Nonstationary Density Estimation and Kernel Autoregression

An asymptotic theory is developed for the kernel density estimate of a random walk and the kernel regression estimator of a nonstationary first order autoregression. The kernel density estimator provides a consistent estimate of the local time spent by the random walk in the spatial vicinity of a point that is determined in part by the argument of the density and in part by initial conditions. The kernel regression estimator is shown to be consistent and to have a mixed normal limit theory. The limit distribution has a mixing variate that is given by the reciprocal of the local time of a standard Brownian motion. The permissible range for the bandwidth parameter h_{n} includes rates which may increase as well as decrease with the sample size n, in contrast to the case of a stationary autoregression. However, the convergence rate of the kernel regression estimator is at most n^{1/4}, and this is slower than that of a stationary kernel autoregression, in contrast to the parametric case. In spite of these differences in the limit theory and the rates of convergence between the stationary and nonstationary cases, it is shown that the usual formulae for confidence intervals for the regression function still apply when h_{n} -> 0.Brownian sheet, kernel regression, local time, martingale embedding, mixture normal, nonstationary density, occupation time, quadratic variation, unit root autoregression

Nonstationary Binary Choice

This paper develops an asymptotic theory for time series binary choice models with nonstationary explanatory variables generated as integrated processes. Both logit and probit models are covered. The maximum likelihood (ML) estimator is consistent but a new phenomenon arises in its limit distribution theory. The estimator consists of a mixture of two components, one of which is parallel to and the other orthogonal to the direction of the true parameter vector, with the latter being the principal component. The ML estimator is shown to converge at a rate of n^{3/4} along its principal component but has the slower rate of n^{1/4} convergence in all other directions. This is the first instance known to the authors of multiple convergence rates in models where the regressors have the same (full rank) stochastic order and where the parameters appear in linear forms of these regressors. It is a consequence of the fact that the estimating equations involve nonlinear integrable transformations of linear forms of integrated processes as well as polynomials in these processes, and the asymptotic behavior of these elements are quite different. The limit distribution of the ML estimator is derived and is shown to be a mixture of two mixed normal distributions with mixing variates that are dependent upon Brownian local time as well as Brownian motion. It is further shown that the sample proportion of binary choices follows an arc sine law and therefore spends most of its time in the neighbourhood of zero or unity. The result has implications for policy decision making that involves binary choices and where the decisions depend on economic fundamentals that involve stochastic trends. Our limit theory shows that, in such conditions, policy is likely to manifest streams of little intervention or intensive intervention.Binary choice model, Brownian motion, Brownian local time, dual convergence rates, Integrated time series, maximum likelihood estimation

Nonlinear Econometric Models with Cointegrated and Deterministically Trending Regressors

This paper develops an asymptotic theory for a general class of nonlinear nonstationary regressions, extending earlier work by Phillips and Hansen (1990) on linear cointegrating regressions. The model considered accommodates a linear time trend and stationary regressors, as well as multiple I(1) regressors. We establish consistency and derive the limit distribution of the nonlinear least squares estimator. The estimator is consistent under fairly general conditions but the convergence rate and the limiting distribution are critically dependent upon the type of the regression function. For integrable regression functions, the parameter estimates converge at a reduced n^{1/4} rate and have mixed normal limit distributions. On the other hand, if the regression functions are homogeneous at infinity, the convergence rates are determined by the degree of the asymptotic homogeneity and the limit distributions are non-Gaussian. It is shown that nonlinear least squares generally yields inefficient estimators and invalid tests, just as in linear nonstationary regressions. The paper proposes a methodology to overcome such difficulties. The approach is simple to implement, produces efficient estimates and leads to tests that are asymptotically chi-square. It is implemented in empirical applications in much the same way as the fully modified estimator of Phillips and Hansen.Nonlinear regressions, integrated time series, nonlinear least squares, Brownian motion, Brownian local time

Asymptotic Equivalence of OLS and GLS in Regressions with Integrated Regressors

In the multiple regression model y t = x’ t β + u t where { u t } is stationary and x t is an integrated m -vector process it is shown that the asymptotic distributions of the ordinary least squares (OLS) and generalized least squares (GLS) estimators of β are identical. This generalizes a recent result obtained by Kramer (1986) for simple two variate regression. Our approach makes use of a multivariate invariance principle and yields explicit representations of the asymptotic distributions in terms of fuctionals of vector Brownian motion. Some useful assumption results for hypothesis tests in the model are also provided

