashi and Shi [23]. Detailed discussion of Dirac’s delta function can befound in textbooks of the Fourier transformation, for example in Bracewell [4].

The alternative hypothesis with jumps can be expressed as a nonlinear state space model as follows:

fλ(yt|et, θt) = 1√2πσ2

t

exp

(− (yt − et)2

2σ2t

), (3.9)

f (θt+1|θt) = 1√2πσ2

exp

(− (θt+1 − α− βθt)2

2σ2

), (3.10)

where fλ denotes a density under the alternative hypothesis, namely in the presence of jumps. Then the density functionof y1, . . . , yT with jumps in levels is expressed as

fλ(y1, y2, . . . , yT )

=∫ ∞

−∞...∫ ∞

−∞fλ(yT |θT )f (θT |θT−1)fλ(yT−1|θT−1)· · ·f (θ2|θ1)fλ(y1|θ1)f (θ1)dθ1· · ·dθT , (3.11)

where

fλ(yt|θt) =∫ ∞

−∞fλ(yt|et, θt)fλ(et)det. (3.12)

We here obtain the Lagrange multiplier (LM) test statistic for the null hypothesis λ = 0 in (3.3) against the alternativemodel (3.1) and (3.3). The derivation of the test statistic is essentially similar to that of Cox [7] and Dean [10], whoconsidered the test for the presence of overdispersion in a count data analysis and random effects. In our paper thedegenerate density of the jump distribution or dispersion distribution under the null is expressed by Dirac’s deltafunction, which can be regarded as a formalization of Cox’s idea and hence is applicable more widely in time seriesanalysis, as well as in count-data analysis. We also show that the jump probability p, which cannot be estimated underthe null hypothesis, is not included in the LM test statistic, so that p is no longer an obstacle in deriving the distributionof the test statistic.

The Lagrange multiplier test rejects the null hypothesis of the absence of jumps in levels, namely λ = 0, when

∂ log fλ(y1, y2, . . . , yT )

∂λ

∣∣∣∣λ=0

(3.13)

differs sufficiently from zero, because it would be distributed with mean zero under the null hypothesis λ = 0. Thistest is useful when the restricted model is easier to estimate than the unrestricted model, because only the estimatesof the restricted model are used to construct the LM test statistic. For more detail of the LM test, see Davidson andMcKinnon [8], Breusch and Pagan [5] and Engle [12].


We have only to evaluate

∂

∂λfλ(y1, y2, . . . , yT ) =

∫. . .

∫∂fλ(yT |θT )

∂λf (θT |θT−1)fλ(yT−1|θT−1)· · ·fλ(y1|θ1)f (θ1)dθ1· · ·dθT

+∫. . .

∫fλ(yT |θT )f (θT |θT−1)

∂fλ(yT−1|θT−1)

∂λ· · ·fλ(y1|θ1)f (θ1)dθ1· · ·dθT

+∫. . .

∫fλ(yT |θT )f (θT |θT−1)fλ(yT−1|θT−1)· · ·∂fλ(y1|θ1)

∂λf (θ1)dθ1· · ·dθT .

(3.14)

under the null hypothesis. First, we have that

∂fλ(yt|θt)∂λ

∣∣∣∣λ=0

= p

∫fλ(yt|et, θt)∂φλ(et)

∂λdet

∣∣∣∣λ=0

= p

∫fλ(yt|et, θt)1

2

∂2φλ(et)

∂e2t

det

∣∣∣∣et=0

, (3.15)

using the equalities

∂φλ(et)

∂λ=(

1

2

)φλ(et)

(e2t

λ2 − 1

λ

)=(

1

2

)∂2φλ(et)

∂e2t

(3.16)

∂fλ(et)

∂λ= p

∂φλ(et)

∂λ. (3.17)

Then, we have that

∂

∂λfλ(yt|θt)

∣∣∣∣λ=0

= p

2

∂2

∂e2t

fλ(yt|et, θt)∣∣∣∣et=0

= p

2Atf (yt|θt), (3.18)

where

At = (yt − et)2

σ4t

− 1

σ2t

, (3.19)

using the formula (3.8) of Dirac’s delta function, since the normal density with variance λ, namely φλ(et), can beregarded as Dirac’s delta function when λ is infinitely small, In fact, the use of Dirac’s delta function is not essentialin this simplest case where ut is normally distributed, since fλ(yt|θt) in (3.12) can be expressed in a closed formby integrating out et analytically. However, the use of Dirac’s delta function is essential in the general case wherethe integral cannot be evaluated. We will consider the case where the observation error ut and the jump et follow tdistribution.

