General Asymptotic Theory Research Articles

Trending Multiple Time Series: Editor's Introduction

One of the more obvious empirical characteristics of macroeconomic time series is their tendency to grow, or trend, over time. Dealing with this trendnonstationarity in models of multiple time series has been a major agenda of econometric research for much of the last decade and has produced an enormous literature. Equally, the goal of developing a general asymptotic theory of inference for stochastic processes has been a long-standing concern of probabilists and statisticians. Finally, understanding and modeling trend processes and cyclical activity lie at the nerve center of much of modern macroeconomics. As a consequence, research on nonstationary time series has brought statisticians, econometricians, and macroeconomists close together in productive ways that simply could not have been anticipated 10 years ago.The focus of this symposium issue of Econometric Theory is inference from multiple time series data with trends, and the symposium brings together researchers with these diverse interests. The papers included in the issue were, with two exceptions, presented at a conference called “Trending Multiple Time Series,” held at Yale University in the fall of 1993 under the financial sponsorship of the National Science Foundation. All of the papers were written by conference participants. The conference was the fourth in a series of small conferences at Yale on the general theme of “Applications of Functional Limit Theory to Econometrics and Statistics.”

Asymptotic Normality of a Class of Adaptive Statistics with Applications to Synthetic Data Methods for Censored Regression

Motivated by regression analysis of censored survival data, we develop herein a general asymptotic distribution theory for estimators defined by estimating equations of the form ∑ni=1ξ (wi, θ, Ĝn) = 0, in which wi represents observed data, θ is an unknown parameter to be estimated, and Ĝn represents an estimate of some unknown underlying distribution. This general theory is used to establish asymptotic normality of synthetic least squares estimates in censored regression models and to evaluate the covariance matrices of the limiting normal distributions.

Conditional Rank Tests for the Two-Sample Problem Under Random Censorship

For the two-sample problem with randomly censored data, there exists a general asymptotic theory of rank statistics which are functionals of stochastic integrals with respect to certain empirical martingales. In the present paper a conditional counterpart of this theory is developed. The conditional martingales are versions of the original ones reduced to the unit interval having their jumps at fixed lattice points. The resulting conditional tests are strictly distribution free under the null hypothesis of randomness if the censoring distributions in both samples are equal and are asymptotically equivalent to their unconditional counterparts even if the censoring distributions are different. Simulations for linear rank statistics and Kolmogorov-Smirnov-type statistics show superiority of the conditional versions over their unconditional counterparts with respect to size and robustness under unequal censoring in both samples. At the same time the power of the conditional and unconditional tests is very similar in most cases.

Acoustic wave scattering from thin shell structures.

The response of heavily fluid-loaded thin shells in the mid-frequency regime is studied using a general asymptotic theory. The technique employs the ideas of matched asymptotic expansions and splits the total shell response into the outer or background component and the inner or resonant contribution. This formulation is applicable to thins shells with varying material and geometrical properties. The background field satisfies an impedance condition that does not correspond to either the so-called soft or rigid conditions. The approach is to expand both the outer and inner fields in inverse powers of 1/kR, where R is a typical radius of curvature of the shell. The leading term in these expansions is selected so as to reflect the dominance of the tangential motion near a compressional resonance and the stronger radial motions in the outer region. Also derived are uniform asymptotic representations. The asymptotic method is applied here to study scattering from various nonseparable two-dimensional geometries. The numerical procedure is to first compute the outer solution, which requires solving for a particularly simple impedance condition on the surface. Then the inner resonances are added to give the total response. The numerical method employed in finding the outer solution is MOOT or the method of optimal truncation. Comparisons are made with the solution to the full set of coupled equations. [Work supported by ONR.]