Statistical Inference in Regressions with Integrated Processes: Part 2

This paper continues the theoretical investigation of Park and Phillips [7]. We develop an asymptotic theory of regression for multivariate linear models that accommodates integrated processes of different orders, nonzero means, drifts, time trends and cointegrated regressors. The framework of analysis is general but has a common architecture that helps to simplify and codify what would otherwise be a myriad of isolated results. A good deal of earlier research by the authors and by others comes within the new framework. Special models of some importance are considered in detail, such as VAR systems with multiple lags and cointegrated variants

On the Formulation of Wald Tests of Nonlinear Restrictions

This paper utilizes asymptotic expansions to investigate alternative forms of the Ward set of nonlinear restrictions. Some formulae for the asymptotic expansion of the distribution of the Wald statistic are provided for a general case. When specialized to the simple cases that have been studied recently in the literature, these formulae are found to explain rather well the discrepancies in sampling behavior that have been observed by other authors. It is further shown how the correction delivered by the Edgeworth expansion may be used to find transformations of the restrictions which accelerate convergence to the asymptotic distribution

Probing the two light Higgs scenario in the NMSSM with a low-mass pseudoscalar

In this article we propose a simultaneous collider search strategy for a pair of scalar bosons in the NMSSM through the decays of a very light pseudoscalar. The massive scalar has a mass around 126 GeV while the lighter one can have a mass in the vicinity of 98 GeV (thus explaining an apparent LEP excess) or be much lighter. The successive decay of this scalar pair into two light pseudoscalars, followed by leptonic pseudoscalar decays, produces clean multi-lepton final states with small or no missing energy. Furthermore, this analysis offers an alternate leptonic probe for the 126 GeV scalar that can be comparable with the ZZ* search channel. We emphasize that a dedicated experimental search for multi-lepton final states can be an useful probe for this scenario and, in general, for the NMSSM Higgs sector. We illustrate our analysis with two representative benchmark points and show how the LHC configuration with 8 TeV center-of-mass energy and 25 fb−1 of integrated luminosity can start testing this scenario, providing a good determination of the light pseudoscalar mass and a relatively good estimation of the lightest scalar mass

Statistical Inference in Regressions with Integrated Processes: Part 1

This paper develops a multivariate regression theory for integrated processes which simplifies and extends much earlier work. Our framework allows for both stochastic and certain deterministic regressors, vector autoregressions and regressors with drift. The main focus of the paper is statistical inference. The presence of nuisance parameters in the asymptotic distributions of regression F -tests is explored and new transformations are introduced to deal with these dependencies. Some specializations of our theory are considered in detail. In models with strictly exogenous regressors we demonstrate the validity of conventional asymptotic theory for appropriately constructed Wald tests. These tests provide a simple and convenient basis for specification robust inferences in this context. Single equation regression tests are also studied in detail. Here it is shown that the asymptotic distribution of the Wald test is a mixture of the chi square of conventional regression theory and the standard unit root theory. The new result accommodates both extremes and intermediate cases

A Layered-Manufacturing Process For the Fabrication of Glass-Fiber-Reinforced Composites

In this paper, we present a rapid manufacturing process for the layered fabrication of polymer-based composite parts using short discontinuous fibers as reinforcements. In the recent past, numerous research efforts, similar to ours, have been made to produce fiber-reinforced plastic parts via layered manufacturing methods. However, most of these attempts have not resulted in the development of an effective commercially-viable manufacturing process. Our proposed fabrication process on the other hand has been experimentally verified to yield composite parts comparable in quality to pure polymer parts manufactured on a commercial stereolithography system.Mechanical Engineerin

Testing for a Unit Root in the Presence of a Maintained Trend

This paper develops statistics for detecting the presence of a unit root in time series data against the alternative stationarity. Unlike most existing procedures, the new tests allow for deterministic trend polynomials in the maintained hypothesis. They may be used to discriminate between unit root nonstationarity and processes which are stationary around a deterministic polynomial trend. The tests allow for both forms of nonstationarity under the null hypothesis. Moreover, the tests allow for a wide class of weakly dependent and possibly heterogenously distributed procedures. We illustrate the use of the new tests by applying them to a number a models of macroeconomic behavior