Comparison of SML and GMM estimators for the random coefficient logit model using aggregate data

A simulated maximum likelihood (SML) estimator for the random coefficient logit model using aggregate data is found to be more efficient than the widely used generalized method of moments estimator (GMM) of Berry et?al. (Econometrica 63:841?C890, 1995). In particular, the SML estimator is better than the GMM estimator in recovery of heterogeneity parameters which are often of central interest in marketing research. With the GMM estimator, the analyst must determine what moment conditions to use for parameter identification, especially the heterogeneity parameters. With the SML estimator, the moment conditions are automatically determined as the gradients of the log-likelihood function, and these are the most efficient ones if the model is correctly specified. Another limitation of the GMM estimator is that the product market shares must be strictly positive while the SML estimator can handle zero market share observations. Properties of the SML and GMM estimators are demonstrated in simulated data and in data from the US photographic film market.     

Moment Estimation of the Probit Model with an Endogenous Continuous Regressor

We propose a generalized method of moments (GMM) estimator with optimal instruments for a probit model that includes a continuous endogenous regressor. This GMM estimator incorporates the probit error and the heteroscedasticity of the error term in the first‐stage equation in order to construct the optimal instruments. The estimator estimates the structural equation and the first‐stage equation jointly and, based on this joint moment condition, is efficient within the class of GMM estimators. To estimate the heteroscedasticity of the error term of the first‐stage equation, we use the k‐nearest neighbour (k‐nn) non‐parametric estimation procedure. Our Monte Carlo simulation shows that in the presence of heteroscedasticity and endogeneity, our GMM estimator outperforms the two‐stage conditional maximum likelihood estimator. Our results suggest that in the presence of heteroscedasticity in the first‐stage equation, the proposed GMM estimator with optimal instruments is a useful option for researchers.     

Econometricians Have Their Moments: GMM at 32

The 2013 Nobel Prize for Economics was awarded to Eugene Fama, Lars Hansen and Robert Shiller for their work on empirical asset pricing. Hansen's primary contribution to the cited work was the development of the generalised method of moments (GMM), a statistical method that has proved such a valuable tool for testing the validity of empirical asset pricing models. The public announcement of the award also acknowledges the wider impact of GMM on empirical analysis in economics and beyond, referring to the 1982 Econometrica paper in which Hansen introduced the method as ‘one of the most influential in econometrics’. In this paper, we reflect on how the GMM‐based inference framework has evolved since 1982, reviewing developments on four main issues: model diagnostic testing, moment selection, identification and inference in misspecified models. We also illustrate the broader influence of GMM on econometrics by briefly exploring the connections between GMM and three other estimation methods: indirect inference, moment inequality based techniques, and a group of techniques that can be presented equivalently within either the generalised empirical likelihood or info‐metric frameworks.     

Population and economic growth: a simultaneous equation perspective

This article examines the relationship between Population Growth (PG) and Economic Growth (EG) in the framework of simultaneous structural equation models. Based on Lewbel (2012), the structural parameters can be estimated using the Generalized Method of Moments (GMM). Identification requires a heteroscedastic covariance restriction that appears in some models of endogeneity, measurement errors and panel data. This study obtains several findings. First, the current and lagged variables of PG negatively and positively affect EG in the short run. Second, PG does not significantly influence EG in the long run. Third, the reverse relations running from EG to PG are weak in both the short and long run, regardless of economic development conditions.     

Convenient estimators for the panel probit model: Further results

Bertschek and Lechner (1998) propose several variants of a GMM estimator based on the period specific regression functions for the panel probit model. The analysis is motivated by the complexity of maximum likelihood estimation and the possibly excessive amount of time involved in maximum simulated likelihood estimation. But, for applications of the size considered in their study, full likelihood estimation is actually straightforward, and resort to GMM estimation for convenience is unnecessary. In this note, we reconsider maximum likelihood based estimation of their panel probit model then examine some extensions which can exploit the heterogeneity contained in their panel data set. Empirical results are obtained using the data set employed in the earlier study. Helpful comments and suggestions by Irene Bertschek and Michael Lechner are gratefully acknowledged. This paper has also benefited from comments by two anonymous referees and from seminar participants at the Center for Health Economics at the University of York. Any remaining errors are the responsibility of the author.     

Structural interactions in spatial panels

Until recently, considerable effort has been devoted to the estimation of panel data regression models without adequate attention being paid to the drivers of interaction amongst cross-section and spatial units. We discuss some new methodologies in this emerging area and demonstrate their use in measurement and inferences on cross-section and spatial interactions. Specifically, we highlight the important distinction between spatial dependence driven by unobserved common factors and those based on a spatial weights matrix. We argue that purely factor-driven models of spatial dependence may be inadequate because of their connection with the exchangeability assumption. The three methods considered are appropriate for different asymptotic settings; estimation under structural constraints when N is fixed and T → ∞, whilst the methods based on GMM and common correlated effects are appropriate when T ≫ N → ∞. Limitations and potential enhancements of the existing methods are discussed, and several directions for new research are highlighted.     

