Truncated Maximum Likelihood, Goodness of Fit Tests and Tail Analysis

Goodness of Fit Tests and Tail Analysis

Christian Gourieroux Joanna Jasiak^y This version: March 11, 1998 (First draft: September 28, 1997)

CRESTandCEPREMAP,e-mail^:gouriero@ensae.fr.

y

YorkUniversity,e-mail^:jasiakj@yorku.ca

Abstract

Truncated Maximum Likelihood, Goodness of Fit Tests and Tail Analysis.

We propose a new method of tail analysis for data featuring a high degree of lep- tokurtosis. Heavy tails can typically be found in nancial series, like for example, the stock returns or durations between arrivals of trades. In our framework, the shape of tails can be assessed by tting some selected pseudo-models to extremely valued observations in the sample. A global examination of the density tails is performed using a sequence of truncated pseudo-maximum likelihood estimators, called the tail parameter function (t.p.f.). In practice, data often exhibit local irregularities in the tail behaviour. To detect and approximate these patterns we introduce the local parameter function (l.p.f.), a method of tail analysis involving a selected interval of extreme observations. An immediate extension of the pseudo-value based approach to density analysis yields a new procedure for testing the goodness of t. We also de- velop a new nonparametric estimator of the density function. The method is applied to unequally spaced high frequency data. We study an intradaily series of returns on the Alcatel stock, one among the most frequently traded stocks on the Paris Stock Exchange.

Keywords: Goodness of t test, tail analysis, truncated maximum likelihood, kernel, Hill estimator.

Resume

Maximum de vraisemblance tronque, tests d'adequation et analyse de queues.

Nous proposons une nouvelle methode d'analyse des queues de distribution. Un e et leptokurtique important est generalement mis en evidence sur la plupart des series nancieres, qu'il s'agisse de series de rendements ou de durees entre echanges. Dans notre approche la forme des queues est etudiee en ajustant divers pseudo-models aux donnees extr^emes de l'echantillon. Un examen global des densites de queues peut ^etre e ectue en utilisant une suite d'estimateurs du pseudo-maximum de vraisemblance tronque, appelee fonction de parametre de queue t.p.f.]. Une approche alternative repose sur des analyses locales de la queue l.p.f.]. Ces analyses permettent aussi d'introduire des tests d'ajustement et un nouvel estimateur a noyau de la densite.

L'approche est appliquee a une serie de rendements du titre Alcatel, l'un de plus echanges a la bourse de Paris.

Mots cles: Tests d'adequation, analyse de queues, maximum de vraisemblance tronque, noyau, estimateur de Hill.

1 Introduction

Since the seminal work of Mandelbrot (1963) on cotton prices, the presence of heavy tails in the marginal distributions of nancial returns has been conrmed by a large number of empirical studies. This phenomenon has inspired a number of researchers seeking to provide a plausible explanation. Mandelbrot advanced a hypothesis of an underlying stable non-normal distribution of returns in a random walk model. Clark (1973) and Harris (1989) suggested that returns are generated by a mixture of normal distributions, whith the rate of new information arrival acting as a stochastic mixing variable. The heavy tails may also be viewed as a consequence of conditional heteroscedasticity, due to time varying volatility of returns Engle (1982)]. In the limiting case of strong persistence and presence of a unit root in the volatility equation, the marginal variance may even not exist Nelson (1990)]. The presence of heavy tails was also documented in the conditional distributions of returns for some nancial assets. This feature is crucial for risk management and should be accounted for in the risk control rules adopted by nancial institutions. Indeed, standard rules such as the Value at Risk ] are implicitely based on a mean-variance approach, and are inappropriate when data frequently admit extreme values. The leptokurtosis of conditional distributions may also explain some other characteristics of the series of nancial returns such as the long range of temporal dependence Resnick, Samorodnitsky (1997)].

In general, we say that a probability distribution features a heavy tail if asymptotically it behaves like the Pareto distributionPY > y] =y^; L(y) >0 y >1, where L is slowly varying:

lim^t!1L(ty)L(t) = 1. The statistical methods for tail analysis focus on the estimation of the tail index . In the case of i.i.d. observations, the estimators are generally computed from the order statistics: Y⁽¹⁾ ::: Y⁽ⁿ⁾. Two well-known estimators of are the Hill estimator Hill (1975)]:

1^ ^{= 1}k

k

X

i=1

log Y⁽ⁱ⁾ Y^{(k +1)}

and the de Haan's moment estimator de Haan (1970)]:

1^ ^{= 1}k

k

X

i=1

log Y⁽ⁱ⁾ Y^{(k +1)}

r:

In this paper we propose a new method of tail analysis. We introduce a parametric family of distributions^F =^ff(y) ^2gand determine the value of theparameter such thatf(y) is at the minimum distance from the empirical distribution of the observedy's, for values suciently largey c, or for y lying in a neighbourhood of largec. The tail behavior of the distribution is revealed by the behavior ofwith respect to c.

The formulas of estimators and test statistics, and their properties are derived under the as- sumption of i.i.d. observationsY¹:::Yⁿ. We present the asymptotic results under general regu- larity conditions.

