THEMATIC PROGRAMS

April 19, 2014

April 23-24, 2010
Workshop on Financial Econometrics

Talk Titles and Abstracts

Yacine Aït-Sahalia (with Julio Cacho-Diaz and Roger Laeven)
Modeling Financial Contagion Using Mutually Exciting Jump Processes

Abstract: Adverse shocks to stock markets propagate across the world, with a jump in one region of the world seemingly causing an increase in the likelihood of a different jump in another region of the world. To capture this effect mathematically, we introduce a model for asset return dynamics with a drift component, a volatility component and mutually exciting jumps known as Hawkes processes. In the model, a jump in one region of the world or one segment of the market increases the intensity of jumps occurring both in the same region (self-excitation) as well as in other regions (cross-excitation). The model generates the type of jump clustering that is observed empirically. Jump intensities then mean-revert until the next jump. We develop and implement an estimation procedure for this model. Our estimates provide evidence for self-excitation both in the US market as well as in other world markets. Furthermore, we find that US jumps tend to get reflected quickly in most other markets, while statistical evidence for the reverse transmission is much less pronounced. Implications of the model for measuring market stress, risk management and optimal portfolio choise are also investigated.

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Marco Bonomo (with René Garcia, Nour Meddahi and Roméo Tédongap)
Generalized Disappointment Aversion, Volatility Long-Run Risk and Asset Prices

Abstract: We propose an asset pricing model where preferences display generalized disappointment aversion (as in Routledge and Zin 2009) and the endowment process involves long-run volatility risk. Those preferences, which are embedded in Epstein and Zin recursive utility framework, overweight disappointing results as compared to expected utility, and display relatively larger risk aversion for small gambles. Our endowment process has only one of the two sources of long-run risks proposed by Bansal and Yaron (2004) (BY): the volatility risk. We approximate the endowment process with a Markov switching model. This enables us to derive closed formula solutions for all returns moments and predictability regressions.The model produces asset returns moments and predictability patterns in line with the data. Compared to BY we generate: i) more predictability of excess returns by price-dividend ratios; ii) less predictability of consumption growth rates by price-dividend ratios. Differently from BY model, our results do not depend on IES being greater than one: similar results may be obtained with IES lower than one. Our results are not due to overparametrization of preferences either: simple disappointment averse with two paramters, where risk aversion comes only from disappointment aversion generates similar implications.

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Robert Engle
Long Term Skewness and Systemic Risk

Abstract: Financial risk management has generally focused on short term risks rather than long term risks and arguably this is an important component of the current financial crisis. Econometric approaches to measuring long term risk are investigated by testing for measures of long term skewness associated with asymmetric volatility models. This skewness in a market factor leads to default correlations even far in the future. Investors concerned about long term risks can hedge exposure as in the ICAPM. Such hedging will affect asset prices and can be tested directly with volatility models. Using estimates from VLAB, evidence is found for several types of hedge portfolios including volatility, long bonds, term spread, credit spread and gold.

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Jianqing Fan
Vast Volatility Matrix Estimation using High Frequency Data for Portfolio Selection

Abstract: Portfolio allocation with gross-exposure constraint is an effective method to increase the efficiency and stability of selected portfolios among a vast pool of assets, as demonstrated in Fan, Zhang and Zhang (2008).  The required high-dimensional volatility matrix can be estimated by using high frequency financial data. This enables us to better adapt the local volatilities and local correlations among vast assets and to increase significantly the sample size for estimating the volatility matrix.  This paper studies the volatility matrix estimation using high-dimensional high-frequency data from the perspective of portfolio selection. Specifically, we propose the use of ``pairwise-refresh time" and ``all-refresh-time" methods  for estimation vast covariance matrix and compare their merits in the portfolio selection.  We also establish the large deviation results of the estimates, which guarantee good properties of the estimated volatility matrix in vast asset allocation with gross exposure constraints.  Extensive numerical studies are made via carefully designed simulation studies.  Comparing with the methods based on low frequency daily data, our methods can capture the most recent trend of the time varying volatility and correlation, hence provide more accurate guidance of the portfolio allocation of the next time period. The advantage of use high-frequency data is significant in our simulation and empirical studies, which consist of 30 Dow-Jones industrial stocks.

