Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models

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Models and Stochastic Volatility Models

Valentine Genon-Catalot, Thierry Jeantheau, Catherine Laredo

To cite this version:

Valentine Genon-Catalot, Thierry Jeantheau, Catherine Laredo. Conditional Likelihood Estimators

for Hidden Markov Models and Stochastic Volatility Models. Scandinavian Journal of Statistics,

Wiley, 2003, 30, pp.297-316. �hal-02676701�

Conditional Likelihood Estimators for Hidden Markov Models

and Stochastic Volatility Models

VALENTINE GENON-CATALOT Universite´ de Marne la Valle´e

THIERRY JEANTHEAU Universite´ de Marne-la-Valle´e

CATHERINE LAREDO INRA-Jouy-en-Josas

ABSTRACT. This paper develops a new contrast process for parametric inference of general hidden Markov models, when the hidden chain has a non-compact state space. This contrast is based on the conditional likelihood approach, often used for ARCH-type models. We prove the strong consistency of the conditional likelihood estimators under appropriate conditions.

The method is applied to the Kalman filter (for which this contrast and the exact likeli- hood lead to asymptotically equivalent estimators) and to the discretely observed stochastic volatility models.

Key words:conditional likelihood, diffusion processes, discrete time observations, hidden Markov models, parametric inference, stochastic volatility

1. Introduction

Parametric inference for hidden Markov models (HMMs) has been widely investigated, es- pecially in the last decade. These models are discrete-time stochastic processes including classical time series models and many other non-linear non-Gaussian models. The observed process ðZ

Þ ismodelled via an unobserved Markov chain ðU

Þ such that, conditionally on ðU

Þ, the ðZ

Þsare independent and the distribution of Z

dependsonly on U

. When studying the statistical properties of HMMs, a difficulty arises since the exact likelihood cannot be explicitly calculated. Asa consequence, a lot of papershave been concerned with approxi- mations by means of numerical and simulation techniques (see, e.g. Gourie´roux et al., 1993;

D

1 urbin & Koopman, 1998; Pitt & Shephard, 1999; Kim et al., 1998; Del Moral & Miclo, 2000;

Del Moral et al., 2001).

The theoretical study of the exact maximum likelihood estimator (m.l.e.) is a difficult problem and has only been investigated in the following cases. Leroux (1992) has proved the consistency when the unobserved Markov chain has a finite state space. Asymptotic nor- mality isproved in Bickel et al. (1998). The extension of these properties to a compact state space for the hidden chain can be found in Jensen & Petersen (1999) and Douc & Matias (2001).

In previouspapers(see Genon-Catalot et al., 1998, 1999, 2000a), we have investigated some

statistical properties of discretely observed stochastic volatility models (SV). In particular,

when the sampling interval is fixed, the SV models are HMMs, for which the hidden chain has

a non-compact state space. Using ergodicity and mixing properties, we have built empirical

moment estimators of unknown parameters. Other types of empirical estimators are given in

Gallant

2 et al. (1997) using the efficient moment method or by Sørensen (2000) considering the class of prediction-based estimating functions.

In this paper, we propose a new contrast for general HMMs, which is theoretical in nature but seems close to the exact likelihood. The contrast is based on the conditional likelihood method, which is the estimating method used in the field of ARCH-type models (see Jeantheau, 1998). The method isapplied to examples. Particular attention ispaid throughout this paper to the well-known Kalman filter which lightens our approach. It is a special case of HMM with non-compact state space for the hidden chain. All computations are explicit and detailed herein. The minimum contrast estimator based on the conditional likelihood is as- ymptotically equivalent to the exact m.l.e. In the case of the SV models, when the unobserved volatility is a positive ergodic diffusion, the method is also applicable. It requires that the state space of the hidden diffusion is open, bounded and bounded away from zero. Contrary to moment methods, only the finiteness of the first moment of the stationary distribution is needed.

In Section 2, we recall definitions, ergodic properties and expressions for the likelihood (Propositions 1 and 2) for HMMs. We define and study in Section 3 the conditional like- lihood (Theorem 1 and Definition 2) and build the associated contrasts. Then, we state a general theorem of consistency to study the related minimum contrast estimators (Theorem 2). We apply this method to examples, especially to the Kalman filter and to GARCH(1,1) model. Then, we specialize these results to SV models in Section 4. We consider the con- ditional likelihood (Proposition 4) and study its properties (Proposition 5). Finally, we consider the case of mean reverting hidden diffusion. The proof of Theorem 2 is given in the appendix.

2. Hidden Markov models

2.1. Definition and ergodic properties

The formal definition of an HMM can be taken from Leroux (1992) or Bickel & Ritov (1996).

Definition 1. A stochastic process ðZ

; nP1Þ, with state space ðZ; BðZÞÞ, isa hidden Markov model if:

(i) (Hidden chain) We are given (but do not observe) a time-homogeneous Markov chain U

; U

; . . . ; U

; . . . with state space ðU; BðUÞÞ.

(ii) (Conditional independence) For all n, given ðU

; U

; . . . ;U

Þ, the Z

;i ¼ 1; . . . n are con- ditionally independent, and the conditional distribution of Z

only dependson U

. (iii) (Stationarity) The conditional distribution of Z

given U

¼ u doesnot depend on i.

Above, Z and U are general Polish spaces equipped with their Borel sigma-fields. There- fore, thisdefinition extendsthe one given in Leroux (1992) or Bickel & Ritov (1996) since we do not require a finite state space for the hidden chain.

We give now some examples of HMMs. They are all included in the following general framework. We set

Z

¼ GðU

; e

Þ; ð1Þ

where ð Þ e

is a sequence of i.i.d. random variables, ð U

Þ

isa Markov chain with state

space U, and these two sequences are independent.

Example 1 (Kalman filter). The above class includes the well-known discrete Kalman filter

Z

¼ U

þ e

; ð2Þ

where ðU

Þ is a stationary real AR(1) Gaussian process and ðe

Þ are i.i.d. Nð0; c

Þ.

Example 2 (Noisy observations of a discretely observed Markov process). In (1), take U

¼ V

, where ðV

Þ

is any Markov process independent of the sequence ð Þ e

.

Example 3 (Stochastic volatility models). These models are given in continuous time by a two-dimensional process ðY

; V

Þ with

dY

¼ r

dB

;

and V

¼ r

(the so-called volatility) is an unobserved Markov process independent of the brownian motion ðB

Þ. Then, a discrete observation with sampling interval D istaken. We set

Z

¼ 1 ffiffiffiffi D p

Z

r

dB

¼ 1 ffiffiffiffi D

p ðY

Y

Þ: ð3Þ Conditionally on ðV

; s 0Þ, the random variables ðZ

Þ are independent and Z

has distribution Nð0; V V

Þ with

V V

¼ 1

D Z

ðn1ÞD

V

ds: ð4Þ

Noting that U

¼ ð V V

; V

Þ isMarkov, we set, in accordance with (1), Z

¼ GðU

; e

Þ ¼ V V

e

:

Such models have been first proposed by Hull & White (1987), with ðV

Þ a diffusion process.

