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## Empirical properties of asset returns: stylized facts and statistical issues (2001)

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Venue: | Quantitative Finance |

Citations: | 332 - 4 self |

### Citations

3136 |
A wavelet tour of signal processing
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Citation Context ... for which the scaling in equation (30) holds exactly, the Legendre transform (31) may be inverted to obtain D(α) from ζ(q). The technique was subsequently refined [62,99] using the wavelet transfor=-=m [92]-=-, who proposed an algorithm for determining the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods provide a framework to investigat... |

1717 |
Efficient capital markets: a review of theory and empirical work
- Fama
- 1970
(Show Context)
Citation Context ...for τ � 15 minutes it can be safely assumed to be zero for all practical purposes [21]. The absence of significant linear correlations in price increments and asset returns has been widely document=-=ed [43,102] and-=- is often cited as support for the ‘efficient market hypothesis’ [44]. The absence of correlation is intuitively easy to understand: if price changes exhibit significant correlation, this correlat... |

1151 | The Behavior of Stock Market Prices - Fama |

1063 |
The Variation of Certain Speculative Prices
- Mandelbrot
- 1963
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Citation Context ... estimator of the density of 30 minute price increments. S&P 500 index futures. The probability density function (PDF) is then defined as its derivative fT = F ′ T . As early as the 1960s, Mandelbro=-=t [80]-=- pointed out the insufficiency of the normal distribution for modelling the marginal distribution of asset returns and their heavy-tailed character. Since then, the non-Gaussian character of the distr... |

854 | The Fractional Brownian Motions, Fractional Noises and Applications - Mandelbrot, Ness - 1968 |

820 |
Techniques in fractal geometry
- Falconer
- 1997
(Show Context)
Citation Context ...y point basis. The interest of physicists 3 The Hausdorff–Besicovich dimension is one of the numerous mathematical notions corresponding to the general concept of ‘fractal’ dimension. For detail=-=s see [40]-=-.sQ UANTITATIVE F INANCE Empirical properties of asset returns: stylized facts and statistical issues and empirical researchers in singularity spectra was ignited by the work of Parisi and Frisch [101... |

805 | Applied Nonparametric Regression - Härdle - 1990 |

544 | A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices. Econometrica 41:135–56
- Clark
- 1973
(Show Context)
Citation Context ... chosen to correct for seasonalities observed in calendar time and is therefore usually a cumulative measure of market activity: the number of transactions (tick time) [3], the volume of transactions =-=[19]-=- or a sample-based measure of market activity (see work by Dacorogna and coworkers [97, 105] and also [2]). 3.2. Ergodicity While stationarity is necessary to ensure that one can mix data from differe... |

428 |
Efficient capital markets
- Fama
- 1991
(Show Context)
Citation Context ...oses [21]. The absence of significant linear correlations in price increments and asset returns has been widely documented [43,102] and is often cited as support for the ‘efficient market hypothesis=-=’ [44]-=-. The absence of correlation is intuitively easy to understand: if price changes exhibit significant correlation, this correlation may be used to conceive a simple strategy with positive expected earn... |

334 |
Stochastic Limit Theory
- DAVIDSON
- 1994
(Show Context)
Citation Context ...r the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assuming that the noise terms (innovations) in the return process are ‘weakly’ depend=-=ent [30]-=-. In order to obtain confidence intervals for finite samples, one often requires the residuals to be IID and some of their higher-order (typically fourth order) moments to be well defined (finite). As... |

293 |
Statistical aspects of ARCH and stochastic volatility
- Shephard
- 1996
(Show Context)
Citation Context ...markets and time periods, has been observed by independent studies and classified as ‘stylized facts’. We present here a pedagogical overview of these stylized facts. With respect to previous revi=-=ews [10, 14, 16, 50, 95, 102, 109]-=- on the same subject, the aim of the present paper is to focus more on the properties of empirical data than on those of statistical models and introduce the reader to some new insights provided by me... |

279 | Market Volatility - Shiller - 1989 |

270 | Intermittent turbulence in self similar cascades: Divergence of high moments and dimension of the carrier - Mandelbrot - 1974 |

260 | Trading volume and serial correlation in stock returns, Quarterly - Campbell, Grossman, et al. - 1993 |

