Unit root quantile autoregression testing using covariates

This paper extends unit root tests based on quantile regression proposed by Koenker and Xiao [Koenker, R., Xiao, Z., 2004. Unit root quantile autoregression inference, Journal of the American Statistical Association 99, 775–787] to allow stationary covariates and a linear time trend. The limiting distribution of the test is a convex combination of Dickey–Fuller and standard normal distributions, with weight determined by the correlation between the equation error and the regression covariates. A simulation experiment is described, illustrating the finite sample performance of the unit root test for several types of distributions. The test based on quantile autoregression turns out to be especially advantageous when innovations are heavy-tailed. An application to the CPI-based real exchange rates using four different countries suggests that real exchange rates are not constant unit root processes.     

Bayesian analysis of a Tobit quantile regression model

This paper develops a Bayesian framework for Tobit quantile regression. Our approach is organized around a likelihood function that is based on the asymmetric Laplace distribution, a choice that turns out to be natural in this context. We discuss families of prior distributions on the quantile regression vector that lead to proper posterior distributions with finite moments. We show how the posterior distribution can be sampled and summarized by Markov chain Monte Carlo methods. A method for comparing alternative quantile regression models is also developed and illustrated. The techniques are illustrated with both simulated and real data. In particular, in an empirical comparison, our approach out-performed two other common classical estimators.     

Properties and estimation of asymmetric exponential power distribution

The new distribution class, Asymmetric Exponential Power Distribution (AEPD), proposed in this paper generalizes the class of Skewed Exponential Power Distributions (SEPD) in a way that in addition to skewness introduces different decay rates of density in the left and right tails. Our parametrization provides an interpretable role for each parameter. We derive moments and moment-based measures: skewness, kurtosis, expected shortfall. It is demonstrated that a maximum entropy property holds for the AEPD distributions. We establish consistency, asymptotic normality and efficiency of the maximum likelihood estimators over a large part of the parameter space by dealing with the problems created by non-smooth likelihood function and derive explicit analytical expressions of the asymptotic covariance matrix; where the results apply to the SEPD class they enlarge on the current literature. Also we give a convenient stochastic representation of the distribution; our Monte Carlo study illustrates the theoretical results. We also provide some empirical evidence for the usefulness of employing AEPD errors in GARCH type models for predicting downside market risk of financial assets.     

Markov switching quantile autoregression

This paper considers the location‐scale quantile autoregression in which the location and scale parameters are subject to regime shifts. The regime changes in lower and upper tails are determined by the outcome of a latent, discrete‐state Markov process. The new method provides direct inference and estimate for different parts of a non‐stationary time series distribution. Bayesian inference for switching regimes within a quantile, via a three‐parameter asymmetric Laplace distribution, is adapted and designed for parameter estimation. Using the Bayesian output, the marginal likelihood is readily available for testing the presence and the number of regimes. The simulation study shows that the predictability of regimes and conditional quantiles by using asymmetric Laplace distribution as the likelihood is fairly comparable with the true model distributions. However, ignoring that autoregressive coefficients might be quantile dependent leads to substantial bias in both regime inference and quantile prediction. The potential of this new approach is illustrated in the empirical applications to the US inflation and real exchange rates for asymmetric dynamics and the S&P 500 index returns of different frequencies for financial market risk assessment.     

Realized extreme quantile: A joint model for conditional quantiles and measures of volatility with EVT refinements

We propose a new framework exploiting realized measures of volatility to estimate and forecast extreme quantiles. Our realized extreme quantile (REQ) combines quantile regression with extreme value theory and uses a measurement equation that relates the realized measure to the latent conditional quantile. Model estimation is performed by quasi maximum likelihood, and a simulation experiment validates this estimator in finite samples. An extensive empirical analysis shows that high‐frequency measures are particularly informative of the dynamic quantiles. Finally, an out‐of‐sample forecast analysis of quantile‐based risk measures confirms the merit of the REQ.     

Quantile cointegrating regression

Quantile regression has important applications in risk management, portfolio optimization, and asset pricing. The current paper studies estimation, inference and financial applications of quantile regression with cointegrated time series. In addition, a new cointegration model with quantile-varying coefficients is proposed. In the proposed model, the value of cointegrating coefficients may be affected by the shocks and thus may vary over the innovation quantile. The proposed model may be viewed as a stochastic cointegration model which includes the conventional cointegration model as a special case. It also provides a useful complement to cointegration models with (G)ARCH effects. Asymptotic properties of the proposed model and limiting distribution of the cointegrating regression quantiles are derived. In the presence of endogenous regressors, fully-modified quantile regression estimators and augmented quantile cointegrating regression are proposed to remove the second order bias and nuisance parameters. Regression Wald tests are constructed based on the fully modified quantile regression estimators. An empirical application to stock index data highlights the potential of the proposed method.     

