Journal of the Chinese
Statistical Association
Vol. 48, (2010) 31–44
ON EMPIRICAL DISTRIBUTION OF
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
Hwai-Chung Ho1 and Fang-I Liu2
1 Institute
of Statistical Science, Academia Sinica
2 Department
of Finance, National Taiwan University
ABSTRACT
Theoretical properties of the empirical distribution are investigated in the context
of long-memory stochastic volatility (LMSV). The LMSV process is a stationary time
series which is formed by martingale differences that contain a latent volatility process
exhibiting long memory. We show that in spite of the fact that the LMSV process is
martingale differences, its empirical distribution may converge to a normal distribution
√
with a rate slower than n.
Key words and phrases: Empirical distribution, long-memory stochastic volatility model.
JEL classification: C50.
1. Introduction
Due to the wide range of its applications in statistics, from parametric to nonparametric estimation, the theory of empirical processes has always been fundamentally important in statistical inference. Let Yt , t ∈ Z = {. . . , −1, 0, +1, . . .}, be realvalued strictly stationary sequence (time series). Given observations Y1 , . . . , Yn , the
classical Glivenko-Cantelli theorem shows that empirical distribution function Fn (y) =
n
1 P
I (Yt < y) converges a.s. to the marginal distribution function FY (y) = P (Yt < y)
n
t=1
uniformly in y ∈ R. The magnitude of Fn (y) − F (y) can be very different depend-
ing on the dependence properties and the structure of the underlying process {Yt }.
32
HWAI-CHUNG HO AND FANG-I LIU
For independent observations, the normalized error
√
n(Fn (y) − FY (y)) approaches a
Gaussian process U (y) with independent increments, with zero mean and the covariance
Cov(U (x), U (y)) = FY (x∧ y)− FY (x)FY (y). The central limit theorem also holds if the
stationary sequence Yt is weakly dependent and satisfies φ-mixing condition (Billingsley, 1968). See Andrews and Pollard (1994) and Arcones and Yu (1994) for surveys of
empirical processes for mixing sequences. Under suitable mixing rates, results of this
sort usually assert that empirical processes behaves as if the observations were i.i.d. The
asymptotic behavior of the empirical process changes dramatically in the case of longmemory dependent sequence. For long-memory observations, the empirical distribution
√
converges to the true distribution function at a rate slower than n. One commonly
seen class of long-memory processes is in the form of infinite moving averages,
Zt =
∞
X
i=1
ai εt−i ,
ai ∼ c · i−β ,
(1)
where the i.i.d. innovations {εi } have zero mean and finite variance σε2 , c is some positive
constant, β ∈ (1/2, 1), and gn ∼ hn signifying lim gn /hn = 1. The term “longn→∞
P
memory” is referred to the property of i |ai | = ∞ or the fact that the autocovariance
P
P 2β−1
2
function γ(j) of {Zt } is not summable since
= ∞. Note
j |γ(j)| ∼ c
jj
that we do not assume εi to be Gaussian. Two important models of long-memory
linear process (1) is the fractional autoregressive integrated moving average (ARFIMA)
process of Adenstedt (1974), Granger and Joyeux (1980) and Hosking (1981) and the
fractional Gaussian processes (FGN) of Mandelbrot and van Ness (1968). Ho and
Hsing (1996) obtain an asymptotic expansion of the empirical process of long-memory
moving averages and develop a new approach to the study of nonlinear functional
of non-Gaussian moving averages which in some sense replaces the method of Hermite
expansions in the Gaussian case. They derive an asymptotic expansion of Fn (y) similar
in spirit to a Taylor expansion. While there is extensive literature on the study of
empirical process for stationary sequences, the corresponding work for an important
class of nonlinear time series remains largely unaccomplished. The development of
this class of time series can be traced back to the clustering volatility property about
speculative returns. The evidence has been observed that strong serial correlation exists
in some nonlinear transformation of many financial returns, such as square, logarithm
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
33
of square, and absolute value, whereas the return series itself behaves almost like white
noise (see, e.g., Taylor (1986), and Ding, Granger, and Engle (1993)). The so-called
clustering volatility property has a profound implication. Stationary models that have
been proposed to described the properties mentioned above include the ARCH (or
GARCH) family (Engle (1982) and Bollerslev (1986)), and the stochastic volatility
(SV) model (see, for example, Taylor (1986) and Harvey, Ruiz, and Shephard (1994)).