Acoustic wave scattering from thin shell structures

A general asymptotic theory is developed to describe the acoustic response of heavily fluid-loaded thin shells in the midfrequency regime between the ring and coincidence frequencies. The method employs the ideas of matched asymptotic expansions and represents the total response as the sum of an outer, or background response, plus an inner or resonant contribution. The theory is developed for thin shells with smoothly varying material and geometrical properties. First, a suitable background field is found which satisfies neither the rigid nor the soft boundary conditions that have been typically employed, but corresponds to an impedance boundary condition. The background field is effective throughout the midfrequency as well as the strong bending regimes. The corresponding inner or resonance field is also valid in the same range. The approach taken is to represent these fields as inverse power series in the asymptotically small parameter 1/kR, where R is a typical radius of curvature of the shell and k is the fluid wave number. The leading-order terms in the series differ in the inner and outer expansions, in such a way that the displacement tangential to the surface is negligible in the outer (background) region, but dominates the scattering near resonances. The resonances can therefore be associated with compressional and shear waves in the shell. A uniform asymptotic solution is derived from the combined outer and inner fields. Numerical results are presented for the circular cylinder and the sphere and comparisons are made with exact results for these canonical geometries. The results indicate that the method is particularly effective in the midfrequency range. The strong bending regime is also well represented, especially for cylindrical scatterers.

Parameter-based asymptotics

In certain cases statistical methods based on standard maximum likelihood asymptotics become valid as the true parameter value approaches a boundary of the parameter space. Examples are given which motivate a general parameter-based asymptotic theory, and a result is obtained which covers such situations. Of particular interest are applications to stochastic process models.

Spectral based testing of the martingale hypothesis

This paper proposes a method of testing whether a time series is a martingale. A general asymptotic theory is developed for the spectral distribution function of the first differences. Under the null hypothesis, the spectral distribution function is shaped as a straight line. Several tests are developed which determine whether the sample spectral distribution function possesses this shape. These tests are consistent against all MA alternatives. Additional tests are developed to analyze subsets of frequencies, which can enhance power against particular alternatives. Application of the test to stock prices finds evidence against the random walk theory.

The theory of magnetohydrodynamic wave generation by localized sources. I - General asymptotic theory

The magnetohydrodynamic wave emission from several localized, periodic, kinematically specified fluid velocity fields are calculated using Lighthill's method for finding the far-field wave forms. The waves propagate through an isothermal and uniform plasma with a constant B field. General properties of the energy flux are illustrated with models of pulsating flux tubes and convective rolls. Interference theory from geometrical optics is used to find the direction of minimum fast-wave emission from multipole sources and slow-wave emission from discontinuous sources. The distribution of total flux in fast and slow waves varies with the ratios of the source dimensions l to the acoustic and Alfven wavelengths.

Testing that a Gaussian Process is Stationary

A class of procedures is proposed for testing the stationarity of a Gaussian process or the homogeneity of independent processes. Requiring very limited prior knowledge of model structure, the methods can detect changes or differences in mean, in variance, in covariances and even in law. Although the theory of the stationarity test is worked out only for processes whose realizations are stationary over "epochs" separated by known change points, Monte Carlo evidence indicates that it can be useful also in detecting more general forms of nonstationarity. The test statistic is a quadratic form in differences among epoch means of certain "sensing" functions, the choice of which governs sensitivity to specific forms of nonstationarity or inhomogeneity. The applicability of the general asymptotic theory of the test is verified for two specific forms of sensing function, and small-sample properties of tests of each form are studied by means of simulation.

A Nonlinear Analysis on the Behavior of a Small Amount of Inert Gas during Evaporation and Condensation of a Vapor between the Parallel Surfaces

The behavior of an inert gas contained slightly in a vapor is studied analytically together with the behavior of the vapor itself, when the evaporation and condensation process is taking place between the two parallel plane condensed phases. The analysis is based on a general asymptotic theory for small Knudsen numbers which has been derived systematically from the kinetic equation of B-G-K type under the diffusive boundary condition. The results obtained here are valid for finite values of the Reynolds number based on the flow due to evaporation and condensation. The inert gas, in this case, does not affect the behavior of the vapor because of its smallness in its amount, it is greatly affected by the vapor motion and exhibits strong nonlinearity in its pressure and density variations. Nonlinear effects due to the presence of the flow also manifest themselves distinctly on the temperature of the vapor. The well-known negative temperature gradient and negative mass flow phenomena are also observed.

