THE STABILITY OF THE AR(1) PROCESS WITH AN AR(1) COEFFICIENT

Abstract. In this paper we consider a simple time varying coefficient ARMA process:the AR (1) process with an AR (1) coefficient. A basic requirement of the process is that the output has finite variance, and we derive a condition on the parameters for this to be satisfied. The analysis is complicated by the interaction between the equations for the data and the varying coefficient.     

On Uniform Asymptotic Normality of Sequential Estimators for the Parameters in a Stable AR(1)

Abstract

A sequential procedure for estimating two autoregressive parameters is constructed. The uniform asymptotic normality of estimators is established.     

ON THE UNIMODALITY OF THE EXACT LIKELIHOOD FUNCTION FOR NORMAL AR(2) SERIES

Abstract. For stationary second-order autoregressive normal processes, the conjecture of uniqueness of the solution of the exact likelihood equations is examined. A sufficient condition for uniqueness is given; this condition is satisfied with very high probability if the number of observations is not extremely small. Moreover, it is shown that not more than two maxima may exist. Examples of data which actually produce a likelihood function with two local maxima are given.     

A NOTE ON THE THRESHOLD AR(1) MODEL WITH CAUCHY INNOVATIONS

Abstract. A threshold autoregressive process of the first order with one threshold r and with Cauchy innovations is investigated in the paper. An explicit formula for the stationary density of such process is derived for the special case that r = 0 and that the autoregressive parameters have the same absolute value.     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

ON THE STABILITY OF A THRESHOLD AR(1) WITHOUT INTERCEPTS

Abstract. Binary polynomials are used to determine the stability of a threshold AR(1) without intercepts.     

CONFIDENCE REGIONS FOR PARAMETERS IN THE AR(1) MODEL

Abstract. The construction of approximate joint and marginal confidence regions for parameters in the first-order autoregressive time series model is discussed. These regions are based on the large sample distributions of the likelihood ratio (and approximations to it), of the maximum likelihood estimates and of the score statistics. All these approaches are illustrated using a well-known example from Box and Jenkins (Time Series Analysis:Forecasting and Control, revised edn. San Francisco:Holden Day, 1976) and some simulated series. In addition, a simulation study is provided for comparing the coverage properties of the various procedures.     

ON THE CONSISTENCY OF LEAST SQUARES ESTIMATORS FOR A THRESHOLD AR(1) MODEL

Abstract. For the SETAR (2; 1,1) model

where {a_t(i)} are i.i.d. random variables with mean 0 and variance σ²(i), i = 1,2, and {a_t(l)} is independent of {a_t(2)}, we consider estimators of φ₁, φ ₂ and r which minimize weighted sums of the sum of squares functions for σ²(1) and σ²(2). These include as a special case the usual least squares estimators. It is shown that the usual least squares estimators of φ₁, φ₂ and r are consistent. If σ²(1) ≠σ²(2) conditions on the weights are found under which the estimators of r and φ₁ or φ₂ are not consistent.     

Bayesian Unit Root Test in Nonnormal AR(1) Model

In this paper, we approximate the distribution of disturbances by the Edgeworth series distribution and propose a Bayesian analysis in a nonnormal AR(1) model. We derive the posterior distribution of the autocorrelation and the posterior odds ratio for unit roots hypothesis in the AR(1) model when the first four cumulants of the Edgeworth series distribution are finite and the higher order cumulants are negligible. We also apply the posterior analysis to eight real exchange rates and investigate whether these exchange rates behave like a random walk or not.     

An Adaptive Estimator of the Autocorrelation Coefficient in Regression Models with Autoregressive Errors

In this paper an adaptive estimator of the autocorrelation coefficient is constructed in regression models whose error variables follow a stationary autoregressive process of order 1. Examples of nonparametric, additive and semiparametric regression models are discussed.     

A NOTE ON SOME STATISTICS USEFUL IN IDENTIFYING THE ORDER OF AUTOREGRESSIVE MOVING AVERAGE MODEL

Abstract. Several recursive relations concerning some statistics useful in identifying the order of autoregressive moving average are derived and the asymptotic behaviour of these statistics are studied.     

CONTEMPORANEOUS AGGREGATION OF TRIANGULAR ARRAY OF RANDOM‐COEFFICIENT AR(1) PROCESSES

We discuss contemporaneous aggregation of independent copies of a triangular array of random‐coefficient processes with i.i.d. innovations belonging to the domain of attraction of an infinitely divisible law W. The limiting aggregated process is shown to exist, under general assumptions on W and the mixing distribution, and is represented as a mixed infinitely divisible moving average in (4). Partial sums process of is discussed under the assumption EW² < ∞ and a mixing density regularly varying at the ‘unit root’ x = 1 with exponent β > 0. We show that the previous partial sums process may exhibit four different limit behaviors depending on β and the Lévy triplet of W. Finally, we study the disaggregation problem for in spirit of Leipus et al. (2006) and obtain the weak consistency of the corresponding estimator of ϕ(x) in a suitable L₂ space.     