Recently, models other than ARCH family have been seen to provide better fitting for
empirical data. Lobato and Savin (1998) examines the S&P 500 index returns for the
period of July 1962 to December 1994 and report that the squared daily returns exhibit
the genuine long-memory effect which ARCH process cannot produce (see also Ding et
al., 1993). Following the work of Lobato and Savin (1998), Breidt, Crato, and De Lima
(1998) suggest the following long-memory stochastic volatility model (LMSV):
Yt = µ + σt ut ,
σt = σ̄ exp(Zt /2),
(2)
with where σ̄ > 0, {ut } is an i.i.d. sequence with Eut = 0 and Eu2t = 1, and is independent of the latent volatility component {Zt }, which is a linear process defined
as equation (1). The LMSV process is a stationary time series formed by martingale
differences that contain a latent volatility process exhibiting long memory and can better fit the decay rate of autocorrelations of returns’ volatility than some other popular
models such as IGARCH(1,1) and GARCH(1.1).
In this paper we derive the asymptotic distribution of empirical distribution for
the LMSV model. The result can be applied to making inference on the distribution
function of financial returns. Knowledge of distribution functions of asset returns is an
essential piece of information for both the researchers and practitioners when studying
important issues in quantitative finance like the mean-variance analysis of portfolios
and equity market efficiency. Especially, there is strong evidence of departure from
normality for the distribution function of speculative returns, as revealed by the nonnegligible probability of large market movements.
The rest of the paper is organized as follows. A central limit theorem for the
empirical distribution based on LMSV process is presented in the next section. In
section 3 we conduct a simulation study to evaluate the finite-sample performance of
34
HWAI-CHUNG HO AND FANG-I LIU
the theory developed. An empirical analysis based on the S&P 500 return series is
provided in section 4. Proofs of the theorem is postponed to the Appendix.
2. Main Results
In the sequel the returns are modeled by LMSV process (2) with mean µ and
distribution function FY (·). Define ξ ≡ y − µ. Consider the empirical distribution
function of {Yt },
n
1X
I(Yt < y)
n
t=1
 n
n
P
P
1


I(σt ut < ξ)I(ut > 0) +
I(ut < 0) , if ξ > 0,
n
t=1
t=1
=
n
P

1

I(σt ut < ξ)I(ut < 0) ,
if ξ < 0.
n
Fn (y) ≡
(3)
t=1
We only state in the case of ξ > 0 because the proofs are similar for ξ < 0. Taking
logarithm in both side of I(σt ut < ξ), the implicit long-memory dependent process {Zt }
appears. By applying the idea of the main results in Ho and Hsing (1996), asymptotic
normality for empirical distribution of financial returns which follows LMSV process is
established. We find that the implicit long-memory effect in volatility dominates the
asymptotic behaviors of the empirical distribution.
Theorem. Suppose y − µ ≡ ξ > 0. Let {Yt } be a LMSV process defined by (1) and
(2). Then
n1−H [Fn (y) − FY (y)] → N 0, α2 η ,
i
h
(1)
where α ≡ E FZ (2(ln ξ − ln ut · I(ut > 0) − ln σ̄)) and η =
(4)
c
.
H(2H − 1)
Remark. (i) Regarding ξ < 0, the above asymptotic normality in (4) also holds with
i
h
(1)
α = E FZ (2(ln(−ξ) − ln(−ut ) · I(ut < 0) − ln σ̄)) . (ii) In the Theorem above the
σZ2 Γ(1 − d)
when Zt is ARFIMA(0, d, 0), or
Γ(d)
σZ2 d(1 + 2d) when Zt is FGN, where σZ2 denotes the variance of the linear process. (iii)
constant c of the limiting variance equals
Note that H = 1/2 + d. In the sequel both H and d will interchangeably be used to
denote the memory parameter.
