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.     

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.     

On Hypotheses Testing for the Selection of Spatio‐Temporal Models

Abstract. Several models have been proposed in recent years for analysing spatial data and also, to some extent, spatio‐temporal data. One of the important problems, namely the choice of an appropriate model for describing real data sets, remains unsolved. Here we consider the analysis of spatio‐temporal processes from which observations over space and time are available. We propose statistical tests for discriminating between space–time autoregressive processes and multivariate autoregressive processes. The sampling properties of the proposed tests are considered. We illustrate the methods with a real example. We use the above tests to find the best model to describe spatio‐temporal variations of hourly carbon monoxide measurements at four locations in London in January 2004.     

Fractional integration and structural breaks at unknown periods of time

Abstract. This article deals with the analysis of structural breaks in the context of fractionally integrated models. We assume that the break dates are unknown and that the different sub‐samples possess different intercepts, slope coefficients and fractional orders of integration. The procedure is based on linear regression models using a grid of values for the fractional differencing parameters and least squares estimation. Several Monte Carlo experiments conducted across the study show that the procedure performs well if the sample size is large enough. Two empirical applications are described at the end of the article.     

Testing for co‐integration and nonlinear adjustment in a smooth transition error correction model

This article introduces a testing procedure for cointegration and nonlinear adjustment in a smooth transition vector error correction model. To overcome the unidentified parameters problem under the null of no‐cointegration, the Wald statistic is optimized over the unidentified parameter space. The asymptotic distribution of the test statistic is shown to be non‐standard but nuisance parameter‐free and hence critical values are obtained by simulations, Simulations show that the proposed test outperforms the alternatives in small sample sizes both in terms of size and power. Application to the exchange rate‐monetary fundamentals relationship show that the proposed test works considerably well. This article also finds that nonlinear adjustment dynamics are symmetric for some currencies and therefore the speed of adjustment depends on the size of the deviations and is asymmetric for others, hence, the adjustment dynamics depend not only on the size but also on the sign of the deviations.     

Likelihood inference for discriminating between long‐memory and change‐point models

We develop a likelihood ratio (LR) test procedure for discriminating between a short‐memory time series with a change‐point (CP) and a long‐memory (LM) time series. Under the null hypothesis, the time series consists of two segments of short‐memory time series with different means and possibly different covariance functions. The location of the shift in the mean is unknown. Under the alternative, the time series has no shift in mean but rather is LM. The LR statistic is defined as the normalized log‐ratio of the Whittle likelihood between the CP model and the LM model, which is asymptotically normally distributed under the null. The LR test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006) . Moreover, the LR test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change‐in‐mean. We show its good performance in simulations and apply it to two data examples.     

Test for the null hypothesis of cointegration with reduced size distortion

Abstract. This article considers a single‐equation cointegrating model and proposes the locally best invariant and unbiased (LBIU) test for the null hypothesis of cointegration. We derive the local asymptotic power functions and compare them with the standard residual‐based test, and show that the LBIU test is more powerful in a wide range of local alternatives. Then, we conduct a Monte Carlo simulation to investigate the finite sample properties of the tests and show that the LBIU test outperforms the residual‐based test in terms of both size and power. The advantage of the LBIU test is particularly patent when the error is highly autocorrelated. Furthermore, we point out that finite sample performance of existing tests is largely affected by the initial value condition while our tests are immune to it. We propose a simple transformation of data that resolves the problem in the existing tests.     

Testing for the Cointegrating Rank of a VAR Process with Level Shift and Trend Break

Abstract. A test for the cointegrating rank of a vector autoregressive (VAR) process with a possible shift and broken linear trend is proposed. The break point is assumed to be known. Our test is not a likelihood ratio test but the deterministic terms including the broken trends are removed first by a generalized least squares procedure. Then, a likelihood ratio‐type test is applied to the adjusted series. The asymptotic null distribution of the test is derived and it is shown by a Monte Carlo experiment that the test has better small‐sample properties in many cases than a corresponding Gaussian likelihood ratio test for the cointegrating rank. Moreover, response surface techniques can be used to easily obtain p‐values of the test for any possible break date.     

