Generalized empirical likelihood tests in time series models with potential identification failure

We introduce test statistics based on generalized empirical likelihood methods that can be used to test simple hypotheses involving the unknown parameter vector in moment condition time series models. The test statistics generalize those in Guggenberger and Smith [2005. Generalized empirical likelihood estimators and tests under partial, weak and strong identification. Econometric Theory 21 (4), 667–709] from the i.i.d. to the time series context and are alternatives to those in Kleibergen [2005a. Testing parameters in GMM without assuming that they are identified. Econometrica 73 (4), 1103–1123] and Otsu [2006. Generalized empirical likelihood inference for nonlinear and time series models under weak identification. Econometric Theory 22 (3), 513–527]. The main feature of these tests is that their empirical null rejection probabilities are not affected much by the strength or weakness of identification. More precisely, we show that the statistics are asymptotically distributed as chi-square under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright [2000. GMM with weak identification. Econometrica 68 (5), 1055–1096]. We also introduce a modification to Otsu's (2006) statistic that is computationally more attractive. A Monte Carlo study reveals that the finite-sample performance of the suggested tests is very competitive.     

Moment convergence of M‐estimators

This study extends the rate of convergence theorem of M‐estimators presented by van der Vaart and Wellner (weak convergence and empirical processes: with applications to statistics, Springer‐Verlag, Newyork, 1996) who gave a result of the form r  to a result of the form sup_nE | r , for any p≥1. This result is useful for deriving the moment convergence of the rescaled residual. An application to maximum likelihood estimators is discussed.     

An invariance principle for progressively truncated likelihood ratio statistics

Summary In the context of time-sequential tests based on likelihood ratio statistics, weak convergence (in the Skorokhod as well as the sup-norm metric) of progressively truncated likelihood ratio statistics (to appropriate Wiener processes) is established through the construction of a continuous time-parameter martingale process. The case of local (contiguous) alternative hypotheses is also treated. The results are extended to the multiparameter case as well.Work partially supported by the Air Force Office of Scientific Research, U.S.A.F., A.F.S.C., Grant No. 74-2736 and partially by the National Heart, Lung and Blood Institute, Contract NIHNHLBI-71-2243 from the National Institutes of Health.     

Bootstrap inference in systems of single equation error correction models

We analyze OLS-based tests of long-run relationships, weak exogeneity and short-run dynamics in conditional error correction models. Unweighted sums of single equation test statistics are used for hypothesis testing in pooled systems. When model errors are (conditionally) heteroskedastic tests of weak exogeneity and short run dynamics are affected by nuisance parameters. Similarly, on the pooled level the advocated test statistics are no longer pivotal in presence of cross-sectional error correlation. We prove that the wild bootstrap provides asymptotically valid critical values under both conditional heteroskedasticity and cross-sectional error correlation. A Monte-Carlo study reveals that in small samples the bootstrap outperforms first-order asymptotic approximations in terms of the empirical size even if the asymptotic distribution of the test statistic does not depend on nuisance parameters. Opposite to feasible GLS methods the approach does not require any estimate of cross-sectional correlation and copes with time-varying patterns of contemporaneous error correlation.     

Cointegration in Panel Data with Structural Breaks and Cross‐Section Dependence

The power of standard panel cointegration statistics may be affected by misspecification errors if structural breaks in the parameters generating the process are not considered. In addition, the presence of cross‐section dependence among the panel units can distort the empirical size of the statistics. We therefore design a testing procedure that allows for both structural breaks and cross‐section dependence when testing the null hypothesis of no cointegration. The paper proposes test statistics that can be used when one or both features are present. We illustrate our proposal by analysing the pass‐through of import prices on a sample of European countries. Copyright © 2013 John Wiley & Sons, Ltd.     