241 | Long-term memory in stock market prices - Lo - 1991 |

224 |
Fractal measures and their singularities: The characterization of strange sets
- Halsey, Kadanoff, et al.
- 1986
(Show Context)
Citation Context ...first the continuous time (�t → 0) limit for determining the local Hölder exponents and second the determination of the Hausdorff dimension of the sets �(α) which, as remarked already by Halse=-=y et al [58], -=-may be intertwined fractal sets with complex structures and impossible to separate on a point by point basis. The interest of physicists 3 The Hausdorff–Besicovich dimension is one of the numerous m... |

212 | Modeling volatility persistence of speculative returns: A new approach - Ding, Granger - 1996 |

197 | What Moves Stock Prices
- Cutler, Poterba, et al.
- 1989
(Show Context)
Citation Context ...market analysts has been and remains an event-based approach in which one attempts to ‘explain’ or rationalize a given market movement by relating it to an economic or political event or announcem=-=ent [27]. -=-From this point of view, one could easily imagine that, since different assets are not necessarily influenced by the same events or information sets, price series obtained from different assets and—... |

185 | Heteroskedasticity in Stock Return Data: Volume versus GARCH effects - Lamoureux, Lastrapes - 1990 |

173 |
The econometrics of financial markets
- Pagan
- 1996
(Show Context)
Citation Context ...markets and time periods, has been observed by independent studies and classified as ‘stylized facts’. We present here a pedagogical overview of these stylized facts. With respect to previous revi=-=ews [10, 14, 16, 50, 95, 102, 109]-=- on the same subject, the aim of the present paper is to focus more on the properties of empirical data than on those of statistical models and introduce the reader to some new insights provided by me... |

135 | New insights into smile, mispricing and value at risk
- Eberlein, Keller, et al.
- 1998
(Show Context)
Citation Context ...pastime: there are dozens of parametric models proposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution [9, 72], hyperbolic distributions =-=[37, 104]-=-, normal inverse Gaussian distributions [7], exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From the empirical features described abov... |

126 |
Mandelbrot and the Stable Paretian Hypothesis
- Fama
- 1963
(Show Context)
Citation Context ...ge rate returns. Time scale: ticks. attraction but the tail index is found to be larger than two— which means that the variance is finite and the tails lighter than those of stable Lévy distributio=-=ns [41], -=-but compatible with a power-law (Pareto) tail with (the same) exponent α(T ) = 1/ξ. These studies seem to validate the power-law nature of the distribution of returns, with an exponent around three,... |

120 |
Statistical methods for data with long-range dependence
- Beran
- 1992
(Show Context)
Citation Context ... in which one computes the likelihood of a model to hold given the value of a statistic and rejects/accepts the model by comparing the test statistics to a threshold value. With a few exceptions (see =-=[8, 87, 88]-=-), the large majority of statistical tests are based on a central limit theorem for the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assumin... |

120 | On the singularity structure of fully developed turbulence, in: Turbulence and Predictability - Parisi, Frisch - 1985 |

110 |
The asymptotic distribution of extreme stock market returns
- Longin
- 1996
(Show Context)
Citation Context ... precisely with the probabilities of extreme events. To our knowledge, the first application of extreme value theory to financial time series was given by Jansen and de Vries [70], followed by Longin =-=[76], Dacorogn-=-a et al [28], Lux [77] and others. Given a series of n non-overlapping returns r(t, �t), t = 0,�t,2�t,...,n�t, the extremal (minimal and maximal) returns are defined as: mn(�t) = min{r(t + k... |

103 | The dependence between hourly prices and trading volume - JAIN, JOH - 1988 |

77 |
High frequency data in financial markets: Issues and applications* 1
- GOODHART, O'HARA
- 1997
(Show Context)
Citation Context ...tive autocorrelations at the first lag in bid or ask prices themselves, suggesting a fast mean reversion of the price at the tick level. This feature may be attributed to the action of a market maker =-=[47]. -=-229sR Cont Q UANTITATIVE F INANCE The absence of autocorrelation does not seem to hold systematically when the time scale �t is increased: weekly and monthly returns do exhibit some autocorrelation.... |

59 |
When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models
- Mandelbrot
- 1971
(Show Context)
Citation Context ...esent the time the market takes to react to new information. This correlation time is typically several minutes for organized futures markets and even shorter for foreign exchange markets. Mandelbrot =-=[85] exp-=-ressed this property by stating that ‘arbitrage tends to whiten the spectrum of price changes’. This property implies that traditional tools of signal processing which are based on second-order pr... |