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Christian Gourieroux (with Patrick Gagliardini)
Approximate Derivative Pricing for Large Class of Homogeneous Assets with Systematic Risk

Abstract: We consider an homogeneous class of assets, whose returns are driven by an unobservable factor representing systematic risk. We derive approximated pricing formulas for the future factor values and their proxies, when the size n of the class is large. Up to order 1=n, these closed form approximations involve well-chosen summary statistics of the basic asset returns, but not the current and lagged factor values. The potential of the closed form approximation formulas seems quite large, especially for credit risk analysis, which considers large portfolios of individual loans or corporate bonds, and for longevity risk analysis, which involves large portfolios of life insurance contracts.

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Kaddour Hadri (with Ruijun Bu)
Modelling Multivariate Interest Rates using Time-Varying Copulas and Reducible Non-Linear Stochastic Differential Equations

Abstract: We propose a new approach for modelling non-linear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modelling of the marginal processes, we consider a class of non-linear SDEs that are reducible to Ornstein-Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a non-linear transformation function. The main advantage of this approach is that these SDEs can account for non-linear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of non-linear constant elasticity volatility (CEV) processes, denoted OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed form likelihood functions. The transition density, the conditional distribution function, the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain more flexible functional form over time, we allow the transformation function to be time-varying. Results from our study of US and UK short term interest rates suggest that the new models outperform existing parametric models with closed form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behaviour of interest rate series, we propose flexible non-linear multivariate models by joining univariate non-linear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying Symmetrized Joe-Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates.

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Lars Hansen
Nonlinear Filtering and Learning Dynamics (with Nick Polson and Thomas Sargent)

Abstract: We develop and apply two refinements of particle filtering methods to be used in characterizing the learning behavior of individual agents within an economic model. One refinement extends the use of sufficient statistics conditioned on hidden states and a subset of parameters as a device to induce randomization in the parameters within the algorithm. This allows us to replenish particles and extend the number of time periods to which the numerical results remain reliable. The other refinement focuses the accuracy of the particle filtering algorithm on the portions of the filtered distribution that are more germane to decision problems of the individual agents. We illustrate these methods in an equilibrium model with investors that make robust decisions implemented through the use of exponential tilting.

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Stan Hurn
Quasi-maximum Likelihood Estimation of the Parameters of Multivariate Diffusions

Abstract: This paper develops a quasi-maximum likelihood procedure for estimating the parameters of multi-dimensional stochastic differential equations. The transitional density is taken to be a time-varying multivariate Gaussian where the first two moments of the distribution are approximately the true moments of the unknown transitional density. For affine drift and diffusion functions, the moments are shown to be exactly those of the true transitional density and for nonlinear drift and diffusion functions the approximation is extremely good. The estimation procedure is easily generalizable to models with latent factors, such as the stochastic volatility class of model, thereby avoiding the need to use proxies.

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Jean Jacod
Testing for Functional Relationships between Log-price and Volatility (with C. Kluppelberg and G. Muller)

Abstract: In many models for asset prices, the (stochastic) volatility jumps at the same times as the price itself, and moreover the two jumps are related by a functional relationship: this is in particular the case for the so-called COGARCH (continuous-time GARCH) models. In this paper we give a method allowing to
test whether a specific relationship between the price and volatility jumps is satisfied.

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Robert Kimmel
On Estimation of Risk Premia in Linear Factor Models (with Kewei Hou)

Abstract: We examine theoretical and econometric issues in the estimation of risk premia in a linear factor model, when the model is possibly misspecified. Common empirical methodologies can produce very misleading results. With unspanned factors and possible model misspecification, there are problems not just in estimating the risk premia, but even in defining them unambiguously. We show that, for a given set of test assets, the risk premium of an unspanned factor is very sensitive to the choice of other factors in the model. However, the risk premium of the projection of the unspanned factor onto the asset space is robust to the choice of other factors. The problem is greatly exacerbated in the presence of model misspecification, and can occur even when the unspanned components of the factors are very small (relative to the spanned components). These results highlight the importance of using factor-mimicking portfolios, rather than unspanned factors, in estimation of linear factor models.

