Finite Sample Theory of QMLE in ARCH Models with Dynamics in the Mean Equation

Abstract. We provide simulation and theoretical results concerning the finite‐sample theory of quasi‐maximum‐likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)–ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t‐tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean‐squared error of the maximum‐likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)‐ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1–10] invariance properties for the biases in these situations.     

The restricted likelihood ratio test for autoregressive processes

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (?1, 1] × (?1, 1)^p?1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.     

A Bayesian regime‐switching time‐series model

This article provides a new Bayesian approach for AR(2) time‐series models with multiple regime‐switching points. Our formulation of the regime‐switching model involves a binary discrete variable that indicates the regime change. This variable is specified to be detected by data in each regime. The model is estimated using Stochastic approximation Monte Carlo method proposed by Liang et al. [JASA (2007)]. This methodology is quite useful since it allows for fitting of more complex regime‐switching models without transition constraint. The proposed model is illustrated using simulated and real data such as GNP and US interest rate data.     

Inference in Autoregression under Heteroskedasticity

Abstract. A scalar pth‐order autoregression (AR(p)) is considered with heteroskedasticity of the unknown form delivered by a transition function of time. A limit theory is developed and three heteroskedasticity‐robust test statistics are proposed for inference, one of which is based on the nonparametric estimation of the variance function. The performance of the resulting testing procedures in finite samples is compared in simulations and some suggestions for practical application are given.     

Stability of nonlinear AR‐GARCH models

Abstract. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR(p)] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.     

Non‐stationary autoregressive processes with infinite variance

Consider an AR(p) process , where {?_t} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0<α<2. This time series {Y_t} is said to be a non‐stationary AR(p) process if at least one of its characteristic roots lies on the unit circle. The limit distribution of the least squares estimator (LSE) of for {Y_t} with infinite variance innovation {?_t} is established in this paper. In particular, by virtue of the result of Kurtz and Protter (1991) of stochastic integrals, it is shown that the limit distribution of the LSE is a functional of integrated stable process. Simulations for the estimator of β and its limit distribution are also given.     

Thermally stimulated depolarization current studies on PC/PTBF blends

Thermally stimulated depolarization current (TSDC) studies have been carried out on blends of polycarbonate (PC) and poly(p-t-butyl phenolformaldehyde) (PTBF) using electric poling at temperatures ranging from 348°K to 383°K. The PC/PTBF blends poled at identical electric field (E_p) and temperature (T_p) exhibit a continuous distribution of polarizability (in general, in the range 300°K to 450°K) with a blend composition dependent single peak (T_M). With increasing E_p and T_p, the TSDC peak of a blend shifts toward higher temperature with increasing peak current (I_M) and charge (Q) associated with the peak. The effects of polarization field and temperature indicate that the polarization in the blend system is due to induced dipole formation. The activation energy decreases with increasing PTBF content in the blend, indicating shallow traps in PC/PTBF electret. The present blend electrets, however, comparative to its two components PC and PTBF, store more charge but decay faster. © 1994 John Wiley & Sons, Inc.     

Computer Algebra Derivation of the Bias of Linear Estimators of Autoregressive Models

Abstract. A symbolic method which can be used to obtain the asymptotic bias and variance coefficients to order O(1/n) for estimators in stationary time series is discussed. Using this method, the large‐sample bias of the Burg estimator in the AR(p) for p = 1, 2, 3 is shown to be equal to that of the least squares estimators in both the known and unknown mean cases. Previous researchers have only been able to obtain simulation results for the Burg estimator's bias because this problem is too intractable without using computer algebra. The asymptotic bias coefficient to O(1/n) of Yule–Walker as well as least squares estimates is also derived in AR(3) models. Our asymptotic results show that for the AR(3), just as in the AR(2), the Yule–Walker estimates have a large bias when the parameters are near the nonstationary boundary. The least squares and Burg estimates are much better in this situation. Simulation results confirm our findings.     

Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

Abstract. Here we obtain difference equations for the higher order moments and cumulants of a time series {X_t} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method – the Whittle criterion – to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient.     

Estimating a change point in the long memory parameter

We propose an estimator of change point in the long memory parameter d of an ARFIMA(p, d, q) process using the sup Wald test. We derive the consistency and the rate of convergence of the estimator for the time of change. The convergence rate of our change point estimator depends on the magnitude of a shift. Furthermore, we obtain the limiting distribution of our change point estimator without depending on the distribution of the process. Therefore, we can construct confidence intervals for the change point. Simulations show the validity of the asymptotic theory of our estimator if the sample size is large enough. We apply our change point estimator to the yearly Nile river minimum water level.     