53 | Forecasts of Future Prices, Unbiased Markets, and 'Martingale - Mandelbrot - 1966 |

50 |
Stylized facts on the temporal and distributional properties of daily data from speculative markets. UCSD Working Paper
- Granger, Ding
- 1995
(Show Context)
Citation Context ...e way one can study autocorrelation functions of various powers of the returns: Cα(τ) = corr(|r(t + τ, �t)| α , |r(t,�t)| α ). (16) Comparing the decay of Cα for various values of α, Ding a=-=nd Granger [34, 35] rem-=-arked that, for a given lag τ, this correlation is highest for α = 1, which means that absolute returns are more predictable than other powers of returns. Several authors [11,20–22,54,55,59,105] h... |

49 |
Fractals and Scaling
- Mandelbrot
- 1997
(Show Context)
Citation Context ...on may have a finite tail index α without being a power-law distribution. Measuring the tail index of a distribution gives a measure of how heavy the tail is. A simple method, suggested by Mandelbrot=-= [80,89]-=-, is to represent the sample moments (or cumulants) as a function of the sample size n. If the theoretical moment is finite then the sample moment will eventually settle down to a region defined aroun... |

48 | Estimating the extremal index - Smith, Weissman - 1994 |

43 | Volatility and Correlation - Rebonato - 1999 |

42 | The multifractal nature of Lévy processes
- Jaffard
- 1999
(Show Context)
Citation Context ...unately, this turns out not to be the case: it has been shown that, for large classes of stochastic processes, the singularity spectrum is the same for almost all sample paths. Results due to Jaffard =-=[68] -=-show that a large class of Lévy processes verifies this property. As defined above, the singularity spectrum of a function does not appear to be of any practical use since its definition involves fir... |

40 |
Long memory continuous time models
- Comte, Renault
- 1996
(Show Context)
Citation Context ...ity factor whose dynamics should be specified to match the empirically observed dependences. Examples of models in this direction are GARCH models [10,39] and long-memory stochastic volatility models =-=[20,59,100]. Note howe-=-ver that in this decomposition the volatility variable σ (t, �t) is not directly observable, only the returns r(t,�t) are. Therefore, the definition of ‘volatility’ is model dependent and ‘... |

34 |
Normal inverse Gaussian distributions and the modelling of stock returns
- Barndorff-Nielsen
- 1997
(Show Context)
Citation Context ...oposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution [9, 72], hyperbolic distributions [37, 104], normal inverse Gaussian distributions =-=[7]-=-, exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From the empirical features described above, one can conclude that, in order for a pa... |

33 | Intra-day market activity - Gourieroux, Jasiak, et al. - 1999 |

21 |
Long memory in stochastic volatility. In: Forecasting Volatility
- Harvey
- 1998
(Show Context)
Citation Context ...ity factor whose dynamics should be specified to match the empirically observed dependences. Examples of models in this direction are GARCH models [10,39] and long-memory stochastic volatility models =-=[20,59,100]. Note howe-=-ver that in this decomposition the volatility variable σ (t, �t) is not directly observable, only the returns r(t,�t) are. Therefore, the definition of ‘volatility’ is model dependent and ‘... |

20 |
Klüppelberg C and Mikosch T 1997 Modelling Extremal Events for Insurance and Finance
- Embrecht
(Show Context)
Citation Context ...�t) and Mn(�t) as the sample size n increases. If such a limit exists, then it is described by the Fisher–Tippett theorem in the case where the returns are IID. Extreme value theorem for IID seq=-=uence [38]. Assume-=- the log returns (r(t, �t))t�0 form an IID sequence with distribution F�t. If there exist normalizing constants (λn,σn) and a non-degenerate limit distribution H for the normalized maximum ret... |

19 | Coherent and random sequences in financial fluctuations",Physica A - Vandewalle, Ausloos - 1997 |

19 |
Multifractal formalism for functions. I. Results valid for all functions. II. Self-similar functions
- Jaffard
- 1997
(Show Context)
Citation Context ...sey et al [58] who, in different contexts , proposed a formalism for empirically computing the singularity spectrum from sample paths of the process. This formalism, called the multifractal formalism =-=[58, 66, 67, 101], enab-=-les the singularity spectrum to be computed from sample moments (‘structure functions’) of the increments. More precisely, if the sample moments of the returns verify a scaling property 〈|r(t,T)... |