Reporting errors, ability heterogeneity, and returns to schooling in China

Abstract. This study is aimed at providing a more accurate estimate of the returns to schooling in China by controlling for unobserved ability heterogeneity and measurement errors. We identify a unique instrument to correct for the omitted ability bias. We find that the attenuation bias caused by measurement error dominates the omitted ability bias in the OLS estimation. Based on the GMM estimation, for young workers in China, the return to schooling is 15.0% overall and 16.9% for women.     

EFFICIENCY GAINS OF SYSTEM GMM AND MDE OVER INDIVIDUAL EQUATION ESTIMATION*

In the econometric literature it is known that, under certain conditions, estimating a system of equations together is more efficient than estimating each equation separately. This finding has been proved, however, only under the assumption of a known parametric form of heteroskedasticity (including homoskedasticity) or non‐random regressors/instruments. This note shows that an analogous finding holds for GMM under heteroskedasticity of unknown form and random regressors/instruments. Specifically, I provide a necessary condition for the efficiency gain of the system GMM over the single‐equation GMM. An analogous necessary condition for the efficiency gain is also shown to hold for minimum‐distance (or χ2) estimation (MDE).     

Redundancy of Moment Conditions in Restricted GMM Estimation

Using a novel approach to calculating the rank of the difference of two asymptotic variance matrices, The author derives the necessary and sufficient conditions for an extra set of moment conditions to be redundant given a set of moment conditions in GMM estimation with general nonlinear restrictions. The necessary and sufficient conditions derived in this paper include as a special case the redundancy of moment conditions for GMM estimation without restrictions that was first derived by Breusch et al. (1999). Therefore this paper advances the research on redundancy of moment conditions from unrestricted GMM estimation to a larger class of GMM estimation. To show their usefulness, the main results of the current paper are applied to instrumental variables estimation of linear regression models and the efficient estimation of seemingly unrelated regressions models, subject to restrictions.     

Rolling Regression Analysis of the Pástor-Stambaugh Model: Evidence from Robust Instrumental Variables

The capital asset pricing model (CAPM), Fama-French (FF), and Pástor-Stambaugh (PS) factor models are examined using a new dynamic rolling regression version of the generalized method of moments (GMM) method. This rolling regression framework not only allows us to investigate phases of the business cycle, but also permits regression estimates to vary through time due to changes in the development and efficiency of the sectors. The principal reasons for using the dynamic GMM with robust instruments is that some of these factors are measured with errors and the disturbances may be non-spherical. The CAPM appears as the most parsimonious model to explain the FF sector returns. Furthermore, the rolling GMM approach is clearly more sensitive to dynamic financial episodes than the ordinary least squares approach. In particular, liquidity has some anticipatory power, as it is able to forecast the 2007–2009 crises with heightened volatility starting in late 2005.     

Uncovered interest parity puzzle: does it really exist?

Most studies indicate the violation of the Uncovered Interest Parity (UIP) restrictions, and indeed, there are many who find a statistically significant and negative interest rate differential slope. This finding has now become known in international economics as the UIP puzzle. Using recent data on four major currencies vis-à-vis the US dollar and employing the Generalized Method of Moments (GMM) technique, our estimation results indicate the absence of any relationship between the interest rate differential and the expected change in the exchange rate, rather than the presence of the UIP puzzle.     

MACROECONOMIC POLICY REFORMS AND PRODUCTIVITY GROWTH IN AFRICAN AGRICULTURE

This article employs a two‐stage procedure to investigate the impact of macroeconomic policy reforms on the agricultural productivity growth of 33 African countries from 1981 to 2001. In the first stage, we measure agricultural productivity using a nonparametric Malmquist productivity index. In the second stage, we build a generalized method of moments (GMM) model with a measure of structural adjustment program (SAP) intensity as a key instrument for macroeconomic policy reforms. We also control for the effects of globalization, civil violence, level of development of physical and financial infrastructure, and other economic variables as well as natural resource factors that directly affect agricultural productivity. Our results indicate a strong positive correlation between the extent of SAP intensity and agricultural productivity, suggesting that the macroeconomic policy reforms improved agricultural productivity growth in the sample countries. (JEL E6, O13, O41)     

Adjustment of inputs and measurement of technical efficiency: A dynamic panel data analysis of the Egyptian manufacturing sectors

The purpose of this article is to construct a dynamic stochastic production frontier incorporating the sluggish adjustment of inputs, to measure the speed of adjustment of output, and to compare the technical efficiency estimates from this dynamic model to those from a static model. By assuming instantaneous adjustment of all inputs, a static model may underestimate technical efficiency of a production unit in the short-run. However, in this article I show that under the assumption of similar adjustment speed for all inputs, a linear partial adjustment scheme for output characterizes the dynamic production frontier. The dynamic frontier with time-invariant technical efficiency is estimated using the system GMM (generalized method of moments) estimator. Applying the model and estimation method on a panel data set spanning 9?years of data on private manufacturing establishments in Egypt, I find that (1) the speed of adjustment of output is significantly lower than unity, (2) the static model underestimates technical efficiency by 4.5 percentage points on average, and (3) the ranking of production units based on their technical efficiency measures changes when the lagged adjustment process of inputs is taken into account.     