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
35
An important message delivered by Theorem is that even if the LMSV sequence
itself is a sequence of martingale differences having no lag correlation, the empirical
distribution converges at the same rate as the sample mean of a long-memory process.
In other words, when applying to the LMSV sequence a particular transformation such
as the indicator function, the memory contained in the latent volatility component
emerges and makes the transformed sequence highly dependent.
3. Numerical Studies
In this section we describe the design and results of a simulation study to examine the finite sample performance of confidence intervals for FY (y). We generate 2000
replications of time series of length n = 2520, 7560 based on (2) with µ = 0.00027 and
σ̄ = 1; the long-memory process {Zt } is FGN with unit variance or FARIMA(0, d, 0)
given by (1 − B)d Zt = εt , where {εt } is Gaussian white noise with unit variance σε2 = 1
and independent of {ut }. Three values of d (0.3, 0.4 and 0.45) are considered. We study
finite sample performance of confidence intervals for FY (y) = 0.95, 0.9, 0.2, and 0.15.
Recall that Yt = µ+σt ut and Fe (·) denote the distribution function of et = σt ut . Denote
FY (y) = τ . To get the τ -th quantile y of the distribution function FY (y) = τ , we only
need to compute the τ -th quantile ξ of the distribution function
Fe and use the relation
Z
∞
FZ (2 log(ξ/σ̄y))fu (y)dy
y = µ + ξ. ξ is obtained by solving the equation τ = 0.5 +
0
Z 0
FZ (2 log(ξ/σ̄y))fu (y)dy for negative ξ,
numerically for positive ξ and τ = 1 −
−∞
where FZ and fu are the distribution function of Zt and the density function of
ut , respectively. The approximation error is controlled to be less than 0.0001 (i.e.
Fe (ξ) − τ ≤ 0.0001). To investigate the performance of (4), we plus the solved ξ into
i
h
(1)
E FZ (2(ln ξ − ln u · I(u > 0) − ln σ̄)) , and then solve the expectation by numerical
integration. The coverage probabilities of 95% confidence interval for distribution function are reported in Tables 1 and 2. The two tables serve three purpose. First, we
attempt to analyze the coverage probability of confidence interval for FY (y) with the
given d and µ of the data generating process (DGP) and name it by Theoretical DGP.
Second, we analyze the performance of confidence intervals for FY (y) with µ̂ and dˆ
36
HWAI-CHUNG HO AND FANG-I LIU
which are estimated from the corresponding simulated series, where µ̂ is the sample
mean. It is named by Theoretical DGP. We obtain dˆ through log periodogram regression using the first m Fourier frequencies and set m = n0.5 without omission of a block
of frequencies near the origin, which is considered in Deo and Hurvich (2001)’s work.
At the same time, σZ2 is roughly estimated by σ̂Z2 , which is the corresponding sample
variance of log(Yt − µ̂)2 . Third, instead of estimating d and η separately, we treat the
normalizing constants n1/2−d · η as one parameter and employ the subsampling method
introduced in Hall, Jing, and Lahiri (1998) and Nordman and Lahiri (2005) to estimate
it. It is named by Subsampling. We focus on the observed sequence of square returns,
2
Yt . Ho and Liu (2009) have shown that the sampling window method applied to
2
Yt will produces a consistent estimate of the normalizing constant n1/2−d · η. Thus,
three 95% confidence intervals are formed. Overall, the theoretical confidence interval
of FY (y) with long memory parameter d given in DGP produces quite satisfactory results. The confidence intervals for FY (y) with GPH estimator dˆ performs better than
subsampling confidence intervals.
4. Empirical Studies
In this section, the coverage accuracy of confidence interval for distribution function
is examined over daily log-return series of S&P 500 index. We assume that the returns
follow the LMSV process. We use series starting on the first trading day of January
1950 and ending on the last trading day of December 2008, which consists of fifty-nine
years of data. The mean of the daily log returns is 0.027%, and the daily volatility using
the same data is 1%. The return series and its sample autocorrelation functions are
displayed in Figures 1 and 2, respectively. Figure 3 shows the sample autocorrelations
for log-squared returns. The sample autocorrelation exhibits a slow decay, remaining
non-negligible for hundred of lags, which indicates substantial amount of dependence
in volatility.