In section 2 we introduce a partition of the real line into non overlapping intervals and allow the parameter to vary independently between the intervals. The comparison of the full sample maximum likelihood estimators of with the maximum likelihood estimators truncated over the intervals yields various goodness of t tests. In section 3 we consider the pseudo maximum likeli- hood (henceforth P.M.L.) estimation of theparameter from the data truncated byc. The limits of the P.M.L. estimators dene a function ofc, called the tail parameter function (t.p.f.). We show that the tail parameter function is constant only for a distribution in the ^F family. We discuss conditions under which the t.p.f. indicates the true underlying distribution. In section 4, we intro- duce an estimator of the t.p.f., derive its asymptotic distributional properties and discuss the links with the Hill estimator. In section 5 we develop a local pseudo-maximum likelihood method for estimation of density functions. The basic idea consists in smoothing the estimator of censored over an interval using a kernel. Next, the log-derivative of the unknown density function at a given point is approximated by the log-derivative of the pseudo density evaluated at the local estimator of. The empirical results are discussed in section 6, where various methods proposed in the paper are applied to a series of nancial returns. Some technical issues are are explained in Appendices 1,2 and 3. Section 7 concludes the paper.

2 Truncated Maximum Likelihood and Goodness of Fit Tests

2.1 Decomposition of the Log-Likelihood Function

Let us consider a family ^F of distributions on ^{Y 2} R, parametrized by ² R^p. The distributions in^F have positive p.d.f. f(:) with respect to a dominating measure. We denote byF(:) andS(:) the corresponding cumulative function and survivor function, respectively.

For a random variable Y with the distribution function f(:), andA¹:::A^K a partition of

Y, we introduce the qualitative variableZ indicating the element of the partition containingY: Z = (Z¹:::Z^K)⁰ = 1^A1(Y):::1^AK(Y)]⁰ (2.1) where 1^A(Y) = 1, ifY ²A and 1^A(Y) = 0, otherwise.

The marginal distribution of the vectorZ is multinomialM(1 F(A¹):::F(A^K)), with the probability function PZ =z] =h(z). We denote by f^k(y) the conditional density function ofY givenZ^k = 1, (i.e. Y ²A^k).

By applying the Bayes theorem we obtain the following decomposition of the conditional density function ofY:

log f(y) = logh(z) +^XK

k =1

z^k logf^k(y) (2.2) which implies:

;

@²logf(y)

@@^{0 =;}@²log h(z)

@@^{0 ;}

K

X

k =1

z^k @²logf^k(y )

@@⁰ : (2.3)

The expectations of all terms in (2.3) with respect to the distribution f(:) yield the following decomposition of the information matrix:

J^Y() =J^Z() +^XK

k =1

J^Yjk() (2.4)

where

J^Y() = E^;@²logf(Y)

@@⁰

J^Z() = E^;@²logh(Z)

@@⁰

J^Yjk() = E

;Z^k@²log f^k(Y)

@@⁰

:

Let us also introduce the matrix:

J^YjZ() =J^Y()^;J^Z() =^XK

k =1

J^Yjk() (2.5)

which is the information matrix based on the conditional distribution ofY givenZ.

2.2 Extended Model

The model ^F may be nested in a larger model^F , allowing for an independent variation of the parameters appearing in the marginal distribution ofZ and in the conditional distributions ofZ givenY ²A^k. We now consider the model^F , where

the marginal distribution of Z ish(z),

the conditional distribution ofY givenY ²A^k isf^k(y^k),k= 1:::K,

¹:::^K are unconstrained with values in R^p.

The parameters of this extended model can consistently be estimated from a sampleY¹:::Yⁿ of i.i.d. variables by the maximum likelihood. The log-likelihood function to be maximized:

L^Y(¹:::^K) =^Xn

i=1

logh(zⁱ) +^XK

k =1

(^Xn

i=1

z^{k i}logf^k(y^ik)) (2.6)

consists of two additive terms involving separately the parametersand. This allows for distinct optimizations with respect to these parameters and ensures an asymptotic independence of the estimators of and¹:::^K.

Property 2.1

:

LetY¹:::Yⁿbe i.i.d. variables with a distribution in^F . Under standard regularity conditions, the unconstrained maximum likelihood estimators ~^~1:::^~K are consistent, asymptotically normal, and asymptotically independent:

pn~^;] ^!d N0J^Z()^;1]

pn~^k;k] ^!d N0J^Yjk(^k)^;1] k= 1:::K:

2.3 Goodness of Fit Tests

In this subsection we present a procedure involving various pseudo-true value based tests to evaluate the t of the model. Let us introduce the hypotheses: H =^f1:::^K, unconstrained ^g, H⁰ =

f¹=:::=^Kg,H⁰=^f1=:::=^K =^g.