Then, substituting (3.19) into (3.14), the score function with respect to λis expressed as

∂ log fλ(y1, . . . , yT )

∂λ

∣∣∣∣λ=0

= ∂fλ(y1, . . . , yT )/∂λ|λ=0

f (y1, . . . , yT )= p

2

T∑t=1

∫. . .

∫Atf (θ1, . . . , θT |y1, . . . , yT )dθ1· · ·dθT ,

(3.20)

and the conditional density of θ1, . . . , θT given y1, . . . , yT is defined by

f (θ1, . . . , θT |y1, . . . , yT ) = f (yT |θT )f (θT |θT−1)f (yT−1|θT−1)· · ·f (θ2|θ1)f (y1|θ1)f (θ1)

f (y1, . . . , yT ). (3.21)

Note that the distribution of y1, y2, . . . , yT is independent of p if λ = 0 and we have that

fλ(y1, . . . , yT )|λ=0 = f (y1, . . . , yT ). (3.22)

because the jump probability p is a nuisance parameter unidentified under the null hypothesis in testing λ = 0, becausewe cannot know the jump probability if the jump size is identically zero.


The multiple integral in (3.20) is simplified to a one-dimensional integral, since At depends only upon θt and wecan express the score function with resect to λ under the null, say �T , as

�T = ∂ log fλ(y1, . . . , yT )

∂λ

∣∣∣∣λ=0

= p

2

T∑t=1

∫Atf (θt|y1, . . . , yT )dθt. (3.23)

The conditional density function f (θt|y1, . . . , yT ) can be obtained as the smoothed density of θt in (2.8) and hence theintegral in (3.23) can be evaluated numerically. See Watanabe [28] for a detailed algorithm of smoothing and filteringof the stochastic volatility model.

The LM test statistic for λ = 0 is defined by

LM1 = �2T I11, (3.24)

where I11 is the (1,1) element of the inverse of the Fisher information matrix I with respect to (ζ1, . . . , ζ4) =(λ, α, β, σ2). In this paper the Fisher information matrix I is estimated by the BHHH method (Berndt et al. [3]),as suggested by Hamilton [15], whose (i, j) th element is estimated as

Ii,j =T∑t=1

∂ log fλ(yt|y1, . . . , yt−1)

∂ζi

∣∣∣∣λ=0

∂ log fλ(yt|y1, . . . , yt−1)

∂ζj

∣∣∣∣λ=0

. (3.25)

Then, it can be expressed as

I = F′F, (3.26)

where F is a T-by-4 matrix whose (t, i) th element is

Fti = ∂ log fλ(yt|y1, . . . , yt−1)

∂ζi

∣∣∣∣λ=0

. (3.27)

The first derivative function of the log likelihood of yt conditional on y1, . . . , yt−1 with respect to λ can be evaluatedby differencing the log likelihood as

∂ log fλ(yt|y1, . . . , yt−1)

∂λ

∣∣∣∣λ=0

= ∂ log fλ(y1, . . . , yt)

∂λ

∣∣∣∣λ=0

− ∂ log fλ(y1, . . . , yt−1)

∂λ

∣∣∣∣λ=0

, (3.28)

where the derivative of the log likelihood for y1, . . . , yt for t < T on the right-hand side can be evaluated iteratively byapplying the same integration routine used in obtaining (3.23) for t = 1, . . . , T . This step is the most time consumingpart of the calculation of the test statistic. The first derivative of the conditional log likelihood with respect to α, β,and σ2 in (3.25) can be evaluated easily by differentiating numerical the conditional log likelihood function of the SVmodel evaluated in (2.7).