18 | Nonlinear Time Series, Complexity Theory, and Finance
- Brock, Lima
- 1996
(Show Context)
Citation Context ...markets and time periods, has been observed by independent studies and classified as ‘stylized facts’. We present here a pedagogical overview of these stylized facts. With respect to previous revi=-=ews [10, 14, 16, 50, 95, 102, 109]-=- on the same subject, the aim of the present paper is to focus more on the properties of empirical data than on those of statistical models and introduce the reader to some new insights provided by me... |

18 | Characterization of Self-Similar Multifractals with Wavelet Maxima
- Hwang, Mallat
- 1993
(Show Context)
Citation Context ...n the case of multifractal processes for which the scaling in equation (30) holds exactly, the Legendre transform (31) may be inverted to obtain D(α) from ζ(q). The technique was subsequently refine=-=d [62,99]-=- using the wavelet transform [92], who proposed an algorithm for determining the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods ... |

17 | The Distribution of Futures Prices: A test of the Stable Paretian and Mixture of Normals Hypothesis - Hall, Brorsen, et al. - 1989 |

17 |
2000): Limit Theory for the Sample Autocorrelations and Extremes of a GARCH(1,1) Process,Annals of Statistics 28
- Mikosch, StAricA
(Show Context)
Citation Context ... These criticisms can be quantified if one considers analogous quantities for some time series models with fattailed marginals, such as GARCH. In a critical study of GARCH models, Mikosch and Starica =-=[96]-=- show that the ACF of the squared returns in GARCH(1,1) models can have non-standard sample properties and generate large confidence bands, which raises serious questions about the methods used to fit... |

17 | Why non-linearities can ruin the heavy tailed modeler’s day. In A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions
- RESNICK
- 1997
(Show Context)
Citation Context ...meaning and relevance of such confidence intervals. As we will discuss below, this can have quite an impact on the significance and interpretation of commonly used estimators (see also discussions in =-=[1, 31, 107]-=-). 4. The distribution of returns: a tale of heavy tails Empirical research in financial econometrics in the 1970s mainly concentrated on modelling the unconditional distribution of returns, defined a... |

14 |
Gonedes (1974), ‘A comparison of the stable and student distributions as statistical models for stock prices
- Blattberg, J
(Show Context)
Citation Context ...rice changes has become a popular pastime: there are dozens of parametric models proposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution =-=[9, 72]-=-, hyperbolic distributions [37, 104], normal inverse Gaussian distributions [7], exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From t... |

14 | Statistical Properties of Daily Exchange Rate: 1974-1983 - Hsieh - 1988 |

13 | Modeling economic randomness: statistical mechanics of market phenomena - Cont - 1999 |

13 |
The limiting extremal behavior of speculative returns: An analysis of intradaily data from the Frankfurt stock exchange. Applied financial economics
- Lux
- 2001
(Show Context)
Citation Context ...s of extreme events. To our knowledge, the first application of extreme value theory to financial time series was given by Jansen and de Vries [70], followed by Longin [76], Dacorogna et al [28], Lux =-=[77] and others. Given-=- a series of n non-overlapping returns r(t, �t), t = 0,�t,2�t,...,n�t, the extremal (minimal and maximal) returns are defined as: mn(�t) = min{r(t + k�t, �t), k ∈ [1,n]}, (6) Mn(�t) ... |

12 |
Vries (1991): “On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective,”Review of Economics and Statistics
- Jansen, de
(Show Context)
Citation Context ...robability theory dealing precisely with the probabilities of extreme events. To our knowledge, the first application of extreme value theory to financial time series was given by Jansen and de Vries =-=[70], follow-=-ed by Longin [76], Dacorogna et al [28], Lux [77] and others. Given a series of n non-overlapping returns r(t, �t), t = 0,�t,2�t,...,n�t, the extremal (minimal and maximal) returns are defined... |

12 | Joah, joseph and operational hydrology. Water Resources Research 4 - B, Wallis - 1968 |

11 |
Stochastic volatility and transaction time: an activity-based volatility estimator
- Ané, Geman
- 1999
(Show Context)
Citation Context ...narity. This time deformation is chosen to correct for seasonalities observed in calendar time and is therefore usually a cumulative measure of market activity: the number of transactions (tick time) =-=[3]-=-, the volume of transactions [19] or a sample-based measure of market activity (see work by Dacorogna and coworkers [97, 105] and also [2]). 3.2. Ergodicity While stationarity is necessary to ensure t... |

