\section{Introduction}

This paper is concerned with the particular \emph{hidden stochastic model}:

\begin{equation}\label{mod1}
\left\lbrace\begin{array}{ll}
Z_{i}=X_{i}+\varepsilon_{i}\\
X_{i+1}=b_{\phi_0}(X_{i})+\eta_{i+1},
\end{array}
\right.
\end{equation}
where $(\varepsilon_{i})_{i\geq 1}$ and $(\eta_{i})_{i\geq 1}$ are two independent sequences of independent and identically distributed (i.i.d) centered random variables with variance $\sigma^{2}_{\epsilon}$ and $\sigma^{2}_{0}$. It is assumed that the variance $\sigma^{2}_{\epsilon}$ is known. The terminology \emph{hidden} comes from the unobservable character of the process $(X_i)_{i\geq 1}$ since the only available observations are $Z_1,\cdots, Z_n$.\\ 

The dynamics of the process $X_i$ is described by a measurable function $b_{\phi_0}$ which depends on an unknown parameter $\phi_0$ and by a sequence of i.i.d centered random variables with unknown variance $\sigma_0^2$. We denote by $\theta_0=(\phi_0, \sigma^{2}_{0})$  the vector of parameters governing the process $X_i$ and suppose that the model is correctly specified: that is, $\theta_0$ belongs to the parameter space $\Theta \subset \mathbb{R}^{r}$, with $r \in \mathbb{N}^{*}$. The question of estimating such a model is a real challenge and has been studied by many authors. In particular the field of applications is very large: radar models, genomic models, financial models...(see \cite{cape} for a partial review).\\

When $b$ is linear in $\phi_0$ and $X_i$, the model (\ref{mod1}) is an autoregressive model with measurement error. K.C.Chanda provided in \cite{KCD95, KCD96} an asymptotically normal estimator for the vector of parameters  $\theta_0$ by using modified Yule Walker equation. When $b$ is non linear, an efficient estimator of  $\theta_0$ is given by the \emph{Maximisation Likelihood Estimator} (MLE). Recently, in \cite{Dou11}, the authors show that such an estimator is consistent and asymptotically normal. In practice, the main difficulty arises from the unobservable character of the process $X_i$. The likelihood is then intractable: it is only available in the form of a multiple integral. Exact likelihood methods require simulations and have therefore an intensive computational cost. In many case, the MLE has to be approximated.\\
 
	A popular approach to approximate the MLE consists in using Monte Carlo Markov Chain (MCMC) simulation techniques and importance sampling (IS). Thanks to the development of these methods, MLE have known a huge progress and Bayesian estimations have received more attention (see \cite{SaRg93}). Another method for performing MLE estimator consists in using the Expectation-Maximization (EM) algorithm proposed by Dempster et al. in 1977 (see \cite{dempster}). Nevertheless, since $X_i$ is unobservable, this method requires to introduce a MCMC or IS method in the Expectation step. Although these methods are used in practice, it can be very expensive from a computational point of view. Some authors have proposed Sequential Monte Carlo algorithms (SMC) which allow to reduce the computational cost by a recursive construction. We refer to the book of Doucet et al.\cite{DoFr01} for a complete review of these methods.\\



We propose here an approach based on M-estimation: It consists in the optimisation of a well-chosen contrast function (see \cite{van} chapter p.41 for a partial review) and deconvolution strategy. The deconvolution problem is encountered in many statistical situations where the observations are collected with random errors. This approach is much less greedy from a computational point of view. In the regression framework: $Z_{i}=X_{i}+\varepsilon_{i},\ X_{i+1}=b_{\phi_0} (X_{i})+\eta_{i+1}$, a method for estimating the parameter $\phi_0$ has been proposed by F. Comte and M. Taupin \cite{comte3}. Their procedure of estimation is based on a modified least squared minimization. In the same perspective, J. Dedecker, A. Samson and M-L. Taupin in \cite{RLA11} propose also the same procedure of estimation based on a weighted least squared estimation: Their assumptions on the process $X_i$ are less restrictive than those proposed by F. Comte and M. Taupin and they provide consistent estimation of the parameter $\phi_0$ with a parametric rate of convergence in a very general framework. Their general estimator is based on the introduction of a kernel deconvolution density and depends on the choice of a weight function. The approach proposed here is different: it is not based on a weighted least squared estimation so that the choice of the weight function is not encountered in this paper. Moreover, it allows to estimate both the parameters $\phi_0$ and $\sigma^{2}_0$.\\

 Our principle of estimation relies on the Nadaraya-Watson strategy and is proposed by F. Comte et al. in \cite{CfLc11} in a non parametric case to estimate the function $b_{\phi}$ as a ratio of an estimate of $l_{\theta}=b_{\phi}f_{\theta}$ and an estimate of $f_{\theta}$, where $f_{\theta}$ represents the invariant density of the $X_i$. We propose to adapt their approach in the parametric context and suppose that the form of the stationary density $f_{\theta}$ is known up to some unknown parameter $\theta$. The first purpose of the paper is to present our estimator and its statistical properties: Under weak assumptions, we show that it is a consistent and asymptotically normal estimator. Our work is purely parametric but we go further in this direction by proposing an analytical expression of the asymptotic variance matrix $\Sigma(\hat{\theta}_{n})$ which allows to construct confidence interval at level $\alpha$ (see Corollary \ref{lele}). 


\paragraph{The models:}

	In this paper, we focus on financial applications. In the popular Black \& Sholes model (1973) (see \cite{BS73}), the returns of an asset price are modelled as a geometrical Brownian motion with constant volatility. Since then, empirical works have shown that this assumption is not satisfied. In particular, it appears that the volatility of an asset return has also to be considered as a stochastic process. Many authors have therefore proposed \emph{Stochastic Volatility models} which are widely used in finance and econometric. The real challenge is that the practitioner does not observe any realisation of the volatility process: only the assets prices are observed making Stochastic Volatility models belong to the class of hidden Markov models. Although the scope of our method is general, we have chosen to focus on the so-called SV model in this paper. We also investigate the behavior of the method on the simpler autoregressive process AR(1) with measurement noise which has been widely studied and on which our method can be more easily understood and compared with other ones. Our procedure allows to estimate the parameters of a large class of discrete Stochastic Volatility models (SV, ARCH-E model,Vasicek(1977), Merton(1973)...), which is a real challenge in financial application.\\


\emph{ (i) Gaussian Autoregressive AR(1) with measurement noise:} It has the following form:
\begin{equation}\label{gaussintro}
\left\lbrace\begin{array}{ll}
Z_{i+1}=X_{i+1}+\varepsilon_{i+1}\\
X_{i+1}=\phi_0 X_{i}+\eta_{i+1},
\end{array}
\right.
\end{equation}	
where $\varepsilon_{i+1}$ and  $\eta_{i+1}$ are two centered Gaussian random variables with variance $\sigma_{\epsilon}^{2}$ assumed to be known and $\sigma^{2}_{0}$ assumed to be unknown. Additionally, we assume that $|\phi_0|<1$ which implies the stationary and ergodic property of the process $X_i$ \cite{Do94}.\\



