Linear Trend with Fractionally Integrated Errors

We consider the estimation of linear trend for a time series in the presence of additive long-memory noise with memory parameter d ∈[0, 1.5). Although no parametric model is assumed for the noise, our assumptions include as special cases the random walk with drift as well as linear trend with stationary invertible autoregressive moving-average errors. Moreover, our assumptions include a wide variety of trend-stationary and difference-stationary situations. We consider three different trend estimators: the ordinary least squares estimator based on the original series, the sample mean of the first differences and a class of weighted (tapered) means of the first differences. We present expressions for the asymptotic variances of these estimators in the form of one-dimensional integrals. We also establish the asymptotic normality of the tapered means for d ∈[0, 1.5) −{0.5} and of the ordinary least squares estimator for d ∈ (0.5, 1.5). We point out connections with existing theory and present applications of the methodology.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

Semiparametric Inference in Seasonal and Cyclical Long Memory Processes

Several semiparametric estimates of the memory parameter in standard long memory time series are now available. They consider only local behaviour of the spectrum near zero frequency, about which the spectrum is symmetric. However long-range dependence can appear as a spectral pole at any Nyqvist frequency (reflecting seasonal or cyclical long-memory), where the spectrum need display no such symmetry. We introduce Seasonal/Cyclical Asymmetric Long Memory (SCALM) processes that allow differing rates of increase on either side of such a pole. To estimate the two consequent memory parameters we extend two semiparametric methods that were proposed for the standard case of a spectrum diverging at the origin, namely the log-periodogram and Gaussian or Whittle methods. We also provide three tests of symmetry. Monte Carlo analysis of finite sample behaviour and an empirical application to UK inflation data are included. Our models and methods allow also for the possibility of negative dependence, described by a possibly asymmetric spectral zero.     

Broadband Semiparametric Estimation of the Memory Parameter of a Long-Memory Time Series Using Fractional Exponential Models

We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long-memory time series. The estimator is constructed by fitting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other finite-parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is sufficiently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke–Porter-Hudak (GPH) and Gaussian semiparametric estimators (GSE), attaining an asymptotic mean squared error proportional to (log n )/ n , where n is the sample size. In a simulation study, we demonstrate the merits of using a finite-sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows' C_L criterion.     

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.     

Deconvolution of fractional brownian motion

We show that a fractional Brownian motion with H'∈(0,1) can be represented as an explicit transformation of a fractional Brownian motion with index H ∈(0,1). In particular, when H'=½, we obtain a deconvolution formula (or autoregressive representation) for fractional Brownian motion. We work both in the `time domain' and the `spectral domain' and contrast the advantages of one domain over the other.     

STATIONARY TANGENT: THE DISCRETE AND NON-SMOOTH CASE

Abstract. In Keich (2000 ),we define a stationary tangent process, or a locally optimal stationary approximation, to a real non-stationary smooth Gaussian process. This paper extends the idea by constructing a discrete tangent – a `locally' optimal stationary approximation – for a discrete time, real Gaussian process. Analogously to the smooth case, our construction relies on a generalization of the recursion formula for the orthogonal polynomials of the spectral distribution function. More precisely, we use a generalization of the Schur parameters to identify the stationary tangent. By way of discretizing, we later demonstrate how this tangent can be used to obtain `good' local stationary approximations to non-smooth continuous time, real Gaussian processes. Further, we demonstrate how, analogously to the curvatures in the smooth case, the Schur parameters can be used to determine the order of stationarity of a non-smooth process.     

Consistent Estimation for Non-Gaussian Non-Causal Autoregessive Processes

Traditional estimation based on least squares or Gaussian likelihood cannot distinguish between causal and non-causal representation of a stationary autoregressive (AR) process. Breidt et al . (Maximum likelihood estimation for non-causal autoregressive processes. J. Multivariate Anal. 36 (1991), 175–98) proved the existence of a consistent likelihood estimation of possibly non-causal AR processes; however, in this case an existence result is not very useful since the likelihood function generally exhibits multiple maxima. Moreover the method assumes full knowledge of the distribution of the innovation process. This paper shows a constructive proof that a modified L ₁ estimate is consistent if the innovation process has a stable law distribution with index α∈ (1, 2). It is also shown that neither non-Gaussianity nor infinite variance is sufficient to ensure consistency.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.     

