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.     

Tests against stationary and explosive alternatives in vector autoregressive models

Abstract. The article proposes new tests for the number of unit roots in vector autoregressive models based on the eigenvalues of the companion matrix. Both stationary and explosive alternatives are considered. The limiting distributions of test statistics depend only on the number of unit roots. Size and power are investigated, and it is found that the new test against some stationary alternatives compares favourably with the widely used likelihood ratio test for the cointegrating rank. The powers are prominently higher against explosive than against stationary alternatives. Some empirical examples are provided to show how to use the new tests with real data.     

Efficient estimation and inference in cointegrating regressions with structural change

Abstract. This paper investigates an efficient estimation method for a cointegrating regression model with structural change. Our proposal is that we first estimate the break point by minimizing the sum of squared residuals and then, by replacing the break fraction with the estimated one, we estimate the regression model by the canonical cointegrating regression (CCR) method proposed by Park [ Econometrica (1992 ) Vol. 60, pp. 119–143]. We show that the estimator of the break fraction has the same convergence rate as obtained in Bai, Lumsdaine and Stock [ Review of Economic Studies (1998 ) Vol. 65, pp. 395–432] and that the CCR estimator with the estimated break fraction has the same asymptotic property as the estimator with the known break point. However, we also show that our method breaks down when the magnitude of structural change is very small. Simulation experiments reveal how the finite sample distribution approaches the limiting distribution as the magnitude of the break and or the sample size increases.     

Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are non‐stationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non‐fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a Chi‐squared‐test.     

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.     

Consistent Monitoring of Cointegrating Relationships: The US Housing Market and the Subprime Crisis

We propose a consistent monitoring procedure for structural change in a cointegrating relationship. The procedure is inspired by Chu et al. (1996) by being based on parameter estimation on a prebreak ‘calibration’ period. We use three modified least squares estimators to obtain nuisance parameter‐free limiting distributions. We study the asymptotic and finite sample properties of the procedures and finally apply the approach to monitor two‐fundamentals‐driven US housing prices cointegrating relationships over the period 1976:Q1–2010:Q4 using the data of Anundsen (2015). Depending on the relationship considered and the estimation method used, a break point is detected as early as 2003:Q2, that is, well before US housing prices started to fall in 2007.     

Improvement of the quasi‐likelihood ratio test in ARMA models: some results for bootstrap methods

Abstract. Quasi‐likelihood ratio tests for autoregressive moving‐average (ARMA) models are examined. The ARMA models are stationary and invertible with white‐noise terms that are not restricted to be normally distributed. The white‐noise terms are instead subject to the weaker assumption that they are independently and identically distributed with an unspecified distribution. Bootstrap methods are used to improve control of the finite sample significance levels. The bootstrap is used in two ways: first, to approximate a Bartlett‐type correction; and second, to estimate the p‐value of the observed test statistic. Some simulation evidence is provided. The bootstrap p‐value test emerges as the best performer in terms of controlling significance levels.     

Length to diameter ratio of extrudates in catalyst technology I. Modeling catalyst breakage by impulsive forces

Natural or forced catalyst extrudate breakage is an important phenomenon during catalyst manufacture. Here, a two‐parameter model for predicting the reduction in the length to diameter ratio of catalyst extrudates due to breakage by impulsive forces as experienced in a laboratory drop test is developed. Part II will show how both parameters can be correlated with the strength of the extrudates and the severity of the drop test. For breakage by impulsive forces, the model reveals that extrudates are reduced in length to diameter ratio according to a pseudosecond‐order break law. Also, a tie‐in exists with the well‐known Golden Ratio that is famous for its inherent esthetic value. Applying the model to cases of “severity sequencing” and “severity conditioning” reveals the nonlinear behavior of the length to diameter ratio and yields results that are often nonintuitive and hard to get without this engineering analysis. © 2015 American Institute of Chemical Engineers AIChE J, 62: 639–647, 2016     

Local Likelihood for non‐parametric ARCH(1) models

Abstract. We propose a non‐parametric local likelihood estimator for the log‐transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non‐parametric estimator is constructed within the likelihood framework for non‐Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real‐data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.     