It is intuitive that

LM1 = �2T I11 = (�2

T /p2)I11p2

is independent of p, since �T /p is independent of p, as shown in (3.23), and 1/(I11p2) is the asymptotic variance of�T /p, and hence independent of p. This independency is shown more rigorously as follows. First, we see that from(3.23) and (3.28) that p−1�T and FP are independent of the jump probability p, where

P = diagonal(p−1, 1, 1, 1), (3.29)

because p appears in multiplicative form in �T and the conditional log likelihood with respect to λ, and hence in thefirst column of F; the elements of the other columns, namely the derivatives of log fλ(yt|y1, . . . , yt−1) with respect toα, β, σ2 evaluated under the null are independent of p, because they are based upon the log likelihood of the SV modelwithout jumps. Then, since �T /p and FP are independent of p, we see that

�2T I11 = �2

T (1, 0, 0, 0)(F′F)−1(1, 0, 0, 0)′ = �2T (1, 0, 0, 0)P[(FP)′(FP)]−1P(1, 0, 0, 0)′

= (�T /p)2(1, 0, 0, 0)[(FP)′(FP)]−1(1, 0, 0, 0)′ (3.30)

is independent of p.


It is supposed thatLM1 followsχ2 distribution with one degree of freedom asymptotically under the null hypothesis,whose proof is not given here. The actual distribution of the test statistic is examined by Monte Carlo experiments.

We can summarize the result as follows.

Proposition 1. The Lagrange multiplier test statistic for λ = 0 in the SV model defined by (3.1)–(3.3)is expressed as

LM1 = �2T I11, (3.31)

where �T is defined by (3.23) and I11 by the (1, 1) element of the inverse of I defined by (3.26), and the null distributionof the test statistic LM1 is independent of the jump probability p, which is an unidentified parameter under the null.

We can obtain the one-sided version of the LM test statistic as follows:

Proposition 2. The one-sided Lagrange multiplier test statistic for λ = 0 the SV model defined by (3.1)–(3.3)isexpressed as

LM2 = �T

√I11 (3.32)

where �T is defined in (3.23) and I11 is the (1, 1) element of I−1 defined by (3.26), and the null distribution of the teststatistic LM2 is independent of the jump probability p, which is an unidentified parameter under the null.

The null distribution of LM2 is supposed to be the standard normal distribution, which is examined by Monte Carloexperiments in the next section. This one-sided test has the rejection region only in the upper tail are a, namely the nullshould be rejected only if

∂ log fλ(y1, . . . , yT )

∂λ

∣∣∣∣λ=0

has a sufficiently large positive value, since the null hypothesis λ = 0 should be rejected against the alternative λ > 0and

∂ log fλ(y1, . . . , yT )

∂λ

∣∣∣∣λ=0

< 0

implies the negative ML estimate of λ and hence is not an evidence of λ > 0. The asymptotic critical value of the teststatistic at level α is the upper 100 × α percentile of the standard normal distribution. It is easy to see that this one-sidedtest statistic is more powerful than the ordinary Lagrange multiplier test statistic defined by (3.24). See Tanaka [27],Honda [17], and Rogers [26] for more examples where the one-sided version of the Lagrange multiplier test statisticis available using nonnegative restriction in testing for a single parameter.

4. Monte Carlo experiment and empirical examples

In this section, the actual distribution of the LM test statistic under the null and alternative hypotheses is examinedby Monte Carlo experiments in the case where the error term follows normal distribution. The sample size is 1000and the number of iteration is 550 in each case. In our Monte Carlo experiment one iteration under the null hypothesistakes approximately 1 min using GAUSS 7.0 on a PC with Pentium M, but it takes much longer under the alternativehypothesis. The number of intervals to be used for the numerical integration in nonlinear filtering and smoothing is101, and the coefficient β is estimated indirectly using the logistic transformation

β = 2

1 + exp(−γ)− 1

and hence the ML estimation of γ ensures that the estimated β always satisfies the “stationarity” condition−1 < β < 1.

In our experiment the parameter value of α is set at 0 without loss of generality, since the sample variance of theseries can be standardized to be 1 by dividing the standard deviation, which corresponds to α = 0 approximately. Thevalues of the other parameters are set near to the estimates of the empirical analysis reported later.


