LONG-RANGE DEPENDENCE AND MIXING FOR DISCRETE TIME FRACTIONAL PROCESSES

Abstract. A large class of discrete time stationary processes, an extension of the well-known fractionally integrated autoregressive moving-average models, is investigated. For a suitable choice of parameters, these processes are long-range dependent. After a detailed study of the asymptotic behaviour of their correlations, we investigate their mixing properties and then give some simulated examples.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.     

LAGRANGE MULTIPLIER TESTS FOR FRACTIONAL DIFFERENCE

Abstract. This paper develops Lagrange multiplier tests of ARMA( p, q ) models against fractional ARIMA( p, d, q ) alternatives. The performance of the tests is investigated for moderate-sized samples. It is concluded that fractional difference will be difficult to detect when the orders ( p, q ) are over-specified in an autoregressive moving-average (ARMA) analysis. The importance of distinguishing between the mean known and mean estimated cases in fractional difference models is illustrated in the context of these tests.     

ON LINEAR PROCESSES WITH GIVEN MOMENTS

Abstract. Let X _t = c ₀ Y _t + c ₁ Y _{t -1}+… be a linear process with known coefficients c _k , where Y _t is a strict white noise. Let m ₁, …, m _2r be given numbers. A method is presented to determine whether there exists a distribution of Y _t such that EX ^k _t = m _k for k = 1, …, 2 r . In the positive case, such a distribution of Y _t is described. Some explicit formulas for AR(1) and AR(2) models are derived. The results can be used for simulating a process with given moments of its stationary distribution. The procedure also enables proof that some stationary distributions cannot belong to the given linear process.     

STUDENTIZING WEIGHTED SUMS OF LINEAR PROCESSES

This article presents a general method for studentizing weighted sums of a linear process where weights are arrays of known real numbers and innovations form a martingale difference sequence. Asymptotical normality for such sums was established in Abadir et al. (2013). This article centres on the estimation of the standard deviation, to make the normal approximation operational. The proposed studentization is easy to apply and robust against unknown types of dependence (short range and long range) in the observations. It does not require the estimation of the parameters controlling the dependence structure. A finite‐sample Monte Carlo simulation study shows the applicability of the proposed methodology for moderate sample sizes. Assumptions for studentization are satisfied by the Nadaraya–Watson kernel type weights used for inference in non‐parametric regression settings.     

ON THE ASYMPTOTIC DISTRIBUTION OF THE GENERALIZED PARTIAL AUTOCORRELATION FUNCTION IN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. It has been conjectured and illustrated that the estimate of the generalized partial autocorrelation function (GPAC), which has been used for the identification of autoregressive moving-average (ARMA) models, has a thick-tailed asymptotic distribution. The purpose of this paper is to investigate the asymptotic behaviour of the GPAC in detail. It will be shown that the GPAC can be represented as a ratio of two functions, known as the θ function and the Λ function, each of which itself has a useful pattern for ARMA model identification. We shall show the consistencies of the extended Yule-Walker estimates of the three functions and present their asymptotic distributions.     

ESTIMATION OF THE FRACTIONAL DIFFERENCE PARAMETER IN THE ARIMA(p, d, q) MODEL USING THE SMOOTHED PERIODOGRAM

Abstract. In recent work on time series analysis considerable interest has been focused on series having the property of long memory. Long memory is a characteristic of time series in which the dependence between distant observations is not negligible. The model that has been most frequently studied, which in some situations shows properties of long memory, is based on the autoregressive integrated moving-average ARIMA( p, d, q ) process. Hosking (Fractional differencing, Biometrika 68 (1) (1981), 165–76) generalized this model by permitting the degree of differencing d to take fractional values. He then demonstrated that for d in the range 0 < d < 0.5 the process is stationary and possesses the long memory property. Our study is based on the ARIMA( p, d, q ) model when d takes any real non-integer value in the interval (-0.5, 0.5). The main aim of our study is to examine methods for estimating the parameters of this model. For estimating d we suggest an estimator based on the smoothed periodogram. Using an empirical approach we compare this estimator with other which are well known in the literature of long memory models, e.g. the raw periodogram regression method and the Hurst coefficient method.     

A GENERALIZED FRACTIONALLY INTEGRATED AUTOREGRESSIVE MOVING-AVERAGE PROCESS

Abstract. This paper considers the long memory Gegenbauer autoregressive movingaverage (GARMA) process that generalizes the fractionally integrated ARMA (ARFIMA) process to allow for hyperbolic and sinusoidal decay in autocorrelations. We propose the conditional sum of squares method for estimation (which is asymptotically equivalent to the maximum likelihood estimation) and develop the asymptotic theory. Many results are in sharp contrast to those of the ARFIMA model. Simulations are conducted to assess the performance of the proposed estimators in small sample applications. Two applications to the sunspot data and the US inflation rates based on the wholesale price index are provided.     

MODELLING FOR THE WAVELET COEFFICIENTS OF ARFIMA PROCESSES

We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale.     