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Suzanne Lee
Jumps and Information Flow in Financial Markets

Abstract: I propose a new two-stage semi-parametric test to investigate the predictability of stochastic jump arrivals in asset prices. The test allows us to pin down relevant information for jump prediction up to the intra-day level. Based on the test, I find that systematic jumps in U.S. individual equity markets are likely to occur shortly after macroeconomic information release such as Fed's announcements, market jumps, employment reports, or initial jobless claims. I also present firm-specific jump predictors along with the jump clustering effect. Evidence suggests that systematic jump intensity has increased in recent years.

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Haitao Li
Exploring Statistical Arbitrage Opportunities in the Term Structure of CDS Spreads

Abstract: The rapid growth of the CDS market makes it possible to speculate on the relative pricing of the credit risk of a company across a wide range of maturities. Based on a reduced-form model of credit risk, we explore “statistical” arbitrage opportunities in the term structure of CDS spreads of a large number of companies in North America. Specifically, we estimate an affine model for the term structure of CDS spreads of a given company and identify mis-valued CDS contracts along the credit curve. We trade market-neutral portfolios of mis-valued CDS contracts relative to our model, betting that the mis-valuation will disappear over time. Empirical analysis shows that our “arbitrage” strategy can be very profitable. For most firms, the Sharpe ratios are higher than one, and for some firms, the Sharpe ratios are even above two.

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Jia Li
A Local-to-continuity Theory for the Pre-averaging Method

Abstract: This paper develops a local asymptotic theory for certain functionals of moving averages of Ito semi-martingales plus noise when the semimartingales are nearly continuous. The model provides a more complete interface between the continuity and jump asymptotics developed in (Jacod, Podolskij, and
Vetter 2009). Simulation suggests that our theory provides an improvement over the existing theory in a
practically relevant setting. The theory has applications to the estimation of functionals of jumps and to
the analysis of the local asymptotic power of tests for jumps with noisy high frequency data.

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Yingying Li (with Yacine Aït-Sahalia and Jianqing Fan)
Studying the Leverage Effect Based on High-frequency Data

Abstract: We show how high-frequency data can be used to detect the leverage effect, and explain why extra caution has to be used when one studies the leverage effect based on the asymptotic results of the high-frequency volatility estimators.

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Andrew Lo (with Mark T. Mueller)
WARNING: Physics Envy May Be Hazardous To Your Wealth

The quantitative aspirations of economists and financial analysts have for many years been based on the belief that it should be possible to build models of economic systems---and financial markets in particular---that are as predictive as those in physics. While this perspective has led to a number of important breakthroughs in economics, "physics envy" has also created a false sense of mathematical precision in some cases. We speculate on the origins of physics envy, and then describe an alternate perspective of economic behavior based on a new taxonomy of uncertainty. We illustrate the relevance of this taxonomy with two concrete examples: the classical harmonic oscillator with some new twists that make physics look more like economics, and a quantitative equity market-neutral strategy. We conclude by offering a new interpretation of tail events, proposing an "uncertainty checklist" with which our taxonomy can be implemented, and considering the role that quants played in the current financial crisis.

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Cecilia Mancini
Test for the Presence of Noise in Observed Data

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Gael Martin (with Brendan McCabe and David Harris)
Optimal Probabilistic Forecasts for Counts

Abstract: Optimal probabilistic forecasts of integer-valued random variables are derived. The optimality is achieved by estimating the forecast distribution nonparametrically over a given broad model class and proving asymptotic efficiency in that setting. The ideas are demonstrated within the context of the integer autoregressive class of models, which is a suitable class for any count data that can be interpreted as a queue, stock, birth and death process or branching process. The theoretical proofs of asymptotic optimality are supplemented by simulation results which demonstrate the overall superiority of the nonparametric method relative to a misspecified parametric maximum likelihood estimator, in large but finite samples. The method is applied to counts of wage claim benefits, stock market iceberg orders and civilian deaths in Iraq, with bootstrap methods used to quantify sampling variation in the estimated forecast distributions. Illustration of the method using the stock market order data will be emphasized in the presentation.

