This paper considers dynamic time series binary choicemodels. It proves near epoch dependence and strongmixing for the dynamic binary choice model withcorrelated errors. Using this result, it shows in atime series setting the validity of the dynamicprobit likelihood procedure when lags of thedependent binary variable are used as regressors,and it establishes the asymptotic validity ofHorowitz’s smoothed maximum score estimation ofdynamic binary choice models with lags of thedependent variable as regressors. For thesemiparametric model, the latent error is explicitlyallowed to be correlated. It turns out that nolong-run variance estimator is needed for thevalidity of the smoothed maximum score procedure inthe dynamic time series framework.

A new class of large-sample covariance and spectraldensity matrix estimators is proposed based on thenotion of flat-top kernels. The new estimators areshown to be higher-order accurate when higher-orderaccuracy is possible. A discussion on kernel choiceis presented as well as a supporting finite-samplesimulation. The problem of spectral estimation undera potential lack of finite fourth moments is alsoaddressed. The higher-order accuracy of flat-topkernel estimators typically comes at the sacrificeof the positive semidefinite property. Nevertheless,we show how a flat-top estimator can be modified tobecome positive semidefinite (even strictly positivedefinite) while maintaining its higher-orderaccuracy. In addition, an easy (and consistent)procedure for optimal bandwidth choice is given;this procedure estimates the optimal bandwidthassociated with each individual element of thetarget matrix, automatically sensing (and adaptingto) the underlying correlation structure.

This paper studies the properties of naive blockbootstrap tests that are scaled by zero frequencyspectral density estimators (long-run varianceestimators). The naive bootstrap is a bootstrapwhere the formula used in the bootstrap world tocompute standard errors is the same as the formulaused on the original data. Simulation evidence showsthat the naive bootstrap can be much more accuratethan the standard normal approximation. The largerthe HAC bandwidth, the greater the improvement. Thisimprovement holds for a large number of popularkernels, including the Bartlett kernel, and it holdswhen the independent and identically distributed(i.i.d.) bootstrap is used and yet the data areserially correlated. Using recently developedfixed-b asymptotics for HACrobust tests, we provide theoretical results thatcan explain the finite sample patterns. We show thatthe block bootstrap, including the special case ofthe i.i.d. bootstrap, has the same limitingdistribution as the fixed-basymptotic distribution. For the special case of alocation model, we provide theoretical results thatsuggest the naive bootstrap can be more accuratethan the standard normal approximation depending onthe choice of the bandwidth and the number of finitemoments in the data. Our theoretical results lay thefoundation for a bootstrap asymptotic theory that isan alternative to the traditional approach based onEdgeworth expansions.

This paper proposes a nonparametric simultaneous testfor parametric specification of the conditional meanand variance functions in a time series regressionmodel. The test is based on an empirical likelihood(EL) statistic that measures the goodness of fitbetween the parametric estimates and thenonparametric kernel estimates of the mean andvariance functions. A unique feature of the test isits ability to distribute natural weightsautomatically between the mean and the variancecomponents of the goodness-of-fit measure. To reducethe dependence of the test on a single pair ofsmoothing bandwidths, we construct an adaptive testby maximizing a standardized version of theempirical likelihood test statistic over a set ofsmoothing bandwidths. The test procedure is based ona bootstrap calibration to the distribution of theempirical likelihood test statistic. We demonstratethat the empirical likelihood test is able todistinguish local alternatives that are differentfrom the null hypothesis at an optimal rate.

New notions of tail and nontail dependence are used tocharacterize separately extremal and nonextremalinformation, including tail log-exceedances andevents, and tail-trimmed levels. We prove that nearepoch dependence (McLeish, 1975; Gallant and White,1988) andL0-approximability(Pötscher and Prucha, 1991) are equivalent for tailevents and tail-trimmed levels, ensuring a Gaussiancentral limit theory for important extreme value androbust statistics under general conditions. We applythe theory to characterize the extremal andnonextremal memory properties of possibly veryheavy-tailed GARCH processes and distributed lags.This in turn is used to verify Gaussian limits fortail index, tail dependence, and tail-trimmed sumsof these data, allowing for Gaussian asymptotics fora new tail-trimmed least squares estimator forheavy-tailed processes.

In this comment we identify a lacuna in a proof in thepaper by M. Caner published in 1997 inEconometric Theory concerning theweak limit behavior of various expressions involvingheavy-tailed multivariate vectors and theconvergence of stochastic integrals. In a laterpaper (Caner, 1998) the results for these limitrelations are used to formulate tests forcointegration with infinite variance errors.

We generalize the cumulative sum of squares (CUSQ) testto the case of nonstationary autoregressivedistributed lag models with deterministic timetrends. The test may be implemented with eitherordinary least squares residuals or standardizedforecast errors. In explosive cases the asymptotictheory applies more generally for the least squaresresiduals-based test. Preliminary simulations of thetests suggest a very modest difference between thetests and a very modest variation with nuisanceparameters. This supports the use of the tests inexplorative analysis.