Sílvia Gonçalves

Bootstrapping Realized Volatility

We propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases. We consider the independent and identically distributed (i.i.d.) bootstrap and the wild bootstrap (WB), and prove their first-order asymptotic validity under general assumptions on the log-price process that allow for drift and leverage effects. We derive Edgeworth expansions in a simpler model that rules out these effects. The i.i.d. bootstrap provides a second-order asymptotic refinement when volatility is constant, but not otherwise. The WB yields a second-order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the WB data appropriately. None of these methods provides third-order asymptotic refinements. Both methods improve upon the first-order asymptotic theory in finite samples. Copyright 2009 The Econometric Society.

Maximum likelihood and the bootstrap for nonlinear dynamic models

We provide a unified framework for analyzing bootstrapped extremum estimators of nonlinear dynamic models for heterogeneous dependent stochastic processes. We apply our results to the moving blocks bootstrap of Kunsch (Ann. Stat. 17 (1989) 1217) and Liu and Singh (in: R. Lepage, L. Billiard (Eds.), Exploring the Limits of the Bootstrap, Wiley, New York, 1992) and prove the first-order asymptotic validity of the bootstrap approximation to the true distribution of quasi-maximum likelihood estimators. We also consider bootstrap testing. In particular, we prove the first-order asymptotic validity of the bootstrap distribution of suitable bootstrap analogs of Wald and Lagrange Multiplier statistics for testing hypotheses.

Bootstrap Standard Error Estimates for Linear Regression

Standard errors of parameter estimates are widely used in empirical work. The bootstrap often can provide a convenient means of estimating standard errors. The conditions under which bootstrap standard error estimates are theoretically justified have not received much attention, however. This article establishes conditions for the consistency of the moving blocks bootstrap estimators of the variance of the least squares estimator in linear dynamic models with dependent data. We discuss several applications of this result, in particular, the use of bootstrap standard error estimates for bootstrapping Studentized statistics. A simulation study shows that inference based on bootstrap standard error estimates may be considerably more accurate in small samples than inference based on closed-form asymptotic estimates.

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.

Predictable Dynamics in the S&P 500 Index Options Implied Volatility Surface

One key stylized fact in the empirical option pricing literature is the existence of an implied volatility surface (IVS). The usual approach consists of fitting a linear model linking the implied volatility to the time to maturity and the moneyness, for each cross section of options data. However, recent empirical evidence suggests that the parameters characterizing the IVS change over time. In this paper we study whether the resulting predictability patterns in the IVS coefficients may be exploited in practice. We propose a two-stage approach to modeling and forecasting the S&P 500 index options IVS. In the first stage we model the surface along the cross-sectional moneyness and time-to-maturity dimensions, similarly to Dumas et al. (1998). In the second-stage we model the dynamics of the cross-sectional first-stage implied volatility surface coefficients by means of vector autoregression models. We find that not only the S&P 500 implied volatility surface can be successfully modeled, but also that its movements over time are highly predictable in a statistical sense. We then examine the economic significance of this statistical predictability with mixed findings. Whereas profitable delta-hedged positions can be set up that exploit the dynamics captured by the model under moderate transaction costs and when trading rules are selective in terms of expected gains from the trades, most of this profitability disappears when we increase the level of transaction costs and trade multiple contracts off wide segments of the IVS. This suggests that predictability of the time-varying S&P 500 implied volatility surface may be not inconsistent with market efficiency.

