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.     

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.     

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.     

Tests Based on Simplicial Depth for AR(1) Models With Explosion

We propose outlier a robust and distribution‐free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. The asymptotic distribution of the test statistic is given by a specific Gaussian process. Conditions for the consistency are given, and the power of the test at finite samples is compared with alternative tests. The new test outperforms these tests in the case of skewed errors and outliers. Finally, we apply the method to crack growth data and compare the results with an OLS approach.     

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.     

Likelihood ratio tests for the structural change of an AR(p) model to a Threshold AR(p) model

This article considers the likelihood ratio (LR) test for the structural change of an AR model to a threshold AR model. Under the null hypothesis, it is shown that the LR test converges weakly to the maxima of a two‐parameter vector Gaussian process. Using the approach in Chan and Tong (1990)and Chan (1991), we obtain a parameter‐free limiting distribution when the errors are normal. This distribution is novel and its percentage points are tabulated via a Monte Carlo method. Simulation studies are carried out to assess the performance of the LR test in the finite sample and a real example is given.     

Temporal Aggregation of Lognormal AR processes

We consider temporal aggregation of lognormal autoregressive (AR) processes. More specifically, we develop a novel moment‐matching approximation for temporally aggregated lognormal AR processes. In addition, we show that our approximation provides the closest lognormal AR process in terms of Kullback–Leibler divergence. Moreover, we perform a simulation study to compare our proposed approximation with two competing approximations. This study shows that in terms of L¹‐ and L²‐norm distances our approximation provides superior results. Our results have an important practical application and one main practical implication. In terms of practical application, our approximation can provide possible candidate solutions for simulation‐based algorithms such as the Metropolis–Hastings algorithm. The practical implication gives support to common practice: when the original fine‐level process follows a lognormal AR process but only aggregated data are available, then instead of assuming a Gaussian process it is better to assume a lognormal AR process at the aggregated level. Finally, we illustrate the utility of our results with two applications. The first example considers a simulated dataset whereas the second example examines the number of yearly sunspots in the period 1700–1984.     

Gaussian inference in general AR(1) models based on difference

This article develops a simple difference transformation for estimation and inference in general AR(1) models. As in Paparoditis and Politis (2000, Test 9, 487–509) and Phillips and Han (2008, Econometric Theory 24, 631–650), a Gaussian limit theory with a convergence rate of is available, whether a unit root is present in the process. Yet the novelty of our limit results is that the same weak convergence applies to the models with or without a trend, unlike those established in the literature. The merits promise usefulness of the difference transformation in applications to dynamic panels.     

Testing for structural change of AR model to threshold AR model

The purpose of this article is to develop the likelihood ratio test for the structural change of an AR model to a threshold AR model. It is shown that the log‐likelihood ratio test converges to the maxima of a two‐parameter Gaussian process in distribution. This limiting distribution is novel and we tabulate the critical values. Some simulations are carried out to examine the finite‐sample performance of this test statistic. This article also includes a weak convergence of a two‐parameter marked empirical process, which is of independent interest.     

Some Results on Specification Search and Pre-testing in an ARMA(1,1) Process

In this study we consider simple autoregressive moving-average (ARMA) models of order at most 1. Pre-testing, on the moving-average coefficient θ, is used to choose between an ARMA(1,1) and an AR(1) in a Monte Carlo design. We find that the pre-test estimator is not always dominated by the others, and that the bias and the mean square error of the estimate of the autoregressive coefficient φ very often depend on the sign of the autoregressive and moving-average parameters of the ARMA(1,1) model in the data-generating process. Further, we note that the degrees of size and power distortion of the t test on φ, after pre-testing for θ, are generally associated with model misspecification.     

TESTING FOR A UNIT ROOT IN AN AR(1) TIME SERIES USING IRREGULARLY OBSERVED DATA

Abstract. For an AR(1) model having a unit root with nonconsecutively observed or missing data we consider the ordinary least squares estimator, the one-step Newton-Raphson estimator and an ordinary least squares type estimator which is a simple approximation of the Newton-Raphson estimator. It is shown that the limiting distributions of these estimators of the unit root are the same as those of the regression estimators as tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74 (1979), 427–31) for the complete data situation. Simulation results show that our proposed unit root tests perform very well for small samples.     

