Consistent Estimation of Linear and Non-linear Autoregressive Models with Markov Regime

An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time-point is given by a (non-observable) Markov chain. We examine maximum likelihood estimation for such models and show consistency of a conditional maximum likelihood estimator. Also identifiability issues are discussed     

Reversible Jump Markov Chain Monte Carlo Strategies for Bayesian Model Selection in Autoregressive Processes

Abstract.  This paper addresses the problem of Bayesian inference in autoregressive (AR) processes in the case where the correct model order is unknown. Original hierarchical prior models that allow the stationarity of the model to be enforced are proposed. Obtaining the quantities of interest, such as parameter estimates, predictions of future values of the time series, posterior model-order probabilities, etc., requires integration with respect to the full posterior distribution, an operation which is analytically intractable. Reversible jump Markov chain Monte Carlo (MCMC) algorithms are developed to perform the required integration implicitly by simulating from the posterior distribution. The methods developed are evaluated in simulation studies on a number of synthetic and real data sets.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

Bootstrap Tests for an Autoregressive Unit Root in the Presence of Weakly Dependent Errors

This paper examines bootstrap tests of the null hypothesis of an autoregressive unit root in models that may include a linear rend and/or an intercept and which are driven by innovations that belong to the class of stationary and invertible linear processes. Our approach makes use of a sieve bootstrap procedure based on residual resampling from autoregressive approximations, the order of which increases with the sample size at a suitable rate. We show that the sieve bootstrap provides asymptotically valid tests of the unit-root hypothesis and demonstrate the small-sample effectiveness of the method by means of simulation.     

A Joint Regression Variable and Autoregressive Order Selection Criterion

Abstract.  In linear regression models with autocorrelated errors, we apply the residual likelihood approach to obtain a residual information criterion (RIC), which can jointly select regression variables and autoregressive orders. We show that RIC is a consistent criterion. In addition, our simulation studies indicate that it outperforms heuristic selection criteria – the Akaike information criterion and the Bayesian information criterion – when the signal-to-noise ratio is not weak.     

On Asymptotic Theory for ARCH (∞) Models

Autoregressive conditional heteroskedasticity (ARCH)() models nest a wide range of ARCH and generalized ARCH models including models with long memory in volatility. Existing work assumes the existence of second moments. However, the fractionally integrated generalized ARCH model, one version of a long memory in volatility model, does not have finite second moments and rarely satisfies the moment conditions of the existing literature. This article weakens the moment assumptions of a general ARCH( ) class of models and develops the theory for consistency and asymptotic normality of the quasi‐maximum likelihood estimator.     

Testing for a Unit Root in Noncausal Autoregressive Models

This work develops maximum likelihood‐based unit root tests in the noncausal autoregressive (NCAR) model with a non‐Gaussian error term formulated by Lanne and Saikkonen (2011, Journal of Time Series Econometrics 3, Issue 3, Article 2). Finite‐sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.     

Empirical likelihood confidence intervals for the mean of a long‐range dependent process

Abstract. This paper considers blockwise empirical likelihood for real‐valued linear time processes which may exhibit either short‐ or long‐range dependence. Empirical likelihood approaches intended for weakly dependent time series can fail in the presence of strong dependence. However, a modified blockwise method is proposed for confidence interval estimation of the process mean, which is valid for various dependence structures including long‐range dependence. The finite‐sample performance of the method is evaluated through a simulation study and compared with other confidence interval procedures involving subsampling or normal approximations.     

基于区域协调分析的约束内模控制方法与研究

针对多变量系统控制输入变量受到约束的严格限制时,一般工业上都是用预测控制来显式的处理这些约束条件和变量之间的关联耦合作用,而用内模控制来解决这方面的问题还处在探索中这一现状,初步性地利用内模控制的思想,用静态优先级来协调变量之间的耦合关联,并结合模型预测理论知识,利用区域分析法判断,使控制量处在约束范围内。仿真结果表明这种设计方法对控制效果具有一定的可实现性和意义。     

Estimation of the Dominating Frequency for Stationary and Nonstationary Fractional Autoregressive Models

This paper was motivated by the investigation of certain physiological series for premature infants. The question was whether the series exhibit periodic fluctuations with a certain dominating period. The observed series are nonstationary and/or have long-range dependence. The assumed model is a Gaussian process X _t whose m th difference Y_t = (1 − B ) ^m X_t is stationary with a spectral density f that may have a pole (or a zero) at the origin. the problem addressed in this paper is the estimation of the frequency ω_max where f achieves the largest local maximum in the open interval (0, π). The process X_t is assumed to belong to a class of parametric models, characterized by a parameter vector θ, defined in Beran (1995). An estimator of ω_max is proposed and its asymptotic distribution is derived, with θ being estimated by maximum likelihood. In particular, m and a fractional differencing parameter that models long memory are estimated from the data. Model choice is also incorporated. Thus, within the proposed framework, a data driven procedure is obtained that can be applied in situations where the primary interest is in estimating a dominating frequency. A simulation study illustrates the finite sample properties of the method. In particular, for short series, estimation of ω_max is difficult, if the local maximum occurs close to the origin. The results are illustrated by two of the data examples that motivated this research.     