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Nour Meddahi (with Peter Christoffersen, Bruno Feunou, and Kris Jacobs)
The Economic Value of Realized Volatility

Abstract: Many existing studies have documented that daily realized volatility estimates based on intraday data provide volatility forecasts that are superior to forecasts constructed from daily data only. Some studies also find that density forecasts based on realized volatility are superior to those based on daily data. We investigate whether these forecasting improvements translate into economic value added. In order to address this question we develop a new class of discrete-time option valuation models that use daily returns as well as realized volatility, and that nest the daily Heston and Nandi (2000) GARCH model as a special case. We derive closed-form option valuation formulas and we assess the option valuation properties using S&P500 return and option data. We find that realized volatility reduces the pricing errors of the benchmark model significantly across moneyness, maturity and volatility levels.

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Joon Park
Asymptotic Theory of Maximum Likelihood Estimator for Diffusion Model

Abstract: We derive the asymptotics of the maximum likelihood estimators for diffusion models. The models considered in the paper are very general, including both stationary and nonstationary diffusions. For such a broad class of diffusion models, we establish the consistency and find the limit distributions of the exact maximum likelihood estimator, and also the quasi and approximate maximum likelihood estimators based on various versions of approximated transition densities. Our asymptotics are two dimensional, allowing the sampling interval to decrease as well as the time span of sample to increase. The two dimensional asymptotics provide a unifying framework for the development of statistical theories for the stationary and nonstationary diffusion model. More importantly, they yield the asymptotic expansions that are very useful to analyze the exact, quasi and approximate maximum likelihood estimators of the diffusion models, if the samples are collected at high frequency intervals over modest lengths of sampling horizons as in the case of many practical applications.

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Eckhard Platen
Empirical Properties of a Well Diversified Global Stock Index

Abstract: Most of the papers that study the distributional and fractal properties of financial instruments focus on stock prices or exchange rates. This leads typically to mixed results concerning the distributions of log-returns and some multi-fractal properties of exchange rates, stock prices, and regional indices. It will be suggested to use a very well diversified world stock index in various denominations as the main object of empirical analysis. Such index has been formed using daily and intraday data. It aggregates, in principle, the non-diversifiable risk of the stock market. Compared to other global stock market indices it has extremely low volatility and, thus, a high signal to noise ratio when denominated in a currency. Furthermore, by diversification such an index can be shown to approximate the growth optimal portfolio or numeraire portfolio, which is the central object of the, so called benchmark approach. The paper will demonstrate that the above mentioned diversified index is an ideal object for studying the statistical properties of given securities. For instance, when denominating the savings account of a currency in units of this diversified global world index, one observes the movements of the currency against the entire market. This provides a practically undisturbed observation of the currency dynamics against the whole of the market. In this manner, one can conveniently disentangle, e.g., the superposition of the characteristic properties of the two currencies generating a given exchange rate. The exchange rate is then obtained as the ratio of the two currency denominations of the benchmark.

The proposed benchmark approach to the empirical analysis of financial data allows one to establish remarkable stylized facts. For instance, the log-returns of a well diversified global stock index, when denominated in a currency, are with high significance Student t distributed with about four degrees of freedom. The repeatedly documented multi-fractal appearance of financial time series turns out to be only very weak when analysed for a well diversified global index. The Hurst exponent of the observed mono-fractal behavior assumes typical values between 0.55 and 0.65. Accordingly, the quadratic variation vanishes asymptotically when reducing the observation time step size. These results can be contrasted with the mixed findings on empirical properties of FX rates or stock prices. A range of further empirical facts can be expected to be identifiable when using a well diversified index in the denomination of a given security as the object of study.

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Eric Renault (with Thijs van der Heijdeny and Bas J.M. Werker)
A Structural Autoregressive Conditional Duration Model

Abstract: We propose a structural model for durations between events and associated marks. Our model is structural in the sense that both durations and marks are generated by an underlying Brownian motion. In particular, we model the durations as the successive passage times of this Brownian motion relative to in itself random boundaries. Additional Brownian motions serve as processes generating the marks, whose conditional distribution is a mixture of normals. Multivariate Brownian motions allow us to incorporate a vector of marks combined with a single duration generating process. Our model embeds in particular the standard autoregressive conditional duration model. Applied to high-frequency financial data, we derive the conditional distributions of the durations and the vector of price changes. A first empirical illustration, using transaction level data on a NYSE.

