Table Of ContentHandbook of Statistics
Series Editor
C.R. Rao
C.R. Rao AIMSCS, University of Hyderabad Campus,
Hyderabad, India
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Contributors
NumbersinParenthesesindicatethepagesonwhichtheauthor’scontributionsbegin.
Kris Boudt(193),Department of Economics,Ghent University, Ghent;Solvay
BusinessSchool,VrijeUniversiteitBrussel,Brussel,Belgium;SchoolofBusiness
and Economics,Vrije Universiteit Amsterdam, Amsterdam, The Netherlands
Sebastia´nCano-Berlanga (281),Universitat Auto`noma deBarcelona and CREIP,
Catalonia, Spain
Manfred Deistler(145),TU Wien,Vienna,Austria
Jean-MarieDufour(3),DepartmentofEconomics,McGillUniversity,Montr(cid:1)eal,QC,
Canada
Alexios Galanos(193), Amazon,Seattle,WA, UnitedStates
Jose(cid:1)-ManuelGime(cid:1)nez-Go´mez (281),Universitat Rovirai Virgili andCREIP,
Tarragona, Spain
Chirok Han(119),Professor of EconomicsatKorea University, Seoul,Republic of
Korea
Yong Li (81), HanqingAdvanced InstituteofEconomics and Finance,Renmin
University ofChina,Beijing, China
Julien Neves (3),Cornell University, Ithaca, NY, UnitedStates
Scott Payseur (193),Amazon, Seattle,WA, UnitedStates
PeterC.B.Phillips(119),SterlingProfessorofEconomicsandProfessorofStatistics
atYale University, New Haven,CT, UnitedStates
Joaquim Jose SantosRamalho(245),Department ofEconomics and BRU-IUL,
Instituto Universita´riode Lisboa(ISCTE-IUL), Lisboa, Portugal
Wolfgang Scherrer (145),TU Wien,Vienna,Austria
Cori Vilella(281),Universitat RoviraiVirgili and CREIP,Tarragona, Spain
Hrishikesh D.Vinod(33),FordhamUniversity, Bronx, NY, UnitedStates
KennethD.West(65),DepartmentofEconomics,UniversityofWisconsin-Madison,
Madison, WI, UnitedStates
JunYu(81),SchoolofEconomicsandLeeKongChianSchoolofBusiness,Singapore
Management University, Singapore,Singapore
Tao Zeng(81),Schoolof Economics,Academyof Financial Research, andInstitute
for FiscalBig-Data & Policyof Zhejiang University, ZhejiangUniversity,
Zhejiang, China
xi
xii Contributors
ZifengZhao(65),DepartmentofInformationTechnology,AnalyticsandOperations,
MendozaCollege ofBusiness, University ofNotre Dame, NotreDame, IN,
UnitedStates
EricZivot(193),Amazon; University ofWashington,Seattle, WA, UnitedStates
Preface
As with earlier volumes in this series, volume 41 of Handbook of Statistics
with the subtitle “Conceptual Econometrics Using R” and a companion vol-
ume 42 with the subtitle “Financial, Macro and Micro Econometrics Using
R” provide state-of-the-art information on important topics in Econometrics,
a branch of Economics concerned with quantitative methods. This handbook
covers a great many conceptual topics of practical interest to quantitative
scientists, especially in Economics and Finance.
The book has uniquely broad coverage with all chapter authors providing
practical R software tools for implementing their research results. Despite
some overlap, we divide the chapters into three parts. We list the three parts
while retaining the nine chapter numbers as:
1. Statistical Inference
(1) Jean-Marie Dufour and Julien Neves propose new simulation-based
exact finite sample inference methods implemented in their R package
MaxMC. Its toolkit includes overcoming nuisance parameters.
(2) Hrishikesh D. Vinod discusses new tools from his R package
“generalCorr”forinferringexogeneityandcausalpathsfrompassively
observed data, citing applications in diverse fields.
(3) Zifeng Zhao provides new tools for bias reduction in h-step ahead
forecast when h is large.
(4) YongLi,JunYuandTaoZengreviewMCMCbasedfrequentistinfer-
ence methods which avoid using Bayes Factors.
2. Multivariate Models
(5) Peter C. B. Phillips and Chirok Han provide efficient R tools for
dynamic panel data models including difference GMM, system GMM,
and within group estimation.
(6) Wolfgang Scherrer and Manfred Deistler provide tools for avoiding
inappropriate VAR models by using multivariate ARMA and state-
space models. They consider identification issues, Hankel matrices
and reduced rank regressions.
(7) Kris Boudt, Alexios Galanos, Scott Payseur, and Eric Zivot survey
multivariate GARCH models for large data sets and outlier-robust
MGARCH and evaluations of cokurtosis and coskewness.
xiii
xiv Preface
3. Miscellaneous Topics
(8) Joaquim Ramalho considers estimation and inference for direct and
marginal effects in regressions where the dependent variable is
restricted to the range [0,1], such as when it is a ratio.
