Spatial correlation robust inference with errors in location or distance

This paper presents results from a Monte Carlo study concerning inference with spatially dependent data. We investigate the impact of location/distance measurement errors upon the accuracy of parametric and nonparametric estimators of asymptotic variances. Nonparametric estimators are quite robust to such errors, method of moments estimators perform surprisingly well, and MLE estimators are very poor. We also present and evaluate a specification test based on a parametric bootstrap that has good power properties for the types of measurement error we consider.     

Indirect inference with time series observed with error

We propose the indirect inference estimator as a consistent method to estimate the parameters of a structural model when the observed series are contaminated by measurement error by considering the noise as a structural feature. We show that the indirect inference estimates are asymptotically biased if the error is neglected. When the condition for identification is satisfied, the structural and measurement error parameters can be consistently estimated. The issues of identification and misspecification of measurement error are discussed in detail. We illustrate the reliability of this procedure in the estimation of stochastic volatility models based on realized volatility measures contaminated by microstructure noise.     

Nonparametric transformation to white noise

We consider a semiparametric distributed lag model in which the “news impact curve” m is nonparametric but the response is dynamic through some linear filters. A special case of this is a nonparametric regression with serially correlated errors. We propose an estimator of the news impact curve based on a dynamic transformation that produces white noise errors. This yields an estimating equation for m that is a type two linear integral equation. We investigate both the stationary case and the case where the error has a unit root. In the stationary case we establish the pointwise asymptotic normality. In the special case of a nonparametric regression subject to time series errors our estimator achieves efficiency improvements over the usual estimators, see Xiao et al. [2003. More efficient local polynomial estimation in nonparametric regression with autocorrelated errors. Journal of the American Statistical Association 98, 980–992]. In the unit root case our procedure is consistent and asymptotically normal unlike the standard regression smoother. We also present the distribution theory for the parameter estimates, which is nonstandard in the unit root case. We also investigate its finite sample performance through simulation experiments.     

Variable selection for additive partially linear models with measurement error

Variable selection for additive partially linear models with measurement error is considered. By the backfitting technique, we first propose a variable selection procedure for the parametric components based on the smoothly clipped absolute deviation (SCAD) penalization, and one-step spare estimates for parametric components are also presented. The resulting estimates perform asymptotic normality as well as an oracle property. Then, two-stage backfitting estimators are also presented for the nonparametric components by using the local linear method, and the structures of asymptotic biases and covariances of the proposed estimators are the same as those in partially linear model with measurement error. The finite sample performance of the proposed procedures is illustrated by simulation studies.     

ERRORS IN VARIABLES IN ECONOMETRICS: NEW DEVELOPMENTS AND RECURRENT THEMES*

The paper reviews some old and new approaches to the analysis of linear models with errors in variables. The emphasis is on the identification problems that usually arise in errors–in–variables models and on the various types of additional information that econometricians have invoked to be able to estimate parameters consistently. The approaches discussed include instrumental variables, grouping, simultaneous equations, multiple equations and bounds on measurement error variances.     

A simple estimator for nonlinear error in variable models

We propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data, such as validation data or double measurements, are available. We assume that the marginal distributions of the measurement errors are Laplace (double exponential) with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other. Under these assumptions, we derive simple revised moment conditions in terms of the observed variables. They are used to make inference about the model parameters and the variance of the measurement error. The results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest. Our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute. Our estimation method is particularly useful in situations where no additional data are available, which is the case in many economic data sets. Simulation study demonstrates good finite sample properties of our proposed estimator. We also examine the performance of the estimator in the case where the error distribution is misspecified.     

Spline regression for hazard rate estimation when data are censored and measured with error

In this paper, we study an estimation problem where the variables of interest are subject to both right censoring and measurement error. In this context, we propose a nonparametric estimation strategy of the hazard rate, based on a regression contrast minimized in a finite‐dimensional functional space generated by splines bases. We prove a risk bound of the estimator in terms of integrated mean square error and discuss the rate of convergence when the dimension of the projection space is adequately chosen. Then we define a data‐driven criterion of model selection and prove that the resulting estimator performs an adequate compromise. The method is illustrated via simulation experiments that show that the strategy is successful.     

