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on Econometric Time Series |
| By: | Thomas Busch; Thomas Busch; Bent Jesper Christensen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We study the forecasting of future realized volatility in the stock, bond, and for- eign exchange markets, as well as the continuous sample path and jump components of this, from variables in the information set, including implied volatility backed out from option prices. Recent nonparametric statistical techniques of Barndor¤-Nielsen & Shephard (2004, 2006) are used to separate realized volatility into its continuous and jump components, which enhances forecasting performance, as shown by Andersen, Bollerslev & Diebold (2005). We generalize the heterogeneous autoregressive (HAR) model of Corsi (2004) to include implied volatility as an additional regressor, and to the separate forecasting of the realized components. We also introduce a new vector HAR (VecHAR) model for the resulting simultaneous system, controlling for possible endogeneity issues in the forecasting equations. We show that implied volatility con- tains incremental information about future volatility relative to both continuous and jump components of past realized volatility. Indeed, in the foreign exchange market, implied volatility completely subsumes the information content of daily, weekly, and monthly realized volatility measures, when forecasting future realized volatility or its continuous component. In addition, implied volatility is an unbiased forecast of future realized volatility in the foreign exchange and stock markets. Perhaps surprisingly, the jump component of realized return volatility is, to some extent, predictable, and options appear to be calibrated to incorporate information about future jumps in all three markets. |
| Keywords: | Bipower variation, HAR, Heterogeneous Autoregressive Model, implied volatility, jumps, options, realized volatility, VecHAR, volatility forecasting |
| JEL: | C22 C32 F31 G1 |
| Date: | 2007–06–06 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-09&r=ets |
| By: | Bent Jesper Christensen; Morten Ørregaard Nielsen; Jie Zhu (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We extend the fractionally integrated exponential GARCH (FIEGARCH) model for daily stock return data with long memory in return volatility of Bollerslev and Mikkelsen (1996) by introducing a possible volatility-in-mean effect. To avoid that the long memory property of volatility carries over to returns, we consider a filtered FIEGARCH-in-mean (FIEGARCH-M) effect in the return equation. The filtering of the volatility-in-mean component thus allows the co-existence of long memory in volatility and short memory in returns. We present an application to the S&P 500 index which documents the empirical relevance of our model. |
| Keywords: | FIEGARCH, financial leverage, GARCH, long memory, risk-return tradeoff, stock returns, volatility feedback |
| JEL: | C22 |
| Date: | 2007–06–12 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-10&r=ets |
| By: | Michael Jansson (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper derives asymptotic power envelopes for tests of the unit root hypothesis in a zero-mean AR(1) model. The power envelopes are derived using the limits of experiments approach and are semiparametric in the sense that the underlying error distribution is treated as an unknown infinitedimensional nuisance parameter. Adaptation is shown to be possible when the error distribution is known to be symmetric and to be impossible when the error distribution is unrestricted. In the latter case, two conceptually distinct approaches to nuisance parameter elimination are employed in the derivation of the semiparametric power bounds. One of these bounds, derived under an invariance restriction, is shown by example to be sharp, while the other, derived under a similarity restriction, is conjectured not to be globally attainable. |
| Keywords: | Unit root testing, semiparametric efficiency |
| JEL: | C14 C22 |
| Date: | 2007–06–25 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-12&r=ets |
| By: | Tim Bollerslev; Michael Gibson; Hao Zhou (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper proposes a method for constructing a volatility risk premium, or investor risk aversion, index. The method is intuitive and simple to implement, relying on the sample moments of the recently popularized model-free realized and option-implied volatility measures. A small-scale Monte Carlo experiment confirms that the procedure works well in practice. Implementing the procedure with actual S&P500 option-implied volatilities and high-frequency five-minute-based realized volatilities indicates significant temporal dependencies in the estimated stochastic volatility risk premium, which we in turn relate to a set of macro-finance state variables. We also find that the extracted volatility risk premium helps predict future stock market returns. |
| Keywords: | Stochastic Volatility Risk Premium, Model-Free Implied Volatility, Model-Free Realized Volatility, Black-Scholes, GMM Estimation, Return Predictability |
| JEL: | G12 G13 C51 C52 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-16&r=ets |
| By: | Tim Bollerslev; Hao Zhou (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We find that the difference between implied and realized variation, or the variance risk premium, is able to explain more than fifteen percent of the ex-post time series variation in quarterly excess returns on the market portfolio over the 1990 to 2005 sample period, with high (low) premia predicting high (low) future returns. The magnitude of the return predictability of the variance risk premium easily dominates that afforded by standard predictor variables like the P/E ratio, the dividend yield, the default spread, and the consumption-wealth ratio (CAY). Moreover, combining the variance risk premium with the P/E ratio results in an R2 for the quarterly returns of more than twenty-five percent. The results depend crucially on the use of “model-free”, as opposed to standard Black-Scholes, implied variances, and realized variances constructed from high-frequency intraday, as opposed to daily, data. Our findings suggest that temporal variation in both risk-aversion and volatility-risk play an important role in determining stock market returns. |
| Keywords: | Return Predictability, Implied Variance, Realized Variance, Equity Risk Premium, Variance Risk Premium, Time-Varying Risk Aversion |
| JEL: | G12 G14 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-17&r=ets |
| By: | Tim Bollerslev; Tzuo Hann Law; George Tauchen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We test for price discontinuities, or jumps, in a panel of high-frequency intraday returns for forty large-cap stocks and an equiweighted index from these same stocks. Jumps are naturally classified into two types: common and idiosyncratic. Common jumps affect all stocks, albeit to varying degrees, while idiosyncratic jumps are stock-specific. Despite the fact that each of the stocks has a of about unity with respect to the index, common jumps are virtually never detected in the individual stocks. This is truly puzzling, as an index can jump only if one or more of its components jump. To resolve this puzzle, we propose a new test for cojumps. Using this new test we find strong evidence for many modest-sized common jumps that simply pass through the standard jump detection statistic, while they appear highly significant in the cross section based on the new cojump identification scheme. Our results are further corroborated by a striking within-day pattern in the non-diversifiable cojumps. |