(9) Sebastia´n Cano-Berlanga, Jos(cid:1)e-Manuel, Gim(cid:1)enez-Go´mez and Cori
Vilella discuss cooperative game theory including transferable utility,
“punctualsolutions,”votingpowerindexand“claimsproblems”while
providingtoolsforsharingofbenefitsamonginterdependent(economic)
agents.
All chapters are authored by distinguished researchers. Most senior authors
have received professional honors, such as being elected “Fellows” of the
Journal of Econometrics or of the Econometric Society.
The intended audience is not only students, teachers, and researchers in
variousindustriesandsciencesbutalsoprofitandnonprofitbusinessdecision
makers and government policymakers. The wide variety of applications of
statistical methodology should be of interest to researchers in all quantitative
fields in both natural and social sciences and engineering.
A unique feature of this volume is that all included chapters provide not
only a review of the newer theory but also describe ways of implementing
authors’ new ideas using free R software. Also, the writing style is user-
friendly and includes descriptions and links to resources for practical imple-
mentationsonafreeopensourceR,allowingreaderstonotonlyusethetools
ontheirowndatabutalsoprovidingajumpstartforunderstandingthestateof
the art. Open source allows reproducible research andopportunity for anyone
to extend the toolbox.
According to a usage dating back to Victorian England, the phrase “The
threeR’s”describesbasicskillstaughtinschools:Reading,wRiting,andaRith-
metic. Inthe 21st century,we should add R software asthe fourth R, which is
fast becoming an equally basic skill. Unfortunately, some economists are
continuing to rely on expensive copyrighted commercial software which not
onlyneedsexpensiveupdatingbutalsohidesmanyinternalcomputationalalgo-
rithmsfromcriticalpublicevaluationforrobustness,speed,andaccuracy.Users
of open source software routinely work with the latest updated versions. This
savestime,resources,andeffortneededindecidingwhethertheimprovements
in the latest update are worth the price and arranging to pay for it.
Inteachingundergraduatestatisticsclassesoneofus(Vinod)introducesstu-
dentstoRasaconvenientcalculator,wheretheycannamenumericalvectoror
matrixobjectsforeasymanipulationbyname.Startingwiththeconvenienceof
not having to use Normal or Binomial tables, students begin to appreciate and
enjoy the enormous power of R for learning and analyzing quantitative data.
There are over 14,686 free R packages, contributed and maintained by
researchers from around the world, which can be searched at https://mran.
microsoft.com/packages. In short, R has a huge and powerful ecosystem.
Preface xv
Studentssoonlearnthatifastatisticaltechniqueexists,thereismostlikelyan
R package which has already implemented it. The plotting functions in R are
excellent and easy to use, with the ability to create animations, interactive
chartsandsuperimposestatisticalinformationongeographicalmaps,including
theabilitytoindicatedynamicallychangingfacts.Risabletoworkwithother
programminglanguagesincludingFortran,Java,C++,andothers.Risaccessi-
ble in the sense that one does not need to have formal training in computer
science to write R programs for general use.
For reviewing the papers we thank: Peter R. Hansen (University of North
Carolina at Chapel Hill), Shujie Ma (University of California, Riverside),
Aaron Smith (University of California, Davis), Tayyeb Shabbir (California
State University Dominguez Hills, Carson, CA), Andreas Bauer (IMF Senior
ResidentRepresentative,NewDelhi,India),Jos(cid:1)eDiasCurto(ISCTE-Instituto
Universitario de Lisboa, Portugal), Ruey S. Tsay (Booth School of Business,
University of Chicago), Alessandro Magrini (University of Florence, Italy),
Jae H. Kim (La Trobe University, Australia), In Choi (Sogang University,
Korea), among others.
Acommonthreadinallchaptersinthishandbookisthatallauthorsofthis
volumehavetakenextraefforttomaketheirresearchimplementableinR.We
are grateful to our authors as well as many anonymous researchers who have
refereed the papers and made valuable suggestions to improve the chapters.
We also thank Peter Llewellyn, Kari Naveen, Vignesh Tamilselvvanignesh,
Arni S.R. Srinivasa Rao, Sam Mahfoudh, Alina Cleju, and others connected
with Elsevier’s editorial offices.
Hrishikesh D. Vinod
C.R. Rao
Chapter 1
Finite-sample inference
and nonstandard asymptotics
with Monte Carlo tests and R
Jean-Marie Dufoura,* and Julien Nevesb
aDepartmentofEconomics,McGillUniversity,Montre(cid:1)al,QC,Canada
bCornellUniversity,Ithaca,NY,UnitedStates
*Correspondingauthor:e-mail:[email protected]
Abstract
WereviewtheconceptofMonteCarlo testasasimulation-basedinferenceprocedure
which allows one to construct tests with provably exact levels in situations where the
distributionofateststatisticisdifficulttoestablishbutcanbesimulated.Thenumber
of simulations required can be extremely small, as low as 19 to run a test with level
0.05. We discuss three extensions of the method: (1) a randomized tie-breaking tech-
niquewhichallowsonetouseteststatisticswithdiscretenulldistributions,withoutfur-
ther information on the mass points; (2) an extension (maximized Monte Carlo tests)
which yields provably valid tests when the test statistic depends on a (finite) number
of nuisance parameters;(3)an asymptotic version which allows one to getasymptoti-
callyvalidtestswithoutanyneedtoestablishanasymptoticdistribution.Asthemethod
iscomputerintensive,wedescribeanR package (MaxMC) thatallowsone toimple-
mentthistypeofprocedure.Anumberofspecialcasesandapplicationsarediscussed.
