Asymptotics of self-weighted M-estimators for autoregressive models

In this paper, we consider a stationary autoregressive AR(p) time series \(y_t=\phi _0+\phi _1y_{t-1}+\cdots +\phi _{p}y_{t-p}+u_t\). A self-weighted M-estimator for the AR(p) model is proposed. The asymptotic normality of this estimator is established, which includes the asymptotic properties under the innovations with finite or infinite variance. The result generalizes and improves the known one in the literature.     

Bemerkung zur Lösung der Yule-Walker-Gleichungen

The covariance matrix and the firstp autocovariance matrices of a stationary vectorAR(p) process can be determined uniquely from the firstp + 1 Yule-Walker-equations. A simple proof of this result is given.      

Coherent forecasting for count time series using Box–Jenkins's AR(p) model

During the last three decades, integer‐valued autoregressive process of order p [or INAR(p)] based on different operators have been proposed as a natural, intuitive and maybe efficient model for integer‐valued time‐series data. However, this literature is surprisingly mute on the usefulness of the standard AR(p) process, which is otherwise meant for continuous‐valued time‐series data. In this paper, we attempt to explore the usefulness of the standard AR(p) model for obtaining coherent forecasting from integer‐valued time series. First, some advantages of this standard Box–Jenkins's type AR(p) process are discussed. We then carry out our some simulation experiments, which show the adequacy of the proposed method over the available alternatives. Our simulation results indicate that even when samples are generated from INAR(p) process, Box–Jenkins's model performs as good as the INAR(p) processes especially with respect to mean forecast. Two real data sets have been employed to study the expediency of the standard AR(p) model for integer‐valued time‐series data.     

Zur Lösung der Yule-Walker-Gleichungen

Summary It is shown that the stationarity condition is sufficient to determine uniquely the variance and the firstp covariances of a stationary AR(p)-process by means of the Yule-Walker equations. A formula for the determinant of the equations’ system depending on the zeros of the operator polynomial is given.      

Bartlett correction in the stable second‐order autoregressive model with intercept and trend

This paper derives the Bartlett factors that can be used to obtain higher‐order improvements for testing hypotheses about the autoregressive (AR) parameters in the stable AR(2) model with possible intercept and linear trend. The factors are obtained for testing hypotheses about individual parameters (φ₁ and φ₂) as well as their sum. Moreover, the effect of deterministic terms on the correction factors is found explicitly. All corrections are non‐decreasing in the AR parameters. Furthermore, the Bartlett corrections for φ₁ and φ₂ tend to infinity as φ₂ approaches 1, whereas the correction for φ₁ + φ₂ tends to infinity as φ₁ + φ₂ is close to 1. The effectiveness of these Bartlett corrections in finite samples is evaluated by simulations.     

The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors

We consider the normalized least squares estimator of the parameter in a nearly integrated first-order autoregressive model with dependent errors. In a first step we consider its asymptotic distribution as well as asymptotic expansion up to order O_p(T⁻¹). We derive a limiting moment generating function which enables us to calculate various distributional quantities by numerical integration. A simulation study is performed to assess the adequacy of the asymptotic distribution when the errors are correlated. We focus our attention on two leading cases: MA(1) errors and AR(1) errors. The asymptotic approximations are shown to be inadequate as the MA root gets close to −1 and as the AR root approaches either −1 or 1. Our theoretical analysis helps to explain and understand the simulation results of Schwert (1989) and DeJong, Nankervis, Savin, and Whiteman (1992) concerning the size and power of Phillips and Perron's (1988) unit root test. A companion paper, Nabeya and Perron (1994), presents alternative asymptotic frameworks in the cases where the usual asymptotic distribution fails to provide an adequate approximation to the finite-sample distribution.     

Nonparametric discrimination of time series data

A normality assumption is usually made for the discrimination between two stationary time series processes. A nonparametric approach is desirable whenever there is doubt concerning the validity of this normality assumption. In this paper a nonparametric approach is suggested based on kernel density estimation firstly on (p+1) sample autocorrelations and secondly on (p+1) consecutive observations. A numerical comparison is made between Fishers linear discrimination based on sample autocorrelations and kernel density discrimination for AR and MA processes with and without Gaussian noise. The methods are applied to some seismological data.     

Generalized least squares inference in panel and multilevel models with serial correlation and fixed effects

In this paper, I consider generalized least squares (GLS) estimation in fixed effects panel and multilevel models with autocorrelation. The presence of fixed effects complicates implementation of GLS as estimating the fixed effects will typically render standard estimators of the covariance parameters necessary for obtaining feasible GLS estimates inconsistent. I focus on the case where the disturbances follow an AR(p) process and offer a simple to implement bias-correction for the AR coefficients. The usefulness of GLS and the derived bias-correction for the parameters of the autoregressive process is illustrated through a simulation study which uses data from the Current Population Survey.     