| Keywords: | risk, diversification |
| JEL: | C12 C32 C33 G12 G14 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-19&r=ets |
| By: | Torben G. Andersen; Tim Bollerslev; Francis X. Diebold; Clara Vega (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | Using a unique high-frequency futures dataset, we characterize the response of U.S., German and British stock, bond and foreign exchange markets to real-time U.S. macroeconomic news. We find that news produces conditional mean jumps, hence high-frequency stock, bond and exchange rate dynamics are linked to fundamentals. Equity markets, moreover, react differently to news depending on the stage of the business cycle, which explains the low correlation between stock and bond returns when averaged over the cycle. Hence our results qualify earlier work suggesting that bond markets react most strongly to macroeconomic news, in particular, when conditioning on the state of the economy, the equity and foreign exchange markets appear equally responsive. Finally, we also document important contemporaneous links across all markets and countries, even after controlling for the effects of macroeconomic news. |
| Keywords: | Asset Pricing, Macroeconomic News Announcements, Financial Market Linkages, Market Microstructure, High-Frequency Data, Survey Data, Asset Return Volatility, Forecasting |
| JEL: | F3 F4 G1 C5 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-20&r=ets |
| By: | Torben G. Andersen; Tim Bollerslev; Per Houmann Frederiksen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We provide an empirical framework for assessing the distributional properties of daily specu- lative returns within the context of the continuous-time modeling paradigm traditionally used in asset pricing finance. Our approach builds directly on recently developed realized variation measures and non-parametric jump detection statistics constructed from high-frequency intra- day data. A sequence of relatively simple-to-implement moment-based tests involving various transforms of the daily returns speak directly to the import of different features of the under- lying continuous-time processes that might have generated the data. As such, the tests may serve as a useful diagnostic tool in the specification of empirically more realistic asset pricing models. Our results are also directly related to the popular mixture-of-distributions hypoth- esis and the role of the corresponding latent information arrival process. On applying our sequential test procedure to the thirty individual stocks in the Dow Jones Industrial Average index, the data suggest that it is important to allow for both time-varying diffusive volatility, jumps, and leverage effects in order to satisfactorily describe the daily stock price dynamics. At a broader level, the empirical results also illustrate how the realized variation measures and high-frequency sampling schemes may be used in eliciting important distributional features and asset pricing implications more generally. |
| Keywords: | Return distributions, continuous-time models, mixture-of-distributions hypothesis, financial-time sampling, high-frequency data, volatility signature plots, realized volatilities, jumps, leverage and volatility feedback effects |
| JEL: | C1 G1 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-21&r=ets |
| By: | Tim Bollerslev; Uta Kretschmer; Christian Pigorsch; George Tauchen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We develop an empirically highly accurate discrete-time daily stochastic volatility model that explicitly distinguishes between the jump and continuoustime components of price movements using nonparametric realized variation and Bipower variation measures constructed from high-frequency intraday data. The model setup allows us to directly assess the structural inter-dependencies among the shocks to returns and the two different volatility components. The model estimates suggest that the leverage effect, or asymmetry between returns and volatility, works primarily through the continuous volatility component. The excellent fit of the model makes it an ideal candidate for an easyto- implement auxiliary model in the context of indirect estimation of empirically more realistic continuous-time jump diffusion and L´evy-driven stochastic volatility models, effectively incorporating the interdaily dependencies inherent in the high-frequency intraday data. |
| Keywords: | Realized volatility, Bipower variation, Jumps, Leverage effect, Simultaneous equation model |
| JEL: | C1 C3 C5 G1 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-22&r=ets |
| By: | Torben G. Andersen; Oleg Bondarenko (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | The notion of model-free implied volatility (MFIV), constituting the basis for the highly publicized VIX volatility index, can be hard to measure with accuracy due to the lack of precise prices for options with strikes in the tails of the return distribution. This is reflected in practice as the VIX index is computed through a tail-truncation which renders it more compatible with the related concept of corridor implied volatility (CIV). We provide a comprehensive derivation of the CIV measure and relate it to MFIV under general assumptions. In addition, we price the various volatility contracts, and hence estimate the corresponding volatility measures, under the standard Black-Scholes model. Finally, we undertake the first empirical exploration of the CIV measures in the literature. Our results indicate that the measure can help us refine and systematize the information embedded in the derivatives markets. As such, the CIV measure may serve as a tool to facilitate empirical analysis of both volatility forecasting and volatility risk pricing across distinct future states of the world for diverse asset categories and time horizons. |
| Keywords: | Model-Free Implied Volatility, Corridor Implied Volatility, Realized Volatility, VIX, Volatility Forecasting, Risk-Neutral Density |
| JEL: | C53 G12 G13 |
| Date: | 2007–09–17 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-24&r=ets |
| By: | Mark Podolskij; Daniel Ziggel (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We propose a new test for the parametric form of the volatility function in continuous time diffusion models of the type dXt = a(t,Xt)dt + s(t,Xt)dWt. Our approach involves a range-based estimation of the integrated volatility and the integrated quarticity, which are used to construct the test statistic. Under rather weak assumptions on the drift and volatility we prove weak convergence of the test statistic to a centered mixed Gaussian distribution. As a consequence we obtain a test, which is consistent for any fixed alternative. Moreover, we present a parametric bootstrap procedure which provides a better approximation of the distribution of the test statistic. Finally, it is demonstrated by means of Monte Carlo study that the range-based test is more powerful than the return-based test when comparing at the same sampling frequency. |