When ðV

Þ isan ergodic diffusion, it isproved in Genon-Catalot et al. (2000a)

3 that ðZ

Þ isan

HMM. In Barndorff-Nielsen & Shephard (2001), the model is analogous, with ðV

Þ an Ornstein Uhlenbeck Levy process.

An HMM hasthe following properties.

Proposition 1

(1) The process ððU

; Z

Þ; nP1Þ is a time-homogeneous Markov chain.

(2) If the hidden chain ðU

; n 1Þ is strictly stationary, so is ððU

; Z

Þ; nP1Þ.

(3) If, moreover, the hidden chain ðU

; nP1Þ is ergodic, so is ððU

; Z

Þ; nP1Þ.

(4) If, moreover, ðU

; nP1Þ is a-mixing, then ððU

; Z

Þ; nP1Þ is also a-mixing, and a

ðnÞOa

ðnÞOa

ðnÞ:

The first two points are straightforward, the third is proved in Leroux (1992), and the mixing property isproved in Genon-Catalot et al. (2000a, b) and Sørensen (1999). This mixing property holdsfor the previousexamples.

Example 1 (continued) (Kalman filter). We set

U

¼ aU

þ g

; ð5Þ

where ðg

Þ are i.i.d. Nð0;b

Þ. When jaj < 1, and U

haslaw Nð0; s

¼ b

=ð1 a

ÞÞ, U

is a-mixing.

Examples 2 and 3 (continued). In both cases, whenever ðV

Þ isa diffusion, we have a

ðnÞOa

ððn 1ÞDÞ:

For ðV

Þ a strictly stationary diffusion process, it is well known that a

ðtÞ ! 0 as t ! þ1:

For details(such asthe rate of convergence of the mixing coefficient), we refer to Genon- Catalot et al. (2000a) and the referencestherein.

The above properties have interesting statistical consequences. When the unobserved chain dependson unknown parameters, the ergodicity and mixing propertiesof ðZ

Þ can be used to derive the consistency and asymptotic normality of empirical estimators of the form 1=n P

i¼0

uðZ

; . . . ; Z

Þ. Thisleadsto variousproceduresthat have been already applied to models[Moment method, GMM (Hansen, 1982), EMM (Tauchen et al., 1996; Gallant et al., 1997) or prediction-based estimating equations

4 (Sørensen, 2000)]. These methods yield esti-

matorswith rate p ffiffiffi n

. However, to be accurate, they require high moment conditionsthat may be not fulfilled, and often lead to biased estimators and/or to cumbersome computations.

Another central question lies in the fact that they may be very far from the exact likelihood method.

2.2. The exact likelihood

We recall here general propertiesof the likelihood in an HMM model. Such likelihoodshave already been studied but mainly under the assumption that the state space of the hidden chain is finite (see, e.g. Leroux, 1992; Bickel & Ritov, 1996). Since this is not the case in the examples above, we focus on the properties which hold without this assumption.

Consider a general HMM as in Definition 1 and let us give some more notations and assumptions.

• (H1) The conditional distribution of Z

given U

¼ u isgiven by a density fðz=uÞ with respect to a dominating measure lðdzÞ on ðZ; BðZÞÞ.

• (H2) The transition operator P

of the hidden chain ðU

Þ dependson an unknown parameter h 2 H R

; pP1, and the transition probability is specified by a density pðh; u; tÞ with respect to a dominating measure mðdtÞ on ðU; BðUÞÞ.

• (H3) For all h, the transition P

admits a stationary distribution having density with respect to the same dominating measure, denoted by gðh; uÞ. The initial variable U

hasthissta- tionary distribution.

• (H4) For all h, the chain ðU

Þ with marginal distribution gðh; uÞmðduÞ isergodic.

Under these assumptions, the process ðU

; Z

Þ is strictly stationary and ergodic. Let us point out that we do not introduce unknown parametersin the density of Z

given U

¼ u, since, in our examples, all unknown parameters will come from the hidden chain.

With these notations, the distribution of the initial variable ðU

; Z

Þ of the chain ðU

; Z

Þ has density gðh; uÞf ðz=uÞ with respect to mðduÞ lðdzÞ; the transition density is given by

pðh; u; tÞf ðz=tÞ: ð6Þ

Now, if we denote by p

ðh; z

; . . . ; z

Þ the density of ðZ

; . . . ; Z

Þ with respect to dlðz

Þ dlðz

Þ, we have

p

ðh; z

Þ ¼ Z

U

gðh; uÞf ðz

=uÞ dmðuÞ; ð7Þ

p

ðh; z

; . . . ; z

Þ ¼ Z

Uⁿ

gðh; u

Þf ðz

=u

Þ Y

i¼2

pðh; u

; u

Þf ðz

=u

Þ dmðu

Þ dmðu

Þ: ð8Þ In formula (8), the function under the integral sign is the density of ðU

; . . . ; U

; Z

; . . . ; Z

Þ.

A more tractable expression for p

ðh; z

; . . . ; z

Þ is obtained from the classical formula:

p

ðh; z

; . . . ; z

Þ ¼ p

ðh; z

Þ Y

i¼2

t

ðh; z

=z

; . . . ; z

Þ; ð9Þ

where t

ðh; z

=z

; . . . ; z

Þ isthe conditional density of Z

given Z

¼ z

; . . . ; Z

¼ z

. The interest of this representation appears below.

Proposition 2

(1) For all iP2, we have (see (9)) t

ðh;z

=z

; . . . ; z

Þ ¼

Z

U

g

ðh;u

=z

; . . . ; z

Þf ðz

=u

Þ dmðu

Þ; ð10Þ where g

ðh; u

=z

; . . . ; z

Þ is the conditional density of U

given Z

¼ z

; . . . ; Z

¼ z

. (2) If g g ^

ðh; u

=z

; . . . ; z

Þ denotes the conditional density of U

given Z

¼ z

; . . . ; Z

¼ z

, then it is

given by

^ g

g

ðh; u

=z

; . . . ; z

Þ ¼ g

ðh; u

=z

; . . . ; z

Þf ðz

=u

Þ t

ðh; z

=z

; . . . ; z

Þ : (3) For all iP1,

g

ðh; u

=z

; . . . ; z

Þ ¼ Z

U

^ g

g

ðh; u

=z

; . . . ; z

Þpðh; u

; u

Þ dmðu

Þ: ð11Þ The result of Proposition 2 is standard in the field of filtering theory (see, e.g. Liptser &

Shiryaev, 1978, and for details, Genon-Catalot et al., 2000b). It leads to a recursive expression of the exact likelihood function. Now, we are faced with two kindsof problems, a numerical one and a theoretical one.

