Time-varying leverage effects (with Roberto Reno') (paper)

Journal of Econometrics, 2012, 169, 94-113

Vast empirical evidence points to the existence of a negative correlation, named "leverage effect", between shocks to variance and shocks to returns. We provide a nonparametric theory of leverage estimation in the context of a continuous-time stochastic volatility model with jumps in returns, jumps in variance, or both. Leverage is defined as a flexible function of the state of the firm, as summarized by the spot variance level. We show that its point-wise functional estimates have asymptotic properties (in terms of rates of convergence, limiting biases, and limiting variances) which crucially depend on the likelihood of the individual jumps and co-jumps as well as on the features of the jump size distributions. Empirically, we find economically important time-variation in leverage with more negative values associated with higher variance levels.

Nonparametric stochastic volatility (with Roberto Reno') (paper)

Econometric Theory, 2018, 34, 1207-1255

We provide nonparametric methods for stochastic volatility modelling. 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. 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. We present a complete asymptotic theory under recurrence, thereby accommodating the persistence properties of volatility in finite samples.

Short-term interest rate dynamics: a spatial approach (paper)

Journal of Financial Economics, 2002, 65, 73-110

We use new fully functional methods to describe and study the dynamics of the short-term interest rate process in continuous time. The suggested procedure exploits the spatial properties, embodied in the local time process of the diffusion of interest, and is robust against deviations from stationarity. Our results indicate that the misspecification of a standard constant elasticity of variance model with linear mean-reverting drift cannot be attributed to the nonlinear behavior of the infinitesimal first moment of the short-term interest rate process at high rates. Rather, it should be attributed to the martingale nature of the process over most of its empirical range (i.e., between 3% and about 15%).

A simple approach to the parametric estimation of potentially nonstationary diffusions (with Peter C.B. Phillips) (paper)

Journal of Econometrics, 2007, 137, 354-395

A simple and robust approach is proposed for the parametric estimation of scalar homogeneous stochastic differential equations. We specify a parametric class of diffusions and estimate the parameters of interest by minimizing criteria based on the integrated squared difference between kernel estimates of the drift and diffusion functions and their parametric counterparts. The procedure does not require simulations or approximations to the true transition density and has the simplicity of standard nonlinear least-squares methods in discrete time. A complete asymptotic theory for the parametric estimates is developed. The limit theory relies on infill and long span asymptotics and is robust to deviations from stationarity, requiring only recurrence.

On the functional estimation of multivariate diffusion processes (with Guillermo Moloche) (paper)

Econometric theory, 2018, 34, 896-946

We propose a nonparametric estimation theory for the drift vector and the diffusion matrix of multivariate diffusion processes. The estimators are sample analogues to infinitesimal conditional expectations constructed as Nadaraya-Watson kernel averages. Minimal assumptions are imposed on the statistical properties of the multivariate system to obtain limiting results. Harris recurrence is all that we require to show strong consistency and asymptotic (mixed) normality of the functional estimates. The estimation method and asymptotic theory apply to both stationary and nonstationary multivariate diffusion processes of the recurrent type.

On the functional estimation of jump-diffusion models (with Thong Nguyen) (paper)

Journal of Econometrics, 2003, 116, 293-328

We provide a general asymptotic theory for the fully functional estimates of the infinitesimal moments of continuous-time models with discontinuous sample paths of the jump–diffusion type. Minimal requirements are placed on the dynamic properties of the underlying jump–diffusion process, i.e., stationarity is not required. Our theoretical framework justifies consistent (in a statistical sense) nonparametric extraction of the parameters and functions that drive the dynamic evolution of the process of interest (i.e., the potentially non-affine and level-dependent intensity of the jump arrival being an example) from the estimated infinitesimal conditional moments as suggested in Johannes, 2003 (The statistical and economic role of jumps in continuous-time interest rate models, Journal of Finance, forthcoming).

Fully nonparametric estimation of scalar diffusion models (with Peter C.B. Phillips) (paper)

Econometrica, 2003, 71, 241-283

We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens. We prove almost sure consistency and weak convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying diffusion process, that is the random time spent by the process in the vicinity of a generic spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes.

Nonstationary continuous-time processes (with Peter C.B. Phillips) (paper)

Handbook of Financial Econometrics, Elsevier Science, 2009

This chapter illustrates the important role that is played by local nonparametricmethods along with the assumption of recurrence. The focus has been on estimation procedures, which are general both in terms of model specification and in terms of statistical assumptions needed for identification. Local nonparametric methods achieve the former by being robust (at the cost of an efficiency loss) to model misspecifications. Recurrence is a promising avenue to achieve the latter. Similar arguments in favor of minimal conditions on the underlying statistical structure of the process of interest may, however, be put forward when dealing with parametric models and discrete-time series. Sometimes empirical researchers may be a lot more comfortable avoiding restrictions like stationarity or arbitrary mixing conditions on the processes they are modeling. In the same circumstances, it might also seem inappropriate to impose explicit nonstationary behavior in the specification. Indeed, many practical situations arise where neither stationarity nor nonstationarity can be safely ruled out in advance, and in such situations, the assumption of recurrence appears to be a suitable alternative condition that permits a wide range of plausible sample behaviors and includes both stationary and nonstationary processes.