Workshop on Recent Progressin Time Series:
in honour of Peter Robinson

Program and Abstracts
May 29-30, 2019
Large meeting room on 4F
Graduate School of Economics and ManagementTohoku University
financially supported by
Grant-in-Aid for Scientific Research(B) 17H01701, (B) 17H02508

Program
Wednesday, May 29, 201913:00-14:00

Peter M Robinson (LSE): Long-Range Dependent Curve TimeSeries’ (joint work with Degui Li and Han Lin Shang)

14:15-15:45

E. Kurozumi (Hitotsubashi Univ.): Asymptotic Properties ofBubble Monitoring Tests
T. Kurita (Fukuoka Univ.)：Modelling and forecasting the dollarpound exchange rate in the presence of structural breaks(written with Jennifer L. Castle, University of Oxford)
Y. Yajima (Tohoku Univ.): On Gaussian semiparametric estimaiton
for two-dimeinsional intrinsic stationary random fields

16:00-17:10

Y. Hosoya (Tohoku Univ.)：Extending the concepts ofinterdependence beyond the stationary time series
M. Shintani (Univ. Tokyo)：Frequency-wise causality analysis ininfinite order vector autoregressive processes

BanquetWednesday, May 30, 2019
9:30-10:30

Y. Konishi (RIETI)：Zipf’s and Gibrat’s Laws for Tourist Flows inJapan: 2011-2017
Y. Nishiyama (Kyoto Univ.): Sequential Unit Root Test
Joint work with Kohtaro Hitomi and Keiji Nagai

10:45-11:45

H. Nishino (Hiroshima Univ.)：Estimation for ARMA models with$t$-distributed innovations
Y. Liu (Kyoto Univ.)：Testing structure of dependence for highdimensional time series based on bootstrap

12:00-13:00

M. Matsui (Nanzan Univ.)：Applications of DistanceCorrelation to Time Series
Y. Matsuda (Tohoku Univ.)：Bivariate CARMA random fields

Abstracts
Peter M Robinson: London School of Economics‘Long-Range Dependent Curve Time Series’ (joint work with Degui Li and Han LinShang)
Abstract:
We introduce methods and theory for functional or curve time series with long range
dependence. The temporal sum of the curve process is shown to be asymptoticallynormally distributed, the conditions for this covering a functional version of fractionallyintegrated autoregressive moving averages. We also construct an estimate of the longrun covariance function, which we use, via functional principal component analysis, inestimating the orthonormal functions spanning the dominant sub-space of the curves. Ina semiparametric context, we propose an estimate of the memory parameter andestablish its consistency. A Monte-Carlo study of finite sample performance is included,
along with two empirical applications. The first ofthese finds a degree of stability and persistence in intra-day stock returns. The second
finds similarity in the extent of long memory in incremental age-specific fertility rates
across some developed nations.
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Eiji Kurozumi, Hitotsubashi University
Asymptotic Properties of Bubble Monitoring Tests
Abstract:
We investigate bubble monitoring tests by extending Phillips, Wu, and Yu (2011) andPhillips, Shi, and Yu's (2015) sup-ADF and generalized sup-ADF tests to the monitoring
scheme. We also consider applying the CUSUM detector as proposed by Homm andBreitung (2012). We derive the limiting distributions of the detecting statistics under thenull hypothesis, the moderate deviation alternative, and the local alternative. We findthat although the moderate deviation alternative shows the difference in the divergence
rates of the ADF- and CUSUM-type detectors, the local asymptotic theory is more
useful for understanding their differences in detail. We also conduct finite sample

simulations and confirm that the local asymptotic theory approximates the finite sample
properties of the tests very well.
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Takamitsu Kurita, Fukuoka University
Modelling and forecasting the dollar-pound exchange rate in the presence of structural
breaks (written with Jennifer L. Castle, University of Oxford)
Abstract:
We employ a newly-developed partial cointegration system allowing for level shifts to
examine whether economic fundamentals form the long-run determinants of the dollarpound exchange rate in an era of structural change. The paper uncovers a class of local
data generation mechanisms underlying long-run and short-run dynamic features of theexchange rate using a set of economic variables that explicitly reﬂect the central banks’monetary policy stances and the inﬂuence of a forward exchange market. The impact ofthe Brexit referendum is evaluated by examining forecasts when the dollar-pound
exchange rate fell substantially around the vote.
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Yoshihiro Yajima, Tohoku UniversityOn Gaussian semiparametric estimaiton for two-dimeinsional intrinsic stationary
random fields
Abstract:
We propose a Gaussian semiparametric estimator for semiparametric models of twodimensional intrinsic stationary random fields (ISRFs) observed on a regular grid andderive its asymptotic properties.
Originally this estimator is an approximate likelihood estimator in a frequency
domain for long memory models of stationary and nonstationary timeseries(Robinson(1995.Ann.Statist.); Velasco(1999.J.Time Ser.
Anal.)). We apply it to two dimensional ISRFs. These ISRFs include a fractional
Brownian field, which is a Gaussian random field and is used to model many physicalprocesses in space.
The estimator is consistent and has the limiting normal distribution as the sample size

