Do high-frequency measures of volatility improve forecasts of return distributions?

Many finance questions require the predictive distribution of returns. We propose a bivariate model of returns and realized volatility (RV), and explore which features of that time-series model contribute to superior density forecasts over horizons of 1 to 60 days out of sample. This term structure of density forecasts is used to investigate the importance of: the intraday information embodied in the daily RV estimates; the functional form for log(RV)$log\left(RV\right)$ dynamics; the timing of information availability; and the assumed distributions of both return and log(RV)$log\left(RV\right)$ innovations. We find that a joint model of returns and volatility that features two components for log(RV)$log\left(RV\right)$ provides a good fit to S&P 500 and IBM data, and is a significant improvement over an EGARCH model estimated from daily returns.     

Realized volatility forecasting and market microstructure noise

We extend the analytical results for reduced form realized volatility based forecasting in ABM (2004) to allow for market microstructure frictions in the observed high-frequency returns. Our results build on the eigenfunction representation of the general stochastic volatility class of models developed byMeddahi (2001). In addition to traditional realized volatility measures and the role of the underlying sampling frequencies, we also explore the forecasting performance of several alternative volatility measures designed to mitigate the impact of the microstructure noise. Our analysis is facilitated by a simple unified quadratic form representation for all these estimators. Our results suggest that the detrimental impact of the noise on forecast accuracy can be substantial. Moreover, the linear forecasts based on a simple-to-implement ‘average’ (or ‘subsampled’) estimator obtained by averaging standard sparsely sampled realized volatility measures generally perform on par with the best alternative robust measures.     

Estimating covariation: Epps effect,microstructure noise

This paper is about how to estimate the integrated covariance _〈X,Y〉T${〈X,Y〉}_T$ of two assets over a fixed time horizon [0,T]$[0,T]$, when the observations of X$X$ and Y$Y$ are “contaminated” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previous-tick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a two scales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.     

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model _Yi=exp(_Xi/2)_ηi$Y_i=exp(X_i/2){\eta }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where only (_Yi)$\left(Y_i\right)$ is observed. The model is rewritten as a particular hidden model: _Zi=_Xi+_εi$Z_i=X_i+{\epsilon }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where (_ξi)$\left({\xi }_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent sequences of i.i.d. noise. Moreover, the sequences (_Xi)$\left(X_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent and the distribution of ε$\epsilon$ is known. Then, our aim is to estimate the functions b$b$ and σ²${\sigma }^2$ when only observations _Z1,…,_Zn$Z_1,\dots ,Z_n$ are available. We propose to estimate bf$bf$ and (b²+σ²)f$(b^2+{\sigma }^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b$b$ and σ²${\sigma }^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.     

A consistent characteristic function-based test for conditional independence

Y is conditionally independent of Z given X   if Pr{f(y|X,Z)=f(y|X)}=1$\{f(y|X,Z)=f(y|X)\}=1$ for all y on its support, where f(·|·)$f(·|·)$ denotes the conditional density of Y   given (X,Z)$(X,Z)$ or X.$X.$ This paper proposes a nonparametric test of conditional independence based on the notion that two conditional distributions are equal if and only if the corresponding conditional characteristic functions are equal. We extend the test of Su and White (2005. A Hellinger-metric nonparametric test for conditional independence. Discussion Paper, Department of Economics, UCSD) in two directions: (1) our test is less sensitive to the choice of bandwidth sequences; (2) our test has power against deviations on the full support of the density of (X,Y,Z$X,Y,Z$). We establish asymptotic normality for our test statistic under weak data dependence conditions. Simulation results suggest that the test is well behaved in finite samples. Applications to stock market data indicate that our test can reveal some interesting nonlinear dependence that a traditional linear Granger causality test fails to detect.     

Dominating estimators for minimum-variance portfolios

In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets d≥4$d\ge 4$ and number of observations n≥d+2$n\ge d+2$. The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed d$d$ but n→∞$n\to \infty$ and n,d→∞$n,d\to \infty$ but n/d→q≤∞$n/d\to q\le \infty$ are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.     

