Optimal convergence rates in non‐parametric regression with fractional time series errors

Consider the estimation of g^(ν), the νth derivative of the mean function, in a fixed‐design non‐parametric regression model with stationary time series errors ξ_i. We assume that , ξ_i are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξ_i has the form as λ → 0 with constants c_f > 0 and α ∈ (?1,1). Under regularity conditions, the optimal convergence rate of is shown to be with r = (1 ? α)(k ? ν)/(2k+1 ? α). This rate is achieved by local polynomial fitting. Moreover, in spite of including long memory and antipersistence, the required conditions on the innovation distribution turn out to be the same as in non‐parametric regression with iid errors.     

Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion

Abstract. Fractional Brownian motion is a mean‐zero self‐similar Gaussian process with stationary increments. Its covariance depends on two parameters, the self‐similar parameter H and the variance C. Suppose that one wants to estimate optimally these parameters by using n equally spaced observations. How should these observations be distributed? We show that the spacing of the observations does not affect the estimation of H (this is due to the self‐similarity of the process), but the spacing does affect the estimation of the variance C. For example, if the observations are equally spaced on [0, n] (unit‐spacing), the rate of convergence of the maximum likelihood estimator (MLE) of the variance C is . However, if the observations are equally spaced on [0, 1] (1/n‐spacing), or on [0, n²] (n‐spacing), the rate is slower, . We also determine the optimal choice of the spacing Δ when it is constant, independent of the sample size n. While the rate of convergence of the MLE of C is in this case, irrespective of the value of Δ, the value of the optimal spacing depends on H. It is 1 (unit‐spacing) if H = 1/2 but is very large if H is close to 1.     

Testing for parameter constancy in general causal time‐series models

We consider a process belonging to a large class of causal models including AR(∞), ARCH(∞), TARCH(∞),… processes. We assume that the model depends on a parameter and consider the problem of testing for change in the parameter. Two statistics and are constructed using quasi‐likelihood estimator of the parameter. Under the null hypothesis that there is no change, it is shown that each of these two statistics weakly converges to the supremum of the sum of the squares of independent Brownian bridges. Under the alternative of a change in the parameter, we show that the test statistic diverges to infinity. Some simulation results for AR(1), ARCH(1), GARCH(1,1) and TARCH(1) models are reported to show the applicability and the performance of our procedure with comparisons to some other approaches.     

High‐frequency sampling of a continuous‐time ARMA process

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.     

A Shrinked Forecast in Stationary Processes Favouring Percentage Error

Abstract. In stationary time‐series forecasting, the commonly used criterion for selecting a proper forecast is the mean square error (MSE), which is minimized by the conditional expectation of future observation given the entire past known as a minimum MSE forecast. In this paper, mean square percentage error (MSPE) instead of is used to forecast autoregressive moving average (ARMA)(p,q) series. The suggested forecast takes the form of or (CV_t+1 is the coefficient of variation for one step ahead) times the minimum MSE forecast, which performs better not only in MSPE, but also in mean absolute percentage error (MAPE) than the ordinary MSE forecast in simulation studies. A real data example also supported this result. We conclude that, if percentage error is a prime concern, this shrinked version of MSE forecast performs better than the ordinary forecast in the stationary ARMA(p,q) model.     

On robust tail index estimation for linear long‐memory processes

We consider robust estimation of the tail index α for linear long‐memory processes with i.i.d. innovations ε_j following a symmetric α‐stable law (1 < α < 2) and coefficients a_j ～ c·j^?β. Estimates based on the left and right tail respectively are obtained together with a combined statistic with improved efficiency, and a test statistic comparing both tails. Asymptotic results are derived. Simulations illustrate the finite sample performance.     

Limit theory for a general class of GARCH models with just barely infinite variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ?_t = σ_tη_t and σ_t = f (σ_t?1, σ_t?2,…, σ_t?p, ?_t?1, ?_t?2,…, ?_t?q), when {?_t} is a process with just barely infinite variance, that is, {?_t} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.     

