Yacine Ait-Sahalia (Chicago): Maximum-Likelihood Estimation of
Discretely-Sampled Diffusions: A Closed-Form Approach.
When a continuous-time diffusion is observed only at discrete dates, not
necessarily close together, the likelihood function of the observations is in
most cases not explicitly computable. Researchers have relied on simulations
of sample paths in between the observation points, or numerical solutions of
partial differential equations, to obtain estimates of the function to be
maximized. By contrast, we construct a sequence of fully explicit functions
which we show converge under very general conditions, including non-ergodicity,
to the true (but unknown) likelihood function of the discretely-sampled
diffusion. We document that the rate of convergence of the sequence is
extremely fast for a number of examples relevant in finance. We then show
that maximizing the sequence instead of the true function results in an
estimator which converges to the true maximum-likelihood estimator and shares
its asymptotic properties of consistency, asymptotic normality and efficiency.
Applications to the valuation of derivative securities are also discussed.
Mikkel Baadsgaard (Copenhagen): Estimation of Continuous Time Models
of the Term Structure of Interest Rates Using Second Order Filtering.
This paper presents an econometric analysis of continuous time models of
the term structure of interest rates, using the prices of coupon bonds as
the observed entity. The model considered is a three factor model of the
term structure of interest rates. In the model the prices of coupon bonds
depends on the following state variables
1) the current short rate, 2) the short term mean of the short rate and 3) the
current volatility of the short. The three state variables are assumed to
be stochastic and the processes described by appropriate SDEs.
In order to obtain a reasonable estimate of the entire term structure a panel
of bond prices with different maturities are used in the estimation.
As it is assumed that the observed bond prices are encumbered with
measurement noise, a second order filtering approach is used to estimate the
unobservable state variables, and the parameters are estimated by a quasi
maximum likelihood method. Monte Carlo simulation as well as empirical
results based on the Danish bond market are presented.
Bo Martin Bibby (Copenhagen): Simplified Estimating Functions for
Diffusion Models with a High-Dimensional Parameter.
A satisfactory description of complex dynamical systems often leads to a
parametric model with a high-dimensional parameter.
We consider observations at discrete time-points of a diffusion process
where the parameter of interest is of high dimension. We propose to estimate
part of the parameter using a simple estimating function and to use a
martingale estimating function to estimate the remaining part of the
parameter.
We consider the asymptotic properties of the resulting estimator.
In the talk data from an experiment involving the monitoring of wind speed at
the west coast of Jutland will be presented as a motivating example. The
estimation procedure will also be considered for other simple examples of
diffusion models.
Ola Elerian (Oxford): Likelihood Inference for Discretely Observed
Non-linear Diffusions. Co-authors: Neil Shephard (Nuffield) and Siddhartha
Chib (John M. Olin School of Business, Washington University)
This paper is concerned with the Bayesian estimation of
non-linear stochastic differential equations when only discrete observations
are available. The estimation is carried out using MCMC methods, in
particular the Metropolis-Hastings algorithm, by introducing auxiliary
points and using the Euler-Maruyama discretisation scheme. We develop
efficient simulation routines and show that naive MCMC methods can perform
dramatically poorly in this situation. Techniques for computing the
likelihood function, the marginal likelihood and diagnostic measures (all
based on the MCMC output) are presented. Examples using simulated and
real data are presented and discussed in detail.
Keywords:
Bayes estimation, nonlinear diffusion, Euler-Maruyama approximation, Maximum
Likelihood, Markov chain Monte Carlo, Metropolis Hastings algorithm, missing
data, Simulation, Stochastic Differential Equation.
Bjørn Eraker (Bergen): Bayesian Analysis for Discretely Observed
Diffusions with Unobserved State Variables.
In this paper a new method is proposed for estimation of parameters
in diffusion processes from discrete observations. The proposed
simulation based MCMC methodology applies to a wide class of models
including systems with unobservable state variables and
non-linearities. We apply the method to the estimation of
parameters in one-factor interest rate models of the CEV class and
to a generalization of this model to a two-factor model with a
stochastic volatility component. The small sample properties of the
estimator are studied trough sampling experiments for the
stochastic volatility model and the results indicate that the
method provides accurate estimates at moderate sample sizes.
Alexander Gushchin (Moscow): On Convergence
to Exponential-Type Statistical Models.
In this talk we discuss conditions for a sequence of filtered statistical
experiments to converge to an experiment generated by an exponential-type
family of probability measures. Our approach covers many practically
important models. The non-filtered case is also considered.
Ernst Hansen (Copenhagen): Pulse Dimension.
