Understanding uncertainty in estimating risk measures is important in modern financial risk management. In this paper we consider a nonparametric framework that incorporates auxiliary information available in covariates and propose a family of inferential methods for the value at risk, expected shortfall, and related risk measures. A two-step generalized empirical likelihood test statistic is constructed and is shown to be asymptotically pivotal without requiring variance estimation. We also show its validity when applied to a semiparametric index model. Asymptotic theories are established allowing for serially dependent data. Simulations and an empirical application to Canadian stock return index illustrate the finite sample behavior of the methodologies proposed.

This paper proposes a nonparametric test for conditional independence that is easy to implement, yet powerful in the sense that it is consistent and achieves n−1/2 local power. The test statistic is based on an estimator of the topological “distance” between restricted and unrestricted probability measures corresponding to conditional independence or its absence. The distance is evaluated using a family of Generically Comprehensively Revealing (GCR) functions, such as the exponential or logistic functions, which are indexed by nuisance parameters. The use of GCR functions makes the test able to detect any deviation from the null. We use a kernel smoothing method when estimating the distance. An integrated conditional moment (ICM) test statistic based on these estimates is obtained by integrating out the nuisance parameters. We simulate the critical values using a conditional simulation approach. Monte Carlo experiments show that the test performs well in finite samples. As an application, we test an implication of the key assumption of unconfoundedness in the context of estimating the returns to schooling.

A semiparametric model is proposed in which a parametric filtering of a nonstationary time series, incorporating fractionally differencing with short memory correction, removes correlation but leaves a nonparametric deterministic trend. Estimates of the memory parameter and other dependence parameters are proposed, and shown to be consistent and asymptotically normally distributed with parametric rate. Tests with standard asymptotics for I(1) and other hypotheses are thereby justified. Estimation of the trend function is also considered. We include a Monte Carlo study of finite-sample performance.

This paper addresses the problem of semiparametric efficiency bounds for conditional moment restriction models with different conditioning variables. We characterize such an efficiency bound, that in general is not explicit, as a limit of explicit efficiency bounds for a decreasing sequence of unconditional (marginal) moment restriction models. An iterative procedure for approximating the efficient score when this is not explicit is provided. Our theoretical results provide new insight for the theory of semiparametric efficiency bounds literature and open the door to new applications. In particular, we investigate a class of regression-like (mean regression, quantile regression,…) models with missing data, an example of a supply and demand simultaneous equations model and a generalized bivariate dichotomous model.

We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles, if and only if the underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets.

In this paper we propose primitive conditions under which a projection of a conditional density onto a set defined by conditional moment restrictions exists and is unique. Moreover, we provide an analytic expression of the obtained projection. The range of applications where conditional density projections are used is wide. The derived results are potentially useful in a variety of areas including: semiparametric efficient estimation and optimal testing in (conditional) moment models, Bayesian prior determination and inference in semiparametric models, density forecasting, and simulation-based econometric analysis.

Regarding existence, we propose three different combinations of assumptions that are all sufficient to show that the projection exists and is unique. The proposed conditions exhibit a clear trade off between restrictions put on the divergence between the conditional densities and on the moment function which defines the projection set. Depending on the nature of the application, the researcher can pick and choose which set of conditions to use. Our second set of results characterizes the projection. The expression for the projected density is new though not surprising given the previously obtained results for the unconditional case. The projection is characterized by the dual of the original projection problem. In establishing the strong duality, however, we work with a constraint qualification condition that is weaker than that used by Borwein and Lewis (1991a, 1992a, 1993 in their seminal work concerning the unconditional case.

This paper proposes a semiparametric generalized method of moments estimator (GMM) estimator for a partially parametric spatial model with endogenous spatially dependent regressors. The finite-dimensional estimator is shown to be consistent and root-n asymptotically normal under some reasonable conditions. A spatial heteroscedasticity and autocorrelation consistent covariance estimator is constructed for the GMM estimator. The leading application is nonlinear spatial autoregressions, which arise in a wide range of strategic interaction models. To derive the asymptotic properties of the estimator, the paper also establishes a stochastic equicontinuity criterion and functional central limit theorem for near-epoch dependent random fields.