An approximate likelihood function for panel data with a mixed ARMA(p, q) remainder disturbance model

Abstract. An approximate likelihood function for panel data with an autoregressive moving‐average (ARMA)(p, q) model remainder disturbance is presented and Whittle's approximate maximum likelihood estimator (MLE) is used to yield an asymptotic estimator. Although an asymptotic approach, the power test is quite successful for estimating and testing. In this approach, we do not need to calculate the transformation matrix in exact form. Through the Riemann sum approach, we can construct a simple approximate concentrated likelihood function. In addition, the model is also extended to the restricted maximum likelihood (REML) function, in which the package of Gilmour, Thompson and Cullis [Biometrics (1995) Vol. 51, pp. 1440–1450] is applied without difficulty. In the case study, we implement the model on the characteristic line for the investment analysis of Taiwanese computer motherboard makers.     

DETERMINING THE NUMBER OF REGIMES IN MARKOV SWITCHING VAR AND VMA MODELS

We give stable finite‐order vector autoregressive moving average (p ^* ,q ^* ) representations for M‐state Markov switching second‐order stationary time series whose autocovariances satisfy a certain matrix relation. The upper bounds for p ^* and q ^* are elementary functions of the dimension K of the process, the number M of regimes, the autoregressive and moving‐average orders of the initial model. If there is no cancellation, the bounds become equalities, and this solves the identification problem. Our classes of time series include every M‐state Markov switching multi‐variate moving‐average models and autoregressive models in which the regime variable is uncorrelated with the observable. Our results include, as particular cases, those obtained by Krolzig (1997) and improve the bounds given by Zhang and Stine (2001) and Francq and Zakoïan (2001) for our classes of dynamic models. A Monte Carlo experiment and an application on foreign exchange rates complete the article. Copyright © 2013 John Wiley & Sons, Ltd.     

Time-varying autoregressions with model order uncertainty

We explore some aspects of the analysis of latent component structure in non-stationary time series based on time-varying autoregressive (TVAR) models that incorporate uncertainty on model order. Our modelling approach assumes that the AR coefficients evolve in time according to a random walk and that the model order may also change in time following a discrete random walk. In addition, we use a conjugate prior structure on the autoregressive coefficients and a discrete uniform prior on model order. Simulation from the posterior distribution of the model parameters can be obtained via standard forward filtering backward simulation algorithms. Aspects of implementation and inference on decompositions, latent structure and model order are discussed for a synthetic series and for an electroencephalogram (EEG) trace previously analysed using fixed order TVAR models.     

Gas Separation Performance of Polycarbonate Membranes Modified With Multifunctional Low Molecular‐Weight Additives

Abstract

The gas permeation performances and various physical properties of dense polycarbonate (PC) membranes containing low molecular‐weight additives (LMWA) with functional groups were evaluated. The selected LMWAs were p‐nitroaniline, 4‐amino 3‐nitro phenol, catechol and 2‐hydroxy 5‐methyl aniline, and LMWA concentration in the membrane was changed between 1 and 10% (w/w). The PC/LMWA membranes had higher ideal selectivities but lower permeabilities than pure PC membrane. The most effective LMWA which provided the highest selectivity and the lowest permeability in membranes was p‐nitroaniline and the least effective one was catechol. The selected LMWAs antiplasticized the PC matrix and improved the separation performances even at low concentrations. The glass transition temperature (T_g) of PC/LMWA membranes was lower than that of pure PC membrane. Reduction of T_gs as a function of p‐nitroaniline concentration obeyed to Gordon‐Taylor model which indicates antiplasticization type interaction. FTIR spectra of pure and blend membranes also demonstrated the existence of interaction.     

Off-Line Detection of Multiple Change Points by the Filtered Derivative with p-Value Method

Abstract

This article deals with off-line detection of change points, for time series of independent observations, when the number of change points is unknown. We propose a sequential analysis method with linear time and memory complexity. Our method is based, on a filtered derivative method that detects the right change points as well as false ones. We improve the filtered derivative method by adding a second step in which we compute the p-values associated to every single potential change point. Then, we eliminate false alarms; that is, the change points that have p-values smaller than a given critical level. Next, we apply our method and penalized least squares criterion procedure to detect change points on simulated data sets and then we compare them. Eventually, we apply the filtered derivative with p-value method to the segmentation of heartbeat time series, and the detection of change points in the average daily volume of financial time series.     