Under the hypothesis H⁰, the common parameter value of and can be estimated by the maximum likelihood based on the entire sample ofnobservations:

^= argmax

n

X

i=1

log f(yⁱ): (2.7)

UnderH⁰, this estimator is consistent and asymptotically normal:

pn(^^;)^!d N0J^Y()^;1]: (2.8) Under the hypothesis H⁰, two estimation methods can be considered. One consists again on maximizing the likelihood function and yields the estimator of =^k k= 1:::K. Alternatively, we can use the equivalent approximation:

~ = ^XK

k =1

J^Yjk(~^k)]^;1XK

k =1

J^Yjk(~^k)~^k: (2.9)

Under the hypothesis H⁰, ~ and ~ are consistent estimators of , asymptotically normal and independent, with the limiting variance matrices:

V^aspn(~^;)] = J^Z()^;1 (2.10) V^aspn(~ ^;)] = J^YjZ()^;1: (2.11)

Let us now consider the sequence of nested hypotheses:

H⁰H⁰ H =^ff ^{2F g}

and introduce test statistics ^H0jH, ^H0jH0, ^H0

jH for testingH⁰ against H, H⁰ against H⁰, H⁰ againstH respectively.

The test statistics can be formulated as the following likelihood ratios:

^H0jH = 2L^Y(^)^;L^Y(~ ~¹:::^~K)]

^H0jH0 = 2L^Z(^)^;L^Z(~)]

^H0

jH = 2L^YjZ(^)^;L^YjZ(~¹:::^~K)]

whereL^YL^ZL^YjZ are the log-likelihood functions corresponding to the observations ofYZ and to the conditional model of Ygiven Z respectively. Standard results on likelihood ratio tests Gourieroux, Monfort (1997)] imply the following property:

Property 2.2

:

Under the null hypothesisH⁰, the statistics^H0jH0 and^H0

jH are asymptotically independent, with asymptotic distributions: ²(p) and²((K^;1)p), respectively.

The statistic^H0jH =^H0jH0 +^H0

jH follows asymptotically a²(Kp) distribution.

It is possible to replace these test statistics by their asymptotic equivalents which are easier to compute. Such equivalent statistics are:

^H0jH0 = (^^;~)⁰^V(^) + ^V(^ )]^;1(^^;~) (2.12) ^H0

jH = ^XK

k =1

(~^k;~)⁰(^V(~^k))^;1(~^k;~ ): (2.13) The extended model^F may itself be nested in a more general model^F , where the marginal dis- tribution ofZis multinomial with parametersp¹:::p^Kconstrained only by p^k 0⁸k^{PKk =1}p^k= 1, and where the conditional distributions ofY givenZ are as in^F . The likelihood ratio statistic for testingH =^ff ^{2F g}againstH =^ff ^{2F g}is asymptotically equivalent to the standard chi-square statistic of the goodness of t test. Under the null hypothesis H⁰, the statistic is asymptotically independent of^H0jH0,^H0

jH,^H0jH and follows the asymptotic²(K^;p^;1) dis- tribution. In some sense the statistics^H0jH0,^H0

jH are complementary to the standard chi-square goodness of t test.

3 The Tail Parameter Function

In section 2 we showed the pseudo-true value based test statistics to assess the t of the model.

We now extend this approach and introduce a new method of tail analysis which instead of the pseudo-true values based on the entire sample involves tail based pseudo-true values.

3.1 Denition

Let us consider a parametric family of density functions: ^F =^ff(:)²R^pg, with respect to the Lebesgue measure on ^Y = R⁺ = (0+¹. We assume that the density functions in this family are positive.

For any probability distribution, with density function f⁰(:), we can write the optimization problem:

max

2

E⁰

1^Yc log f(Y) S(c )

(3.1)

wherec²R⁺ andE⁰ is the expectation with respect tof⁰. We assume:

Assumption

A.1.

The optimization problem (3.1) has a unique solution(cf^0F) for anyc 0.

The parameter value(cf^0F) is such that the parametrized distributionf(:) provides the best approximation of the tail of the true distribution dened by the levelc.

Denition 3.1

Let us consider a distribution f⁰ satisfying A.1. The tail parameter function associated withf⁰ is the mapping(: f^0F) fromR⁺ on .

For notational convenience we further denote the tail parameter function (t.p.f.) by (: f⁰), omiting ^F. Under standard regularity conditions, (cf⁰) is the limit of a pseudo maximum likelihood estimator based on the pseudo model ^F and a sample Y¹:::Yⁿ with distributionf⁰, and left truncated byc:

(c f⁰) = lim

n!1

^ⁿ(c) (3.2)

where:

^ⁿ(c) =argmax

2 n

X

i=1

f1^yiclog f(yⁱ)

S(c)^g: (3.3)

3.2 Characterization of a Distribution of the Pseudo Model

In this subsection we show that the t.p.f. is constant only for distributions in^F.

Property 3.2

^:

Let us assume that the density functions are rst order dierentiable with respect toand y, and that the parameteris identiable. The densityf⁰ belongs to the family^F if and only if the tail parameter function is constant over c.