We get the value of Fn (y) at the given daily return y = 0.001 (0.0002) to con
p
struct the two-sided 95% confidence intervals, Fn (y) − nd−1/2 · Φ−1 (0.025) · α2 η ,
37
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
Table 1
The coverage probabilities for 95% confidence interval of distribution function
when DGP is Yt = 0.00027 + exp(Zt /2) · ut , where Zt ∼ ARFIMA(0, d, 0).
ARFIMA
n
τ = 0.95
τ = 0.9
τ = 0.20
τ = 0.15
Theoretical DGP Theoretical GPH Subsampling
d = 0.45
2520
(ξ = 2.99, α = 0.037) 7560
0.92
0.93
0.76
0.82
0.69
0.76
d = 0.4
2520
(ξ = 2.38, α = 0.045) 7560
0.94
0.94
0.80
0.86
0.76
0.80
d = 0.3
2520
(ξ = 2.08, α = 0.051) 7560
0.92
0.92
0.80
0.84
0.76
0.78
d = 0.45
2520
(ξ = 1.76, α = 0.057) 7560
0.94
0.94
0.75
0.81
0.68
0.75
d = 0.4
2520
(ξ = 1.55, α = 0.071) 7560
0.96
0.96
0.81
0.87
0.75
0.81
d = 0.3
2520
(ξ = 1.43, α = 0.083) 7560
0.94
0.95
0.83
0.86
0.80
0.82
d = 0.45
2520
(ξ = −0.84, α = 0.072) 7560
0.97
0.96
0.75
0.80
0.61
0.72
d = 0.4
2520
(ξ = −0.84, α = 0.085) 7560
0.95
0.94
0.81
0.86
0.74
0.78
d = 0.3
2520
(ξ = −0.85, α = 0.100) 7560
0.93
0.94
0.82
0.86
0.78
0.80
d = 0.45
2520
(ξ = −1.19, α = 0.068) 7560
0.96
0.96
0.75
0.80
0.61
0.73
d = 0.4
2520
(ξ = −1.12, α = 0.083) 7560
0.96
0.95
0.82
0.86
0.75
0.80
d = 0.3
2520
(ξ = −1.09, α = 0.099) 7560
0.95
0.96
0.83
0.88
0.81
0.82
NOTE: The coverage probabilities of 95% confidence intervals are approximated by an
average over 2000 similation runs. The confidence intervals are constructed from (4)
where the value of α is obtained by numerical integration for each τ and d and given
in parentheses.
38
Table 2
HWAI-CHUNG HO AND FANG-I LIU
The coverage probabilities for 95% confidence interval of distribution function
when DGP is Yt = 0.00027 + exp(Zt /2) · ut , where Zt ∼ FGN.
FGN
n
Theoretical DGP Theoretical GPH Subsampling
τ = 0.95
d = 0.45 2520
(ξ = 1.95, α = 0.054)
7560
0.92
0.92
0.67
0.76
0.56
0.74
d = 0.4 2520
7560
0.93
0.93
0.77
0.83
0.67
0.76
d = 0.3 2520
7560
0.90
0.91
0.81
0.83
0.74
0.75
τ = 0.9
d = 0.45 2520
(ξ = 1.38, α = 0.090)
7560
0.96
0.96
0.70
0.76
0.59
0.64
d = 0.4 2520
7560
0.96
0.96
0.80
0.84
0.72
0.80
d = 0.3 2520
7560
0.90
0.95
0.81
0.87
0.74
0.81
τ = 0.2
d = 0.45 2520
(ξ = −0.86, α = 0.113)
7560
0.98
0.98
0.72
0.80
0.61
0.70
d = 0.4 2520
7560
0.97
0.97
0.83
0.87
0.75
0.80
d = 0.3 2520
7560
0.95
0.95
0.80
0.88
0.82
0.84
τ = 0.15
d = 0.45 2520
(ξ = −1.07, α = 0.11)
7560
0.99
0.98
0.78
0.80
0.61
0.70
d = 0.4 2520
7560
0.98
0.98
0.83
0.87
0.75
0.80
d = 0.3 2520
7560
0.96
0.97
0.81
0.90
0.82
0.84
NOTE: The coverage probabilities of 95% confidence intervals are approximated by an
average over 2000 similation runs. The confidence intervals are constructed from (4)
where the value of α is obtained by numerical integration for each τ and d and given
in parentheses.