The Limit Distribution of the Bootstrap for the Unit Root Test Statistic when the Residuals are Dependent

In this article, the unit root test for the AR(1) model with dependent residuals is considered. We adopt a bootstrap procedure to bootstrap the residuals with bootstrap sample size m less than the size n of the original sample. Under the assumptions that m → ∞ and m/n → 0, the convergence in probability of the bootstrap distribution function is established. Research supported by National Natural Science Foundation of China (No. 10471126)     

Sequential numerical integration in nonlinear state space models for microeconometric panel data

This paper discusses the estimation of a class of nonlinear state space models including nonlinear panel data models with autoregressive error components. A health economics example illustrates the usefulness of such models. For the approximation of the likelihood function, nonlinear filtering algorithms developed in the time‐series literature are considered. Because of the relatively simple structure of these models, a straightforward algorithm based on sequential Gaussian quadrature is suggested. It performs very well both in the empirical application and a Monte Carlo study for ordered logit and binary probit models with an AR(1) error component. Copyright © 2008 John Wiley & Sons, Ltd.     

The performance of alternative forecasting methods for SETAR models

We compare a number of methods that have been proposed in the literature for obtaining h-step ahead minimum mean square error forecasts for self-exciting threshold autoregressive (SETAR) models. These forecasts are compared to those from an AR model. The comparison of forecasting methods is made using Monte Carlo simulation. The Monte-Carlo method of calculating SETAR forecasts is generally at least as good as that of the other methods we consider. An exception is when the disturbances in the SETAR model come from a highly asymmetric distribution, when a Bootstrap method is to be preferred.An empirical application calculates multi-period forecasts from a SETAR model of US gross national product using a number of the forecasting methods. We find that whether there are improvements in forecast performance relative to a linear AR model depends on the historical epoch we select, and whether forecasts are evaluated conditional on the regime the process was in at the time the forecast was made.     

Distribution-free tests for time series models specification

We consider a class of time series specification tests based on quadratic forms of weighted sums of residuals autocorrelations. Asymptotically distribution-free tests in the presence of estimated parameters are obtained by suitably transforming the weights, which can be optimally chosen to maximize the power function when testing in the direction of local alternatives. We discuss in detail an asymptotically optimal distribution-free alternative to the popular Box–Pierce when testing in the direction of AR or MA alternatives. The performance of the test with small samples is studied by means of a Monte Carlo experiment.     

An econometric analysis of asymmetric volatility: Theory and application to patents

The purpose in registering patents is to protect the intellectual property of the rightful owners. Deterministic and stochastic trends in registered patents can be used to describe a country's technological capabilities and act as a proxy for innovation. This paper presents an econometric analysis of the symmetric and asymmetric volatility of the patent share, which is based on the number of registered patents for the top 12 foreign patenting countries in the USA. International rankings based on the number of foreign US patents, patent intensity (or patents per capita), patent share, the rate of assigned patents for commercial exploitation, and average rank scores, are given for the top 12 foreign countries. Monthly time series data from the United States Patent and Trademark Office for January 1975 to December 1998 are used to estimate symmetric and asymmetric models of the time-varying volatility of the patent share, namely US patents registered by each of the top 12 foreign countries relative to total US patents. A weak sufficient condition for the consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the univariate GJR(1,1) model is established under non-normality of the conditional shocks. The empirical results provide a diagnostic validation of the regularity conditions underlying the GJR(1,1) model, specifically the log-moment condition for consistency and asymptotic normality of the QMLE, and the computationally more straightforward but stronger second and fourth moment conditions. Of the symmetric and asymmetric models estimated, AR(1)–EGARCH(1,1) is found to be suitable for most countries, while AR(1)–GARCH(1,1) and AR(1)–GJR(1,1) also provide useful insights. Non-nested procedures are developed to test AR(1)–GARCH(1,1) versus AR(1)–EGARCH(1,1), and AR(1)–GJR(1,1) versus AR(1)–EGARCH(1,1).     

Multivariate subset autoregressive modelling with zero constraints for detecting ‘overall causality’

The necessary and sufficient condition to test for ‘overall causality’, i.e., the presence of Granger- causality and instantaneous causal relations, in a bivariate and trivariate autoregressive model with recursive form is discussed. It is argued that the conventional AR model (the reduced form AR) is a more straightforward and effective means of testing for ‘overall causality’. To detect instanta- neous causality it is proposed to select the best subset system in a residual regression system in conjunction with model selection criteria. The Canadian money-income-bank rate system is re-examined in this way and by using a previously proposed algorithm we identify the optimum multivariate subset AR with constraints to detect whether there is ‘overall causality’ in that system.     