| Keywords: | Bipower Variation, Central Limit Theorem, Diffusion Models, Goodness-Of- Fit Testing, High-Frequency Data, Integrated Volatility, Range-Based Bipower Variation, Semimartingale Theory |
| JEL: | C12 C14 |
| Date: | 2007–09–19 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-26&r=ets |
| By: | Mark Podolskij; Mathias Vetter (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate n-1/4. Moreover, we construct estimates which are robust to finite activity jumps. |
| Keywords: | Bipower Variation, Central Limit Theorem, Finite Activity Jumps, High-Frequency Data, Integrated Volatility, Microstructure Noise, Semimartingale Theory, Subsampling |
| JEL: | C10 C13 C14 |
| Date: | 2007–09–19 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-27&r=ets |
| By: | Michael Sørensen; Julie Lyng Forman (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating func- tions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-likelihood, and non- linear weighted least squares estimation too, and it is discussed how explicit likeli- hood or approximate likelihood inference is possible for the Pearson diffusions. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Special attention is given to a skew t-type distribution. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. The analyti- cal tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions, and stochastic volatility models with Pearson volatility process. For the non-Markov models explicit optimal prediction based estimating functions are found and shown to yield consistent and asymptotically normal estimators. |
| Keywords: | eigenfunction, ergodic diffusion, integrated diffusion, martingale estimating function, likelihood inference, mixing, optimal estimating function, Pearson system, prediction based estimating function, quasi likelihood, spectral methods,stochastic differential equation, stochastic volatility |
| JEL: | C22 C51 |
| Date: | 2007–09–27 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-28&r=ets |
| By: | Søren Johansen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | An analysis of some identification problems in the cointegrated VAR is given. We give a new criteria for identification by linear restrictions on indi- vidual relations which is equivalent to the rank condition. We compare the asymptotic distribution of the estimators of alpha and beta when they are identified by linear restrictions on alpha and when they are identified by linear restrictions on alpha in which case a component of beta is asymptotically Gaussian. Finally we discuss identification of shocks by introducing the contemporaneous and permanent effect of a shock and the distinction between permanent and transi- tory shocks, which allows one to identify permanent shocks from the long-run variance and transitory shocks from the short-run variance |
| Keywords: | Identfication, cointegration, common trends |
| JEL: | C32 |
| Date: | 2007–11–07 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-32&r=ets |
| By: | Søren Johansen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper discusses model based inference in an autoregressive model for fractional processes based on the Gaussian likelihood. We consider the likelihood and its derivatives as stochastic processes in the parameters, and prove that they converge in distribution when the errors are i.i.d. with suitable moment conditions and the initial values are bounded. We use this to prove existence and consistency of the local likelihood estimator, and to .nd the asymptotic distribution of the estimators and the likelihood ratio test of the associated fractional unit root hypothesis, which contains the fractional Brownian motion of type II. |
| Keywords: | Dickey-Fuller test, fractional unit root, likelihood inference |
| JEL: | C22 |
| Date: | 2007–11–07 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-33&r=ets |
| By: | Søren Johansen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | Yule (1926) introduced the concept of spurious or nonsense correlation, and showed by simulation that for some nonstationary processes, that the empirical correlations seem not to converge in probability even if the processes were inde- pendent. This was later discussed by Granger and Newbold (1974), and Phillips (1986) found the limit distributions. We propose to distinguish between empirical and population correlation coefficients and show in a bivariate autoregressive model for nonstationary variables that the empirical correlation and regression coe¢ cients do not converge to the relevant population values, due to the trending nature of the data. We conclude by giving a simple cointegration analysis of two interests. The analysis illustrates that much more insight can be gained about the dynamic behavior of the nonstationary variables then simply by calculating a correlation coefficient. |
| JEL: | C22 |
| Date: | 2007–11–06 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-35&r=ets |
| By: | Dennis Kristensen; Anders Rahbek (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We consider a class of vector nonlinear error correction models where the transfer function (or loadings) of the stationary relation- ships is nonlinear. This includes in particular the smooth transition models. A general representation theorem is given which establishes the dynamic properties of the process in terms of stochastic and deter- ministic trends as well as stationary components. In particular, the behaviour of the cointegrating relations is described in terms of geo- metric ergodicity. Despite the fact that no deterministic terms are included, the process will have both stochastic trends and a linear trend in general. Gaussian likelihood-based estimators are considered for the long- run cointegration parameters, and the short-run parameters. Asymp- totic theory is provided for these and it is discussed to what extend asymptotic normality and mixed normaity can be found. A simulation study reveals that cointegration vectors and the shape of the adjust- ment are quite accurately estimated by maximum likelihood, while at the same time there is very little information about some of the individual parameters entering the adjustment function. |
| Date: | 2007–11–19 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-38&r=ets |
| By: | Søren Johansen; Anders Rygh Swensen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We interpret the linear relations from exact rational expectations models as restrictions on the parameters of the statistical model called the cointegrated vector autoregressive model for non-stationary variables. We then show how reduced rank regression, Anderson (1951), plays an important role in the calculation of maximum likelihood estimation of the restricted parameters. |
| Keywords: | Exact rational expectations, Cointegrated VAR model, Reduced rank regression |
| JEL: | C32 |
| Date: | 2007–12–04 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-41&r=ets |
