American Institute of Mathematical Sciences

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June  2021, 6(2): 139-158. doi: 10.3934/puqr.2021007

Correlated squared returns

Dilip B. Madan 1,, and  King Wang 2,

1. 

Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA
2. 

Derivative Product Strats, Morgan Stanley, New York, NY 10036, USA

Email: dbm@umd.edu

Received  April 17, 2020 Accepted  June 03, 2021 Published  June 2021

Joint densities for a sequential pair of returns with weak autocorrelation and strong correlation in squared returns are formulated. The marginal return densities are either variance gamma or bilateral gamma. Two-dimensional matching of empirical characteristic functions to its theoretical counterpart is employed for dependency parameter estimation. Estimations are reported for 3920 daily return sequences of one thousand days. Path simulation is done using conditional distribution functions. The paths display levels of squared return correlation and decay rates for the squared return autocorrelation function that are comparable to these magnitudes in daily return data. Regressions of log characteristic functions at different time points are used to estimate time scaling coefficients. Regressions of these time scaling coefficients on squared return correlations support the view that autocorrelation in squared returns slows the rate of passage of economic time. An analysis of financial markets for 2020 in comparison with 2019 displays a post-COVID slowdown in financial markets.

Keywords: Correlated gamma processes, Joint characteristic functions, Digital moment estimation, Path simulation.
Mathematics Subject Classification: G11, G13, G17.
Citation: Dilip B. Madan, King Wang. Correlated squared returns. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 139-158. doi: 10.3934/puqr.2021007
References:
[1]

Andersen, T. G., Bollerslev, T., Christoffersen, P. and Diebold, F. X., Practical volatility and correlation modeling for financial market risk management, In: Carey M. and Stulz, R. M. (Eds.), The Risks of Financial Institutions, University of Chicago Press, Chicago, 2007.

[2]

Anderson, T. W. and Darling, D. A., Asymptotic theory of certain’ goodness of fit’ criteria based on stochastic processes, The Annals of Mathematical Statistics, 1952, 23: 193-212. doi: 10.1214/aoms/1177729437.

[3]

Baillie, R. T., and Chung, H., Estimation of garch models from the autocorrelations of the squares of a process, Journal of Time Series Analysis, 1999, 22: 631-650.

[4]

Bodie, Z., On the risk of stocks in the long run, Financial Analysts Journal, 1995, 51: 18-22.

[5]

Bollerslev, T. and Mikkelsen, H. O., Modelling and pricing long memory in stock market volatility, Journal of Econometrics, 1996, 73: 151-184. doi: 10.1016/0304-4076(95)01736-4.

[6]

Buchmann, B., Madan, D. B. and Lu, K., Weak subordination of multivariate Lévy processes and variance generalized gamma convolutions, Bernoulli, 2019, 25: 742-770.

[7]

Carr, P. and Madan, D. B., Joint modeling of VIX and SPX options at a single and common maturity with risk management applications, IIE Transactions, 2014, 46: 1125-1131. doi: 10.1080/0740817X.2013.857063.

[8]

Crato, N. and de Lima, P. J. F., Long range dependence in the conditional variance of stock returns, Economics Letters, 1994, 45: 281-285. doi: 10.1016/0165-1765(94)90024-8.

[9]

Ding, Z. and Granger, C. W. J., Modeling volatility presistence of speculative markets: A new approach, Journal of Econometrics, 1996, 73: 185-215. doi: 10.1016/0304-4076(95)01737-2.

[10]

Ding, Z., Granger, C. W. J. and Engle, R. F., A long memory property of stock returns and a new model, Journal of Empirical Finance, 1993: 83-106.

[11]

Fama, E. F. and French, K. R., Long-horizon returns, Review of Asset Pricing Studies, 2018, 8: 232-252. doi: 10.1093/rapstu/ray001.

[12]

Feuerverger, A. and McDunnough, P., On the efficiency of empirical characteristic function procedures, Journal of the Royal Statictical Society, Series B, Methodological, 1981, 43: 20-27.

[13]

Jondeau, E., Poon, S.H., and Rockinger, M., Financial modeling under non-gaussian distributions, Springer, Berlin, 2007.

[14]

Khintchine, A. Y., Limit laws of sums of independent random variables, ONTI, Moscow, Russian, 1938.

[15]

Küchler, U. and Tappe, S., Bilateral gamma distributions and processes in financial mathematics, Stochastic Processes and Their Applications, 2008, 118: 261-283. doi: 10.1016/j.spa.2007.04.006.

[16]

Lévy, P., Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1937.

[17]

Madan, D. B., Estimating parametric models of probability distributions, Methodology and Computing in Applied Probability, 2015, 17: 823-831. doi: 10.1007/s11009-014-9409-4.

[18]

Madan D., Carr, P. and Chang, E., The variance gamma process and option pricing, European Finance Review, 1998, 2: 79-105. doi: 10.1023/A:1009703431535.