L-Estimation for Linear Models

Abstract Linear combinations of order statistics, or L-estimators, have played an extremely important role in the development of robust methods for the one-sample problem. We suggest analogs of L-estimators for the parameters of the linear model based on the p-dimensional “regression quantiles” proposed by Koenker and Bassett (1978). A uniform, Bahadur-type asymptotic representation of regression quantiles is established, and this yields a general asymptotic theory of L-estimators for the linear model. A leading example of the proposed estimators is an analog of the trimmed mean (TRQ), which is asymptotically equivalent to the trimmed least squares estimator studied by Ruppert and Carroll (1980), but appears to be somewhat less sensitive to influential design observations. This estimator is also asymptotically equivalent to the well-known Huber M-estimator, but offers the significant advantage that it is scale invariant. We illustrate the methods by reconsidering a mid-18th century linear model analyzed b...

Multiple Time Series Regression with Integrated Processes

This paper develops a general asymptotic theory of regression for processes which are integrated of order one. The theory includes vector autoregressions and multivariate regressions amongst integrated processes that are driven by innovation sequences which allow for a wide class of weak dependence and heterogeneity. The models studied cover cointegrated systems such as those advanced recently by Granger and Engle and quite general linear simultaneous equations systems with contemporaneous regressor error correlation and serially correlated errors. Problems of statistical testing in vector autoregressions and multivariate regressions with integrated processes are also studied. It is shown that the asympotic theory for conventional tests involves major departures from classical theory and raises new and important issues of the presence of nuisance parameters in the limiting distribution theory. Unlike many of the time series encountered in the natural sciences, economic time series frequently exhibit characteristics that are widely believed to be intrinsically nonstationary. For example, real macroeconomic variables such as output and consumption typically display a strong secular or growth component as well as cyclical behaviour; and many financial series like common stock prices behave in general as if they had no fixed mean. Recognizing these typical characteristics of economic time series, econometricians have devoted attention to the problem of describing and modelling nonstationarity. In the 1960's important contributions in the area were made by Granger, Hatanaka and their associates in Granger and Hatanaka (1964), Granger and Morgenstern (1963), Brillinger and Hatanaka (1968) and Hatanaka and Suzuki (1967). Later, following the influential work of Box and Jenkins (1976), attention shifted to the role of integrated processes in modelling economic time series. While undoubtedly restricting the class of nonstationary models, integrated processes of the ARIMA type have been found to produce highly satisfactory representations of many observed time series in economics. Quite recently, Nelson and Plosser (1982) have published a detailed empirical study of historical economic time series for the U.S.A. These authors provide some convincing evidence that macroeconomic time series normally thought to be stationary about a time trend are better described as integrated processes with drift. Amongst the latest research in this field have been the studies of cointegration by Granger and Weiss (1983) and Granger and Engle (1985). Two time series are said to be cointegrated if some linear combination of the series has a lower order of integration than the individual series. These authors argue that the notion of (steady state) equilibrium in economics implies the existence of such relationships. Thus, a classical economist's view of the interaction of money growth and price movements would require these series

Theory of Partial Likelihood

A general asymptotic theory is developed for the maximum likelihood estimator based on a partial likelihood. Conditions are given for consistency and asymptotic normality, and a method is provided for the calculation of the asymptotic efficiency of the estimator. The implications of the general theory are examined in special cases such as inference in stochastic processes, Cox regression models, and AR processes with missing segments.

Maximum likelihood methods for complex sample data: logistic regression and discrete proportional hazards models

To estimate model parameters from complex sample data. we apply maximum likelihood techniques to the complex sample data from the finite population, which is treated as a sample from an i nfinite superpopulation. General asymptotic distribution theory is developed and then applied to both logistic regression and discrete proportional hazards models. Data from the Lipid Research Clinics Program areused to illustrate each model, demonstrating the effects on inference of neglecting the sampling design during parameter estimation. These empirical results also shed light on the issue of model-based vs. design-based inferences.

Asymptotic spacings theory with applications to the two‐sample problem

AbstractThe asymptotic distribution theory of test statistics which are functions of spacings is studied here. Distribution theory under appropriate close alternatives is also derived and used to find the locally most powerful spacing tests. For the two‐sample problem, which is to test if two independent samples are from the same population, test statistics which are based on “spacing‐frequencies” (i.e., the numbers of observations of one sample which fall in between the spacings made by the other sample) are utilized. The general asymptotic distribution theory of such statistics is studied both under the null hypothesis and under a sequence of close alternatives.