The numerical computation of the exact m.l.e. of h raises difficulties and there is a large amount of literature devoted to approximation algorithms(Markov chain Monte Carlo methods(MCMC), EM algorithm, particle filter method, etc.). In particular, MCMC methods have been recently considered in econometrics and applied to partially observed diffusions (see, e.g. Eberlein et al., 2001)

5 .

From a theoretical point of view, for HMMs with a finite state space for the hidden chain, results on the asymptotic behaviour (consistency) of the m.l.e. are given in Leroux (1992), Francq & Roussignol (1997) and Bickel & Ritov (1996). Extensions to the case of a compact state space for the hidden chain have been recently investigated (Jensen & Petersen, 1999;

Douc & Matias, 2001). Apart from these cases, the problem is open. Let us stress that, in the SV models, the state space of the hidden chain is not compact.

Nevertheless, there is at least one case where the state space of the hidden chain is non- compact and where the asymptotic behaviour of the m.l.e. is well known: the Kalman filter model defined in (2).

In this paper, we consider and study new estimators that are minimum contrast estimators based on the conditional likelihood. We can justify our approach by the fact that these estimators are asymptotically equivalent to the exact m.l.e. in the Kalman filter.

3. Contrasts based on the conditional likelihood

Our aim is to develop a theoretical tool for obtaining consistent estimators. A specific feature of HMMsisthat the conditional law of Z

given the past observations depends effectively on i and all the observations Z

; . . . ; Z

. In order to recover some stationarity properties, we need to introduce the infinite past of each observation Z

. Thisisthe concern of the conditional likelihood. This estimation method derives from the field of discrete time ARCH models.

Besides, as a tool, it appears explicitly in Leroux (1992, Section 4).

We first introduce and study the conditional likelihood, and then derive associated contrast functions leading to estimators.

3.1. Conditional likelihood

Since the process ðU

; Z

Þ is strictly stationary, we can consider its extension to a process ððU

; Z

Þ; n 2 ZÞ indexed by Z, with the same finite-dimensional distributions. For all i 2 Z, let Z

¼ ðZ

; Z

; . . .Þ be the vector of R

defining the past of the process until i. Below, we prove that the conditional distribution of the sample ðZ

; . . . ;Z

Þ given the infinite past Z

admitsa density with respect to lðdz

Þ lðdz

Þ, which allowsto define the conditional likelihood of ðZ

; . . . ;Z

Þ given Z

.

From now on, let us suppose that Z ¼ R and U ¼ R

, for some kP1. Denote by P

the distribution of ððU

;Z

Þ;n 2 ZÞ on the canonical space, ððU

; Z

Þ; n 2 ZÞ will be the canonical process, and E

¼ E

.

Theorem 1

Under (H1)–(H3), the followingholds:

(1) The conditional distribution, under P

, of U

given the infinite past Z

admits a density with respect to mðduÞ equal to

~

g gðh; u=Z

Þ ¼ E

ðpðh; U

; uÞ=Z

Þ: ð12Þ

The function ~ g gðh; u=Z

Þ is well defined under P

as a measurable function of ðu; Z

Þ, since there exists a regular version of the conditional distribution of U

given Z

.

(2) The conditional distribution, under P

, of Z

given the infinite past Z

admits a density with respect to dlðz

Þ equal to

~

p pðh; z

=Z

Þ ¼ Z

U

~ g

gðh; u

=Z

Þfðz

=u

Þmðdu

Þ: ð13Þ

Proof of Theorem 1. We omit h in all notationsfor the proof. By the Markov property of ðU

; Z

Þ, the conditional distribution of U

given ðU

;Z

Þ; ðU

; Z

Þ; . . . ; ðU

; Z

Þ isiden- tical to the conditional distribution of U

given ðU

; Z

Þ, which iss imply the conditional distribution of U

given U

[see (6)]. Consequently, for u : U ! ½0; 1 measurable,

EðuðU

Þ=U

; Z

; Z

; . . . ; Z

Þ ¼ EðuðU

Þ=U

Þ ¼ PuðU

Þ; ð14Þ with

P uðU

Þ ¼ Z

uðuÞpðU

; uÞmðduÞ: ð15Þ

Hence,

EðuðU

Þ=Z

; Z

; . . . ; Z

Þ ¼ EðPuðU

Þ=Z

; Z

; . . . ;Z

Þ: ð16Þ

By the martingale convergence theorem, we get

EðuðU

Þ=Z

Þ ¼ EðPuðU

Þ=Z

Þ: ð17Þ

Now, using the fact that there is a regular version of the conditional distribution of U

given Z

, say dP

ðu

=Z

Þ, we obtain

EðuðU

Þ=Z

Þ ¼ Z

U

P uðu

Þ dP

ðu

=Z

Þ: ð18Þ

Applying the Fubini theorem yields EðuðU

Þ=Z

Þ ¼

Z

U

uðuÞEðpðU

; uÞ=Z

ÞmðduÞ: ð19Þ

So, the proof of (1) iscomplete.

Using again the Markov property of ðU

; Z

Þ and (6), we get that the conditional distri- bution of Z

given ðU

; Z

; Z

; . . . ;Z

Þ isidentical to the conditional distribution of Z

given U

which issimply f ðz

=U

Þlðdz

Þ. So taking u : Z ! ½0; 1 measurable as above, we get

EðuðZ

Þ=Z

; Z

; . . . ;Z

Þ ¼ EðEðuðZ

Þ=U

Þ=Z

; Z

; . . . ; Z

Þ: ð20Þ So, using the martingale convergence theorem and Proposition 2, we obtain

EðuðZ

Þ=Z

Þ ¼ Z

R

uðz

Þlðdz

Þ Z

U

f ðz

=u

Þ~ g gðu

=Z

Þmðdu

Þ: ð21Þ

Thisachievesthe proof. h

Note that, under P

, by the strict stationarity, for all iP1, the conditional density of Z

given Z

isgiven by

~

p pðh; z

=Z

Þ ¼ Z

U

~

g gðh; u

=Z

Þf ðz

=u

Þmðdu

Þ: ð22Þ Thus, the conditional distribution of ðZ

; . . . ; Z

Þ given Z

¼ z

hasa density given by

~ p

p

ðh;z

; . . . ;z

=z

Þ ¼ Y

i¼1

~

p pðh; z

=z

Þ: ð23Þ

Hence, we may introduce Definition 2.