Pulse dimension is introduced as a tool for discussion of correlation
dimension. It is defined for an arbitrary continuous distribution function
F on [0, \infty) as dim_P(F) = \limsup_{x \to 0^+} \frac {\log F(x)}{\log x},
provided that the limit exists. We propose and investigate a class
of estimators of pulse dimension, IGP_n(p, \xi),
depending on two auxiliary parameters, p and \xi - the index n designates the
number of independent observations from F. These estimators are based on the
empirical quantile function, and they are quite naive, but they
have the advantage from adaption of more common estimators of correlation
dimension, that they are always well defined. Analysis, based on
the asymptotic distribution of IGP_n for fixed choice of the auxiliary
parameters identifies the optimal choice of these parameters.
The optimal parameters depend on the number of observations. It
is shown that asymptotic normality is preserved, if we calculate
the IGP_n not for fixed values of the auxiliary parameters, but for the
optimal choice. Finally, simulation results are presented to justify the
approximations involved in the analysis.
Keywords: Correlation dimension, Grassberger Procaccia estimator,
optimal asymptotic scheme, pulse dimension, tail index.
Peter Honoré (Aarhus): Panel-Data Estimation of Non-Linear
Term-Structure Models.
In this paper we present a maximum likelihood estimator of non-linear
term-structure models based on a panel-data approach, which
facilitates examining a broader class of term-structure models
compared to the majority of recent panel-data literature. It is
assumed that all zero-coupon yields are observed with measurement
error. By imposing linear restrictions on the errors, the underlying
state variables are recovered. Hence, the explicit likelihood function
is directly available without the need for a filtering algorithm.
Furthermore, we use a finite-difference approach to calculate the
yields of zero-coupon bonds. Monte Carlo simulations show the benefit
of adopting the panel-data approach. Empirical results based on the
U.S. term structure of interest rate are presented.
Keywords: Panel-Data Approach, Non-Linear Spot-Rate Diffusion Models,
Maximum Likelihood Estimation, Finite-Difference Method, Linear
Inversion Restriction.
JEL Codes: C15, C23, C60, G12.
Reinhard Höpfner (Paderborn): Nonparametric Estimation in Birth
and Death on a Flow.
Birth and death on a flow (bdf), introduced by \c{C}inlar and Kao (1992), is a model for the random evolution of finite point configurations in Euclidean space. We speak of these points as particles. The process starts at time $t{=}0$ in a finite deterministic initial configuration. New particles appear at random times/positions, modelled by Poisson random measure $\mu (dt,dy)$ with intensity $dt \, \pi (dy)$ for some finite measure $\pi$. During their random life time, particles are transported in space by a stochastic Brownian flow $\Phi$ - independent of $\mu$ - which is the solution flow to an SDE
$
dX_t = b(X_t)dt + \gs (X_t)dW_t
$
with drift $b(.)$ and diffusion coefficient $\gs (.)$; on this flow, particles living at the same time move dependently. Particles die according to a position dependent killing rate $k(.)$ and vanish from the configuration. bdf is the resulting c$\grave {\rm a}$dl$\grave {\rm a}$g process $(\gph _t)_{t\geq 0}$ of finite particle configurations.
A statistical model for bdf is a class of probability laws on a canonical path space defined in terms of a birth measure $\pi$, a drift function $b(.)$ and a death rate function $k(.)$ (we assume $\gs (.)$ to be fixed and known), under certain assumptions on these. For one-dimensional state space $\RR$, we prove asymptotic normality of kernel estimators for $k(.)$ if the bdf process $\gph$ is observed over a long time interval. Key tools are limit theorems for local time of the bdf process.
Valerie Isham (London): Inference for Spatio-Temporal Processes:
A Hydrological Case-Study.
Formal theories of inference for stochastic processes are generally
likelihood-based, but for many spatio-temporal processes, the
likelihood function is not readily available. As an illustration of
some of the problems involved, we consider the case of
spatio-temporal precipitation fields, where the fundamental
underlying binary (wet/dry) structure makes Gaussian-based models
inappropriate. Some models developed for use in addressing specific
problems arising in hydrology will be described; one purpose of this
work is to enable the continuous simulation of rainfall fields over
very long time periods, for input into distributed rainfall-runoff
models. The fitting and assessment of the adequacy of such models
raises many interesting statistical and computational issues which
will be discussed.
Mathieu Kessler (Murcia): Simulations Based Estimating Functions
for a Discretely Observed Diffusion.
