Bootstrapping realized multivariate volatility measures

We propose a bootstrap method for statistics that are a function of multivariate high frequency returns such as realized regression, covariance and correlation coefficients. We show that the finite sample performance of the bootstrap is superior to the existing first-order asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to error heteroskedasticity, the Edgeworth expansion for the pairs bootstrap that we develop here shows that this method is not second-order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.     

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.     

Tikhonov regularization for nonparametric instrumental variable estimators

We study a Tikhonov Regularized (TiR) estimator of a functional parameter identified by conditional moment restrictions in a linear model with both exogenous and endogenous regressors. The nonparametric instrumental variable estimator is based on a minimum distance principle with penalization by the norms of the parameter and its derivatives. After showing its consistency in the Sobolev norm and uniform consistency under an embedding condition, we derive the expression of the asymptotic Mean Integrated Square Error and the rate of convergence. The optimal value of the regularization parameter is characterized in two examples. We illustrate our theoretical findings and the small sample properties with simulation results. Finally, we provide an empirical application to estimation of an Engel curve, and discuss a data driven selection procedure for the regularization parameter.     

On Bayesian analysis and computation for functions with monotonicity and curvature restrictions

Our goal is inference for shape-restricted functions. Our functional form consists of finite linear combinations of basis functions. Prior elicitation is difficult due to the irregular shape of the parameter space. We show how to elicit priors that are flexible, theoretically consistent, and proper. We demonstrate that uniform priors over coefficients imply priors over economically relevant quantities that are quite informative and give an example of a non-uniform prior that addresses this issue. We introduce simulation methods that meet challenges posed by the shape of the parameter space. We analyze data from a consumer demand experiment.     

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.     

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.     

Distribution-free tests of stochastic monotonicity

This article proposes a nonparametric test of monotonicity for conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves. Distinguishing features of our approach are that critical values are pivotal under the null in finite samples and that the test is invariant to any monotonic continuous transformation of the explanatory variable. The test statistic is the sup-norm of the difference between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate. The resulting test is able to detect local alternatives converging to the null at the parametric rate n^−1/2$n^{-1/2}$, with n$n$ the sample size. The finite sample performance of the test is examined by means of a Monte Carlo experiment and an application to testing intergenerational income mobility.     

A consistent nonparametric test of affiliation in auction models

We propose a new nonparametric test of affiliation, a strong form of positive dependence with independence as a special, knife-edge, case. The test is consistent against all departures from the null of affiliation, and its null distribution is standard normal. Like most nonparametric tests, a sample-size dependent input parameter is needed. We provide an informal procedure for choosing the input parameter and evaluate the test’s performance using a simulation study. Our test can be used to test the fundamental assumptions of the auctions literature. We implement our test empirically using the Outer Continental Shelf (OCS) auction data.     

Smooth minimum distance estimation and testing with conditional estimating equations: Uniform in bandwidth theory

To study the influence of a bandwidth parameter in inference with conditional moments, we propose a new class of estimators and establish an asymptotic representation of our estimator as a process indexed by a bandwidth, which can vary within a wide range including bandwidths independent of the sample size. We study its behavior under misspecification. We also propose an efficient version of our estimator. We develop a procedure based on a distance metric statistic for testing restrictions on parameters as well as a bootstrap technique to account for the bandwidth’s influence. Our new methods are simple to implement, apply to non-smooth problems, and perform well in our simulations.     

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.     

Jumps in equilibrium prices and market microstructure noise

Asset prices observed in financial markets combine equilibrium prices and market microstructure noise. In this paper, we study how to tell apart large shifts in equilibrium prices from noise using high frequency data. We propose a new nonparametric test which allows us to asymptotically remove the noise from observable price data and to discover jumps in fundamental asset values. We provide its asymptotic distribution to decide when such jumps occur. In finite samples, our test offers reasonable power for distinguishing between noise and jumps. Empirical evidence indicates that it is necessary to incorporate the presence of jumps in equilibrium prices.     

Testing functional inequalities

This paper develops tests for inequality constraints of nonparametric regression functions. The test statistics involve a one-sided version of _Lp$L_p$-type functionals of kernel estimators (1≤p<∞)$(1\le p<\infty )$. Drawing on the approach of Poissonization, this paper establishes that the tests are asymptotically distribution free, admitting asymptotic normal approximation. In particular, the tests using the standard normal critical values have asymptotically correct size and are consistent against general fixed alternatives. Furthermore, we establish conditions under which the tests have nontrivial local power against Pitman local alternatives. Some results from Monte Carlo simulations are presented.     

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

This paper proposes several tests of restricted specification in nonparametric instrumental regression. Based on series estimators, test statistics are established that allow for tests of the general model against a parametric or nonparametric specification as well as a test of exogeneity of the vector of regressors. The tests’ asymptotic distributions under correct specification are derived and their consistency against any alternative model is shown. Under a sequence of local alternative hypotheses, the asymptotic distributions of the tests are derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases. A Monte Carlo study examines finite sample performance of the test statistics.     

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.     

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.     

A versatile and robust metric entropy test of time-reversibility,and other hypotheses

We examine the performance of a metric entropy statistic as a robust test for time-reversibility (TR), symmetry, and serial dependence. It also serves as a measure of goodness-of-fit. The statistic provides a consistent and unified basis in model search, and is a powerful diagnostic measure with surprising ability to pinpoint areas of model failure. We provide empirical evidence comparing the performance of the proposed procedure with some of the modern competitors in nonlinear time-series analysis, such as robust implementations of the BDS and characteristic function-based tests of TR, along with correlation-based competitors such as the Ljung–Box Q-statistic. Unlike our procedure, each of its competitors is motivated for a different, specific, context and hypothesis. Our evidence is based on Monte Carlo simulations along with an application to several stock indices for the US equity market.