Recent Advances in Estimating Nonlinear Models: With Applications in Economics and Finance PDF
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Jun 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 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013947683 ©SpringerScience+BusinessMediaNewYork2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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
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