On Quantile Estimator of Location parameter in ...

On Quantile Estimator of Location parameter in Stochastic Volatility Model Debajit Dutta, Subhra Sankar Dhar and Amit Mitra Department of Mathematics and Statistics, IIT Kanpur, India

Abstract Stochastic volatility models are of great importance in the field of mathematical finance, especially for explaining the dynamics of financial derivatives. A quantile based estimator for the location parameter, in a stochastic volatility model is obtained by solving an optimization equation. In this article, the asymptotic distribution of the estimator is derived even when the density function of the error is not positive around the corresponding population quantile. Keywords: Optimization equation, L-estimator, Martingale Central Limit Theorem, Non-Gaussian error distribution. 1. Introduction The term “volatility” in mathematical finance means variance. In stochastic volatility models, the variance of a stochastic process is itself randomly distributed. These are in essence the new generation of option pricing models. Strictly speaking, stochastic volatility models are non-linear time series models. Linear models cannot be used to explain processes which have autocorrelation existent among their squares (see Robinson and Za↵aroni (1998)). Engle (1982) introduced ARCH type models to explain these phenomena but with short memory autocorrelation in the squares. Robinson (1991) introduced GARCH models (with ARCH models being

a special case) which incorporates long memory in the squares. Then after a few years, a large class of “product type” models for stochastic volatility, including linear ARCH models with long memory in volatility, were introduced and discussed by Robinson (2001), Robinson and Za↵aroni (1998), Surgailis and Viano (2002), Beran (2006), among others. In this article, we consider the following type of stochastic volatility model: Xt = µ + Yt , t 2 Z,

(1)

where Yt = ✏t Zt , and Z is the set of integers. Here Xt stands for the observed time series, and {✏t } denotes the i.i.d. white noise sequence with zero median and finite variance. Among the other notations, the constant µ is the unknown location parameter of interest. Furthermore, Zt is a covariance stationary process, and ✏t is independent of Zs (s  t). 1 1 P P Specifically, Zt = a + bj (Yt j ), where a 6= 0, b2j < 1 (See Beran (2006)). j=1

j=1

Suppose that the time series data consists of n observations X = {x1 , x2 , . . . , xn }. In

order to find an estimator of µ, the quantile based estimation procedure can be considered as a valid approach. It is to be noted that the L1 estimation of the location parameter forms a special case of the former approach. In the quantile based estimation procedure, the following objective function is minimized with respect to µ. n X t=1

{|xt

µ| + (2↵

1)(xt

µ)},

(2)

where ↵ 2 (0, 1) is the index of the quantile. This is a case of ↵-th regression quantile problem, and the solution is given by µ ˆ↵ , the sample ↵-th quantile obtained from the data X . Alternatively, a straightforward algebra implies that the objective function can n P be re-stated as {⇢(xt µ)}, where ⇢↵ (x) = 2↵xI(x > 0) + 2(↵ 1)xI(x < 0)). Here t=1

I(.) is the indicator function.

2

If µ0 is the true value of µ and FYt 1 (↵) is the ↵-th quantile of Yt , the asymptotic 1

distribution of n 2 (ˆ µ↵

µ0

FYt 1 (↵)) is well-established result in the literature under

the assumption of positiveness of the probability density function in the neighborhood of population quantile. To be precise, when fYt , the probability density function of Yt , is positive around the population quantile, the asymptotic distribution is Gaussian with mean zero and variance

↵(1 ↵) . fYt (FY 1 (↵))fYt (FY 1 (1 ↵)) t

However, this assumption may not hold

t

in practice. In fact, even the checking of positiveness of the density function may not be easily doable. Knight (1998) derived the asymptotic distribution of the L1 estimator, foregoing this assumption in case of a generalized linear model. Wang (2011) considered the distribution of the L1 estimator (i.e., when ↵ = 1/2 in (2)) under the framework of stochastic volatility in a similar context. In this article, the objective is to derive the asymptotic property of the estimator obtained by minimizing (2), ignoring the assumption of a positive probability density function around the population quantile. The rest of the article is organized as follows. Section 2 deals with some assumptions and the main results related to weak convergence of the quantile estimator. Some concluding remarks have been discussed in Section 3. All technical details are provided in the Appendix. 2. Preliminaries and main results In this section, we establish that for a suitable sequence of constants an , an (ˆ µ↵