Proof

^:

Necessary Condition

Iff⁰ belongs to^F, there exists a unique parameter value⁰such thatf⁰(:) =f(:⁰), and the t.p.f. is(cf⁰) =⁰⁸c:

Sucient Condition

Let the common solution of the optimization problems:

= argmax

E⁰1^Y>c log f(Y ) S(c ) ]

= argmax

Z

1

c

logf(y)f⁰(y)dy^;S⁰(c)logS(c)]: The rst order conditions are:

Z

1

c

@logf(y )

@ f⁰(y)dy^;S⁰(c)@logS(c )

@ ^{= 0 8}c:

By dierentiating again with respect to the truncation level c we get:

;

@logf(c )

@ f⁰(c)^;dS⁰(c)

dc @logS(c )

@ ^;S⁰(c)@²logS(c )

@c@ ^{= 0 8}c:

f⁰(c)

;

@log f(c )

@ ⁺@logS(c )

@

;S⁰(c)@²logS(c )

@c@ ^{= 0 8}c:

f⁰(c)

S⁰(c) = @²logS(c )

@c@^j

;

@logf(c )

@^{j +}@logS(c )

@^j

;1 ⁸c j = 1:::p:

The hazard function corresponding to f⁰ is uniquely dened by f(c ) and since the density functionf(c ) also satises the last equation, we conclude thatf⁰(:) =f(: ).

Q.E.D.

3.3 Identifying a Distribution by its Tail Parameter Function

In this subsection we focus on two important questions:

(i) Does there exist pseudo-families^F such that(: f⁰) characterizes anyf⁰?

(ii) For a given pseudo-family ^F, what condition satisfy the distributions identiable by their t.p.f. ?

We give below some preliminary answers to these questions.

Property 3.3

^:

If Y is a positive variable and the pseudo-family is the family of exponential distributions:

F=^ff(y) =exp(^;y) ²R^+g, then

(i) the t.p.f. is the expected residual life : (cf⁰) =E⁰((Y ^;c)^jY > c) (ii) the t.p.f. characterizes the distribution.

Proof

:

(i) The optimization problem max

fE⁰1^Y>c log]^;E⁰1^Y>c(Y ^;c)]^g involves the rst order condition:

1E^0(1Y>c);E⁰1^Y>c(Y ^;c)] = 0 and yields the solution:

(cf⁰) =E⁰((Y ^;c)^jY > c): (ii) Conversely we get:

(cf⁰) = E⁰1^Y>c(Y ^;c)]

E⁰(1^Y>c)

=

R

1

c (y^;c)f⁰(y)dy S⁰(c)

=

R

1

c S⁰(y)dy S⁰(c) :

Therefore (cf⁰)^;1 = ^;dcd log ^Rc1S⁰(y)dy], and, when (:f⁰) is known, the survivor function can be found by integration.

Q.E.D.

Property 3.4

^:

There exists a one to one relationship between the distributionf⁰and its t.p.f. in a neighbour- hood of the parametric pseudo-model^F.

Proof

: The rst order condition dening(cf⁰) is:

8c:

Z

1

c

f⁰(y) @

@^logfy(cf⁰)]dy^;S⁰(c)@logS

@ c(cf⁰)] = 0:

Let us consider a distribution f⁰ close tof(: ) and introduce (cf⁰) = (cf⁰)^;cf(:⁰)]

=(cf⁰)^;0.

A rst order expansion yields:

8c:

Z

1

c

f⁰(y)^;f(y⁰)]@

@^logf(y⁰)dy^;S⁰(c)^;S(c⁰)] @

@^logS(c⁰) + (cf⁰)^{fZ 1}

c

f(y⁰) @²

@^2logf(y⁰)dy^;S⁰(c) @²

@^2logS(c⁰)^g= 0:

Consider now a given discrepancy function (:f⁰). By dierentiating the previous relation with respect toc we obtain a dierential equation of the type:

f⁰(c)^;f(c⁰)]A(c)^;S⁰(c)^;S(c⁰)]B(c) +C(c) = 0

where ABC are unknown functions. This is a rst order linear dierential equation for the functionS⁰(:)^;S(: ⁰). The set of solutions is ane of dimension 1, and the integration constant is uniquely determined by the initial condition: S⁰(^;1)^;S(^{;1 0}) = 1^;1 = 0.

Q.E.D.

The previous property is a local identiability condition of the true density by its t.p.f.

3.4 The Form of the Tail Parameter Function in a Neighborhood of the Pseudo-Family

In this section we show how to evaluate the discrepancy function (:f⁰) introduced in the Property 3.4, in a neighborhood of the pseudo-family. The property 3.5 below follows directly from the expansion of the rst order condition and corresponds to some standard properties of the maximum likelihood estimators under local alternatives.

Property 3.5

:

Let us consider a parametric family: ~^F=^ff(:)()²B^g, and assume^F =^ff(:) = f(:0)^{2 g}, andf⁰=f(:⁰) with small . Then (c f⁰) =⁰ + ~J(c), where ~J(c) is the cross term of the information matrix corresponding to the model ^F truncated at c and evaluated at (⁰0).

4 Tail Analysis

In this section we present the t.p.f. method of tail analysis. We begin with the null hypothesis concerning the tail of the distribution:

H~⁰=^f9c⁰:f⁰(y) =f⁰(y⁰) ⁸y c^0g:

This hypothesis may be tested by applying the procedure presented in section 2, based on a partition of the real line. Else, for an increasing sequence ofc it can be veried if the consistent estimator of the t.p.f. remains approximately constant for largec. The second approach is similar to some classical methods of tail analysis. It provides exploratory plotting techniques, but should be used with caution. Its limitation results from a strong correlation of the estimated values of the t.p.f. at dierent points, due to the overlapping intervals.

