Calibration of the Model
One of the most important issues for the practical applications is the estimation of the three unknown parameters of MRW model (ct, A, L) with the real data.
where At is the scale oflog-returns (e. g, 1-, 5-, 10-, 20-min etc.). The parameter a2 can be then estimated with the linear regression of the Var [1AtXAt [k]] on At as it is shown in Fig. 8.
Estimation of intermittency coefficient A and integral scale L is much more complicated, because they define the unobserved log-volatility process! At [k]. In Bacry et al. (2001) it is shown that the magnitude correlation function
Cp(x,/) = E [|1xX[k C l] |p, |1xX[k] n,
where the pre-factor is defined as
K2p = Lpa2p (2p - 1)!! dui... dUp |u - Uj |
C(r,/) ~-A2 ln^(23)
In other words, magnitude correlation function, for small enough r, has similar behavior to the correlation function of underlying log-volatility process [k]. Thus, regressing C (r, l) on log l one can estimate the parameter A2. Finally, the integral scale L can be obtained as the scale l after which autocorrelation function (23) is indistingushable from noise.
Figure 9 illustrates fits of the A2 and L using relation (23). Measures of the slope and intercept of Cr (l) ~ ln (l) provide good estimate of respectively A2 and L, though the estimation of the integral scale L is typically worse in comparison with estimation of A2. The algorithm of determining L could be summarized as follows:
1. Set the size of small rolling window;
2. Scan values of magnitude function within rolling window;
3. Stop scanning if all elements within rolling window belong to the interval of insignificance;
4. Set L for the index of the middle point of rolling window at its last position.
Fig. 9 Magnitude correlation function C (r, l) increments of MRW process sample of length 221 for A2 = 0.06, a = 7.5 • 10~5, L = 2048 and r = 15. Black solid line represents linear regression (23). Horizontal dashed lines represent insignificance interval and vertical dashed line denotes estimated value of L. The estimated A2 equals to 0.0623 and estimated L is 1905
However, this algorithm strongly depends on the choice of the rolling window size x and requires additional validation of the results.
Alternative way of estimation of intermittency coefficient A2 involves estimation of the multifractal spectrum £q = f (q) of the process. Given the analytical expression (11) one can then estimate A2 with the least squares estimator. Straightforward estimation of £q requires calculation of moments of increments Mq(l) as a function of scale l using the definition (4) and then regressing log Mq(l) on log l for different values of q, implying relation (5). Results of estimation of the multifractal spectrum for MRW process are presented in Fig. 10. One can see good agreement of the empirical spectrum with theoretical prediction up to orders of q = 6. The divergence of analytical and theoretical spectrum for higher values of q results from the insufficient sample size. Alternative methods of estimation of multifractal spectrum are based on the wavelet transform—so - called Wavelet Transform Modulus Maxima (WTMM) (Arneodo et al. 1998a) and detrended fluctuation analysis: Multifractal Detrended Fluctuation Analysis (MF-DFA) (Kantelhardt et al. 2002) and Multifractal Detrended Moving Average (MF-DMA) (Gu and Zhou 2010). However it should be noted that all these methods are subjected to the bias for large values of q and in real cases due to short observed realizations are not efficient with respect to estimation of A2.
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