Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

时空动态半参数变系数随机效应面板模型的估计

研究目标：克服半参数变系数面板模型中扰动项和因变量存在时空动态性问题。研究方法：提出一类更加一般化的时空动态半参数变系数随机效应面板模型,并构建截面似然估计量。研究发现：估计量具有良好的小样本性质,估计误差随着样本总量的提高而减小,在Case空间矩阵下,空间滞后和时空滞后系数的估计精度随空间复杂度的增大而降低,用该方法分析我国外商直接投资、知识产权保护与经济增长关系,进一步证实了模型的适用性。研究创新：证明了估计量满足一致性和渐近正态性,数值模拟考察了估计量的小样本性质。研究价值：拓展了现有半参数变系数空间面板模型的形式,增强了模型的适用性和解释力,有益于经济问题实证研究的开展。     

Maximum Likelihood Estimation of a General Unbalanced Spatial Random Effects Model: a Monte Carlo Study

Abstract

This paper discusses the maximum likelihood estimator of a general unbalanced spatial random effects model with normal disturbances, assuming that some observations are missing at random. Monte Carlo simulations show that the maximum likelihood estimator for unbalanced panels performs well and that missing observations affect mainly the root mean square error. As expected, these estimates are less efficient than those based on the unobserved balanced model, especially if the share of missing observations is large or spatial autocorrelation in the error terms is pronounced.

Estimation de vraisemblance maximale d'un modèle général d'effets aléatoires spatiaux déséquilibré: une étude Monte Carlo

RÉSUMÉ La présente communication se penche sur l'estimateur du maximum de vraisemblance d'un modèle général d'effets aléatoires spatiaux déséquilibré avec des perturbations normales, en supposant l'absence aléatoire de certaines observations. Des simulations de Monte Carlo montrent que des groupes déséquilibrés se comporte bien, et que les observations manquantes affectent principalement l'erreur de la moyenne quadratique. Comme prévu, ces évaluations sont moins efficaces que celles qui sont basées sur le modèle équilibré non observé, notamment si la part des observations manquantes est importantes, ou l'on déclare une autocorrélation spatiale dans les termes d'erreur.

Estimación de la probabilidad máxima de un modelo espacial general desequilibrado de efectos al azar: un estudio de Monte Carlo

RÉSUMÉN Este trabajo discute el estimador de probabilidad máxima de un modelo espacial general desequilibrado de efectos al azar con alteraciones normales, suponiendo que faltan algunas observaciones al azar. Las simulaciones de Monte Carlo muestran que el estimador de probabilidad máxima para los paneles desequilibrados funciona satisfactoriamente, y que las observaciones omisas afectan principalmente al error de la media cuadrática. Como se suponía, estas estimaciones son menos eficientes que las basadas en el modelo equilibrado inadvertido, especialmente si la cantidad de omisiones es grande/o la autocorrelación en los términos de error es pronunciada.

     

Maximum Likelihood Estimation in Panels with Incidental Trends

It is shown that the maximum likelihood estimator of a local to unity parameter can be consistently estimated with panel data when the cross-section observations are independent. Consistency applies when there are no deterministic trends or when there is a homogeneous deterministic trend in the panel model. When there are heterogeneous deterministic trends the panel MLE of the local to unity parameter is inconsistent. This outcome provides a new instance of inconsistent ML estimation in dynamic panels, and, unlike earlier results of this type, applies when both T →∞ and N →∞.     

随机效应Logistic模型的参数估计

在经济计量学中,对面板(panel)数据的研究是一个热门的话题。目前,讨论得较多的是如何运用线性随机效应模型来对它建模。可是,当因变量是二元的数据时,用线性随机效应模型进行建模显然是错误的。这时,比较常用的是随机效应Logistic模型。本文讨论了如何运用EM算法对随机效应Logistic模型进行参数估计。     

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.     

变系数动态面板数据模型的半参数估计

本文把一般的常系数的动态面板数据模型拓广到变系数的情形。对于变系数的动态面板数据模型首先推导出模型所隐含的各种矩条件,然后利用广义矩估计的方法得到了模型中未知参数的半参数广义矩估计,最后对于我们所得到的估计的渐进性和一致性进行证明。     

Fixed and Random Effects in Stochastic Frontier Models

Received stochastic frontier analyses with panel data have relied on traditional fixed and random effects models. We propose extensions that circumvent two shortcomings of these approaches. The conventional panel data estimators assume that technical or cost inefficiency is time invariant. Second, the fixed and random effects estimators force any time invariant cross unit heterogeneity into the same term that is being used to capture the inefficiency. Inefficiency measures in these models may be picking up heterogeneity in addition to or even instead of inefficiency. A fixed effects model is extended to the stochastic frontier model using results that specifically employ the nonlinear specification. The random effects model is reformulated as a special case of the random parameters model. The techniques are illustrated in applications to the U.S. banking industry and a cross country comparison of the efficiency of health care delivery.JEL classification: C1, C4     