10 |
A W Lo and A C McKinlay (1997): The econometrics of financial markets
- Campbell
(Show Context)
Citation Context |

10 | Testing for nonlinear dependence in foreign ex change rates - Hsieh - 1989 |

10 | The Generalized Hyperbolic Model
- Eberlein, Prause
- 2000
(Show Context)
Citation Context ...pastime: there are dozens of parametric models proposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution [9, 72], hyperbolic distributions =-=[37, 104]-=-, normal inverse Gaussian distributions [7], exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From the empirical features described abov... |

8 |
MacKinley (1999): A non-random walk down Wall
- Lo, C
(Show Context)
Citation Context ...tering: large price variations are more likely to be followed by large price variations. Figure 1 illustrates this phenomenon on daily returns of BMW shares. Log prices are therefore not random walks =-=[17, 21]. A qu-=-antity commonly used to measure volatility clustering is the autocorrelation function of the squared returns: C2(τ) = corr(|r(t + τ, �t)| 2 , |r(t,�t)| 2 ). (15) Empirical studies using returns ... |

8 | Statistical properties of financial time series - Cont - 1999 |

7 | Scaling behavior of an economic index, Nature 376 - Mantegna, Stanley - 1995 |

6 |
Clustering and dependence in extreme market returns Extremes: Risk and Safety. Statistical Extreme Value Theory and Applications
- Cont
- 1998
(Show Context)
Citation Context ...ng a relative stability of the tails. However, the IID hypothesis underlying these estimation procedure has to be treated with caution given the dependence present in asset returns (see section 5 and =-=[25]-=-). 5. Dependence properties of returns 5.1. Absence of linear autocorrelation It is a well-known fact that price movements in liquid markets do not exhibit any significant autocorrelation: the Sample ... |

6 |
Multifractal formalism for functions II: selfsimilar functions
- Jaffard
- 1993
(Show Context)
Citation Context ...sey et al [58] who, in different contexts , proposed a formalism for empirically computing the singularity spectrum from sample paths of the process. This formalism, called the multifractal formalism =-=[58, 66, 67, 101], enab-=-les the singularity spectrum to be computed from sample moments (‘structure functions’) of the increments. More precisely, if the sample moments of the returns verify a scaling property 〈|r(t,T)... |

6 | Scaling in financial prices: Tails and dependence. Quantitative Finance - Mandelbrot - 2001 |

5 |
R Y and Kroner K F 1992 ARCH modeling in finance
- Bollerslev, Chou
(Show Context)
Citation Context |

5 |
Bouchaud (1997), “Scaling in stock market data: stable laws and beyond
- Cont, Potters, et al.
- 1014
(Show Context)
Citation Context ...s far from its Gaussian value: typical values for T = 5 minutes are (see table 1): κ � 74 (US$/DM exchange rate futures), κ � 60 (US$/Swiss Franc exchange rate futures), κ � 16 (S&P500 index =-=futures) [16, 21, 22, 102]. -=-One can summarize the empirical results by saying that the distribution f�t tends to be non-Gaussian, sharp peaked and heavy tailed, these properties being more pronounced for Table 1. Descriptive s... |

5 |
Statistical Finance: empirical and theoretical approaches to the statistical modelling of price variations in speculative markets Doctoral Thesis
- Cont
- 1998
(Show Context)
Citation Context ...rs [11,20–22,54,55,59,105] have remarked that the decay of Cα(τ) as τ increases is well reproduced by a power law: Cα(τ) ∼ A τ β (17) with a coefficient β ∈ [0.2, 0.4] for absolute or sq=-=uared returns [21, 22, 74]-=-. This slow decay is sometimes interpreted as a sign of long-range dependence in volatility and motivated the development of models integrating this feature (see below). More generally, one can ask wh... |

5 | Cizeau P, Bouchaud J-P and Potters M 1999 Noise dressing of financial correlation matrices Phys - Laloux |

5 |
Limit theorems on the self normalized range for weakly and strongly dependent processes, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete 31
- Mandelbrot
- 1975
(Show Context)
Citation Context ... in which one computes the likelihood of a model to hold given the value of a statistic and rejects/accepts the model by comparing the test statistics to a threshold value. With a few exceptions (see =-=[8, 87, 88]-=-), the large majority of statistical tests are based on a central limit theorem for the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assumin... |