Definition 2. Let us assume that, P

a.s., the function ~ p pðh; Z

=Z

Þ iswell defined for all h and all n. Then, the conditional likelihood of ðZ

; . . . ; Z

Þ given Z

isdefined by

~ p

p

ðh;Z

Þ ¼ Y

i¼1

~

p pðh; Z

=Z

Þ ¼ ~ p p

ðh; Z

; . . . ; Z

=Z

Þ: ð24Þ Let usset when defined

lðh; z

Þ ¼ log ~ p pðh; z

=z

Þ; ð25Þ

so that

log ~ p p

ðhÞ ¼ X

i¼1

lðh; Z

Þ:

We have Proposition 3.

Proposition 3

Under (H1)–(H4), if E

jlðh; Z

Þj < 1, then, we have, almost surely, under P

, 1

n log ~ p p

ðh; Z

Þ ! E

lðh; Z

Þ:

Proof of Proposition 3. Using Proposition 1 (3), ðZ

Þ isergodic, and the ergodic theorem

may be applied. h

Example 1 (continued) (Kalman filter). For the discrete Kalman filter [see (2)], the suc- cessive distributions appearing in Proposition 2 are explicitely known and Gaussian.

With the notationsintroduced previously, the unknown parametersare a; b

while c

is supposed to be known and jaj < 1. We assume that ðU

Þ isin a s tationary regime. The following properties are well known and are obtained following the steps of Proposition 2.

Under P

, ðh ¼ ða; b

ÞÞ:

(i) the conditional distribution of U

given ðZ

; . . . ; Z

Þ isthe law Nðx

;r

Þ;

(ii) the conditional distribution of U

given ðZ

; . . . ; Z

Þ isthe law Nð^ x x

; r r ^

Þ;

(iii) the conditional distribution of Z

given ðZ

;. . . ; Z

Þ isthe law Nð x x

; r r

Þ, with

v

¼ r r ^

c

v

¼ s

s

þ c

; ð26Þ

^ x

x

¼ a^ x x

þ ðZ

a^ x x

Þv

^ x x

¼ 0; ð27Þ

v

¼ fðv

Þ with f ðvÞ ¼ 1 1 þ b

c

þ a

v

; ð28Þ

x

¼ a^ x x

¼ x x

r

¼ b

þ a

c

v

; ð29Þ

r

r

¼ r

þ c

r r

¼ s

þ c

: ð30Þ With the above notations, the distribution of Z

isthe law Nð x x

; r r

Þ and the exact likelihood of ðZ

; . . . ; Z

Þ is(up to a constant) equal to

p

ðhÞ ¼ p

ðh; Z

; . . . ; Z

Þ ¼ ð r r

. . . r r

Þ

exp X

i¼1

ðZ

x x

Þ

2 r r

!

; ð31Þ

where h ¼ ða;b

Þ, xx

¼ xx

ðhÞ and r r

¼ r r

ðhÞ.

The conditional distribution of Z

given ðZ

; . . . ; Z

Þ is obtained substituting ðZ

; . . . ; Z

Þ by ðZ

; . . . ; Z

Þ in the law Nð x x

; r r

Þ. This sequence of distributions convergesweakly to the conditional distribution of Z

given Z

. Indeed, the deterministic recurrence equation for ðv

Þ convergesto a limit vðhÞ 2 ð0; 1Þ with exponential rate [see (28)].

Using the above equations, we find after some computations that the conditional distribution of Z

given Z

isthe law Nð xxðh; Z

Þ; r r

ðhÞÞ with

xxðh; Z

Þ ¼ aE

ðU

=Z

Þ ¼ avðhÞ X

i¼0

a

ð1 vðhÞÞ

Z

and r r

ðhÞ ¼ ac

v þ b

þ c

: Therefore, the conditional likelihood is(up to a constant) explicitely given by

~ p

p

ðh; Z

Þ ¼ r rðhÞ

exp X

i¼1

ðZ

x xðh; Z

ÞÞ

2 r rðhÞ

!

: ð32Þ

Since the series xxðh; Z

Þ convergesin L

ðP

Þ, thisfunction iswell defined for all h under P

.

Moreover, the assumptions of Proposition 3 are satisfied, and the limit is

E

lðh; Z

Þ ¼ 1

2 log r rðhÞ

þ 1 r

rðhÞ

E

ðZ

x xðh; Z

ÞÞ

( )

: ð33Þ

3.2. Associated contrasts

The conditional likelihood cannot be used directly, since we do not observe Z

. However, the conditional likelihood suggests a family of appropriate contrasts to build consistent estima- tors.

As mentioned earlier, our approach is motivated by the estimation methods used for dis- crete time ARCH models (see Engle, 1982). In these models, the exact likelihood is untract- able, whereasthe conditional likelihood isexplicit and simple. Moreover, the common feature of these series is that they usually do not have even second-order moments. This rules out any standard estimation method based on moments. Elie & Jeantheau (1995) and Jeantheau (1998) have proved an appropriate theorem to deal with this difficulty. We also use this theorem later and recall it. For the sake of clarity, its proof is given in the appendix.

3.2.1. A general result of consistency

For a given known sequence z

of R

, we define the random vector of R

Z

ðz

Þ ¼ ðZ

; . . . ;Z

; z

Þ:

Let f be a real function defined on H R

and set F

ðh; z

Þ ¼ n

X

i¼1

f ðh; z

Þ: ð34Þ

Let usintroduce the random variable h

¼ arg inf

h

F

ðh;Z

Þ ¼ h

ðZ

Þ: ð35Þ

Now, we introduce the estimator defined by the equation h ~

h

ðz

Þ ¼ arg inf

h

F

ðh;Z

ðz

ÞÞ ¼ h

ðZ

ðz

ÞÞ: ð36Þ

The estimator h h ~

ðz

Þ isa function of the observationsðZ

; . . . ; Z

Þ, but also depends on f and z

. Theorem 2 givesconditionson f and z

to obtain strong consistency for this type of estimators.

We have in mind the case of f ¼ log l [see (25)] so that F

ðh; Z

Þ ¼ log ~ p p

ðhÞ. Never- theless, other functions of f could be used.

Asusual, let h

be the true value of the parameter and consider the following conditions:

• C0 H iscompact.

• C1 The function f issuch that

(i) For all n, fðh; Z

Þ ismeasurable on H X and continuousin h, P

a.s.

