Estimation of a nonlinear panel data model with semiparametric individual effects

This paper investigates identification and estimation of a class of nonlinear panel data, single-index models. The model allows for unknown time-specific link functions, and semiparametric specification of the individual-specific effects. We develop an estimator for the parameters of interest, and propose a powerful new kernel-based modified backfitting algorithm to compute the estimator. We derive uniform rates of convergence results for the estimators of the link functions, and show the estimators of the finite-dimensional parameters are root-N$N$ consistent with a Gaussian limiting distribution. We study the small sample properties of the estimator via Monte Carlo techniques.     

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.     

Nonparametric dynamic panel data models: Kernel estimation and specification testing

Motivated by the first-differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogenous regressors. The estimators utilize the additive structure of the first-differenced model—the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the _L2$L_2$ distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well for finite samples. We apply our new method to study the relationships among economic growth, the initial economic condition and capital accumulation, and find a significant nonlinear relation between economic growth and the initial economic condition.     

Tighter bounds in triangular systems

We study a nonparametric triangular system with (potentially discrete) endogenous regressors and nonseparable errors. Like in other work in this area, the parameter of interest is the structural function evaluated at particular values. We impose a global exclusion and exogeneity condition, in contrast to Chesher (2005), but develop a rank condition which is weaker than Chesher’s. The alternative rank condition can be satisfied for binary endogenous regressors, and it often leads to an identified interval tighter than Chesher (2005)’s minimum length interval. We illustrate the potential of the new rank condition using the Angrist and Krueger (1991) data.     

Determining the MSE-optimal cross section to forecast

In this paper, we address the question of which subset of time series should be selected among a given set in order to forecast another series. We evaluate the quality of the forecasts in terms of Mean Squared Error. We propose a family of criteria to estimate the optimal subset. Consistency results are proved, both in the weak (in probability) and strong (almost sure) sense. We present the results of a Monte Carlo experiment and a real data example in which the criteria are compared to some hypothesis tests such as the ones by Diebold and Mariano (1995),  and  and Giacomini and White (2006).     

Semiparametric estimation of Markov decision processes with continuous state space

We propose a general two-step estimator for a popular Markov discrete choice model that includes a class of Markovian games with continuous observable state space. Our estimation procedure generalizes the computationally attractive methodology of Pesendorfer and Schmidt-Dengler (2008) that assumed finite observable states. This extension is non-trivial as the policy value functions are solutions to some type II integral equations. We show that the inverse problem is well-posed. We provide a set of primitive conditions to ensure root-T consistent estimation for the finite dimensional structural parameters and the distribution theory for the value functions in a time series framework.     

Estimation and inference in two-stage,semi-parametric models of production processes

Many papers have regressed non-parametric estimates of productive efficiency on environmental variables in two-stage procedures to account for exogenous factors that might affect firms’ performance. None of these have described a coherent data-generating process (DGP). Moreover, conventional approaches to inference employed in these papers are invalid due to complicated, unknown serial correlation among the estimated efficiencies. We first describe a sensible DGP for such models. We propose single and double bootstrap procedures; both permit valid inference, and the double bootstrap procedure improves statistical efficiency in the second-stage regression. We examine the statistical performance of our estimators using Monte Carlo experiments.     

Local multiplicative bias correction for asymmetric kernel density estimators

We consider semiparametric asymmetric kernel density estimators when the unknown density has support on [0,∞)$[0,\infty )$. We provide a unifying framework which relies on a local multiplicative bias correction, and contains asymmetric kernel versions of several semiparametric density estimators considered previously in the literature. This framework allows us to use popular parametric models in a nonparametric fashion and yields estimators which are robust to misspecification. We further develop a specification test to determine if a density belongs to a particular parametric family. The proposed estimators outperform rival non- and semiparametric estimators in finite samples and are easy to implement. We provide applications to loss data from a large Swiss health insurer and Brazilian income data.     

Estimation of partial differential equations with applications in finance

Linear parabolic partial differential equations (PDE’s) and diffusion models are closely linked through the celebrated Feynman–Kac representation of solutions to PDE’s. In asset pricing theory, this leads to the representation of derivative prices as solutions to PDE’s. Very often implied derivative prices are calculated given preliminary estimates of the diffusion model for the underlying variable. We demonstrate that the implied derivative prices are consistent and derive their asymptotic distribution under general conditions. We apply this result to three leading cases of preliminary estimators: Nonparametric, semiparametric and fully parametric ones. In all three cases, the asymptotic distribution of the solution is derived. We demonstrate the use of these results in obtaining confidence bands and standard errors for implied prices of bonds, options and other derivatives. Our general results also are of interest for the estimation of diffusion models using either historical data of the underlying process or option prices; these issues are also discussed.     

On the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.     

Bayesian estimation approaches to first-price auctions

This paper considers Bayesian estimation strategies for first-price auctions within the independent private value paradigm. We develop an ‘optimization’ error approach that allows for estimation of values assuming that observed bids differ from optimal bids. We further augment this approach by allowing systematic over or underbidding by bidders using ideas from the stochastic frontier literature. We perform a simulation study to showcase the appeal of the method and apply the techniques to timber auction data collected in British Columbia. Our results suggest that significant underbidding is present in the timber auctions.     

Generalizing weak instrument robust IV statistics towards multiple parameters,unrestricted covariance matrices and identification statistics

We generalize the weak instrument robust score or Lagrange multiplier and likelihood ratio instrumental variables (IV) statistics towards multiple parameters and a general covariance matrix so they can be used in the generalized method of moments (GMM). The GMM extension of Moreira's [2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio statistic towards GMM preserves its expression except that it becomes conditional on a statistic that tests the rank of a matrix. We analyze the spurious power decline of Kleibergen's [2002. Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1803, 2005. Testing parameters in GMM without assuming that they are identified. Econometrica 73, 1103–1124] score statistic and show that an independent misspecification pre-test overcomes it. We construct identification statistics that reflect if the confidence sets of the parameters are bounded. A power study and the possible shapes of confidence sets illustrate the analysis.     

