Federico M. Bandi

Fully Nonparametric Estimation of Scalar Diffusion Models

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.

On the functional estimation of jump-diffusion models

Abstract 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 nonaffine 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).

Using High-Frequency Data in Dynamic Portfolio Choice

This article evaluates the economic benefit of methods that have been suggested to optimally sample (in an MSE sense) high-frequency return data for the purpose of realized variance/covariance estimation in the presence of market microstructure noise (Bandi and Russell, 2005a, 2008). We compare certainty equivalents derived from volatility-timing trading strategies relying on optimally-sampled realized variances and covariances, on realized variances and covariances obtained by sampling every 5 minutes, and on realized variances and covariances obtained by sampling every 15 minutes. In our sample, we show that a risk-averse investor who is given the option of choosing variance/covariance forecasts derived from MSE-based optimal sampling methods versus forecasts obtained from 5- and 15-minute intervals (as generally proposed in the literature) would be willing to pay up to about 80 basis points per year to achieve the level of utility that is guaranteed by optimal sampling. We find that the gains yielded by optimal sampling are economically large, statistically significant, and robust to realistic transaction costs.

Short-term interest rate dynamics: a spatial approach

Abstract 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%).

Nonparametric Stochastic Volatility

Using recent advances in the nonparametric estimation of continuous-time processes under mild statistical assumptions as well as recent developments on nonparametric volatility estimation by virtue of market microstructure noise-contaminated high-frequency asset price data, we provide (i) a theory of spot variance estimation and (ii) functional 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, jumps in returns and volatility with possibly state-dependent jump intensities, as well as nonlinear risk-return trade-offs. Our identification approach and asymptotic results apply under weak recurrence assumptions and, hence, accommodate the persistence properties of variance in finite samples. Functional estimation of a generalized (i.e., nonlinear) version of the square-root stochastic variance model with jumps in both volatility and returns for the S&P500 index suggests the need for richer variance dynamics than in existing work. We find a linear specification for the variances diffusive variance to be misspecified (and inferior to a more flexible CEV specification) even when allowing for jumps in the variance dynamics.

Nonstationary Continuous-Time Processes ⁄

Publisher Summary This chapter illustrates the important role that is played by local nonparametric methods 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.

Realized Volatility Forecasting in the Presence of Time-Varying Noise

Observed high-frequency financial prices can be considered as having two components, a true price and a market microstructure noise perturbation. It is an empirical regularity, coherent with classical market microstructure theories of price determination, that the second moment of market microstructure noise is time-varying. We study the optimal, from a finite-sample forecast mean squared error (MSE) standpoint, frequency selection for realized variance in linear variance forecasting models with time-varying market microstructure noise. We show that the resulting sampling frequencies are generally considerably lower than those that would be optimally chosen when time-variation in the second moment of the noise is unaccounted for. These optimal, lower frequencies have the potential to translate into considerable out-of-sample MSE gains. When forecasting using high-frequency variance estimates, we recommend treating the relevant frequency as a parameter and evaluating it jointly with the parameters of the forecasting model. The proposed joint solution is robust to the features of the true price formation mechanism and generally applicable to a variety of forecasting models and high-frequency variance estimators, including those for which the typical choice variable is a smoothing parameter, rather than a frequency.

The Scale of Predictability

We introduce a new stylized fact: the hump-shaped behavior of slopes and coefficients of determination as a function of the aggregation horizon when running (forward/backward) predictive regressions of future excess market returns onto past economic uncertainty (as proxied by market variance, consumption variance, or economic policy uncertainty). To justify this finding formally, we propose a novel modeling framework in which predictability is specified as a property of low-frequency components of both excess market returns and economic uncertainty. We dub this property scale-specific predictability. We show that classical predictive systems imply restricted forms of scale-specific predictability. We conclude that for certain predictors, like economic uncertainty, the restrictions imposed by classical predictive systems may be excessively strong.

The Horizon of Systematic Risk: A New Beta Representation

We show that a business-cycle consumption factor can explain the differences in risk premia across alternative portfolios, including recently-proposed anomalies portfolios. We argue that explicit allowance for a separation between consumption fluctuations with heterogeneous durations is important for interpreting cross-sectional pricing as well as the time-series dynamics of consumption and returns across horizons (i.e., the hump-shaped pricing ability of the covariance between ultimate consumption and returns, the hump-shaped structure of long-run risk premia, the decaying pattern in consumption growth predictability). Using a novel modeling approach relying on a frequency-based decomposition, we formalize the important role that aggregation can play in asset pricing.While contemporaneous consumption growth is known not to price the cross section of stock returns, we nd that suitable sub-components (or details) of consumption growth with periodicities corresponding to the business cycle do. Specically, we disaggregate consumption growth into details with dierent levels of persistence and show that those corresponding to businesscycle scales can explain the dierences in risk premia across book-to-market and size-sorted portfolios. We argue that accounting for persistence heterogeneity in consumption is important for interpreting risk compensations in nancial markets but also for capturing the joint dynamics of consumption and returns across horizons (for instance, the hump-shaped pricing ability of the covariance between ultimate consumption and returns, the hump-shaped structure of long-run risk premia as well as the decaying pattern in consumption growth predictability). Using our proposed scale-based data generating process for consumption growth, we discuss implications for the cross-sectional pricing literature relying on aggregation.

NONPARAMETRIC NONSTATIONARITY TESTS

We propose additive functional-based nonstationarity tests that exploit the different divergence rates of the occupation times of a (possibly nonlinear) process under the null of nonstationarity (stationarity) versus the alternative of stationarity (nonstationarity). We consider both discrete-time series and continuous-time processes. The discrete-time case covers Harris recurrent Markov chains and integrated processes. The continuous-time case focuses on Harris recurrent diffusion processes. Notwithstanding finite-sample adjustments discussed in the paper, the proposed tests are simple to implement and rely on tabulated critical values. Simulations show that their size and power properties are satisfactory. Our robustness to nonlinear dynamics provides a solution to the typical inconsistency problem between assumed linearity of a time series for the purpose of nonstationarity testing and subsequent nonlinear inference.