\emph{(ii) SV model:} The discrete time Stochastic Volatility model (SV in short) was introduced by Taylor in 1982 (see \cite{Ta82}): it is directly connected to the type of diffusion process used in asset-pricing theory ( see \cite{Melino}):
\begin{equation}\label{svintro}
\left\lbrace\begin{array}{ll}
Y_{i+1}=\exp\left(\frac{X_{i+1}}{2}\right)\xi_{i+1},\\
X_{i+1}=\phi_{0}X_{i}+\eta_{i+1}, 
\end{array}
\right.
\end{equation}
where $\xi_{i+1}$ and  $\eta_{i+1} $ are two centered Gaussian random variables with variance $\sigma_{\epsilon}^{2}$ assumed to be known and $\sigma^{2}_{0}$ assumed to be unknown. \\
By applying a log-transformation $Z_{i+1}=\log(Y^{2}_{i+1})-\mathbb{E}[\log(\xi^{2}_{i+1})]$ and $\varepsilon_{i+1}=\log(\xi^{2}_{i+1})-\mathbb{E}[\log(\xi^{2}_{i+1})]$, the SV model  is a particular version of (\ref{mod1}).  We assume that $|\phi_0|<1$ and we refer the reader to \cite{JT00} for the mixing properties of stochastic volatility models.\\ 

Most of the computational problems stem from the assumptions that the innovation of the returns are Gaussian which translates into a logarithmic chi-square distribution when the model is transformed in a linear state space model. Many authors have ignored it in their implementation and many authors used some mixture of Gaussian to approximate the log-chi-square density. For example, in the Quasi-Maximum Likelihood (QML) method implemented by Jacquier, Polson and Rossi in \cite{JaPo94} and in the Simulated Expectation-Maximization Likelihood estimator proposed (SIEMLE) by Kim, Shephard, and Chib in \cite{shep} they used a mixture of Gaussian distribution to approximate the log-chi-square distribution. Harvey \cite{harvey} used the Kalman filter to estimate the likelihood of the transform state space model, hence the model was also assumed to be Gaussian.\\

Consequently, the second purpose of this paper consists in comparing our contrast estimator with different estimations: the QML, the SIEMLE and Bayesian estimators. From a computational point of view, we show that as well as a straightforward implementation, the contrast estimator is faster than the SIEMLE estimator (see Table [\ref{computing_siem}] in Section \ref{comp-time}). Besides, we show that our estimator outperforms the QML and Bayesian estimators.\\


\textbf{Organization of the paper:} The remainder of the paper is organized as follows: the procedure of estimation and its statistical properties are presented in Section \ref{Procedure}. Section \ref{Application} contains the numerical study: In Section \ref{implementation} we give the parameter estimates and the comparison with others ones for simulation data and Section \ref{Application_real_data} contains the study on real data. We compare our contrast estimator with other ones on the SP$\&$500 and FTSE index. Finally all results are proved in Appendix \ref{proof_result} and \ref{appendiceB}.\\


\textbf{Notations:} We denote by: $u^{*}(t)=\int_{}^{}e^{itx}u(x)dx$ the Fourier transform of the function $u(x)$ and $\left \langle u,v \right \rangle=\int_{}^{}u(x)\overline{v}(x)dx$ with $v\overline{v}=|v|^{2}$. We write $||u||_{2}=\left(\int_{}^{} |u(x)|^{2} dx\right)^{1/2}$ the norm of $u(x)$ on the space of functions $\mathbb{L}^{2}(\mathbb{R})$. By property of the Fourier transform, one has $(u^{*})^{*}(x)=2\pi u(-x)$ and $\left\langle u_1,u_2\right\rangle=\frac{1}{2\pi}\left\langle u^{*}_1,u^{*}_2\right\rangle$. The vector of the partial derivatives of $f$ with respect to (w.r.t) $\theta$ is denoted by $\nabla_{\theta}f$ and the Hessian matrix of $f$ w.r.t $\theta$ is denoted by $\nabla^{2}_{\theta}f$ . The Euclidean norm matrix, that is, the square root of the sum of the squares of all its elements will be written by $\left\|A\right\|$. We denote by $\mathbf{Z}_{i}$ the pair $(Z_{i},Z_{i+1})$ and $\mathbf{z}_{i}=(z_{i},z_{i+1})$ is a given realisation of $\mathbf{Z}_i$. \\
In the following, $\mathbb{P}, \mathbb{E}, {\rm Var}$ and ${\rm Cov}$ denote respectively the probability $\mathbb{P}_{\theta_0}$, the expected value $\mathbb{E}_{\theta_0}$, the variance ${\rm Var}_{\theta_0}$ and the covariance ${\rm Cov}_{\theta_0}$ when the true parameter is $\theta_0$. Additionally, we write $\mathbf{P}_n$ (\emph{resp.} $\mathbf{P}$) the empirical expectation (\emph{resp.} theoretical), that is: for any stochastic variable $X$: $\mathbf{P}_{n}(X) = \frac{1}{n}\sum_{i=1}^{n} X_i$ (\emph{resp.} $\mathbf{P}(X)=\mathbb{E}[X]$).\\
$\newline$


	
\subsection{Procedure: Contrast estimator}\label{Procedure}


Hereafter, we propose explicit estimators of the parameter $\theta_0$. This estimator called the contrast estimator is based on minimization of suitable functions of the observations usually called ``contrasts functions". We refer the reader to \cite{van} for a general account on this notion. Furthermore, in this part, we use the contrast function proposed by \cite{CfLcRy10}, that is: 
 
\begin{equation}\label{contraste}
\mathbf{P}_{n}m_{\theta}=\frac{1}{n}\sum_{i=1}^{n}m_{\theta}(\mathbf{Z}_{i}),
\end{equation} with $n$ the number of observations and:
\begin{equation*}
m_{\theta}(\mathbf{z}_i): (\theta, \mathbf{z}_{i})\in (\Theta \times \mathbb{R}^{2})\mapsto m_{\theta}(\mathbf{z}_i)=||l_{\theta}||^{2}_{2}-2z_{i+1}u^{*}_{l_{\theta}}(z_i),
\end{equation*}
where the function $l_{\theta}$ and $u_{v}$ are given by:

\begin{equation}
l_{\theta}(x)=b_{\phi}(x)f_{\theta}(x) \text{ and }\quad u_{v}(x)=\frac{1}{2\pi}\frac{v^{*}(-x)}{f_{\varepsilon}^{*}(x)}
\end{equation}

with $f_{\theta}$ the invariant density of $X_i$.\\

\textbf{Some assumptions}. As our procedure relies on the Fourier deconvolution strategy, in order to construct our estimator, we assume that the density of the noise $\varepsilon_i$, denoted by $f_\varepsilon$, is fully known and  belongs to $\mathbb{L}_2(\mathbb{R})$, and for all $x \in \mathbb{R}$ $f^{*}_{\varepsilon}(x)\neq 0$. Furthermore, we assume that the function $l_{\theta}$ belongs to $\mathbb{L}_1(\mathbb{R})\cap \mathbb{L}_2(\mathbb{R})$, is twice continuously differentiable w.r.t $\theta \in \Theta$ for any $x$ and measurable w.r.t $x$ for all $\theta$ in $\Theta$. Additionally, each coordinate of $\nabla_{\theta}l_{\theta}$ and each coordinate of $\nabla^{2}_{\theta}l_{\theta}$ belong to $\mathbb{L}_1(\mathbb{R})\cap \mathbb{L}_2(\mathbb{R})$. The function $u_{l_{\theta}}$ must be integrable and each coordinate of $u_{\nabla_{\theta}l_{\theta}}$ and each coordinate of $u_{\nabla_{\theta}^{2}l_{\theta}}$ have to be integrable as well.\\
For the statistical study, the key assumption is that the process $(X_i)_{i \geq 1}$ is stationary and ergodic  (see \cite{JT00} for a definition).\\