5 |
A Arneodo (1994) The multifractal formalism revisitied with wavelets
- Muzy, Bacry
(Show Context)
Citation Context ...n the case of multifractal processes for which the scaling in equation (30) holds exactly, the Legendre transform (31) may be inverted to obtain D(α) from ζ(q). The technique was subsequently refine=-=d [62,99]-=- using the wavelet transform [92], who proposed an algorithm for determining the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods ... |

5 |
Bacry (2000), Modelling Fluctuations of Financial Time Series: from Cascade Process to Stochastic Volatility Model
- Muzy, Delour, et al.
(Show Context)
Citation Context ...rgodicity is not uncommon in physical systems exhibiting long-range dependence [12]. This may also be the case for some multifractal processes recently introduced to model highfrequency asset returns =-=[90, 100]-=-, in which case the relation between sample averages and model expectations remains an open question. 3.3. Finite sample properties of estimators Something which seems obvious to any statistician but ... |

4 |
Bollerslev T 1997 Intraday periodicity and volatility persistence in financial markets
- Andersen
(Show Context)
Citation Context ...re of market activity: the number of transactions (tick time) [3], the volume of transactions [19] or a sample-based measure of market activity (see work by Dacorogna and coworkers [97, 105] and also =-=[2]-=-). 3.2. Ergodicity While stationarity is necessary to ensure that one can mix data from different periods in order to estimate moments of the returns, it is far from being sufficient: one also needs t... |

4 |
Delour (2001): Multifractal Random Walks
- Bacry, Muzy, et al.
(Show Context)
Citation Context ...e examples of stochastic processes for which the singularity spectrum resembles the one observed in empirical data are stochastic cascades [90] or their causal versions, the multifractal random walks =-=[6, 100]-=-. One drawback of the singularity spectrum is that its finite sample properties are not well known. Veneziano et al [114] have investigated, in the context of study of width functions of river basins,... |

4 | A and Sornette D 1998 Stock market crashes are outliers Euro - Johansen |

4 |
Durrleman V, Nikeghbali A, Riboulet G and Roncalli T 2000 Copulas for finance: a reading guide and some applications Groupe de Recherche Operationnelle, Credit Lyonnais
- Bouye
(Show Context)
Citation Context ...sets may have extremal correlations while their covariance is zero: covariance does not measure the correlation of extremes. Some recent theoretical work has been done in this direction using copulas =-=[108]-=- and multivariate extreme value theory [64, 112, 113], but a lot remains to be done on empirical grounds. For a recent review with applications to foreign exchange rate data see Hauksson et al [64]. 7... |

3 |
R and Taqqu M (eds) 1996 A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions
- Adler, Feldman
(Show Context)
Citation Context ...meaning and relevance of such confidence intervals. As we will discuss below, this can have quite an impact on the significance and interpretation of commonly used estimators (see also discussions in =-=[1, 31, 107]-=-). 4. The distribution of returns: a tale of heavy tails Empirical research in financial econometrics in the 1970s mainly concentrated on modelling the unconditional distribution of returns, defined a... |

3 |
Muzy J F and Sornette D 1998 Causal cascade in the stock market from the infrared to the ultraviolet Euro
- Arnéodo
(Show Context)
Citation Context ...the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods provide a framework to investigate pathwise regularity of price trajectories =-=[4,26,45,100]. A -=-first surprising result is that the shape of the singularity spectrum does not depend on the asset considered: all series exhibit the same, ‘inverted parabola’ shape also observed by Fisher et al ... |

3 |
Matacz A and Potters M 2001 The leverage effect in financial markets: retarded volatility and market panic Preprint http://xxx.lpthe.jussieu.fr/abs/cond-mat/0101120
- Bouchaud
(Show Context)
Citation Context ...s is the so-called ‘leverage effect’: the correlation of returns with subsequent squared returns defined by L(τ) = corr(|r(t + τ, �t)| 2 ,r(t,�t)) (20) starts from a negative value and decay=-=s to zero [13, 102], sugges-=-ting that negative returns lead to a rise in volatility. However this effect is asymmetric L(τ) �= L(−τ) and in general L(τ) is negligible for τ<0. The existence of such nonlinear dependence, ... |

3 |
M and Bouchaud J-P 2000 Correlations of extreme stock returns within a non-Gaussian one-factor model Science & Finance Working Paper
- Cizeau, Potters
(Show Context)
Citation Context ... correlations once the common factors have been accounted for, one can define conditional correlations by conditioning on an aggregate variable such as the market return before computing correlations =-=[18]-=-. 6.2. Correlations of extreme returns Independently of the significance of its information content, the covariance matrix has been criticized as a tool for measuring dependence because it is based on... |