Tests for Linearity in Star Models: Supwald and Lm‐Type Tests

This article investigates approximation and supremum approaches for testing linearity in smooth transition autoregressive (STAR) models. We show that since the approximation of STAR models by Taylor series expansions may not accurately describe the specific transition dynamic when the process is away from the null, LM‐type tests may fail to detect the form of nonlinearity for which they are designed for. Investigating a supremum approach, the article provides the asymptotic distribution of a SupWald test that is obtained by taking the supremum of a Wald statistic over the Cartesian product of the spaces for the transition and threshold parameters. Simulated asymptotic critical values for the resulting tests are provided for a wide range of autoregressive orders and shown to differ across exponential and logistic STAR (ESTAR and LSTAR) models. Monte Carlo experiments show that SupWald tests for ESTAR and LSTAR models outperform LM‐type tests, compares well relative to the recently developed score‐based tests and each SupWald statistic performs the best against the true alternative for which it is formed. SupWald tests also provide results that are consistent with the findings from (independently) estimating and diagnostic testing of STAR models in real exchange rate data.     

Fault detection of large-scale process control system with higher-order statistical and interpretative structural model

Nonlinear characteristic fault detection and diagnosis method based on higher-order statistical (HOS) is an effec-tive data-driven method, but the calculation costs much for a large-scale process contr...     

基于高阶统计和解释结构模型相结合的大型过程控制系统的故障检测(英文)

Nonlinear characteristic fault detection and diagnosis method based on higher-order statistical (HOS) is an effec-tive data-driven method, but the calculation costs much for a large-scale process control system. An HOS-ISM fault diagnosis framework combining interpretative structural model (ISM) and HOS is proposed:(1) the adja-cency matrix is determined by partial correlation coefficient;(2) the modified adjacency matrix is defined by directed graph with prior knowledge of process piping and instrument diagram;(3) interpretative structural for large-scale process control system is built by this ISM method;and (4) non-Gaussianity index, nonlinearity index, and total nonlinearity index are calculated dynamical y based on interpretative structural to effectively eliminate uncertainty of the nonlinear characteristic diagnostic method with reasonable sampling period and data window. The proposed HOS-ISM fault diagnosis framework is verified by the Tennessee Eastman process and presents improvement for highly non-linear characteristic for selected fault cases.     

A sequential procedure to determine the number of breaks in trend with an integrated or stationary noise component

Perron and Yabu (2009a) consider the problem of testing for a break occurring at an unknown date in the trend function of a univariate time series when the noise component can be either stationary or integrated. This article extends their work by proposing a sequential test that allows one to test the null hypothesis of, say, l breaks versus the alternative hypothesis of (l + 1) breaks. The test enables consistent estimation of the number of breaks. In both stationary and integrated cases, it is shown that asymptotic critical values can be obtained from the relevant quantiles of the limit distribution of the test for a single break. Monte Carlo simulations suggest that the procedure works well in finite samples.     

On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential Equations*

Abstract. A method for estimating the parameters of stochastic differential equations (SDEs) by simulated maximum likelihood is presented. This method is feasible whenever the underlying SDE is a Markov process. Estimates are compared to those generated by indirect inference, discrete and exact maximum likelihood. The technique is illustrated with reference to a one‐factor model of the term structure of interest rates using 3‐month US Treasury Bill data.     

Estimation of the long‐memory stochastic volatility model parameters that is robust to level shifts and deterministic trends

I provide conditions under which the trimmed FDQML estimator, advanced by McCloskey (2010) in the context of fully parametric short‐memory models, can be used to estimate the long‐memory stochastic volatility model parameters in the presence of additive low‐frequency contamination in log‐squared returns. The types of low‐frequency contamination covered include level shifts as well as deterministic trends. I establish consistency and asymptotic normality in the presence or absence of such low‐frequency contamination under certain conditions on the growth rate of the trimming parameter. I also provide theoretical guidance on the choice of trimming parameter by heuristically obtaining its asymptotic MSE‐optimal rate under certain types of low‐frequency contamination. A simulation study examines the finite sample properties of the robust estimator, showing substantial gains from its use in the presence of level shifts. The finite sample analysis also explores how different levels of trimming affect the parameter estimates in the presence and absence of low‐frequency contamination and long‐memory.     

Improved Tests for Forecast Comparisons in the Presence of Instabilities

Of interest is comparing the out‐of‐sample forecasting performance of two competing models in the presence of possible instabilities. To that effect, we suggest using simple structural change tests, sup‐Wald and UDmax for changes in the mean of the loss differences. It is shown that Giacomini and Rossi ( 2010 ) tests have undesirable power properties, power that can be low and non‐increasing as the alternative becomes further from the null hypothesis. On the contrary, our statistics are shown to have higher monotonic power, especially the UDmax version. We use their empirical examples to show the practical relevance of the issues raised.     

