Hybrid estimators for small diffusion processes based on reduced data

We deal with the Bayes type estimators and the maximum likelihood type estimators of both drift and volatility parameters for small diffusion processes defined by stochastic differential equations with small perturbations from high frequency data. From the viewpoint of numerical analysis, initial Bayes type estimators for both drift and volatility parameters based on reduced data are required, and adaptive maximum likelihood type estimators with the initial Bayes type estimators, which are called hybrid estimators, are proposed. The asymptotic properties of the initial Bayes type estimators based on reduced data are derived and it is shown that the hybrid estimators have asymptotic normality and convergence of moments. Furthermore, a concrete example and simulation results are given.     

Estimation of semiparametric locally stationary diffusion models

This paper proposes a class of locally stationary diffusion processes. The model has a time varying but locally linear drift and a volatility coefficient that is allowed to vary over time and space. The model is semiparametric because we allow these functions to be unknown and the innovation process is parametrically specified, indeed completely known. We propose estimators of all the unknown quantities based on long span data. Our estimation method makes use of the property of local stationarity. We establish asymptotic theory for the proposed estimators as the time span increases, so we do not rely on infill asymptotics. We apply this method to interest rate data to illustrate the validity of our model. Finally, we present a simulation study to provide the finite-sample performance of the proposed estimators.     

Parameter estimation and bias correction for diffusion processes

This paper considers parameter estimation for continuous-time diffusion processes which are commonly used to model dynamics of financial securities including interest rates. To understand why the drift parameters are more difficult to estimate than the diffusion parameter, as observed in previous studies, we first develop expansions for the bias and variance of parameter estimators for two of the most employed interest rate processes, Vasicek and CIR processes. Then, we study the first order approximate maximum likelihood estimator for linear drift processes. A parametric bootstrap procedure is proposed to correct bias for general diffusion processes with a theoretical justification. Simulation studies confirm the theoretical findings and show that the bootstrap proposal can effectively reduce both the bias and the mean square error of parameter estimates, for both univariate and multivariate processes. The advantages of using more accurate parameter estimators when calculating various option prices in finance are demonstrated by an empirical study.     

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.     

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.     

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.     

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.     

Nearly-singular design in GMM and generalized empirical likelihood estimators

Nearly-Singular design relaxes the nonsingularity assumption of the limit weight matrix in GMM, and the nonsingularity of the limit variance matrix for the first order conditions in GEL. The sample versions of these matrices are nonsingular, but in large samples we assume these sample matrices converge to a singular matrix. This can result in size distortions for the overidentifying restrictions test and large bias for the estimators. This nearly-singular design may occur because of the similar instruments in these matrices. We derive the large sample theory for GMM and GEL estimators under nearly-singular design. The rate of convergence of the estimators is slower than root n$n$.     

Alternative approximations of the bias and MSE of the IV estimator under weak identification with an application to bias correction

We provide analytical formulae for the asymptotic bias (ABIAS) and mean-squared error (AMSE) of the IV estimator, and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage F-statistic to converge to a finite (possibly small) positive limit as the number of instruments approaches infinity. Our analytical formulae can be viewed as generalizing the bias and MSE results of [Richardson and Wu 1971. A note on the comparison of ordinary and two-stage least squares estimators. Econometrica 39, 973–982] to the case with nonnormal errors and stochastic instruments. Our approximations are shown to compare favorably with approximations due to [Morimune 1983. Approximate distributions of k$k$-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51, 821–841] and [Donald and Newey 2001. Choosing the number of instruments. Econometrica 69, 1161–1191], particularly when the instruments are weak. We also construct consistent estimators for the ABIAS and AMSE, and we use these to further construct a number of bias corrected OLS and IV estimators, the properties of which are examined both analytically and via a series of Monte Carlo experiments.     

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.     

Measuring correlations of integrated but not cointegrated variables: A semiparametric approach

Many macroeconomic and financial variables are integrated of order one (or I(1)) processes and are correlated with each other but not necessarily cointegrated. In this paper, we propose to use a semiparametric varying coefficient approach to model/capture such correlations. We propose two consistent estimators to study the dependence relationship among some integrated but not cointegrated time series variables. Simulations are used to examine the finite sample performances of the proposed estimators.     