Keywords:R,Exactinference,Testlevel,Testsize,Discretedistribution,Randomized
tie-breaker,Nonstandardasymptoticdistribution,MonteCarlotest,MaximizedMonte
Carlo, MMC, Simulated annealing, Genetic algorithm, Particle swarm, Bootstrap,
Kolmogorov–Smirnov,Behrens–Fisher,Autoregressive model,SingularWaldtest
1 Introduction
One ofthe central problems ofstatistical methodology consists in findingcriti-
cal values for performing tests and building confidence sets. However, it is
often the case that analytical formulae are not available. The dominant model
where finite-sample methods are available is the classical linear model with
fixed(orstrictlyexogenous)regressorsandindependentlyidenticallydistributed
HandbookofStatistics,Vol.41.https://doi.org/10.1016/bs.host.2019.05.001
©2019ElsevierB.V.Allrightsreserved. 3
4 PART I StatisticalInference
(i.i.d.)Gaussiandisturbances.Asaresult,statisticalinferenceistypicallybased
on large-sample approximations—which may be quite unreliable in finite
samples—or bootstrapping. The bootstrap usually provides improvements over
theuse of limitingdistributions,butit is alsobasedonlarge-sample arguments
throughademonstrationthattheasymptoticdistributionofteststatisticandthe
bootstrap distribution are identical in large samples; for reviews, see Efron
(1982), Beran and Ducharme (1991), Efron and Tibshirani (1993), Hall
(1992), Jeong and Maddala (1993), Vinod (1993), Shao and Tu (1995),
Davison and Hinkley (1997), Chernick (1999), and Horowitz (1997).
In this paper, we focus on the method of Monte Carlo tests, which can
deliver tests whose size (or level) is controlled in finite samples, without the
need to establish analytically the distribution of the test statistic. The number
ofsimulationsrequiredcanbeextremelysmall,aslowas19torunatestwith
level 0.05. This feature allows one to use computationally expensive test sta-
tistics. We also emphasize that the approach can yield asymptotically valid
tests in many situations where the limiting distribution of the test statistic is
nonstandard or may not exist.
ThetechniqueofMonteCarlotestsactuallypredatesbootstrappingandwas
originallysuggested byDwass (1957)inorder toimplementpermutation tests.
Another variant was later proposed by Barnard (1963), Hope (1968), and
Birnbaum(1974), inviewofperformingtestsbased onteststatisticswith con-
tinuousdistributionsunderthenullhypothesis.Otherearlyworkonthismethod
is available in Besag and Diggle (1977), Marriott (1979), Edgington (1980),
Foutz (1980), Ripley (1981), Edwards (1985), J€ockel (1986), and Edwards
and Berry (1987). These results typically rely on special assumptions on the
formofthedistributionsoftheteststatistics(continuousordiscreteinaspecific
way) and do not allow for the presence of nuisance parameters. A general
theory of Monte Carlo tests is presented in Dufour (2006) and includes three
main extensions. For other discussions and applications, see Kiviet and
Dufour (1997), Dufour et al. (1998, 2003, 2010), Dufour and Kiviet (1998),
Dufour and Khalaf (2001, 2002), Dufour and Farhat (2002), Dufour and
Jouini (2006), Beaulieu et al. (2007, 2013), and Coudin and Dufour (2009).
The first extension allows for pivotal (nuisance-parameter-free) test statis-
tics withotherwise arbitrary distributions—which may becontinuous, discrete,
ormixed (e.g., mixtures of continuous and discrete distributions). This is done
inparticularbyexploitingatechniqueofrandomizedranksforbreakingtiesin
rank tests (see Ha´jek, 1969), which is simple to implement with exchangeable
replications[asopposedtoindependentandidenticallydistributed(i.i.d.)repli-
cations]. No information onthe probabilitiesofmasspoints (if any)isneeded.
Thesecondextensioninvolvesteststatisticswhosenulldistributiondepends
onnuisanceparameters.Thisisdonebyconsideringasimulatedp-valuefunc-
tion which depends on nuisance parameters (under the null hypothesis). Maxi-
mizingthelatterwithrespecttothenuisanceparametersthenyieldsatestwith
provablyexactlevel,irrespectiveofthesamplesizeandthenumberreplications
used. We call such tests maximized Monte Carlo (MMC) tests.