AN EXPLICIT NEARLY UNBIASED ESTIMATE OF THE AR(1) PARAMETER FOR REPEATED MEASUREMENTS

Abstract. It is assumed that n ( n ≥ 1) independent time series, each of length T. have the same autocorrelation function of the AR(1) type, but they may differ in mean value, with the mean value of the i th series equal to a linear combination of a set of covariates associated with the series. To estimate the common autoregressive parameter, Daniels' method is extended to the present case. As, for small T , this gives a severely biased estimate, a formula for its mean value is obtained. A modified estimate which has a substantially smaller bias is found using this formula.     

Improved Prediction Limits For AR(p) and ARCH(p) Processes

Abstract. A new simulation‐based prediction limit that improves on any given estimative d‐step‐ahead prediction limit for a Markov process is described. This improved prediction limit can be found with almost no algebraic manipulations. Nonetheless, it has the same asymptotic coverage properties as the Barndorff‐Nielsen and Cox [Inference and Asymptotics (1994) Chapman and Hall, London] and Vidoni [Journal of Time Series Analysis Vol. 25, pp. 137–154.] (2004) improved prediction limits. The new simulation‐based prediction limit is ideally suited to those Markov process models for which the algebraic manipulations required for the latter improved prediction limits are very complicated. We illustrate the new method by applying it in the context of one‐step‐ahead prediction for a zero‐mean Gaussian AR(2) process and an ARCH(2) process.     

ON THE CHOICE OF THE ORDER OF AUTOREGRESSIVE MODELS: A RANKING AND SELECTION APPROACH

Abstract. The problem of estimating the order of autoregressive models is considered from the point of view of ranking and selection procedures. This approach offers a formulation to many problems more realistic than that of classical hypothesis testing or of criteria based on estimation theory (e.g., AIC). In the method considered here, sampling variations are taken into account and the experimenter is also allowed to incorporate any a priori knowledge of the true order (e.g., lower bound as well as upper bound).     

RESULTS ON ESTIMATION AND TESTING FOR A UNIT ROOT IN THE NONSTATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODEL

Abstract. We review the limiting distribution theory for Gaussian estimation of the univariate autoregressive moving-average (ARMA) model in the presence of a unit root in the autoregressive (AR) operator, and present the asymptotic distribution of the associated likelihood ratio (LR) test statistic for testing for a unit root in the ARMA model. The finite sample properties of the LR statistic as well as other unit root test procedures for the ARMA model are examined through a limited simulation study. We conclude that, for practical empirical work that relies on standard computations, the LR test procedure generally performs better than other standard procedures in the presence of a substantial moving-average component in the ARMA model.     

A FREQUENCY DOMAIN APPROACH TO LAGRANGE MULTIPLIER TEST FOR AUTOREGRESSIVE MOVING AVERAGE MODELS

Abstract. The paper derives a goodness of fit test for autoregressive moving average models using the frequency domain approximation to the log likelihood and the Lagrange multiplier approach. The test statistic is based on the sample autocovariances and can be quickly computed through a recursive procedure.     

SOME PROPERTIES OF CONDITIONAL QUASI-LIKELIHOOD FUNCTIONS FOR TIME SERIES MODEL FITTING

Abstract. We consider fitting a parametric model to a time series and obtain the maximum likelihood estimates of unknown parameters included in the model by regarding the time series as a Gaussian process satisfying the model. We evaluate the asymptotic value of the conditional quasi-likelihood function when the number of observations tends to infinity. We show what properties of the time series we can find by examining the behaviour of the conditional quasi-likelihood function, even when the time series does not necessarily satisfy the model and is not necessarily Gaussian.     

ON EMBEDDING A DISCRETE-PARAMETER ARMA MODEL IN A CONTINUOUS-PARAMETER ARMA MODEL

Abstract. It is shown that a real-valued discrete-parameter Gaussian ARMA ( p. q ) model with q < p can be embedded in a real-valued continuous-parameter Gaussian ARMA( p', q' ) model with q' < p' . The problem of embedding a real-valued discrete-parameter Gaussian AR( p ) into a real-valued continuous-parameter Gaussian AR( p ) is also discussed.     

ADDITIVE ELEMENTS OF ARMA MODELS

Abstract. The additive elements of the likelihood function (Whittaker, 1984) are a natural generalization of the regression elements of Newton and Spurrell (1967) and serve to simplify the analysis of several competing models. The aim of this note is to extend their application to time series analysis.