Katsuto Tanaka

In this paper we discuss some distributional properties of quadratic functionals of the ordinary and fractional Brownian motions (fBms). As far as the ordinary Brownian motion (Bm) is concerned, those properties have been established extensively. A transition from the Bm to the fBm is not straightforward. Some difficulties associated with dealing with the fBm are explained, and a way to solving the problem is indicated, and some conjectures are given. © 2008 Elsevier B.V. All rights reserved.

In this paper we discuss some distributional properties of quadratic functionals of the ordinary and fractional Brownian motions (fBms). As far as the ordinary Brownian motion (Bm) is concerned, those properties have been established extensively. A transition from the Bm to the fBm is not straightforward. Some difficulties associated with dealing with the fBm are explained, and a way to solving the problem is indicated, and some conjectures are given. (C) 2008 Elsevier B.V. All rights reserved.

This paper deals with nonstationary autoregressive (AR) models with complex roots on the unit circle. We examine the asymptotic properties of the least squares estimators (LSEs) in the model. We also extend the model to the case where the error term follows a stationary linear process. We show that the limiting distribution of the LSE of the unit root parameter has a property comparable to that of the LSE in the standard nonstationary seasonal model with period two. Percent points and moments of the limiting distribution are computed by numerical integration.

This paper deals with nonstationary autoregressive (AR) models with complex roots on the unit circle. We examine the asymptotic properties of the least squares estimators (LSEs) in the model. We also extend the model to the case where the error term follows a stationary linear process. We show that the limiting distribution of the LSE of the unit root parameter has a property comparable to that of the LSE in the standard nonstationary seasonal model with period two. Percent points and moments of the limiting distribution are computed by numerical integration.

Abstract This paper discusses the estimation problems associated with signal plus noise models, where the signal is assumed to follow a stationary or nonstationary long-memory process, whereas the noise is assumed to be an independent process. Moreover, the signal and noise are independent of each other. We take up various frequency domain and wavelet-based estimators, and examine finite sample properties of these estimators. It is found that frequency domain estimators perform better for stationary cases, whereas they tend to become worse for nonstationary cases. To overcome this deficiency we consider frequency domain estimators based on differenced series. It is also found that wavelet-based estimators perform well for both stationary and nonstationary cases. Introduction Let us consider the model where only yt is observable, xt is a stochastic signal and ut is noise or a measurement error. We assume that xt and ut are independent of each other. Moreover, ut is assumed to be normally independent and identically distributed with the mean 0 and variance ρσ2, which is abbreviated as NID(0, ρσ2) hereafter, where ρ is a nonnegative constant while σ2 is a positive constant that is the variance of the innovation driving the signal xt. The signal process xt is assumed to be of the form: where εt follows NID(0, σ2), whereas the differencing parameter d is assumed to take any positive value, which yields a long-memory stationary or nonstationary process. The model (4.2) is often referred to as the ARFIMA(0, d, 0) model.

The measurement error problem that we consider in this paper is concerned with the situation where time series data of various kinds - short memory, long memory, and random walk processes - are contaminated by white noise. We suggest a unified approach to testing for the existence of such noise. It is found that the power of our test crucially depends on the underlying process.

The measurement error problem that we consider in this paper is concerned with the situation where time series data of various kinds-short memory, long memory, and random walk processes-are contaminated by white noise. We suggest a unified approach to testing for the existence of such noise. It is found that the power of our test crucially depends on the underlying process.

Dealing with integrated and near-integrated processes, this paper investigates the validity of regression on deterministic trends of K terms as K becomes large. It is found that the regression tends to be valid in spite of the true process being free from deterministic trends, which implies that the distinction between stochastic and deterministic trends disappears in K-asymptotics of the integrated and near-integrated processes. It is also shown that in K-asymptotics the usual unit root test, based on the model with deterministic trends of K terms, becomes useless against near-integration since the unit root distribution remains unchanged.

Dealing with integrated and near-integrated processes, this paper investigates the validity of regression on deterministic trends of K terms as K becomes large. It is found that the regression tends to be valid in spite of the true process being free from deterministic trends, which implies that the distinction between stochastic and deterministic trends disappears in K-asymptotics of the integrated and near-integrated processes. It is also shown that in K-asymptotics the usual unit root test, based on the model with deterministic trends of K terms, becomes useless against near-integration since the unit root distribution remains unchanged.

This paper deals with a scalar l(d) process {y(j)}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {y(j)}, assuming that d is known and is greater than or equal to 1/2. Note that {y(j)} becomes stationary when d < 1/2, whose case is not our concern here. It turns out that the case of d = 1/2 needs a separate treatment from d > 1/2. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality, Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d greater than or equal to 1/2. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.

An alternative approach is taken to the asymptotic theory of cointegration. The present approach gives a different expression for the limiting distributions of statistics associated with cointegration, which enables us to compute accurately the distribution functions. Alternative interpretations of cointegration are given and a notion of near cointegration is introduced. We then devise tests which take cointegration as the null and discuss the limiting local power under the alternative of near cointegration.

A unified approach which I call the Fredholm approach is suggested for the study of asymptotic behavior of estimators and“ test statistics arising from nonstationary and/or noninvertible time series models. Some limit theorems are given concerning the distribution of (the ratio of) quadratic (plus linear) forms in random variables generated by a linear process that is not necessarily stationary. Especially, the limiting characteristic function is derived explicitly via the Fredholm determinant and resolvent of a given kernel. Some examples are also shown to illustrate our methodology. © 1990, Cambridge University Press. All rights reserved.