Table Of ContentJun Ma
Mark Wohar Editors
Recent Advances
in Estimating
Nonlinear Models
With Applications in Economics
and Finance
Recent Advances in Estimating Nonlinear Models
Jun Ma • Mark Wohar
Editors
Recent Advances in
Estimating Nonlinear Models
With Applications in Economics and Finance
123
Editors
JunMa MarkWohar
DepartmentofEconomics,Finance DepartmentofEconomics
andLegalStudies UniversityofNebraska–Omaha
UniversityofAlabama Omaha,USA
Tuscaloosa,USA
ISBN978-1-4614-8059-4 ISBN978-1-4614-8060-0(eBook)
DOI10.1007/978-1-4614-8060-0
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Preface
Inthepastfewdecadestherehasbeenasurgeofinterestinmodelingofeconomic
relationships using nonlinear models. Economic theory often suggests that the
relationships between major economic and financial variables are nonlinear. For
example, the law of one price starts to apply only when the deviation from it
generatesenoughprofitsto exceedthe involvedtransactionortransportationcosts
inthearbitrageactivities(see,e.g.,Michael,Nobay,andPeel(1997)LoandZivot
(2001)).Itisalsooftenarguedthatbusinesscyclesareasymmetricwithrecessions
and expansions having different statistical features (see, e.g., Kim and Nelson
(1999)).TherecentGreatRecessionintheUSAandthemuchdebatedeffectiveness
offiscalandmonetarypoliciesalsoledtoimportantstudiesthatattempttodocument
asymmetriceffectsofthepoliciesdependingoneconomicslack(seee.g.,Fazzari,
Morley,andPanovska(2012)).
Despite such consensus for a need of nonlinear models in modeling economic
and financial time series, economic theory cannot tell us the exact form of the
nonlinear model that is needed to best describe a particular set of data. As a
result, numerouscompeting nonlinearmodels have been developedto explain the
movements in economic and financial time series. The most popular nonlinear
modelsthathavebeenusedineconomicsandfinancearetheThresholdandSmooth
Transition model (Terasvirta 1994) and the Markov Switching model (Hamilton
1989). Threshold Autoregressive (TAR) models have as their variants forms of
Smooth Transition Autoregressive (STAR) models, which include specifically the
popular exponential STAR (ESTAR) and logistic STAR (LSTAR) models. The
centralideaoftheThresholdandSmoothTransitionmodelsisthatlargedeviations
fromtheequilibriumarecorrectedbutsmall,permanentdeviationsarepossibledue
to market frictions such as transaction or transportation costs. One feature of this
type of model is that it is “self-exciting” in the sense that once the variable of
interest passes a certain threshold there is a change of regime and its time series
propertychanges.Ontheotherhand,theMarkovSwitchingmodelemploysalatent
discretestatevariabletodeterminewhichregimethevariableofinterestisoperating
in. The inference of the state or regime can be made by utilizing corresponding
filteringtechniques.OneimportantmeritoftheMarkovSwitchingapproachisthat
theregimechangesarestochasticbyitsnature.
v
vi Preface
Problems withNonlinearModels
One primary challenge in applying the nonlinear models in practice is about how
to determine which model to estimate among the many alternative models. Since
differentmodelshavedifferentstatisticalfeaturestheydonotfitdataequallywell.
It proves challenging to choose the appropriate nonlinear model given a set of
observations as the selection is often heavily influenced by outliers or influential
observations(see, e.g., Ahmadand Glosser (2011)).This is an empiricalquestion
that researchers need to carefully address given a particular set of economic or
financialtimeseriesdata.
Often nonlinear models are employed not only to shed light on the underlying
relationship between the economic and financial variables but also to provide the
out-of-sample forecasts of these variables. Although it is computationally more
complexandtimeconsumingtogenerateout-of-sampleforecastsfromthenonlinear
models, it is not clear, as indicated by Kilian and Taylor (2003), whether the
nonlinear models can provide a better out-of-sample forecast than simple linear
models.
Recently, the literature also starts to pay more attention to the identification
issue with some commonly employed nonlinear models. For example, Ma and
Nelson (2010) find that a broad class of nonlinear models including the widely
usedstate-spacemodelandGARCH modelissubjecttoZILC(Zero-Information-
Limit-Condition)asformulatedbyNelsonandStartz(2007),andthereforeweakly
identifiedastheidentifyingparameterapproachesacertainvalue.Inparticular,Ma
andNelson(2010)pointoutthatinthestate-spacemodelwhenthesignalissmall
relativetonoise,thestate-spacemodelbecomesweaklyidentifiedandthestandard
error of the persistence parameter would appear much smaller than it actually is,
leading to oversized standard Wald-type test. To resolve this issue they propose a
reduced-formtestbasedonalinearapproximationtotheexacttestofFieller(1954)
andshow thatthis test hasa muchbetter finite sample performance.Furthermore,
Andrews and Cheng (2012) develop asymptotic results for models considered in
MaandNelson(2010)inthepresenceofweakidentificationandtheyalsoconsider
thethresholdmodels.Finally,Heinen,Michael,andSibbertsen(2013)focusonthe
ESTAR model and find that given a certain value of the error variance term the
model is weakly identified and the resulting estimators are strongly biased. They
proposeanalternativemodeltomitigatethisissue.