3 |
Müller U A, Pictet O V and de Vries C G 1992 The distribution of extremal foreign exchange rate returns in large data sets Olsen and Associates Internal document UAM
- Dacorogna
- 1992
(Show Context)
Citation Context ...obabilities of extreme events. To our knowledge, the first application of extreme value theory to financial time series was given by Jansen and de Vries [70], followed by Longin [76], Dacorogna et al =-=[28], Lux [77] and o-=-thers. Given a series of n non-overlapping returns r(t, �t), t = 0,�t,2�t,...,n�t, the extremal (minimal and maximal) returns are defined as: mn(�t) = min{r(t + k�t, �t), k ∈ [1,n]}, (... |

3 |
P and Gourieroux Ch 1999 Kernel based nonlinear canonical analysis CREST Working Paper
- Darolles, Florens
(Show Context)
Citation Context ...nutes 80 Figure 8. Behaviour of some nonlinear correlation functions of price changes. the object of ‘canonical correlation analysis’, can yield more insight into the dependence properties of retu=-=rns [29]-=-. Some examples of autocorrelations of different nonlinear transforms of returns are compared in figure 8. These autocorrelations are actually weighted sums of covariances of various integer powers of... |

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A and Mikosch T 1998 Limit theory for the sample ACF of stationary process with heavy tails with applications to ARCH Ann
- Davis
(Show Context)
Citation Context ...meaning and relevance of such confidence intervals. As we will discuss below, this can have quite an impact on the significance and interpretation of commonly used estimators (see also discussions in =-=[1, 31, 107]-=-). 4. The distribution of returns: a tale of heavy tails Empirical research in financial econometrics in the 1970s mainly concentrated on modelling the unconditional distribution of returns, defined a... |

3 | A and Mikosch T 1999 The sample autocorrelations of financial time series models EURANDOM Working Paper 99–039 - Davis |

3 | A C and Cox D R 1989 Some simple properties of sums of random variables having long range dependence Proc - Davison |

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CWJandEngle R F 1983 A long memory property of stock market returns and a new model
- Ding, Granger
(Show Context)
Citation Context ...e way one can study autocorrelation functions of various powers of the returns: Cα(τ) = corr(|r(t + τ, �t)| α , |r(t,�t)| α ). (16) Comparing the decay of Cα for various values of α, Ding a=-=nd Granger [34, 35] rem-=-arked that, for a given lag τ, this correlation is highest for α = 1, which means that absolute returns are more predictable than other powers of returns. Several authors [11,20–22,54,55,59,105] h... |

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Amaral LANandStanley H E 1998 Inverse cubic law for the distribution of stock price variations Euro
- Gopikrishnan, Meyer
(Show Context)
Citation Context ... exponent α(T ) = 1/ξ. These studies seem to validate the power-law nature of the distribution of returns, with an exponent around three, using a direct log–log regression on the histogram of retu=-=rns [52].-=- Note however that these studies do not allow us to pinpoint the exponent with more than a single significant digit. Also, a positive value of ξ does not imply power-law tails [12] but is compatible ... |

3 | CWJandDing Z 1996 Varieties of long memory models - Granger |

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Dacorogna M M, Domenig T, Müller U and Samorodnitsky G 2001 Multivariate extremes, aggregation and risk estimation Quantitative Finance 1
- Hauksson
(Show Context)
Citation Context ...d well behaved [38]. These methods, when applied to daily returns of stocks, market indices and exchange rates, yield a positive value of ξ between 0.2 and 0.4, which means a tail index 2 < α(T ) ��=-=� 5 [64, 70, 76, 77]. In-=- all cases, ξ is bounded away from zero, indicating heavy tails belonging to the Fréchet domain of Table 2. Limit distributions for extreme values. Here 1x>0 and 1x�0 are indicator functions. Gumb... |

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Models of stock returns: a comparison J. Finance XXXIX 147–65
- Kon
- 1984
(Show Context)
Citation Context ...rice changes has become a popular pastime: there are dozens of parametric models proposed in the literature, starting with the normal distribution, stable distributions [80], the Student distribution =-=[9, 72]-=-, hyperbolic distributions [37, 104], normal inverse Gaussian distributions [7], exponentially truncated stable distributions [11,21] are some of the parametric models which have been proposed. From t... |