Testing and imposing Slutsky symmetry in nonparametric demand systems

Maximization of utility implies that consumer demand systems have a Slutsky matrix which is everywhere symmetric. However, previous non- and semi-parametric approaches to the estimation of consumer demand systems do not give estimators that are restricted to satisfy this condition, nor do they offer powerful tests of this restriction. We use nonparametric modeling to test and impose Slutsky symmetry in a system of expenditure share equations over prices and expenditure. In this context, Slutsky symmetry is a set of nonlinear cross-equation restrictions on levels and derivatives of consumer demand equations. The key insight is that due to the differing convergence rates of levels and derivatives and due to the fact that the symmetry restrictions are linear in derivatives, both the test and the symmetry restricted estimator behave asymptotically as if these restrictions were (locally) linear. We establish large and finite sample properties of our methods, and show that our test has advantages over the only other comparable test. All methods we propose are implemented with Canadian micro-data. We find that our nonparametric analysis yields statistically significantly and qualitatively different results from traditional parametric estimators and tests.     

Semi-nonparametric estimation and misspecification testing of diffusion models

Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [Kristensen, D., 2010. Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. Journal of Econometrics 156, 239-259] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.     

Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models

Two classes of semiparametric diffusion models are considered, where either the drift or the diffusion term is parameterized, while the other term is left unspecified. We propose a pseudo-maximum likelihood estimator (PMLE) of the parametric component that maximizes the likelihood with a preliminary estimator of the unspecified term plugged in. It is demonstrated how models and estimators can be used in a two-step specification testing strategy of semiparametric and fully parametric models, and shown that approximate/simulated versions of the PMLE inherit the properties of the actual but infeasible estimator. A simulation study investigates the finite sample performance of the PMLE.     

An analysis of Hansen–Scheinkman moment estimators for discretely and randomly sampled diffusions

We derive closed-form expansions for the asymptotic distribution of Hansen and Scheinkman [1995. Back to the future: generating moment implications for continuous-time Markov processes. Econometrica 63, 767–804] moment estimators for discretely, and possibly randomly, sampled diffusions. This result makes it possible to select optimal moment conditions as well as to assess the efficiency of the resulting parameter estimators relative to likelihood-based estimators, or to an alternative type of moment conditions.     

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.     

Statistical inference for moving‐average Lévy‐driven processes: Fourier‐based approach

We consider a new method of semiparametric statistical estimation for the continuous‐time moving‐average Lévy processes. We derive the convergence rates of the proposed estimators and show that these rates are optimal in minimax sense.     

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.     

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.     

Matching and semi-parametric IV estimation, a distance-based measure of migration, and the wages of young men

Our paper estimates the effect of US internal migration on wage growth for young men between their first and second job. Our analysis of migration extends previous research by: (i) exploiting the distance-based measures of migration in the National Longitudinal Surveys of Youth 1979 (NLSY79); (ii) allowing the effect of migration to differ by schooling level and (iii) using propensity score matching to estimate the average treatment effect on the treated (ATET) for movers and (iv) using local average treatment effect (LATE) estimators with covariates to estimate the average treatment effect (ATE) and ATET for compliers.We believe the Conditional Independence Assumption (CIA) is reasonable for our matching estimators since the NLSY79 provides a relatively rich array of variables on which to match. Our matching methods are based on local linear, local cubic, and local linear ridge regressions. Local linear and local ridge regression matching produce relatively similar point estimates and standard errors, while local cubic regression matching badly over-fits the data and provides very noisy estimates.We use the bootstrap to calculate standard errors. Since the validity of the bootstrap has not been investigated for the matching estimators we use, and has been shown to be invalid for nearest neighbor matching estimators, we conduct a Monte Carlo study on the appropriateness of using the bootstrap to calculate standard errors for local linear regression matching. The data generating processes in our Monte Carlo study are relatively rich and calibrated to match our empirical models or to test the sensitivity of our results to the choice of parameter values. The estimated standard errors from the bootstrap are very close to those from the Monte Carlo experiments, which lends support to our using the bootstrap to calculate standard errors in our setting.From the matching estimators we find a significant positive effect of migration on the wage growth of college graduates, and a marginally significant negative effect for high school dropouts. We do not find any significant effects for other educational groups or for the overall sample. Our results are generally robust to changes in the model specification and changes in our distance-based measure of migration. We find that better data matters; if we use a measure of migration based on moving across county lines, we overstate the number of moves, while if we use a measure based on moving across state lines, we understate the number of moves. Further, using either the county or state measures leads to much less precise estimates.We also consider semi-parametric LATE estimators with covariates (Frölich 2007), using two sets of instrumental variables. We precisely estimate the proportion of compliers in our data, but because we have a small number of compliers, we cannot obtain precise LATE estimates.