| By: | Ole E. Barndorff-Nielsen; José Manuel Corcuera; Mark Podolskij (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We develop the asymptotic theory for the realised power variation of the processes X = f • G, where G is a Gaussian process with stationary increments. More specifically, under some mild assumptions on the variance function of the increments of G and certain regularity condition on the path of the process f we prove the convergence in probability for the properly normalised realised power variation. Moreover, under a further assumption on the H¨older index of the path of f, we show an associated stable central limit theorem. The main tool is a general central limit theorem, due essentially to Hu & Nualart (2005), Nualart & Peccati (2005) and Peccati & Tudor (2005), for sequences of random variables which admit a chaos representation. |
| Keywords: | Central Limit Theorem, Chaos Expansion, Gaussian Processes, High-Frequency Data, Multiple Wiener-Itô Integrals, Power Variation |
| JEL: | C10 C13 C14 |
| Date: | 2007–12–07 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-42&r=ets |
| By: | Jean Jacod; Yingying Li; Per A. Mykland; Mark Podolskij; Mathias Vetter (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper presents a generalized pre-averaging approach for estimating the integrated volatility. This approach also provides consistent estimators of other powers of volatility – in particular, it gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. We show that our approach, which possess an intuitive transparency, can generate rate optimal estimators (with convergence rate n-1/4). |
| Keywords: | consistency, continuity, discrete observation, Itô process, leverage effect, pre-averaging, quarticity, realized volatility, stable convergence |
| JEL: | C10 C13 C14 |
| Date: | 2007–12–10 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-43&r=ets |
| By: | Charles S. Bos; Siem Jan Koopman; Marius Ooms (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We investigate changes in the time series characteristics of postwar U.S. inflation. In a model-based analysis the conditional mean of inflation is specified by a long memory autoregressive fractionally integrated moving average process and the conditional variance is modelled by a stochastic volatility process. We develop a Monte Carlo maximum likelihood method to obtain efficient estimates of the parameters using a monthly dataset of core inflation for which we consider different subsamples of varying size. Based on the new modelling framework and the associated estimation technique, we find remarkable changes in the variance, in the order of integration, in the short memory characteristics and in the volatility of volatility. |
| Keywords: | Time varying parameters, Importance sampling, Monte Carlo simulation, Stochastic Volatility, Fractional Integration |
| JEL: | C15 C32 C51 E23 E31 |
| Date: | 2007–12–21 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-44&r=ets |
| By: | James Davidson; Nigar Hashimzade (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper considers the asymptotic distribution of the covariance of a nonstationary frac- tionally integrated process with the stationary increments of another such process - possibly, itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analysed in a previous paper, and the construction derived from moving average representations in the time domain. The limiting integrals are shown to be expressible in terms of functionals of Itô integrals with respect to two distinct Brownian motions. Their mean is nonetheless shown to match that of the harmonic rep- resentation, and they satisfy the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulae are valid for the full range of the long memory parameters, and extend to non-Gaussian processes. |
| Keywords: | Stochastic integral, weak convergence, fractional Brownian motion |
| JEL: | C22 C32 |
| Date: | 2007–12–21 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-45&r=ets |
| By: | Michael Sørensen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | A general theory of efficient estimation for ergodic diffusions sampled at high fre- quency is presented. High frequency sampling is now possible in many applications, in particular in finance. The theory is formulated in term of approximate martingale estimating functions and covers a large class of estimators including most of the pre- viously proposed estimators for diffusion processes, for instance GMM-estimators and the maximum likelihood estimator. Simple conditions are given that ensure rate optimality, where estimators of parameters in the diffusion coefficient converge faster than estimators of parameters in the drift coefficient, and for efficiency. The conditions turn out to be equal to those implying small delta-optimality in the sense of Jacobsen and thus gives an interpretation of this concept in terms of classical sta- tistical concepts. Optimal martingale estimating functions in the sense of Godambe and Heyde are shown to be give rate optimal and efficient estimators under weak conditions. |
| Keywords: | Approximate martingale estimating functions, discrete time observation of a diffusion, efficiency, Euler approximation, generalized method of moments, optimal estimating function, optimal rate, small delta-optimality |
| JEL: | C22 C32 |
| Date: | 2008–01–22 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-46&r=ets |
| By: | Peter Reinhard Hansen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We derive an identity for the determinant of a product involving non-squared matrices. The identity can be used to derive the maximum likelihood estimator in reduced-rank regres- sions with Gaussian innovations. Furthermore, the identity sheds light on the structure of the estimation problem that arises when the reduced-rank parameters are subject to additional constraints. |
| Keywords: | Determinant Identity, Reduced Rank Regression, Least Squares |
| JEL: | C3 C32 |
| Date: | 2008–01–15 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-02&r=ets |
| By: | Søren Johansen; Katarina Juselius; Roman Frydberg; Michael Goldberg (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper discusses a number of likelihood ratio tests on long-run relations and common trends in the I(2) model and provide new results on the test of overidentifying restrictions on beta’xt and the asymptotic variance for the stochas- tic trends parameters, alpha 1: How to specify deterministic components in the I(2) model is discussed at some length. Model specification and tests are illustrated with an empirical analysis of long and persistent swings in the foreign exchange market between Germany and USA. The data analyzed consist of nominal exchange rates, relative prices, US in.ation rate, two long-term interest rates and two short-term interest rates over the 1975-1999 period. One important aim of the paper is to demonstrate that by structuring the data with the help of the I(2) model one can achieve a better understanding of the empirical regularities underlying the persistent swings in nominal exchange rates, typical in periods of floating exchange rates. |
| Keywords: | PPP puzzle, Forward premium puzzle, cointegrated VAR, likelihood inference |
| JEL: | C32 C52 F41 |
| Date: | 2008–01–15 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-03&r=ets |