4.1 Statistics Based on the Truncated Pseudo Maximum Likelihood

Let us consider a sample of i.i.d. variables Y¹:::Yⁿ with an unknown distribution f⁰. We are interested in testing the null hypothesis ~H⁰. Since the t.p.f. is constant forc c⁰, it is natural to expect that a consistent estimator of(: f⁰) is constant as well. Such an estimator is:

^ⁿ(c) =argmax

n

X

i=1

1^yic logf(yⁱ)]^;logS(c)^Xn

i=1

1^yic (4.1)

cvarying. This functional estimator is a stepwise function with jumps at each observed value. To see that, let us introduce the order statistics corresponding to the observations:

y⁽¹⁾ y⁽²⁾ :::: y⁽ⁿ⁾: The jumps arise at the valuesy^{(k +1)},kvarying, and we get:

^^nk = ^ⁿ(y^{(k +1)}) (4.2)

= argmax

k

X

i=1

logf(y⁽ⁱ⁾ )^;logS(y^{(k +1)})k: (4.3)

Example 4.1: When the pseudo-family ^F is exponential family ^F = ^ff(y) = exp(^;y), ²R^+g, the estimator is:

^1 ^{nk = 1}k

k

X

i=1

y^(i);y^{(k +1)}]:

Example 4.2 : When the pseudo-family is Pareto family, ^F =^ff(y) =y^{;;1 2} R^+g, we get:

^1 ^{nk = 1}k

k

X

i=1

log

y⁽ⁱ⁾ y^{(k +1)}

which is the Hill estimator Hill (1975)], with properties in the i.i.d. case discussed by Mason (1982).

4.2 Asymptotic Properties of the Estimated t.p.f. under the Null Hy- pothesis

The asymptotic behavior of ^ⁿ(c¹):::^{^n}(c^J) for a nite set of truncation points: c^j j = 1:::J with c^j c⁰⁸j follows from the standard results on the maximum likelihood estimators. We denote byJ(c⁰) the information matrix of the model truncated by c.

Property 4.1

:

Under the null hypothesis ~H⁰:

(i) the estimator ^ⁿ(c¹):::^{^n}(c^J)]⁰ converges to (⁰:::⁰) and is asymptotically normal:

pn^ⁿ(c¹)^;0:::^{^n}(c^J)^;0]^0!d N(0), where =A^;1BA^;1, A=

2

6

4

J(c¹⁰) 0 0 ... J(c^J0)

3

7

5

B= J(sup(cⁱ c^j)⁰)]: (ii) In particular, for a truncation pointc we get:

pn^ⁿ(c)^;0]^!d N0J(c⁰)^;1]:

5 Local P.M.L. Analysis of the Density Function

5.1 The Estimation Method

An alternative application of the approach based on approximating the unknown density function f⁰ by f(: ~^k) on the interval A^k (see, section 2) is a non parametric estimation of the density

functionf⁰. Let us consider an intervalA= c^;h c+h]. The approximation of theparameter is the solution of:

~^ch = argmax

E⁰1^c;h<Y<c+hlog f(Y )]^;E⁰1^c;h<Y<c+h] log

Z

1c;h<y<c+hf(y)dy

= argmax

E⁰

1

2h^{1c;h<Y<c+h log}f(Y)

;E⁰

1

2h^1c;h<Y<c+h

log

Z 1

2h¹c;h<y<c+hf(y)dy ~^ch=argmax

E⁰

1

hK Y ^;c h

log f(Y)

;E⁰

1

hK Y ^;c h ^log

Z 1

hK y^;c h

f(y)dy (5.1) whereK(u) = ¹²1^;11](u).

From the last formula, we derive a non parametric estimation method for the density function f⁰. We consider a kernel satisfying the following assumptions:

Assumptions concerning the kernel:

A.2

^RK(u)du= 1

A.3

^RK(u)udu= 0

A.4

^RK(u)u²du=², (exists).

We propose to use the estimator dened by : ~ⁿ(c) =argmax

"

n

X

i=1

hK1 y^i;c h

log f(yⁱ)^;Xn

i=1

hK1 y^i;c h

log^Z 1

hK y^;c h

f(y )dy

#

(5.2) to estimate the p.d.f. f⁰from the pseudo distribution after substituting ~ⁿ(c) for. This approach is based on a local approximation of the p.d.f. by an element of the pseudo family ^F and is an analogue of the locally weighted least square regression developped by Cleveland (1979) and Hardle (1990) in the context of non parametric estimation of the regression function see also Gourieroux, Monfort, Tenreiro (1994) for the extension to kernel M-estimators].