The first-order approach when the cost of effort is money

We provide sufficient conditions for the first-order approach in the principal-agent problem when the agent’s utility has the nonseparable form u(y−c(a))$u(y-c\left(a\right))$ where y$y$ is the contractual payoff and c(a)$c\left(a\right)$ is the money cost of effort. We first consider a decision-maker facing prospects which cost c(a)$c\left(a\right)$ and with distributions of returns y$y$ that depend on a$a$. The decision problem is shown to be concave if the primitive of the cdf of returns is jointly convex in a$a$ and y$y$, a condition we call Concavity of the Cumulative Quantile (CCQ) and which is satisfied by many common distributions. Next we apply CCQ to the distribution of outcomes (or their likelihood-ratio transforms) in the principal-agent problem and derive restrictions on the utility function that validate the first-order approach. We also discuss another condition, log-convexity of the distribution, and show that it allows binding limited liability constraints, which CCQ does not.     

A complete asymptotic series for the autocovariance function of a long memory process

An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q$p,d,q$) model. The leading term of the expansion is of the order O(1/k^1−2d)$O(1/k^{1-2d})$, where k$k$ is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k^3−2d)$O(1/k^{3-2d})$. The derivation uses Erdélyi’s [Erdélyi, A., 1956. Asymptotic Expansions. Dover Publications, Inc, New York] expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2π}$\{0,2\pi \}$. Numerical evaluations show that the expansion is accurate even for small k$k$ in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k$k$. The approximations are easy to compute across a variety of parameter values and models.     

Risk-parameter estimation in volatility models

This paper introduces the concept of risk parameter in conditional volatility models of the form _?t=_σt(_θ0)_ηt${?}_t={\sigma }_t\left({\theta }_0\right){\eta }_t$ and develops statistical procedures to estimate this parameter. For a given risk measure r$r$, the risk parameter is expressed as a function of the volatility coefficients _θ0${\theta }_0$ and the risk, r(_ηt)$r\left({\eta }_t\right)$, of the innovation process. A two-step method is proposed to successively estimate these quantities. An alternative one-step approach, relying on a reparameterization of the model and the use of a non Gaussian QML, is proposed. Asymptotic results are established for smooth risk measures, as well as for the Value-at-Risk (VaR). Asymptotic comparisons of the two approaches for VaR estimation suggest a superiority of the one-step method when the innovations are heavy-tailed. For standard GARCH models, the comparison only depends on characteristics of the innovations distribution, not on the volatility parameters. Monte-Carlo experiments and an empirical study illustrate the superiority of the one-step approach for financial series.     

Testing for a change in persistence in the presence of non-stationary volatility

In this paper we consider tests for the null of (trend-) stationarity against the alternative of a change in persistence at some (known or unknown) point in the observed sample, either from I(0)$I\left(0\right)$ to I(1)$I\left(1\right)$ behaviour or vice versa, of, inter alia, [Kim, J., 2000. Detection of change in persistence of a linear time series. Journal of Econometrics 95, 97–116]. We show that in circumstances where the innovation process displays non-stationary unconditional volatility of a very general form, which includes single and multiple volatility breaks as special cases, the ratio-based statistics used to test for persistence change do not have pivotal limiting null distributions. Numerical evidence suggests that this can cause severe over-sizing in the tests. In practice it may therefore be hard to discriminate between persistence change processes and processes with constant persistence but which display time-varying unconditional volatility. We solve the identified inference problem by proposing wild bootstrap-based implementations of the tests. Monte Carlo evidence suggests that the bootstrap tests perform well in finite samples. An empirical illustration using US price inflation data is provided.     

Estimating first-price auctions with an unknown number of bidders: A misclassification approach

In this paper, we consider nonparametric identification and estimation of first-price auction models when N^∗$N^{\ast }$, the number of potential bidders, is unknown to the researcher, but observed by bidders. Exploiting results from the recent econometric literature on models with misclassification error, we develop a nonparametric procedure for recovering the distribution of bids conditional on the unknown N^∗$N^{\ast }$. Monte Carlo results illustrate that the procedure works well in practice. We present illustrative evidence from a dataset of procurement auctions, which shows that accounting for the unobservability of N^∗$N^{\ast }$ can lead to economically meaningful differences in the estimates of bidders’ profit margins.     