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
Figure 1
Time series plot of S&P 500 log-returns.
Figure 2
Figure 3
ACF for S&P 500 returns.
ACF for log-squared S&P 500 returns.
39
40
HWAI-CHUNG HO AND FANG-I LIU
Table 3
Coverage probabilities of 95% confidence intervals for distribution function
FY (y), which is examined over daily log-return series of S&P 500 index.
length of
learning samples
10 years
y = 0.0002
y = 0.001
Theoretical GPH Subsampling
Theoretical GPH Subsampling
0.95
0.90
0.98
0.93
NOTE: The real value of distribution function FY (y) is estimated by the corresponding
empirical distribution function over daily returns from 1950 through 2008, which equals
to 0.53 for y = 0.001 and 0.48 for y = 0.0002.
Fn (y) + nd−1/2 · Φ−1 (0.025) ·
p
α2 η . Two 95% confidence intervals for distribution
function are formed. One is theoretical confidence interval, in which we assume the
daily return follows (2) with Zt ∼ ARFIMA(0, d, 0). The long-memory parameter is
estimated by GPH estimator dˆ (e.g., Geweke and Porter-Hudak (1983)). Alternatively,
instead of estimating the long-memory parameter d, sampling window method provided
by Ho and Liu (2009) is used to estimate the normalizing constant n1/2−d ·η and named
by Subsampling confidence interval. We start with the 10-year period of January 1950
to December 1959 and move it forward one trading day at a time to construct confidence intervals. To evaluate the performance of the confidence intervals, the real value
of distribution function FY (0.001) (FY (0.0002)) is estimated by the corresponding empirical distribution function over daily returns from 1950 through 2008 and checked if
it falls in the intervals established before. The results are summarized in Table 3. We
find that the coverage accuracy of theoretical and subsampling confidence intervals are
quite satisfactory and similar to each other.
Appendix
Proof of Theorem:
Denote et ≡ σt ut . Recall that y ≡ µ + ξ. We only state in the case of ξ > 0 because
the proving process is similar for ξ < 0. In the sequel we define 2(ln ξ − ln ut I(ut >
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
41
0)− ln σ̄) by a(ξ, ut , σ̄). Taking logarithm in both side of I(et < ξ), we have the identity
I(et ≤ ξ) = I σ̄eZt /2 ut ≤ ξ I(ut > 0) + I σ̄eZt /2 ut ≤ ξ I(ut ≤ 0)
= I(Zt ≤ a(ξ, ut , σ̄)) + I(ut ≤ 0).
Define Zt,0 = 0 and Zt,j =
P
1≤i≤j
(5)
ai εt−i , Z̃t,j = Zt − Zt,j , j ≥ 1. Denote Ft =
σ({us , s ≤ t} ∪ {εs , s ≤ t}). Let Fj be the distribution functions of Zt,j conditional
on {ut }. For fixed y, write I(Yt < y) = K(Zt , ut ). For j ≥ 0, define Kj Z̃t,j , ut =
R R
E(K(Zt , ut )|Ft−j−1 ) = K Z̃t,j + x, ut dFj (x), K∞ (Zt , ut ) = K(Zt + x, ut )dF (x)
whenever they are well defined. From (5), we have
Fn (y) − FY (y)
#
" n
1 X
=
(I(Zt ≤ a(ξ, ut , σ̄)) − E[I(Zt ≤ a(ξ, ut , σ̄))])
n
t=1
#
" n
1 X
(I(ut ≤ 0) − E[I(ut ≤ 0)])
+
n t=1
n
=
1X
[I(Zt ≤ a(ξ, ut , σ̄)) − K∞ (0, ut )]
n
t=1
n
1X
[K∞ (0, ut ) − E(I(Zt ≤ a(ξ, ut , σ̄))) + I(ut ≤ 0) − E(I(ut ≤ 0))]
n t=1


n
∞
1 X X Kj−1 Z̃t,j−1 , ut − Kj Z̃t,j , ut 
=
n
+
t=1
+
1
n
j=1
n
X
t=1
[K∞ (0, ut ) − E(I(Zt ≤ a(ξ, ut , σ̄))) + I(ut ≤ 0) − E(I(ut ≤ 0))].