Testing for structural change in regression quantiles

Most studies in the structural change literature focus solely on the conditional mean, while under various circumstances, structural change in the conditional distribution or in conditional quantiles is of key importance. This paper proposes several tests for structural change in regression quantiles. Two types of statistics are considered, namely, a fluctuation type statistic based on the subgradient and a Wald type statistic, based on comparing parameter estimates obtained from different subsamples. The former requires estimating the model under the null hypothesis, and the latter involves estimation under the alternative hypothesis. The tests proposed can be used to test for structural change occurring in a pre-specified quantile, or across quantiles, which can be viewed as testing for change in the conditional distribution with a linear specification of the conditional quantile function. Both single and multiple structural changes are considered. We derive the limiting distributions under the null hypothesis, and show they are nuisance parameter free and can be easily simulated. A simulation study is conducted to assess the size and power in finite samples.     

Repeat tourism in Uruguay: modelling truncated distributions of count data

This paper studies the determinants of repeat visiting in Uruguay, where loyal visitors are a relevant part of the total. From a statistical point of view, the number of times a visitor has been to a place constitutes count data. In this regard available information on Uruguay presents relevant limitations. Count data is in fact reported only for those who visited the country up to five times, whereas records about the most frequent visitors are collapsed into one residual category. This implies that the classic models for count data such as Poisson or negative binomial cannot be put into consideration. The paper suggests instead modelling the available part of the empirical distribution through quantile count data regression. It is a model based on measures of location rather than mean values, which allows estimating tourists’ behaviour as the number of visits increases. A set of explanatory variables related to budgetary constraints, socioeconomic, trip-related and psychographic characteristics are taken as regressors to the considered count data.     

Do firms share the same functional form of their growth rate distribution? A statistical test

We propose a hypothesis testing procedure to investigate whether the same growth rate distribution is shared by all the firms in a balanced panel or, more generally, whether they share the same functional form for this distribution, without necessarily sharing the same parameters. We apply the test to panels of US and European Union publicly quoted manufacturing firms, both at the sectoral and at the subsectoral NAICS level. We consider the following null hypotheses about the growth rate distribution of the individual firms: (i) an unknown shape common to all firms, with all the firms sharing also the same parameters, or with the firm variance related to its firm size through a scaling relationship, and (ii) several functional shapes described by the Subbotin family of distributions. Our empirical results indicate that firms do not have a common shape of the growth rate distribution at the sectorial NAICS level, whereas firms may typically be described by the same shape of the distribution at the subsectorial level, even if the specific shape may not be the same for different subsectors.     

Robust Hierarchical Bayes Small Area Estimation for the Nested Error Linear Regression Model

Standard model‐based small area estimates perform poorly in presence of outliers. Sinha & Rao ( 2009 ) developed robust frequentist predictors of small area means. In this article, we present a robust Bayesian method to handle outliers in unit‐level data by extending the nested error regression model. We consider a finite mixture of normal distributions for the unit‐level error to model outliers and produce noninformative Bayes predictors of small area means. Our modelling approach generalises that of Datta & Ghosh ( 1991 ) under the normality assumption. Application of our method to a data set which is suspected to contain an outlier confirms this suspicion, correctly identifies the suspected outlier and produces robust predictors and posterior standard deviations of the small area means. Evaluation of several procedures including the M‐quantile method of Chambers & Tzavidis ( 2006 ) via simulations shows that our proposed method is as good as other procedures in terms of bias, variability and coverage probability of confidence and credible intervals when there are no outliers. In the presence of outliers, while our method and Sinha–Rao method perform similarly, they improve over the other methods. This superior performance of our procedure shows its dual (Bayes and frequentist) dominance, which should make it attractive to all practitioners, Bayesians and frequentists, of small area estimation.     

A unified approach to validating univariate and multivariate conditional distribution models in time series

Modeling conditional distributions in time series has attracted increasing attention in economics and finance. We develop a new class of generalized Cramer–von Mises (GCM) specification tests for time series conditional distribution models using a novel approach, which embeds the empirical distribution function in a spectral framework. Our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags, which is particularly useful for non-Markovian processes. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally downweights higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events. Unlike the existing methods in the literature, the proposed GCM tests cover both univariate and multivariate conditional distribution models in a unified framework. They exploit the information in the joint conditional distribution of underlying economic processes. Moreover, a class of easy-to-interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications. Distinct from conventional CM and Kolmogorov–Smirnov (KS) tests, which are also based on the empirical distribution function, our GCM test statistics follow a convenient asymptotic $N(0,1)$ distribution and enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance.     

Regression Revisited

Sir Francis Galton introduced median regression and the use of the quantile function to describe distributions. Very early on the tradition moved to mean regression and the universal use of the Normal distribution, either as the natural ‘error’ distribution or as one forced by transformation. Though the introduction of ‘quantile regression’ refocused attention on the shape of the variability about the line, it uses nonparametric approaches and so ignores the actual distribution of the ‘error’ term. This paper seeks to show how Galton's approach enables the complete regression model, deterministic and stochastic elements, to be modelled, fitted and investigated. The emphasis is on the range of models that can be used for the stochastic element. It is noted that as the deterministic terms can be built up from components, so to, using quantile functions, can the stochastic element. The model may thus be treated in both modelling and fitting as a unity. Some evidence is presented to justify the use of a much wider range of distributional models than is usually considered and to emphasize their flexibility in extending regression models.     

