5. Diseño de un plan de capacitación para la Firma Consulting & Tax de la ciudad de
5.9. Organización de los eventos de capacitación
This section first introduces the reader the concepts of long-range autocorrelation, fractality, self-affinity and scaling exponent (common references are Feder, 1988; Kantelhardt, 2009), then presents the relevant financial literature regarding the long- range dependence detection and analysis.
2.4.1 Long-range autocorrelation and scaling exponent
The dynamics of the times series depicting a complex system is often characterized by scaling laws, over a continuous range of time scales and frequencies (Kantelhardt, 2009). A system characterized by self-similar structures (self affinity) over different time-scales is called afractal, i.e. a magnification of a small part is statistically equiva- lent to the whole (Kantelhardt, 2009). Financial time-series are an example of such complexity: their highly stochastic nature of complex dynamics and their suscep- tibility to exogenous factors often emerges as time-dependent properties or, more generally, non-stationarity. Self affinity and persistence are closely related: persistence holds for all time scales, where self-affinity holds. A key ingredient in determining the degree of persistence is provided by the autocorrelation function. For station- ary data (therefore for data where mean and variance do not change over time), the autocorrelation function for the incrementsΔxi, of some process{xt}t=0,...,N:
C(s) = 1 N−s N−s i=0 ΔxiΔxi+s (2.1)
characterizes two types of correlations: short-range correlation and long-range corre- lation. An exponentially declining autocorrelation function of the typeC(s) =e−s/T catheterizes short-range autocorrelation (examples are found for instance in AR
processes), whereas a power-law decay characterizes long-range autocorrelation:
C(s)∝s−γ (2.2)
with 0< γ <1. The exponent γ is referred to as thescaling exponent, and more generally a scaling law with a scaling exponentα, describes the behaviour of a certain
quantity F in terms of the parameterα: F(s) = sα. A system characterized by a
non-integer scaling exponent is calledfractal. Whether persistence, in general holds at least for a range of time-scales, in long-range correlation the decay is sufficiently slow such that a characteristic correlation time scale (or range) where the persistence holds cannot be defined, i.e.C(s)∝s−γ, at least asymptotically.
Finally the notion of self-affinity. The data is self-affine, whenever for a given factor
athe following scaling relation holds(Kantelhardt, 2009):
x(t)→aHx(at). (2.3)
Re-scaling timet by a factora, requires re-scalingx(t)by a factoraH, to obtain a statistically equivalent magnification (Kantelhardt, 2009). The type of self affinity is
driven by the Hurst exponentH. For long-range correlation,H=1−γ2. Therefore
self affinity with 12 < H < 1 corresponds to long-range autocorrelation. In strict sense, self-affinity encompasses fractality, i.e. the terms are not equivalent. In prac-
tice, literature refers to “fractal” wheneverH can be defined. Similarly long-range
dependence, long-range correlation/autocorrelation and power-law decay are used instinctively, all referring to eq. (2.2).
Several series are not characterized by a unique exponentH, thus they are not self-
affine: we talk of crossovers in the scaling exponent, identified by differentH appli-
cable at different time-scales. Crossovers can be be caused by non-stationarity in the data as well. Violations of either weak stationarity or strong stationarity can impact on the estimate ofH (andγ, or in generalα), therefore methods robust to e.g. trends are needed to correctly estimate the scaling exponent. Traditional approaches for estimating the scaling exponent under stationary time-series are (i) autocorrelation function analysis,not advisable since affected by the generally superimposed noise on the underlying process, and because at large time scales is of high variability (Kantel- hardt, 2009), (ii) Hurst re-scaled range analysis (classical references are Hurst, 1951; Mandelbrot and Wallis, 1969; Feder, 1988), (iii) Spectral analysis (Taqqu et al., 1995;
Rangarajan et al., 2000; Hunt, 1951) and (iv) fluctuation analysis (FA) (Peng, S. V. Buldyrev, Goldberger et al., 1992). Complex financial series can potentially preset non-stationary features (e.g. due to seasonal patterns or structural breaks). There- fore estimation methods robust to non-stationarity are preferable options, among these, there are methods relying on wavelet analysis (Koscielny-Bunde et al., 1998; Kantelhardt, Roman et al., 1995) and the DFA (Peng, S. V. Buldyrev, Havlin et al., 1994). The DFA (a detrended, non-stationarity robust version of FA) constitutes the methodological method of Publication III and constitutes the most used meth- ods for the estimation of the scaling coefficient. Chapter 4 discusses DFA in detail. Comparisons between the methods are provided in a number of studies (among them Heneghan et al., 2000; Delignieres et al., 2006; Mielniczuk et al., 2007; Bashan et al., 2008; Shao et al., 2012).