When dealing with the estimation of a parameter theta in the
coefficients of a stochastic differential equation
from a discrete observation of a trajectory, it is well
known that, since the likelihood is intractable, the mle is not a solution
in practice. The estimating functions turned out to provide a quite
satisfactory way to overcome the difficulty : they are easily implemented
and yield a consistent and asymptotically normal estimator (see Sørensen,
1997). However the expression of the estimating functions usually
involve a quantity that cannot be computed exactly, (typically some moment
of the transition density) and therefore relies on a numerical approximation
through the simulation of a ''large'' number of paths of the solution thanks
to an approximation scheme. This procedure turned out to yield satisfactory
results from a practical point of view but, up to now, there was no
theoretical results about the loss of precision due to these numerical
approximations of the theoretically well behaved estimating functions. In
this work, we address these issues and provide results to assess the choice
of the number of approximating paths as well as the choice of the
approximating scheme.
Sørensen, M. (1997): Estimating functions for discretely observed diffusions: A
review. In Basawa, I.V., Godambe, V.P. and Taylor, R.L. (eds.):
Selected Proceedings of the Symposium on Estimating Functions.
IMS Lecture Notes - Monograph Series, Vol. 32, 305-325.
Henrik Madsen (Copenhagen) : Methods for Estimating Embedded
Parameters in Linear and Non-linear SDEs Using State Filtering
Techniques.
During the last couple of decades state filtering techniques have been
used for estimating parameters in stochastic differential equations
based on discrete time measurements. In the talk some of the methods
will be described, and examples from engineering applications will be
provided. Finally, some of the numerical details used in an
implementation of the methods will be outlined.
Jan Nygaard Nielsen (Copenhagen): Estimation in Continuous-time
Stochastic Volatility Models Using Nonlinear Filters.
Volatility modelling and estimation plays an important role in the
valuation and hedging of financial derivatives.
The stylized facts of stock prices, interest and exchange rates have
lead econometricians to propose stochastic volatility models in both discrete
and continuous time. However, the volatility as a measure of economic
uncertainty is not directly observable in the financial markets.
The objective of the continuous-discrete filtering problem considered
here is to obtain estimates of the stock price and, in particular, the
volatility using discrete-time observations of the stock price.
In general, only approximate solutions to the continuous-discrete
filtering problem exist, under some regularity conditions, in the form of
two ordinary differential equations for the mean and variance of the state
variables. In the present paper a nonlinear, second order filter is
examined for some bivariate stochastic volatility models and the filter
is applied to US stock market data using a maximum likelihood method.
The filter is a generalization of the extended Kalman filter that cannot
handle state-dependent diffusion terms.
Søren Feodor Nielsen (Copenhagen): On Simulated EM
Algorithms.
The EM algorithm is a useful method for finding the maximum likelihood
estimator in incomplete data problems. However, in some cases we cannot
calculate the conditional expectation required in the E-step of the
algorithm. Instead an estimate can be formed by simulation, leading to a
so-called simulated EM algorithm. The simulations can in
principle be done in two ways; we can either draw new random numbers in
each iteration or we can re-use the random numbers in each iteration. This
leads to two rather different algorithms, which will be discussed and
compared during the talk.
Jan Pedersen (Aarhus): Weak Convergence of Generating
Strategies.
The binomial model is a discrete time
complete model. This means that an arbitrary claim
is generated by an appropriate trading strategy. Further,
this strategy is explicitly known and is easy to
represent in terms of stock and claim prices.
In continuous time the Black-Scholes model is complete as
well. However, some generating strategies are not explicitly known.
We show that a weak approximation to a (generally unknown)
continuous time strategy is obtained in terms of suitable
binomial models.
Ulrike Putschke (Berlin): Properties of the ML-Estimator for
Homogeneous Gaussian Diffusions.
For a (0,B,C)-diffusion in finite dimensions the asymptotic
properties of the likelihood-function are studied in the sense of Le Cam
with a special emphasis on the case in which the matrix B is normal.
The essential influence is based on the spectral properties
of the unknown matrix B. They lead to a rich structure of different
cases classified in four categories -- LAN, LAMN, PLAMN and LAQ --
where each of them is diveded into several sub-classes. Applications
to the asymptotic behaviour of the MLE are given.
Anders Rahbek (Copenhagen): Unit-root Inference in
Autoregressive (AR)\ Models with Autoregressive Conditional Heteroscedastic
Innovations (ARCH). Co-authors: Peter Boswijk (Amsterdam) and Anders
Svennesen (Copenhagen).
In unit-root and, in particular, cointegration analysis involving
financial data it is often the case that estimated residuals appear to have
ARCH-like behaviour. Implications and modelling of ARCH innovations are
discussed for the class of AR models with emphasis on unit-root hypotheses.