On bootstrapping panel factor series

This paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series. Two main results are shown. Firstly, a bootstrap Invariance Principle is derived pointwise in i$i$, obtaining an upper bound for the order of truncation of the AR polynomial that depends on n$n$ and T$T$. Consistent estimation of the long run variances is also studied for (n,T)→∞$(n,T)\to \infty$. Secondly, joint bootstrap asymptotics is also studied, investigating the conditions under which the bootstrap is valid. In particular, the extent of cross sectional dependence which can be allowed for is investigated. Whilst we show that, for general forms of cross dependence, consistent estimation of the long run variance (and therefore validity of the bootstrap) is fraught with difficulties, however we show that “one-cross-sectional-unit-at-a-time” resampling schemes yield valid bootstrap based inference under weak forms of cross-sectional dependence.     

Semiparametric binary regression models under shape constraints with an application to Indian schooling data

We consider estimation of the regression function in a semiparametric binary regression model defined through an appropriate link function (with emphasis on the logistic link) using likelihood-ratio based inversion. The dichotomous response variable Δ$\Delta$ is influenced by a set of covariates that can be partitioned as (X,Z)$(X,Z)$ where Z$Z$ (real valued) is the covariate of primary interest and X$X$ (vector valued) denotes a set of control variables. For any fixed X$X$, the conditional probability of the event of interest (Δ=1$\Delta =1$) is assumed to be a non-decreasing function of Z$Z$. The effect of the control variables is captured by a regression parameter β$\beta$. We show that the baseline conditional probability function (corresponding to X=0$X=0$) can be estimated by isotonic regression procedures and develop a likelihood ratio based method for constructing asymptotic confidence intervals for the conditional probability function (the regression function) that avoids the need to estimate nuisance parameters. Interestingly enough, the calibration of the likelihood ratio based confidence sets for the regression function no longer involves the usual χ²${\chi }^2$ quantiles, but those of the distribution of a new random variable that can be characterized as a functional of convex minorants of Brownian motion with quadratic drift. Confidence sets for the regression parameter β$\beta$ can however be constructed using asymptotically χ²${\chi }^2$ likelihood ratio statistics. The finite sample performance of the methods are assessed via a simulation study. The techniques of the paper are applied to data sets on primary school attendance among children belonging to different socio-economic groups in rural India.     

On the distribution of estimated technical efficiency in stochastic frontier models

We consider a stochastic frontier model with error ε=v−u$\epsilon =v-u$, where v$v$ is normal and u$u$ is half normal. We derive the distribution of the usual estimate of u,E(u|ε)$u,E\left(u\right|\epsilon )$. We show that as the variance of v$v$ approaches zero, E(u|ε)−u$E\left(u\right|\epsilon )-u$ converges to zero, while as the variance of v$v$ approaches infinity, E(u|ε)$E\left(u\right|\epsilon )$ converges to E(u)$E\left(u\right)$. We graph the density of E(u|ε)$E\left(u\right|\epsilon )$ for intermediate cases. To show that E(u|ε)$E\left(u\right|\epsilon )$ is a shrinkage of u towards its mean, we derive and graph the distribution of E(u|ε)$E\left(u\right|\epsilon )$ conditional on u$u$. We also consider the distribution of estimated inefficiency in the fixed-effects panel data setting.     

Nonlinearity and temporal dependence

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: ρ-mixing$\rho \text{-}mixing$, β-mixing$\beta \text{-}mixing$ and α-mixing$\alpha \text{-}mixing$. Stationary diffusions that are ρ-mixing$\rho \text{-}mixing$ have mixing coefficients that decay exponentially to zero. When they fail to be ρ-mixing$\rho \text{-}mixing$, they are still β-mixing$\beta \text{-}mixing$ and α-mixing$\alpha \text{-}mixing$; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state dependent, Poisson sampling alters the temporal dependence.