Maximum likelihood estimation of monotone and concave production frontiers

In this paper we bring together the previously separate parametric and nonparametric approaches to production frontier estimation by developing composed error models for maximum likelihood estimation from nonparametrically specified classes of frontiers. This approach avoids the untestable restrictions of parametric functional forms and also provides a statistical foundation for nonparametric frontier estimation. We first examine the single output setting and then extend our formulation to the multiple output setting. The key step in developing the estimation problems is to identify operational constraint sets to ensure estimation from the desired class of frontiers. We also suggest algorithms for solving the resulting constrained likelihood function optimization problems.The refereeing process of this paper was handled through R. Robert Russell. Helpful comments from Bob Russell and two anonymous referees are gratefully acknowedged. We are, of course, solely responsible for any remaining errors or omissions.     

Logarithmic calibration for multiplicative distortion measurement errors regression models

In this article, we propose a new identifiability condition by using the logarithmic calibration for the distortion measurement error models, where neither the response variable nor the covariates can be directly observed but are measured with multiplicative measurement errors. Under the logarithmic calibration, the direct-plug-in estimators of parameters and empirical likelihood based confidence intervals are proposed, and we studied the asymptotic properties of the proposed estimators. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Simulation studies demonstrate the performance of the proposed procedure and a real example is analyzed to illustrate its practical usage.     

Method of moments estimation and identifiability of semiparametric nonlinear errors-in-variables models

This paper deals with a nonlinear errors-in-variables model where the distributions of the unobserved predictor variables and of the measurement errors are nonparametric. Using the instrumental variable approach, we propose method of moments estimators for the unknown parameters and simulation-based estimators to overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals. Both estimators are consistent and asymptotically normally distributed under fairly general regularity conditions. Moreover, root-n consistent semiparametric estimators and a rank condition for model identifiability are derived using the combined methods of the nonparametric technique and Fourier deconvolution.     

Estimating house price appreciation: A comparison of methods

Several parametric and nonparametric methods have been advanced over the years for estimating house price appreciation. This paper compares five of these methods in terms of predictive accuracy, using data from Montgomery County, Pennsylvania. The methods are evaluated on the basis of the mean squared prediction error and the mean absolute prediction error. A statistic developed by Diebold and Mariano is used to determine whether differences in prediction errors are statistically significant. We use the same statistic to determine the effect of sample size on the accuracy of the predictions. In general, parametric methods of estimation produce more accurate estimates of house price appreciation than nonparametric methods. And when the mean absolute prediction error is used as the criterion of accuracy, the repeat sales method produces the most accurate estimate among the parametric methods we tested. Finally, of the five methods we tested, the accuracy of the repeat sales method is least diminished by a reduction in sample size.     

Nonparametric estimation and testing of fixed effects panel data models

In this paper we consider the problem of estimating nonparametric panel data models with fixed effects. We introduce an iterative nonparametric kernel estimator. We also extend the estimation method to the case of a semiparametric partially linear fixed effects model. To determine whether a parametric, semiparametric or nonparametric model is appropriate, we propose test statistics to test between the three alternatives in practice. We further propose a test statistic for testing the null hypothesis of random effects against fixed effects in a nonparametric panel data regression model. Simulations are used to examine the finite sample performance of the proposed estimators and the test statistics.     

Adjusting for measurement error in the value added model: evidence from Portugal

The multilevel value added approach to measuring school effectiveness is now widely used. We propose a method to adjust for measurement error to investigate the extent to which this changes school effect estimates. It is applied to longitudinal data collected in the region of Cova da Beira (NUT III) for 1st, 3rd, 5th, 7th and 8th grades. Three different variance component models are considered, depending on the predictor variables included. Assuming measurement error occurs in explanatory and/or response variables, corrections are made for different values of the coefficient of reliability. Moreover, models are fitted under the assumption of either independent or correlated measurement errors.     

Testing and detecting jumps based on a discretely observed process

We propose a new nonparametric test for detecting the presence of jumps in asset prices using discretely observed data. Compared with the test in Aït-Sahalia and Jacod (2009), our new test enjoys the same asymptotic properties but has smaller variance. These results are justified both theoretically and numerically. We also propose a new procedure to locate the jumps. The jump identification problem reduces to a multiple comparison problem. We employ the false discovery rate approach to control the probability of type I error. Numerical studies further demonstrate the power of our new method.     

Volatility clustering and herding agents: does it matter what they observe?

