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.     

Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances

This study develops a methodology of inference for a widely used Cliff–Ord type spatial model containing spatial lags in the dependent variable, exogenous variables, and the disturbance terms, while allowing for unknown heteroskedasticity in the innovations. We first generalize the GMM estimator suggested in  and  for the spatial autoregressive parameter in the disturbance process. We also define IV estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GMM estimator. Much of the theory is kept general to cover a wide range of settings.     

Estimating high-dimensional demand systems in the presence of many binding non-negativity constraints

Two econometric issues arise in the structural estimation of consumer or producer demand systems in the presence of many binding non-negativity constraints. Firstly, most existing methods entail the evaluation of multivariate probability integrals. Secondly, the issue of statistical coherency must be addressed. We circumvent both of these issues using Gibbs’ Sampling, along with data augmentation and rejection sampling. We illustrate our method using several simulated data sets.     

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.     

Inference in regression models with many regressors

We investigate the behavior of various standard and modified F$F$, likelihood ratio (LR$LR$), and Lagrange multiplier (LM$LM$) tests in linear homoskedastic regressions, adapting an alternative asymptotic framework in which the number of regressors and possibly restrictions grows proportionately to the sample size. When the restrictions are not numerous, the rescaled classical test statistics are asymptotically chi-squared, irrespective of whether there are many or few regressors. However, when the restrictions are numerous, standard asymptotic versions of classical tests are invalid. We propose and analyze asymptotically valid versions of the classical tests, including those that are robust to the numerosity of regressors and restrictions. The local power of all asymptotically valid tests under consideration turns out to be equal. The “exact” F$F$ test that appeals to critical values of the F$F$ distribution is also asymptotically valid and robust to the numerosity of regressors and restrictions.     

Instrumental variable estimation based on grouped data

The paper considers the estimation of the coefficients of a single equation in the presence of dummy intruments. We derive pseudo ML and GMM estimators based on moment restrictions induced either by the structural form or by the reduced form of the model. The performance of the estimators is evaluated for the non-Gaussian case. We allow for heteroscedasticity. The asymptotic distributions are based on parameter sequences where the number of instruments increases at the same rate as the sample size. Relaxing the usual Gaussian assumption is shown to affect the normal asymptotic distributions. As a result also recently suggested new specification tests for the validity of instruments depend on Gaussianity. Monte Carlo simulations confirm the accuracy of the asymptotic approach.     

A central limit theorem for endogenous locations and complex spatial interactions

We provide a new central limit theorem (CLT) for spatial processes under weak conditions that are plausible for many economic applications in which location is endogenous. In particular, our CLT is designed for problems that have some, but not necessarily all, of the following features: (i) Agents choose the locations of observations to maximize profits, welfare, or some other objective. (ii) The objects that are chosen (e.g., stores or brands) interact with one another. For example, they can be substitutes or complements. (iii) Interaction can be complex. In particular, interaction between i and j need not depend only on the distance between the locations of i and j, but can also depend on distance to or location of other observations k, or possibly on the number of other such observations.     

Nonparametric estimation of distributional policy effects

This paper proposes a fully nonparametric procedure to evaluate the effect of a counterfactual change in the distribution of some covariates on the unconditional distribution of an outcome variable of interest. In contrast to other methods, we do not restrict attention to the effect on the mean. In particular, our method can be used to conduct inference on the change of the distribution function as a whole, its moments and quantiles, inequality measures such as the Lorenz curve or Gini coefficient, and to test for stochastic dominance. The practical applicability of our procedure is illustrated via a simulation study and an empirical example.     

Estimation of technical and allocative inefficiency: A primal system approach

The estimation of technical and allocative inefficiencies using a flexible (translog) cost system is found to be quite difficult, especially when both the inefficiencies are random. In this paper we use the alternative primal system consisting of the production function (translog) and the first-order conditions of cost minimization. The estimation of the primal system is more straightforward and it enables us to estimate observation-specific technical and allocative inefficiencies, and their impact on input demand and cost. We use data on steam-electric generating plants from the U.S. to estimate the model using both Cobb–Douglas and translog production functions.     

Selection of estimation window in the presence of breaks

In situations where a regression model is subject to one or more breaks it is shown that it can be optimal to use pre-break data to estimate the parameters of the model used to compute out-of-sample forecasts. The issue of how best to exploit the trade-off that might exist between bias and forecast error variance is explored and illustrated for the multivariate regression model under the assumption of strictly exogenous regressors. In practice when this assumption cannot be maintained and both the time and size of the breaks are unknown, the optimal choice of the observation window will be subject to further uncertainties that make exploiting the bias–variance trade-off difficult. To that end we propose a new set of cross-validation methods for selection of a single estimation window and weighting or pooling methods for combination of forecasts based on estimation windows of different lengths. Monte Carlo simulations are used to show when these procedures work well compared with methods that ignore the presence of breaks.     