基于贝叶斯结构突变的随机系数动态面板模型研究

针对目前随机系数动态面板模型中存在内生变量初始值固定、个体自回归系数平稳以及不存在结构突变的种种限制,本文提出用分层贝叶斯方法首次检测和估计了含未知结构突变的随机系数动态面板模型。容许初始值与个体相关,自回归系数服从logitnormal分布保证平稳性,得到了未知结构突变和随机系数的后验密度估计。对1995年到2012年中国五省市出口总值月度数据进行实证分析,检测出四个结构突变,分析突变前后的情况表明出口总值存在三大特征：呈现稳定增长态势,但省市间差距逐渐扩大;重大的外部需求冲击对出口有显著影响;出口总值的结构突变有明显的季节特征.     

Random Effects Misspecification Can Have Severe Consequences for Random Effects Inference in Linear Mixed Models

There has been considerable and controversial research over the past two decades into how successfully random effects misspecification in mixed models (i.e. assuming normality for the random effects when the true distribution is non‐normal) can be diagnosed and what its impacts are on estimation and inference. However, much of this research has focused on fixed effects inference in generalised linear mixed models. In this article, motivated by the increasing number of applications of mixed models where interest is on the variance components, we study the effects of random effects misspecification on random effects inference in linear mixed models, for which there is considerably less literature. Our findings are surprising and contrary to general belief: for point estimation, maximum likelihood estimation of the variance components under misspecification is consistent, although in finite samples, both the bias and mean squared error can be substantial. For inference, we show through theory and simulation that under misspecification, standard likelihood ratio tests of truly non‐zero variance components can suffer from severely inflated type I errors, and confidence intervals for the variance components can exhibit considerable under coverage. Furthermore, neither of these problems vanish asymptotically with increasing the number of clusters or cluster size. These results have major implications for random effects inference, especially if the true random effects distribution is heavier tailed than the normal. Fortunately, simple graphical and goodness‐of‐fit measures of the random effects predictions appear to have reasonable power at detecting misspecification. We apply linear mixed models to a survey of more than 4 000 high school students within 100 schools and analyse how mathematics achievement scores vary with student attributes and across different schools. The application demonstrates the sensitivity of mixed model inference to the true but unknown random effects distribution.     

Estimating Nonlinear Models with Multiple Fixed Effects: A Computational Note*

In this paper we consider estimation of nonlinear panel data models that include multiple individual fixed effects. Estimation of these models is complicated both by the difficulty of estimating models with possibly thousands of coefficients and also by the incidental parameters problem; that is, noisy estimates of the fixed effects when the time dimension is short contaminate the estimates of the common parameters due to the nonlinearity of the problem. We propose a simple variation of existing bias‐corrected estimators, which can exploit the additivity of the effects for numerical optimization. We exhibit the performance of the estimators in simulations.     

A Note on Estimated Coefficients in Random Effects Probit Models

This note points out to applied researchers what adjustments are needed tothe coefficient estimates in a random effects probit model in order to make valid comparisons in terms of coefficient estimates and marginal effects across different specifications. These adjustments are necessary because of the normalization that is used by standard software in order to facilitate easyestimation of the random effects probit model.     

Dickey–Fuller Type of Tests against Nonlinear Dynamic Models*

In this paper, we introduce several test statistics testing the null hypothesis of a random walk (with or without drift) against models that accommodate a smooth nonlinear shift in the level, the dynamic structure and the trend. We derive analytical limiting distributions for all the tests. The power performance of the tests is compared with that of the unit‐root tests by Phillips and Perron [Biometrika (1988), Vol. 75, pp. 335–346], and Leybourne, Newbold and Vougas [Journal of Time Series Analysis (1998), Vol. 19, pp. 83–97]. In the presence of a gradual change in the deterministics and in the dynamics, our tests are superior in terms of power.     

Hedonic Housing Prices in Paris: An Unbalanced Spatial Lag Pseudo‐Panel Model with Nested Random Effects

This paper estimates a hedonic housing model based on flats sold in the city of Paris over the period 1990–2003. This is done using maximum likelihood estimation, taking into account the nested structure of the data. Paris is historically divided into 20 arrondissements, each divided into four quartiers (quarters), which in turn contain between 15 and 169 blocks (îlot, in French) per quartier. This is an unbalanced pseudo?panel data containing 156,896 transactions. Despite the richness of the data, many neighborhood characteristics are not observed, and we attempt to capture these neighborhood spillover effects using a spatial lag model. Using likelihood ratio tests, we find significant spatial lag effects as well as significant nested random error effects. The empirical results show that the hedonic housing estimates and the corresponding marginal effects are affected by taking into account the nested aspects of the Paris housing data as well as the spatial neighborhood effects.Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation in Two-Stage Models with Heteroscedasticity