Data features1

This article attempts to provide a formal framework for a data based inference which explicitly and consistently recognizes the approximate nature of probability models. It is based on the idea that a stochastic model is adequate if samples generated under the model are very much like the sample actually obtained. The formalization is based on the concept of data feature. Examples are given of applying the ideas to different areas of statistics including location-scale models, densities, non-parametric regression, interlaboratory test, auto-regressive processes and the analysis of variance. The four cornerstones of the approach are direct comparison, approximation, weak topologies and parsimony. The approach is contrasted to that of much of conventional statistics many of whose concepts are pathologically discontinuous with respect to the topology of data analysis and common sense.     

An improved empirical Bayes test for positive exponential families

We exhibit an empirical Bayes test δ^* _n for a decision problem using a linear error loss in a class of positive exponential families. This empirical Bayes test δ^* _n possesses the asymptotic optimality, and its associated regret converges to zero with rate n ⁻¹(ln n )⁶ This rate of convergence improves the previous results in the literature in the sense that a faster rate of convergence is achieved under much weaker conditions. Examples are presented to illustrate the performance of the empirical Bayes test δ^* _n     

A note on invariance principles for v. Mises' statistics

Summary We extend Filippova's result on weak convergence of v. Mises' functionals and prove a weak invariance principle. Applications toU-statistics are given and extensions to contiguity and weakly dependent processes are briefly discussed.The work of the third author has been supported by the Deutsche Forschungsgemeinschaft.     

Segmenting mean-nonstationary time series via trending regressions

In this paper, we provide a segmentation procedure for mean-nonstationary time series. The segmentation is obtained by casting the problem into the framework of detecting structural breaks in trending regression models in which the regressors are generated by suitably smooth functions. As test statistics we propose to use the maximally selected likelihood ratio statistics and a related statistics based on partial sums of weighted residuals. The main theoretical contribution of the paper establishes the extreme value distribution of these statistics and their consistency. To circumvent the slow convergence to the extreme value limit, we propose to employ a version of the circular bootstrap. This procedure is completely data-driven and does not require knowledge of the time series structure. In an empirical part, we show in a simulation study and applications to air carrier traffic and S&P 500 data that the finite sample performance is very satisfactory.     

新凯恩斯菲利普斯曲线的弱识别检验和稳健估计

对基于GMM大样本统计量的检验和估计,弱识别会带来点估计不一致和置信区间规模不正确的问题。本文讨论了弱识别问题产生的原因、影响、检验和相关弱识别稳健统计量。用Cragg-Donald F统计量对NKPC重新进行了检验,发现的确存在弱识别问题,本文进而对NKPC的参数进行了弱识别稳健的估计,并且用CLR统计量构造了相应规模正确的置信区间。实证表明中国的通货膨胀具有明显的向前和向后行为,而向前行为从数量上来说要大于向后行为的影响,但是向后行为在数量上来说也是不可忽视的;子样本的检验表明通涨的向后行为有增强的趋势。     

The random-time binomial model

In this paper we study a binomial model with random time steps and explain how to calculate values for European and American call and put options. We prove both weak convergence of the discrete processes to the Black–Scholes setup and convergence of the values for European and American put options. Computational experiments exhibit a smooth convergence structure and suggest that we can obtain a quadratic order of convergence via an extrapolation procedure. Approximations to jump-diffusions are straightforward.     

Insights into the Process Dynamics of Innovation Implementation

Abstract

In France, the differences between public and private organizations seem to have been reduced by the ubiquity of some management tools and approaches, and, at the same time, highlighted by the political environment, stereotypes, and negative perceptions of French public management. Given this confusion, systematic empirical studies are needed to determine the degree of convergence, if any, between management in the public and private sectors. The present study analyzed a survey of 908 managers, in order to determine whether differences in managerial practices still exist. Our results show that some differences do persist, although they are relatively small.     

Local inference for locally stationary time series based on the empirical spectral measure

The time varying empirical spectral measure plays a major role in the treatment of inference problems for locally stationary processes. The properties of the empirical spectral measure and related statistics are studied — both when its index function is fixed or when dependent on the sample size. In particular we prove a general central limit theorem. Several applications and examples are given including semiparametric Whittle estimation, local least squares estimation and spectral density estimation.     