 Let us explain the choice of the contrast function and how the strategy of deconvolution works. Obviously, as the model (\ref{mod1}) shows, the $\mathbf{Z}_i$ are not i.i.d. However, by assumption, they are stationary ergodic, so the convergence of $\mathbf{P}_{n}m_{\theta}$ to $\mathbf{P}m_{\theta}=\mathbb{E}\left[m_{\theta}(\mathbf{Z}_1)\right]$ as $n$ tends to the infinity is provided by the Ergodic Theorem. Moreover, the limit $\mathbb{E}\left[m_{\theta}(\mathbf{Z}_1)\right]$ of the contrast function can be explicitly computed: 

\begin{equation*}
\mathbb{E}\left[m_{\theta}(\mathbf{Z}_1)\right]=\left\|l_{\theta}\right\|_{2}^{2}-2\mathbb{E}\left[Z_{2}u_{l_{\theta}}^{*}(Z_{1})\right].
\end{equation*} By Eq.(\ref{mod1}) and under the independence assumptions of the noise $(\varepsilon_2)$ and $(\eta_2)$, we have:

\begin{equation*}
\mathbb{E}\left[ Z_2  u^*_{l_{\theta}}(Z_1)\right] = \mathbb{E}\left[ b_{\phi_{0}}(X_1)  u^*_{l_{\theta}}(Z_1)\right].
\end{equation*} Using Fubini's Theorem and Eq.(\ref{mod1}), we obtain:

\begin{eqnarray*}
\mathbb{E}\left[ b_{\phi_{0}}(X_1)  u^*_{l_{\theta}}(Z_1)\right] &=& \mathbb{E}\left[b_{\phi_{0}}(X_1) \int e^{iZ_1y} u_{l_{\theta}}(y) dy  \right]\nonumber\\
&=&\mathbb{E}\left[b_{\phi_{0}}(X_1) \int \frac{1}{2\pi}\frac{1}{f_{\varepsilon}^*(y)}e^{iZ_1y} (l_{\theta}(-y))^*dy  \right]\nonumber\\
&=&\frac{1}{2\pi} \int \mathbb{E}\left[b_{\phi_{0}}(X_1)e^{i(X_1+\varepsilon_1)y} \right] \frac{1}{f_{\varepsilon}^*(y)} (l_{\theta}(-y))^* dy \nonumber\\
&=&\frac{1}{2\pi} \int \frac{\mathbb{E}\left[ e^{i\varepsilon_1y}\right]}{f_{\varepsilon}^*(y)} \mathbb{E}\left[b_{\phi_{0}}(X_1)e^{iX_1y}\right] (l_{\theta}(-y))^* dy\nonumber\\
&=&\frac{1}{2\pi} \mathbb{E}\left[ b_{\phi_{0}}(X_1) \int e^{iX_1y} (l_{\theta}(-y))^* dy \right]\nonumber\\
&=&\frac{1}{2\pi} \mathbb{E}\left[ b_{\phi_{0}}(X_1) \left((l_{\theta}(-X_1))^{*}\right)^*\right]\nonumber\\
&=& \mathbb{E}\left[ b_{\phi_{0}}(X_1)l_{\theta}(X_1)\right].\nonumber\\
&=&\int b_{\phi_{0}}(x)f_{\theta_{0}}(x)b_{\phi}(x)f_{\theta}(x)dx \nonumber\\
&=&\left\langle l_{\theta}, l_{\theta_{0}} \right\rangle.
\end{eqnarray*} Then, 

\begin{eqnarray}\label{pmtheta}
\mathbb{E}\left[m_{\theta}(\mathbf{Z}_1)\right]&=&\left\|l_{\theta}\right\|_{2}^{2}-2\left\langle l_{\theta}, l_{\theta_{0}} \right\rangle,\\
&=&\left\|l_{\theta}-l_{\theta_0}\right\|_{2}^{2}-\left\|l_{\theta_{0}}\right\|_{2}^{2}. 
\end{eqnarray} The computation above is the key of the deconvolution procedure. Note that, in a smooth framework (uniqueness assumption), this quantity is minimal when $\theta$=$\theta_{0}$.\\


Hence, the associated minimum-contrast estimators $\widehat{\theta}_n$ is defined as any solution of:  

\begin{equation}\label{min}
\widehat{\theta}_n=\arg\min_{\theta \in \Theta}\mathbf{P}_{n}m_{\theta}.
\end{equation}

\begin{remarque} 
We refer the reader to \cite{Do94} for the proof that if $X_i$ is an ergodic process then the process $Z_{i}$, which is the sum of an ergodic process with an i.i.d. noise, is again stationary ergodic. Furthermore, by the definition of an ergodic process, if $Z_i$ is an ergodic process then the couple $\mathbf{Z}_i=(Z_i, Z_{i+1})$ inherits the property (see \cite{JT00}).\\
\end{remarque}

\subsection{Asymptotic properties of the Contrast estimator}

Our proof holds under the following assumptions. For the reader convenience, we denote by \textbf{(E)} (\emph{resp.} \textbf{(C)} and \textbf{(T)}) the assumptions which serve us for the existence (\emph{resp.} Consistency and Central Limit Theorem). If the same assumption is needed for two results, for example for the existence and the consistency, it is denoted by \textbf{(EC)}.\\

\begin{flushleft}
\textbf{(ECT):} The parameter space $\Theta$ is a compact subset of $\mathbb{R}^{r}$ and $\theta_0$ is an element of the interior of $\Theta$.\\
\textbf{(C):} (Local dominance): $\mathbb{E}\left[\sup_{\theta \in \Theta}\left|Z_{2}u^{*}_{l_{\theta}}(Z_{1})\right|\right]<\infty$.\\
\textbf{(CT):} The application $\theta \mapsto \mathbf{P}m_{\theta}$ admits an unique minimum and its Hessian matrix denoted by $V_{\theta}$ is non-singular in $\theta_0$.\\ 
\textbf{(T):} (Moment condition): For some $\delta >0$ and for $j \in \left\{1,\cdots,r\right\}$: $\mathbb{E}\left[|Z_{2}u^*_{\frac{\partial l_{\theta}}{\partial \theta_j}}(Z_1)|^{2+\delta}\right]<\infty$.\\
\qquad (Hessian Local dominance): For some neighbourhood $\mathcal{U}$ of $\theta_0$ and for $j,k \in \left\{1,\cdots,r\right\}$:
$$\mathbb{E}\left[\sup_{\theta \in \mathcal{U}}|Z_{2}u^{*}_{\frac{\partial^2 l_{\theta}}{\partial \theta_j\partial \theta_k}}(Z_{1})|\right]<\infty.$$
\end{flushleft}