Some Results Concerning the Asymptotic Distribution of Sample Fourier Transforms and Periodograms for a Discrete-time Stationary Process with a Continuous Spectrum

We consider a stationary process ( X_t , t = 0, ±1, ...) with a continuous spectrum. Denote by D_n (λ) a tapered Fourier transform of ( X ₀, X ₁, ..., X _{n −1}) at (angular) frequency λ. We obtain the asymptotic distribution of D_n (λ) and the joint asymptotic distribution of { D_n (λ _j ), 1 ≤ j ≤ k } with continuity of the spectral density f (.) at the relevant frequencies as the only assumption concerning the second-order structure of ( X_t ); all other assumptions required are easily stated. The results are extended to processes for which f (.) is continuous except at λ = 0, with lim_λ←0 f (λ)λ^{2 d} = K , a constant, where 0 < d < ½, as is typical of certain types of processes with long-range dependence. Results for the sample periodogram, proportional to | D_n (λ)|², follow immediately.     

A Test of Linearity for Functional Autoregressive Models

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR( p ) model on ℜ ^d of the form
X _n = f ( X _{n −1}, ..., X _{n − p} ) + ξ _n ( n ∈ N).
The testing procedure is based on a suitably normalized sum of quadratic deviations between two different estimates of the function f evaluated at q distinct points of ℜ ^dp . The estimators are f^ _n , a recursive version of the non-parametric kernel estimator of f , and  _n , a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ² limit distribution under the null hypothesis. A similar result is derived under the alternative hypothesis for the test statistic corrupted by a non-linear term. Our simulations indicate that our asymptotic results hold for moderate sample sizes when the testing procedure is used carefully     

Bootstrap-assisted Goodness-of-fit Tests in the Frequency Domain

The validity of a stationary time series model may be measured by the goodness of fit of the spectral distribution function. Anderson (Technical Report 27, 1991; Technical Report 309, 1995; Stanford University) has worked out the closed-form characteristic functions for the Cramer–von Mises criterion for general linear processes, under the condition that all values of the parameters are specified. The asymptotic approach is not easily implemented and usually requires a case by case analysis. In this paper we propose a bootstrap goodness-of-fit test in the frequency domain. By properly resampling the residuals, we can consistently estimate the p values for many weakly dependent semiparametric models with unspecified parameter values. This is the content of the main theorem that we try to explain. A group of simulations is conducted, showing consistent significance level and good power. The special tests are applied to the lynx data and reveal structure unexplained by the AR(1) model fitted by Tong ( J. R. Stat. Soc. A 140 (1977), 432–36). A possible generalization with application to financial data analysis is also discussed.     

Averaged Periodogram Spectral Estimation with Long-memory Conditional Heteroscedasticity

The empirical relevance of long-memory conditional heteroscedasticity has emerged in a variety of studies of long time series of high frequency financial measurements. A reassessment of the applicability of existing semiparametric frequency domain tools for the analysis of time dependence and long-run behaviour of time series is therefore warranted. To that end, in this paper the averaged periodogram statistic is analysed in the framework of a generalized linear process with long-memory conditional heteroscedastic innovations according to a model specification first proposed by Robinson (Testing for strong serial correlation and dynamic conditional heteroscedasticity in multiple regression. J. Economet. 47 (1991), 67–84). It is shown that the averaged periodogram estimate of the spectral density of a short-memory process remains asymptotically normal with unchanged asymptotic variance under mild moment conditions, and that for strongly dependent processes Robinson's averaged periodogram estimate of long memory (Semiparametric analysis of long memory time series. Ann. Stat. 22 (1994), 515–39) remains consistent.     

Simulating a class of stationary Gaussian processes using the Davies–Harte algorithm, with application to long memory processes

We demonstrate that the fast and exact Davies–Harte algorithm is valid for simulating a certain class of stationary Gaussian processes – those with a negative autocovariance sequence for all non-zero lags. The result applies to well known classes of long memory processes: Gaussian fractionally differenced (FD) processes, fractional Gaussian noise (fGn) and the nonstationary fractional Brownian Motion (fBm).     

Robust Automatic Bandwidth for Long Memory

The choice of bandwidth, or number of harmonic frequencies, is crucial to semiparametric estimation of long memory in a covariance stationary time series as it determines the rate of convergence of the estimate, and a suitable choice can insure robustness to some non-standard error specifications, such as (possibly long-memory) conditional heteroscedasticity. This paper considers mean squared error minimizing bandwidths proposed in the literature for the local Whittle, the averaged periodogram and the log periodogram estimates of long memory. Robustness of these optimal bandwidth formulae to conditional heteroscedasticity of general form in the errors is considered. Feasible approximations to the optimal bandwidths are assessed in an extensive Monte Carlo study that provides a good basis for comparison of the above-mentioned estimates with automatic bandwidth selection.     