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Correlations in economic time series Physica A
- Liu, Cizeau, et al.
- 1997
(Show Context)
Citation Context ...rs [11,20–22,54,55,59,105] have remarked that the decay of Cα(τ) as τ increases is well reproduced by a power law: Cα(τ) ∼ A τ β (17) with a coefficient β ∈ [0.2, 0.4] for absolute or sq=-=uared returns [21, 22, 74]-=-. This slow decay is sometimes interpreted as a sign of long-range dependence in volatility and motivated the development of models integrating this feature (see below). More generally, one can ask wh... |

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Fisher A and Calvet L 1997 The multifractal model of asset returns Cowles Foundation for Economic Research Working Paper
- Mandelbrot
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Citation Context ...rgodicity is not uncommon in physical systems exhibiting long-range dependence [12]. This may also be the case for some multifractal processes recently introduced to model highfrequency asset returns =-=[90, 100]-=-, in which case the relation between sample averages and model expectations remains an open question. 3.3. Finite sample properties of estimators Something which seems obvious to any statistician but ... |

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Handbook of Statistics: Statistical Methods in Finance vol 14
- Maddala, Rao
- 1997
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Citation Context |

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Dacorogna M, Davé R D, Olsen R
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(Show Context)
Citation Context ... a cumulative measure of market activity: the number of transactions (tick time) [3], the volume of transactions [19] or a sample-based measure of market activity (see work by Dacorogna and coworkers =-=[97, 105]-=- and also [2]). 3.2. Ergodicity While stationarity is necessary to ensure that one can mix data from different periods in order to estimate moments of the returns, it is far from being sufficient: one... |

3 | Dacorogna M A and Pictet O V 1996 Heavy tails in high-frequency financial data Olsen and Associates - Müller |

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Dacorogna M, Müller U A, Olsen R B and Ward J R 1997 Statistical study of foreign exchange rates, empirical evidence of a price change scaling law and intraday analysis
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(Show Context)
Citation Context ... a cumulative measure of market activity: the number of transactions (tick time) [3], the volume of transactions [19] or a sample-based measure of market activity (see work by Dacorogna and coworkers =-=[97, 105]-=- and also [2]). 3.2. Ergodicity While stationarity is necessary to ensure that one can mix data from different periods in order to estimate moments of the returns, it is far from being sufficient: one... |

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Calvet L and Mandelbrot B 1998 Multifractal analysis of USD/DM exchange rates Yale University Working Paper
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(Show Context)
Citation Context ...the singularity spectrum from its wavelet transform [66, 67]. 7.3. Singularity spectra of asset price series These methods provide a framework to investigate pathwise regularity of price trajectories =-=[4,26,45,100]. A -=-first surprising result is that the shape of the singularity spectrum does not depend on the asset considered: all series exhibit the same, ‘inverted parabola’ shape also observed by Fisher et al ... |

2 | et al 1996 Turbulent cascades in foreign exchange markets Nature 381 - Ghashghaie |

2 | CWJandMorgenstern O 1970 - Granger |

2 | CWJ1977 Long term dependence in common stock returns - Granger - 1970 |

2 | Dacorogna M M, Davé RR,Müller U A, Olsen R B and Pictet O V 1997 From the birds eye to the microscope: a survey of new stylized facts of the intra-day foreign exchange markets Finance Stochastics 1 95–130 - Guillaume |

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Taqqu M 1979 Robust R/S analysis of long serial correlation Bull
- Mandelbrot
(Show Context)
Citation Context ... in which one computes the likelihood of a model to hold given the value of a statistic and rejects/accepts the model by comparing the test statistics to a threshold value. With a few exceptions (see =-=[8, 87, 88]-=-), the large majority of statistical tests are based on a central limit theorem for the estimator from which the asymptotic normality is obtained. This central limit theorem can be obtained by assumin... |

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Amaral LANand Stanley H E 1999 Universal and nonuniversal properties of cross correlations in financial time series Phys
- Plerou, Gopikrishnan, et al.
(Show Context)
Citation Context ...e been usually interpreted in economic terms as factors of randomness underlying market movements. In a recent empirical study of the covariance matrix of 406 NYSE assets, Laloux et al [78] (see also =-=[103]-=-) showed that among the 406 available eigenvalues and principal components, apart from the highest eigenvalue (whose eigenvector roughly corresponds to the market index) and the next few (ten) highest... |