Table 1Summary of the ML estimates under the null.

Parameter values Mean Standard deviation (S.D.)

α β σ α β σ α β σ

0.0 0.75 0.2 −0.005 0.606 0.226 0.032 0.339 0.1000.0 0.75 0.4 −0.003 0.721 0.408 0.023 0.120 0.0890.0 0.75 0.6 −0.002 0.736 0.607 0.025 0.062 0.0810.0 0.75 0.8 −0.002 0.741 0.804 0.032 0.049 0.0770.0 0.95 0.2 0.000 0.941 0.207 0.008 0.027 0.0440.0 0.95 0.4 −0.0007 0.944 0.408 0.015 0.016 0.0470.0 0.95 0.6 0.0005 0.947 0.599 0.021 0.013 0.0480.0 0.95 0.8 −0.0009 0.946 0.796 0.026 0.013 0.059

Only the one-sided version of the LM test statistic, namely LM2 in (3.32), is considered here, since the distributionof the two-sided version of the test statistic, LM1 in (3.24), is supposed to be χ2 with one degree of freedom under thenull, and can be obtained from that of LM2, which is supposed to be N(0, 1). Table 1 summarizes the distribution ofthe parameter estimates of the SV model and Table 2 shows the distribution of the one-sided version of the LM teststatistic (3.32) under the null hypothesis. The Jarque–Bera [19] statistic in Table 2 rejects the normality assumptionat 5 percent significance level in the case of (σ, β) = (0.2, 0.75), (0.6, 0.95), (0.8, 0.95). As far as the ratio of LM2exceeding the upper 5 percentile of N(0, 1) in Table 2 shows, however, the size distortion of the test is not seriouswhen the data generating process has parameter values (β, σ) = (0.75, 0.6), (0.95, 0.2), which are near to the empiricalestimates in Table 4.

Table 3 shows the actual power of the test when the parameter values of the data generating process are p =0.02, 0.05,

√λ = 5, 10, (β, σ) = (0.75, 0.6), (0.95, 0.2), under the assumption that the standard deviation of jumps,√

λ, five and ten times as large as the of the SV process is not unrealistic. The choice of combinations of (β, σ) simulatesthe actual estimates given in Table 4. Table 3 shows that the presence of jumps is detected with higher probability asthe jump size and probability increase.

We have tested the presence of jumps in the US dollar–Japanese yen exchange rate returns in the New York marketusing the daily series from 22 November 1989 to 25 October 2001, which is divided into three periods. The series isillustrated in Fig. 1. The null hypothesis of no jumps is rejected only for the third period (5 November 1997–25 October2001), where the test statistic LM2 exceeds 1.645, namely the 95 percentile of N(0, 1). It is supposed that our test hasdetected the turmoil of the Asian financial crisis started in 1997.

Table 2Normality of the one-sided LM test under the null.

Parameter values Summary of the distribution of LM2

α β σ Mean S.D. Skewness Kurtosis Jarque–Bera Pr(LM2 > z0.95)

0.0 0.75 0.2 −0.251 1.146 −0.200 3.32 6.06 0.04360.0 0.75 0.4 −0.111 1.045 −0.162 2.68 4.70 0.03450.0 0.75 0.6 0.015 0.988 −0.006 2.99 0.005 0.04910.0 0.75 0.8 0.060 1.023 −0.085 3.09 0.837 0.04910.0 0.95 0.2 0.087 1.000 −0.19 3.17 3.82 0.04550.0 0.95 0.4 0.145 0.988 −0.040 3.35 3.00 0.06360.0 0.95 0.6 −0.015 1.092 −0.557 4.32 68.23 0.04910.0 0.95 0.8 −0.352 1.321 −0.878 4.27 107.67 0.0345

Note: the number of iteration is 550 and the sample size is 1000. The Jarque–Bera [19] test statistic for normality is asymptotically distributed asχ2 with 2 degrees of freedom under the null hypothesis. The critical value of the Jarque–Bera test statistic at size 0.05 is 5.99. The 95 percentile ofN(0, 1), denoted as z0.95, is 1.645.