goes to infinity. We conduct a computational simulation to compare the performance of itwith those of different estimators.
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Yuzo Hosoya, Tohoku University
Extending the concepts of interdependence beyond the stationary time series
Abstract:
The measures of one-way effect, reciprocity, and association are defined in thefrequency domain typically for vector stationary time series ([3]). The basic idea iselicitation of a one-way effect component of a supposedly causing series and expressing
its contribution in reduction of the prediction error by means of the Szegö representation
formula of the one-step ahead prediction error of stationary processes. To define apartial version of the measures of interdependence, the paper [4] suggested the thirdeffect elimination of the one-way effect component of the third series from a pair ofsubject-matter series so that temporal sequence character of interdependence between
series in question is least distorted. The present paper proposes extension of thoseconcepts and allied measures to processes such as:(i)
Nonstationary cointegrated processes ([5]),(ii)Locally stationary processes introduced by ([1]),(iii) Doubly-parametrized stationary time series and stationary fields ([2], [6]).
Extension to (i) is enabled by introduction of an auxiliary generating process. Regarding(ii), the measures of interdependence is generalized to locally stationary vector
processes by extension of Dahlhaus’ block shifting method and the allied asymptotics.
The extension to the case (iii) and hence to nontemporal processes is enabled byapplication of the Szegö formulas of the prediction error in prediction based on the halfplane and the allied subspace for higher-dimensional fields.
References:[1] Dahlhaus, R. (1996). Stochastic processes and their Applications, 62, 139-164.
[2] Helson, H. and Lowdenslager, D. (1958). Acta Mathematics, 99, 165-202.
[3] Hosoya, Y. (1991). Probability Theory and Related Fields, 88, 429-444.
[4] Hosoya, Y. (2001). Journal of Time Series Analysis, 22, 537-554.
[5] Hosoya, Y. et al. (2017). Characterizing Interdependencies of Multiple Time Series:
Theory and Application, Springer.
[6] Whittle, P. (1954). Biometrika, 41, 434-449.

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Mototsugu Shintani, The University of Tokyo
Frequency-wise causality analysis in infinite order vector autoregressive processes
Abstract:
This paper derives the asymptotic properties of frequency-domain causality measureusing the vector autoregressive model of infinite order and proposes a test for causality
at a particular frequency. Simulation results confirm that our procedure works wellwith sample size typically available in practice. We illustrate the usefulness of ourmethod via an application to financial data.
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Yoko Konishi, RIETI
Zipf’s and Gibrat’s Laws for Tourist Flows in Japan: 2011-2017
Abstract:
Since 2012, importance of tourism industry in Japan has been increasing due to an
unprecedented inbound tourism boom. This paper examines the size, rank and growth
rates and study whether Zipf’s and Gibrat’s laws strike Japan’s tourist flows. Our analysis
reveals that both the ranks and the sizes for Japanese travelers are stable, while thenumbers of inbound travelers of each region have higher growth rates and fluctuate inrank order. From Gibrat’s regression, we find regions with smaller number of inboundtravelers tend to attract more foreign tourists.
This indicates that they keep growing toward a steady state.
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Yoshihiko Nishiyama, Kyoto University
Sequential Unit Root Test
K. Hitomi, K. Nagai and Y. Nishiyama
Abstract:
Motivated by Lai and Siegmund (1983), we consider unit root tests under sequential

