Let us do the twist again

Krämer (Sankhy $\bar{\mathrm{a }}$ 42:130–131, 1980) posed the following problem: “Which are the $\mathbf{y}$ , given $\mathbf{X}$ and $\mathbf{V}$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $\mathbf{y}$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $\varvec{\beta }$ coincide under the general Gauss–Markov model $\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $\varvec{\beta }$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $\varvec{\beta }$ , we consider the estimation of the systematic part $\mathbf{X} \varvec{\beta }$ , which is a natural consequence of relaxing the assumption that $\mathbf{X}$ and $\mathbf{V}$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $\mathbf{X} \varvec{\beta }$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.     

Nonlinear IV panel unit root testing under structural breaks in the error variance

The paper examines the behavior of a generalized version of the nonlinear IV unit root test proposed by Chang (2002) when the series’ errors exhibit nonstationary volatility. The leading case of such nonstationary volatility concerns structural breaks in the error variance. We show that the generalized test is not robust to variance changes in general, and illustrate the extent of the resulting size distortions in finite samples. More importantly, we show that pivotality is recovered when using Eicker-White heteroskedasticity-consistent standard errors. This contrasts with the case of Dickey-Fuller unit root tests, for which Eicker-White standard errors do not produce robustness and thus require computationally costly corrections such as the (wild) bootstrap or estimation of the so-called variance profile. The pivotal versions of the generalized IV tests – with or without the correct standard errors – do however have no power in $1/T$ -neighbourhoods of the null. We also study the validity of panel versions of the tests considered here.     

On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

Let \(X_1 ,X_2 ,\ldots ,X_n \) be a sequence of Markov Bernoulli trials (MBT) and \(\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })\) be a random vector where \(X_{n,k_i } \) represents the number of occurrences of success runs of length \(k_i \,( {i=1,2,\ldots ,r})\) . In this paper the joint distribution of \(\underline{X}_n \) in the sequence of \(n\) MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least \(k\) , overlapping runs, runs of exact length \(k\) and \(\ell \) -overlapping runs (i.e. \(\ell \) -overlapping counting scheme), \(0\le \ell are considered. The pgf of joint distribution of \(\underline{X}_n \) is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of \(\underline{X}_n \) such as in evaluation of the reliability of \(( {n,f,k})\!\!:\!\!G\) and \(\!:\!\!G\) system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed.     

Testing for a unit root in a nonlinear quantile autoregression framework

The nonlinear unit root test of Kapetanios, Shin, and Snell (2003 Kapetanios, G., Shin, Y., Snell, A. (2003). Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 112:359–379.[Crossref], [Web of Science ®] , [Google Scholar]) (KSS) has attracted much recent attention. However, the KSS test relies on the ordinary least squares (OLS) estimator, which is not robust to a heavy-tailed distribution and, in practice, the test suffers from a large power loss. This study develops three kinds of quantile nonlinear unit root tests: the quantile t-ratio test; the quantile Kolmogorov–Smirnov test; and the quantile Cramer–von Mises test. A Monte Carlo simulation shows that these tests have significantly better power when an innovation follows a non-normal distribution. In addition, the quantile t-ratio test can reveal the heterogeneity of the asymmetric dynamics in a time series. In our empirical studies, we investigate the unit root properties of U.S. macroeconomic time series and the real effective exchange rates for 61 countries. The results show that our proposed tests reject the unit roots more often, indicating that the series are likely to be asymmetric nonlinear reverting processes.     

Panel Unit Root Tests in the Presence of Cross-Sectional Dependencies: Comparison and Implications for Modelling

Several panel unit root tests that account for cross-section dependence using a common factor structure have been proposed in the literature recently. Pesaran's (2007 Pesaran , M. H. ( 2007 ). A simple panel unit root test in the presence of cross section dependence . Journal of Applied Econometrics 22 : 265 – 312 .[Crossref], [Web of Science ®] , [Google Scholar]) cross-sectionally augmented unit root tests are designed for cases where cross-sectional dependence is due to a single factor. The Moon and Perron (2004 Moon , H. R. , Perron , B. (2004). Testing for a unit root in panels with dynamic factors. Journal of Econometrics 122:81–126.[Crossref], [Web of Science ®] , [Google Scholar]) tests which use defactored data are similar in spirit but can account for multiple common factors. The Bai and Ng (2004a Bai , J. , Ng , S. ( 2004a ). A PANIC attack on unit roots and cointegration . Econometrica 72 : 1127 – 1177 .[Crossref], [Web of Science ®] , [Google Scholar]) tests allow to determine the source of nonstationarity by testing for unit roots in the common factors and the idiosyncratic factors separately. Breitung and Das (2008 Breitung , J. , Das , S. ( 2008 ). Testing for unit roots in panels with a factor structure . Econometric Theory 24 : 88 – 108 .[Crossref], [Web of Science ®] , [Google Scholar]) and Sul (2007 Sul , D. ( 2007 ) . Panel Unit Root Tests Under Cross Section Dependence with Recursive Mean Adjustment . Mimeo : University of Auckland . [Google Scholar]) propose panel unit root tests when cross-section dependence is present possibly due to common factors, but the common factor structure is not fully exploited.