DiscussionofArticles in OurBook
Thelast25yearshasseenatremendousincreaseinthenumberofpapersexamining
the nonlinear dynamics of both economic and financial time series. Some of the
morepopularmodelsincludeSmoothTransition(ESTARandLSTAR),Threshold
andMarkovSwitchingmodels,artificialneuralnetworkmodels,randomcoefficient
Preface vii
models,generalizedautoregressivemodels,andmultipleadaptiveregressivespline
models.Modelingandselectingtheappropriatemodelcontinuestobeachallenge
for researchers. The following 13 papers present some of the frontier techniques
employedintheareaofnonlinearmodeling.
Levant,Ma,andWoharexaminethecorrelationbetweennominalstockreturns
and inflation. Theory suggests that if stocks are a perfect hedge against inflation,
then real stock returns should, ex-ante, not be correlated with inflation. A large
bodyofempiricalworkhasfoundanegativecorrelationbetweenrealstockreturns
andinflation.Theydevelopastate-spacemodelthatallowsthemtodecomposethe
realizedvalueof realstock returnsandinflationinto their expectedvaluesas well
asnewsshocks.Theyfindthatexpectedreturnsandexpectedinflationratesare,in
general, negatively correlated. In addition, they find that the expected returns and
expected inflation rate appear persistent. However, the small signal-to-noise ratio
implies weak identification and potentially a great deal of uncertainty around the
estimates.Theyillustratehowtoconstructaconfidencebandfortheestimatedstate
variables to account for both the parameter and filter uncertainties based on the
approachsuggestedbyHamilton(1986)andshowthattheresultingtotaluncertainty
issubstantial,inparticularforthereturnsprocess.
Kim and Swanson discuss the use of mixed frequency models and diffusion
approximation methods in the context of prediction. They first discuss recent
specification and estimation methods. Economic time series datasets containing
variablesmeasuredat differentfrequencieshave been used in the macroeconomic
literature. One of the most popular approaches is the mixed data sampling or
MIDAS. An alternative approach involves extracting common factors (called a
diffusion index), where all variables are measured at the same frequency, from
large-scale mixed frequency datasets. The idea is to extract a small number of
“common” factors that drive the dynamics of a set of variables. These kinds
of “factor augmenting forecasting models” have been found to outperform a
number of alternative forecasting models. The existing common factor papers
employ datasets where variables are of the same frequency. Kim and Swanson
allow for the constructionof diffusionindexes(factors) formedusing variablesof
differentfrequencies.Theyask the question whetherthe combinationof diffusion
indexesbasedon mixedfrequenciesproducesimprovedforecasts.Theirempirical
illustration looks at a large-scale dataset and a small mixed frequency dataset in
order to construct diffusion indexes to be used to forecast US unemployment.
They employ both classical principle components-based diffusion indexes and a
combinationofdiffusionindexesandfactorsformedusingsmallmixedfrequency
datasets. Their evidence indicates that mixed frequency-basedforecasting models
yieldimprovementsoverstandardfixedfrequencymodels.
There are many papers that compare tests for linearity. Lee, Xi and Zhang
investigate the artificial neural network model. The artificial neural network that
testsforneglectednonlinearityisaconditionalmomenttestwhosenullhypothesis
consistsofconditionalmomentconditionsthatholdifthelinearmodeliscorrectly
specified for the conditional mean. It differs from other tests by the choice of
the “test function” that is selected to be the artificial neural network’s hidden
viii Preface
layeractivations.Theadvantageofusinganartificialneuralnetworkmodeltotest
nonlinearity is that the model inherits the flexibility as a universal approximator
ofunknownfunctionalform.Astheestimationofartificialneuralnetworkmodels
is difficult, it has been suggested that one could activate the hidden units of the
artificial neural network model through randomly drawn activation parameters.
To robustify the random activations, a large number of activations are desirable.
This leads to a situation in which regularization of dimensionality is needed for
techniques such as principal componentanalysis, Lasso pretest, and Partial Least
Squares, among others. Lee, Xi, and Zhang demonstrate that while supervising
regularization methods may be useful for forecasting, they may not be useful for
testing becausethe supervisingregularizationwill createpost-sampleinferenceor
post-selection inference (PoSI). They employ Monte Carlo simulations and show
that the PoSI problem is especially severe with Partial Least Squares and Pretest,
whileitseemsrelativelymildornegligentwithLasso.Thepaperalsodemonstrates
that the use of principal components does not lead to the PoSI problem. Lee, Xi,
andZhang’sartificialneuralnetworktestusestheresidualsfromalinearmodeland
examinestheir correlationwith the hiddenunit activationfunctionof the artificial
neuralnetwork.