5.2 The Gaussian Pseudo-family

As an illustration of the method described above we examine a special case where the estimator ~ⁿ(c) admits an explicit form and can easily be interpreted. Let us consider a gaussian pseudo- family with the mean :

f(y) =(y^;)

where is the p.d.f. of the standard normal distribution. Let us also use a gaussian kernel:

K(:) =(:). We get:

Z 1

h y^;c h

f(y)dy = ^Z 1

h c^;y h

(y^;)dy

= 1

p1 +h² c^;

p1 +h²

and the estimator is:

~ⁿ(c) = argmax

n

X

i=1

h 1 y^i;c h

;

12 log 2^;(y^i;)² 2

; n

X

i=1

h 1 y^i;c h

;

12 log 2^;1_{2 log(1 +}h²)^; (c^;)² 2(1 +h²)

: It is equal to :

~ⁿ(c) = 1 +h² h²

P

n

i=1 1

h^;yih;cyⁱ

P

n

i=1 1

h^{;yi;ch ;}h¹²c:

We nd that:

~ⁿ(c) = 1 +h²

h² m^~n(c)^; 1 h²c where :

m~ⁿ(c) =

;P

n

i=1 1

h^;yi;ch yⁱ

;P

n

i=1 1

h^;yih;c

is the Nadaraya - Watson estimator of the regression function: m(c) =EY^jY =c] =c, Nadaraya (1964), Watson (1964) ].

Moreover:

~ⁿ(c)^;c = 1 +h²

h^{2 ( ~}mⁿ(c)^;c)

= 1 +h² h²

P

n

i=1 1

h(y^i;c)^;yi;ch

P

n

i=1 1

h^;yi;ch

= (1 +h²)^@c@

P

n

i=1 1

h^;yih;c

P

n

i=1 1

h^;yi;ch

= (1 +h²)@log

@c f^~n(c)

where ~fⁿ(c) is the (gaussian) kernel estimator of the density functionf⁰. Then, the asymptotic properties of ~ⁿ(c)^;c are deduced directly from the asymptotic properties of ~fⁿ(c) and ^@f~@cn(c) Silverman (1986)]. In particular ~ⁿ(c) converges toc+^@@clogf⁰(c) when ntends to innity andh tends to zero at appropriate rates.

5.3 The Local Parameter Function

Before presenting the regularity conditions ensuring the consistency of the local pseudo-maximum likelihood estimator when the number of observationsn tends to innity whereas the bandwidth htends to zero, we derive the possible limits for ~^ch whenhtends to zero.

Property 5.1

^:

Let us assume the conditions A1-A4 and

A.5

The density functionsf(y ) andf⁰(y) are positive and third order dierentiable with respect to y.

A.6

^{For h} small and any c, the following integrals exist: ^R K(u)logf(c+uh)f⁰(c+uh)du,

R K(u)f⁰(c+uh)du, ^RK(u)f(c+uh)du, and are twice dierentiable under the integral sign with respect toh.

i) Whenhtends to zero, the objective function : A^h() =E⁰hK¹ Y ^;c

h

logf(Y)

;E⁰hK¹ Y ^;c h

log

Z 1

hK y^;c h

f(y)dy is equivalent to :

A^h() = ²h² 2

"

2@logf(c)

@y @logf⁰(c)

@y ^;

@logf(c)

@y

2

#

+o(h²):

ii) The local parameter function (l.p.f.) ~(cf⁰) = lim^h!0~^ch is the solution of the equation:

@logf(c ~(cf⁰))

@y ⁼ @logf⁰(c)

@y :

Proof

^:

i) is shown in Appendix 1.

ii) The objective function is a locally quadratic function with respect to

@log f(c)

@y

which implies the rst order condition:

@logf(c)

@y ⁼@log f⁰(c)

@y : Q.E.D.

We deduce from the previous property the following corollary:

Corollary 5.1

The local parameter function characterizes the distribution.

Proof:

Indeed, if ~(:f⁰) is known, we also know the log-derivative of the density function, since

@logf⁰(c)

@y ⁼ @logf(c ~(cf⁰))

@y ⁸c and by integration we nd the density functionf⁰itself.

Q.E.D.

Corollary 5.2

The local parameter function is constant if and only if the distribution belongs to the pseudo-family

F.

Proof:

Iff⁰(y) =f(y⁰), we see immediately that ~(cf⁰)) =⁰ is constant. The converse part results from Corollary 5.1.

Q.E.D.

The Corollary 5.1 may be used to derive the functional estimators of the log-derivative of the density function. Indeed if ~ⁿ(c) is a consistent functional estimator of ~(cf⁰), then

@log f(c ~ⁿ(c))

@y

is a consistent functional estimator of ^@log@yf0(c), whereas ^@log@yf(c) is continuous with respect to . Also, the Corollary 5.2 may be used for testing the goodness of t in the family ^F.

Example 5.1:

For the gaussian family^F=^ff(y) =(y^;) ²R^g, we have:

@logf(c)

@y ⁼ @log(c^;)

@y ^=;c:

We deduce that:

~(cf⁰) =c+@logf⁰(c)

@y : Q.E.D.