This article makes four contributions: (1) it compares the testing procedures in terms of similarities and differences in the data generation process, tests, null, and alternative hypotheses considered, (2) using Monte Carlo results it compares the small sample properties of the tests in models with up to two common factors, (3) it provides an application which illustrates the use of the tests, and (4) finally, it discusses the use of the tests in modelling in general.     

A Monte Carlo Investigation of Unit Root Tests and Long Memory in Detecting Mean Reversion in I(0) Regime Switching,Structural Break,and Nonlinear Data

The potential observational equivalence between various types of nonlinearity and long memory has been recognized by the econometrics community since at least the contribution of Diebold and Inoue (2001 Diebold, F., Inoue, A. (2001). Long memory and regime switching. Journal of Econometrics 105:131–159.[Crossref], [Web of Science ®] , [Google Scholar]). A large literature has developed in an attempt to ascertain whether or not the long memory finding in many economic series is spurious. Yet to date, no study has analyzed the consequences of using long memory methods to test for unit roots when the “truth” derives from regime switching, structural breaks, or other types of mean reverting nonlinearity. In this article, I conduct a comprehensive Monte Carlo analysis to investigate the consequences of using tests designed to have power against fractional integration when the actual data generating process is unknown. I additionally consider the use of tests designed to have power against breaks and threshold nonlinearity. The findings are compelling and demonstrate that the use of long memory as an approximation to nonlinearity yields tests with relatively high power. In contrast, misspecification has severe consequences for tests designed to have power against threshold nonlinearity, and especially for tests designed to have power against breaks.     

EWMA charts: ARL considerations in case of changes in location and scale

Widely spread tools within the area of Statistical Process Control are control charts of various designs. Control chart applications are used to keep process parameters (e.g., mean \(\mu \) , standard deviation \(\sigma \) or percent defective \(p\) ) under surveillance so that a certain level of process quality can be assured. Well-established schemes such as exponentially weighted moving average charts (EWMA), cumulative sum charts or the classical Shewhart charts are frequently treated in theory and practice. Since Shewhart introduced a \(p\) chart (for attribute data), the question of controlling the percent defective was rarely a subject of an analysis, while several extensions were made using more advanced schemes (e.g., EWMA) to monitor effects on parameter deteriorations. Here, performance comparisons between a newly designed EWMA \(p\) control chart for application to continuous types of data, \(p=f(\mu ,\sigma )\) , and popular EWMA designs ( \(\bar{X}\) , \(\bar{X}\) - \(S^2\) ) are presented. Thus, isolines of the average run length are introduced for each scheme taking both changes in mean and standard deviation into account. Adequate extensions of the classical EWMA designs are used to make these specific comparisons feasible. The results presented are computed by using numerical methods.     

A Note on Two-Way ECM Estimation of SUR Systems on Unbalanced Panel Data

This article considers the two-way error components model (ECM) estimation of seemingly unrelated regressions (SUR) on unbalanced panel by generalized least squares (GLS). As suggested by Biørn (2004 Biørn , E. ( 2004 ). Regression systems for unbalanced panel data: a stepwise maximum likelihood procedure . Journal of Econometrics 122 : 281 – 291 . [Google Scholar]) for the one-way case, in order to use the standard results for the balanced case the individuals are arranged in groups according to the number of times they are observed. Thus, the GLS estimator can be interpreted as a matrix weighted average of the group specific GLS estimators with weights equal to the inverse of their respective covariance matrices.     

On tests of radial symmetry for bivariate copulas

The unique copula of a continuous random pair \((X,Y)\) is said to be radially symmetric if and only if it is also the copula of the pair \((-X,-Y)\) . This paper revisits the recently considered issue of testing for radial symmetry. Three rank-based statistics are proposed to this end which are asymptotically equivalent but simpler to compute than those of Bouzebda and Cherfi (J Stat Plan Inference 142:1262–1271, 2012). Their limiting null distribution and its approximation using the multiplier bootstrap are discussed. The finite-sample properties of the resulting tests are assessed via simulations. The asymptotic distribution of one of the test statistics is also computed under an arbitrary alternative, thereby correcting an error in the recent work of Dehgani et al. (Stat Pap 54:271–286, 2013).     

Conditional MSE-based discrimination of the sample mean and the post-stratification estimator in population sampling

Whenever a random sample is drawn from a stratified population, the post-stratification estimator $\tilde X$ usually is preferred to the sample mean $\tilde X$ , when the population mean is to be estimated. This is due to the fact that the variance of $\tilde X$ is asymptotically smaller than that of $\tilde X$ , while both estimators are asymptotically unbiased. However, this only holds looking at post-stratification unconditionally, when strata sample sizes are random. Conditioned on the realized sample sizes, the MSE of $\tilde X$ can be higher than that of $\tilde X$ which means that $\tilde X$ should be preferred to $\tilde X$ , even if it is biased. The conditional MSE difference of $\tilde X$ and $\tilde X$ is estimated, and using this estimation and its variance a heuristic test based on the Vysochanskiî-Petunin inequality is derived.     

