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Nonlinear Cointegrating Regression Under Weak Identi…cation

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nonlinear29.pdfIntroductionThe Model and Basic AssumptionsNonlinear Least Squares Estimation NLS for Integrable FunctionsNLS for Asymptotically Homogeneous FunctionsConfidence Intervals Confidence Intervals with Integrable FunctionsConfidence Intervals with Asymptotically Homogeneous FunctionsConclusionAuxiliary LemmasProof of the TheoremsProof of the Main LemmasProof of the Auxiliary LemmasNONLINEAR COINTEGRATING REGRESSION UNDER WEAK IDENTIFICATION By Xiaoxia Shi and Peter C. B. Phillips September 2010 COWLES FOUNDATION DISCUSSION PAPER NO. 1768 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/Nonlinear Cointegrating RegressionUnder Weak Identi…cation Xiaoxia ShiyPeter C. B. PhillipszJune, 2010AbstractAn asymptotic theory is developed for a we akly identi…ed cointegrating regressionmodel in which the regressor is a nonlinear transformation of an integrated process.Weak identi…cation arises from the presence of a loading coe¢ cient for the nonlinearfunction that may be close to zero. In that case, standard nonlinear cointegratinglimit theory does not provide good approximations to the …nite sample distributionsof nonlinear least squares estimators, resulting in potentially misleading inference. Anew local limit theory is developed that approximates the …nite sample distributionsof the estimators uniformly well irrespective of the strength of the identi…cation. Animportant technical component of this theory involves new results showing the uniformweak convergence of sample covariances involving nonlinear functions to mixed normaland stochastic integral limits. Based on these asymptotics, we construct con…denceintervals for the loading coe¢ cient and the nonlinear transformation parameter andshow that these con…dence intervals have correct asymptotic size. As in other casesof nonlinear estimation with integrated processes and unlike stationary process as-ymptotics, the properties of the nonlinear transformations a¤ect the asymptotics and,in particular, give rise to parameter dependent rates of convergence and d i¤erencesbetween the limit results for integrable and asymptotically homogeneous functions. This paper originated in a 2008 Yale take home examination. The …rst complete draft was circulatedin December 2009.yYale Un iversity. Support from the Cowles Fo undation via a Carl Arvid Anderson Fellowship is gra te-fully acknowledged. Email: [email protected] Unive rsity, University of Auckland, University of Southampton and Singapore Management Uni-versity. Support from the NSF und er Grant Nos. SES 06 -47086 and SES 09 -56687 is gra tefully acknowl-edged. Email: [email protected]: Integrated process; Local time; Nonlinear regression; Uniform weak con-vergence; Weak identi…cation.JEL classi…cation: C13; C221 IntroductionNonlinear models provide an important means of extending the conventional linear coin-tegrating structures that are now commonly used in applied work. Nonlinearities providea mechanism for controlling and modifying the random wandering characteristics of unitroot time series, leading to a much wider range of possible response functions in regres-sions with such time series. For instance, integrable transformations of integrated timeseries attenuate outliers rather than proportionately transmit their e¤ects as in linear coin-tegrating systems. Transformations of this type are valuable in modeling uneven outputresponses to economic fundamentals such as those that can occur in the p resen ce of marketinterventions or regulatory regimes like exchange rate target zones.Another useful property of nonlinear transformations is that they can modify the char-acteristics of nonstationary series, including their memory attributes. Modi…cations of thistype are helpful in modeling time series like asset retu rns, which have near martingaledi¤erence characteristics, in terms of economic fundamentals that may behave much morelike integrated time series. In such cases, the e¤ects of the stochastic trend in the funda-mentals is su¢ ciently attenuated to be n egligible, except perhaps over long time periodswhere the drift in asset returns becomes perceptible. A useful mechanism for capturingsuch e¤ects is to utilize loading coe¢ cients on the nonlinear response functions that are al-lowed to be local to zero. The cointegrating e¤ects then become “small”and they are onlyweakly identi…ed. This approach gives ‡exibility in modeling the e¤ects of fundamentalson returns and o¤ers the p otential for improvements over linear models in predicting assetreturns using near integrated predictor processes , whose role has recently been emphasizedin the work of Campbell and Yogo (2006) and others.The goal of the present paper is to deal with such formulations and develop an asymp-totic theory that retains its validity for small cointegrating e¤ects. In particular, we studynonlinear cointegration models of the following formYt= g(Xt; ) + ut, (1.1)2where Xtis an I(1) process, Ytis a dependent variable, not necessarily I(1), utis anerror term (to be speci…ed more precisely later), g(x; ) is a nonlinear transformation of xwhose form is known up to a parameter , and  is a loading coe¢ cient that measures theimportance of the nonlinear regression e¤ect.Models like (1.1) have the attractive feature that they can relate processes of di¤erentintegration orders. As intimated above, this feature may be especially appealing in mod-eling and predicting stock market returns. Stock returns commonly behave as martingaledi¤erences, while the variables that are used in prediction are often I(1), as d iscu sse d inMarmer (2008), leading to a potential imbalance in a regression formulation. Accordingly,any relationship between stock return levels and stochastic trend predictors is inevitablyweak bec ause of the e¢ ciency of modern stock markets. In terms of the model (1.1), thisconsideration may be captured for a wide class of possible regression functions simply bypermitting the true value of the loading coe¢ cient to be close to zero. To develop an or-derly asymptotic theory that accommodates this possibility, the model may be formulatedto allow the true parameter, n; to drift to zero as the sample size n ! 1: Then, if Ytdenotes stock returns and Xtdenotes an I(1) regressor embodying economic fundamentals,the


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