Identification of Persistent Cycles in Non‐Gaussian Long‐Memory Time Series

Abstract. Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non‐Gaussian long‐memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long‐memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non‐Gaussian long‐memory moving‐average process and (c_n,k,1 ≤ k ≤ n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms      

Uniform Limit Theory for Stationary Autoregression

Abstract. First order autoregression is shown to satisfy a limit theory which is uniform over stationary values of the autoregressive coefficient ρ = ρ_n ∈ [0, 1) provided (1 ? ρ_n)n → ∞. This extends existing Gaussian limit theory by allowing for values of stationary ρ that include neighbourhoods of unity provided they are wider than O(n^?1), even by a slowly varying factor. Rates of convergence depend on ρ and are at least but less than n. Only second moments are assumed, as in the case of stationary autoregression with fixed ρ.     

Robust estimation of the scale and of the autocovariance function of Gaussian short‐ and long‐range dependent processes

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time‐series modelling. In this article, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Ma and Genton (2000) are established for Gaussian processes, with either short‐ or long‐range dependence. It is shown in the short‐range dependence setting that this robust estimator is asymptotically normal at the rate , where n is the number of observations. An explicit expression of the asymptotic variance is also given and compared with the asymptotic variance of the classical autocovariance estimator. In the long‐range dependence setting, the limiting distribution displays the same behaviour as that of the classical autocovariance estimator, with a Gaussian limit and rate when the Hurst parameter H is less than 3/4 and with a non‐Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when H ∈ (3/4,1). Some Monte Carlo experiments are presented to illustrate our claims and the Nile River data are analysed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.     

Using the HEGY Procedure When Not All Roots Are Present

Abstract. Empirical studies have shown little evidence to support the presence of all unit roots present in the Δ₄ filter in quarterly seasonal time series. This paper analyses the performance of the Hylleberg, Engle, Granger and Yoo [Journal of Econometrics (1990) Vol. 44, pp. 215–238] (HEGY) procedure when the roots under the null are not all present. We exploit the vector of quarters representation and cointegration relationship between the quarters when factors (1 − L), (1 + L), (1 + L²), (1 − L²) and (1 + L + L² + L³) are a source of nonstationarity in a process in order to obtain the distribution of tests of the HEGY procedure when the underlying processes have a root at the zero, Nyquist frequency, two complex conjugates of frequency π/2 and two combinations of the previous cases. We show both theoretically and through a Monte Carlo analysis that the t‐ratios t and t and the F‐type tests used in the HEGY procedure have the same distribution as under the null of a seasonal random walk when the root(s) is (are) present, although this is not the case for the t‐ratio tests associated with unit roots at frequency π/2.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

A p‐Order signed integer‐valued autoregressive (SINAR(p)) model

In this article, we propose an extension of integer‐valued autoregressive INAR models. Using a signed version of the thinning operator, we define a larger class of ‐valued processes, called SINAR, which can have positive as well as negative correlations. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. In particular, it is shown that the autocorrelation function of any real‐valued AR process can be recovered with a SINAR process, which improves INAR modeling.     

ON M‐Estimation Under Long‐Range Dependence in Volatility

Abstract. We consider M‐estimation of a location parameter for processes with zero autocorrelations but long‐range dependence in volatility. The observed process is the product of i.i.d. Gaussian observations and a long‐memory Gaussian process. For nonlinear estimators, the rate of convergence depends on the type of the ψ‐function. For skew‐symmetric ψ‐functions, a central limit theorem with ‐rate of convergence holds, under suitable regularity assumptions. This is not true in general for M‐estimators where the ψ‐function is not skewsymmetric.     

NON‐STATIONARITY AND QUASI‐MAXIMUM LIKELIHOOD ESTIMATION ON A DOUBLE AUTOREGRESSIVE MODEL

This article first studies the non‐stationarity of the first‐order double AR model, which is defined by the random recurrence equation , where γ₀ > 0, α₀ ≥ 0, and {η_t}is a sequence of i.i.d. symmetric random variables. It is shown that the double AR(1) model is explosive under the condition . Based on this, it is shown that the quasi‐maximum likelihood estimator of (φ₀,α₀) is consistent and asymptotically normal so that the unit root problem does not exist in the double AR(1) model. Simulation studies are carried out to assess the performance of the quasi‐maximum likelihood estimator in finite samples.     