Let us introduce the matrix:

\begin{equation*}
\Sigma(\theta)=V_{\theta}^{-1} \Omega(\theta) V_{\theta}^{-1'} \text{ with } \Omega(\theta)=\Omega_{1}(\theta)+2 \sum_{j=2}^{+\infty} \Omega_{j}(\theta).
\end{equation*}

where $\Omega_{1}(\theta)={\rm Var}\left(\nabla_{\theta}m_{\theta}(\mathbf{Z_{1}})\right)$ and $\Omega_{j}(\theta)={\rm Cov}\left(\nabla_{\theta}m_{\theta}(\mathbf{Z_{1}}),\nabla_{\theta}m_{\theta}(\mathbf{Z_{j}})\right)$

\begin{theoreme}\label{MR}
Under our assumptions, let $\widehat{\theta}_{n}$ be the minimum-contrast estimator defined by (\ref{min}). Then:
\begin{equation*}
\widehat{\theta}_{n} \longrightarrow \theta_{0}  \qquad \text{ in probability }\text{ as }n \rightarrow \infty.  
\end{equation*}

Moreover, if $\mathbf{Z}_i$ is geometrically ergodic (see Definition \ref{def_geo} in Appendix \ref{appen}), then:
\begin{equation*}
\sqrt{n}(\widehat{\theta}_{n}-\theta_{0})\rightarrow \mathcal{N}\left(0,\Sigma(\theta_{0})\right)   \qquad  \text{ in law} \text{ as }n \rightarrow \infty. 
\end{equation*}
\end{theoreme}

 The following corollary gives an expression of the matrix $\Omega(\theta_0)$ and $V_{\theta_0}$ of Theorem \ref{MR} for the practical implementation:

\begin{corollaire}\label{lele}
Under our assumptions, the matrix $\Omega(\theta_0)$ is given by:

$$\Omega(\theta_0) = \Omega_{1}(\theta_0)+ 2\sum_{j=2}^{+\infty} \Omega_{j}(\theta_0)$$

where:
\begin{equation*}
\Omega_{1}(\theta_0)= 4\mathbb{E} \left[Z^{2}_{2}\left(u^{*}_{\nabla_{\theta}l_{\theta}}(Z_1)\right)^{2}\right] -4\mathbb{E}\left[b_{\phi_{0}}(X_1) \left(\nabla_{\theta}l_{\theta}(X_1) \right)\right]\mathbb{E}\left[b_{\phi_{0}}(X_1) \left(\nabla_{\theta}l_{\theta}(X_1) \right)\right] '.\\
\end{equation*}

And, the covariance terms are given by:

\begin{equation*}
\Omega_{j}(\theta_0)=4\left[\tilde{C}_j-\mathbb{E}\left[b_{\phi_{0}}(X_1)\left(\nabla_{\theta}l_{\theta}(X_1)\right)\right]\mathbb{E}\left[b_{\phi_{0}}(X_1)\left(\nabla_{\theta}l_{\theta}(X_1)\right)\right] '\right],
\end{equation*}

where $\tilde{C}_{j}=\mathbb{E}\left[b_{\phi_{0}}(X_1)\left(\nabla_{\theta}l_{\theta}(X_1)\right)\left(b_{\phi_{0}}(X_j)\nabla_{\theta}l_{\theta}(X_j)\right)'\right] $.\\

Furthermore, the Hessian matrix $V_{\theta_0}$ is given by:

\begin{eqnarray*}
\left(\left[V_{\theta_0}\right]_{j,k}\right)_{1\leq j,k\leq r}&=&2\left( \left\langle \frac{\partial l_{\theta}}{\partial  \theta_k}, \frac{\partial l_{\theta}}{\partial \theta_j}\right\rangle \right)_{j,k} \text { at the point $\theta_0$.} 
\end{eqnarray*}
\end{corollaire}



Let us now state the strategy of the proof, the full proof is given in Appendix \ref{proof_result}. Clearly, the proof of Theorem \ref{MR} relies on M-estimators properties and on the deconvolution strategy. The existence of our estimator follows from regularity properties of the function $l_{\theta}$ and compactness argument of the parameter space, it is explained in Appendix \ref{EoE}. The key of the proof consists in proving the asymptotic properties of our estimator. This is done by splitting the proof into two parts: we first give the consistency result in Appendix \ref{CoE} and then give the asymptotic normality in Appendix \ref{ANoE}. Let us introduce the principal arguments:\\
	
The main idea for proving the consistency of a M-estimator comes from the following observation: if $\mathbf{P}_{n}m_{\theta}$ converges to $\mathbf{P}m_{\theta}$ in probability, and if the true parameter solves the limit minimization problem, then, the limit of the argminimum $\widehat{\theta}_n$ is $\theta_0$. By using an argument of uniform convergence in probability and by compactness of the parameter space, we show that the argminimum of the limit is the limit of the argminimum. A standard method to prove the uniform convergence is to use \emph{the Uniform Law of Large Numbers} (see Lemma \ref{ULLN} in Appendix \ref{appen}). Combining these arguments with the dominance argument \textbf{(C)} give the consistency of our estimator, and then, the first part of Theorem \ref{MR}.\\

The asymptotic normality follows essentially from Central Limit Theorem for a mixing process (see \cite{Ga04}). Thanks to the consistency, the proof is based on a moment condition of the Jacobian vector of the function $m_{\theta}(\mathbf{z})$ and on a local dominance condition of its Hessian matrix. To refer to likelihood results, one can see these assumptions as a moment condition of the score function and a local dominance condition of the Hessian.\\

	

\section{ Applications}\label{Application}

\subsubsection{ Contrast estimator for the Gaussian AR(1) model with measurement noise:}\label{arr} 


Consider the following autoregressive process AR(1) with measurement noise:
  
\begin{equation}\label{cac}
\left\lbrace\begin{array}{ll}
Z_{i}=X_{i}+\varepsilon_{i}\\
X_{i+1}=\phi_0 X_{i}+\eta_{i+1},
\end{array}
\right.
\end{equation}

The noises $\varepsilon_i$ and $\eta_i$ are supposed to be centered Gaussian randoms with variance respectively $\sigma^2_{\varepsilon}$ and $\sigma^2_0$.  We assume that $\sigma^2_{\varepsilon}$ is known. Here, the unknown vector of parameters is $\theta_0=(\phi_0,\sigma^2_0)$ and for stationary and ergodic properties of the process $X_i$, we assume that the parameter $\phi_0$ satisfies $|\phi_0|<1$ \cite{Do94}. The functions $b_{\phi}$ and $l_{\theta}$ are defined by:

\begin{eqnarray*}
&& b_{\phi}(x):\ (x,\theta) \in (\mathbb{R}\times \Theta) \mapsto b_{\phi}(x)=\phi x,\\ 
&& l_{\theta}(x):\ (x,\theta) \in (\mathbb{R}\times \Theta) \mapsto l_{\theta}(x)=b_{\phi}(x)f_{\theta}(x)=\frac{\phi}{\sqrt{2\pi\gamma^{2}}} x \exp\left(-\frac{1}{2\gamma^{2}}x^{2}\right),
\end{eqnarray*}
where $\gamma^{2}=\frac{\sigma^{2}}{1-\phi^{2}}$. The vector of parameter $\theta$ belongs to the compact subset $\Theta$ : $\Theta = [-1+r; 1-r]\times[\sigma^{2}_{min}; \sigma^{2}_{max}]$ with $\sigma^{2}_{min}\geq \sigma_{\varepsilon}^2+\overline{r}$ where $r$,  $\overline{r}$, $\sigma^{2}_{min}$ and $\sigma^{2}_{max}$ are positive real constants. We consider this subset since by stationary of $X_i$, the parameter $|\phi|<1$ and by construction the function $u^{*}_{l_{\theta}}$ is well defined for $\sigma^{2}> \sigma_{\varepsilon}^2(1-\phi^2)$ with $\phi \in [-1+r; 1-r]$ which implies that $\sigma^{2}>\sigma_{\varepsilon}^2$. The contrast estimator defined in (\ref{Procedure}) has the following form:

