Saddlepoint approximations for the sum of independent non‐identically distributed binomial random variables

We discuss saddlepoint approximations to the distribution of the sum of independent non‐identically distributed binomial random variables. We examine the accuracy of the saddlepoint methods for a sum of 10 binomials with different sets of parameter values. The numerical results indicate that the saddlepoint approximations provide very accurate estimates for the probability mass function and the right‐tail probabilities for the cumulative distribution function of the sum.     

Some remarks on the construction of independent identically distributed random variables,Markov chains and martingales

In this note, we make some remarks on the construction of sequences of independent identically distributed random variables and of Markov chains concretely on a probability space (Ω,A,P). We also show that there are non non-trivial martingales of exponential type.     

Robust feature screening for varying coefficient models via quantile partial correlation

This article is concerned with feature screening for varying coefficient models with ultrahigh-dimensional predictors. We propose a new sure independence screening method based on quantile partial correlation (QPC-SIS), which is quite robust against outliers and heavy-tailed distributions. Then we establish the sure screening property for the QPC-SIS, and conduct simulations to examine its finite sample performance. The results of simulation study indicate that the QPC-SIS performs better than other methods like sure independent screening (SIS), sure independent ranking and screening, distance correlation-sure independent screening, conditional correlation sure independence screening and nonparametric independent screening, which shows the validity and rationality of QPC-SIS.     

Identification with latent variables

This is an essay on a unified approach to the identifiability problem in static models with and without hidden endogenous variables. As is well known, when some of these variables are unobserved, the prior information requirements for models when all endogenous variables are observed, are still there. In addition, extra prior information that takes the place of the means and covariances of the missing variables will have to be supplied directly or indirectly by the statistical researcher. In the paper we characterize the quality and quantity of the required information for the general linear static model and apply it when the model is i) an econometric demand and supply model with missing observations on the quantity transacted, ii) a factor analysis model with observed characteristics of the test takers and iii) a LISREL Model without fixed exogenous variables. With unknown true parameters, the exact rank conditions are seldom verifiable but we do recommend an implementable check-list that is adequate for almost all parameters.     

Consistent estimation of a regression with errors in the variables

Summary As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.     

Implementation with partial verification

This paper examines the implementability of social choice functions when only partial verification of private information is possible. Green and Laffont (1986) used this framework to derive a necessary and sufficient condition for the revelation principle to continue to hold with partial verification. We provide economically interesting characterizations of this condition, which suggest that it may be too restrictive. This leads us to consider implementation (not necessarily truthful) in general, when there is partial verification. We consider the case where compensatory transfers are allowed, giving the mechanism designer further leeway. We show how partial verification may allow efficient implementation of bilateral trade, where it would otherwise not be possible. Received: 1 August 1998 / Accepted: 5 September 2000     

Optimal pollution control with distributed delays

We present a model of optimal stock pollution control with general distributed delays in the stock accumulation dynamics. Using generic functional forms and a distribution structure covering a wide range of distributions, we solve analytically the complex dynamic system that arises from the introduction of these distributed delays. From a theoretical standpoint, our contribution extends the dynamic optimization literature that focused on single discrete delays and develops an original method to address control problems written as mixed type functional differential equations with general kernels. Our results show the qualitative impact of acknowledging these distributed delays on the optimal pollution paths dynamics. We study analytically the properties of the dynamics and we identify the conditions for the occurrence of limit cycles. This theoretical work contributes to the design of efficient environmental policies in the presence of complex delays.     

Estimating the LQAC model with I(2) variables

This paper derives a method for estimating and testing the Linear Quadratic Adjustment Cost (LQAC) model when the target variable and some of the forcing variables follow I(2) processes. Based on a forward-looking error-correction formulation of the model it is shown how to obtain strongly consistent estimates of the structural parameters from both a linear and a non-linear cointegrating regression where first-differences of the I(2) variables are included as regressors (multicointegration). Further, based on the estimated parameter values, it is shown how to test and evaluate the LQAC model using a VAR approach. A simple easy interpretable metric for measuring the model fit is suggested. In an empirical application using UK money demand data, the non-linear multicointegrating regression delivers an economically plausible estimate of the adjustment cost parameter. However, the restrictions implied by the exact LQAC model under rational expectations are strongly rejected and the metric for model fit indicates a substantial noise component in the model. Copyright © 1999 John Wiley & Sons, Ltd.     

Simulation-based estimation of models with lagged latent variables

We extend here our earlier work (Laroque-Salanié, 1989) and propose a dynamic simulated pseudo-maximum likelihood method to deal with a very general class of dynamic non-linear models, including models with lagged latent variables. We test this method on Monte Carlo-generated data for a canonical disequilibrium model. It appears to provide very satisfactory estimates at little computational cost. However, accurate estimation of the standard errors of the estimates may require some care in non-differentiable models.     

Techniques for discriminant analysis with discrete variables

7. Summary Techniques for discriminant analyses are given if some or all discriminators are discrete. An application of the techniques to a previously carried out validation study is presented.     

Estimating the validity of administrative variables

Administrative data have become more important for both official statistics and academic research. One possible problem with such data is that they are biased and have a low validity. Although this problem is often mentioned in a qualitative respect, the validity is seldom quantitatively measured. This article presents a method to estimate the validity of administrative variables. By applying the classical test theory, the validity can be determined by using linked survey and administrative data which should measure the same concepts. This idea is elaborated with an empirical example in which the construct validity of age, gender, educational attainment and wages is determined simultaneously. A linear structural equations model with a measurement component is used to compute the construct validity. The analyses reveal that educational attainment and wages show some bias, but not higher than the bias found in the survey.     