Administrative Records and Surveys as Basis for Statistics on International Labour Migration

This paper discusses possible sources for statistics to be used for describing and analysing the number, structure, situation, development and impact of migrant workers. The discussion is focused on key, intrinsic features of the different sources, important for the understanding of their strengths and weaknesses, and draws the reader's attention to features which may tend to undermine the quality of the statistics produced as well as ways in which the impact of such features can be evaluated and, if possible, reduced. This discussion thereby provides concrete illustrations of many of the methodological issues referred to in (Hoffmann, 1995). The paper is organized around three key groups of migrant workers:

• (a) Persons who are arriving in a country to work there, i.e. the inflow of foreign workers;
• (b) Persons who are leaving their country to find work abroad, i.e. the outllow of migrant workers
• (c) Stock of foreign workers in the country.

Definitions of these groups are given in section 2. Sections 3 to 5 review, for each key group, main possible sources of statistics and the extent to which they can cover the group and identify separately important sub-groups. The discussion in each section distinguishes between administrative registrations and statistical surveys, and deals with the capacity of a source to provide estimates of the size of the group, and whether it can provide estimates of the amount of change in the group over time, or at least indications of the direction of such changes. Section 6 outlines problems related to other data quality dimensions, such as frequency and timeliness of the statistics as well as the validity, reliability and consistency of the registration of variables used to describe migrant workers and where they come from. Section 7 presents strategies for the development of statistics on migrant workers based on two model scenarios for “countries” of different geographic circumstances and institutional capacities. Summary recommendations on how to organize cooperation, for the production of statistics on migrant workers, between the relevant agencies and between them and the potential users are proposed in the concluding section. The overall conclusion is that trying to piece together a coherent statistical picture of any of the above key groups is similar to trying to put together a large puzzle based on photographs of a rapidly changing reality, with important pieces missing and many of the available photographs out of focus.     

Parametric bootstrap tests for continuous and discrete distributions

Statistical procedures based on the estimated empirical process are well known for testing goodness of fit to parametric distribution families. These methods usually are not distribution free, so that the asymptotic critical values of test statistics depend on unknown parameters. This difficulty may be overcome by the utilization of parametric bootstrap procedures. The aim of this paper is to prove a weak approximation theorem for the bootstrapped estimated empirical process under very general conditions, which allow both the most important continuous and discrete distribution families, along with most parameter estimation methods. The emphasis is on families of discrete distributions, and simulation results for families of negative binomial distributions are also presented.     

Factor‐Based Identification‐Robust Interference in IV Regressions

Robust methods for instrumental variable inference have received considerable attention recently. Their analysis has raised a variety of problematic issues such as size/power trade‐offs resulting from weak or many instruments. We show that information reduction methods provide a useful and practical solution to this and related problems. Formally, we propose factor‐based modifications to three popular weak‐instrument‐robust statistics, and illustrate their validity asymptotically and in finite samples. Results are derived using asymptotic settings that are commonly used in both the factor and weak‐instrument literature. For the Anderson–Rubin statistic, we also provide analytical finite‐sample results that do not require any underlying factor structure. An illustrative Monte Carlo study reveals the following. Factor‐based tests control size regardless of instruments and factor quality. All factor‐based tests are systematically more powerful than standard counterparts. With informative instruments and in contrast to standard tests: (i) power of factor‐based tests is not affected by k even when large; and (ii) weak factor structure does not cost power. An empirical study on a New Keynesian macroeconomic model suggests that our factor‐based methods can bridge a number of gaps between structural and statistical modeling. Copyright © 2015 John Wiley & Sons, Ltd.     