Specification tests based on the heterogeneous generalized gamma model of duration: With an application to Kennan's strike data

In this paper, tests for neglected heterogeneity and functional form misspecification of some commonly used parametric distributions are derived within a heterogeneous generalized gamma model. It is argued that the conventional test of heterogeneity may not be valid when the underlying hazard function is misspecified. Hence, if the estimated hazard function is deemed restrictive, tests for functional form misspecification should accompany any test of heterogeneity. An empirical illustration based on Kennan's (1985) model of strikes is used to show that incorrect inferences may be drawn, as in a number of previous analyses, if the relevant restrictions are not tested jointly.     

Characterizations of normal distributions and EDF goodness-of-fit tests

Two characterizations of a normal distribution with an unknown variance based on the corresponding UMVU estimators of the density functions are given, depending on whether its mean is known, or unknown. Applications of these characterization results in the procedures to construct empirical distribution function (EDF) goodness-of-fit tests for normal distributions are mentioned. Received April 2000/Revised April 2002     

一种新的收入差距研究的计量方法——基于分布函数的半参数化估计

收入差距扩大的成因分解方法的最新研究强调描述整个收入分布,并且对收入分布的变化进行分解,进而得到各个统计量的分解,这类方法的优点是可以显示收入差距的变化主要集中在哪些收入群体中。本文着重介绍了两类对收入分布函数进行成因分解的半参数化方法:一类是Lemieux及其合作者们发展起来的权重重置法;另一类是Machado和Mata首创的基于分位数回归的分解方法。     

Conditional empirical likelihood estimation and inference for quantile regression models

This paper considers two empirical likelihood-based estimation, inference, and specification testing methods for quantile regression models. First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667–1714] and Zhang and Gijbels [2003. Sieve empirical likelihood and extensions of the generalized least squares. Scandinavian Journal of Statistics 30, 1–24] to quantile regression models. Second, to avoid practical problems of the CEL method induced by the discontinuity in parameters of CEL, we propose a smoothed counterpart of CEL, called smoothed conditional empirical likelihood (SCEL). We derive asymptotic properties of the CEL and SCEL estimators, parameter hypothesis tests, and model specification tests. Important features are (i) the CEL and SCEL estimators are asymptotically efficient and do not require preliminary weight estimation; (ii) by inverting the CEL and SCEL ratio parameter hypothesis tests, asymptotically valid confidence intervals can be obtained without estimating the asymptotic variances of the estimators; and (iii) in contrast to CEL, the SCEL method can be implemented by some standard Newton-type optimization. Simulation results demonstrate that the SCEL method in particular compares favorably with existing alternatives.     

A strategic predictive distribution for tests of probabilistic calibration

Forecasts of probability distributions are needed to support decision making in many applications. The accuracy of predictive distributions should be evaluated by maximising sharpness subject to calibration. Sharpness relates to the concentration of the predictive distributions, while calibration concerns their statistical consistency with the data. This paper focuses on calibration testing. It is important that a calibration test cannot be gamed by forecasts that have been strategically designed to pass the test. The widely used tests of probabilistic calibration for predictive distributions are based on the probability integral transform. Drawing on previous results for quantile prediction, we show that strategic distributional forecasting is a concern for these tests. To address this, we provide a simple extension of one of the tests. We illustrate ideas using simulated data.     

New unit root asymptotics in the presence of deterministic trends

Recent work by Phillips (Econometrica 66 (1998) 1299) has shown that stochastic trends can be validly represented in empirical regressions in terms of deterministic functions of time. These representations offer an alternative mechanism for modelling stochastic trends. It is shown here that the alternate representations affect the asymptotics of all commonly used unit root tests in the presence of trends. In particular, the critical values of unit root tests diverge when the number of deterministic regressors K→∞ as the sample size n→∞. When they are appropriately recentered and standardized, unit root limit distributions are shown to be normal as K→∞.     

Measuring Agreement in Ordered Rating Scales

Ordered rating scales are one of the most frequently used question formats in large-scale surveys. Analysts of the responses to such questions often find themselves in need of describing the degree of agreement (concentration, consensus) of the answers to such questions. For that purpose they commonly use standard deviations of the response distributions, or measures based on these (such as the coefficient of consensus defined by Granberg and Holmberg, 1988), or the coefficient of variability, etc. This paper demonstrates that such measures are inappropriate for this purpose because they misrepresent what they are supposed to measure: the `peakedness' of a distribution. As an alternative a measure of agreement A is proposed. This measure is a weighted average of the degree of agreement that exists in the simple component parts – layers – into which any frequency distribution can be disaggregated, and for which agreement can be expressed in a straightforward and unequivocal way.     

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.