A threshold stochastic volatility model with explanatory variables

In this paper, we introduce a threshold stochastic volatility model with explanatory variables. The Bayesian method is considered in estimating the parameters of the proposed model via the Markov chain Monte Carlo (MCMC) algorithm. Gibbs sampling and Metropolis–Hastings sampling methods are used for drawing the posterior samples of the parameters and the latent variables. In the simulation study, the accuracy of the MCMC algorithm, the sensitivity of the algorithm for model assumptions, and the robustness of the posterior distribution under different priors are considered. Simulation results indicate that our MCMC algorithm converges fast and that the posterior distribution is robust under different priors and model assumptions. A real data example was analyzed to explain the asymmetric behavior of stock markets.     

Estimation of the sum of statistical variables by using classes with random limits

Summary A method is discussed, which makes use of pre-generated random criteria to obtain an unbiased estimate of the mean value of the variate on the condition that the available information on each element of the universe is limited to the possibility to determine in which of any given ranges the value is contained. A formula for the standard deviation of this estimate is also derived. This would greatly facilitate the answering of questionnaires and may therefore have very practical aspects in censuses and surveys.     

单整变量之间Granger因果关系的一种检验程序

本文系统研究了含有单整变量的变量之间Granger因果关系基于OLS估计的检验方法,将适用于存在（1,1）阶协整关系的I(1)变量之间Granger因果关系检验的Engle和Granger（1987）两步程序,扩展到了存在协整关系的高阶单整变量的情形,并提出了含有单整变量的变量之间Granger因果关系检验的一般程序。     

Forecasting economic and financial variables with global VARs

This paper considers the problem of forecasting economic and financial variables across a large number of countries in the global economy. To this end a global vector autoregressive (GVAR) model, previously estimated by Dees, di Mauro, Pesaran, and Smith (2007) and Dees, Holly, Pesaran, and Smith (2007) over the period 1979Q1–2003Q4, is used to generate out-of-sample forecasts one and four quarters ahead for real output, inflation, real equity prices, exchange rates and interest rates over the period 2004Q1–2005Q4. Forecasts are obtained for 134 variables from 26 regions, which are made up of 33 countries and cover about 90% of the world output. The forecasts are compared to typical benchmarks: univariate autoregressive and random walk models. Building on the forecast combination literature, the effects of model and estimation uncertainty on forecast outcomes are examined by pooling forecasts obtained from different GVAR models estimated over alternative sample periods. Given the size of the modelling problem, and the heterogeneity of the economies considered–industrialised, emerging, and less developed countries–as well as the very real likelihood of possibly multiple structural breaks, averaging forecasts across both models and windows makes a significant difference. Indeed, the double-averaged GVAR forecasts perform better than the benchmark competitors, especially for output, inflation and real equity prices.     

Models with latent variables: LISREL versus PLS

Summary: LISREL and PLS are two different ways of modelling latent variables and their relations to each other within a set of manifest variables. These two models are contrasted with each other. In the special case of two groups of manifest variables the relations that exist between corresponding parameters and latent variables of both types of models are revealed.     

Path analysis with partial association measures

This paper discusses the use of partial association measures for carrying out path analyses on categorical data. The measures considered are essentially PRE (proportion of reduction in error of prediction) measures for nominal variables and concordance-discordance indices for ordinal ones. These measures provide a natural way to evaluate the strength of the path linking a non-measurable response variable to one of several categorical explanatory factors. Concerning the decomposition of raw association into direct and indirect effects, it is shown, however, that they do not share the properties of conventional path coefficients for measurable variables. Especially purely nominal association measures need to be interpreted with care. The scope of the partial measures for path analysis is illustrated through a study of the relationships between the educational styles experienced by swiss adolescents and their selfesteem.     

On the maxima of heterogeneous gamma variables with different shape and scale parameters

In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let \(X_{1}\) and \(X_{2}\) be two independent gamma random variables with \(X_{i}\) having shape parameter \(r_{i}>0\) and scale parameter \(\lambda _{i}\) , \(i=1,2\) , and let \(X^{*}_{1}\) and \(X^{*}_{2}\) be another set of independent gamma random variables with \(X^{*}_{i}\) having shape parameter \(r_{i}^{*}>0\) and scale parameter \(\lambda _{i}^{*}\) , \(i=1,2\) . Denote by \(X_{2:2}\) and \(X^{*}_{2:2}\) the corresponding maxima, respectively. It is proved that, among others, if \((r_{1},r_{2})\) majorize \((r_{1}^{*},r_{2}^{*})\) and \((\lambda _{1},\lambda _{2})\) weakly majorize \((\lambda _{1}^{*},\lambda _{2}^{*})\) , then \(X_{2:2}\) is stochastically larger that \(X^{*}_{2:2}\) in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.     

On the properties of the coefficient of determination in regression models with infinite variance variables

We examine the asymptotic properties of the coefficient of determination, R²$R^2$, in models with α-stable$\alpha \text{-stable}$   random variables. If the regressor and error term share the same index of stability α<2$\alpha <2$, we show that the R²$R^2$  statistic does not converge to a constant but has a nondegenerate distribution on the entire [0,1]$[0,1]$ interval. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable, and we show that the density function is unbounded at 0 and 1. If the indices of stability of the regressor and error term are unequal, we show that the coefficient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability, irrespective of the value of the slope coefficient. In an empirical application, we revisit the Fama and MacBeth (1973) two-stage regression and demonstrate that in the infinite-variance case the R²$R^2$  statistic of the second-stage regression converges to 0 in probability even if the slope coefficient is nonzero. We deduce that a small value of the R²$R^2$  statistic should not, in itself, be used to reject the usefulness of a regression model.