Page 146 - Contributed Paper Session (CPS) - Volume 1

CPS1216 Teppei O.
We will study the asymptotic theory of a maximum-likelihood-type estimator
for misspecified model. In this setting, the original maximum-likelihood-type
estimator cannot attain the optimal convergence rate $n^{-1/4}$ due to the
asymptotic bias. We construct a new estimator which attains the optimal rate
by using a bias correction and show the asymptotic mixed normality.

1. Introduction
Forecasting variances of stocks and covariances of stock pairs is an
important task to control the loss from stock assets for many financial
institutions which hold huge amount of stocks. Statistical analysis of stock
price data and data of financial statements is useful for this purpose.
Nowadays, we can easily get intraday stock prices data such as all transactions
of a stock in a day. Then, the study of high-frequency data becomes more
important because huge information of high-frequency data enable us to
forecast stock variances and covariances more accurately. However, there are
two problems on statistical analysis of high-frequency data. The first one is
market microstructure noise: when we model stock prices by using diffusion
processes, some empirical facts suggest the existence of additional noise. The
second one is nonsynchronous observations: We observe stock prices when
transactions occur. So observation times must be different for different stocks.
In this paper, we study parametric inference under the existence of market
microstructure noise and nonsynchronous observations. We study maximum-
likelihood-type estimation for parametric diffusion processes with noisy,
nonsynchronous observations, assuming that the true model is contained in
the parametric family. We further study the case that this assumption is not
satisfied. Such model is called a misspecified model. Ogihara [3] studied a
parametric statistical model that a stochastic process Yt is given by

dY = (, X ) + (, X , )W (1.1)

∗

for some unknown value ∗ of a parameter with noisy, nonsynchronous
observations of . Maximum-likelihood- and Bayes-type estimators were
constructed by using a quasi-likelihood function, and their asymptotic
normality were shown. Asymptotic efficiency of the estimators was also proved
by showing local asymptotic normality when the diffusion coefficients are
deterministic and noises follow normal distributions. In this model, we assume
that the true model is contained in the parametric family.
In practice for high-frequency data, to satisfy the assumption that the true
model is contained in the parametric family, we need to choose the parametric
family carefully so that it accurately captures microstructure of stock prices.
This is a difficult task because several empirical facts of a stock market (intra-
day seasonality, volatility clustering, complicated dependence structure of

135 | I S I W S C 2 0 1 9