Table 3Power of the one-sided LM test statistic under the alternative.

Parameter values Summary of LM2

α β σ p√λ Mean S.D. Pr(LM2 > z0.95)

0.0 0.75 0.6 0.02 5 1.17 1.01 0.3250.0 0.75 0.6 0.05 5 1.97 1.07 0.6220.0 0.75 0.6 0.02 10 2.70 1.10 0.8200.0 0.75 0.6 0.05 10 3.40 1.13 0.9310.0 0.95 0.2 0.02 5 2.17 1.16 0.6780.0 0.95 0.2 0.05 5 3.20 1.09 0.9200.0 0.95 0.2 0.02 10 4.03 1.12 0.9850.0 0.95 0.2 0.05 10 4.49 1.20 1.00

Note: the number of iterations is 550 and the sample size is 1000. The 95 percentile of N(0, 1), denoted as z0.95, is 1.645.

Table 4Testing of jumps in US dollar–Japanese yen exchange rate returns.

Period Sample size Parameter estimates Test for jumps

α β σ LM2

22 November 1989–15 November 1993 1000 −0.084 0.726 0.548 −1.04216 November 1993–4 November 1997 1000 −0.108 0.735 0.599 1.3085 November 1997–25 October 2001 1000 −0.011 0.961 0.183 3.023

5. Extension to the SV model with t-distributed errors

In this section we generalize the results of Propositions 1 and 2 to the case where the observation error term andjump size have t-distribution, which has a fatter tail than normal distribution. We next see that the test statistic can bederived in the case where the observation equation error and transition equation error are correlated. The derivation ofthese test statistic is by far more time-consuming than the normal case considered in the previous sections so that theactual distribution of the test statistic cannot be examined by Monte Carlo experiments here.

The derivation of the test statistic in the case of t-distributed errors is similar to that of the normality case, so thatonly major differences from the normality case are illustrated in this section. It is assumed that the observation errorterm ut in (3.1) follows t distribution with unknown degrees of freedom, say μ, and the density of yt conditional onlog volatility θt and jump et is expressed as

fλ(yt|θt, et) = � ((μ+ 1)/2)√πσ2

t μ� (μ/2)

(1 + (yt − et)2

σ2t μ

)−(1/2)(μ+1)

. (5.1)

Fig. 1. US dollars–Japanese yen exchange rate return (daily, 22 November 1989–25 October 2001, New York).


The density of the jump variable et is assumed to be

fλ(et) = pψλ(et ;m) + (1 − p)δ(et), (5.2)

where ψλ(et ;m) is the density of t distribution with m degrees of freedom and scale parameter√λ/m, which is

expressed as

ψλ(et ;m) = � ((m+ 1)/2)√πλ� (m/2)

(1 + e2

t

λ

)−(1/2)(m+1)

. (5.3)

The degrees of freedom parameter m is a nuisance parameter unidentified under the null, because it is unobservable ifthe jump size is zero, namely if λ = 0. We show that this nuisance parameter, as well as λ, is cancelled out in the teststatistic. For this jump size density we have the equality which corresponds to (3.16) as

1

2(m− 2)

∂2ψλ(et ;m− 2)

∂e2t

= ∂ψλ(et ;m)

∂λ, (5.4)

since we have that

∂ψλ(et ;m)

∂λ= 1

2

�((m+ 1)/2)√λπ�(m/2)

[(m+ 1)

(1 + e2

t

λ

)−1e2t

λ4 − 1

λ

](1 + e2

t

λ

)−(1/2)(m+1)

, (5.5)

∂2ψλ(et ;m− 2)

∂e2t

= 2�((m− 1)/2)((m− 1)/2)√πλ�((m− 2)/2)

[(m+ 1)

(1 + e2

t

λ

)−1e2t

λ4 − 1

λ

](1 + e2

t

λ

)−(1/2)(m+1)

, (5.6)

�

(m− 1

2

)m− 1

2= �

(m+ 1

2

), �

(m2

)= �

(m− 2

2

)m− 2

2. (5.7)