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Roberto Renò
Nonparametric Leverage Effects

Abstract: Vast empirical evidence points to the existence of a negative correlation, named "leverage effect," between shocks in volatility and shocks in 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 volatility, or both. Leverage is defined as a flexible function of the state of the firm, as summarized by the spot volatility 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 volatility levels.

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Paul Schneider (with Damir Filipovic and Eberhard Mayerhofer)
Transition Density Approximations for Multivariate Affine Jump Diffusion Processes

Abstract: We develop closed-form transition density approximations for multivariate affine jump diffusion processes using polynomial expansion techniques. The approximations converge in L2 for a fixed time horizon, provided that the processes with support on R+ satisfy non-attainment conditions. Empirical applications in portfolio credit risk, likelihood inference, and option pricing using the (integrated) square-root jump diffusion, and Heston's model indicate that the approximations perform very accurately. The expansions are extremely fast to evaluate and numerically stable compared to Fourier inversion.

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Osnat Stramer
Bayesian Inference of Discretely Sampled Markov Processes with Closed-Form Likelihood Expansions

Abstract: The closed-form (CF) likelihood approximation of Ait-Sahalia (2002, 2008) is commonly used in financial modeling. Bayesian inference requires the use of MCMC and the (unnormalized) CF likelihood can become inaccurate when the parameters are far from the MLE; samplers can become stuck when (typically) in the tails of the posterior distribution. Auxiliary variables have been used in conjunction with MCMC to address intractable normalizers (see Moller et al. (2006)), but choosing such variables is not trivial. We propose a MCMC algorithm that addresses the intractable normalizers in the CF likelihood which 1) is easy to implement, 2) yields a sampler with the correct limiting distribution, and 3) greatly increases the stability of the sampler compared to using the unnormalized CF likelihood in a standard Metropolis-Hastings algorithm. The efficacy of our approach is demonstrated in a simulation study of the Cox-Ingersoll-Ross (CIR) and Heston models, and is applied to two well known real-world datasets.

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George Tauchen (with Viktor Todorov)
The Realized Laplace Transform of Volatility

Abstract: We introduce a new measure constructed from high-frequency financial data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform of the latent stochastic volatility process over a given interval of time. When a long span of data is used, i.e., under joint long-span and fill-in asymptotics, it is an estimate of the volatility Laplace transform. The asymptotic behavior of the statistic depends on the small scale behavior of the driving martingale. We derive the asymptotics both in the case when the latter is known and when it needs to be inferred from the data. When the underlying process is a jump-diffusion our statistic is robust to jumps and when the process is pure-jump it is robust to presence of less active jumps. We apply our results to simulated and real financial data.

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Allan Timmerman
What is the Shape of the Risk-Return Relation?

Abstract: Using a flexible modeling approach that avoids imposing restrictive parametric assumptions, we find evidence of a clear non-monotonic relation between conditional volatility and expected stock returns: At low-to-medium levels of conditional volatility there is a positive trade-off between risk and expected returns, but this relationship gets inverted at high levels of conditional volatility as observed during the recent financial crisis. We next propose a novel measure of risk based on the conditional covariance between daily observations on a broad economic activity index and stock returns. Using this measure, we find clear evidence of a monotonically increasing risk-return trade-off. Our finding that the conditional volatility-expected return relation is non-monotonic, while the conditional covariance-expected return is monotonically rising helps explain why some empirical studies find a negative risk-return relation, while others find a positive risk-return trade-off and also suggests that a positive risk-return relationship can be established once a better measure of risk is used.

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Viktor Todorov (with Tim Bollerslev)
Estimation of Jump Tails

Abstract: We consider the problem of estimating the jump tails of an Ito semimartingale. The new estimation strategy developed in the paper is based on in-fill asymptotic arguments and a method-of-moments type procedure that explicitly utilizes the weak assumption of regular variation in the jump tails. On implementing the new procedures with actual high-frequency data for the aggregate market portfolio, we find strong evidence for richer and much more complex dynamic dependencies in the jump tails than hitherto considered in the literature.

