Generalised residuals and heterogeneous duration models: With applications to the Weilbull model

This paper deals with models for the duration of an event that are misspecified by the neglect of random multiplicative heterogeneity in the hazard function. This type of misspecification has been widely discussed in the literature [e.g., Heckman and Singer (1982), Lancaster and Nickell (1980)], but no study of its effect on maximum likelihood estimators has been given. This paper aims to provide such a study with particular reference to the Weibull regression model which is by far the most frequently used parametric model [e.g., Heckman and Borjas (1980), Lancaster (1979)]. In this paper we define generalised errors and residuals in the sense of Cox and Snell (1968, 1971) and show how their use materially simplifies the analysis of both true and misspecified duration models. We show that multiplicative heterogeneity in the hazard of the Weibull model has two errors in variables interpretations. We give the exact asymptotic inconsistency of M.L. estimation in the Weibull model and give a general expression for the inconsistency of M.L. estimators due to neglected heterogeneity for any duration model to O(σ²), where σ² is the variance of the error term. We also discuss the information matrix test for neglected heterogeneity in duration models and consider its behaviour when σ²>0.     

Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.     

Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n ^?1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ^?1 A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$ ). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.     

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.     

Rangvariable bei Vorliegen von Bindungen

Summary The family of distributions of the random sampleX ₁,...,X _N not being specified solution of different statistical problems is obtained by methods based on ranks, because these methods are distribution-free under the assumption ofQ:=P(X _i =X _j )=0. Frequently, in case ofQ>0, the use of midranks is proposed, though only few is known about their distribution depending on the (usually unknown) distribution of the random sample. Here general formulas of variance and covariance of midranks are derived, after the importance of the termQ determining variance and covariance has been briefly pointed out.     

Some Decompositions of OLSEs and BLUEs Under a Partitioned Linear Model

We consider in this paper a partitioned linear model { y , X ₁ β ₁ + X ₂ β ₂ , σ ² σ } and two corresponding small models { y , X ₁ β ₁ , σ ² σ } and { y , X ₂ β ₂ , σ ² σ } . We derive necessary and sufficient conditions for (i) the ordinary least squares estimator under the full model to be the sum of the ordinary least squares estimators under the two small models; (ii) the best linear unbiased estimator under the full model to be the sum of the best linear unbiased estimators under the two small models; (iii) the best linear unbiased estimator under the full model to be the sum of the ordinary least squares estimators under the two small models. The proofs of the main results in this paper also demonstrate how to use the matrix rank method for characterizing various equalities of estimators under general linear models.     

Tests of significance for the mean of a finite lot

Let ξ₁, ξ₂, ξ₃, ... be independent identically distributed random variables each with normal distribution with mean μ and variance σ². Tests for the process mean μ are well-known elements of statistical analysis: the Gauß test under known process variance σ², Student’st-test under unknown process variance σ². Let the process be partitioned in lots (ξ₁, ..., ξ _N ), (ξ _N+1, ..., ξ_2N ), ... of sizeN. Consider (ξ₁, ..., ξ _N ) as a stochastic representative of this lot sequence and let the lot be characterized by the lot mean $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } $ . The lot mean can be considered as a parameter of the joint conditional distribution function of the lot variables under $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ . The present paper investigates the analogies of the Gauß test and Student’st-test for the lot situation, i.e. tests of significance for the lot meanz under known and unknown process variance σ². This approach is of special interest for the statistical control of product quality in situations where the quality of a lot of items 1, 2, ...,N with quality characteristics ξ₁, ξ₂, ..., ξ _N is identified with the lot average $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ .     

NONPARAMETRIC AND SEMIPARAMETRIC PANEL DATA MODELS: RECENT DEVELOPMENTS

In this paper, we provide an intensive review of the recent developments for semiparametric and fully nonparametric panel data models that are linearly separable in the innovation and the individual-specific term. We analyze these developments under two alternative model specifications: fixed and random effects panel data models. More precisely, in the random effects setting, we focus our attention in the analysis of some efficiency issues that have to do with the so-called working independence condition. This assumption is introduced when estimating the asymptotic variance–covariance matrix of nonparametric estimators. In the fixed effects setting, to cope with the so-called incidental parameters problem, we consider two different estimation approaches: profiling techniques and differencing methods. Furthermore, we are also interested in the endogeneity problem and how instrumental variables are used in this context. In addition, for practitioners, we also show different ways of avoiding the so-called curse of dimensionality problem in pure nonparametric models. In this way, semiparametric and additive models appear as a solution when the number of explanatory variables is large.     