Effect of drawing on creep fracture of polypropylene fibers

The ultimate properties of polypropylene fibers, prepared by drawing the same undrawn fibers to different draw ratios, were determined by the creep fracture method at temperatures from 40 to 120°C. The composite curves of the engineering stress at break, the true stress at break, the ultimate extension ratio were constructed as a function of reduced time to break by assuming time–temperature superposition. The temperature dependence of the shift factor a_T used in the superposition could be represented by a single equation of the Williams, Landel, and Ferry form. The composite curves of the true stress at break for the samples with different draw ratios were almost the same independently of draw ratio. The composite curves of the ultimate extension ratio for these samples could be reduced to a single curve by assuming that the ultimate extension ratio evaluated with respect to the unit length of the undrawn fibers were constant independently of draw ratio if the time to break and temperature conditions of the measurement were fixed constant. A possible fracture mechanism is discussed on the basis of the fracture mechanisms proposed by Samuels and Peterlin.     

Trend Adjustment Prior to Testing for the Cointegrating Rank of a Vector Autoregressive Process

Testing the cointegrating rank of a vector autoregressive process that may have a deterministic linear trend is considered. Previous proposals for dealing with such a situation are either to allow for a deterministic trend term in computing a suitable test statistic or else to remove the linear trend first and then derive the test statistic from the trend-adjusted data. In this study the latter approach is considered and a new method for trend removal is proposed that is based on estimating the trend parameters under the null hypothesis. Likelihood ratio and Lagrange multiplier type test statistics are derived on the basis of the trend-adjusted data and their asymptotic distributions are considered under the null hypothesis and under local alternatives. A simulation comparison with other proposals is performed.     

Inference about long run canonical correlations

This article proposes methods for testing the null hypothesis that a number of so‐called long run canonical correlations (LRCCs) are zero. Two test statistics are proposed and their limiting distributions are derived under the null hypothesis. The finite sample properties of the tests are illustrated via a number of simulation studies that reveal the asymptotic theory provides a good guidance to behaviour in moderate or large sized samples. It is shown that the statistics provide a natural way for testing the asymptotic independence of two standardized sums. The usefulness of the tests is illustrated via the following examples: inference about cointegrating vector in a particular cointegration model; inference about break points in a cointegration model; moment estimation; parameter estimation in Generalized Method of Moments estimation.     

Small‐b and Fixed‐b Asymptotics for Weighted Covariance Estimation in Fractional Cointegration

In a standard cointegrating framework, Phillips (1991) introduced the weighted covariance (WC) estimator of cointegrating parameters. Later, Marinucci (2000) applied this estimator to fractional circumstances and, like Phillips (1991), analysed the so‐called small‐b asymptotic approximation to its sampling distribution. Recently, an alternative limiting theory (fixed‐b asymptotics) has been successfully employed to approximate sampling distributions. With the purpose of comparing both approaches, we derive here the fixed‐b limit of WC estimators in a fractional setting, filling also some gaps in the traditional (small‐b) theory. We also provide some Monte Carlo evidence that suggests that the fixed‐b limit is more accurate.     

VECTOR AUTOREGRESSIVE MODELS WITH UNIT ROOTS AND REDUCED RANK STRUCTURE:ESTIMATION. LIKELIHOOD RATIO TEST, AND FORECASTING

Abstract. The nonstationary multivariate autoregressive (AR) model Φ ( L ) Y _t =ε _t is considered for an m -dimensional process { Y _t }, where it is assumed that det {Φ( L )}= 0 has d < m unit roots and all other roots are outside the unit circle, and also that rank {Φ(1)}= r ( r = m – d ). Limiting distribution results obtained by Ahn and Reinsel for the least-squares and the Gaussian reduced rank (unit roots imposed) estimators for this AR model are extended to a model where the AR parameters possess additional structure such as nested reduced rank, and based on these results the asymptotic distribution of the likelihood ratio test statistic for testing the number d of unit roots is obtained. An analysis of three US monthly interest rate series is presented to illustrate the testing and estimation procedures. A small simulation study is also performed to examine the finite-sample properties of the likelihood ratio test and the prediction performance of models which impose different numbers of unit roots.     