(ii) Let Bðh;qÞ be the open ball of centre h and radius q, and set for i 2 Z, f

ðh; q;Z

Þ ¼ infff ðh

; Z

Þ; h

2 Bðh;qÞ \ Hg. Then, 8h 2 H; E

ðf

ðh; q; Z

ÞÞ > 1 [with the notation a

¼ infða; 0Þ] .

• C2 The function h ! F ðh

; hÞ ¼ E

ð fðh; Z

Þ Þ hasa unique (finite) minimum at h

.

• C3 The function f and z

are such that

(i) For all n, fðh; Z

ðz

ÞÞ ismeasurable on H X and continuousin h, P

a.s.

(ii) f ðh; Z

Þ f ðh; Z

ðz

ÞÞ ! 0 as n ! 1; P

a:s:; uniformly in h.

Theorem 2

Assume (H1)–(H4). Then, under (C0)–(C2), the random variable h

defined in (35) converges P

a.s. to h

when n ! 1. Under (C0)–(C3), h h ~

ðz

Þ converges P

a.s. to h

.

Let us make some comments on Theorem 2. It holds not only for HMMs, but for any strictly stationary and ergodic process under condition (C0)–(C3). It is an extension of a result of Pfanzagl (1969) proved for i.i.d. data and for the exact likelihood. Itsmain interest isto obtain strongly consistent estimators under weaker assumptions than the classical ones. In particular, (C1) and (C2) are weak moment and regularity conditions. Condition (C3) appears as the most difficult one. We give below examples where these conditions may be checked. Let us stress that in econometric literature, (C3) is generally not checked and only h

isconsidered and treated as the standard estimator. Note also that (C3) may be weakened into

1 n

X

i¼1

f ðh; q; Z

Þ f ðh;q; Z

ðz

ÞÞ

ð Þ ! 0 as n ! 1;

uniformly in h, in P

-probability. Thisleadsto a convergence in P

-probability of h h ~

ðz

Þ to h

.

Example 4 (ARCH-type models). Consider observations Z

given by Z

¼ U

e

and U

¼ uðh; Z

Þ;

where ðe

Þ isa sequence of i.i.d. random variableswith mean 0 and variance 1, and with I

¼ rðZ

; kOnÞ, e

is I

-measurable and independent of I

. Although U

isa Markov chain, Z

is not an HMM in the sense of Definition 1. Still, under appropriate assumptions, Z

is strictly stationary and ergodic. If the e

s are Gaussian, we choose f ðh; Z

Þ ¼ 1

2 log uðh;Z

Þ þ Z

uðh;Z

Þ

;

which corresponds to the conditional loglikelihood. If the e

s are not Gaussian, we may still consider the same function.

To clarify, consider the special case of a GARCH(1,1) model, given by U

¼ x þ aZ

þ bU

¼ x þ ðae

þ bÞU

;

where x; a and b are positive real numbers. It is well known that, if Eðlogðae

þ bÞÞ < 0, there exists a unique stationary and ergodic solution, and, almost surely,

U

¼ x

1 b þ a X

iP1

b

Z

:

Let us remark that this solution does not even have necessarily a first-order moment.

However, it is enough to add the assumption xPc > 0 to prove that assumptions (C1)–(C2) hold. Fixing Z

¼ z

isequivalent to fixing U

¼ u

, and (C3) can be proved (see Elie &

Jeantheau, 1995).

3.2.2. Identifiability assumption

Condition (C2) is an identifiability assumption. The most interesting case is when f directly comesfrom the conditional likelihood, that isto say

f ðh; z

Þ ¼ lðh; z

Þ ¼ log ~ p pðh; z

=z

Þ:

In this case, the limit appearing in (C2) admits a representation in terms of a Kullback–Leibler

information. Consider the random probability measure

P ~

P

ðdzÞ ¼ ~ p pðh; z=Z

ÞlðdzÞ ð37Þ

and the assumptions

• (H5) 8h

; h; E

jlðh; Z

Þj < 1

• (H6) ~ P P

¼ P P ~

; P

a.s. ¼) h ¼ h

. Set

Kðh

; hÞ ¼ E

Kð P P ~

; P P ~

Þ; ð38Þ

where Kð P P ~

; P P ~

Þ denotesthe Kullback information of P P ~

with respect to P P ~

.

Lemma 1

Assume (H1)–(H6), we have, almost surely, under P

, 1

n ðlog ~ p p

ðh

; Z

Þ log ~ p p

ðh; Z

ÞÞ ! Kðh

;hÞ;

and (C2) holds for f ¼ l.

Proof of Lemma 1. Under (H5), the convergence isobtained by the ergodic theorem.

Conditioning on Z

, we get E

log ~ p pðh; Z

=Z

Þ ¼ E

Z

R

log ~ p pðh; z=Z

Þ d P P ~

ðzÞ:

Therefore,

E

½ lðh; Z

Þ lðh

; Z

Þ ¼ Kðh

; hÞ:

Thisquantity isnon negative [s ee (38)] and, by (H6), equal to 0 if and only if

~ P

P

¼ P P ~

P

a.s. h

Example 1 (continued) (Kalman filter). Recall that [see (33)] the conditional likelihood is based on the function

f ðh; Z

Þ ¼ lðh; Z

Þ ¼ 1

2 log r rðhÞ

þ 1

r rðhÞ

ðZ

x xðh;Z

ÞÞ

( )

: ð39Þ

Condition (C1) is immediate. To check (C2), we use Lemma 1. Assumption (H5) is also immediate. Let uscheck (H6). We have seen that

P ~

P

¼ Nð xxðh; Z

Þ; r r

ðhÞÞ:

Thus, P ~

P

¼ P P ~

P

a.s.

isequivalent to

xxðh; Z

Þ ¼ xxðh

; Z

Þ P

a.s. and r rðhÞ

¼ r rðh

Þ

: ð40Þ The first equality in (40) writes P

iP0

k

Z

¼ 0 P

a.s. with k

¼ avðhÞa

ð1 vðhÞÞ

a

vðh

Þa

ð1 vðh

ÞÞ

:

Suppose that the k

sare not all equal to 0. Denote by i

the smallest integer such that k

6¼ 0.

Then, Z

becomes a deterministic function of its infinite past. This is impossible. Hence, for all

i, k

¼ 0. Thus, we get avðhÞ ¼ a

vðh

Þ and að1 vðhÞÞ ¼ a

ð1 vðh

ÞÞ. Since 0 < vðhÞ; vðh

Þ < 1, we get a ¼ a

and vðhÞ ¼ vðh

Þ. The second equality in (40) yields b ¼ b

. 3.2.3. Stability assumption

Condition (C3) can be viewed as a stability assumption, since it states an asymptotic forgetting of the past. But, here, the stability condition has only to be checked on the specific function f and point z

chosen to build the estimator. This holds for Example 4 (ARCH-type models).