\begin{equation}\label{contraste_application}
\widehat{\theta}_n= \arg \min_{\theta \in \Theta}\left\{ \frac{\phi^{2}\gamma}{4\sqrt{\pi}}-\sqrt{\frac{2}{\pi}}\frac{\phi \gamma^{2}}{n(\gamma^{2}-\sigma^{2}_{\epsilon})^{3/2}}\sum_{j=1}^{n} Z_{j+1} Z_{j}\exp-\left(\frac{1}{2}   \frac{Z^{2}_{j}}{(\gamma^{2}-\sigma^{2}_{\epsilon})}\right)\right\}
\end{equation} 

with $n$ the number of observations. Theorem \ref{MR} applies here and the corresponding result for the  Gaussian AR(1) model is given in Appendix \ref{AppGauss}. As we already mentioned, Corollary \ref{lele} allows to compute confidence intervals: For all $i=1,2$:

\begin{equation*}
\mathbb{P}\left(\hat{\theta}_{n,i}-z_{1-\alpha/2}\sqrt{\frac{\mathbf{e}_{i}'\Sigma (\hat{\theta}_{n})\mathbf{e}_{i}}{n}}\leq \theta_{0,i}\leq \hat{\theta}_{n,i}+z_{1-\alpha/2}\sqrt{\frac{\mathbf{e}_{i}'\Sigma(\hat{\theta}_{n})\mathbf{e}_{i}}{n}}\right)\rightarrow 1-\alpha, \quad \text{ as } n \rightarrow \infty
\end{equation*}
where $z_{1-\alpha/2}$ is the $1-\alpha/2$ quantile of the Gaussian law, $\theta_{0,i}$ is the $i^{th}$ coordinate of $\theta_0$ and $\mathbf{e}_{i}$ is the $i^{th}$ coordinate of the vector of the canonical basis of $\mathbb{R}^2$. The covariance matrix $\Sigma(\hat{\theta}_{n})$ is computed in Lemma \ref{hessienne_application} in Appendix \ref{EFCM}.\\


\subsubsection{ Contrast estimator for the SV model:}\label{svv}

We consider the following SV model:

\begin{equation}\label{svintro}
\left\lbrace\begin{array}{ll}
Y_{i+1}=\exp\left(\frac{X_{i+1}}{2}\right)\xi_{i+1}^{\beta},\\
X_{i+1}=\phi_{0}X_{i}+\eta_{i+1}, 
\end{array}
\right.
\end{equation}

with $\beta$ a positive constant. The noises $\xi_{i+1}$ and  $\eta_{i+1} $ are two centered Gaussian random variables with standard variance $\sigma^{2}_{\varepsilon}$ and $\sigma^{2}_{0}$. We assume that $|\phi_0|<1$ and we refer the reader to \cite{JT00} for the mixing properties of this model.\\
 
 In the original version of the SV model introduced by Taylor in \cite{Ta82}, the constant $\beta$ is equal to $1$. In this case, by applying a log-transformation $Z_{i+1}=\log(Y^{2}_{i+1})-\mathbb{E}[\log(\xi^{2}_{i+1})]$ and $\varepsilon_{i+1}=\log(\xi^{2}_{i+1})-\mathbb{E}[\log(\xi^{2}_{i+1})]$, the log-transform SV model is given by:
 
 \begin{equation}\label{svappl}
\left\lbrace\begin{array}{ll}
Z_{i+1}=X_{i+1}+\varepsilon_{i+1}\\
X_{i+1}=\phi_{0}X_{i}+\eta_{i+1}, 
\end{array}
\right.
\end{equation}
 
The Fourier transform of the noise $\varepsilon_{i+1}$ is given by:

$$f^{*}_{\varepsilon}(x)=\frac{1}{\sqrt{\pi}}2^{ix}\Gamma(\frac{1}{2}+ix)e^{-i\mathcal{E}x}$$ where $\mathcal{E}=\mathbb{E}[\log(\xi^{2}_{i+1})]$ and ${\rm Var}[\log(\xi^{2}_{i+1})]$= $\sigma^{2}_{\varepsilon}=\frac{\pi^2}{2}$. Here, $\Gamma$ represents the gamma function given by:

\begin{equation*}
\Gamma: u\rightarrow \int_{0}^{+\infty}t^{u-1}e^{-t}dt \qquad \forall u \in \mathbb{C} \text { such that } \mathcal{R}_{e}(u)>0.
\end{equation*}

The vector of parameters $\theta=(\phi,\sigma^2)$ belongs to the compact subset $\Theta=[-1+r; 1-r]\times[ \sigma^{2}_{min} ;  \sigma^{2}_{max}]$ with $r$, $\sigma^{2}_{min}$ and $\sigma^{2}_{max}$ positive real constants. \\

Our contrast estimator (\ref{Procedure}) is given by:

\begin{equation}\label{contraste_application2}
\widehat{\theta}_n= \arg \min_{\theta \in \Theta}\left\{ \frac{\phi^{2}\gamma}{4\sqrt{\pi}}-\frac{2}{n}\sum_{i=1}^{n}Z_{i+1}u^{*}_{l_{\theta}}(Z_{i})\right\},
\end{equation} with $u_{l_{\theta}}(y)=\frac{1}{2\sqrt{\pi}}\left(\frac{-i\phi y\gamma^2\exp\left(\frac{-y^2}{2}\gamma^2\right)}{\exp\left(-i\mathcal{E}y\right)2^{i y}\Gamma\left(\frac{1}{2}+i y\right)}\right)$.\\

Theorem \ref{MR} applies and by Slutsky's Lemma we also obtain confidence intervals. We refer the reader to Appendix \ref{AppSV} for the proof.