A Meta‐Analysis of β‐Convergence: the Legendary 2%

Abstract.  The topic of convergence is at the heart of a wide‐ranging debate in the growth literature, and empirical studies of convergence differ widely in their theoretical backgrounds, empirical specifications, and in their treatment of cross‐sectional heterogeneity. Despite these differences, a rate of convergence of about 2% has been found under a variety of different conditions, resulting in the widespread belief that the rate of convergence is a natural constant. We use meta‐analysis to investigate whether there is substance to the 'myth' of the 2% convergence rate and to assess several unresolved issues of interpretation and estimation. Our data set contains approximately 600 estimates taken from a random sample of empirical growth studies published in peer‐reviewed journals. The results indicate that it is misleading to speak of a natural convergence rate since estimates of different growth regressions come from different populations, and we find that correcting for the bias resulting from unobserved heterogeneity in technology levels leads to higher estimates of the rate of convergence. We also find that correcting for endogeneity of the explanatory variables has a substantial effect on the estimates and that measures of financial and fiscal development are important determinants of long‐run differences in per capita income levels. We show that although the odds of a study being published is not uniform for studies with different p ‐values, publication bias has no significant effect on the conclusions of the analysis.     

Exponential convergence properties of linear estimators under exponential and uniform distribution

Summary Certain measures of asymptotic efficiency of test statistics are based on exponential convergence properties of the underlying error probabilities (Bahadur-, Hodges-Lehmann-efficiency). From a general large deviation theorem, that is specified to weighted sums of independent identically distributed (i.i.d.) random variables, such exponential convergence properties are derived for test statistics which are linear functions of order statistics of i.i.d. random variables under exponential and uniform distribution. For that purpose some smoothness-conditions for the weights have to be established. In a series of examples it is shown that these conditions are fulfilled for certain robust linear estimators of location or scale parameters. With the help of some numerical results two of them, namely Winsorized and trimmed mean, are compared with regard to the asymptotic relative efficiency against each other.

---

Zusammenfassung Bestimmte asymptotische Effizienzbegriffe für Tests basieren auf einem exponentiellen Konvergenzverhalten der zugrundeliegenden Fehlerwahrscheinlichkeiten (Bahadur-, Hodges-Lehmann-Effizienz). Mit Hilfe eines allgemeinen Satzes üver Wahrscheinlichkeiten großer Abweichungen, der spezialisiert wird auf gewichtete Summen unabhängiger, identisch verteilter (i.i.d.) Zufallsvariablen mit momenterzeugenden Funktionen, wird ein solches exponentielles Konvergenzverhalten nachgewiesen für Linearkombinationen von order statistics von i.i.d. Zufallsvariablen unter Exponential- und Rechteckverteilung. Dazu sind bestimmte Bedingungen an die Gewichte zu stellen. In einigen Beispielen wird gezeigt, daß solche Gewichtsbedingungen für eine Reihe von robusten Schätzern erfüllt sind. Zwei spezielle, nämlich das Winsorisierte und getrimmte Mittel, werden mit Hilfe einiger numerischer Ergebnisse hinsichtlich ihrer asymptotischen Effizienz miteinander verglichen.

     

Applications of the Likelihood Theory in Finance: Modelling and Pricing

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.     

Efficient tests for the presence of a pair of complex conjugate unit roots in real time series

We introduce a framework which allows us to draw a clear parallel between the test for the presence of seasonal unit roots and that for unit root at frequency 0 (or π$\pi$). It relies on the properties of the complex conjugate integrated of order one processes which are implicitly at work in the real time series. In the same framework as that of Phillips and Perron (Biometrica 75 (1988) 335), we derive tests for the presence of a pair of conjugate complex unit roots. The asymptotic distribution we obtain are formally close to those derived by these authors but expressed with complex Wiener processes. We then introduce sequences of near-integrated processes which allow us to study the local-to-unity asymptotic of the above test statistics. We state a result on the weak convergence of the partial sum of complex near-random walks which leads to complex Orstein–Uhlenbeck processes. Drawing on Elliott et al. (Econometrica 64 (1996) 813) we then study the design of point-optimal invariant test procedures and compute their envelope employing local-to-unity asymptotic approximations. This leads us to introduce new feasible and near efficient seasonal unit root tests. Their finite sample properties are investigated and compared with the different test procedures already available (J. Econometrics 44 (1991) 215; 62 (1994) 415; 85 (1998) 269) and those introduced in the first part of the paper.