We can also check it for the Kalman filter.

Example 1 (continued) (Kalman filter). Let usprove that (C3) holdswith z

¼ ð0; 0;. . .Þ ¼ 0 and f given by (39). Note that for abirtrary z

in R

, x xðh; z

Þ may be undefined. For z

¼ 0, we have

f ðh; Z

Þ f ðh; Z

ð0ÞÞ ¼ 1

2 r rðhÞ

ð x xðh; Z

Þ x xðh; Z

ð0ÞÞ Þ ð 2Z

x xðh; Z

Þ xxðh; Z

ð0ÞÞ Þ:

Now,

x xðh; Z

Þ x xðh; Z

ð0ÞÞ ¼ a

ð1 vðhÞÞ

xðh; Z

Þ:

Using that H iscompact, we easily deduce (C3) for thisexample.

Remark

To conclude, let us stress that it is possible to compare the exact m.l.e. and the minimum contrast estimator h h ~

ð0Þ in the Kalman filter example. Indeed, ðZ

Þ isa stationary ARMA(1,1) Gaussian process. The exact likelihood requires the knowledge of Z

R

Z where R

isthe covariance matrix of ðZ

;. . . ; Z

Þ. To avoid the difficult computation of R

, two approxi- mations are classical. The first one is the Whittle approximation which consists in computing

~ Z

Z

R

Z Z, where ~ R

isthe covariance matrix of the infinite vector ðZ

; n 2 ZÞ and

~ Z

Z ¼ ð. . . ;0; 0; Z

; . . . ; Z

; 0; 0; . . .Þ. The second one is the case described here. It corresponds to computing ~ Z Z

R

Z Z ~ with ~ Z Z ¼ Z

ð0Þ ¼ ð. . . ; 0; 0; Z

; . . . ; Z

Þ. It iswell known that the three estimators are asymptotically equivalent. It is also classical to use the previous estimators even for non-Gaussian stationary processes (for details, see Beran, 1995).

4. Stochastic volatility models

In this section, we give more details on Example 3, in the case where the volatility V

isa strictly stationary diffusion process.

4.1. Model and assumptions

We consider for t 2 R , ðY

; V

Þ defined by, for sOt Y

Y

¼

Z

r

dB

; ð41Þ

V

¼ r

and V

V

¼ Z

s

bðh;V

Þ du þ Z

s

aðh; V

Þ dW

: ð42Þ

For positive D, we observe a discrete sampling ðY

; i ¼ 1;. . . ; nÞ of (41) and the problem isto

estimate the unknown h 2 H R

of (42) from thisobservation.

We assume that

• (A0) ðB

;W

Þ

isa s tandard Brownian motion of R

defined on a probability space ðX; A; PÞ.

Equation (42) defines a one-dimensional diffusion process indexed by t 2 R. We make now the standard assumptions on functions bðh; uÞ and aðh;uÞ ensuring that (42) admits a unique strictly stationary and ergodic solution with state space ðl; rÞ included is ð0; 1Þ.

• (A1) For all h 2 H; bðh; vÞ and aðh; vÞ are continuous(in v) real functionson R, and C

functionson ðl; rÞ such that

9k > 0; 8v 2 ðl; rÞ; b

ðh; vÞ þ a

ðh;vÞOkð1 þ v

Þ and 8v 2 ðl;rÞ; aðh; vÞ > 0:

For v

2 ðl; rÞ, define the derivative of the scale function of diffusion ðV

Þ, sðh;vÞ ¼ exp 2

Z

bðh; uÞ a

ðh;uÞ du

: ð43Þ

• (A2) For all h 2 H, Z

lþ

sðh; vÞ dv ¼ þ1;

Z

sðh; vÞ dv ¼ þ1;

Z

dv

a

ðh; vÞsðh; vÞ ¼ M

< þ1:

Under (A0)–(A2), the marginal distribution of ðV

Þ is p

ðdvÞ ¼ pðh;vÞdv, with pðh; vÞ ¼ 1

M

1

a

ðh;vÞsðh;vÞ 1

: ð44Þ

In order to study the conditional likelihood, we consider the additional assumptions

• (A3) For all h 2 H; R

l

vp

ðdvÞ < 1.

• (A4) 0 < l < r < 1.

Let us stress that (A3) is a weak moment condition. The condition l > 0 iscrucial. Intui- tively, it is natural to consider volatilities bounded away from 0 in order to estimate their parametersfrom the observation of ðY

Þ.

Let C ¼ CðR; R

Þ be the space of continuous functions on R and R

-valued, equipped with the Borel r-field C associated with the uniform topology on each compact subset of R. We shall assume that ðY

; V

Þ is the canonical diffusion solution of (41) and (42) on ðC; CÞ, and we keep the notation P

for its distribution. For given positive D, we observe ðY

Y

; i ¼ 1; . . . ; nÞ and we define ðZ

Þ asin (3). Asrecalled in Example 3, ðZ

Þ isan HMM with hidden chain U

¼ ðV

; V

Þ.

Setting t ¼ ðv; vÞ 2 ðl; rÞ

, the conditional distribution of Z

given U

¼ t is the Gaussian law Nð0;v), so that lðdzÞ ¼ dz is the Lebesgue measure on R and

f ðz=tÞ ¼ f ðz=vÞ ¼ 1

ð2pvÞ

exp z

2v

: ð45Þ

For the transition density of the hidden chain U

¼ ðV

; V

Þ, it isnatural to have, as dominating measure, the Lebesgue measure

mðdtÞ ¼ 1

ðv; vÞdv dv: ð46Þ

Actually, it amounts to proving that the two-dimensional diffusion ð R

0

V

ds;V

Þ admitsa transition density. Two points of view are possible. In some models, a direct proof may be feasible. Otherwise, this will be ensured under additional regularity assumptions on functions

aðh; :Þ and bðh; :Þ (see, e.g. Gloter, 2000). For sake of clarity, we introduce the following assumption:

• (A5) The transition probability distribution of the chain ðU

Þ isgiven by

pðh; u; tÞmðdtÞ; ð47Þ

where ðu; tÞ 2 ðl; rÞ

ðl; rÞ

and mðdtÞ is the Lebesgue measure (46).

This has several interesting consequences. First, note that this transition density has a special form. Setting u ¼ ða; aÞ 2 ðl; rÞ

,

pðh; u; tÞ ¼ pðh; a; tÞ ð48Þ

only dependson a and isequal to the conditional density of U

¼ ðV

; V

Þ given V

¼ a.