sampling for an AR(1) process against both stationary and explosive alternatives. We
can straightforwardly use the uniform asymptotic normality of their sequential OLS
estimator for the test. We find that it is also possible to use the stopping time for thispurpose. From this finding, we propose three kinds of test using t value, stopping time
and their combination by Bonferroni. To examine the statistical properties, we obtain
their weak joint limit by approximating the processes in D[0,¥infty) and a DDS(Dambis and Dubins-Schwarz) Brownian motion. To the best of our knowledge, this is
the first attempt to derive the joint weak limit of the estimator and the stopping time insequential analysis. The distribution of the stopping time is characterized by a Bessel
process of dimension 3/2 with and without drift, while the estimator is asymptoticallynormally distributed as Lai and Siegmund (1983) show. We implement Monte Carlosimulations to examine the small sample properties of the tests.
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Haruhisa Nishino, Hiroshima University
Estimation for ARMA models with $t$-distributed innovations
Abstract:
The standard ARMA model assumes that its innovations are white noise processes with0 mean and a constant variance. That is, the ARMA model is a second-order stationary
process characterised by its autocovariance function. The white noise process has noinformation about its fourth-order moment. To estimate the ARMA model, we assume a
Gaussian process and a Gaussian likelihood. On the other hand, the literature offinancial time series tells that the financial returns have fatter tails than Gaussian ones.
The Student $t$-distribution is a typical fat-tailed distribution. The fat-tailed property isrelated to the fourth order moment. We thus consider that estimation for ARMA models
with $t$-distributed innovations is useful for analysing financial time series. If we know
the degree of freedom of the $t$-distribution of the model, can we estimate theparameters of the ARMA model more efficiently than the Gaussian likelihood? Besides,
the talk proposes a preliminary estimate for the degree of freedom based on the methodof moments, since the application of MLE for the degree of freedom has a severe
problem.
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Yan Liu, Kyoto University

Testing structure of dependence for high-dimensional time series based on bootstrap
Abstract:
The problem of testing the sphericity hypothesis for the covariance matrix of highdimensional time series is considered. It has been shown that the test statistic forsphericity under the null hypothesis is asymptotically normal for high-dimensional timeseries models. However, the asymptotic variances of the test statistic are different fromeach other if the time series models are different. To alleviate the computational
complexity and model dependence, we propose a bootstrap procedure for the teststatistic. The validity and the consistency of the proposed method are justified from theasymptotic theory. The performance is illustrated by simulation studies.
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Muneya Matsui, Nanzan University
Applications of Distance Correlation to Time Series
Abstract:
Applications of empirical characteristic functions have been investigated from 70s,
which include estimations in certain parametric models and goodness of fit tests ofvarious distributions. More recently, they have been used to measure dependence and totest independence. The distance covariance and correlation, developed by Szekely et al.
(2007) and Szekely and Rizzo (2009) are perhaps the best known statistics for such
purpose. Our talk is about applications of these ideas to stationary univariate andmultivariate time series to measure lagged auto- and cross-dependence in a time series.
Assuming strong mixing, we establish the relevant asymptotic theory for the sample
auto- and cross-distance correlation functions. We also apply the auto-distancecorrelation function (ADCF) to the residuals of an autoregressive processes as a test of
goodness of fit. Under the null that an autoregressive model is true, the limit distributionof the empirical ADCF can differ markedly from the corresponding one based on an iidsequence.
Time permitting, we also explain the basic idea of applying the distancecovariance/correlation to continuous-time processes.
This is joint work with Richard Davis (Columbia), Thomas Mikosch (Copenhagen) andPhyllis Wan (Rotterdam).
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Yasumasa Matsuda, Tohoku University
Bivariate CARMA random fields
Abstract:
Brockwell and Matsuda (2017) extended CARMA models for time series to those for
random fields, which we call as "CARMA random fields". In this talk, we consider abivariate extension of CARMA random fields to analyze spatially scattered bivariate
observations. After defining bivariate CARMA random fields, we introduce Whittle
likelihoods to estimate the parameters with applications to imputation for missing
components of bivariate observations. There is a gap between discrete observations andcontinuous models that needs to be accounted to conduct the imputation. We employ
Bayesian ways to fill the gap. We demonstrate them by applying bivariate CARMA
random fields to precipitation and temperature data observed at around 7000 irregularly
scattered points in US continent.

List of ParticipantsPeter M Robinson (London School of Economics)
Eiji Kurozumi (Hitotsubashi University)
Takamitsu Kurita (Fukuoka University)
Yoshihiro Yajima (Tohoku University)
Yuzo Hosoya (Tohoku University)
Mototsugu Shintani (The University of Tokyo)
Yoko Konishi (RIETI)
Yoshihiko Nishiyama (Kyoto University)
Haruhisa Nishino (Hiroshima University)
Yan Liu (Kyoto University)
Muneya Matsui (Tohoku University)
Yasumasa Matsuda (Tohoku University)