Properties of the Sieve Bootstrap for Fractionally Integrated and Non-Invertible Processes

In this paper we will investigate the consequences of applying the sieve bootstrap under regularity conditions that are sufficiently general to encompass both fractionally integrated and non-invertible processes. The sieve bootstrap is obtained by approximating the data generating process by an autoregression whose order h increases with the sample size T. The sieve bootstrap may be particularly useful in the analysis of fractionally integrated processes since the statistics of interest can often be non-pivotal with distributions that depend on the fractional index d. The validity of the sieve bootstrap is established and it is shown that when the sieve bootstrap is used to approximate the distribution of a general class of statistics admitting an Edgeworth expansion then the error rate achieved is of order O (T  β+d-1 ), for any β > 0. Practical implementation of the sieve bootstrap is considered and the results are illustrated using a canonical example.Autoregressive approximation, fractional process, non-invertibility, rate of convergence, sieve bootstrap.

On The Identification and Estimation of Partially Nonstationary ARMAX Systems

This paper extends current theory on the identification and estimation of vector time series models to nonstationary processes. It examines the structure of dynamic simultaneous equations systems or ARMAX processes that start from a given set of initial conditions and evolve over a given, possibly infinite, future time horizon. The analysis proceeds by deriving the echelon canonical form for such processes. The results are obtained by amalgamating ideas from the theory of stochastic difference equations with adaptations of the Kronecker index theory of dynamic systems. An extension of these results to the analysis of unit-root, partially nonstationary (cointegrated) time series models is also presented, leading to straightforward identification conditions for the error correction, echelon canonical form. An innovations algorithm for the evaluation of the exact Gaussian likelihood is given and the asymptotic properties of the approximate Gaussian estimator and the exact maximum likelihood estimator based upon the algorithm are derived. Examples illustrating the theory are discussed and some experimental evidence is also presented.ARMAX, partially nonstationary, Kronecker index theory identification.

The Finite-Sample Properties of Autoregressive Approximations of Fractionally-Integrated and Non-Invertible Processes

This paper investigates the empirical properties of autoregressive approximations to two classes of process for which the usual regularity conditions do not apply; namely the non-invertible and fractionally integrated processes considered in Poskitt (2006). In that paper the theoretical consequences of fitting long autoregressions under regularity conditions that allow for these two situations was considered, and convergence rates for the sample autocovariances and autoregressive coefficients established. We now consider the finite-sample properties of alternative estimators of the AR parameters of the approximating AR(h) process and corresponding estimates of the optimal approximating order h. The estimators considered include the Yule-Walker, Least Squares, and Burg estimators.Autoregression, autoregressive approximation, fractional process,

Description Length and Dimensionality Reduction in Functional Data Analysis

In this paper we investigate the use of description length principles to select an appropriate number of basis functions for functional data. We provide a flexible definition of the dimension of a random function that is constructed directly from the Karhunen-Loève expansion of the observed process. Our results show that although the classical, principle component variance decomposition technique will behave in a coherent manner, in general, the dimension chosen by this technique will not be consistent. We describe two description length criteria, and prove that they are consistent and that in low noise settings they will identify the true finite dimension of a signal that is embedded in noise. Two examples, one from mass-spectroscopy and the one from climatology, are used to illustrate our ideas. We also explore the application of different forms of the bootstrap for functional data and use these to demonstrate the workings of our theoretical results.Bootstrap, consistency, dimension determination, Karhunen-Loève expansion, signal-to-noise ratio, variance decomposition

Approximating the Distribution of the Instrumental Variables Estimator when the Concentration Parameter is Small.

This paper presents a new approximation to the exact sampling distribution of the instrumental variables estimator in simultaneous equations models. It differs from many of the approximations currently available, Edgeworth expansions for example, in that it is specifically designed to work well when the concentration parameter is small. The approximation is remarkable for the simplicity of its final form, for its accuracy and for its ability to capture those stylized facts that characterize lack of identification and weak instrument scenarios. The development leading to the approximation is also novel in that it introduces techniques of some independent interest not seen in this literature hitherto.concentration parameter, IV estimator, simultaneous equations model, t approximation, weak instruments.

Bias Reduction of Long Memory Parameter Estimators via the Pre-filtered Sieve Bootstrap

This paper investigates the use of bootstrap-based bias correction of semi-parametric estimators of the long memory parameter in fractionally integrated processes. The re-sampling method involves the application of the sieve bootstrap to data pre-filtered by a preliminary semi-parametric estimate of the long memory parameter. Theoretical justification for using the bootstrap techniques to bias adjust log-periodogram and semi-parametric local Whittle estimators of the memory parameter is provided. Simulation evidence comparing the performance of the bootstrap bias correction with analytical bias correction techniques is also presented. The bootstrap method is shown to produce notable bias reductions, in particular when applied to an estimator for which analytical adjustments have already been used. The empirical coverage of confidence intervals based on the bias-adjusted estimators is very close to the nominal, for a reasonably large sample size, more so than for the comparable analytically adjusted estimators. The precision of inferences (as measured by interval length) is also greater when the bootstrap is used to bias correct rather than analytical adjustments.Comment: 38 page

Assessing the Magnitude of the Concentration Parameter in a Simultaneous Equations Model

Poskitt and Skeels (2003) provide a new approximation to the sampling distribution of the IV estimator in a simultaneous equations model. This approximation is appropriate when the concentration parameter associated with the reduced form model is small and a basic purpose of this paper is to provide the practitioner with a method of ascertaining when the concentration parameter is small, and hence when the use of the Poskitt and Skeels (2003) approximation is appropriate. Existing procedures tend to focus on the notion of correlation and hypothesis testing. Approaching the problem from a different perspective leads us to advocate a different statistic for use in this problem. We provide exact and approximate distribution theory for the proposed statistic and show that it satisfies various optimality criteria not satisfied by some of its competitors. Rather than adopting a testing approach we suggest the use of p-values as a calibration device.Concentration parameter, simultaneous equations model, alienation coefficient, Wilks-lambda distribution, admissible invariant test.

Higher-Order Improvements of the Sieve Bootstrap for Fractionally Integrated Processes

This paper investigates the accuracy of bootstrap-based inference in the case of long memory fractionally integrated processes. The re-sampling method is based on the semi-parametric sieve approach, whereby the dynamics in the process used to produce the bootstrap draws are captured by an autoregressive approximation. Application of the sieve method to data pre-filtered by a semi-parametric estimate of the long memory parameter is also explored. Higher-order improvements yielded by both forms of re-sampling are demonstrated using Edgeworth expansions for a broad class of statistics that includes first- and second-order moments, the discrete Fourier transform and regression coefficients. The methods are then applied to the problem of estimating the sampling distributions of the sample mean and of selected sample autocorrelation coefficients, in experimental settings. In the case of the sample mean, the pre-filtered version of the bootstrap is shown to avoid the distinct underestimation of the sampling variance of the mean which the raw sieve method demonstrates in finite samples, higher order accuracy of the latter notwithstanding. Pre-filtering also produces gains in terms of the accuracy with which the sampling distributions of the sample autocorrelations are reproduced, most notably in the part of the parameter space in which asymptotic normality does not obtain. Most importantly, the sieve bootstrap is shown to reproduce the (empirically infeasible) Edgeworth expansion of the sampling distribution of the autocorrelation coefficients, in the part of the parameter space in which the expansion is valid