µ0

FYt 1 (↵)) converges weakly to a specified random variable under some conditions that will be described later. In order to determine the asymptotic distribution of the quantile estimators, one may define the following objective function. Zn,↵ ( ) =

an p n

n P

t=1

an 1 )

{⇢↵ (Yt

3

⇢↵ (Yt )},

(3)

where ⇢↵ (.) is defined as in Section 1. Here it is appropriate to point out that the aforementioned objective function will be minimized with respect to µ0

at ˆ↵ = an (ˆ µ↵

FYt 1 (↵)). In order to derive the asymptotic distribution of the quantile estimator, we need the

following assumptions: (A1){✏t } are i.i.d with median 0 and is symmetric about it. (A2) Yt is the stationary solution of (2). (A3) For each k 2 N, define ⌘i,k (i 2 N) to be i.i.d. random variables with characteristic 1 Q function ⇥,k (t) = E[eit⇥i,k ] = k (bj t), where k (t) = E[exp(it✏1 ✏2 ....✏k )]. Then as j=1

= op (1). Here N is the set of natural numbers.  1 (A4) For each k 2 N , lim E max ⇥i,k 2k = 0. n!1 n 1in n R 1 P p a (A5) For each , n1 an n[ 0 n {I(Yt  s) I(Yt  0)}ds (2↵ 1)an 1 /2] converges t=1 Z an 1 n 1X p in probability to ⌧↵ ( ), where ⌧↵ ( ) = lim an n[ {FYt (s) FYt (0)}ds (2↵ n!1 n 0 t=1 1)an 1 /2] = lim n,↵ ( )(say). n ! 1,

1 max1in ⇥i,k 2k n

n!1

Remark 1: Assumption (A1) will be satisfied for several continuous distributions having center of symmetry = 0. (A2) is common across quantile based estimator. (A3) and (A4) are well-known assumption for applying the Martingale Central Limit theorem (see Billingsley (1995)). Along with these, (A5) (see Knight (1998)) will also be needed to prove the asymptotic convergence of a part of Zn,↵ ( ). Note that, in contrast to previous attempts (see Serfling (1980)), it is not assumed that the density function of the error random variables is positive around the population quantile. 2.1. Main results The asymptotic distribution of the quantile estimator of the location parameter has been investigated in this section. At first, we study the asymptotic properties of Zn,↵ ( ),

4

and afterwards, we make a conclusion on the asymptotic properties of the minimizer of Zn,↵ ( ). Theorem 1: Under the conditions (A1)-(A5) stated in Section 2, Zn,↵ ( ) converges in distribution to Z↵ ( ) =

⇣ + 2⌧↵ ( ), where ⇣ is a standard normal random variable,

and ⌧↵ ( ) is same as defined in (A5). Theorem 1 asserts that Zn ( ), a strict random convex function, converges to a specified random variable. The strict convexity of Zn ( ) ensures the unique minima of it with respect to . Next theorem establishes the limiting distribution of the quantile estimator (i.e., the unique minima of Zn ( )) after appropriate normalization. Theorem 2: Let µ0

d

FY 1 (↵)) !

?

?

be the unique minima of

⇣ + 2⌧↵ ( ), then we have an (ˆ µ↵

, where ⇣ is a standard normal random variable.

Remark 2: The assertion in Theorem 2 implies that the asymptotic distribution of ?

may be a non-normal distribution when the probability density of the error random

variables may not be positive around the population quantile. To be precise, the nature of that asymptotic distribution will be controlled by the nature ⌧↵ ( ). For instance, the asymptotic distribution of an (ˆ µ↵

µ0

d

FYt 1 (↵)) !

?

will be Gaussian when ⌧↵ ( ) is

a linear function of . Also, as we discussed in the introduction, the quantile estimator will follow asymptotically Gaussian distribution after appropriate normalization when the density function is positive around the population quantile, and this fact is inevitably a special case of the results described in Theorem 2. The verification of the special case (i.e., the density function is positive around the population quantile) is as follows. n P p p Ra 1 Consider an = n when ↵ = 1/2, and it leads to n,1/2 ( ) = n1 an n[ 0 n {FYt (s) t=1

n p R P FYt (0)}ds] = n[ 0 {FYt (s/an ) FYt (0)}ds]. If now FY0 t (µ0 ) is assumed to be post=1 p itive then, lim n[FYt (s/an ) FYt (0)] = sFY0 t (0), and consequently, lim n,1/2 ( ) = 1 n