Fracture properties of epoxy/poly(styrene‐co‐allylalcohol) blends

The epoxy/polystyrene system is characterized by a poor adhesion between the constituent phases, which determines its mechanical properties. The adhesion can be improved via blends based on epoxy resin and random copolymers, poly(styrene‐co‐allylalcohol) (PS‐co‐PA). In this work, the influence of PS‐co‐PA content and the good adhesion between the phases on the tensile properties and the fracture toughness achieved through instrumented Charpy tests have been investigated. The tensile strength and the deformation at break showed an increase in the PS‐co‐PA content while the Young's modulus remained the same. The tensile fracture surfaces revealed that the improvement of these magnitudes was mainly due to a crack deflection mechanism. Also, the fracture toughness of the blends was superior to that of the pure epoxy resin. The main operating toughening mechanism was crack deflection. The fractographic analysis showed that ～ 80% of the particles were broken, and the crack tended to divert from its original path through the broken PS‐co‐PA particles. The remaining particles were detached from the epoxy resin, and the holes left suffered plastic deformation. Analytical models were used to predict successfully the toughness due to these mechanisms. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007     

Bartlett Correction of Empirical Likelihood for Non‐Gaussian Short‐Memory Time Series

Bartlett correction, which improves the coverage accuracies of confidence regions, is one of the desirable features of empirical likelihood. For empirical likelihood with dependent data, previous studies on the Bartlett correction are mainly concerned with Gaussian processes. By establishing the validity of Edgeworth expansion for the signed root empirical log‐likelihood ratio statistics, we show that the Bartlett correction is applicable to empirical likelihood for short‐memory time series with possibly non‐Gaussian innovations. The Bartlett correction is established under the assumptions that the variance of the innovation is known and the mean of the underlying process is zero for a single parameter model. In particular, the order of the coverage errors of Bartlett‐corrected confidence regions can be reduced from O(n^?1) to O(n^?2).     

A Comparison of two sequential tests forµ/σ

Either the sequential probability ratio test or the sequential likelihood ratio test can be used to test one—sided hypotheses regarding the ratio λ = µ/σ, Both tests are functions of the usual t—statistic. I discuss and compare empirical and theoretical properties of these two tests, and provide tables of critical and truncation values for a modified version of the sequential likelihood ratio test     

A FIXED‐ b TEST FOR A BREAK IN LEVEL AT AN UNKNOWN TIME UNDER FRACTIONAL INTEGRATION

In this paper, we propose a test for a break in the level of a fractionally integrated process when the timing of the putative break is not known. This testing problem has received considerable attention in the literature in the case where the time series is weakly autocorrelated. Less attention has been given to the case where the underlying time series is allowed to be fractionally integrated. Here, valid testing can only be performed if the limiting null distribution of the level break test statistic is well defined for all values of the fractional integration exponent considered. However, conventional sup‐Wald type tests diverge when the data are strongly autocorrelated. We show that a sup‐Wald statistic, which is standardized using a non‐parametric kernel‐based long‐run variance estimator, does possess a well‐defined limit distribution, depending only on the fractional integration parameter, provided the recently developed fixed‐b asymptotic framework is applied. We give the appropriate asymptotic critical values for this sup‐Wald statistic and show that it has good finite sample size and power properties.     

Fully bio‐based polyesters derived from 2,5‐furandicarboxylic acid (2,5‐FDCA) and dodecanedioic acid (DDCA): From semicrystalline thermoplastic to amorphous elastomer

A serials of fully bio‐based poly(ethylene dodecanedioate‐2,5‐furandicarboxylate) (PEDF) were synthesized from Dodecanedioic acid (DDCA), 2,5‐Furandicarboxylic acid (2,5‐FDCA), and ethylene glycol through a two‐step procedure consisted of transesterification and polycondensation. After their chemical structures were confirmed by Nuclear Magnetic Resonance and Fourier Transform Infrared Spectroscopy, their thermal, mechanical, and biodegradation properties were investigated in detail. Results showed that the chemical composition of PEDFs could be easily controlled by the feeding mole ratio of DDCA to FDCA and they possessed the characteristic of random copolyester with the intrinsic viscosity ranged from 0.82 to 1.2 dL/g. With the varied mole ratio of DDCA to FDCA, PEDFs could be changed from semicrystalline thermoplastic to the completely amorphous elastomer, indicated by the elongation at break ranged from 4 for poly(ethylene 2,5‐furandicarboxylate) to 1500% for amorphous PEDF‐40. The amorphous PEDF‐30 and PEDF‐40 showed satisfactory shape recovery after cyclic tensile test, which was the typical behavior for elastomer. Enzymatic degradation test indicated that all the PEDFs were biodegradable and the degradation rate was heavily affected by their chemical compositions. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018 , 135, 46076.     

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.