A NOTE ON THE EMBEDDING OF DISCRETE-TIME ARMA PROCESSES

Abstract. Let {X_n, n= 0, 1, 2,…} be a discrete-time ARMA(p, q) process with q < p whose autoregressive polynomial has r (not necessarily distinct) negative real roots. According to a recent result of He and Wang (On embedding a discrete-parameter ARMA model in a continuous-parameter ARMA model. J. Time Ser. Anal. 10 (1989), 315–23) there exists a continuous-time ARMA (p', q') process {Y(t), t≥0} with q' < p'=p+r such that {Y(n), n= 0, 1, 2,…} has the same autocorrelation function as {X_n}. In this paper we show that this result is false by considering the case when {X_n} is a discrete-time AR(2) process whose autoregressive polynomial has distinct complex conjugate roots. We identify the proper subset of such processes which are embeddable in a continuous-time ARMA(2, 1) process. We show that every discrete-time AR(2) process with distinct complex conjugate roots can be embedded in either a continuous-tie ARMA(2, 1) process or a continuous-time ARMA(4, 2) process, or in some cases both. We derive an expression for the spectral density of the process obtained by sampling a general continuous-time ARMA(p, q) process (with distinct autoregressive roots) at arbitrary equally spaced time points. The expression clearly shows that the sampled process is a discrete-time ARMA (p', q') process with q' < p.     

Local asymptotic normality and efficient estimation for INAR(p) models

Abstract. Integer‐valued autoregressive (INAR) processes have been introduced to model non‐negative integer‐valued phenomena that evolve in time. The distribution of an INAR(p) process is determined by two parameters: a vector of survival probabilities and a probability distribution on the non‐negative integers, called an immigration distribution. This paper provides an efficient estimator of the parameters, and in particular, shows that the INAR(p) model has the Local Asymptotic Normality property.     

Hydrothermal synthesis and piezoelectric property of Ta-doping K0.5Na0.5NbO3 lead-free piezoelectric ceramic

Ta-doping K_0.5Na_0.5Nb_1−xTa_xO₃ (x = 0.1, 0.2, 0.3, 0.4) powder was synthesized by hydrothermal approach and its ceramics were prepared after sintering and polarizing treatment in this work. The K_0.5Na_0.5Nb_0.7Ta_0.3O₃ ceramics near morphotropic phase boundary (MPB), which exhibited optimum piezoelectric properties of d₃₃ = 210 pC/N and good electromechanical coupling factors of K_p = 0.3. The domain structure has been observed from TEM images which indicates that the K_0.5Na_0.5Nb_0.7Ta_0.3O₃ ceramics have good piezoelectric and ferroelectric properties for it is near the MPB.     

BAYES SEQUENTIAL ESTIMATION OF POISSON MEAN UNDER A LINEX LOSS FUNCTION

This paper considers Bayes sequential estimation of the mean of a Poisson distribution using a LINEX loss function and a cost c > 0 for each observation. Under a Gamma prior distribution, it is shown that an asymptotically pointwise optimal rule is asymptotically non-deficient in the sense of Woodroofe (1981).     

RESIDUAL AUTOCOVARIANCES AND UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLE ESTIMATORS FROM TIME SERIES REGRESSION MODELS

Abstract. Hall (Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989), 49–56; Joint hypothesis tests for a random walk based on instrumental variable estimators. J. Time Ser. Anal. 13 (1992), 29–45), Pantula and Hall (Testing for unit roots in autoregressive moving average models:an instrumental variable approach. J. Econometrics 48 (1991), 325–53) and Lee and Schmidt (Unit root tests based on instrumental variable estimation. Int. Econ. Rev. 39 (1994), 449–62) proposed instrumental variable (IV) based tests for a unit root in an ARMA(p+ 1, q) time series. To perform the tests it is essentially necessary to know (p, q) but in many cases this information is unknown. In practice a natural solution to this problem is to estimate (p, q) from the data using a strategy based on the residual autocovariances from the IV regression. In this paper we examine the properties of these residual autocovariances under various assumptions about the true nature of the time series. This analysis allows us to propose a model selection procedure which has desirable asymptotic and finite sample properties whether the time series is stationary or possesses a unit root. A sideproduct of our analysis is that we extend Box and Pierce's (Distribution of residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65 (1970), 1509–26) analysis of the least squares residual autocorrelations to the residual autocovariances from IV regressions.