Implementing Residual‐Based KPSS Tests for Cointegration with Data Subject to Temporal Aggregation and Mixed Sampling Frequencies

We show how different data types (stocks and flows) and temporal aggregation affect the size and power of the dynamic ordinary least squares residual‐based Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test of the null of cointegration. Size may be more effectively controlled by setting the minimum number of leads equal to one – as opposed to zero – when selecting the lag/lead order of the dynamic ordinary least squares regression using aggregated data, but at a cost to power. If high‐frequency data for one or more series are available – that is, the model has mixed sampling frequencies – we show how to effectively utilize the high‐frequency data to increase power while controlling size.     

Local power of likelihood‐based tests for cointegrating rank: Comparative analysis of full and partial systems

This note investigates local power properties of likelihood‐based cointegrating rank tests for partial and full vector autoregressive systems. The asymptotic distributions of partial likelihood‐based tests under local alternatives are derived, depending on various specifications of deterministic terms. A simulation study is then performed using both the full and partial systems. It is demonstrated that the rank tests based on the partial system, if a required parametric condition is fulfilled, can be more powerful than those based on the full system. This finding encourages testing cointegrating rank using a partial system as well as a full system, in such circumstances as the parametric condition could be satisfied.     

Testing for the Presence of a Random Walk in Series with Structural Breaks

We consider tests for the presence of a random walk component in a stationary or trend stationary time series and extend them to series that contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution is obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its asymptotic distribution is not dependent on the location of the break point and its form is that of the generalized Cramer–von Mises distribution, with degrees of freedom depending on the number of break points. The performance of this modified test is shown, via some simulation experiments, to be comparable with that of the LBI test. An unconditional test, based on the assumption that there is a single break at an unknown point, is also examined. The use of the tests is illustrated with data on the flow of the Nile and US gross national product.     

Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients

We focus on output feedback control of distributed processes whose infinite dimensional representation in appropriate Hilbert subspaces can be decomposed to finite dimensional slow and infinite dimensional fast subsystems. The controller synthesis issue is addressed using a refined adaptive proper orthogonal decomposition (APOD) approach to recursively construct accurate low dimensional reduced order models (ROMs) based on which we subsequently construct and couple almost globally valid dynamic observers with robust controllers. The novelty lies in modifying the data ensemble revision approach within APOD to enlarge the ROM region of attraction. The proposed control approach is successfully used to regulate the Kuramoto‐Sivashinsky equation at a desired steady state profile in the absence and presence of uncertainty when the unforced process exhibits nonlinear behavior with fast transients. The original and the modified APOD approaches are compared in different conditions and the advantages of the modified approach are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4595–4611, 2013     

FE coupled to SPH numerical model for the simulation of high-velocity impact on ceramic based ballistic shields

Predictive models are an important tool in the design and optimization of ballistic shields. Indeed, several authors in the literature have developed numerical models for simulating high-velocity impact on ceramic-based ballistic shields which are based on the finite element method. Element erosion is usually implemented in finite element models simulating impact to remove excessively distorted elements but, it leads to energy loss, which in turns may lead to the production of incorrect results. Due to the absence of a fixed mesh, the smoothed particle hydrodynamics method is well suited for large deformation problems, overcoming the limitations of the finite element method. On the other hand, the smoothed particle hydrodynamics method is computationally more expensive than the finite element method. Thus, a numerical model combining the lower computational cost of finite elements and the capability of smoothed particle hydrodynamics of dealing with crack formation and fracturing would be an interesting solution for the simulation of high-velocity impact on ceramics. The aim of this work is therefore to develop a finite element coupled to smoothed particle hydrodynamics numerical model for the simulation of high-velocity impact on ceramic-based ballistic shields. High-velocity impact tests were performed on Al₂O₃ tiles and the experimental results were used for the calibration of the numerical models; furthermore, high-velocity impact test were performed on multilayer targets with Al₂O₃ front layer and AA6061-T6 backing layer for the validation of the numerical models. This study proved that this approach is more appropriate for the simulation of the response of ceramic materials rather the common finite element model.