Towards a Unified Approach for Proving Geometric Ergodicity and Mixing Properties of Nonlinear Autoregressive Processes

Abstract. In this paper we attempt to establish unified sufficient conditions for geometric ergodicity of autoregressive models. It is shown that there is a close relationship between geometric ergodicity and mixing properties. The case of nonstationary time series is incorporated into the investigations. Several time series models including threshold and EXPARCH‐models are examined with respect to geometric ergodicity. In some cases we obtain regions of geometric ergodicity in the parameter space, which are larger than that known from the literature.     

Empirical likelihood intervals for conditional Value‐at‐Risk in ARCH/GARCH models

Value‐at‐Risk (VaR) is a simple, but useful measure in risk management. When some volatility model is employed, conditional VaR is of importance. As autoregressive conditional heteroscedastic (ARCH) and generalized ARCH (GARCH) models are widely used in modelling volatilities, in this article, we propose empirical likelihood methods to obtain an interval estimation for the conditional VaR with the volatility model being an ARCH/GARCH model.     

Joint Determination of the State Dimension and Autoregressive Order for Models with Markov Regime Switching

Abstract. This paper is concerned with the problem of joint determination of the state dimension and autoregressive order of models with Markov‐switching parameters. A model selection procedure is proposed which is based on optimization of complexity‐penalized likelihood criteria. The efficacy of the procedure is evaluated by means of Monte Carlo experiments.     

A Note on the Information Matrix for Multiplicative Seasonal Autoregressive Moving‐Average Models

Abstract. It is shown that there is an invariance property for each of the elements of the information matrix of a multiplicative seasonal autoregressive moving‐average time‐series model, which enables the integral specification of Whittle (1953a,b) to be solved in a straightforward way. The resulting non‐iterative closed procedure shares the property possessed by the piecemeal approach of Godolphin and Bane (2006) of being independent of the seasonal period, but our procedure is preferable if one or more orders of the seasonal components of the model are greater than unity. The procedure is therefore simpler, in general, than the iterative method of Klein and Mélard (1990) that depends necessarily on the seasonal period. In the strictly non‐seasonal case this invariance property prescribes a non‐iterative closed procedure for evaluating the information matrix which improves on the methods of Godolphin and Unwin (1983) , Friedlander (1984) , McLeod (1984) and Klein and Spreij (2003) . Three illustrations of the approach are given.     

A Unified Model for the Mixing of Non‐Newtonian Fluids in the Laminar,Transition, and Turbulent Region

Summary: Non‐Newtonian fluid behavior has significant influence on quantities in chemical engineering like power input, mixing time, heat transfer etc. In the laminar flow region, the concept of effective viscosity by Metzner and Otto is well established. In the transition region between laminar and turbulent flow, the existing concepts use three and even more empirical parameters to determine the specific power input. Here, a unified and general but simple approach is introduced to calculate the power input for shear thinning fluids over the whole flow region using just one empirical parameter. The Metzner‐Otto relation is obtained as a limiting case for the laminar region. The empirical parameter of the new approach is related to the Metzner‐Otto constant. The concept is validated for eight different stirrer systems. Mixing time and maximum shear rate and heat transfer can also be calculated using this approach. The new concept presented should also be applicable for other apparatuses, e.g., static mixers.

Comparison of experimental data and a curve calculated according to the new method (solid line).     

QMLE for Quadratic ARCH Model with Long Memory

We discuss parametric quasi‐maximum likelihood estimation for quadratic autoregressive conditional heteroskedasticity (ARCH) process with long memory introduced in Doukhan emphet al. (2016) and Grublyt? and ?karnulis (2016) with conditional variance involving the square of inhomogeneous linear combination of observable sequence with square summable weights. The aforementioned model extends the quadratic ARCH model of Sentana ( 1995 ) and the linear ARCH model of Robinson ( 1991 ) to the case of strictly positive conditional variance. We prove consistency and asymptotic normality of the corresponding quasi‐maximum likelihood estimates, including the estimate of long memory parameter 0 < d < 1/2. A simulation study of empirical mean‐squared error is included.     

A Novel Synthetic Antibody Library with Complementarity-Determining Region Diversities Designed for an Improved Amplification Profile

Antibody discovery by phage display consists of two phases, i.e., the binding phase and the amplification phase. Ideally, the selection process is dominated by the former, and all the retrieved clones are amplified equally during the latter. In reality, the amplification efficiency of antibody fragments varies widely among different sequences and, after a few rounds of phage display panning, the output repertoire often includes rapidly amplified sequences with low or no binding activity, significantly diminishing the efficiency of antibody isolation. In this work, a novel synthetic single-chain variable fragment (scFv) library with complementarity-determining region (CDR) diversities aimed at improved amplification efficiency was designed and constructed. A previously reported synthetic scFv library with low, non-combinatorial CDR diversities was panned against protein A superantigen, and the library repertoires before and after the panning were analyzed by next generation sequencing. The enrichment or depletion patterns of CDR sequences after panning served as the basis for the design of the new library. Especially for CDR-H3 with a higher and more random diversity, a machine learning method was applied to predict potential fast-amplified sequences among a simulated sequence repertoire. In a direct comparison with the previous generation library, the new library performed better against a panel of antigens in terms of the number of binders isolated, the number of unique sequences, and/or the speed of binder enrichment. Our results suggest that the amplification-centric design of sequence diversity is a valid strategy for the construction of highly functional phage display antibody libraries.     

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.