| By: | Annastiina Silvennoinen; Timo Teräsvirta (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | In this paper we propose a multivariate GARCH model with a time-varying conditional correlation structure. The new Double Smooth Transition Conditional Correlation GARCH model extends the Smooth Transition Conditional Correlation GARCH model of Silvennoinen and Ter¨asvirta (2005) by including another variable according to which the correlations change smoothly between states of constant correlations. A Lagrange multiplier test is derived to test the constancy of correlations against the DSTCC–GARCH model, and another one to test for another transition in the STCC–GARCH framework. In addition, other specification tests, with the aim of aiding the model building procedure, are considered. Analytical expressions for the test statistics and the required derivatives are provided. The model is applied to a selection of world stock indices, and it is found that time is an important factor affecting correlations between them. |
| Keywords: | Multivariate GARCH, Constant conditional correlation, Dynamic conditional correlation, Return comovement, Variable correlation GARCH model, Volatility model evaluation |
| JEL: | C12 C32 C51 C52 G1 |
| Date: | 2008–01–28 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-05&r=ets |
| By: | Annastiina Silvennoinen; Timo Teräsvirta (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This article contains a review of multivariate GARCH models. Most common GARCH models are presented and their properties considered. This also includes nonparametric and semiparametric models. Existing specification and misspecification tests are discussed. Finally, there is an empirical example in which several multivariate GARCH models are fitted to the same data set and the results compared. |
| Keywords: | Multivariate GARCH, Volatility |
| JEL: | C32 C51 C52 |
| Date: | 2008–01–28 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-06&r=ets |
| By: | Changli He; Annastiina Silvennoinen; Timo Teräsvirta (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | In this paper we consider the third-moment structure of a class of time series models. It is often argued that the marginal distribution of financial time series such as returns is skewed. Therefore it is of importance to know what properties a model should possess if it is to accommodate unconditional skewness. We consider modelling the unconditional mean and variance using models that respond nonlinearly or asymmetrically to shocks. We investigate the implications of these models on the third-moment structure of the marginal distribution as well as conditions under which the unconditional distribution exhibits skewness and nonzero third-order autocovariance structure. In this respect, an asymmetric or nonlinear specification of the conditional mean is found to be of greater importance than the properties of the conditional variance. Several examples are discussed and, whenever possible, explicit analytical expressions provided for all third-order moments and cross-moments. Finally, we introduce a new tool, the shock impact curve, for investigating the impact of shocks on the conditional mean squared error of return series. |
| Keywords: | Asymmetry, GARCH, Nonlinearity, Shock Impact Curve, Time series, Unconditional skewness |
| JEL: | C22 |
| Date: | 2008–01–28 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-07&r=ets |
| By: | Christina Amado; Timo Teräsvirta (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the conditional variance to have a smooth time-varying structure of either ad- ditive or multiplicative type. The suggested parameterizations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. A modelling strategy for these new time-varying parameter GARCH models is developed. It relies on a sequence of Lagrange multiplier tests, and the adequacy of the estimated models is investigated by Lagrange multiplier type misspecification tests. Finite-sample properties of these procedures and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance. |
| Keywords: | Conditional heteroskedasticity, Structural change, Lagrange multiplier test, Misspecification test, Nonlinear time series, Time-varying parameter model |
| JEL: | C12 C22 C51 C52 |
| Date: | 2008–01–28 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-08&r=ets |
| By: | Peter Christoffersen; Kris Dorion; Yintian Wang (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | Recent work by Engle and Lee (1999) shows that allowing for long-run and short-run components greatly enhances a GARCH model’s ability fit daily equity return dynamics. Using the risk-neutralization in Duan (1995), we assess the option valuation performance of the Engle-Lee model and compare it to the standard one-component GARCH(1,1) model. We also compare these non-affine GARCH models to one- and two- component models from the class of affine GARCH models developed in Heston and Nandi (2000). Using the option pricing methodology in Duan (1999), we then compare the four conditionally normal GARCH models to four conditionally non-normal versions. As in Hsieh and Ritchken (2005), we find that non-affine models dominate affine models both in terms of fitting return and in terms of option valuation. For the affine models we find strong evidence in favor of the component structure for both returns and options, but for the non-affine models the evidence is much less strong in option valuation. The evidence in favor of the non-normal models is strong when fitting daily returns, but the non-normal models do not provide much improvement when valuing options. |
| Keywords: | Volatility, Component Model, GARCH, Long Memory, Option Valuation, Affine, Normality |
| JEL: | C22 G13 |
| Date: | 2008–02–06 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-10&r=ets |
| By: | Jie Zhu (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We apply the fractionally integrated exponential GARCH with volatility-in-mean (FIEGARCH-M) model of Christensen, Nielsen & Zhu (2007) to estimate the risk premium after different crises occurred in major stock markets during the past two decades. The model allows keeping the long memory property in volatility and a filtered volatility-in-mean component is used as a proxy for the risk factor. The esti- mation results show that the 1987 stock market crash and September 11, 2001 attack have persistent effects on stock markets. A significant risk factor is found for both crises in most crisis-hit markets, and it is nonmonotic for different markets. Either volatility feedback or risk premium is a possible explanation for the risk factor. On the contrary, Asian financial crisis and other market-specific crises have no persistent impact on most markets. |
| Keywords: | FIEGARCH-M, international stock market crisis, 1987 stock market crash, dotcom bubble, Asian crisis, 9/11 attack, country-specific crisis |
| JEL: | C22 F36 G15 |
| Date: | 2008–03–05 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-16&r=ets |
| By: | Almut Veraart (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | Recent research has focused on modelling asset prices by Itô semimartingales. In such a modelling framework, the quadratic variation consists of a continuous and a jump component. This paper is about inference on the jump part of the quadratic variation, which can be estimated by the difference of realised variance and realised multipower variation. The main contribution of this paper is twofold. First, it provides a bivariate asymptotic limit theory for realised variance and realised multipower variation in the presence of jumps. Second, this paper presents new, consistent estimators for the jump part of the asymptotic variance of the estimation bias. Eventually, this leads to a feasible asymptotic theory which is applicable in practice. Finally, Monte Carlo studies reveal a good finite sample performance of the proposed feasible limit theory. |