A practical development of engineering simulation-assisted educational AR environments

Engineering simulations have opened several gates for today’s chemical engineers. They are powerful tools to provide technical content as physics-based numerical solvers. Augmented reality (AR) and virtual reality (VR), on the other hand, are already underway to digitize environments in many fields. The combination of AR/VR environments and simulations in engineering education has been attracting widespread interest. Literature has demonstrated a massive amount of digital educational environments in several contexts as being complementary to conventional educational methods. Nevertheless, hosting technical content produced by engineering simulations with educational AR/VR is still challenging and requires expertise from multiple disciplines throughout the technical development. Present work provides a facile and agile methodology for low-cost hardware but content-wise rich AR software development. Inspired by the Covid-19 pandemic, a case study is developed to teach chemical-engineering concepts using a liquid-soap synthesis process. Accordingly, we assess and conclude the digital development process to guide inexperienced developers for the digitalization of teaching content. The present contribution serves as an example of the power of integrating AR/VR with traditional engineering simulations for educational purposes. The digital tool developed in this work is shared in the online version.     

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.     

Biosorption potential of Neurospora crassa cells for decolorization of Acid Red 57 (AR57) dye

Biosorption of Acid Red 57 (AR57) on to Neurospora crassa was studied with variation of pH, contact time, biosorbent and dye concentrations and temperature to determine equilibrium and kinetic models. The AR57 biosorption was fast and equilibrium was attained within 40 min. Langmuir, Freundlich and Dubinin–Radushkevich (D–R) isotherm models were applied to experimental equilibrium data for AR57 biosorption at various temperatures. The equilibrium data fitted very well to all the equilibrium models in the studied concentration range of AR57. Maximum biosorption capacity (q_max) of AR57 on to N. crassa was 2.16 × 10^?4 mol g^?1 at 20 °C. The kinetics of biosorption of AR57 were analyzed and rate constants were derived. The overall biosorption process was best described by a pseudo‐second‐order kinetic model. The changes in Gibbs free energy, enthalpy and entropy of biosorption were also evaluated for the biosorption of AR57 on to N. crassa. The results indicate that the biosorption was spontaneous and exothermic in nature. Copyright © 2006 Society of Chemical Industry     

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.     

Solutions of Yule‐Walker equations for singular AR processes

A study is presented on solutions of the Yule‐Walker equations for singular AR processes that are stationary outputs of a given AR system. If the Yule‐Walker equations admit more than one solution and the order of the AR system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial has either only stable zeros (implying that only one stationary output exists for this system and it is linearly regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices, and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of unit circle zeros and the most number of stable zeros over all possible solutions.     

Functional Coefficient Autoregressive Models: Estimation and Tests of Hypotheses

In this paper, we study nonparametric estimation and hypothesis testing procedures for the functional coefficient AR (FAR) models of the form X_t = f ₁( X _{t − d} ) X _{t − 1}+ ... + f_p ( X _{t − d} ) X _{t − p} +ε _t , first proposed by Chen and Tsay (1993). As a direct generalization of the linear AR model, the FAR model is a rich class of models that includes many useful parametric nonlinear time series models such as the threshold AR models of Tong (1983) and exponential AR models of Haggan and Ozaki (1981). We propose a local linear estimation procedure for estimating the coefficient functions and study its asymptotic properties. In addition, we propose two testing procedures. The first one tests whether all the coefficient functions are constant, i.e. whether the process is linear. The second one tests if all the coefficient functions are continuous, i.e. if any threshold type of nonlinearity presents in the process. The results of some simulation studies as well as a real example are presented.     

A Median-Unbiased Estimator of the AR(1) Coefficient

A proof is given that the median of the ratios of consecutive observations of a stationary first-order autoregressive process X_t = α X _{t −1} + Y_t with P ( Y_t ≥ 0) = P ( Y_t ≤ 0) = 1/2 and P ( X_t = 0) = 0 is a median-unbiased estimator of α.     

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.     

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.