BLOCK BOOTSTRAP HAC ROBUST TESTS: THE SOPHISTICATION OF THE NAIVE BOOTSTRAP

This paper studies the properties of naive block bootstrap tests that are scaled by zero frequency spectral density estimators (long-run variance estimators). The naive bootstrap is a bootstrap where the formula used in the bootstrap world to compute standard errors is the same as the formula used on the original data. Simulation evidence shows that the naive bootstrap can be much more accurate than the standard normal approximation. The larger the HAC bandwidth, the greater the improvement. This improvement holds for a large number of popular kernels, including the Bartlett kernel, and it holds when the independent and identically distributed (i.i.d.) bootstrap is used and yet the data are serially correlated. Using recently developed fixed- b asymptotics for HAC robust tests, we provide theoretical results that can explain the finite sample patterns. We show that the block bootstrap, including the special case of the i.i.d. bootstrap, has the same limiting distribution as the fixed- b asymptotic distribution. For the special case of a location model, we provide theoretical results that suggest the naive bootstrap can be more accurate than the standard normal approximation depending on the choice of the bandwidth and the number of finite moments in the data. Our theoretical results lay the foundation for a bootstrap asymptotic theory that is an alternative to the traditional approach based on Edgeworth expansions.

Bootstrapping factor-augmented regression models

This paper proposes and theoretically justifies bootstrap methods for regressions where some of the regressors are factors estimated from a large panel of data. We derive our results under the assumption that T/N→c, where 0≤c<∞ (N and T are the cross-sectional and the time series dimensions, respectively), thus allowing for the possibility that the factor estimation error enters the limiting distribution of the OLS estimator as an asymptotic bias term (as was recently discussed by Ludvigson and Ng (2011)). We consider general residual-based bootstrap methods and provide a set of high-level conditions on the bootstrap residuals and on the idiosyncratic errors such that the bootstrap distribution of a rotated OLS estimator is consistent. We subsequently verify these conditions for a simple wild bootstrap residual-based procedure.

THE MOVING BLOCKS BOOTSTRAP FOR PANEL LINEAR REGRESSION MODELS WITH INDIVIDUAL FIXED EFFECTS

In this paper we propose a bootstrap method for panel data linear regression models with individual fixed effects. The method consists of applying the standard moving blocks bootstrap of Kunsch (1989, Annals of Statistics 17, 1217–1241) and Liu and Singh (1992, in R. LePage & L. Billiard (eds.), Exploring the Limits of the Bootstrap ) to the vector containing all the individual observations at each point in time. We show that this bootstrap is robust to serial and cross-sectional dependence of unknown form under the assumption that n (the cross-sectional dimension) is an arbitrary nondecreasing function of T (the time series dimension), where T → ∞, thus allowing for the possibility that both n and T diverge to infinity. The time series dependence is assumed to be weak (of the mixing type), but we allow the cross-sectional dependence to be either strong or weak (including the case where it is absent). Under appropriate conditions, we show that the fixed effects estimator (and also its bootstrap analogue) has a convergence rate that depends on the degree of cross-section dependence in the panel. Despite this, the same studentized test statistics can be computed without reference to the degree of cross-section dependence. Our simulation results show that the moving blocks bootstrap percentile- t intervals have very good coverage properties even when the degree of serial and cross-sectional correlation is large, provided the block size is appropriately chosen.

Consistency of the stationary bootstrap under weak moment conditions

Abstract We prove the first order asymptotic validity of the stationary bootstrap of Politis and Romano [J. Am. Statistics Assoc. 89 (1994) 1303] under the existence of only slightly more than second moments. Our results improve upon previous results in the literature, which assumed finite sixth moments.

Bootstrap Prediction Intervals for Factor Models

We propose bootstrap prediction intervals for an observation h periods into the future and its conditional mean. We assume that these forecasts are made using a set of factors extracted from a large panel of variables. Because we treat these factors as latent, our forecasts depend both on estimated factors and estimated regression coefficients. Under regularity conditions, asymptotic intervals have been shown to be valid under Gaussianity of the innovations. The bootstrap allows us to relax this assumption and to construct valid prediction intervals under more general conditions. Moreover, even under Gaussianity, the bootstrap leads to more accurate intervals in cases where the cross-sectional dimension is relatively small as it reduces the bias of the ordinary least-squares (OLS) estimator.