Entry,exit and productivity growth: Spanish manufacturing during the eighties

This paper analyses the role played by both competitive pressure (increasing imports) and the restructuring of industries through entry and exit in productivity growth of Spanish manufacturing during the eighties, the key period of its accession to the EEC economy. A GMM panel estimation of the determinants of corrected Solow residual for 75 manufacturing during 1979-1990, shows that these forces accounted for 80% of productivity growth, playing an important role the displacement of inefficient firms by competitive entry.JEL Classification: D24, C33We are grateful to S. Bentolila, F. Bourguignon, T. Coelli, J. Dolado, A. de la Fuente, J. García, L. Rodríguez, J. Segura, X. Vives and an anonymous referee for useful comments on previous versions of this paper. All remaining errors are our own.     

A generalized method of moments estimator for a spatial model with moving average errors, with application to real estate prices

This paper proposes a new GMM estimator for spatial regression models with moving average errors. Monte Carlo results are given which suggest that the GMM estimates are consistent and robust to non-normality, and the Bootstrap method is suggested as a way of testing the significance of the moving average parameter. The estimator is applied in a model of English real estate prices, in which the concepts of displaced demand and displaced supply are introduced to derive the spatial lag of prices, and the moving average error process represents spatially autocorrelated unmodelled variables.      

Do Political Factors Cause the Regional Inequality in the Reform‐Era China?

We construct a political variable—the number of ministers of all national government departments born in each province weighted by provincial population—to examine its effect on economic growth in China during 1981–2010. We find that the lagged per capita political variable has a significant, positive effect on economic growth in both ordinary least squares (OLS) estimation and system generalized method of moments (GMM) estimation. We also find that the lagged per capita political variable has a significant, positive effect on financial deregulation policies distributed across provinces, which offers a possible mechanism for the political variable to impact growth.     

扁平证券市场线和融资限制——中国股票市场的实证检验

资本资产定价模型（CAPM)在国际金融市场的实证检验结果往往呈现扁平证券市场线现象。一些学者认为融资限制是该现象的产生原因。本文旨在研究扁平证券市场线现象是否存在于中国股市,同时分析融资限制对中国股市证券市场线的影响。本文采用投资组合分析法对A股主板市场进行实证检验。本文研究发现：（1)扁平的证券市场线现象同样存在于中国股票市场中;（2)在中国股票市场中,融资限制与证券市场线的斜率呈负相关,与截距呈正相关,这符合融资限制理论;（3)以上两点发现同样存在于多因子定价模型中,这表明本文研究结果具有稳健性。       

Generalized Method of Moment estimation of multivariate multifractal models

Multifractal processes have recently been introduced as a new tool for modeling the stylized facts of financial markets and have been found to consistently provide certain gains in performance over basic volatility models for a broad range of assets and for various risk management purposes. Due to computational constraints, multivariate extensions of the baseline univariate multifractal framework are, however, still very sparse so far. In this paper, we introduce a parsimoniously designed multivariate multifractal model, and we implement its estimation via a Generalized Methods of Moments (GMM) algorithm. Monte Carlo studies show that the performance of this GMM estimator for bivariate and trivariate models is similar to GMM estimation for univariate multifractal models. An empirical application shows that the multivariate multifractal model improves upon the volatility forecasts of multivariate GARCH over medium to long forecast horizons.     

Long-run Study of Residential Water Consumption

The estimation of dynamic models and themeasure of long-run effects arerare in residential water demand studies. Weshow in this paper that a dynamicmodel of water consumption can be derived froma structural optimisation programsolved by local communities. Thisnonlinear model is estimated on asample of French municipalities and is foundasymptotically equivalent to a dynamic panel data model that is linear in theparameters. The latter includes anoriginal error-component structure that allowsfor a flexible heterogeneity pattern, including both the usual idiosyncraticeffect, and an additional individualeffect affected by a multiplicative time-varyingparameter. As usual GMM estimators for panel data are not consistent inthis case, we propose a new GMMprocedure that yields consistent and efficientestimates of short- and long-runprice elasticities (respectively −0.26 and−0.40).     

Engineering robust instruments for GMM estimation of panel data regression models with errors in variables: a note

Econometricians have long recognized the need to account in some way for measurement errors, specification errors and endogeneity to ensure that the ordinary least squares estimator is consistent. This article introduces a new generalized method of moments estimator that relies on robust instruments to estimate panel data regression models containing errors in variables. We show how this GMM approach can be generalized for the panel data framework using higher moments and cumulants as instruments. The new instruments, engineered for greater robustness, are proposed to tackle the pervasive problem of weak instruments.