5.4 Local Parameter Function and Tail Index

In this section we show that the local parameter function can be used to nd the tail index >0 of the true unknown distribution f⁰. From the Karamata's theorem Karamata (1962), Resnick (1987), Corollary 1.12], the true distribution is such that : 1^;F⁰(y) =y^; L(y) y >1, with >0, if and only if :

1^;F⁰(y) =c(y)exp^{;Z y}

1

(t)

t dt ^(5.3)

where thecand functions satisfy:

lim

y!1

c(y) =c >0 lim

y!1

(y) = >0: (5.4)

Let us assume thatcand are twice dierentiable and satisfy the additional limiting conditions:

lim

y!1

yc⁰(y) = lim

y!1

y²c⁰(y) = lim

y!1

y⁰(y) = 0: (5.5)

We get from (5.3):

;y@^logf⁰(y)

@y ⁼(y) + 1 + yc⁰(y)(y) +c(y)y⁰(y)^;yc⁰(y)^;y²c⁰⁰(y) c(y)(y)^;yc⁰(y) which implies:

lim

y!1

;y@^logf⁰(y)

@y ⁼+ 1: (5.6)

Therefore the tail index may be directly derived from the limiting behaviour of^;y^@log@yf0(y). Let us now consider the asymptotic behaviour of the local parameter function, which clearly depends on the tail behaviour of the pseudo-familyf(y).

i) If any distribution f(y) admits a tail index() in a one to one continuous relationship with, we deduce from the equation dening ~(cf⁰) that:

;c@fc ~(cf⁰)]

@y ^=;c@^logf⁰(c)

@y

and by taking the limit whenc tends to innity we nd that:

lim

c!1

~(cf⁰)] + 1 = lim

c!1

~(cf⁰)] + 1 =+ 1:

Therefore the local parameter function tends to a limit such that the tail index of the estimated pseudo-distribution coincides with the index of the true distribution.

ii) For a family of distributions with thin tails, the term ~(cf⁰) may diverge to innity when ctends to innity. The examples below illustrate this feature.

Example 5.2: If the pseudo-family is the gaussian family indexed by the mean, we get:

;y@^logf(y)

@y ⁼y(y^;) and by Property 5.1:

~(yf⁰) =y^;1yy@logf⁰(y)

@y y

for largey.

5.5 Asymptotic Properties of the Local Pseudo-Maximum Likelihood Estimator

The asymptotic properties of the local P.M.L. estimator ofare derived along the following lines.

We rst nd the asymptotically equivalent formula of the objective function and of the estimator, which only depend on a limited number of kernel estimators. Then, we deduce the properties of the local P.M.L. estimator from the properties of these basic kernel estimators. We only detail the additional assumptions which are necessary for the asymptotic equivalence to hold, since the set

of assumptions for the existence and asymptotic normality of the basic kernel estimators are quite standard see Parakasa-Rao (1983), Bosq- Lecoutre(1987), Hardle (1990)].

Property 5.2

The local pseudo-maximum likelihood estimator ~ⁿ(c) exists and is a strongly consistent estimator of the local parameter function ~(cf⁰) under A.1 - A.6 and the following additional assumptions:

A.7

: the parameter set is an open set

A.8

: there exists a unique solution in of the equality:

@logf(c)

@y ⁼@log f⁰(c)

@y

A.9

: the following kernel estimators are strongly consistent:

(i) ^n1Pni=1h1K^{;yi;ch a:s:!} f⁰(c)

(ii) ^1nPni=1h1K^{;yi;ch ;yi;ch 2a:s:! 2}f⁰(c)

(iii) ^{nh1 Pni=1 1h}K^{;yih;c;yih;ca:s:! 2}f⁰(c)^@log@yf0(c)

A.10

In any neighbourhood of, the third order derivative @³logf(y)]=@y³ is dominated by a functiona(y) such thaty³a(y) is integrable.

Proof:

See Appendix 2(i).

Property 5.3

Under assumptions A.1 - A.10 the local pseudo-maximum likelihood estimator is asymptotically equivalent to the solution ~~ⁿ(c) of the equation:

@log fc ~~ⁿ(c)]

@y ^{= 12}h^{12( ~}mⁿ(c)^;c) where:

m~ⁿ(c) =

P

n

i=1K^;yi;ch yⁱ

P

n

i=1K^;yi;ch

is the Nadaraya - Watson estimator ofm(c) =E(Y^jY =c) =c based on the kernelK.

Proof:

See Appendix 2(ii).

Therefore the asymptotic distribution of ~ⁿ(c) may be deduced from the standard properties of m~ⁿ(c)^;c. We only consider below the pointwise convergence in distribution A functional theorem for ~ⁿ(:) may only be derived if there exists a functional theorem for ~mⁿ(c) at second order].

Under standard regularity conditions Prakasa-Rao (1983), Hardle (1990), Bosq, Lecoutre (1987)] the numerator and denominator of 1=h²( ~mⁿ(c)^;c) have the following asymptotic proper- ties:

A.11

: Ifn^!1,h^!0,nh^3!1,nh^5!0, we have the limiting distribution:

pnh^3hnh13Pni=1K^;yih;c(y^i;c)^{;2@f0(c)@y i}

pnh^{nh1 Pni=1}K^{;yi;ch ;}f⁰(c)

!

d

! N 0f⁰(c)

R

u²K²(u)du ^R uK²(u)du

R uK²(u)du ^RK²(u)du

:

The formulas of the rst and second order asymptotic moments are easily veried see Appendix 3].