Therefore, the (unconditional) density of U

is(with t ¼ ðv; vÞ) gðh; tÞ ¼

Z

pðh; a; tÞpðh; aÞ da; ð49Þ

where pðh; aÞ da, defined in (44), is the stationary distribution of the hidden diffusion ðV

Þ. Of course, gðh; tÞ is the stationary density of the chain ðU

Þ. The densities of V

and Z

are, therefore [see (45)],

pðh; vÞ ¼ Z

l

gðh; ðv; vÞÞ dv; ð50Þ

p

ðh; z

Þ ¼ Z

l

fðz

=vÞpðh; vÞ dv: ð51Þ

Second, the conditional distribution of V

given Z

¼ z

; . . . ; Z

¼ z

hasa density with respect to the Lebesgue measure on ðl; rÞ, s ay

p

ðh; v

=z

; . . . ; z

Þ: ð52Þ

So, applying Proposition 2, we can integrate with respect to the second coordinate V

of U

¼ ðV

; V

Þ to obtain that the conditional density of Z

given Z

¼ z

; . . . ; Z

¼ z

, is equal to

t

ðh;z

=z

; . . . ; z

Þ ¼ Z

l

p

ðh; v

=z

; . . . ; z

Þf ðz

=v

Þ dv

ð53Þ for all iP2 [see (9)]. Therefore, (53) is a variance mixture of Gaussian distributions, the mixing distribution being p

ðh; v

=z

; . . . ; z

Þ dv

.

Let us establish some links between the likelihood and a contrast previously used in the case of the small sampling interval (see Genon-Catalot et al., 1999). The contrast method is based on the property that the random variables ðZ

Þ behave asymptotically (as D ¼ D

goesto zero) as a sample of the distribution

qðh; zÞ ¼ Z

l

pðh; vÞf ðz=vÞ dv:

The same property of variance mixture of Gaussian distributions appears in (53), with a change of mixing distribution [see also (54)].

4.2. Conditional likelihood

Applying Theorem 1 and integrating with respect to the second coordinate V

of

U

¼ ðV

; V

Þ, we obtain the following proposition:

Proposition 4

Assume (A0)–(A2) and (A5). Then, under P

:

(1) The conditional distribution of V

given Z

¼ z

admits a density with respect to the Lebesgue measure on ðl;rÞ, which we denote by p p ~

ðv=z

Þ.

(2) The conditional distribution of Z

given Z

¼ z

has the density

~

p pðh; z

=z

Þ ¼ Z

l

~ p

p

ðv=z

Þf ðz

=vÞ dv: ð54Þ

Hence, the conditional likelihood of ðZ

; . . . ; Z

Þ given Z

isgiven by

~ p

p

ðh;Z

Þ ¼ Y

i¼1

~

p pðh; Z

=Z

Þ:

Therefore, the distribution given by (54) is a variance mixture of Gaussian distributions, the mixing distribution being now the conditional distribution p p ~

ðv=Z

Þ dv of V

given Z

[com- pare with (53)].

In accordance with Definition 2, let us assume that, P

a.s., the function p p ~

ðv=Z

Þ iswell defined for all h and isa probability density on ðl; rÞ. We keep the following notations:

f ðh; Z

Þ ¼ log ~ p pðh; Z

=Z

ÞÞ and P P ~

ðdzÞ ¼ ~ p pðh;z=Z

Þ dz:

Then, we have

Proposition 5 Under (A0)–(A5):

(1) 8h

;h, E

jf ðh;Z

Þj < 1:

(2) We have, almost surely, under P

, 1

n ðlog ~ p p

ðh

; Z

Þ log ~ p p

ðh; Z

ÞÞ ! Kðh

; hÞ ¼ E

Kð ~ P P

; P P ~

Þ;

see (38).

Proof of Proposition 5. Using (A4) and the fact that p p ~

ðv=Z

Þ isa probability density over ðl; rÞ, we get

1 ffiffiffiffiffiffiffi p 2pr exp z

2l O~ p pðh; z

=Z

ÞÞO 1 ffiffiffiffiffiffiffi

p 2pl : ð55Þ

So, for some constant C (independent of h, involving only the boundaries l; r), we have

jf ðh; Z

ÞjOCð1 þ Z

Þ: ð56Þ

By (A3), E

Z

¼ E

V

< 1. Therefore, we get the first part. The second follows from the

ergodic theorem. h

So, we have checked (H5). We do not know how to check the identifiability assumption (H6). However, in statistical problems for which the identifiability assumption contains ran- domness, this assumption can rarely be verified. Hence, if we know that regularity condition (C1) holds, we get that h

convergesa.s. to h

. Condition (C3) remainsto be checked (see, in thisrespect, our commentsafter Theorem 2).

To conclude, the above results on the SV models are of theoretical nature but clarify the difficulties of the statistical inference in this model and enlight the set of minimal assumptions.

4.3. Mean revertinghidden diffusion

In the SV models, we cannot have explicit expressions for the conditional densities. Therefore, we must use other functions f ðh; Z

Þ to build estimators. To illustrate this, let us consider mean-reverting volatilities, that is, models of the form

dV

¼ aðb V

Þ dt þ aðV

Þ dW

; ð57Þ

where a > 0, b > 0 and aðV

Þ may also depend on unknown parameters. Due to the mean reverting drift, these models present some special features. In particular, many authors have remarked that the covariance structure of the process ðV

Þ is simple (see, e.g. Genon-Catalot et al., 2000a; Sørensen, 2000).

Assume that the above hidden diffusion ðV

Þ satisfies (A1) and (A2), and that EV

isfinite.

Then, EV

¼ EV

¼ b, EV

2

¼ b

þ VarðV

Þ 2ðaD 1 þ e

Þ

a

D

ð58Þ

and for kP1,

EV

V

¼ b

þ VarðV

Þ ð1 e

Þ

a

D

e

: ð59Þ The previousformulae allow to compute the covariance function of ðZ

; iP1Þ.

Proposition 6

Assume (A0)–(A2) and that EV

is finite. Then, the process defined for iP1 by

X

¼ Z

b e

ðZ

bÞ ð60Þ

satisfies, for jP2, CovðX

; X

Þ ¼ 0. Hence, ððZ

bÞ; iP1Þ is centred and ARMA(1,1).