Efficient estimation of the semiparametric spatial autoregressive model

Efficient semiparametric and parametric estimates are developed for a spatial autoregressive model, containing non-stochastic explanatory variables and innovations suspected to be non-normal. The main stress is on the case of distribution of unknown, nonparametric, form, where series nonparametric estimates of the score function are employed in adaptive estimates of parameters of interest. These estimates are as efficient as the ones based on a correct form, in particular they are more efficient than pseudo-Gaussian maximum likelihood estimates at non-Gaussian distributions. Two different adaptive estimates are considered, relying on somewhat different regularity conditions. A Monte Carlo study of finite sample performance is included.     

Central limit theorems and uniform laws of large numbers for arrays of random fields

Over the last decades, spatial-interaction models have been increasingly used in economics. However, the development of a sufficiently general asymptotic theory for nonlinear spatial models has been hampered by a lack of relevant central limit theorems (CLTs), uniform laws of large numbers (ULLNs) and pointwise laws of large numbers (LLNs). These limit theorems form the essential building blocks towards developing the asymptotic theory of M-estimators, including maximum likelihood and generalized method of moments estimators. The paper establishes a CLT, ULLN, and LLN for spatial processes or random fields that should be applicable to a broad range of data processes.     

HAC estimation in a spatial framework

We suggest a non-parametric heteroscedasticity and autocorrelation consistent (HAC) estimator of the variance–covariance (VC) matrix for a vector of sample moments within a spatial context. We demonstrate consistency under a set of assumptions that should be satisfied by a wide class of spatial models. We allow for more than one measure of distance, each of which may be measured with error. Monte Carlo results suggest that our estimator is reasonable in finite samples. We then consider a spatial model containing various complexities and demonstrate that our HAC estimator can be applied in the context of that model.     

Specification testing in nonparametric instrumental variable estimation

In nonparametric instrumental variable estimation, the function being estimated is the solution to an integral equation. A solution may not exist if, for example, the instrument is not valid. This paper discusses the problem of testing the null hypothesis that a solution exists against the alternative that there is no solution. We give necessary and sufficient conditions for existence of a solution and show that uniformly consistent testing of an unrestricted null hypothesis is not possible. Uniformly consistent testing is possible, however, if the null hypothesis is restricted by assuming that any solution to the integral equation is smooth. Many functions of interest in applied econometrics, including demand functions and Engel curves, are expected to be smooth. The paper presents a statistic for testing the null hypothesis that a smooth solution exists. The test is consistent uniformly over a large class of probability distributions of the observable random variables for which the integral equation has no smooth solution. The finite-sample performance of the test is illustrated through Monte Carlo experiments.     

Likelihood estimation and inference in threshold regression

This paper studies likelihood-based estimation and inference in parametric discontinuous threshold regression models with i.i.d. data. The setup allows heteroskedasticity and threshold effects in both mean and variance. By interpreting the threshold point as a “middle” boundary of the threshold variable, we find that the Bayes estimator is asymptotically efficient among all estimators in the locally asymptotically minimax sense. In particular, the Bayes estimator of the threshold point is asymptotically strictly more efficient than the left-endpoint maximum likelihood estimator and the newly proposed middle-point maximum likelihood estimator. Algorithms are developed to calculate asymptotic distributions and risk for the estimators of the threshold point. The posterior interval is proved to be an asymptotically valid confidence interval and is attractive in both length and coverage in finite samples.     

Partial rank estimation of duration models with general forms of censoring

In this paper we propose estimators for the regression coefficients in censored duration models which are distribution free, impose no parametric specification on the baseline hazard function, and can accommodate general forms of censoring. The estimators are shown to have desirable asymptotic properties and Monte Carlo simulations demonstrate good finite sample performance. Among the data features the new estimators can accommodate are covariate-dependent censoring, double censoring, and fixed (individual or group specific) effects. We also examine the behavior of the estimator in an empirical illustration.     

GMM estimation of social interaction models with centrality

This paper considers the specification and estimation of social interaction models with network structures and the presence of endogenous, contextual, correlated, and group fixed effects. When the network structure in a group is captured by a graph in which the degrees of nodes are not all equal, the different positions of group members as measured by the Bonacich (1987) centrality provide additional information for identification and estimation. In this case, the Bonacich centrality measure for each group can be used as an instrument for the endogenous social effect, but the number of such instruments grows with the number of groups. We consider the 2SLS and GMM estimation for the model. The proposed estimators are asymptotically efficient, respectively, within the class of IV estimators and the class of GMM estimators based on linear and quadratic moments, when the sample size grows fast enough relative to the number of instruments.