Two invariance principles hold. The first implies that likelihood ratio
tests for unit-roots in AR models with ARCH\ innovations have non-standard
asymptotic distributions. These may be characterized in terms of two,
possibly dependent, brownian motions and stochastic integrals in terms of
these. The second invariance principle implies that unit-root likelihood
ratio tests derived in AR models with iid. gaussian distributed innovations,
have the same asymptotic distribution whether or not the innovations are
iid. or ARCH.
Tina Rydberg (Oxford): A Modelling
Framework for the Prices and Times of Trades Made on the NYSE.
We set down a framework for the modelling of the price and
time of each trade made on a particular stock on the New York Stock
Exchange (NYSE). The model has two main characteristics: (i) Prices only
occur on a non-negative lattice of points separated by 1/8 of a dollar,
(ii) the times of the trades occur randomly and the time between each
consecutive trade has distinct intra-daily patterns and are serially
correlated around the pattern. We model the time between trades as a Cox
process and the price movements as being very close to being a compound
Poisson process. Our models have the advantage that they have no direct
latent variable and so likelihood inference is straightforward. There is
no time deformation or stochastic volatility component which makes the model
easy to simulate.
Key words: Complete models, Compound Poisson processes,
Cox processes, Randomly spaced observations.
Tobias Rydén (Lund): Bayesian Inference in Hidden Markov
Models through Reversible Jump Markov Chain Monte Carlo.
A hidden Markov model (HMM) is a bivariate stochastic process
$\{(X_k,Y_k)\}$ such that (i) $\{X_k\}$ is a finite state Markov
chain (ii) given $\{X_k\}$, the process $\{Y_k\}$ is a sequence of
conditionally independent random variables with the conditional
distribution of $Y_n$ depending on $X_n$ only. The chain $\{X_k\}$
is generally not observable, hence the word `hidden', so that
inference has to be based on $\{Y_k\}$alone.
HMMs have during the last decade become widely spread for modelling
sequences of weakly dependent random variables with applications in
areas like speech processing, communication networks, biochemistry,
biology, medicine, econometrics, environmetrics, etc. Sometimes the
hidden Markov chain $\{X_k\}$ does indeed exist, so that the physical
nature of the problem suggests the use of an HMM, in other cases HMMs
just provide a good fit to data.
One of the most difficult problems in inference for HMM is to estimate
the number of states, $d$ say, of $\{X_k\}$. Classical approaches to
this problem include likelihood ratio tests and penalized likelihoods
(AIC/BIC). In this talk we present a Bayesian approach: by placing a
prior on the unknown $d$ we obtain a posterior distribution for $d$ and
the other parameters of the model. This distribution is analytically
untractable but can be explored using jump Markov chain Monte Carlo
algorithms. Finally an application to stock market data is presented.
Vladimir Spokoiny (Berlin): Adaptive Estimation for Non-stationary
Stochastic Systems.
We consider a stochastic system which is perturbed at some unknown moments.
Each perturbation may drastically change characteristics of the system.
The goal is to estimate adaptively the current system parameters
from the observations of this system.
Anders Stockmarr (Copenhagen): Asymptotic Behavior of the MLE's in
Models for Multivariate Time-homogeneous Gaussian Diffusions.
The asymptotic behavior of the Maximum Likelihood estimators
in these models have proven to be very complex and hard to describe, and
though the behaviour is well-known in the stationary case, a general
result has not yet been obtained. The talk will go a bit of the way via an
approach based on the Jordan decomposition of the linear drift parameter.
Helgi Tomasson (Reykjavik): Estimation of Market Value When Trading
is Infrequent.
The value process of a stock market is assumed to evolve
continuously. When trading takes place a noisy observation, (the
price process) of
the value process is obtained. Individual stocks have different
trading intensity. A computational algorithm based on the Kalman
filter for estimating the value process based on observations on
the price process is given. This is implemented on Icelandic data
to get an on-line estimate of a market value index.
Esko Valkeila (Helsinki): Stock Prices Driven by Fractional Brownian
Motion.
It is common to use geometric Brownian motion as a model
for stock prices. In this talk we look, what happens, when
the driving Brownian motion is replaced by a fractional Brownian
motion.
Jeanette Wörner (Freiburg): Optimal Estimation for Discretely
Observed Diffusion Processes.
In the ergodic case we prove LAN for a discretely observed diffusion
process and apply the results to martingale estimating functions.
Furthermore, we derive a relation between the LAN property and the
optimality in the sense of Godambe and Heyde.