Order selection in finite mixtures of linear regressions

Finite mixture models can adequately model population heterogeneity when this heterogeneity arises from a finite number of relatively homogeneous clusters. An example of such a situation is market segmentation. Order selection in mixture models, i.e. selecting the correct number of components, however, is a problem which has not been satisfactorily resolved. Existing simulation results in the literature do not completely agree with each other. Moreover, it appears that the performance of different selection methods is affected by the type of model and the parameter values. Furthermore, most existing results are based on simulations where the true generating model is identical to one of the models in the candidate set. In order to partly fill this gap we carried out a (relatively) large simulation study for finite mixture models of normal linear regressions. We included several types of model (mis)specification to study the robustness of 18 order selection methods. Furthermore, we compared the performance of these selection methods based on unpenalized and penalized estimates of the model parameters. The results indicate that order selection based on penalized estimates greatly improves the success rates of all order selection methods. The most successful methods were \(MDL2\) , \(MRC\) , \(MRC_k\) , \(ICL\) – \(BIC\) , \(ICL\) , \(CAIC\) , \(BIC\) and \(CLC\) but not one method was consistently good or best for all types of model (mis)specification.     

Comparing Several Regression Models with Unequal Variances

We consider the problem of comparing k regression models, when the variances are not assumed to be equal. For this problem, the classical F test can lead to misleading results, and there is no simple test which adequately controls the size when the sample sizes are small. For k = 2, the most widely used test is the “weighted F test,” also known as the “asymptotic Chow test.” But this test does not work well for small samples, and various modifications have been proposed in the literature. For k > 2, few tests are available and only the parametric-bootstrap (PB) test of Tian et al. (2009) Tian, L., Ma, C., Vexler, A. (2009). A parametric bootstrap test for comparing heteroscedastic regression models. Communications in Statistics—Simulation and Computation, 38, 1026–1036.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] controls the size fairly adequately. In this article, we propose three fairly simple F tests which can easily be applied in the general case, k ? 2, and avoid the complications of the PB test. Our simulations indicate that these tests have satisfactory performance. Also, our simulations confirm that the power properties of our proposed tests are similar to the PB test. Therefore, our proposed tests provide simple alternatives to the PB test, which can easily be used by practitioners who may not be familiar with the PB.     

On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .     

Testing for Unit Roots and the Impact of Quadratic Trends,with an Application to Relative Primary Commodity Prices

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modeled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated nonlinear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this article, we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modeled, we show that the quasi-differenced detrended Dickey–Fuller-type test of Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this article. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether the data admit a unit root.     

Testing for cross-sectional dependence in a panel factor model using the wild bootstrap F test

This paper considers testing for cross-sectional dependence in a panel factor model. Based on the model considered by Bai (Econometrica 71: 135–171, 2003), we investigate the use of a simple $F$ test for testing for cross-sectional dependence when the factor may be known or unknown. The limiting distributions of these $F$ test statistics are derived when the cross-sectional dimension and the time-series dimension are both large. The main contribution of this paper is to propose a wild bootstrap $F$  test which is shown to be consistent and which performs well in Monte Carlo simulations especially when the factor is unknown.     

Equalities between OLSE,BLUE and BLUP in the linear model

We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation.     

A unified method for constructing expectation tolerance intervals

Given a random sample of size \(n\) with mean \(\overline{X} \) and standard deviation \(s\) from a symmetric distribution \(F(x; \mu , \sigma ) = F_{0} (( x- \mu ) / \sigma ) \) with \(F_0\) known, and \(X \sim F(x;\; \mu , \sigma )\) independent of the sample, we show how to construct an expansion \( a_n^{\prime } = \sum _{i=0}^\infty \ c_i \ n^{-i} \) such that \(\overline{X} - s a_n^{\prime } < X < \overline{X} + s a_n^{\prime } \) with a given probability \(\beta \) . The practical value of this result is illustrated by simulation and using a real data set.     

Bispectral-Based Goodness-of-Fit Tests of Gaussianity and Linearity of Stationary Time Series

Spectral domain tests for time series linearity typically suffer from a lack of power compared to time domain tests. We present two tests for Gaussianity and linearity of a stationary time series. The tests are two-stage procedures applying goodness-of-fit techniques to the estimated normalized bispectrum. We illustrate the performances of the tests are competitive with time domain tests. The new tests typically outperform Hinich's (1982 Hinich , M. J. ( 1982 ). Testing for Gaussianity and linearity of a stationary time series . J. Time Ser. Anal. 3 : 169 – 176 .[Crossref] , [Google Scholar]) bispectral based test, especially when the length of the time series is not large.     

Principal points for an allometric extension model

A set of \(n\) -principal points of a \(p\) -dimensional distribution is an optimal \(n\) -point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space \(R^p\) for \(n\) -principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components.