2

2

n!1

= ⌧1/2 ( ), where

n!1

= FY0 (µ0 ). These facts imply that Z1/2 ( ) of Theorem 1 becomes

5

⇣+

2

, and hence, minimizing Z1/2 ( ) with respect to

gives

min

=

⇣ 2

. In view of

the fact that ⇣ is a standard normal random variable, we have, the minimum of Z1/2 ( ) follows a normal distribution with mean zero and variance

1 . 4{fYt (µ0 )}2

Arguing in a similar

way, one can show it for any ↵ 2 (0, 1) also. 3. Concluding remarks Asymptotic properties of L-estimator: L-estimator (see Serfling (1980)) is wellknown estimator for location parameter. It is defined as the linear combination of order statistics, and this construction gives us a robust analogue of the classical estimators of the location parameter. For example, the ↵-trimmed mean and the sample median are well-known examples of L-estimator. Since an order statistic has a one-to-one corresponding with a specified quantile, one may express an L-estimator as a linear combination of certain quantiles. It is expected that one can derive the asymptotic distribution of an Lestimator of the location parameter in (1) by applying the results described in Theorems 1 and 2. Quantile based measure in descriptive statistics: For stochastic volatility model, one can define di↵erent quantile based estimators to measure the scale, namely, inter quartile range or it might be possible to introduce quantile based measure of asymmetry of distribution. Apart from the point estimation, it is possible to construct an appropriate confidence interval based on the estimated quantiles of the location parameter in the stochastic volatility model. It is needless to mention that the results in Theorems 1 and 2 will help us to investigate the asymptotic properties for the aforementioned cases. Non-Gaussian error distribution: When errors follow any heavy tailed distribution like Cauchy or t-distribution with 2 degrees of freedom, the classical least squares estimator fails to perform satisfactorily. It is expected that any estimator based on quantile like median or trimmed mean based estimators would perform better than the mean based 6

estimator in the stochastic volatility model also. Main results obtained in this article: As it is discussed in the introduction, one does not need to assume that the density function of the error random variables is positive around the population quantile to derive the asymptotic distribution of the quantile of the location parameter in the stochastic volatility model. Here we would like to point out that an application of Bahadur (1966)’s linearization technique along with the central limit theorem enables us to derive the asymptotic distribution of the quantiles when the density function is positive around the population quantile. However, one cannot use such type of linearization technique when the density function is not positive. In this article, we tackle such situation for the quantile estimator of the location parameter in the stochastic volatility model. 4. Appendix: Technical details To prove Theorem 1, we need Lemma 1. Lemma 1: For any x 6= 0 and y 2 R, we have |x y| |x| = y[I(x > 0) I(x < Ry 0)] + 2 0 [I(x  s) I(x  0)]ds, where I(A) is the indicator function of set A. Ry Proof of Lemma 1: It follows from the expression 2 0 [I(x  s) I(x  0)]ds that s  y. We consider di↵erent cases on the position of x and establish the equality for each case. Case 1: x  0  s  y.

Ry x ( x) = y and y[I(x > 0) I(x < 0)] + 2 0 [I(x  s) I(x  Ry 1) + 2 0 (1 1)ds = y. The last fact follows in view of x  0 and x  s

Here |x y| |x| = y 0)]ds =

y(0

as well.

Case 2: 0  x  s  y. Here |x

y|

|x| = y

x

x=y

2x and

7

y[I(x > 0) (since s

I(x < 0)] + 2

x) =

y + 2y

Z

y 0

[I(x  s)

2x = y

I(x  0)]ds =

y(1

0) + 2

2x.

Z

y

(1

0)ds

x

Case 3: 0  s  x  y. Here |x

y|

|x| = (y

x = y 2x and Z y I(x < 0)] + 2 [I(x  s)

y[I(x > 0)

x)

0

(since I(x  s) = 0 if s < x) = y + 2y

I(x  0)]ds = 2x = y

y(1

0) + 2

2x.

Z

y

ds x

Case 4: x

y.

Here |x

y|

|x| = x

0)]ds =

y; since I(x > 0) = 1, I(x < 0) = 0, I(x  s) = 0 and I(x  0) = 0 when

x

y

x=

y and

y[I(x > 0)

I(x < 0)] + 2

Ry 0

[I(x  s)

I(x 

y > 0. This completes the proof.