A unified framework is proposed for tests of unobserved heterogeneity in parametric statistic models based on Neyman’s C(α) approach. Such tests are irregular in the sense that the first order derivative of the log likelihood with respect to the heterogeneity parameter is identically zero, and consequently the conventional Fisher information about the parameter is zero. Nevertheless, local asymptotic optimality of the C(α) tests can be established via LeCam’s differentiability in quadratic mean and the limit experiment approach. This leads to local alternatives of order n−1/4 . The scalar case result is already familiar from existing literature and we extend it to the multidimensional case. The new framework reveals that certain regularity conditions commonly employed in earlier developments are unnecessary, i.e. the symmetry or third moment condition imposed on the heterogeneity distribution. Additionally, the limit experiment for the multidimensional case suggests modifications on existing tests for slope heterogeneity in cross sectional and panel data models that lead to power improvement. Since the C(α) framework is not restricted to the parametric model and the test statistics do not depend on the particular choice of the heterogeneity distribution, it is useful for a broad range of applications for testing parametric heterogeneity.

This paper develops a method for performing inference using spatially dependent data. We consider test statistics formed using nonparametric covariance matrix estimators that account for heteroskedasticity and spatial correlation (spatial HAC). We provide distributions of commonly used test statistics under “fixed-b” asymptotics, in which HAC smoothing parameters are proportional to the sample size. Under this sequence, spatial HAC estimators are not consistent but converge to nondegenerate limiting random variables that depend on the HAC smoothing parameters, the HAC kernel, and the shape of the spatial region in which the data are located. We illustrate the performance of the “fixed-b” approximation in the spatial context through a simulation example.

This article is concerned with semiparametric estimation of a partially linear transformation model under conditional quantile restriction with no parametric restriction imposed either on the link functional form or on the error term distribution. We describe for the finite-dimensional parameter a $\sqrt n$-consistent estimator which combines the features of Chen (2010)’s maximum integrated score estimator as well as Lee (2003)’s average quantile regression. We show the remaining two infinite-dimensional unknown functions in the model can be separately identified and propose estimators for these functions based on the marginal integration method. Furthermore, a simple approach is proposed to estimate the average partial quantile effect. Two important extensions, i.e., random censoring as well as estimating a transformation model with an endogenous regressor are also considered.

As an alternative to quasi-maximum likelihood, targeting estimation is a much applied estimation method for univariate and multivariate GARCH models. In terms of variance targeting estimation, recent research has pointed out that at least finite fourth moments of the data generating process is required, if one wants to perform inference in GARCH models by relying on asymptotic normality of the estimator. Such moment conditions may not be satisfied in practice for financial returns, highlighting a potential drawback of variance targeting estimation. In this paper, we consider the large-sample properties of the variance targeting estimator for the multivariate extended constant conditional correlation GARCH model when the distribution of the data generating process has infinite fourth moments. Using nonstandard limit theory, we derive new results for the estimator stating that, under suitable conditions, its limiting distribution is multivariate stable. The rate of consistency of the estimator is slower than $\sqrt T$ and depends on the tail shape of the data generating process. A simulation study illustrates the derived properties of the targeting estimator.

This paper considers the problem of coverage of the elements of the identified set in a class of partially identified econometric models with a prespecified probability. In order to conduct inference in partially identified econometric models defined by moment (in)equalities, the literature has proposed three methods: bootstrap, subsampling, and asymptotic approximation. The objective of this paper is to compare these methods in terms of the rate at which they achieve the desired coverage level, i.e., in terms of the rate at which the error in the coverage probability (ECP) converges to zero.

Under certain conditions, we show that the ECP of the bootstrap and the ECP of the asymptotic approximation converge to zero at the same rate, which is a faster rate than that of the ECP of subsampling methods. As a consequence, under these conditions, the bootstrap and the asymptotic approximation produce inference that is more precise than subsampling. A Monte Carlo simulation study confirms that these results are relevant in finite samples.

