Stochastic cointegration: estimation and inference

This paper considers the estimation of a stochastically cointegrating regression within the stochastic cointegration modelling framework introduced in McCabe et al. (Stochastic cointegration: testing, 2001). A stochastic cointegrating regression allows some or all of the variables to be conventionally or heteroscedastically integrated. This generalizes Hansen's (J. Econom. 54 (1992) 139) heteroscedastic cointegrating regression model, where the dependent variable is heteroscedastically integrated, but all the regressor variables are restricted to being conventionally integrated. In contrast to conventional and heteroscedastic cointegrating regression, ordinary least-squares (OLS) estimation is shown to be inconsistent, in general, in a stochastically cointegrating regression. As a solution, a new instrumental variables (IVs) estimator is proposed and is shown to be consistent. Under a suitable exogeneity assumption, standard asymptotic inference on the stochastic cointegrating vector can be carried out based on the IV estimator. The finite sample properties of the test statistics, including their robustness to the exogeneity assumption, are examined by simulation.     

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.     

On kernel nonparametric regression designed for complex survey data

In this article, we consider nonparametric regression analysis between two variables when data are sampled through a complex survey. While nonparametric regression analysis has been widely used with data that may be assumed to be generated from independently and identically distributed (iid) random variables, the methods and asymptotic analyses established for iid data need to be extended in the framework of complex survey designs. Local polynomial regression estimators are studied, which include as particular cases design-based versions of the Nadaraya–Watson estimator and of the local linear regression estimator. In this paper, special emphasis is given to the local linear regression estimator. Our estimators incorporate both the sampling weights and the kernel weights. We derive the asymptotic mean squared error (MSE) of the kernel estimators using a combined inference framework, and as a corollary consistency of the estimators is deduced. Selection of a bandwidth is necessary for the resulting estimators; an optimal bandwidth can be determined, according to the MSE criterion in the combined mode of inference. Simulation experiments are conducted to illustrate the proposed methodology and an application with the Canadian survey of labour and income dynamics is presented.     

线性回归在航材订货预测中的应用

线性回归法是一项重要的预测技术,它在相关分析中有着重要的应用。航材的订货数量如果能应用该技术进行预测就可以避免大量的浪费,从而既能保证正常的飞行训练,又能节约大量的经费。     

Quantile models and estimators for data analysis

Quantile regression is used to estimate the cross sectional relationship between high school characteristics and student achievement as measured by ACT scores. The importance of school characteristics on student achievement has been traditionally framed in terms of the effect on the expected value. With quantile regression the impact of school characteristics is allowed to be different at the mean and quantiles of the conditional distribution. Like robust estimation, the quantile approach detects relationships missed by traditional data analysis. Robust estimates detect the influence of the bulk of the data, whereas quantile estimates detect the influence of co-variates on alternate parts of the conditional distribution. Since our design consists of multiple responses (individual student ACT scores) at fixed explanatory variables (school characteristics) the quantile model can be estimated by the usual regression quantiles, but additionally by a regression on the empirical quantile at each school. This is similar to least squares where the estimate based on the entire data is identical to weighted least squares on the school averages. Unlike least squares however, the regression through the quantiles produces a different estimate than the regression quantiles.     

Non-parametric maximum likelihood estimation of censored regression models

This paper presents a consistent estimator of a censored linear regression model which does not require knowledge of the distribution of the error term. The estimator considered here applies Duncan's (1982) suggestion that the likelihood function for the censored regression model be treated as a functional of both the unknown regression vector and the unknown error distribution. Our estimator is the majorizing regression vector for this non-parametric likelihood functional. We find conditions which ensure the consistency of the NPMLE. The paper concludes with the results of Monte Carlo experiments which show the NPMLE to be more efficient than Powell's Least Absolute Deviations (LAD) estimator, particularly when the fraction of censored observations is large and the sample size is small.     

Three Sides of Smoothing: Categorical Data Smoothing, Nonparametric Regression, and Density Estimation

The past forty years have seen a great deal of research into the construction and properties of nonparametric estimates of smooth functions. This research has focused primarily on two sides of the smoothing problem: nonparametric regression and density estimation. Theoretical results for these two situations are similar, and multivariate density estimation was an early justification for the Nadaraya-Watson kernel regression estimator.
A third, less well-explored, strand of applications of smoothing is to the estimation of probabilities in categorical data. In this paper the position of categorical data smoothing as a bridge between nonparametric regression and density estimation is explored. Nonparametric regression provides a paradigm for the construction of effective categorical smoothing estimates, and use of an appropriate likelihood function yields cell probability estimates with many desirable properties. Such estimates can be used to construct regression estimates when one or more of the categorical variables are viewed as response variables. They also lead naturally to the construction of well-behaved density estimates using local or penalized likelihood estimation, which can then be used in a regression context. Several real data sets are used to illustrate these points.     