An Efficient Taper for Potentially Overdifferenced Long-memory Time Series

We propose a new complex-valued taper and derive the properties of a tapered Gaussian semiparametric estimator of the long-memory parameter d ε (−0.5, 1.5). The estimator and its accompanying theory can be applied to generalized unit root testing. In the proposed method, the data are differenced once before the taper is applied. This guarantees that the tapered estimator is invariant with respect to deterministic linear trends in the original series. Any detrimental leakage effects due to the potential noninvertibility of the differenced series are strongly mitigated by the taper. The proposed estimator is shown to be more efficient than existing invariant tapered estimators. Invariance to k th order polynomial trends can be attained by differencing the data k times and then applying a stronger taper, which is given by the k th power of the proposed taper. We show that this new family of tapers enjoys strong efficiency gains over comparable existing tapers. Analysis of both simulated and actual data highlights potential advantages of the tapered estimator of d compared with the nontapered estimator.     

Dynamic stationary subspace analysis based on Gaussian mixture models for ironmaking process monitoring

Blast furnace ironmaking process monitoring is an important and challenging task. Due to the influence of hot blast stove switching and large fluctuations in the quality of raw materials, the measurements of ironmaking processes show obvious non-stationary characteristics, and in addition, the observed data are also characterized by time-series dynamic and non-Gaussian characteristics. In this paper, a dynamic stationary subspace analysis method based on the Gaussian mixture model (DSSA–GMM) is proposed to address the difficulties in blast furnace ironmaking process monitoring. The time-series dynamic relationship of the data is conducted by introducing a sliding time window. The Gaussian mixture model (GMM) is used to deal with the non-Gaussian characteristics of the data, and the parameters of the GMMs are estimated using the expectation–maximization algorithm. The stationary projection matrix is obtained by optimizing the Kullback–Leibler (K–L) divergence between GMMs of different periods to realize the stationary subspace separation. Finally, the convex hull of the stationary subspace is established for fault detection, thus realizing the monitoring for non-stationary and non-Gaussian dynamic processes. The effectiveness of the DSSA–GMM method is verified by a numerical simulation and a dataset collected from an actual blast furnace ironmaking process.     

TESTING CHANGE-POINTS IN THE EXPLOSIVE GAUSSIAN AUTOREGRESSIVE PROCESSES

Abstract. A functional limit theorem with a particular function class and topology is derived for non-ergodic type time series. This limit theorem allows us to study the asymptotic law of the associated likelihood ratio test (LRT) statistic for testing the presence of a change in the covariance parameter in the explosive Gaussian autoregressive model. We show that the level of the LRT cannot be approximated without introducing appropriate normalization. The limit law of a particular weighted likelihood ratio test is examined through a simulation study and is compared with the well-known Kolmogorov distribution obtained in the stationary case; we conclude that for practical applications when the root is really close to unity one can use the same thresholds as in the stationary case. This procedure is applied to the study of three real time series known to be non-stationary.     

An Electrochemical Theory for Oxygen Reboil

Oxygen reboil often occurs when glass is melted in air in platinum containers. Oxygen blisters develop even when well-fined glass is remelted under similar conditions. Experiments show that this reboil is prevented or greatly reduced by: (1) Eliminating electron conductors from the system, (2) removing oxygen from the atmosphere over the melt, (3) applying an external bucking potential, (4) reversing concentration gradients, or (5) removing temperature gradients. These results suggest that oxygen reboil, which occurs at a platinum-glass interface, results from the discharge of a concentration cell or a thermal cell. The proposed mechanism is: O²⁻→½ O₂+ 2e⁻ at the anode and ½ O₂+ 2e⁻→ O²⁻ at the cathode, with electron transport in the platinum and, to complete the circuit, alkali ions in the glass.     

Polyvariograms and their Asymptotes

A definition of a polyvariogram (PV) γ _b ( h )( h = 1, 2, ...) of order b ( b ≥ max(0, d − 1) is suggested for time series { Z ( t })} satisfying {∇ ^d ( Z ( t ) = W ( t ) (where d is a non-negative integer and { W ( t )} is a second-order stationary time series and is not over-differenced). When b = 0, 1 and 2, this definition corresponds to Cressie's ( J. Am. Stat. Assoc. 83 (1988), 1108–16; 85 (1990), 272) semivariogram linvariogram and quadvariogram respectively and is simpler. Under very general conditions about { W ( t )}, we obtain the relationship between γ _b ( h ) and the autocovariance function of { W ( t )} and show that the asymptote of γ _b ( h ) is a straight line having a positive slope when b = d − 1 and levelling out when b ≥ d .
A definition of a sample polyvariogram (SPV) of order b is given and is shown to be an unbiased and consistent estimate of the PV; and further, some uniformly (in h ) almost sure convergence rates are obtained. These properties provide theoretical support for using the SPV to replace the practically unknown PV and generalize the guidelines for identifying d given by Cressie, where { W ( t )} was restricted to a white noise and b ≤ 2. Some further asymptotic theorems and avenues for using them for statistically testing d and parameters of models for { W ( t )} are briefly introduced.