\subsection{Comparison with the others methods}

\subsubsection{QML Estimator}\label{qml}

For the SV model, the QML estimator, proposed independently by Nelson(1988) (see \cite{Nelson}) and Harvey et al.(1994) (see \cite{harvey}) is based on the log-transform model given in (\ref{svappl}). Making as if the $\varepsilon_i$ were Gaussian in the log-transform of the SV model, the Kalman filter \cite{ka} can be applied in order to obtain the quasi likelihood function of $Z_{1:n}=(Z_1,\cdots,Z_n)$ where $n$ is the sample data length. For the AR(1) and  the log-transform of the SV model, the log-likelihood $l(\theta)$ is given by: 

\begin{equation*}
l(\theta)=\log f_{\theta}(Z_{1:n})=-\frac{n}{2}\log(2\pi)-\frac{1}{2}\sum_{i=1}^{n}\log F_i-\frac{1}{2}\sum_{i=1}^{n}\frac{\nu_{i}^{2}}{F_i},
\end{equation*} where $\nu_{i}$ is the one-step ahead prediction error for $Z_i$, and $F_i$ is the corresponding mean square error. More precisely, the two quantities are given by:

\begin{eqnarray*}
\nu_i=(Z_i-\hat{Z}_i^{-}) \text{ and } F_i={\rm Var}_{\theta}[\nu_i]=P_i^{-}+\sigma^{2}_{\varepsilon},
\end{eqnarray*} where $\hat{Z}_i^{-}=\mathbb{E}_{\theta}[Z_i| Z_{1:i-1}]$ is the one-step ahead prediction for $Z_i$ and $P_i^{-}={\rm Var}_{\theta}[(X_i-\hat{X}_{i}^{-})^{2}]$ is the one-step ahead error variance for $X_i$.\\

Hence, the associated estimator of $\theta_0$ is defined as a solution of:
\begin{equation*}
\hat{\theta}_n=\arg\max_{\theta \in \Theta}l(\theta).
\end{equation*}


Note that this procedure can be inefficient: the method does not rely on the exact likelihood of the $Z_{1:n}$ and approximating the true log-chi-square density by a normal density  can be rather inappropriate (see Figure [\ref{approxim}] below).


$$\includegraphics[bb=0 0 1039 411 , scale=0.35]{approximation_chideu_normal.jpg}$$
\caption{Approximation of the $\log$-chi-square density (Red) by a Gaussian density with mean $\mathcal{E}=-1.27$ and variance $\sigma^{2}_{\varepsilon}=\frac{\pi^2}{2}$ (Black). }\label{approxim}




\subsubsection{Sequential Bayesian Estimators: Bootstrap, APF and KSAPF }\label{bay}


In the Bayesian approach, the vector of parameters $\theta=(\phi,\sigma^2)$ is supposed random obeying the prior distribution assumed to be known. We propose to use the Kitagawa and al.'s approach (see \cite{DoFr01} chapter 10 p.189) in which the parameters are supposed time-varying: $\theta_{i+1}=\theta_{i}+\mathcal{G}_{i+1}$ where $\mathcal{G}_{i+1}$ is a centered Gaussian random with a variance matrix $Q$ supposed to be known. Now, we consider the augmented state vector $\tilde{X}_{i+1}=(X_{i+1}, \theta_{i+1})'$ where $X_{i+1}$ is the hidden state variable and $\theta_{i+1}$ the unknown vector of parameters. In this paragraph, we use the terminology of the particle filtering method, that is: we call particle a random variable. The sequential particle estimation of the vector $\tilde{X}_{i+1}$ consists in a combined estimation of $X_{i+1}$ and $\theta_{i+1}$. For initialisation the distribution of $X_1$ \footnote{ To avoid confusion between the true value $\theta_0$ and the initial value $\theta_1$ in the Bayesian algorithms, we start the algorithms with $i=1$.} conditionally to $\theta_1$ is given by the stationary density $f_{\theta_1}$.  \\

For the comparison with our contrast estimator (\ref{Procedure}), we use the three methods: the Bootstrap filter, the Auxiliary Particle filter (APF) and the Kernel Smoothing Auxiliary Particle filter (KSAPF). We refer the reader
to \cite{DoFr01}, \cite{PiSh07} and \cite{liu} for a complete revue of these methods.


\begin{remarque} Let us underline some particularity of  the combined state and parameters estimation: For the Bootstrap and APF estimator, an important issue concerns the choice of the parameter variance $Q$ since the parameter is itself unobservable. If one can choose an optimal variance $Q$ the APF estimator could be a very good estimator since with arbitrary variance the results are acceptable (see Table [\ref{compar2}]). In practice, Q is chosen by an empirical optimization. The KSAPF is an enhanced version of the APF and depends on a smooth factor $0<h<1$ (see \cite{liu}). Therefore, the choice of $h$ is another problem in practice. 
\end{remarque}



\subsection{Application to Simulated Data}\label{implementation}

 For the AR(1) and SV model, we sample the trajectory of the $X_i$ with the parameters $\phi_0=0.7$ and $\sigma_0^{2}=0.3$. Conditionally to the trajectory, we sample the variables $Z_i$  for $i=1\cdots n$ where $n$ represents the number of observations. We take $n=1000$ and $\sigma^{2}_{\varepsilon}=0.1$ for the two models. This means that we take $\beta=\frac{1}{\sqrt{5}\pi}$ for the SV model given in (\ref{svintro}).  In this case, the Fourier transform of $\varepsilon_{i+1}$ is given by: $f^{*}_{\varepsilon}(y)=\exp\left(-i\tilde{\mathcal{E}}y\right)\frac{2^{i\beta y}}{\sqrt{\pi}}\Gamma\left(\frac{1}{2}+i\beta y\right)$ with $\tilde{\mathcal{E}}=\beta\mathbb{E}[\log(\xi^{2}_{i+1})]$(see Appendix \ref{AppSV}).\\

Since particle filters are advanced Monte Carlo algorithms, we take a large number of particles $M$. For the three methods , we take $M$ equal to $5000$.  Note that for the Bayesian procedure (Bootstrap, APF and KSAPF), we need a prior on $\theta$, and this only at the first step. The prior for $\theta_1$ is taken to be the Uniform law and conditionally to $\theta_1$ the distribution of $X_1$ is the stationary law:

\begin{equation*}
\left\lbrace\begin{array}{ll}
p(\theta_1)=\mathcal{U}(0.5, 0.9)\times \mathcal{U}(0.1, 0.4)\\
f_{\theta_1}(X_1)=\mathcal{N}\left(0, \frac{\sigma^{2}_{1}}{1-\phi_1^2}\right)
\end{array}
\right.
\end{equation*}

We take $h=0.1$ for the KSAPF and $Q=\begin{pmatrix}
0.6.10^{-6} & 0\\
0 & 0.1.10^{-6} 
\end{pmatrix}$ for the APF and Bootstrap filter. 	

\begin{remarque}By choosing the stationary law and the Uniform law around the true parameters we bias favourably the Bayesian algorithms.  
\end {remarque}	 

\subsection{Numerical Results}

In the numerical section we compare the different estimations: the QML estimator defined in Section \ref{qml}, the Bayesian estimators defined in Section \ref{bay} and our contrast estimator defined in Section \ref{Procedure}. For the comparison of the computing time, we also compare our contrast estimator with the SIEMLE proposed by Kim, Shepard and Chib (see Appendix \ref{siem} and \cite{shep}  for more general details).