Because of the incidental parameters problem, the fixed effects maximum likelihood estimator in a nonlinear panel data model is in general inconsistent when the time series length T is short and fixed. Even if T approaches infinity but at a rate not faster than the cross sectional sample size n, the fixed effects estimator is still asymptotically biased. This paper proposes using the standard bootstrap and k-step bootstrap to correct the bias. We establish the asymptotic validity of the bootstrap bias corrections for both model parameters and average marginal effects. Our results apply to static models as well as some dynamic Markov models. Monte Carlo simulations show that our procedures are effective in reducing the bias of the fixed effects estimator and improving the coverage accuracy of the associated confidence interval.

We propose a consistent functional estimator for the occupation time of the spot variance of an asset price observed at discrete times on a finite interval with the mesh of the observation grid shrinking to zero. The asset price is modeled nonparametrically as a continuous-time Itô semimartingale with nonvanishing diffusion coefficient. The estimation procedure contains two steps. In the first step we estimate the Laplace transform of the volatility occupation time and, in the second step, we conduct a regularized Laplace inversion. Monte Carlo evidence suggests that the proposed estimator has good small-sample performance and in particular it is far better at estimating lower volatility quantiles and the volatility median than a direct estimator formed from the empirical cumulative distribution function of local spot volatility estimates. An empirical application shows the use of the developed techniques for nonparametric analysis of variation of volatility.

In this paper, we propose a simple nonparametric test for testing the null hypothesis of constant coefficients against nonparametric smooth coefficients in a semiparametric varying coefficient model with integrated time series. We establish the asymptotic distributions of the proposed test statistic under both null and alternative hypotheses. Moreover, we derive a central limit theorem for a degenerate second order U-statistic, which contains a mixture of stationary and nonstationary variables and is weighted locally on a stationary variable. This result is of independent interest and useful in other applications. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed test.

Detecting and modeling structural changes in GARCH processes have attracted increasing attention in time series econometrics. In this paper, we propose a new approach to testing structural changes in GARCH models. The idea is to compare the log likelihood of a time-varying parameter GARCH model with that of a constant parameter GARCH model, where the time-varying GARCH parameters are estimated by a local quasi-maximum likelihood estimator (QMLE) and the constant GARCH parameters are estimated by a standard QMLE. The test does not require any prior information about the alternatives of structural changes. It has an asymptotic N(0,1) distribution under the null hypothesis of parameter constancy and is consistent against a vast class of smooth structural changes as well as abrupt structural breaks with possibly unknown break points. A consistent parametric bootstrap is employed to provide a reliable inference in finite samples and a simulation study highlights the merits of our test.

This paper develops a fully modified OLS (FM-OLS) estimator for cointegrating polynomial regressions, i.e., regressions that include as explanatory variables deterministic variables, integrated processes, and integer powers of integrated processes. The stationary errors are allowed to be serially correlated and the regressors are allowed to be endogenous. The paper extends the fully modified estimator of Phillips and Hansen (1990) from cointegrating regressions to cointegrating polynomial regressions. The FM-OLS estimator has a zero-mean Gaussian mixture limiting distribution that allows for standard asymptotic inference. Wald and LM specification tests as well as a KPSS-type test for cointegration are derived. The theoretical analysis is complemented by a simulation study which shows that this FM-OLS estimator, as well as tests based upon it, perform well in the sense that the performance advantages over OLS are largely similar to the performance advantages of FM-OLS over OLS in standard cointegrating regressions.

The semiparametric GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) model of Yang (2006, Journal of Econometrics 130, 365–384) has combined the flexibility of a nonparametric link function with the dependence on infinitely many past observations of the classic GARCH model. We propose a cubic spline procedure to estimate the unknown quantities in the semiparametric GARCH model that is intuitively appealing due to its simplicity. The theoretical properties of the procedure are the same as the kernel procedure, while simulated and real data examples show that the numerical performance is either better than or comparable to the kernel method. The new method is computationally much more efficient than the kernel method and very useful for analyzing large financial time series data.