Proof of proposition 6. The process ðZ

; iP1Þ is strictly stationary and ergodic. Straight- forward computationslead to EZ

¼ b,

VarðZ

Þ ¼ 2EV

2

þ VarðV

Þ ð61Þ

and for jP1,

CovðZ

; Z

Þ ¼ CovðV

; V

Þ: ð62Þ

Then, the computation of CovðX

;X

Þ easily follows from (58)–(60). h Estimation by the Whittle approximation of the likelihood is, therefore, feasible as sug- gested by Barndorff-Nielsen & Shephard (2001). To apply our method, as in the Kalman filter, we can use the linear projection of Z

b on its infinite past to build a Gaussian conditional likelihood. To be more specific, let us set h ¼ ða; b; c

Þ with c

¼ VarV

and define, under P

,

c

ð0Þ ¼ Var

X

; c

ð1Þ ¼ Cov

ðX

; X

Þ:

Straightforward computationsshow that c

ð1Þ < 0. The L

ðP

Þ-projection of Z

b on the linear space spanned by (Z

b; iP0) hasthe following form:

Z

b ¼ X

iP0

a

ðhÞðZ

bÞ þ Uðh; Z

Þ;

where, for all iP0,

E

ððZ

bÞUðh; Z

ÞÞ ¼ 0 ð63Þ

and the one-step prediction error is

r

ðhÞ ¼ E

ðU

ðh; Z

ÞÞ: ð64Þ The coefficients ða

ðhÞ; iP0Þ can be computed using c

ð0Þ and c

ð1Þ asfollows . By the canonical representation of ðX

Þ as an MA(1)-process, we have,

X

¼ W

bðhÞW

;

where ðW

Þ is a centred white noise (in the wide sense), the so-called innovation process. It is such that jbðhÞj < 1 and

r

ðhÞ ¼ Var

W

:

Therefore, the spectral density f

ðkÞ satisfies

f

ðkÞ ¼ c

ð0Þ þ 2c

ð1Þ cosðkÞ ¼ r

ðhÞð1 þ b

ðhÞ 2bðhÞ cosðkÞÞ:

Since, for all k, f

ðkÞ > 0, we get that c

ð0Þ 4c

ð1Þ > 0. Now, using that c

ð1Þ < 0, the following holds

bðhÞ ¼ c

ð0Þ c

ð0Þ 4c

ð1Þ

2c

ð1Þ ; r

ðhÞ ¼ c

ð1Þ bðhÞ : Then, setting aðhÞ ¼ exp ðaDÞ, vðhÞ ¼ 1 ½bðhÞ=aðhÞ,

a

ðhÞ ¼ aðhÞvðhÞ ð aðhÞð1 vðhÞÞ Þ

: ð65Þ

Now, we can define

f ðh; Z

Þ ¼ log r

ðhÞ þ 1

r

ðhÞ Z

b X

iP0

a

ðhÞðZ

bÞ

!

:

Easy computations using (63)–(65) yield that E

ðf ðh;Z

Þ f ðh

; Z

ÞÞ ¼ r

ðh

Þ

r

ðhÞ 1 log r

ðh

Þ

r

ðhÞ þ ðb

bÞ

r

ðhÞ

1 aðhÞ 1 bðhÞ

þ 1

r

ðhÞ E

X

iP0

ða

ðh

Þ a

ðhÞÞðZ

b

Þ

!

:

Hence, all the conditionsof Theorem 2 may be checked and h can be identified by thismethod.

5. Concluding remarks

The conditional likelihood method is classical in the field of ARCH-type models. In this paper, we have shown that it can be used for HMMs, and in particular for SV models. The approach is theoretical but enlightens the minimal assumptions needed for statistical inference. From this point of view, these assumptions do not require the existence of high-order moments for the hidden Markov process. This is consistent with financial data that usually exhibit fat tailed marginals.

In order to illustrate on an explicit example the conditional likelihood method, we revisit in full detail the Kalman filter. SV modelswith mean-reverting volatility provide another example where the method can be used.

This method may be applied to other classes of models for financial data: models including leverage effects(see, e.g. Tauchen et al. 1996); complete models with SV (see, e.g. Hobson &

Rogers, 1998; Jeantheau, 2002).

6. Acknowledgements

The authors were partially supported by the research training network ‘Statistical Methods for Dynamical Stochastics Models’ under the Improving Human Potential programme, financed by the Fifth Framework Programme of the European Commission. The authors would like to thank the referees and the associate editor for helpful comments.

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Received March 2000, in final form July 2002

Catherine Lare´do, I.N.R.A., Labratoire de Biome´trie, 78350, Jouy-en-Josas, France.

E-mail: Catherine.Laredo@jouy.inra.fr

Appendix

Thisappendix isdevoted to prove Theorem 2 due to Elie & Jeantheau (1995). Recall that we have set a

¼ supða; 0Þ and a

¼ infða; 0Þ, so that a ¼ a

þ a

. The following proof holdsfor any strictly stationary and ergodic process under (C0)–(C3).

Proof. First, let us introduce the random variable defined by the equation h

¼ arg inf

h

F

ðh; Z

Þ:

Note that h

is not an estimator, since it is a function of the infinite past ðZ

Þ. The first part of the proof isdevoted to show that, under (H0)–(H2), h

convergesto h

a.s.

By the continuity assumption on f, f ðh; q; Z

Þ ismeas urable. Moreover, under (C1), E

ð f

ðhÞ; Z

Þ > 1, therefore F ðh

; hÞ iswell defined, but may be equal to þ1.

For all h 2 H and h

2 Bðh; qÞ \ H, the function h

! fðh

;Z

Þ f

ðh; q; Z

Þ isnon-neg- ative. Using (C1) and the continuity of f with respect to h, Fatou’sLemma implies lim inf

F ðh

; h

ÞPF ðh

;hÞ. Therefore, F islower semicontinuousin h.

Applying the monotone convergence theorem to f

ðh; q; Z

Þ and the Lebesgue dominated convergence theorem to f

ðh; q;Z

Þ, we get

q!0

lim E

ðf ðh; q; Z

ÞÞ ¼ E

ðf ðh; Z

ÞÞ ¼ F ðh

; hÞ: ð66Þ Let e > 0 and consider the compact set K

¼ H \ ð Bðh

; eÞ Þ

. By (C2) and the lower semi- continuity of F, there exists a real g > 0 such that:

8h 2 K

; F ðh

; hÞ F ðh

; h

Þ > g: ð67Þ

Consider h 2 K

. If F ðh

; hÞ < þ1, using (66), there exists qðhÞ > 0 such that 0OF ðh

; hÞ E

ð f

ðh; qðhÞ; Z

Þ Þ < g=2:

Combining the above inequality with (67), we obtain

E

ð f ðh;qðhÞ; Z

Þ Þ F ðh

; h

Þ > g=2: ð68Þ If F ðh

; hÞ ¼ þ1, since F ðh

; h

Þ is finite, using (66), we can also associate qðhÞ > 0 such that (68) holds. So we cover the compact set K

with a finite number L of balls, say fB ð h

; qðh

Þ Þ; k ¼ 1; . . . ; Lg.