STATIONARITY AND CENTRAL LIMIT THEOREM ASSOCIATED WITH BILINEAR TIME SERIES MODELS

Abstract. Consider the general bilinear times series model where {X_t; t= 0, L1, …} is a p-variate process, C (p x (s+ 1)), A (p x p). B _t(p x p) (1 ≤j≤q) are arbitrary matrices of constants, εT=[εt,…εt-q+1] and {εt; t=0, ±1, …} is a strictly stationary ergodic sequence of random variables. We investigate a set of minimal regularity conditions (on C, A, B _j and {ε_t}) under which we can establish the existence and causality of X _t and the asymptotic normality of the sample mean derived from { X _t}.     

Estimation for non‐negative time series with heavy‐tail innovations

For moving average processes where the coefficients are non‐negative and the innovations are positive random variables with a regularly varying tail at infinity, we provide estimates for the coefficients based on the ratio of two sample values chosen with respect to an extreme value criteria. We then apply this result to obtain estimates for the parameters of non‐negative ARMA models. Weak convergence results for the joint distribution of our estimates are established and a simulation study is provided to examine the small sample size behaviour of these estimates.     

Polymer melt elongation—methods,results, and recent developments

For film blowing of polyethylene it has been shown previously that melt elongation is very powerful for polymer characterization. With two types of rheometers, simple (also called “uniaxial”) elongational tests as well as creep tests can be performed homogeneously. In simple elongation, the melts of branched polyethylene show a remarkable strain hardening. With respect to their advantages and disadvantages, these rheometers complement each other. For multiaxial elongations the various modes of deformation can be performed by means of the rotary clamp technique. With the strain rate components ordered such that \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \varepsilon $\end{document}₁₁ ? \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \varepsilon $\end{document}₂₂ ≥ \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \varepsilon $\end{document}₃₃, the ratio m = \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \varepsilon $\end{document}₂₂/\documentclass{article}\pagestyle{empty}\begin{document}$ \dot \varepsilon $\end{document}₁₁ characterizes the test mode. The Stephenson definition of the elongational viscosities makes use of the linear viscoelastic material equation and proves to be very efficient because the linear shear viscosity (t) (“stressing” viscosity) can act as the reference for the nonlinear behavior in elongation. Results are given for polyisobutylene measured not only in simple, equibiaxial, and planar elongations, but also in new test modes with a change of m during the deformation. This allows one to investigate the consequences of a deformation-induced anisotropy of the rheological behavior.     

Simulation and optimal design of seeded continuous emulsion polymerization process

Simulation and optimal design of the reactor for the seeded continous emulsion polymerization process have been done in this work. An internal mixer (Toray Hi-Mixer) as seeder connected with a stirred tank is designed to correlate conversion, molecular weight, and MWD with the model simulation proposed. An optimal mean residence time of seeder \documentclass{article}\pagestyle{empty}\begin{document}$(\bar \theta _s)_c$\end{document} is found to lie between \documentclass{article}\pagestyle{empty}\begin{document}$(\bar \theta _1)$\end{document} and t_in, where \documentclass{article}\pagestyle{empty}\begin{document}$(\bar \theta _1)_{{\rm opt}} = (3aS_0 /2r\eta N_a \alpha)^{3/5}$\end{document} and t_in = 1.57(aS₀/r_iηN_A)^3.5. The optimal design of the process is performed according to the above relations under several polymerization conditions. In general, the increase in number of stages inside the seeder can reduced the volume of CSTR for a required production. Molecular weight of products is increased by increasing the number of stages inside the seeder, by decrasing the concentration of the initiator, and by increasing the concentration of the emulsifier under the optimal conditons.