ON q-CONDITIONED PARTIAL CORRELATIONS

Abstract. In attempting to develop a procedure for fitting linear multiple autoregressive-moving average models to observed data, perhaps the most difficult problem is to achieve a reasonable initial model selection. A recent paper by Jenkins and Alavi suggests, as one possibility, the examination of so-called q -conditioned partial correlations. We show that the sampling properties of these statistics are such as to render them of dubious value for this purpose.     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

A NOTE ON RELATIONS BETWEEN SEASONALLY ADJUSTED VARIABLES

Abstract. This paper examines, with reference to some well known data, possible difficulties in the fitting of simple non-seasonal models to seasonally adjusted time series data.     

MULTIPLICATIVE EXPONENTIAL MODELS FOR STATIONARY TIME SERIES

Abstract. A class of models for one dimensional time series is presented. The spectrum of such a model is obtained by raising the spectrum of a known parameterized model to an exponent, allowed to attain arbitrary real values. For a moving average model this for example means that the roots of the moving average operator are allowed to have any real order. This method adds a further flexibility to the model which for example allows us to model long memory time series using only a few parameters. The exponent is parameterized in a special way to make the estimation of the parameter determining the exponent asymptotically independent of the estimation of the other model-parameters. The asymptotic distribution of the estimators is derived. The idea is also used for multiplicative models with an exponent for each seasonal factor. In this case the estimators are only approximately independent for a large season length. Finally an application of the model is given using the Beveridge wheat price index.     

ON THE EXISTENCE OF STATIONARY THRESHOLD AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. Conditions for the existence of causal and strictly stationary solutions of the equations defining a self-exciting threshold autoregressive moving-average (SETARMA) model are derived. For threshold autoregressive models we allow the autoregressive coefficients to be random and derive sufficient conditions for geometric ergodicity and the existence of strictly and weakly stationary solutions of the defining equations.     

Asymptotics for the Conditional‐Sum‐of‐Squares Estimator in Multivariate Fractional Time‐Series Models

This article proves consistency and asymptotic normality for the conditional‐sum‐of‐squares estimator, which is equivalent to the conditional maximum likelihood estimator, in multivariate fractional time‐series models. The model is parametric and quite general and, in particular, encompasses the multivariate non‐cointegrated fractional autoregressive integrated moving average (ARIMA) model. The novelty of the consistency result, in particular, is that it applies to a multivariate model and to an arbitrarily large set of admissible parameter values, for which the objective function does not converge uniformly in probability, thus making the proof much more challenging than usual. The neighbourhood around the critical point where uniform convergence fails is handled using a truncation argument.     

ON THE SPECTRAL DENSITY MATRIX OF A PERIODIC ARMA PROCESS

Abstract. After reviewing the spectral representation theorems for periodic stationary process, we derive a parametric formula for the spectral density of a periodic ARMA process via a new approach. The equivalence with the existing approach is shown.     

ON BOOTSTRAP PREDICTIVE INFERENCE FOR AUTOREGRESSIVE PROCESSES

Abstract. In this paper we consider bootstrap-based predictive inference for autoregressive processes of order p. We consider both unconditional inference and inference conditional on the last p observed values. We make two contributions. Our first contribution is to point out the best way to apply the bootstrap to unconditional predictive inference when the process is Gaussian. Now, it may be argued that predictive inference for autoregressive processes of order p should be carried out conditional on the last p observed values. When the process is Gaussian, a bootstrap predictive inference conditional on the last p observed values is conveniently computed by 'running' the same autoregressive process backwards in time. This procedure is inappropriate for non-Gaussian autoregressive processes. Our second (and more important) contribution is to present a method (which is not computationally burdensome) for the computation of a bootstrap predictive inference for a non-Gaussian autoregressive process of order p conditional on the last p observed values.     

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.     

ON PREDICTION WITH FRACTIONALLY DIFFERENCED ARIMA MODELS

Abstract. This paper considers some extended results associated with the predictors of long-memory time series models. These direct methods of obtaining predictors of fractionally differenced autoregressive integrated moving-average (ARIMA) processes have advantages from the theoretical point of view.     

Gaussian Semi-parametric Estimation of Fractional Cointegration

Abstract. We analyse consistent estimation of the memory parameters of a nonstationary fractionally cointegrated vector time series. Assuming that the cointegrating relationship has substantially less memory than the observed series, we show that a multi-variate Gaussian semi-parametric estimate, based on initial consistent estimates and possibly tapered observations, is asymptotically normal. The estimates of the memory parameters can rely either on original (for stationary errors) or on differenced residuals (for nonstationary errors) assuming only a convergence rate for a preliminary slope estimate. If this rate is fast enough, semi-parametric memory estimates are not affected by the use of residuals and retain the same asymptotic distribution as if the true cointegrating relationship were known. Only local conditions on the spectral densities around zero frequency for linear processes are assumed. We concentrate on a bivariate system but discuss multi-variate generalizations and show the performance of the estimates with simulated and real data.