2.4.2 Applications in finance
The earliest discussion about long-range dependence properties in financial prices is motivated by the empirical evidence that the very early theories were failing to capture phenomena such as non-normality of returns, their fat tails, correlation in prices, cyclical patterns, and general evidence that prices do not behave as random walks. (Mandelbrot, 1997) is a collection and discussion over the early contributions to the development of a following wide literature in this direction. Following analyses are for instance those of (Mandelbrot, 1971), where investigating the problem of market inefficiency the slow-decaying correlation of in the price process is discussed. However, is much later when a financial literature based on these early findings exploded, i.e. when the early contributions to economic literature in the field of time dependence, cyclical pattern, and speed of the autocorrelation decay, met the math-physics literature on fractals (which goes back to e.g. Hurst, 1951; Mandelbrot and Wallis, 1969), also thanks to new methodological developments, such as (Peng, S. V. Buldyrev, Havlin et al., 1994; Taqqu et al., 1995). The following reviews some of the applications in finance, while the latter part focuses on DFA and analyses related to duration time-series, since relevant with respect to Publication III.
Mantegna et al., 1995 analyze six-years S&P 500 index data, finding that the scaling exponent of the power-law is constant over the same period and, that the distribution
of the differences in the index is not Gaussian. Returns have been analyzed in (Grau- Carles, 2000), where a number of methods are utilized to conclude that little temporal correlation is observed in return’s series. Similar conclusion on the returns’ scaling properties are drawn in e.g. (Grau-Carles, 2001; G. Oh et al., 2006; W.-X. Zhou, 2009), confirming the intuition that financial returns are of difficult predictability, neither weakly dependent nor related to most of the economic variables (Cont, 2005), while (Carbone et al., 2004) expands the complexity in the discussion by devising variability in the scaling exponent on high-frequency returns from the German market. On the other hand, (Grau-Carles, 2000) finds strong long-range dependence arising in the volatility series, aligned with a number of later analyses (e.g. Cizeau et al., 1997; Y. Liu et al., 1999; Yamasaki et al., 2005), Y. Wang et al., 2009 point out that under long- range dependence, standard econometric models such as GARCH and EGARCH are inadequate. Early application on exchange rates are those of Vandewalle, Ausloos and Boveroux, 1997; Vandewalle and Ausloos, 1997. More recently (G.-J. Wang et al., 2013) expanded the analysis to cross-correlation between a basket of currencies as well, while G. Cao, Xu et al., 2012 study the cross-correlation between Chinese stock and exchange markets. Applications in studying market efficiency via DFA include, e.g. Stoši´c et al., 2015; Tiwari et al., 2017; Y. Wang et al., 2009. Interestingly, (Lillo et al., 2004) shows that although the long-range correlation found in order placement in London Stock Exchange would stem for market inefficiency, there are long-range anti-correlations in trade size and liquidity whose overall netting effect drives the market closer to efficiency.Related analyses on coexisting factors implying long-range correlations and anti-correlations within the order book have also been proposed in (Bouchaud, Gefen et al., 2004). Furthermore, (e.g. Lillo et al., 2004; Bouchaud, Gefen et al., 2004; Bouchaud, Farmer et al., 2009; Tóth et al., 2011), discussed long- range autocorrelations, in particular those found in trade signs, relating it to price impact and on its persistence. Long-range correlation analyses has also concern business cycles, market periods (Czarnecki et al., 2008; Qian et al., 2004) and can identify forthcoming crashes (e.g. Grech et al., 2004). Connections with the common econometrics time-series literature are discusses for instance in (Torre et al., 2007;
Podobnik et al., 2008).Further references on long-range autocorrelation analyses can
be found in (e.g. Lillo et al., 2004, Section II), while to (e.g. Baillie, 1996) for a general review on the econometric approach for long-range correlation modelling.
2.4.3 Applications in duration analysis
While the above is a general overview of the different application of long-range analyses in finance, the following focuses on the most relevant literature related to inter-event long-range correlation analyses, concerning Publication III.
Inter-trade times for a three-year sample of 30 stocks extracted from the TAQ database, were analyzed in (Ivanov, Yuen, Podobnik et al., 2004), which analyzes the scaling properties of the density function of inter-trades duration, finds that the inter-trade times exhibit power-law correlated behavior within a trading day, devising the possi- bility of universal scaling patterns common to different industry sectors. Importantly, their results provide the first crossover evidence in the scaling exponent extracted via DFA. Inter-trade durations were also considered in (Ivanov, Yuen and Perakakis, 2014) for stocks traded at NYSE and NASDAQ, power-law correlations in inter-trade times are influenced by the market structure and coupled with the power-law correlations of absolute returns and volatility, motivating the association analysis in Publication III. For inter-trade durations, also (Jiang et al., 2009) finds strong evidence of crossovers between two different power-scaling regimes from 23 Chinese. Fractal properties of inter-cancellations durations have been analyzed for 18 stocks in Shenzhen exchange in (Gu et al., 2014), using different variants of standard DFA, and in (Ni et al., 2010), both confirming long-range autocorrelation in cancellation series.