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Giovanni Urga (with Ana-Maria Dumitru)
Identifying Jumps in Financial Assets with a Comparison between Nonparametric Jump Tests

Abstract: We perform a comprehensive Monte Carlo comparison between five procedures available in the literature to detect jumps in financial assets-Andersen et al. (2007), Lee and Mykland (2008), the Aït-Sahalia and Jacod (2008), the Barndorff-Nielsen and Shephard (2006a), the Jiang and Oomen (2008), and the Podolskij and Ziggel (2008). We evaluate size and power properties of the procedures under alternative sampling frequencies, levels of volatility, persistence in volatility, degree of contamination with microstructure noise, jump size and intensity. Using high frequency data for US Treasury bonds, we compare the performance of the alternative tests. Though overall the best performance is showed by the Lee and Mykland (2008) and Andersen et al. (2007) intraday procedures, however we show the validity to use reunion and intersection across procedures and across sampling frequencies for potential users of the tests to minimise spurious jump detection.

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Rossen Valkanov
Robust Measure of Time-Varying Skewness at Short and Long Horizons

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Liuren Wu (with Laurent Calvet and Adlai Fisher)
A Multifrequency Theory of the Interest Rate Term Structure

Abstract: By applying power law scaling, we propose an extremely parsimonious modeling framework to capture the interest rate term structure movements across all frequencies. We estimate a model with merely five parameters on the U.S. dollar LIBOR from one to 12 months and swap rates from two to 30 years. Due to the extreme parsimony, the five model parameters are estimated with strong statistical significance. Meanwhile, by capturing movements of all frequencies, the model prices all interest rate term structures to near perfection, with the mean absolute pricing error averaging around half a basis point. The model also generates much better out-of-sample forecasting performance on the short-term interest rates than either the random walk assumption or an autoregressive specification. Further specification analysis shows that the power law scaling assumption matches well with data.

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Dacheng Xiu
Quasi-Maximum Likelihood Estimation of Volatility with High Frequency Data

Abstract: This paper investigates the properties of the well-known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise, by extending the classic asymptotic results of quasi-maximum likelihood estimation. When trying to estimate the integrated volatility and the variance of noise, this parametric approach remains consistent, efficient and robust as a quasi-estimator under misspecified assumptions. Moreover, it shares the model-free feature with nonparametric alternatives, for instance realized kernels, while being advantageous over them in terms of finite sample performance. Comparisons with a variety of implementations of the Tukey-Hanning 2 kernel are provided using Monte Carlo simulations, and an empirical study with the Euro/US Dollar future illustrates its application in practice.

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Jialin Yu
Option Value of Cash

Abstract: This paper presents a dynamic model of heterogeneous beliefs (where investors agree to disagree) to study the positive price-volume correlation during a housing downturn. It shows: (i) beliefs may diverge, which prevents some pessimists from buying; (ii) in the case that beliefs cross (i.e., buyers become more optimistic than the sellers), home sales occur but are delayed due to the buyers' option to sell cash higher (using house as numeraire) if the downturn worsens. Such option to wait also has implications for the velocity of money during deflation, troubled assets since 2007, takeover bids, IPO waves, and fire sales.

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Zhibiao Zhao
Nonparametric Model Validations for Hidden Markov Models with Applications in Financial Econometrics

Abstract: Nonparametric model validation under dependence has been an important yet difficult problem. We address this problem for hidden Markov models with partially observable variables and unobservable or hidden states. We achieve this goal by constructing nonparametric simultaneous confidence envelope for transition density function of the observable variables and checking whether the parametrically implied density estimate is entirely contained within such an envelope. Our specification test procedure is motivated by a functional connection between the transition density of the observable variables and the Markov transition kernel of the unobservable states. We show that our approach is applicable for a variety of models widely used in financial econometrics, including continuous-time diffusion models, hyperbolic Levy motions, stochastic volatility models, nonlinear time series models, multivariate stochastic regression models, and models with measurement errors among others. The finite sample performance of the proposed method is studied through simulations.

 

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