| Keywords: | Quadratic variation, Itô semimartingale, stochastic volatility, jumps, realised variance, realised multipower variation, high–frequency data |
| JEL: | C13 C14 G10 G12 |
| Date: | 2008–03–31 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-17&r=ets |
| By: | Michael Sørensen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | A review is given of parametric estimation methods for discretely sampled mul- tivariate diffusion processes. The main focus is on estimating functions and asymp- totic results. Maximum likelihood estimation is briefly considered, but the emphasis is on computationally less demanding martingale estimating functions. Particular attention is given to explicit estimating functions. Results on both fixed frequency and high frequency asymptotics are given. When choosing among the many estima- tors available, guidance is provided by simple criteria for high frequency efficiency and rate optimality that are presented in the framework of approximate martingale estimating functions. |
| Keywords: | Asymptotic results, discrete time observation of a diffusion, efficiency, eigenfunctions, explicit inference, generalized method of moments, likelihood infer- ence, martingale estimating functions, high frequency asymptotics, Pearson diffu- sions. |
| JEL: | C22 C32 |
| Date: | 2008–04–04 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-18&r=ets |
| By: | Anne Péguin-Feissolle; Birgit Strikholm; Timo Teräsvirta (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | In this paper we propose a general method for testing the Granger noncausality hypothesis in stationary nonlinear models of unknown functional form. These tests are based on a Taylor expansion of the nonlinear model around a given point in the sample space. We study the performance of our tests by a Monte Carlo experiment and compare these to the most widely used linear test. Our tests appear to be well-sized and have reasonably good power properties. |
| Keywords: | Hypothesis testing, causality |
| JEL: | C22 C51 |
| Date: | 2008–04–25 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-19&r=ets |
| By: | Ole E. Barndorff-Nielsen; José Manuel Corcuera; Mark Podolskij; Jeannette H.C. Woerner (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati and others. |
| Keywords: | Bipower Variation, Central Limit Theorem, Chaos Expansion, Gaussian Processes, Multiple Wiener-Itô Integrals. |
| Date: | 2008–05–08 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-21&r=ets |
| By: | Mark Podolskij; Daniel Ziggel (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We propose a new test for the parametric form of the volatility function in continuous time diffusion models of the type dXt = a(t;Xt)dt + (t;Xt)dWt. Our approach involves a range-based estimation of the integrated volatility and the integrated quarticity, which are used to construct the test statistic. Under rather weak assumptions on the drift and volatility we prove weak convergence of the test statistic to a centered mixed Gaussian distribution. As a consequence we obtain a test, which is consistent for any fixed alternative. We also provide a test for neighborhood hypotheses. Moreover, we present a parametric bootstrap procedure which provides a better approximation of the distribution of the test statistic. Finally, it is demonstrated by means of Monte Carlo study that the range-based test is more powerful than the return-based test when comparing at the same sampling frequency. |
| Keywords: | Bipower Variation, Central Limit Theorem, Diffusion Models, Goodness-Of- Fit Testing, High-Frequency Data, Integrated Volatility, Range-Based Bipower Variation; Semimartingale Theory |
| JEL: | C12 C14 |
| Date: | 2008–05–14 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-22&r=ets |
| By: | Silja Kinnebrock; Mark Podolskij (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper introduces a new estimator to measure the ex-post covariation between high-frequency financial time series under market microstructure noise. We provide an asymptotic limit theory (including feasible central limit theorems) for standard methods such as regression, correlation analysis and covariance, for which we obtain the optimal rate of convergence. We demonstrate some positive semidefinite estimators of the covariation and construct a positive semidefinite estimator of the conditional covariance matrix in the central limit theorem. Furthermore, we indicate how the assumptions on the noise process can be relaxed and how our method can be applied to non-synchronous observations. We also present an empirical study of how high-frequency correlations, regressions and covariances change through time. |
| Keywords: | Central Limit Theorem, Diffusion Models, Market Microstructure Noise, Non-synchronous Trading, High-Frequency Data, Semimartingale Theory |
| Date: | 2008–05–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-23&r=ets |
| By: | Matias D. Cattaneo; Richard K. Crump; Michael Jansson (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper proposes (apparently) novel standard error formulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels and the standard errors are "robust" in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the finite sample coverage rates of confidence intervals constructed using the standard errors developed in this paper coincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths. |
| Keywords: | Semiparametric estimation, density-weighted average derivatives |
| JEL: | C14 C21 |
| Date: | 2008–05–20 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-24&r=ets |
| By: | Mark Podolskij; Mathias Vetter (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We consider a new class of estimators for volatility functionals in the setting of frequently observed Itô diffusions which are disturbed by i.i.d. noise. These statistics extend the approach of pre-averaging as a general method for the estimation of the integrated volatility in the presence of microstructure noise and are closely related to the original concept of bipower variation in the no-noise case. We show that this approach provides efficient estimators for a large class of integrated powers of volatility and prove the associated (stable) central limit theorems. In a more general Itô semimartingale framework this method can be used to define both estimators for the entire quadratic variation of the underlying process and jump-robust estimators which are consistent for various functionals of volatility. As a by-product we obtain a simple test for the presence of jumps in the underlying semimartingale. |
| Keywords: | Bipower Variation, Central Limit Theorem, High-Frequency Data, Microstructure Noise, Quadratic Variation, Semimartingale Theory, Test for Jumps |
| JEL: | C10 C13 C14 |
| Date: | 2008–05–26 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-25&r=ets |
| By: | Martin Møller Andreasen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | The presence of i) stochastic trends, ii) deterministic trends, and/or iii) stochastic volatil- ity in DSGE models may imply that the agents' objective functions attain infinite values. We say that such models do not have a valid micro foundation. The paper derives sufficient condi- tions which ensure that the objective functions of the households and the firms are finite even when various trends and stochastic volatility are included in a standard DSGE model. Based on these conditions we test the validity of the micro foundation in six DSGE models from the literature. The models of Justiniano & Primiceri (American Economic Review, forth- coming) and Fernández-Villaverde & Rubio-Ramírez (Review of Economic Studies, 2007) do not satisfy these sufficient conditions, or any other known set of conditions ensuring finite values for the objective functions. Thus, the validity of the micro foundation in these models remains to be established. |
| Keywords: | Deterministic trends, DSGE models, Error distributions, Moment generating functions, Stochastic trends, Stochastic volatility, Unit-roots |
| JEL: | E10 E30 |
| Date: | 2008–05–26 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-26&r=ets |
| By: | Frank S. Nielsen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper extends the local polynomial Whittle estimator of Andrews & Sun (2004) to fractionally integrated processes covering stationary and non-stationary regions. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the local polynomial Whittle estimator to the non-stationary region. By approximating the short-run component of the spectrum by a polynomial, instead of a constant, in a shrinking neighborhood of zero we alleviate some of the bias that the classical local Whittle estimators is prone to. A simulation study illustrates the performance of the proposed estimator compared to the classical local Whittle estimator and the local polynomial Whittle estimator. The empirical justification of the proposed estimator is shown through an analysis of credit spreads. |
| Keywords: | Bias reduction, fractional integration, local polynomial, local Whittle estimation, long memory. |
| JEL: | C22 |
| Date: | 2008–06–02 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-28&r=ets |
| By: | Per Frederiksen; Frank S. Nielsen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We propose a semiparametric local polynomial Whittle with noise (LPWN) estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the spectrum of the perturbation as well as that of the short-memory component of the signal by two separate polynomials. Furthermore, an empirical investigation of the 30 DJIA stocks shows that this estimator indicates stronger persistence in volatility than the standard local Whittle estimator. |
| Keywords: | Bias reduction, local Whittle, long memory, perturbed fractional process, semiparametric estimation, stochastic volatility |
| JEL: | C22 |
| Date: | 2008–06–09 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-29&r=ets |
| By: | Mika Meitz; Pentti Saikkonen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a functional coefficient autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors. |
| Keywords: | AR-GARCH, asymptotic normality, consistency, nonlinear time series, quasi maximum likelihood estimation |
| JEL: | C13 C22 |
| Date: | 2008–06–09 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-30&r=ets |
| By: | Ingmar Nolte; Valeri Voev (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We propose a unified framework for estimating integrated variances and covariances based on simple OLS regressions, allowing for a general market microstructure noise specification. We show that our estimators can outperform, in terms of the root mean squared error criterion, the most recent and commonly applied estimators, such as the realized kernels of Barndorff-Nielsen, Hansen, Lunde & Shephard (2006), the two-scales realized variance of Zhang, Mykland & Aït-Sahalia (2005), the Hayashi & Yoshida (2005) covariance estimator, and the realized variance and covariance with the optimal sampling frequency derived in Bandi & Russell (2005a) and Bandi & Russell (2005b). For a realistic trading scenario, the efficiency gains resulting from our approach are in the range of 35% to 50%. |
| Keywords: | High frequency data, Realized volatility and covariance, Market microstructure |
| JEL: | G10 F31 C32 |
| Date: | 2008–06–10 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-31&r=ets |
| By: | Torben G. Andersen; Tim Bollerslev; Xin Huang (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | Building on realized variance and bi-power variation measures constructed from high-frequency financial prices, we propose a simple reduced form framework for effectively incorporating intraday data into the modeling of daily return volatility. We decompose the total daily return variability into the continuous sample path variance, the variation arising from discontinuous jumps that occur during the trading day, as well as the overnight return variance. Our empirical results, based on long samples of high-frequency equity and bond futures returns, suggest that the dynamic dependencies in the daily continuous sample path variability is well described by an approximate long-memory HAR-GARCH model, while the overnight returns may be modelled by an augmented GARCH type structure. The dynamic dependencies in the non-parametrically identified significant jumps appear to be well described by the combination of an ACH model for the time-varying jump intensities coupled with a relatively simple log-linear structure for the jump sizes. Lastly, we discuss how the resulting reduced form model structure for each of the three components may be used in the construction of out-of-sample forecasts for the total return volatility. |
| Keywords: | Stochastic Volatility, Realized Variation, Bipower Variation, Jumps, Hazard Rates, Overnight Volatility |
| JEL: | C1 G1 C2 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-14&r=ets |
| By: | Viktor Todorov; Tim Bollerslev (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We provide a new theoretical framework for disentangling and estimating sensitivity towards systematic diffusive and jump risks in the context of factor pricing models. Our estimates of the sensitivities towards systematic risks, or betas, are based on the notion of increasingly finer sampled returns over fixed time intervals. In addition to establish- ing consistency of our estimators, we also derive Central Limit Theorems characterizing their asymptotic distributions. In an empirical application of the new procedures using high-frequency data for forty individual stocks and an aggregate market portfolio, we find the estimated diffusive and jump betas with respect to the market to be quite dif- ferent for many of the stocks. Our findings have direct and important implications for empirical asset pricing finance and practical portfolio and risk management decisions. |
| Keywords: | Factor models, systematic risk, common jumps, high-frequency data, realized variation |
| JEL: | C13 C14 G10 G12 |
| Date: | 2007–08–16 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2007-15&r=ets |