Jones and Enders consider the problems that exist for series that may possess
anunknownnumberofsmoothbreaksinadatageneratingprocess.Evenifbreaks
areabrupt,itisdifficulttoestimatethebreakdatesalongwiththeotherparameters
ofthemodel.Inaddition,itisdifficulttodetermineasmoothbreakifitexists.Itis
quitepossiblethatamisspecificationofthesebreaksmaycausemoreproblemsthan
ignoringthe existenceof the breaksaltogether.The authorssummarizethe results
ofanumberofpapersthatemployavariantoftheflexibleFourierform.Jonesand
Enders illustrate several unit root tests, stationarity tests, and tests for parameter
instabilitythatarebasedonaFourierapproximation.
Morley and Rabah investigate the properties of the Markov Switching model.
WhentestingMarkovSwitchinginmeanorinterceptofanautoregressiveprocess,
it is important to allow for serial correlation under the null of linearity. If this is
not done, then a rejection of linearity could merely reflect misspecification of the
persistence properties of the data, rather than any inherent nonlinearity. Morley
and Rabah conduct Monte Carlo analysis and show that a recently developed
test for MarkovSwitching has low power for empiricallyrelevantdata-generating
processeswhenallowingforserialcorrelationunderthenull.Theauthorsfindthat
aparametricbootstraplikelihoodratiotestforMarkovSwitchinghasmuchhigher
power underthe same conditions. They findthat this parametricbootstrap reveals
strongersupportforaMarkovSwitchingmeaninanapplicationtoanautoregressive
modelofquarterlyUSrealGDP.
AhmadandLoconductnumerousMonteCarlosimulationsinordertoexamine
howaresearchercandistinguishvarioustypesofnonlinearmodels.Thetwomajor
classesofnonlinearmodelstheyinvestigatearethethresholdmodelandtheMarkov
Switching (MS) model. In particular, they first simulate data from the LSTAR,
ESTAR, and MRLSTAR (multiple regime LSTAR) models and then examine the
abilityoftheMSmodeltoapproximatethesesimulateddata.Theyfindthatintheir
Preface ix
benchmarkmodeltheMSmodelcanapproximaterelativelywellthedatasimulated
fromtheLSTARmodelbutnotESTARandMRLSTARmodels,inwhichcasesthe
MSmodelproducessizablebiases.Inanefforttoinvestigatingthispuzzlingresult,
theyfindthatthelowerthepoweroftheLRtestusingthemisspecifiedMSmodel
asanalternative,thelargerthebias.Theythensimulatethedatafromasymmetric
andasymmetricMSmodelandapplythelinearitytestfortheLSTARandESTAR
models to the simulated data. They find that for asymmetric models, the linearity
testhasalowerpower,whilewhenthedifferencesacrossregimesbecomelargerthe
testhasamuchhigherpower.
ChauvetandSu proposea novelMarkovSwitchingmodelto introducea more
flexiblecharacterizationofthepost-warUSrealGDPfluctuation.Thetypeofmodel
theyproposeallowsforaone-timepermanentchangeinbothmeanandvarianceof
the US output that is independent from the other two switching variables, which
capture the short-run business cycle fluctuations of the mean and variance. They
estimate their modelusing the US real GDP data includingthe most recentGreat
Recession periodsand compare their results with the other existing models in the
literature. They find that their model yields interesting findings in terms of both
the long-run changes and the short-run business cycle fluctuations in the output
data. In particular, they find that the average growth rate during expansions has
become much lower after the Great Moderation, while the average growth rate
duringrecessions hasalso becomelower primarilydue to the GreatRecession. In
particular,theyfindthatthevolatilityofUSoutputfluctuationshasbothalong-run
pattern, characterized by a structural break in 1984, and business cycle dynamics
in which high uncertainty states are associated with NBER recessions. They also
find that the dramatic output variance drop after 1984 takes place mainly during
expansionsbutnotasmuchduringrecessions.
Bhatt and Kishor estimate a long-run equilibrium relationship between con-
sumption,disposableincome,andwealthacrossfourcountries:Canada,Germany,
the UK, and the USA. They also examine whether the wealth effect has changed
in these countries over time. Their results indicate that the US and Canadian
economy have become more sensitive to changes in wealth in recent years. They
findthatforeachone-dollarincreaseinwealth,consumptionintheUSAincreases
by 1.8 cents, whereas the corresponding increase in Canada is 2.5 cents. The
response was insignificant for the USA before 1969 and before 1983 in Canada.
ThecorrespondingestimateofthewealtheffectfortheUSAis1centforthepost-
1987timeperiod.TheyfindnowealtheffectinGermanyatanytime.
Kaufmann,Kruse,andSibbertsentacklethechallengingissueofproceduresand
test to distinguish between differenttypes of nonlinearities. They focus on model
selectionbetweensmoothtransitionandthresholdtimeseriesmodels.Theypropose
simple specificationprocedures,basedon standardregressionoutput,to select the
appropriatetransitionfunction.Earlierproceduresaremuchmorecomplicatedand
computer-intensive.Theprocedureisbasedonanauxiliaryregressionofunitroot
tests. It is applicable to a variety of transition functions. In their approach, the
estimationofthecandidatemodelisnotnecessary.Theirapproachreliesentirelyon
OLSestimationofanauxiliaryregression.Theyusestandardinformationcriterion