Adaptive Design and Stochastic Approximation

When y= M(x) + e, where M may be nonlinear, adaptive stochastic approximation schemes for the choice of the levels x 1, x 2, ··· at which y 1, y 2, ··· are observed lead to asymptotically efficient estimates of the value θ of x for which M(θ) is equal to some desired value. More importantly, these schemes make the “cost” of the observations, defined at the nth stage to be Σ 1 n (x i - θ)2, to be of the order of log n instead of n, an obvious advantage in many applications. A general asymptotic theory is developed which includes these adaptive designs and the classical stochastic approximation schemes as special cases. Motivated by the cost considerations, some improvements are made in the pairwise sampling stochastic approximation scheme of Venter.

The theory of ordered spans of unrestricted random walks

The spans of ann-step random walk on a simple cubic lattice are the sides of the smallest rectangular box, with sides parallel to the coordinate axes, that contains the random walk. Daniels first developed the theory in outline and derived results for the simple random walk on a line. We show that the development of a more general asymptotic theory is facilitated by introducing the spectral representation of step probabilities. This allows us to consider the probability density for spans of random walks in which all moments of single steps may be infinite. The theory can also be extended to continuous-time random walks. We also show that the use of Abelian summation simplifies calculation of the moments. In particular we derive expressions for the span distributions of random walks (in one dimension) with single step transition probabilities of the formP(j) ∼ 1/j1+α, where 0<α<2. We also derive results for continuous-time random walks in which the expected time between steps may be infinite.

Asymptotic theory of scattering by a rough surface progressing over an inhomogeneous ocean

A general asymptotic theory of scattering by a moving rough surface is presented. The theory is valid for a slowly varying refractive index, and for ocean-surface wave heights that are small compared with the acoustic wavelength and the ocean-surface correlation length. In contrast with the Kirchhoff or physical optics approximation, this theory is valid when the acoustic wavelength and ocean-surface correlation length are of the same order. The ocean surface is described as a sum of gravity waves of varying frequency progressing in different directions. It is demonstrated that, to a given order of approximation in the asymptotic theory, the average energy associated with signal sideband rays is precisely accounted for by a reduction of intensity of the reflected carrier. Wave number diagrams are introduced which are extremely useful for depicting the directions of the incident and scatteres rays at the surface, demonstrating the effect of varying surface-wave and acoustic frequency, and demonstrating the frequency cutoff of the sideband power spectrum. The phenomenon of Bragg resonance at glancing incidence of the carrier is also easily depicted. The theory can accomodate the effects of ray focusing, both in the carrier and sidebands, through the use of uniformly valid asymptotic expansions which account for the presence of caustics. Subject Classification: [43]30.40, [43]30.20, [43]30.25, [43]30.30.

Steady Radiating Gas Flow Past a Semi-Infinite Flat Plate at a Constant Temperature for an Optically Thick Case

A general asymptotic theory of a radiating gas which has been developed by the same author is applied to the flow past a semi-infinite flat plate at a constant temperature. It is assumed that the global optical thickness of the gas is large and that the convective heat in the flow is of the same order of magnitude as the radiative one. The gas is assumed to have constant properties. The general theory shows that the flow field is composed of an isentropic outer region, a diffusive (Rosseland type) middle layer and a radiative inner layer. It is also shown that the solutions in the inner layer are described by linear combinations as coefficients. Similar solutions of universal functions by using specific values of the middle layer solutions are numerically obtained in the middle layer by making a boundary layer treatment. Solutions in the outer region are obtained for a subsonic flow. Actual analyses are carried out up to the first two orders of expansion with respect to the inverse square root of the glob...

Asymptotic Behavior of Diffusion of Tangential Velocity Discontinuity in Rarefied Gas

A treatment of the diffusion of a tangential discontinuity in flow velocity is given for the linearized Boltzmann-Krook equation, with emphasis on the asymptotic behavior for large values of time. The connection with the general asymptotic theory of Grad is also discussed.