\subsubsection{Computing time}\label{comp-time}

From a theoretical point of view, the MLE is asymptotically efficient. However, in practice since the states $(X_{1}\cdots, X_{n})$ are unobservable and the SV model is non Gaussian, the likelihood is untractable. We have to use numerical methods to approximate it. In this section, we illustrate the SIEMLE which consists in approximating the likelihood and applying the Expectation-Maximisation algorithm introduced by Dempster \cite{dempster} to find the parameter $\theta$.\\
To illustrate the SIEMLE for the SV model, we run an estimator with a number of observations $n$ equal to $1000$.  Although the estimation is good the computing time is very long compared with the others methods (see Table [\ref{computing_siem}] and Figure [\ref{time_gaussian}]). This result illustrates the numerical complexity of the SIEMLE (see Appendix \ref{siem}). Therefore, in the following, we only compare our contrast estimator with the QML and Bayesian estimators. The results are illustrated by Figure [\ref{time_gaussian}]. We can see that our contrast estimator is the fastest for the Gaussian AR(1) model. The QML is the most rapid for the SV model since it assumes that the measurement errors are Gaussian but we show in Figures [\ref{boxplot_compa1}], [\ref{boxplot_compa2}] and [\ref{compar2}] that it is a biased estimator with large mean square error. For our algorithm, for the Gaussian AR(1)  model, the function $u^{*}_{l_{\theta}}$ has an explicit expression but for the SV model, the function $u^{*}_{l_{\theta}}$ is approximated numerically since the Fourier transform of the function $u_{l_{\theta}}$ has not an explicit form. This explains why our algorithm is slower on the SV model than on the Gaussian AR(1) model.\footnote{We use a quadrature method implemented in Matlab to approximate the Fourier transform of $u_{l_{\theta}}(y)$. One can also use the FFT method and we expect that the contrast estimator will be more rapid in this case.}  In spite of this approximation, our contrast estimator is fast and its implementation is straightforward. 


\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{$\phi_0$} &  \multicolumn{1}{c|}{$\sigma^{2}_0$} &  \multicolumn{1}{c|}{$\hat{\phi}_n$} &  \multicolumn{1}{c|}{$\hat{\sigma}^{2}_n$} & \multicolumn{1}{c|}{CPU (sec)}  \\  
\hline
 0.7 & 0.3 & 0.667 & 0.2892& 74300  \\
\hline 
\end{tabular}
\end{center}
\caption{\label{computing_siem} SIEML estimation for the SV model. The number of observations is $n=1000$ and the number of sweeps for the Gibbs sampler is $\tilde{M}=100$ (see Appendix \ref{siem}).  }
\end{table}

$$\includegraphics[bb=0 0 1046 521 , scale=0.35]{temps_calcul_Gaussian.jpg}$$
$\newline$
$$\includegraphics[bb=0 0 1074 304 , scale=0.35]{temps_calcul_SV.jpg}$$
\caption{ \label{time_gaussian} Top: Comparison of the computing time (CPU in seconds) with respect to the number of observations $n=100$ up to $2000$ for the Gaussian AR(1) model. The number of particles in Bayesian estimations is $M=5000$ particles. Bottom: Comparison of the CPU with respect to the number of observations $n=100$ up to $1500$ for the SV model with $M=5000$ particles.}


\subsubsection{Parameter estimates}\label{co}

For the AR(1) Gaussian model, we run $N=1000$ estimates for each method (QML, APF, KSAPF and Bootsrap filter) and $N=500$ for the SV model. The number of observations $n$ is equal to $1000$ for the two models.\\
In order to compare with others the performance of our estimator, we compute for each method the Mean Square Error (MSE) defined by:

\begin{equation}\label{MSE}
MSE=\frac{1}{N}\left(\sum_{j=1}^{N}(\hat{\phi}_{j}-\phi_0)^{2}+(\hat{\sigma}^{2}_{j}-\sigma^{2}_0)^{2}\right),
\end{equation}

We  illustrate by boxplots the different estimates (see Figures [\ref{boxplot_compa1}] and [\ref{boxplot_compa2}]). We also illustrate in Figure [\ref{compar2}] the MSE for each estimator computed by equation(\ref{MSE}). We can see that, for the parameter $\phi_0$, the QML estimator is better for the Gaussian AR(1) model than for the SV model (see Figure [\ref{boxplot_compa1}]). Indeed, the Gaussianity assumption is wrong for the SV model. Moreover, the estimate of $\sigma^{2}_0$ by QML is very bad for the two models (see Figure [\ref{boxplot_compa2}]) and its corresponding boxplots have the largest dispersion meaning that the QML method is not very stable. The Bootstrap, APF and KSAPF have also a large dispersion of their boxplots, in particular for the parameter $\phi_0$ (see Figure [\ref{boxplot_compa1}]). Besides, the Booststrap filter is less efficient than the APF and KSAPF. 
For the Gaussian and SV model, the boxplots of our contrast estimator show  that our estimator is the most stable with respect to $\phi_0$ and we obtain similar results for $\sigma^{2}_0$. The MSE is better for the SV model and the smallest for our contrast estimator. 

$$\includegraphics[bb=0 0 1104 407 , scale=0.2]{compar_boxplot_phiest_ar.jpg}
\includegraphics[bb=0 0 1181 418 , scale=0.2]{BoxplotPhi_QML.jpg}$$
\caption{Boxplot of $\phi$. True value: $\phi_0=0.7$. Left: Gaussian AR(1) model. Right: SV model.}\label{boxplot_compa1}
$$\includegraphics[bb=0 0 1111 410 , scale=0.2]{compar_boxplot_sigmaest_ar.jpg}
\includegraphics[bb=0 0 1176 422 , scale=0.2]{BoxplotSigma_QML.jpg}$$
\caption{Boxplot of $\sigma^2$. True value: $\sigma^{2}_0=0.3$. Left: Gaussian AR(1) model. Right: SV model.}\label{boxplot_compa2}

$$\includegraphics[bb=0 0 882 374 , scale=0.20]{MSE_gaussian.jpg}
\includegraphics[bb=0 0 755 359 , scale=0.20]{MSE_sv.jpg}$$
\caption{MSE computed by Eq.(\ref{MSE}). Left: Gaussian AR(1) model. Right: SV model.}\label{compar2}


\subsubsection{Confidence Interval of the contrast estimator}


To illustrate the statistical properties of our contrast estimator, we  compute for each model the confidence intervals for $N=1$ estimator and the coverages for $N=1000$ with respect to the number of observations. The coverage corresponds to the number of times for which the true parameter $\theta_{0,i}, i=1,2$ belongs to the confidence interval. The results are illustrated by the Figures [\ref{Covera1}]-[\ref{ic1}] and [\ref{ic2}]: for the Gaussian and SV models, the coverage converges to $95\%$ for a small number of observations.  As expected, the confidence interval decreases with the number of observations,. Note that of course a MLE confidence interval would be smaller since the MLE is efficient but the corresponding computing time would be huge. 

$$\includegraphics[bb=0 0 839 287 , scale=0.35]{Coverage_GaussMod.jpg}$$
$$\includegraphics[bb=0 0 778 289 , scale=0.35]{CoveragePhi_ModChi3.jpg}$$
\caption{Coverage with respect to the number of observations $n=100$ up to $5000$ for $N=1000$ estimators . Top: Gaussian AR(1) model. Bottom: SV model.}\label{Covera1}


$$\includegraphics[bb=0 0 1095 426 , scale=0.2]{IC_Nobs_phi4.jpg}
\includegraphics[bb=0 0 1107 425 , scale=0.2]{EmpIC_phi1.jpg}$$
\caption{Confidence interval for the parameter $\phi_0$ with respect to the number of observations $n=100$ up to $5000$ for $N=1$ estimator. Left: Gaussian AR(1) model. Right: SV model.}\label{ic1}
$$\includegraphics[bb=0 0 1107 425 , scale=0.2]{IC_Nobs_sig4.jpg}
\includegraphics[bb=0 0 1105 437, scale=0.2]{EmpIC_sigma1.jpg}$$
\caption{Confidence interval for the parameter $\sigma^2_0$ with respect to the number of observations $n=100$ up to $5000$ for $N=1$ estimator. Left: Gaussian AR(1) model. Right: SV model.}\label{ic2}