The rate of convergence of the numerator is slower than the rate of convergence of the denominator since we study the degenerate case, where the Nadaraya-Watson estimator is applied to a regression with regressor equal to the regressand. We deduce that the asymptotic distribution of

pnh³ h^{12 ~}mⁿ(c)^;c]^;2@logf⁰(c)

@y

coincides with the asymptotic distribution of

pnh³f⁰¹(c)

1 nh³

n

X

i=1

K y^i;c h

(y^i;c)^;2@f⁰(c)

@y

!

i.e. N^h0 ^{f0(c)1 R}u²K²(u)duⁱ.

We deduce directly by the -method the asymptotic distribution of the local pseudo maximum likelihood estimator and of the log derivative of the true p.d.f.

Property 5.4

Under assumptions A.1 - A.11 we have:

i)

pnh³

@logf(c ~~ⁿ(c))

@y ^;@logf⁰(c)

@y

!

d

!N0 ⁴f¹⁰(c)

Z u²K²(u)du:

ii)

pnh^3~~n(c)^;~(cf⁰)^!d N

2

40

@²logf(c ~(cf⁰))

@y@

!

;2 1 ⁴f⁰(c)

Z

u²K²(u)du

3

5: Remark 5. : The functional estimator of the log-derivative ^@log@yf0(c) may be compared to the standard one:

@log ^f⁰(c)

@y ⁼

P

n

i=1 1

hK^0;yih;c

P

n

i=1 1

hK^;yi;ch : The rate of convergence is identical and the asymptotic distribution is:

pnh³

@log ^f⁰(c)

@y ^;@logf⁰(c)

@y

!

d

!N0 ⁴f¹⁰(c)

Z K⁰(u)²du:

The asymptotic distributions of the two kernel estimators of the density function are identical if

jK⁰(u)^j=^juK(u)^j, in particular for a gaussian kernel.

6 Empirical Results

We examine a series of returns for the Alcatel stock covering the period of July and August 1995. Alcatel belongs to the most frequently traded stocks on the Paris Stock Exchange (Paris Bourse). On average the shares of Alcatel are exchanged every 52 seconds. The data consist of 20502 observations recorded intradaily with an accuracy of 1 second. Figure 1 shows the dynamic pattern of the Alcatel series. Due to irregular spacing, the conventional time scale is replaced in Figure 1 by an axis ordered by indices of subsequent trades, or equivalently numbers of observations in the sample. The Alcatel returns feature a tendency for clustering and a time varying volatility, while potential deviations of the mean return from zero can visually not be distinguished. Figure 2 displays the marginal density of returns and indicates the examined tail area. The density function is centered at 4.259E-6, while the variance and standard deviation are 9.771E-7 and 0.000989, respectively. The distribution is slightly asymmetric, with the skewness coecient -0.00246. The high kurtosis 5.329689 is due to the presence of heavy tails stretched between the extreme values of -0.00813 and 0.007255. 90 % of the probability mass is concentrated between -0.0017 and 0.001738.

The interquartile range 0.000454 is 100 times smaller than the overall range 0.01538. The shape of tails, although kernel smoothed in Figure 2, suggests some local irregularities in the rate of tail decay. Slight lobes can clearly be distinguished in both tails. In the right tail we observe a higher probability of returns taking values between 0.0012 and 0.0013 compared to the probability of those of a slightly smaller size, i.e. between 0.0010 and 0.0012. In the left tail we recorded relatively more returns between -0.0014 and -0.0012 than those taking marginally higher values.

We perform the tail analysis using three pseudo-models:

1) a gaussian pseudo-model, 2) a Pareto pseudo-model,

3) an exponential pseudo-model applied to Box-Cox transformed data.

In each case, we apply the methods proposed in the paper. The rst method consists in comput- ing the pseudo maximum likelihood estimator over the interval c^j1) for an increasing sequence (c^j) j = 1:::n. The second method involves maximization of the pseudo likelihood function over disjoint intervals c^j c^j+1] j = 1:::n. These procedures yield estimators of the t.p.f. and the l.p.f., respectively. We present below the optimization criteria for the estimation of the t.p.f for the three pseudo-models considered.

1)

gaussian family :

The t.p.f. estimator consists of a sequence of solutions (^c2c), c=c^j j= 1:::nof the following maximization problems:

max

n

X

i=1

1^{yi>cj ;}1

2 log^2;1_{2 log2}^;(y^i;)²

2^{2 ;log ;}c

forc^j j= 1:::n.

To obtain a concave objective function, we introduce the change of parameters: a= 1= b==.

2)

Pareto family :

For logYⁱ(1) the t.p.f is obtained by solving:

max

n

X

i=1

1^yi>cjlog^;(logy^i;c^j)]: The estimator is:

1^ ⁼

P

n

i=11^yi>cj(logy^i;c^j)

P

n

i=11^yi>cj :

3)

Box - Cox transformed exponential variables :

We introduce a transformed variable X = (Y^i;1)=(1 ). The survivor function is dened by the probability:

S(y) = exp

; y^i;1

while the density of the pseudo model is:

f(y) =y^i;1exp

; y^i;1

: