Econometric Theory

We provide nonparametric methods for stochastic volatility modeling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and diffusion functions, nonlinear leverage effects, and jumps in returns and volatility with possibly state-dependent jump intensities, among other features. In the first stage, we identify spot volatility by virtue of jump-robust nonparametric estimates. Using observed prices and estimated spot volatilities, the second stage extracts the functions and parameters driving price and volatility dynamics from nonparametric estimates of the bivariate process’ infinitesimal moments. For these infinitesimal moment estimates, we report an asymptotic theory relying on joint in-fill and long-span arguments which yields consistency and weak convergence under mild assumptions.

Let x be a transformation of y, whose distribution is unknown. We derive an expansion formulating the expectations of x in terms of the expectations of y. Apart from the intrinsic interest in such a fundamental relation, our results can be applied to calculating E(x) by the low-order moments of a transformation which can be chosen to give a good approximation for E(x). To do so, we generalize the approach of bounding the terms in expansions of characteristic functions, and use our result to derive an explicit and accurate bound for the remainder when a finite number of terms is taken. We illustrate one of the implications of our method by providing accurate naive bootstrap confidence intervals for the mean of any fat-tailed distribution with an infinite variance, in which case currently available bootstrap methods are asymptotically invalid or unreliable in finite samples.

In this article, we investigate the properties of heteroskedasticity and autocorrelation robust (HAR) test statistics in time series regression settings when observations are missing. We primarily focus on the nonrandom missing process case where we treat the missing locations to be fixed as T → ∞ by mapping the missing and observed cutoff dates into points on [0,1] based on the proportion of time periods in the sample that occur up to those cutoff dates. We consider two models, the amplitude modulated series (Parzen, 1963) regression model, which amounts to plugging in zeros for missing observations, and the equal space regression model, which simply ignores the missing observations. When the amplitude modulated series regression model is used, the fixed-b limits of the HAR test statistics depend on the locations of missing observations but are otherwise pivotal. When the equal space regression model is used, the fixed-b limits of the HAR test statistics have the standard fixed-b limits as in Kiefer and Vogelsang (2005). We discuss methods for obtaining fixed-b critical values with a focus on bootstrap methods and find the naive i.i.d. bootstrap with missing dates fixed to be an effective and practical way to obtain the fixed-b critical values.

We establish oracle inequalities for a version of the Lasso in high-dimensional fixed effects dynamic panel data models. The inequalities are valid for the coefficients of the dynamic and exogenous regressors. Separate oracle inequalities are derived for the fixed effects. Next, we show how one can conduct uniformly valid inference on the parameters of the model and construct a uniformly valid estimator of the asymptotic covariance matrix which is robust to conditional heteroskedasticity in the error terms. Allowing for conditional heteroskedasticity is important in dynamic models as the conditional error variance may be nonconstant over time and depend on the covariates. Furthermore, our procedure allows for inference on high-dimensional subsets of the parameter vector of an increasing cardinality. We show that the confidence bands resulting from our procedure are asymptotically honest and contract at the optimal rate. This rate is different for the fixed effects than for the remaining parts of the parameter vector.

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted L2-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the new criteria with alternative existing tests.

This article develops an asymptotic theory for estimators of two parameters in the drift function in the fractional Vasicek model when a continuous record of observations is available. The fractional Vasicek model with long-range dependence is assumed to be driven by a fractional Brownian motion with the Hurst parameter greater than or equal to one half. It is shown that, when the Hurst parameter is known, the asymptotic theory for the persistence parameter depends critically on its sign, corresponding asymptotically to the stationary case, the explosive case, and the null recurrent case. In all three cases, the least squares method is considered, and strong consistency and the asymptotic distribution are obtained. When the persistence parameter is positive, the estimation method of Hu and Nualart (2010) is also considered.

We consider a semiparametric transformation model, in which the regression function has an additive nonparametric structure and the transformation of the response is assumed to belong to some parametric family. We suppose that endogeneity is present in the explanatory variables. Using a control function approach, we show that the proposed model is identified under suitable assumptions, and propose a profile estimation method for the transformation. The proposed estimator is shown to be asymptotically normal under certain regularity conditions. A simulation study shows that the estimator behaves well in practice. Finally, we give an empirical example using the U.K. Family Expenditure Survey.

We examine a higher-order spatial autoregressive model with stochastic, but exogenous, spatial weight matrices. Allowing a general spatial linear process form for the disturbances that permits many common types of error specifications as well as potential ‘long memory’, we provide sufficient conditions for consistency and asymptotic normality of instrumental variables, ordinary least squares, and pseudo maximum likelihood estimates. The implications of popular weight matrix normalizations and structures for our theoretical conditions are discussed. A set of Monte Carlo simulations examines the behaviour of the estimates in a variety of situations. Our results are especially pertinent in situations where spatial weights are functions of stochastic economic variables, and this type of setting is also studied in our simulations.

This article introduces a local Gaussian bootstrap method useful for the estimation of the asymptotic distribution of high-frequency data-based statistics such as functions of realized multivariate volatility measures as well as their asymptotic variances. The new approach consists of dividing the original data into nonoverlapping blocks of M consecutive returns sampled at frequency h (where h−1 denotes the sample size) and then generating the bootstrap observations at each frequency within a block by drawing them randomly from a mean zero Gaussian distribution with a variance given by the realized variance computed over the corresponding block.

Our main contributions are as follows. First, we show that the local Gaussian bootstrap is first-order consistent when used to estimate the distributions of realized volatility and realized betas under assumptions on the log-price process which follows a continuous Brownian semimartingale process. Second, we show that the local Gaussian bootstrap matches accurately the first four cumulants of realized volatility up to o(h), implying that this method provides third-order refinements. This is in contrast with the wild bootstrap of Gonçalves and Meddahi (2009, Econometrica 77(1), 283–306), which is only second-order correct. Third, we show that the local Gaussian bootstrap is able to provide second-order refinements for the realized beta, which is also an improvement of the existing bootstrap results in Dovonon, Gonçalves, and Meddahi (2013, Journal of Econometrics 172, 49–65) (where the pairs bootstrap was shown not to be second-order correct under general stochastic volatility). In addition, we highlight the connection between the local Gaussian bootstrap and the local Gaussianity approximation of continuous semimartingales established by Mykland and Zhang (2009, Econometrica 77, 1403–1455) and show the suitability of this bootstrap method to deal with the new class of estimators introduced in that article. Lastly, we provide Monte Carlo simulations and use empirical data to compare the finite sample accuracy of our new bootstrap confidence intervals for integrated volatility with the existing results.

This article studies two-step sieve M estimation of general semi/nonparametric models, where the second step involves sieve estimation of unknown functions that may use the nonparametric estimates from the first step as inputs, and the parameters of interest are functionals of unknown functions estimated in both steps. We establish the asymptotic normality of the plug-in two-step sieve M estimate of a functional that could be root-n estimable. The asymptotic variance may not have a closed form expression, but can be approximated by a sieve variance that characterizes the effect of the first-step estimation on the second-step estimates. We provide a simple consistent estimate of the sieve variance, thereby facilitating Wald type inferences based on the Gaussian approximation. The finite sample performance of the two-step estimator and the proposed inference procedure are investigated in a simulation study.

We develop a new method for generating dynamics of conditional correlation matrices of asset returns. These correlation matrices are parameterized by a subset of their partial correlations, whose structure is described by a set of connected trees called “vine”. Partial correlation processes can be specified separately and arbitrarily, providing a new family of very flexible multivariate GARCH processes, called “vine-GARCH” processes. We estimate such models by quasi-maximum likelihood. We compare our models with DCC and GAS-type specifications through simulated experiments and we evaluate their empirical performances.

In unit root testing, a piecewise locally stationary process is adopted to accommodate nonstationary errors that can have both smooth and abrupt changes in second- or higher-order properties. Under this framework, the limiting null distributions of the conventional unit root test statistics are derived and shown to contain a number of unknown parameters. To circumvent the difficulty of direct consistent estimation, we propose to use the dependent wild bootstrap to approximate the nonpivotal limiting null distributions and provide a rigorous theoretical justification for bootstrap consistency. The proposed method is compared through finite sample simulations with the recolored wild bootstrap procedure, which was developed for errors that follow a heteroscedastic linear process. Furthermore, a combination of autoregressive sieve recoloring with the dependent wild bootstrap is shown to perform well. The validity of the dependent wild bootstrap in a nonstationary setting is demonstrated for the first time, showing the possibility of extensions to other inference problems associated with locally stationary processes.

We propose a new iterative estimation algorithm for use in semiparametric models where calculation of Z-estimators by conventional means is difficult or impossible. Unlike a Newton–Raphson approach, which makes use of the entire Hessian, this approach only uses curvature information associated with portions of the Hessian that are relatively easy to calculate. Consistency and asymptotic normality of estimators obtained from this algorithm are established under regularity conditions and an information dominance condition. Two specific examples, a quantile regression model with missing covariates and a GARCH-in-mean model with conditional mean of unknown functional form, demonstrate the applicability of the algorithm. This new approach can be interpreted as an extension of the maximization by parts estimation approach to semiparametric models.

This paper considers nonparametric instrumental variable regression when the endogenous variable is contaminated with classical measurement error. Existing methods are inconsistent in the presence of measurement error. We propose a wavelet deconvolution estimator for the structural function that modifies the generalized Fourier coefficients of the orthogonal series estimator to take into account the measurement error. We establish the convergence rates of our estimator for the cases of mildly/severely ill-posed models and ordinary/super smooth measurement errors. We characterize how the presence of measurement error slows down the convergence rates of the estimator. We also study the case where the measurement error density is unknown and needs to be estimated, and show that the estimation error of the measurement error density is negligible under mild conditions as far as the measurement error density is symmetric.

This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e., the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent (NED) functions of mixing processes; they are consistent under the weakest conditions that ensure the original NED process obeys a central limit theorem (CLT), established by De Jong (1997, Econometric Theory 13(3), 353–367). In doing so, this paper extends De Jong’s method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a functional CLT (FCLT) under the same conditions.

The asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is obtained for a wide class of asymmetric GARCH models with exogenous covariates. The true value of the parameter is not restricted to belong to the interior of the parameter space, which allows us to derive tests for the significance of the parameters. In particular, the relevance of the exogenous variables can be assessed. The results are obtained without assuming that the innovations are independent, which allows conditioning on different information sets. Monte Carlo experiments and applications to financial series illustrate the asymptotic results. In particular, an empirical study demonstrates that the realized volatility can be a helpful covariate for predicting squared returns.

This paper studies the weak convergence of renorming volatilities in a family of GARCH(1,1) models from a functional point of view. After suitable renormalization, it is shown that the limiting distribution is a geometric Brownian motion when the associated top Lyapunov exponent γ > 0 and is an exponential functional of the maximum process of a Brownian motion when γ = 0. This indicates that the volatility of the GARCH(1,1)-type model has a completely different random structure according to the sign of γ. The obtained results further strengthen our understanding of volatilities in GARCH-type models. Simulation studies are carried out to assess our findings.

In this article, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of L p functionals of kernel-type estimators (1 ≤ p < ∞) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on the nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating “contact sets.” We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases where the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.