Asymptotic theory for the estimation of nonlinearvector error correction models that exhibitregime-specific short-run dynamics is developed. Inparticular, regimes are determined by the errorcorrection term, and the transition between regimesis allowed to be discontinuous, as in, e.g.,threshold cointegration. Several nonregular problemsare resolved. First of all, consistency—square rootn consistency for thecointegrating vector β—isestablished for the least squares estimation of thisgeneral class of models. Second, the convergencerates are obtained for the least squares ofthreshold cointegration, which aren3/2 andn for β andγ, respectively, whereγ denotes the thresholdparameter. This fast rate for β initself is of practical relevance because, unlike insmooth transition models, the estimation error inβ does not affect the estimationof short-run parameters. We also derive asymptoticdistributions for the smoothed least squaresestimation of threshold cointegration.

A local limit theorem is given for the sample mean of azero energy function of a nonstationary time seriesinvolving twin numerical sequences that pass toinfinity. The result is applicable in certainnonparametric kernel density estimation andregression problems where the relevant quantitiesare functions of both sample size and bandwidth. Aninteresting outcome of the theory in nonparametricregression is that the linear term is eliminatedfrom the asymptotic bias. In consequence and incontrast to the stationary case, the Nadaraya–Watsonestimator has the same limit distribution (to thesecond order including bias) as the local linearnonparametric estimator.

This paper proposes a model specification testingprocedure for parametric specification of theconditional mean function in a nonlinear time seriesmodel with long-range dependent. An asymptoticallynormal test is established even when long-rangedependent is involved. To implement the proposedtest in practice using a simulated example, abootstrap simulation procedure is established tofind a simulated critical value to compute both thesize and power values of the proposed test.

We examine the limit properties of the nonlinear leastsquares (NLS) estimator under functional formmisspecification in regression models with a unitroot. Our theoretical framework is the same as thatof Park and Phillips (2001,Econometrica 69, 117–161). Weshow that the limit behavior of the NLS estimator islargely determined by the relative orders ofmagnitude of the true and fitted models. If theestimated model is of different order of magnitudethan the true model, the estimator converges toboundary points. When the pseudo-true value is on aboundary, standard methods for obtaining rates ofconvergence and limit distribution results are notapplicable. We provide convergence rates and limittheory when the pseudo-true value is an interiorpoint. If functional form misspecification iscommitted in the presence of stochastic trends, theconvergence rates can be slower and the limitdistribution different than that obtained undercorrect specification.

Testing for white noise has been well studied in theliterature of econometrics and statistics. For mostof the proposed test statistics, such as thewell-known Box–Pierce test statistic with fixed lagtruncation number, the asymptotic null distributionsare obtained under independent and identicallydistributed assumptions and may not be valid fordependent white noise. Because of recent popularityof conditional heteroskedastic models (e.g.,generalized autoregressive conditionalheteroskedastic [GARCH] models), which implynonlinear dependence with zero autocorrelation,there is a need to understand the asymptoticproperties of the existing test statistics underunknown dependence. In this paper, we show that theasymptotic null distribution of the Box–Pierce teststatistic with general weights still holds underunknown weak dependence as long as the lagtruncation number grows at an appropriate rate withincreasing sample size. Further applications todiagnostic checking of the autoregressive movingaverage (ARMA) and fractional autoregressiveintegrated moving average (FARIMA) models withdependent white noise errors are also addressed. Ourresults go beyond earlier ones by allowingnon-Gaussian and conditional heteroskedastic errorsin the ARMA and FARIMA models and providetheoretical support for some empirical findingsreported in the literature.

A multivariate extension of the exponential continuoustime GARCH (p, q)model (ECOGARCH) is introduced and studied.Stationarity and mixing properties of the newstochastic volatility model are investigated, andways to model a component-wise leverage effect arepresented.

Asymptotically pivotal structural change tests aredeveloped for simultaneous equations with weaklyidentified parameters, extending the boundedlypivotal tests of Caner (2007, Journal ofEconometrics 137, 28–67) by means of asimple reparametrization of the model.

This paper studies the asymptotic validity of theAnderson–Rubin (AR) test and theJ test for overidentifyingrestrictions in linear models with many instruments.When the number of instruments increases at the samerate as the sample size, we establish that theconventional AR andJ tests are asymptoticallyincorrect. Some versions of these tests, which aredeveloped for situations with moderately manyinstruments, are also shown to be asymptoticallyinvalid in this framework. We propose modificationsof the AR and Jtests that deliver asymptotically correct sizes.Importantly, the corrected tests are robust to thenumerosity of the moment conditions in the sensethat they are valid for both few and manyinstruments. The simulation results illustrate theexcellent properties of the proposed tests.

We analyze the limiting distribution of the Rivers andVuong (2002, Econometrics Journal5, 1–39) statistic for choosing between twocompeting dynamic models based on a comparison ofgeneralized method of moments minimands. It is shownthat (i) if both models are misspecified then thestatistic has a standard normal distribution underthe null hypothesis of equal fit but the rankingcould be determined by the choice of the weightingmatrix; (ii) if both models are correctly specifiedor locally misspecified then the limitingdistribution of the test statistic is nonstandardunder the null.