A surprising number of important problems can be cast in the framework of estimating a mean and variance using data arising from a two-stage structure. The first stage is a random sampling of "units" with some quantity of interest associated with the unit. The second stage produces an estimate of that quantity and usually, but not always, an estimated standard error, which may change considerably across units. Heteroscedasticity in the estimates over different units can arise for a number of reasons, including variation associated with the unit and changing sampling effort over units. This paper presents a broad discussion of the problem of making inferences for the population mean and variance associated with the unobserved true values at the first stage of sampling. A careful discussion of the causes of heteroscedasticity is given, followed by an examination of ways in which inferences can be carried out in a manner that is robust to the nature of the within unit heteroscedasticity. Among the conclusions are that under any type of heteroscedasticity, an unbiased estimate of the mean and the variance of the estimated mean can be obtained by using the estimates as if they were true unobserved values from the first stage. The issue of using the mean versus a weighted average which tries to account for the heteroscedasticity is also discussed. An unbiased estimate of the population variance is given and the variance of this estimate and its covariance with the estimated mean is provided under various types of heteroscedasticity. The two-stage setting arises in many contexts including the one-way random effects models with replication, meta-analysis, multi-stage sampling from finite populations and random coefficients models. We will motivate and illustrate the problem with data arising from these various contexts with the goal of providing a unified framework for addressing such problems.     

Break Date Estimation for Models with Deterministic Structural Change

In this article, we consider estimating the timing of a break in level and/or trend when the order of integration and autocorrelation properties of the data are unknown. For stationary innovations, break point estimation is commonly performed by minimizing the sum of squared residuals across all candidate break points, using a regression of the levels of the series on the assumed deterministic components. For unit root processes, the obvious modification is to use a first differenced version of the regression, while a further alternative in a stationary autoregressive setting is to consider a GLS‐type quasi‐differenced regression. Given uncertainty over which of these approaches to adopt in practice, we develop a hybrid break fraction estimator that selects from the levels‐based estimator, the first‐difference‐based estimator, and a range of quasi‐difference‐based estimators, according to which achieves the global minimum sum of squared residuals. We establish the asymptotic properties of the estimators considered, and compare their performance in practically relevant sample sizes using simulation. We find that the new hybrid estimator has desirable asymptotic properties and performs very well in finite samples, providing a reliable approach to break date estimation without requiring decisions to be made regarding the autocorrelation properties of the data.     

CKLS-JUMP过程驱动的利率动态模型:理论估计与实证模拟

利率期限结构动态模式研究已经成为现代金融领域的一个研究热点,而跳跃扩散过程已经成为模拟存贷款利率最为有效的动态模型。本文主要以商业银行存贷款利率为对象,研究分析利率期限结构的动态变化过程。首先基于存贷款利率的变化特征,建立利率的CKLS-JUMP跳跃扩散模型;其次,运用马尔科夫链蒙特卡罗模拟方法(MCMC)对其参数进行理论估计;最后,以我国商业银行五年期存贷款利率为例进行实证模拟。研究结论认为:CKLS-JUMP模型更加符合我国存贷款利率动态行为;同时MCMC方法比传统估计方法更加准确。     

Estimation and Solution of Models with Expectations and Structural Changes

In this paper, we develop solutions for linearized models with forward‐looking expectations and structural changes under a variety of assumptions regarding agents' beliefs about those structural changes. For each solution, we show how its associated likelihood function can be constructed by using a ‘backward–forward’ algorithm. We illustrate the techniques with two examples. The first considers an inflationary program in which beliefs about the inflation target evolve differently from the inflation target itself, and the second applies the techniques to estimate a new Keynesian model through the Volcker disinflation. We compare our methodology with the alternative in which structural change is captured by switching between regimes via a Markov switching process. We show that our method can produce accurate results much faster than the Markov switching method as well as being easily adapted to handle beliefs departing from reality. Copyright © 2016 John Wiley & Sons, Ltd.     

Identification and Estimation of Club Convergence Models with Spatial Dependence

This article deals with heterogeneity and spatial dependence in economic growth analysis by developing a two‐stage strategy that identifies clubs by a mapping analysis and estimates a club convergence model with spatial dependence. Since estimation of this class of convergence models in the presence of regional heterogeneity poses both identification and collinearity problems, we develop an entropy‐based estimation procedure that simultaneously takes account of ill‐posed and ill‐conditioned inference problems. The two‐step strategy is applied to assess the existence of club convergence and to estimate a two‐club spatial convergence model across Italian regions over the period 1970 to 2000.