Recent agent-based models have demonstrated that agents’ herding behavior causes volatility clustering in stock markets. We examine economies where agents herd on others, yet they have limited sets of information on other agents to imitate. In particular, we conduct experiments on economies with agents with different levels of information sharing where agents can imitate: (1) the strategies of others but with an error, (2) the strategies of only a fraction of agents, or (3) the strategies of others, but update their parameters only by a proportion. In each experiment we change the likelihood that agents make errors to copy the strategy of others, the fraction of agents to herd, or the proportion of the parameter that agents update, in order to examine the effect of the different degrees of information sharing on volatility clustering. We show that volatility clustering tends to disappear when agents have limited information on the strategies of others, and agents need to imitate the strategy details of others in order to generate the clustered volatility.     

Instrumental variable estimation of nonseparable models

There are many environments where knowledge of a structural relationship is required to answer questions of interest. Also, nonseparability of a structural disturbance is a key feature of many models. Here, we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance, which disturbance is independent of instruments. This model leads to conditional quantile restrictions. We give local identification conditions for the structural equations from those quantile restrictions. We find that a modified completeness condition is sufficient for local identification. We also consider estimation via a nonparametric minimum distance estimator. The estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments. We show consistency of this estimator.     

Well-posedness of measurement error models for self-reported data

This paper considers the widely admitted ill-posed inverse problem for measurement error models: estimating the distribution of a latent variable $X^1$ from an observed sample of $X$, a contaminated measurement of $X^1$. We show that the inverse problem is well-posed for self-reporting data under the assumption that the probability of truthful reporting is nonzero, which is supported by empirical evidences. Comparing with ill-posedness, well-posedness generally can be translated into faster rates of convergence for the nonparametric estimators of the latent distribution. Therefore, our optimistic result on well-posedness is of importance in economic applications, and it suggests that researchers should not ignore the point mass at zero in the measurement error distribution when they model measurement errors with self-reported data. We also analyze the implications of our results on the estimation of classical measurement error models. Then by both a Monte Carlo study and an empirical application, we show that failing to account for the nonzero probability of truthful reporting can lead to significant bias on estimation of the latent distribution.     

Estimation with overidentifying inequality moment conditions

This paper derives limit distributions of empirical likelihood estimators for models in which inequality moment conditions provide overidentifying information. We show that the use of this information leads to a reduction of the asymptotic mean-squared estimation error and propose asymptotically uniformly valid tests and confidence sets for the parameters of interest. While inequality moment conditions arise in many important economic models, we use a dynamic macroeconomic model as a data generating process and illustrate our methods with instrumental variable estimators of monetary policy rules. The results obtained in this paper extend to conventional GMM estimators.     

A flexible parametric approach for estimating switching regime models and treatment effect parameters

In this paper, we propose a flexible, parametric class of switching regime models allowing for both skewed and fat-tailed outcome and selection errors. Specifically, we model the joint distribution of each outcome error and the selection error via a newly constructed class of multivariate distributions which we call generalized normal mean–variance mixture distributions. We extend Heckman’s two-step estimation procedure for the Gaussian switching regime model to the new class of models. When the distributions of the outcome errors are asymmetric, we show that an additional correction term accounting for skewness in the outcome error distribution (besides the analogue of the well known inverse mill’s ratio) needs to be included in the second step regression. We use the two-step estimators of parameters in the model to construct simple estimators of average treatment effects and establish their asymptotic properties. Simulation results confirm the importance of accounting for skewness in the outcome errors in estimating both model parameters and the average treatment effect and the treatment effect for the treated.     

Penalty and related estimation strategies in the spatial error model

Spatial autoregressive models are powerful tools in the analysis of data sets from diverse scientific areas of research such as econometrics, plant species richness, cancer mortality rates, image processing, analysis of the functional Magnetic Resonance Imaging (fMRI) data, and many more. An important class in the host of spatial autoregressive models is the class of spatial error models in which spatially lagged error terms are assumed. In this paper, we propose efficient shrinkage and penalty estimators for the regression coefficients of the spatial error model. We carry out asymptotic as well as simulation analyses to illustrate the gain in efficiency achieved by these new estimators. Furthermore, we apply the new methodology to housing prices data and provide a bootstrap approach to compute prediction errors of the new estimators.