Bias Reduction in Dynamic Panel Data Models by Common Recursive Mean Adjustment*

The within‐group estimator (same as the least squares dummy variable estimator) of the dominant root in dynamic panel regression is known to be biased downwards. This article studies recursive mean adjustment (RMA) as a strategy to reduce this bias for AR(p) processes that may exhibit cross‐sectional dependence. Asymptotic properties for N,T→∞ jointly are developed. When ( log ²T)(N/T)→ζ, where ζ is a non‐zero constant, the estimator exhibits nearly negligible inconsistency. Simulation experiments demonstrate that the RMA estimator performs well in terms of reducing bias, variance and mean square error both when error terms are cross‐sectionally independent and when they are not. RMA dominates comparable estimators when T is small and/or when the underlying process is persistent.     

Cooperation strategies in port-oriented bulk supply chains: aligning concept and practice

It seems intuitively obvious that firms in supply chains may have more to gain than to lose from learning to cooperate; but it is now more than two decades since Poirier [1999. Advanced Supply Chain Management. San Francisco, CA: Barrett-Koehler] and others called for cooperation in order to capture mutual gains in supply networks and even now ‘cooperation is neither common nor easy’. The simple fact is that not only are supply chains exceptionally complex but so too is the ‘process of cooperating’ – often in the context of antitrust legislation and competition policy. This paper argues that there is a critical need to rethink the principles and processes of cooperation within the broader framework of the competitive behaviour of firms and business strategy. Particularly, it suggests that the relatively recent thinking of Greenwald and Kahn [2005 Greenwald, B., and J. Kahn. 2005. Competition Demystified A Radically Simplified Approach to Business Strategy. New York, NY: Portfolio, The Penguin Group. [Google Scholar]. Competition Demystified A Radically Simplified Approach to Business Strategy. New York, NY: Portfolio, The Penguin Group] in their ‘radically simplified approach to business strategy’ offers sound conceptual insights into cooperation and cooperative strategies for firms not only in markets but also in chains. Furthermore, it notes that the analytical framework for cooperation and cooperative strategies which the authors develop is far removed from the notion of cooperation as ‘commitment and trust and shared thinking’ and from ‘buyer/seller reciprocity’ and ‘collaborative attitudes’ which tend to underwrite much contemporary thinking and research. The paper also argues that the Greenwald and Kahn framework – its single intelligence model of cooperation and cooperative strategies – resonates with real-world relevance, at least for particular supply chains. The paper focuses attention on research into globally significant export coal chains from major east coast Australian ports and in brief case studies finds substantial alignment between concept and practice.     

Quarterly beta forecasting: An evaluation

Ever since the inception of betas as a measure of systematic risk, the forecast error in relation to this parameter has been a major concern to both academics and practitioners in finance. In order to reduce forecast error, this paper compares a series of competing models to forecast beta. Realized measures of asset return covariance and variance are computed and applied to forecast beta, following the advances in methodology of Andersen, Bollerslev, Diebold and Wu [Andersen, T. G., Bollerslev, T., Diebold, F. X., & Wu, J. (2005). A framework for exploring the macroeconomic determinants of systematic risk. American Economic Review, 95, 398–404; and Andersen, T. G., Bollerslev, T., Diebold, F. X., & Wu, J. (2006). Realized beta: Persistence and Predictability. In T. Fomby & D. Terrell (Eds.), Advances in Econometrics, vol 20B: Econometric Analysis of Economic and Financial Times Series., JAI Press, 1–40.]. This approach is compared with the constant beta model (the industry standard) and a variant, the random walk model. It is shown that an autoregressive model with two lags produces the lowest or close to the lowest error for quarterly stock beta forecasts. In general, the AR(2) model has a mean absolute forecast error half that of the constant beta model. This reduction in forecast error is a dramatic improvement over the benchmark constant model.     

A characterization using the conditional variance

A class of probability distributions is characterized assuming that the conditional variance of a functionh (X), givenX>x, is constant.     

A general class of estimators for estimating population mean using auxiliary information

Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (b−a)p=ρ C _y/C _x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.     

On exact D-optimal designs with 2 two-level factors and n autocorrelated observations

In this paper we consider the exact D-optimal designs for estimation of the unknown parameters in the two factors, each at only two-level, main effects model with autocorrelated errors. The vector of the n random errors in the observed responses is assumed to follow a first-order autoregressive model (AR(1)). The exact D-optimal designs seek the optimal combinations of the design levels as well as the optimal run orders, so that the determinant of the information matrix of BLUEs for the unknown parameters is maximized. Bora-Senta and Moyssiadis (1999) gave some conjectures about the exact D-optimal designs based on their experience of several exhaustive searches. In this paper their conjectures are partially proved to be true.Received: January 2003 / Accepted: October 2003Partially supported by the National Science Council of Taiwan, R.O.C. under grant NSC 91-2115-M-008-013.Supported in part by the National Science Council of Taiwan, R.O.C. under grant NSC 89-2118-M-110-003.