Markov regime-switching tests: asymptotic critical values. (2013) Journal of Econometric Methods 2(1): pp.25-34 (with Doug Steigerwald)

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Supplemental Materials

Empirical research with Markov regime-switching models often requires the researcher not only to estimate the model but also to test for the presence of more than one regime. Despite the need for both estimation and testing, methods of estimation are better understood than are methods of testing. We bridge this gap by explaining, in detail, how to apply the newest results in the theory of regime testing, developed by Cho and White. A key insight in Cho and White is to expand the null region to guard against false rejection of the null hypothesis due to a small group of extremal values. Because the resulting asymptotic null distribution is a function of a Gaussian process, the critical values are not obtained from a closed-form distribution such as the chi-squared. Moreover, the critical values depend on the covariance of the Gaussian process and so depend both on the specification of the model and the specification of the parameter space. To ease the task of calculating critical values, we describe the limit theory and detail how the covariance of the Gaussian process is linked to the specification of both the model and the parameter space. Further, we show that for linear models with Gaussian errors, the relevant parameter space governs a standardized index of regime separation, so one need only refer to the tabulated critical values we present. While the test statistic under study is designed to detect regime switching in the intercept, the test can be used to detect broader alternatives in which slope coefficients and error variances may also switch over regimes.

Asymptotic sufficient statistics in nonparametric regression experiments with correlated noise. (2009) Journal of Probability and Statistics. vol. 2009, Article ID 275308

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We find asymptotically sufficient statistics that could help simplify inference in nonparametric regression problems with correlated errors. These statistics are derived from a wavelet decomposition that is used to whiten the noise process and to effectively separate high-resolution and low-resolution components. The lower-resolution components contain nearly all the available information about the mean function, and the higher-resolution components can be used to estimate the error covariances. The strength of the correlation among the errors is related to the speed at which the variance of the higher-resolution components shrinks, and this is considered an additional nuisance parameter in the model. We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.

Asymptotic equivalence for nonparametric regression experiments with random design. Proceedings of the 2007 Joint Statistical Meetings. (2007)

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In a nonparametric regression problem where the
design points are randomly distributed with an unknown distribution,
there are asymptotically sufficient statistics in
the averages of the data over small subintervals. These statistics can
be used to construct a continuous
Gaussian process experiment that is asymptotically equivalent to the
regression experiment.

Asymptotic approximation of nonparametric regression experiments with unknown variances. (2007) Annals of Statistics. 35 pp. 1644-1673.

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Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean. The result can be extended to regression problems where the variance varies slowly from observation to observation.

A continuous Gaussian process approximation to a nonparametric regression in two dimensions. (2006). Bernoulli. 12(1) pp. 143-156.

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Estimating the mean in a nonparametric regression on a two dimensional regular grid of design points is asymptotically equivalent to estimating the drift of a
continuous Gaussian process on the unit square. In particular, we provide a construction of a Brownian sheet process with a drift that is almost the mean function in the nonparametric regression. This can be used to apply estimation or testing procedures from the continuous process to the regression experiment as in Le Cam's theory of equivalent experiments. Our result is motivated by first looking at the amount of information lost in binning the data in a density estimation problem.

Tusnády's inequality revisited.(2004) *Annals of Statistics*. **32**
pp. 2731-2741 (with David Pollard).

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Tusnády's inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian Bridge. We present an elementary proof of a result that sharpens the Tusnády's inequality, modulo constants. Our method uses the beta integral representation of Binomial tails, simple Taylor expansion, and some novel bounds for the ratios of normal tail probabilities.

Equivalence theory for density estimation, Poisson processes, and Gaussian white noise with drift. (2004) *Annals of Statistics*. **32**
pp. 2074-2097 (with Lawrence Brown, Mark Low, and Cun-Hui Zhang).

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This paper establishes the global asymptotic equivalence between a Poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function. This equivalence is also extended to density estimation models. The asymptotic equivalence is established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that an investigation in one of these nonparametric models automatically yields asymptotically analogous results in the other model.

Deficiency distance between multinomial and multivariate normal experiments.(2002) *Annals
of Statistics*. **30** pp. 708-730.

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The deficiency distance between a multinomial and a multivariate normal experiment is bounded under a condition that the parameters are bounded away from zero.

This result can be used as a key step in establishing asymptotic normal approximations to nonparametric density estimation experiments.

The bound relies on the recursive construction of explicit Markov kernels that can be used to reproduce one experiment from the other. The distance is then bounded using classic local-limit bounds between binomial and normal distributions. Some extensions to other appropriate normal experiments are also presented.