The standardized effect of a compound of dummy variables or polynomial terms

A method is proposed to obtain standardized regression coefficients for composite variables made up of dummy variables or polynomial terms. The method to be described enables the researcher to compare the effect of the composite variable with the effects of other predictor variables. Forming a composite variable is particularly useful in polynomial regression where individual regression coefficients are hard to interpret. A second application is assessing the impact of a compound of dummy variables. An empirical example dealing with the curvilinear relationship between church involvement and prejudice is used to illustrate the approach.     

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.     

Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

The paper discusses the asymptotic validity of posterior inference of pseudo‐Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be under‐recognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.     

Zur Bewertung von Schätzungen bei der gewöhnlichen Regressionsanalyse

Summary In this paper the multiple regression analysis is considered. The regression coefficients obtained by this method are examined with respect to their numerical stability. With simple stability considerations a criteria is given which can be used if an estimator is to evaluate.     

Functional regression of continuous state distributions

In this paper, we consider a regression model to study the distributional relationship between economic variables. Unlike the classical regression dealing exclusively with mean relationship, our model can be used to analyze the entire dependent structure in distribution. Technically, we treat density functions as random elements and represent the regression relationship as a compact linear operator in the Hilbert spaces of square integrable functions. We propose a consistent estimation procedure for our model, and develop a test to investigate the dependent structure of moments. An empirical example is provided to illustrate how our methodology can be implemented in practical applications.     

Pyrrho's lemma,or have it your way

A simple lemma is derived to support the claim that regression models can be manipulated to a very large extent: by simply adding one regressor one can obtain essentially every set of desired regression coefficients and predictions as well ast-values and standard errors. Consequently, if the product of a specification search is not shown to be generalizable, a sceptical attitude towards its validity is well-founded.     

Small sample properties of the mixed regression estimator

In this paper, the small sample properties of the mixed regression estimator are examined when prior information may be biased and when the ration of the variance of the prior restriction errors to the variance of the sample errors is unknown. The mean square error of the mixed regression estimator is derived, and it is shown that the mixed regression estimator gets dominated by the ordinary least squares estimator in terms of the mean square error as the bias of prior information gets larger.     

A comparative study of finite sample properties of band spectrum regression estimators

Monte Carlo methods are used to compared the small sample properties of several approaches to seasonal adjustment when the objective is to estimate regression coefficients. The methods compared are band spectrum regression, the dummy variable method and the moving average method. The results seem to indicate that band spectrum regression will have superior small sample properties compared with the dummy variable method, which again seemto be preferable to ordinary least squares and to the moving average method.     

Logistic Regression, a review

A review is given of the development of logistic regression as a multi-purpose statistical tool.
A historical introduction shows several lines culminating in the unifying paper of Cox (1966), in which theory as developed in the field of bio-assay is shown to be applicable to designs as discriminant-analysis and case-control study. A review is given of several designs all leading to the same analysis. The link is made with epidemiological literature.
Several optimization criteria are discussed that can be used in the case of more observations per cell, namely maximum likelihood, minimum chi-square and weighted regression on the observed logits. Recent literature on the goodness of fit problem is reviewed and finally, comments are made about the non-parametric approach to logistic regression which is still in rapid development.     

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.     

Biased statistics with zero observations in multiple regression analysis of regional data

Observations containing zeroes as the values of all variables, and which are not meaningful, are shown to be likely in certain regional data sets which may be subjected to multiple regression analysis. The biasing effects of such observations on regression statistics are shown and illustrated with a small data set. It is recommended that such zero observation be identified and removed from regional data sets prior to analysis.     

The stepwise regression algorithm seen from the statistician’s point of view

Summary The purpose of the paper is expository. The procedure called Stepwise Regression, and much used in computer regression programmes, is presented and explained in statistical terms. The algorithm used is presented and demonstrated to serve its purpose. Certain well-known properties of Least Squares multiple regression are shown to be easily deducible from the algorithm.