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi–Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.     

Panel data models with spatially correlated error components

In this paper we consider a panel data model with error components that are both spatially and time-wise correlated. The model blends specifications typically considered in the spatial literature with those considered in the error components literature. We introduce generalizations of the generalized moments estimators suggested in Kelejian and Prucha (1999. A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509–533) for estimating the spatial autoregressive parameter and the variance components of the disturbance process. We then use those estimators to define a feasible generalized least squares procedure for the regression parameters. We give formal large sample results for the proposed estimators. We emphasize that our estimators remain computationally feasible even in large samples.     

Principal components estimation and identification of static factors

It is known that the principal component estimates of the factors and the loadings are rotations of the underlying latent factors and loadings. We study conditions under which the latent factors can be estimated asymptotically without rotation. We derive the limiting distributions for the estimated factors and factor loadings when N$N$ and T$T$ are large and make precise how identification of the factors affects inference based on factor augmented regressions. We also consider factor models with additive individual and time effects. The asymptotic analysis can be modified to analyze identification schemes not considered in this analysis.     

Testing for non-nested conditional moment restrictions using unconditional empirical likelihood

We propose non-nested hypothesis tests for conditional moment restriction models based on the method of generalized empirical likelihood (GEL). By utilizing the implied GEL probabilities from a sequence of unconditional moment restrictions that contains equivalent information of the conditional moment restrictions, we construct Kolmogorov–Smirnov and Cramér–von Mises type moment encompassing tests. Advantages of our tests over Otsu and Whang’s (2011) tests are: (i) they are free from smoothing parameters, (ii) they can be applied to weakly dependent data, and (iii) they allow non-smooth moment functions. We derive the null distributions, validity of a bootstrap procedure, and local and global power properties of our tests. The simulation results show that our tests have reasonable size and power performance in finite samples.     

The construction of empirical credit scoring rules based on maximization principles

We examine the econometric implications of the decision problem faced by a profit/utility-maximizing lender operating in a simple “double-binary” environment, where the two actions available are “approve” or “reject”, and the two states of the world are “pay back” or “default”. In practice, such decisions are often made by applying a fixed cutoff to the maximum likelihood estimate of a parametric model of the default probability. Following (Elliott and Lieli, 2007), we argue that this practice might contradict the lender’s economic objective and, using German loan data, we illustrate the use of “context-specific” cutoffs and an estimation method derived directly from the lender’s problem. We also provide a brief discussion of how to incorporate legal constraints, such as the prohibition of disparate treatment of potential borrowers, into the lender’s problem.     

Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers

We consider classes of multivariate distributions which can model skewness and are closed under orthogonal transformations. We review two classes of such distributions proposed in the literature and focus our attention on a particular, yet quite flexible, subclass of one of these classes. Members of this subclass are defined by affine transformations of univariate (skewed) distributions that ensure the existence of a set of coordinate axes along which there is independence and the marginals are known analytically. The choice of an appropriate m-dimensional skewed distribution is then restricted to the simpler problem of choosing m univariate skewed distributions. We introduce a Bayesian model comparison setup for selection of these univariate skewed distributions. The analysis does not rely on the existence of moments (allowing for any tail behaviour) and uses equivalent priors on the common characteristics of the different models. Finally, we apply this framework to multi-output stochastic frontiers using data from Dutch dairy farms.     

The quantilogram: With an application to evaluating directional predictability

We propose a new diagnostic tool for time series called the quantilogram. The tool can be used formally and we provide the inference tools to do this under general conditions, and it can also be used as a simple graphical device. We apply our method to measure directional predictability and to test the hypothesis that a given time series has no directional predictability. The test is based on comparing the correlogram of quantile hits to a pointwise confidence interval or on comparing the cumulated squared autocorrelations with the corresponding critical value. We provide the distribution theory needed to conduct inference, propose some model free upper bound critical values, and apply our methods to S&P500 stock index return data. The empirical results suggest some directional predictability in returns. The evidence is strongest in mid range quantiles like 5–10% and for daily data. The evidence for predictability at the median is of comparable strength to the evidence around the mean, and is strongest at the daily frequency.     

A Bayesian mixed logit–probit model for multinomial choice

In this paper, we introduce a new flexible mixed model for multinomial discrete choice where the key individual- and alternative-specific parameters of interest are allowed to follow an assumption-free nonparametric density specification, while other alternative-specific coefficients are assumed to be drawn from a multivariate Normal distribution, which eliminates the independence of irrelevant alternatives assumption at the individual level. A hierarchical specification of our model allows us to break down a complex data structure into a set of submodels with the desired features that are naturally assembled in the original system. We estimate the model, using a Bayesian Markov Chain Monte Carlo technique with a multivariate Dirichlet Process (DP) prior on the coefficients with nonparametrically estimated density. We employ a “latent class” sampling algorithm, which is applicable to a general class of models, including non-conjugate DP base priors. The model is applied to supermarket choices of a panel of Houston households whose shopping behavior was observed over a 24-month period in years 2004–2005. We estimate the nonparametric density of two key variables of interest: the price of a basket of goods based on scanner data, and driving distance to the supermarket based on their respective locations. Our semi-parametric approach allows us to identify a complex multi-modal preference distribution, which distinguishes between inframarginal consumers and consumers who strongly value either lower prices or shopping convenience.