Pseudo-likelihood estimation and bootstrap inference for structural discrete Markov decision models

This paper analyzes the higher-order properties of the estimators based on the nested pseudo-likelihood (NPL) algorithm and the practical implementation of such estimators for parametric discrete Markov decision models. We derive the rate at which the NPL algorithm converges to the MLE and provide a theoretical explanation for the simulation results in Aguirregabiria and Mira [Aguirregabiria, V., Mira, P., 2002. Swapping the nested fixed point algorithm: A class of estimators for discrete Markov decision models. Econometrica 70, 1519–1543], in which iterating the NPL algorithm improves the accuracy of the estimator. We then propose a new NPL algorithm that can achieve quadratic convergence without fully solving the fixed point problem in every iteration and apply our estimation procedure to a finite mixture model. We also develop one-step NPL bootstrap procedures for discrete Markov decision models. The Monte Carlo simulation evidence based on a machine replacement model of Rust [Rust, J., 1987. Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica 55, 999–1033] shows that the proposed one-step bootstrap test statistics and confidence intervals improve upon the first order asymptotics even with a relatively small number of iterations.     

A two-stage procedure for partially identified models

This paper studies a two-stage procedure for estimating partially identified models, based on Chernozhukov, Hong, and Tamer’s (2007) theory of set estimation and inference. We consider the case where a sub-vector of parameters or their identified set can be estimated separately from the rest, possibly subject to a priori restrictions. Our procedure constructs the second-stage set estimator and confidence set by taking appropriate level sets of a criterion function, using a first-stage estimator to impose restrictions on the parameter of interest. We give conditions under which the two-stage set estimator is a set-valued random element that is measurable in an appropriate sense. We also establish the consistency of the two-stage set estimator.     

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.     

Nonparametric estimation and inference on conditional quantile processes

This paper presents estimation methods and asymptotic theory for the analysis of a nonparametrically specified conditional quantile process. Two estimators based on local linear regressions are proposed. The first estimator applies simple inequality constraints while the second uses rearrangement to maintain quantile monotonicity. The bandwidth parameter is allowed to vary across quantiles to adapt to data sparsity. For inference, the paper first establishes a uniform Bahadur representation and then shows that the two estimators converge weakly to the same limiting Gaussian process. As an empirical illustration, the paper considers a dataset from Project STAR and delivers two new findings.     

Empirical likelihood-based inference for nonparametric recurrent diffusions

This paper provides a new approach to constructing confidence intervals for nonparametric drift and diffusion functions in the continuous-time diffusion model via empirical likelihood (EL). The log EL ratios are constructed through the estimating equations satisfied by the local linear estimators. Limit theories are developed by means of increasing time span and shrinking observational intervals. The results apply to both stationary and nonstationary recurrent diffusion processes. Simulations show that for both drift and diffusion functions, the new procedure performs remarkably well in finite samples and clearly dominates the conventional method in constructing confidence intervals based on asymptotic normality. An empirical example is provided to illustrate the usefulness of the proposed method.     

Nonparametric state price density estimation using constrained least squares and the bootstrap

The economic theory of option pricing imposes constraints on the structure of call functions and state price densities. Except in a few polar cases, it does not prescribe functional forms. This paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions (such as monotonicity and convexity) within a single least squares procedure. The bootstrap is used to produce confidence intervals for the call function and its first two derivatives and to calibrate a residual regression test of shape constraints. We apply the techniques to option pricing data on the DAX.     

Statistical inference on regression with spatial dependence

Central limit theorems are developed for instrumental variables estimates of linear and semiparametric partly linear regression models for spatial data. General forms of spatial dependence and heterogeneity in explanatory variables and unobservable disturbances are permitted. We discuss estimation of the variance matrix, including estimates that are robust to disturbance heteroscedasticity and/or dependence. A Monte Carlo study of finite-sample performance is included. In an empirical example, the estimates and robust and non-robust standard errors are computed from Indian regional data, following tests for spatial correlation in disturbances, and nonparametric regression fitting. Some final comments discuss modifications and extensions.     

Exogeneity in structural equation models

The practical relevance of several concepts of exogeneity of treatments for the estimation of causal parameters based on observational data are discussed. We show that the traditional concepts, such as strong ignorability and weak and super-exogeneity, are too restrictive if interest lies in average effects (i.e. not on distributional effects of the treatment). We suggest a new definition of exogeneity, KL-exogeneity. It does not rely on distributional assumptions and is not based on counterfactual random variables. As a consequence it can be empirically tested using a proposed test that is simple to implement and is distribution-free.     

Regression discontinuity inference with specification error

A regression discontinuity (RD) research design is appropriate for program evaluation problems in which treatment status (or the probability of treatment) depends on whether an observed covariate exceeds a fixed threshold. In many applications the treatment-determining covariate is discrete. This makes it impossible to compare outcomes for observations “just above” and “just below” the treatment threshold, and requires the researcher to choose a functional form for the relationship between the treatment variable and the outcomes of interest. We propose a simple econometric procedure to account for uncertainty in the choice of functional form for RD designs with discrete support. In particular, we model deviations of the true regression function from a given approximating function—the specification errors—as random. Conventional standard errors ignore the group structure induced by specification errors and tend to overstate the precision of the estimated program impacts. The proposed inference procedure that allows for specification error also has a natural interpretation within a Bayesian framework.     