The role of the elasticity of substitution in economic growth: A cross-country investigation

Although the importance of the elasticity of substitution between capital and labor (σ) has long been recognized in several branches of economics, it has not received enough attention in the growth literature. de La Grandville (1989) showed theoretically that at any stage of an economy's development, the growth rate of income per capita is increasing with σ. The higher is σ, the greater the similarity between capital and labor in the production function, and thus diminishing returns set in very slowly. To the best of our knowledge, this is the first paper that tests the hypothesis that growth rate is increasing with the value of σ at the cross-country level. We estimate σ for 90 countries from direct estimation of the normalized CES production function and then include these estimators as an explanatory variable in cross-country growth regression. We investigate the sign and significance of the coefficient of σ conditioning on country characteristics, initial conditions, and a set of policy variables. After accounting for endogeneity and the fact that σ is a “generated” regressor, we find strong support for the hypothesis. The result is robust to both Leamer's (1983) extreme value analysis and Bayesian model averaging. About a fifth to a quarter of the growth rate differential between East Asia and Sub-Saharan Africa can be explained by σ alone.     

Estimation in the pareto distribution

The unique minimum variance unbiased (UMVU) estimate of the probability distribution function of the Pareto distribution is derived. It is shown that the distribution function and ther ^th moment associated with the UMVU estimate are also UMVU estimators. The p.d.f. and its estimator are compared graphically. An estimate of the 100p ^th percentile is given. It is seen that a function of this estimator has a chi-square distribution.     

Approximate inference in multivariate nonlinear functional relationships

An estimation procedure based on estimating equations is presented for the parameters in a multivariate functional relationship model, where all observations are subject to error. The covariance matrix of the observational errors may be parametrized and is allowed to be different for different sets of observations. Estimators are defined for the unknown relation parameters and error parameters.
For linear models (i.e. where the model function is linear in the incidental parameters) the estimators are consistent and asymptotically normal. A consistent expression for the covariance matrix of the estimators is derived. The results are valid for general error distributions.
For nonlinear models the estimators are based on locally linear approximations to the model function. The afore mentioned properties of the estimators are now only approximately valid. The adequacy of the approximate inference, based on asymptotic theory for the linearized model, needs at least informal check. Some examples are given to illustrate the estimation procedure.     

基于EMB多重插补法的线性模型系数估计量的模拟研究

基于EMB多重插补法的线性模型系数估计量,分析其统计性质,并与PMM多重插补法以及DA插补法进行比较。模拟结果显示,随着无回答率增加,系数估计量的偏差绝对值、均方误差呈递增趋势,估计方差的递增趋势相对更显著。在完全随机无回答机制或随机无回答机制下,建议插补重数为15。在依赖被解释变量的非随机无回答机制下,建议插补重数可适当增大。在依赖其他变量的非随机无回答机制下,估计量的均方误差和估计方差的差异大,使用EMB多重插补法要谨慎。     

Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

Nonparametric identification of a binary random factor in cross section data

Suppose V and U are two independent mean zero random variables, where V has an asymmetric distribution with two mass points and U has some zero odd moments (having a symmetric distribution suffices). We show that the distributions of V and U are nonparametrically identified just from observing the sum V+U, and provide a pointwise rate root n estimator. This can permit point identification of average treatment effects when the econometrician does not observe who was treated. We extend our results to include covariates X, showing that we can nonparametrically identify and estimate cross section regression models of the form Y=g(X,D^∗)+U, where D^∗ is an unobserved binary regressor.     

Biases,variances and covariances of raking ratio estimators for marginal and cell totals and averages of observed characteristics

Summary On the basis of a simple random sample from a population, on which a cross-classification is defined with known marginal frequenciesN _i. andN _.j , one wishes to estimate the cell frequenciesN _ij , as well as cell totalsY _ij , marginal totalsY _i. andY _.j , and the grand totalY for characteristics measured on the units. Various authors have discussed so-called raking ratio estimators, which are built up from the estimated cell values by addition. They have given the bias and variance of this estimator ofY. This paper derives biases, variances and covariances for the corresponding estimators of the cell and marginal totals and of the corresponding marginal averages.     

On Maximum Likelihood Estimation of Dynamic Panel Data Models

We analyse the finite sample properties of maximum likelihood estimators for dynamic panel data models. In particular, we consider transformed maximum likelihood (TML) and random effects maximum likelihood (RML) estimation. We show that TML and RML estimators are solutions to a cubic first‐order condition in the autoregressive parameter. Furthermore, in finite samples both likelihood estimators might lead to a negative estimate of the variance of the individual‐specific effects. We consider different approaches taking into account the non‐negativity restriction for the variance. We show that these approaches may lead to a solution different from the unique global unconstrained maximum. In an extensive Monte Carlo study we find that this issue is non‐negligible for small values of T and that different approaches might lead to different finite sample properties. Furthermore, we find that the Likelihood Ratio statistic provides size control in small samples, albeit with low power due to the flatness of the log‐likelihood function. We illustrate these issues modelling US state level unemployment dynamics.     

Estimation of error variance in one-way random model

A preliminary-test estimator for the error variance in the one-way random model is considered. The optimum levels of significance for the preliminary test are obtained based on a regret function. A pooling procedure for estimating the error variance, based on weighting functions, is also considered. A comparison of these estimators is made.