(6)
Because {a(ξ, ut , σ̄)} and {Zt } are independent, in view of argument of formula (2.4) in
Ho and Hsing (1996), we can slightly modify the main results in Ho and Hsing (1996)
to show that under similar regularity conditions,
n
X
[I(Zt ≤ a(ξ, ut , σ̄)) − K∞ (0, ut )]
t=1
n X
(1)
(1)
FZ (a(ξ, ut , σ̄)) − E FZ (a(ξ, ut , σ̄)) Zt
=−
t=1
−E
n
X
(1)
FZ (a(ξ, ut , σ̄))
t=1
Zt + S̃n,1 ,
(7)
42
HWAI-CHUNG HO AND FANG-I LIU
P
2 = Var ( n Z ). In equation (7),
2
with σn,1
where S̃n,1 satisfies V ar S̃n,1 = o σn,1
t=1 t
(r)
we denote FZ = the r-th derivative of the distribution function. Write
( n
i
X h (1)
(1)
FZ (a(ξ, ut , σ̄)) − E FZ (a(ξ, ut , σ̄)) Zt
E1 = E −
t=1
−
2
,
E2 = E S̃n,1
E3 = E
n
X
E
t=1
(1)
FZ (a(ξ, ut , σ̄))
Zt
)2
,
(8)
!2
n
X
[K∞ (0, ut ) − E(I(Zt ≤ a(ξ, ut , σ̄))) + I(ut ≤ 0) − E(I(ut ≤ 0))] ,
t=1
E4 = 2E
E5 = 2E
and E6 = 2E
−
−
n h
X
(1)
FZ (a(ξ, ut , σ̄))
t=1
n
X
t=1
n
X
t=1
E
!
i
(1)
− E FZ (a(ξ, ut , σ̄)) Zt S̃n,1 ,
(1)
FZ (a(ξ, ut , σ̄))
!
Zt S̃n,1 ,
[K∞ (0, ut ) − E(I(Zt ≤ a(ξ, ut , σ̄))) + I(ut ≤ 0) − E(I(ut ≤ 0))]S̃n,1
!
6
P
From the equations (6) and (7), we have Var n1−H (Fn (y) − FY (y)) = n−2H
Ei .
i=1
Based on Holder’s inequality and the argument that the convergent rate of S̃n,1 is
n
1 2
P
Zt
slower than that of Var
i.e. S̃n,1 = o n−H , we have
t=1

n−2H · E5 ≤ 2n−2H · E
−
n
X
t=1
(1)
E FZ (a(ξ, ut , σ̄)) Zt
!2  12

i 1
h 2
2
E S̃n,1
= o(1).
The term E4 (E6 ) is handled in the same way and n−2H · E4 = o(1) n−2H · E6 = o(1) .
Then, we show the property of (8). With independence between {ut } and {Zt }, we
have
n
−2H
· E1 = n
−2H
·E
(
n h
X
(1)
FZ (a(ξ(τ ), ut , σ̄))
t=1
+ δ2 n−2H
i
(1)
− E FZ (a(ξ(τ ), ut , σ̄)) Zt
i2
h (1)
· E FZ (a(ξ(τ ), ut , σ̄))
E
i2
h p
(1)
−→ E FZ (a(ξ(τ ), ut , σ̄))
c
n
X
t=1
Zt
!2
1
1
as n → ∞.
2H − 1 H
)2
.
43
LONG-MEMORY STOCHASTIC VOLATILITY PROCESSES
Thus, we have
1
1
as n → ∞.
2H − 1 H
n
P
Thus, the limiting variance of n1−H (Fn (y) − FY (y)) is determined by that of
Zt and
i2
p h (1)
Var n1−H [Fn (y) − FY (y)] −→ E FZ (a(ξ, ut , σ̄))
c
t=1
the proof of Theorem is accomplished.
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[ Received March 2010; accepted May 2010.]