2

Proof of Theorem 1: Consider Zn,↵ ( ) = Zn,↵ ( )(1) + Zn,↵ ( )(2) , where n

1 X p [I(Yt > 0) n t=1

Zn,↵ ( )(1) =

I(Yt < 0)]

and n

Zn,↵ ( )

(2)

an X = 2p n t=1

"Z

an 0

1

{I(Yt  s)

I(Yt  0)}ds

(2↵

A straightforward application of Lemma 1 leads to Zn,↵ ( )(2) = 2 n1 s)

I(Yt  0)}ds

(2↵

n P

t=1

1)an

1

#

/2 .

p Ra 1 an n[ 0 n {I(Yt 

1)an 1 /2]. Now, (A5) implies that Zn,↵ ( )(2) converges to

2⌧↵ ( ) as n ! 1 (denote it as Fact 1). We now concentrate on the term Zn,↵ ( )(1) . It is to be noted that ✏t has a symmetric 8

distribution and is independent of the random variable Zt . Thus we have, sign(Yt ) = I(Yt > 0)

I(Yt < 0) and E[sign(Yt )] = E[sign(✏t Zt )] = P [✏t Zt > 0]

P [✏t Zt < 0] =

0. Similarly, E(sign(Yt )|Ft 1 ) = E(sign(✏t Zt )|Ft 1 ) = 0, where Ft denotes the -field generated by {sign(Yj ), j  t}. This implies that sign(Yt ) is a martingale di↵erence with respect to the

field Ft .

Further, note that the function sign(Yt ) satisfies the regularity conditions mentioned for Theorem 2 of Beran (2006). Thus, from Theorem 2 of Beran (2006) and Theorem 3.2 of Hall and Heyde (1980) with an application of Martingale Central Limit theorem (see d

Billingsley (1995)), we have Zn,↵ ( )(1) !

⇣, where ⇣ is a standard normal random

variable (denote it as Fact 2). Using Fact 1 and Fact 2, we can conclude that Zn ( ) converges in distribution to Z↵ ( ) =

⇣ + 2⌧↵ ( ). Hence, the theorem is proved.

2

Proof of Theorem 2: In view of convexity of Zn,↵ ’s and the existence of unique minima d

of Z↵ ( ), we have arg min(Zn,↵ )( ) = an (ˆ µ↵ µ0 FYt 1 (↵)) ! arg min(Z↵ ( )) as n ! 1 (see, e.g., Geyer(1996)). This concludes the proof.

2

5. References [1] Bahadur, R. R. (1966) A note on quantiles in large samples. The Annals of Mathematical Statistics, 37, pp. 577–580. [2] Beran, J. (2006) On location estimation for LARCH processes. Journal of Multivariate Analysis, 97, pp. 1766–1782. [3] Billingsley, P. (1995) Probability and Measure. John Wiley & Sons, New York. [4] Engle, R. F. (1982) Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of the United Kingdom. Econometrica, 50, pp. 987–1002. [5] Geyer, C. J.(1996) On the Asymptotics of Convex Stochastic Optimization.(Unpublished manuscript) [6] Hall, P. and Heyde, C. (1980) Martingale Limit Theory and its Applications. Academic Press, New York. [7] Knight, K. (1998) Limiting Distributions for L1 Regression Estimators under General Conditions. Annals of Statistics, 26, pp. 755–770. [8] Robinson, P. M. (1991) Testing for Strong Serial Correlation and Dynamic Conditional Heteroskedasticity in Multiple Regression. Journal of Econometrics, 47, pp. 67–84.

9

[9] Robinson, P. M. (2001) The Memory of Stochastic Volatility. Journal of Econometrics, 101, pp. 195–218. [10] Robinson, P. M. and Za↵aroni, P. (1998) Non-linear Time Series with Long Memory:A Model for Stochastic Volatility. Journal of Statistical Planning and Inference, 68, pp. 359–371. [11] Serfling, R. (1980) Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York. [12] Surgailis, D. and Viano, M. C. (2002) Long Memory Properties and Covariance Structure of the EGARCH Model. ESAIM:Probability and Statistics, 6, pp. 311–329. [13] Wang, L. (2011) L1 Estimation of the Location Parameters for the Stochastic Volatility Model. Mathematical Methods of Statistics, 20, pp. 165–170.

10