| By: | Martin Møller Andreasen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper extends two optimization routines to deal with objective functions for DSGE models. The optimization routines are i) a version of Simulated Annealing developed by Corana, Marchesi & Ridella (1987), and ii) the evolutionary algorithm CMA-ES developed by Hansen, Müller & Koumoutsakos (2003). Following these extensions, we examine the ability of the two routines to maximize the likelihood function for a sequence of test economies. Our results show that the CMA- ES routine clearly outperforms Simulated Annealing in its ability to find the global optimum and in efficiency. With 10 unknown structural parameters in the likelihood function, the CMA-ES routine finds the global optimum in 95% of our test economies compared to 89% for Simulated Annealing. When the number of unknown structural parameters in the likelihood function increases to 20 and 35, then the CMA-ES routine finds the global optimum in 85% and 71% of our test economies, respectively. The corresponding numbers for Simulated Annealing are 70% and 0%. |
| Keywords: | CMA-ES optimization routine, Multimodel objective function, Nelder-Mead simplex routine, Non-convex search space, Resampling, Simulated Annealing |
| JEL: | C61 C88 E30 |
| Date: | 2008–06–19 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-32&r=ets |
| By: | Martin Møller Andreasen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | This paper shows how non-linear DSGE models with potential non-normal shocks can be estimated by Quasi-Maximum Likelihood based on the Central Difference Kalman Filter (CDKF). The advantage of this estimator is that evaluating the quasi log-likelihood function only takes a fraction of a second. The second contribution of this paper is to derive a new particle filter which we term the Mean Shifted Particle Filter (MSPFb). We show that the MSPFb outperforms the standard Particle Filter by delivering more precise state estimates, and in general the MSPFb has lower Monte Carlo variation in the reported log-likelihood function. |
| Keywords: | Multivariate Stirling interpolation, Particle filtering, Non-linear DSGE models, Non-normal shocks, Quasi-maximum likelihood |
| JEL: | C13 C15 E10 E32 |
| Date: | 2008–06–20 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-33&r=ets |
| By: | Mark Podolskij; Daniel Ziggel (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | In this paper we propose a test to determine whether jumps are present in a discretely sampled process or not. We use the concept of truncated power variation to construct our test statistics for (i) semimartingale models and (ii) semimartingale models with noise. The test statistics converge to innity if jumps are present and have a normal distribution otherwise. Our method is valid (under very weak assumptions) for all semimartingales with absolute continuous characteristics and rather general model for the noise process. We nally implement the test and present the simulation results. Our simulations suggest that for semimartingale models the new test is much more powerful then tests proposed by Barndorff-Nielsen and Shephard (2006) and At-Sahalia and Jacod (2008). |
| Keywords: | Central Limit Theorem, High-Frequency Data, Microstructure Noise, Semimartingale Theory, Tests for Jumps, Truncated Power Variation |
| JEL: | C10 C13 C14 |
| Date: | 2008–06–20 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-34&r=ets |
| By: | Per Frederiksen; Morten Ørregaard Nielsen (School of Economics and Management, University of Aarhus, Denmark) |
| Abstract: | We propose to use a variant of the local polynomial Whittle estimator to estimate the memory parameter in volatility for long memory stochastic volatility models with potential nonstation- arity in the volatility process. We show that the estimator is asymptotically normal and capable of obtaining bias reduction as well as a rate of convergence arbitrarily close to the parametric rate, n1=2. A Monte Carlo study is conducted to support the theoretical results, and an analysis of daily exchange rates demonstrates the empirical usefulness of the estimators |
| Keywords: | Bias reduction, local Whittle estimation, long memory stochastic volatility model |
| JEL: | C14 C22 |
| Date: | 2008–06–24 |
| URL: | http://d.repec.org/n?u=RePEc:aah:create:2008-35&r=ets |
| By: | Alexandr Kuchynka (Institute of Economic Studies, Faculty of Social Sciences, Charles University, Prague, Czech Republic; Faculty of Economics, University of West Bohemia in Pilsen; Institute of Information Theory and Automation of the ASCR) |
| Abstract: | This paper focuses on the extraction of volatility of financial returns. The volatility process is modeled as a superposition of two autoregressive processes which represent the more persistent factor and the quickly mean-reverting factor. As the volatility is not observable, the logarithm of the daily high-low range is employed as its proxy. The estimation of parameters and volatility extraction are performed using a modified version of the Kalman filter which takes into account the finite sample distribution of the proxy. |
| Keywords: | volatility, stochastic volatility models, Kalman filter, volatility proxy |
| JEL: | C22 G15 |
| Date: | 2008–06 |
| URL: | http://d.repec.org/n?u=RePEc:fau:wpaper:wp2008_10&r=ets |
| By: | Mark J. Jensen; John M. Maheu |
| Abstract: | This paper extends the existing fully parametric Bayesian literature on stochastic volatility to allow for more general return distributions. Instead of specifying a particular distribution for the return innovation, we use nonparametric Bayesian methods to flexibly model the skewness and kurtosis of the distribution while continuing to model the dynamics of volatility with a parametric structure. Our semiparametric Bayesian approach provides a full characterization of parametric and distributional uncertainty. We present a Markov chain Monte Carlo sampling approach to estimation with theoretical and computational issues for simulation from the posterior predictive distributions. The new model is assessed based on simulation evidence, an empirical example, and comparison to parametric models. |
| Date: | 2008 |
| URL: | http://d.repec.org/n?u=RePEc:fip:fedawp:2008-15&r=ets |
| By: | Marwan Izzeldin; Ana-Maria Fuertes; Elena Kalotychou |
| Abstract: | Several recent studies advocate the use of nonparametric estimators of daily price vari- ability that exploit intraday information. This paper compares four such estimators, realised volatility, realised range, realised power variation and realised bipower variation, by examining their in-sample distributional properties and out-of-sample forecast ranking when the object of interest is the conventional conditional variance. The analysis is based on a 7-year sample of transaction prices for 14 NYSE stocks. The forecast race is conducted in a GARCH framework and relies on several loss functions. The realized range fares relatively well in the in-sample .t analysis, for instance, regarding the extent to which it brings normality in returns. However, overall the realised power variation provides the most accurate 1-day-ahead forecasts. Fore- cast combination of all four intraday measures produces the smallest forecast errors in about half of the sampled stocks. A market conditions analysis reveals that the additional use of intraday data on day t .. 1 to forecast volatility on day t is most advantageous when day t is a low volume or an up-market day. The results have implications for value-at-risk analysis. |
| Keywords: | C53; C32; C14. |
| Date: | 2008 |
| URL: | http://d.repec.org/n?u=RePEc:lan:wpaper:005439&r=ets |