Then, (5.4) implies that the first derivative of the jump size density with respect to λ can be replaced with the secondderivative with respect to et up to a constant 1/(m− 2), though with different degrees of freedom, as in (3.18). Notingthat ψλ(et ;m− 2) can be regarded ad Dirac’s delta function when λ is infinitely small, we have that

∂

∂λfλ(yt|θt)

∣∣∣∣λ=0

= p

2(m− 2)

∂2

∂e2t

f (yt|et, θt)∣∣∣∣et=0

= p

2(m− 2)Btf (yt|θt), (5.8)

where

Bt = μ+ 1

μ

[μ+ 3

μ

(1 + y2

t

σ2t μ

)−2y2t

σ4t

−(

1 + y2t

σ2t μ

)−11

σ2t

](5.9)

using the formula of Dirac’s delta function (3.8).Then, as in (3.23), we can express the score function with respect to λ evaluated under the null, say �T , as

�T = ∂ log fλ(y1, . . . , yT )

∂λ

∣∣∣∣λ=0

= p

2(m− 2)

T∑t=1

∫Btf (θt|y1, . . . , yT )dθt. (5.10)

Then the two-sided and one-sided LM test statistics for the null hypothesis λ = 0 can be expressed as

LM3 = �2T I11, (5.11)

LM4 = �T

√I11, (5.12)

where I11 is the (1, 1) element of the inverse of the Fisher information matrix I with respect to (ζ1, . . . , ζ5) =(λ, α, β, σ2, μ). In this paper it is estimated by

I = F′F, (5.13)


where F is a T-by-5 matrix whose (t, i) th element is

Fti = ∂ log fλ(yt|y1, . . . , yt−1)

∂ζi

∣∣∣∣λ=0

. (5.14)

It can be also shown that LM3 and LM4 are independent of the jump probability p and the degrees of freedom ofthe jump distribution m, by using P = (p/(m− 2), 1, 1, 1, 1)′ in (3.29) and (3.30), instead of P = (p, 1, 1, 1, 1)′,and p/(m− 2) is cancelled out because it is included in multiplicative form in the score function in (5.10), which isanalogous to (3.23).

The above result can be summarized in the following proposition.

Proposition 3. The two-sided and one-sided LM test statistics for jumps in the SV model with the t-distributedobservation errors and jumps with the densities (5.1) and (5.2), respectively, are expressed by (5.11) and (5.12),respectively, and their null distribution is independent of the jump probability p and the degrees of freedom of the jumpdistribution m.

6. Concluding remarks

Theoretically, it is also easy to extend Propositions 1 and 2 to the case of the SV model with correlated errors, whichis considered by Harvey and Shephard [16], Jacquier et al.[18], Yu [29], and Omori et al. [25]. In this specification thetransition equation is

θt+1 = α+ βθt + σvt+1 + ρut, (6.1)

instead of (2.2), and the conditional density of the log volatility is

f (θt+1|θt, yt) = 1√2πσ2

exp

(− (θt+1 − α− βθt − ρyt/σt)2

2σ2

), (6.2)

instead of (2.4). The derivation of the test statistic is exactly the same except for the presence of the additional parameterρ. In this model a shock in levels would increase or decrease the volatility in the next period, according to the signof ρ, and the magnitude of the effect is proportional to ut = yt/σt . The sign of ρ is often supposed to be negative,because a large negative shock in levels might cause a panic in the market and hence an increase of volatility. Thecorrelation between levels and volatility has been regarded as a “stylized fact” in financial econometrics and is referredto by the term “leverage”. Unfortunately, the transition density (6.2) depends upon yt and this change increases thecomputational time drastically and makes the maximum likelihood estimation unstable, so that this extension is notpursued here.

Acknowledgments

The author would like to thank Yasuhiro Omori, seminar participants at Hiroshima University and the Universityof Tokyo, and an anonymous referee for their helpful comments. The financial supports by the Grant-in-Aid forScientific Research (17530239) from the Japan Society for the Promotion of Science, and the Japan Economic ResearchFoundation are gratefully acknowledged.

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testing for jumps in the stochastic volatility models

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