\subsection{Application to Real Data}\label{Application_real_data}

The data consist of daily observations on FTSE stock price index and S$\&$P500 stock price index. The series taken in boursorama.com are closing prices from January, 1, 1973 to November, 2, 1976 for the FTSE and September, 24, 2001 to July, 7, 2005 for the S$\&$P500 leaving a sample of $1000$ observations for the two series.\\
The daily prices $S_i$ are transformed into compounded rates returns centered around their sample mean $c$ for self-normalization (see \cite{mat} and \cite{GeHaRe96}) $y_i=100\times\log\left(\frac{S_i}{S_{i-1}}\right)-c$. We want to model those data by the SV model defined in (\ref{svappl}) leading to :
\begin{eqnarray*}
Z_i&=&\log(y_i^2)-\mathbb{E}[\log(\xi_i^2)]\\
&=&\log(y_i^2)+1.27
\end{eqnarray*}Those data are represented on Figure [\ref{RES}]. 

$$\includegraphics[bb=0 0 899 522, scale=0.5]{donnee_reelle.jpg}$$
\caption{ \label{RES}Top Left: Graph of $Z_i$=FTSE. Top Right:  Graph of $Z_i$=SP500. Bottom Left: Autocorrelation of $Z_i$=FTSE.  Bottom Right: Autocorrelation of $Z_i$=SP500.}




\subsubsection{Parameter Estimates}

In the empirical analysis, we compare the QML, the Bootstrap filter, the APF and the KSAPF estimators. The last one is our contrast estimator. The variance of the measurement noise is $\sigma^2_{\varepsilon}=\frac{\pi^2}{2}$, that is $\beta$ is equal to $1$ (see Section \ref{svv}). Table [\ref{resu}] summarises the parameter estimates and the computing time for the five methods. For initialization of the Bayesian procedure, we take the Uniform law for the parameters $p(\theta_1)=\mathcal{U}(0.4, 0.95)\times \mathcal{U}(0.1, 0.5)$ and the stationary law for the log-volatility process $X_1$, i.e, $f_{\theta_1}(X_1)=\mathcal{N}\left(0, \frac{\sigma^{2}_{1}}{1-\phi_1^2}\right)$.\\


The estimates of $\phi$ are in full accordance with results reported in previous studies of SV models. This parameter is in general close to 1 which implies persistent logarithmic volatility data. Furthermore, by our simulation study, we are inclined to trust the contrast estimator. Therefore, we compute the corresponding confidence intervals at level $5\%$ (see Table [\ref{icc}]). For the SP500, note that the Bootstrap filter and QML are not in the confidence interval for the parameter $\sigma^2$. Furthermore, for the FTSE, the Bootstrap filter is not in the confidence interval for the parameter $\phi$. These results are consistent with the simulations where we proved that both methods were biased (see Section \ref{co}). The coefficients of the autocorrelation for the FTSE are higher than for the S$\&$P500 (see Figure [\ref{RES}]), which explains the large confidence interval for the FTSE. Indeed, the covariance terms of the variance matrix given in Corollary \ref{lele} are large in this case. Note also that as expected the computing time for the QML is the shortest because it assumes Gaussianity which is probably not the case here. Except of QML, the contrast is the fastest method. The results are presented in Table [\ref{resu}] below.

\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|} {Index} & \multicolumn{3}{c|}{FTSE} & \multicolumn{3}{c|}{SP500 }\\
\hline
 &  \multicolumn{1}{c|}{$\hat{\phi}_n$} &  \multicolumn{1}{c|}{$\hat{\sigma}^{2}_n$} &\multicolumn{1}{c|}{CPU} & \multicolumn{1}{c|}{$\hat{\phi}_n$} &  \multicolumn{1}{c|}{$\hat{\sigma}^{2}_n$} &  \multicolumn{1}{c|}{CPU } \\
\hline
Contrast &  0.96 & 0.12 &114 & 0.865 & 0.119 & 108  \\
\hline 
Bootstrap filter &  0.784 & 0.39 & 461.97 & 0.830 & 0.247 & 358  \\
\hline 
APF &  0.90 & 0.317  & 472 & 0.917 & 0.179& 405  \\
\hline
KSAPF &  0.916 & 0.255 & 455 & 0.929 & 0.239 & 502  \\
\hline
QML & 0.906 & 0.279 & 31.67 &  0.895& 0.257 & 104 \\
\hline
\end{tabular}
\end{center}
\caption{\label{resu} Parameter estimates: $n=1000$ and the number of particles $M=5000$ for the Bayesian estimations. The CPU is in seconds. }
\end{table}


\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{1}{|c|} {Index} & \multicolumn{2}{|c|} {Confidence Interval}\\
\hline
  & \multicolumn{1}{|c|}{$\phi$} &  \multicolumn{1}{c|}{$\sigma^{2}$}  \\
\hline
SP500 & [0.791;0.939]& [0.005;0.233]  \\
\hline 
FTSE & [0.822; 0.99]& [ 0.001; 0.57]  \\
\hline 
\end{tabular}
\end{center}
\caption{\label{icc} Confidence interval at level $5\%$.}
\end{table}


\subsection{Summary and Conclusions}

In this paper we propose a new method to estimate an hidden stochastic model on the form (\ref{mod1}). This method is based on the deconvolution strategy and leads to a consistent and asymptotically normal estimator. We empirically study the performance of our estimator for the Gaussian AR(1) model and SV model and we are able to construct a confidence interval (see Figures [\ref{ic1}] and [\ref{ic2}]). As the boxplots [\ref{boxplot_compa1}] and [\ref{boxplot_compa2}] show, only the Contrast, the APF, and the KSAPF estimators are comparable. Indeed the QML and the Bootstrap Filter estimators are biased and their MSE are bad, and in particular, the QML method is the worst estimator (see Figure [\ref{compar2}]). One can see that the QML estimator proposed by Harvey et al. is not suitable for the SV model because the approximation of the log-chi-square density by the Gaussian density is not robust (see Figure [\ref{approxim}]). Furthermore, if we compare the MSE of the three Sequential Bayesian estimation, the KSAPF estimator is the best method. From a Bayesian point of view, it is known that the Bootstrap filter is less efficient than the APF and KSAPF filter since by using the density transition as the importance density, the propagation step of the particles will be made without taking care the observations (see \cite{doucc}).\\

Among the three estimators (Contrast, APF, and KSAPF) which give good results our estimator outperforms the others in a MSE aspect (see Figure [\ref{compar2}]). Moreover, as we already mentioned, in the combined state and parameters estimation the difficulties are the choice of $Q$, $h$ and the prior law since the results depend on these choices. In the numerical section, we have used the stationary law for the variable $X_1$  and this choice yields good results but we expect that the behavior of the Bayesian estimation will be worse for another prior.
The implementation of the contrast estimator is the easiest and it leads to confidence intervals with a larger variance than the SIEMLE but at a smaller computing cost, in particular for the AR(1) Gaussian model (see Figure [\ref{time_gaussian}]). Furthermore, the contrast estimator does not requires an arbitrary choice of parameter in practice. 

\newpage