Efficient information theoretic inference for conditional moment restrictions

The generalised method of moments estimator may be substantially biased in finite samples, especially so when there are large numbers of unconditional moment conditions. This paper develops a class of first-order equivalent semi-parametric efficient estimators and tests for conditional moment restrictions models based on a local or kernel-weighted version of the Cressie–Read power divergence family of discrepancies. This approach is similar in spirit to the empirical likelihood methods of Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restrictions models. Econometrica 72, 1667–1714] and Tripathi and Kitamura [2003. Testing conditional moment restrictions. Annals of Statistics 31, 2059–2095]. These efficient local methods avoid the necessity of explicit estimation of the conditional Jacobian and variance matrices of the conditional moment restrictions and provide empirical conditional probabilities for the observations.     

The zero-information-limit condition and spurious inference in weakly identified models

In many econometric models the asymptotic variance of a parameter estimate depends on the value of another structural parameter in such a way that the data contain little information about the former when the latter is close to a critical value. This paper introduces the zero-information-limit-condition (ZILC) to identify such models where ‘weak identification’ leads to spurious inference. We find that standard errors tend to be underestimated in these cases, but the size of the asymptotic t-test may either be too great (the intuitive case emphasized in the ‘weak instrument’ literature) or too small as in two cases illustrated here.     

A martingale approach for testing diffusion models based on infinitesimal operator

I develop an omnibus specification test for diffusion models based on the infinitesimal operator. The infinitesimal operator based identification of the diffusion process is equivalent to a “martingale hypothesis” for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the “martingale hypothesis” via a multivariate generalized spectral derivative based approach that delivers a N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process is a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performance. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift misspecification plays an important role in such rejections.     

Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities

This paper proposes a computationally simple way to construct confidence sets for a parameter of interest in models comprised of moment inequalities. Building on results from the literature on multivariate one-sided tests, I show how to test the hypothesis that any particular parameter value is logically consistent with the maintained moment inequalities. The associated test statistic has an asymptotic chi-bar-square distribution, and can be inverted to construct an asymptotic confidence set for the parameter of interest, even if that parameter is only partially identified. Critical values for the test are easily computed, and a Monte Carlo study demonstrates implementation and finite sample performance.     

Distribution-free tests for time series models specification

We consider a class of time series specification tests based on quadratic forms of weighted sums of residuals autocorrelations. Asymptotically distribution-free tests in the presence of estimated parameters are obtained by suitably transforming the weights, which can be optimally chosen to maximize the power function when testing in the direction of local alternatives. We discuss in detail an asymptotically optimal distribution-free alternative to the popular Box–Pierce when testing in the direction of AR or MA alternatives. The performance of the test with small samples is studied by means of a Monte Carlo experiment.     

Heteroskedasticity and spatiotemporal dependence robust inference for linear panel models with fixed effects

This paper studies robust inference for linear panel models with fixed effects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator that nests existing estimators as special cases. Our estimator improves upon existing estimators in terms of robustness, efficiency, and adaptiveness. For distributional approximations, we considered two types of asymptotics: the increasing-smoothing asymptotics and the fixed-smoothing asymptotics. Under the former asymptotics, the Wald statistic based on our covariance estimator converges to a chi-square distribution. Under the latter asymptotics, the Wald statistic is asymptotically equivalent to a distribution that can be well approximated by an F distribution. Simulation results show that our proposed testing procedure works well in finite samples.     

Joint and marginal specification tests for conditional mean and variance models

This article proposes a class of joint and marginal spectral diagnostic tests for parametric conditional means and variances of linear and nonlinear time series models. The use of joint and marginal tests is motivated from the fact that marginal tests for the conditional variance may lead to misleading conclusions when the conditional mean is misspecified. The new tests are based on a generalized spectral approach and do not need to choose a lag order depending on the sample size or to smooth the data. Moreover, the proposed tests are robust to higher order dependence of unknown form, in particular to conditional skewness and kurtosis. It turns out that the asymptotic null distributions of the new tests depend on the data generating process. Hence, we implement the tests with the assistance of a wild bootstrap procedure. A simulation study compares the finite sample performance of the proposed and competing tests, and shows that our tests can play a valuable role in time series modeling. Finally, an application to the S&P 500 highlights the merits of our approach.     

Nonparametric tests for conditional symmetry in dynamic models

This article proposes omnibus tests for conditional symmetry around a parametric function in a dynamic context. Conditional moments may not exist or may depend on the explanatory variables. Test statistics are suitable functionals of the empirical process of residuals and explanatory variables, whose limiting distribution under the null is nonpivotal. The tests are implemented with the assistance of a bootstrap method, which is justified assuming very mild regularity conditions on the specification of the center of symmetry and the underlying serial dependence structure. Finite sample properties are examined by means of a Monte Carlo experiment.     

Estimation and inference in unstable nonlinear least squares models

There is compelling evidence that many macroeconomic and financial variables are not generated by linear models. This evidence is based on testing linearity against either smooth nonlinearity or piece-wise linearity, but there is no framework that encompasses both. This paper provides an econometric framework that allows for both breaks and smooth nonlinearity in between breaks. We estimate the unknown break-dates simultaneously with other parameters via nonlinear least-squares. Using new central limit results for nonlinear processes, we provide inference methods on break-dates and parameter estimates and several instability tests. We illustrate our methods via simulated and empirical smooth transition models with breaks.