Kosuke Oya

Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX

This paper proposes a new method for estimating continuous-time stochastic volatility (SV) models for the S&P 500 stock index process using intraday high-frequency observations of both the S&P 500 index and the Chicago Board Options Exchange (CBOE) implied (or expected) volatility index (VIX). Intraday high-frequency observations data have become readily available for an increasing number of financial assets and their derivatives in recent years, but it is well known that attempts to directly apply popular continuous-time models to short intraday time intervals, and estimate the parameters using such data, can lead to nonsensical estimates due to severe intraday seasonality. A primary purpose of the paper is to provide a framework for using intraday high frequency data of both indices, in particular, for improving the estimation accuracy of the leverage parameter, rho, that is, the correlation between the two Brownian motions driving the diffusive components of the price process and its spot variance process, respectively. As a special case, we focus on Hestons (1993) square-root SV model, and propose the realized leverage estimator for rho, noting that, under this model without measurement errors, the “realized leverage,�? or the realized covariation of the price and VIX processes divided by the product of the realized volatilities of the two processes, is in-fill consistent for rho. Finite sample simulation results show that the proposed estimator delivers more accurate estimates of the leverage parameter than do exisiting methods.

Model-Free Implied Volatility: From Surface to Index

We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the markets expectation of future volatility. We utilize the relation between the asset variance and the Black-Scholes implied volatility surface, and discuss the merits of this new model-free approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, and we further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility.

Properties of estimators of count data model with endogenous switching

We examine properties of estimators of count data model with endogenous switching. The estimation of the count data model that accommodates endogenous switching can be accomplished by full information maximum likelihood (FIML). However, FIML estimation requires fully and correctly specified model and is computationally burdensome. Alternative estimation methods do not require fully specified model have been proposed. The typical methods are two-stage method of moments (TSM) and nonlinear weighted least-squares (NWLS). The properties of these estimators have never been studied so far. In this paper, we compared the finite sample properties of these estimators under correct and incorrect model specifications using Monte Carlo experiments. We find that FIML estimator has the smallest standard deviation and TSM estimator has the largest.

Bias-corrected realized variance under dependent microstructure noise

The aim of this study is to develop a bias-correction method for realized variance (RV) estimation, where the equilibrium price process is contaminated with market microstructure noise, such as bid-ask bounces and price-change discreteness. Although RV constitutes the simplest estimator of daily integrated variance, it remains strongly biased, and many estimators proposed in previous studies require prior knowledge about the dependence structure of microstructure noise to ensure unbiasedness and consistency. The dependence structure is unknown however, and needs to be estimated. A bias-correction method based on statistical inference from the general noise dependence structure is thus proposed. The results of Monte Carlo simulation indicate that the new approach is robust with respect to changes in the dependence of microstructure noise.

Dickey–Fuller, Lagrange Multiplier and Combined Tests for a Unit Root in Autoregressive Time Series

In this paper we investigate (augmented) Dickey–Fuller (DF) and Lagrange multiplier (LM) type unit root tests for autoregressive time series through comprehensive Monte Carlo simulations. We consider two sorts of null and alternative hypotheses: a unit root without drift versus level stationarity and a unit root with drift versus trend stationarity. The DF-type coef ficient tests are found to show the best overall performance in both cases, at least if the sample size is sufficiently large. How ever, it is also found that the DF and LM tests are roughly complementary with regard to their finite-sample power. We therefore consider combining these two types of unit root tests to obtain (ad hoc‘but’) ‘robust’ test procedures. Critical values for the proposed tests are provided

Test of random effects with incomplete panel data

This paper examines properties of test statistics for random effects with incomplete panel data. We can divide incomplete panel data into two groups. One group arises from randomly missing or unbalanced data and the other arises from systematically missing data. We focus on the former case. Some statistical properties when there are missing independent variables in regression analysis are well known. A simple approach to treat missing observations is to just discard the missing cases, but such approach may be highly inefficient. In this paper, instead of discarding the missing cases, we consider the missing data to be the outcome of a random variable. The test statistic for random effects with randomly missing panel data is derived. We examine the statistical properties of the derived test statistic and compare it with test statistic derived without randomness. We find that our test statistic is conservative in comparison with the test statistic derived without randomness.

Financial instability and the short-term dynamics of volatility expectations

This study provides new evidence of nonlinearities in the dynamics of volatility expectations during financial crises using Markov regime-switching models of model-free volatility indices. The regimes of changes in implied volatility in international financial markets are defined as function of market sentiment and a realignment process following forecast errors consistent with rational expectations. The results indicate that market returns and changes in forecast errors have indeed the potential of influencing the formation of volatility expectations. But the main force driving the dynamics of volatility expectations during periods of financial instability lies rather in the correlation with returns, reflecting market sentiment. The insignificance of the realignment process may be reflective of consensus beliefs that past information does not provide useful guidance during financial crises. It is forward-looking macroeconomic information and contemporaneous price movements that are more likely to shape the dynamics of volatility expectations.

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints. In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.

The Measures of One-Way Effect, Reciprocity, and Association

To characterize the interdependent structure of a pair of two jointly second-order stationary processes , this chapter introduces the (overall as well as frequency-wise) measures of one-way effect, reciprocity, and association. Section 2.2 defines the Granger and Sims non-causality and establishes their equivalence for a general class of (not necessarily stationary) second-order processes. Sections 2.3 and 2.4 define the overall and frequency-wise one-way effect measures and provide three ways of deriving the frequency-wise measure. One is based on direct canonical factorization of the spectral density matrix. The other two are based on distributed-lag representation and innovation orthogonalization, respectively. Each approach provides a different representation of the same quantity. Section 2.5 introduces the overall and the frequency-wise measures of reciprocity and association.

Representation of the Partial Measures

This chapter extends the measures introduced in the previous chapter to partial measures in the presence of third-series involvement. Third-series intervention is known to sometimes incur phenomena such as spurious or indirect causality attributable to possible feedback from the series. To address the problem, this chapter introduces an operational way to define the partial causality and allied concepts between a pair of processes. The third-effect elimination is of the one-way effect component of the third series from a pair of subject-matter series to preserve the inherent feedback structure of the pair of interest.