[19]

Madan, D. B. and Schoutens, W., Self-similarity in long horizon returns, Mathematical Finance, 2020, 30: 1368-1391.

[20]

Madan, D. B. and Schoutens, W., Nonlinear Valuation and Non-Gaussian Risks, Cambridge University Press, Cambridge, UK forthcoming.

[21]

Madan, D. B., Schoutens, W. and Wang, K., Measuring and monitoring the efficiency of markets, International Journal and Theoretical and Applied Finance, 2017, 20.

[22]

Madan, D. B., Schoutens, W. and Wang, K., Bilateral multiple gamma returns: Their risks and rewards, International Journal of Financial Engineering, 2020, 07(01): 2050008, https://doi.org/10.1142/S2424786320500085.

[23]

Madan D. B. and Seneta, E., The variance gamma (VG) model for share market returns, Journal of Business, 1990, 63: 511-524. doi: 10.1086/296519.

[24]

Madan, D. B. and Wang, K., Asymmetries in financial returns, International Journal of Financial Engineering, 2017.

[25]

Merton, R. C. and Samuelson, P. A., Fallacy of the log-normal approximation to portfolio decision-making over many periods, Journal of Financial Economics, 1974, 1: 67-94. doi: 10.1016/0304-405X(74)90009-9.

[26]

Sato, K., Lévy processes and infinitely divisible distributions, Cambridge Uinversity Press, Cambridge UK, 1999.

[27]

Schoutens, W., Lévy processes in finance, John Wiley and Sons, Hoboken, New Jersey, 2003.

[28]

Shepard, N., Stochastic volatility models, In: Durlauf, S. N. and Blume, L. E. (Eds.), Macro econometrics and Time Series Analysis, The New Palgrave Economics Collection, Palgrave and Macmillan, London, 2010.

[29]

Singleton, K. J., Estimation of affine asset pricing models using the empirical characteristic function, Journal of Econometrics, 2001, 102: 111-141. doi: 10.1016/S0304-4076(00)00092-0.

[30]

Taylor, S. J., Modeling financial time series, Wiley, Chichester, 1986.

show all references

References:
[1]

Andersen, T. G., Bollerslev, T., Christoffersen, P. and Diebold, F. X., Practical volatility and correlation modeling for financial market risk management, In: Carey M. and Stulz, R. M. (Eds.), The Risks of Financial Institutions, University of Chicago Press, Chicago, 2007.

[2]

Anderson, T. W. and Darling, D. A., Asymptotic theory of certain’ goodness of fit’ criteria based on stochastic processes, The Annals of Mathematical Statistics, 1952, 23: 193-212. doi: 10.1214/aoms/1177729437.

[3]

Baillie, R. T., and Chung, H., Estimation of garch models from the autocorrelations of the squares of a process, Journal of Time Series Analysis, 1999, 22: 631-650.

[4]

Bodie, Z., On the risk of stocks in the long run, Financial Analysts Journal, 1995, 51: 18-22.

[5]

Bollerslev, T. and Mikkelsen, H. O., Modelling and pricing long memory in stock market volatility, Journal of Econometrics, 1996, 73: 151-184. doi: 10.1016/0304-4076(95)01736-4.

[6]

Buchmann, B., Madan, D. B. and Lu, K., Weak subordination of multivariate Lévy processes and variance generalized gamma convolutions, Bernoulli, 2019, 25: 742-770.

[7]

Carr, P. and Madan, D. B., Joint modeling of VIX and SPX options at a single and common maturity with risk management applications, IIE Transactions, 2014, 46: 1125-1131. doi: 10.1080/0740817X.2013.857063.

[8]

Crato, N. and de Lima, P. J. F., Long range dependence in the conditional variance of stock returns, Economics Letters, 1994, 45: 281-285. doi: 10.1016/0165-1765(94)90024-8.

[9]

Ding, Z. and Granger, C. W. J., Modeling volatility presistence of speculative markets: A new approach, Journal of Econometrics, 1996, 73: 185-215. doi: 10.1016/0304-4076(95)01737-2.

[10]

Ding, Z., Granger, C. W. J. and Engle, R. F., A long memory property of stock returns and a new model, Journal of Empirical Finance, 1993: 83-106.

[11]

Fama, E. F. and French, K. R., Long-horizon returns, Review of Asset Pricing Studies, 2018, 8: 232-252. doi: 10.1093/rapstu/ray001.

[12]

Feuerverger, A. and McDunnough, P., On the efficiency of empirical characteristic function procedures, Journal of the Royal Statictical Society, Series B, Methodological, 1981, 43: 20-27.

[13]

Jondeau, E., Poon, S.H., and Rockinger, M., Financial modeling under non-gaussian distributions, Springer, Berlin, 2007.

[14]

Khintchine, A. Y., Limit laws of sums of independent random variables, ONTI, Moscow, Russian, 1938.

[15]

Küchler, U. and Tappe, S., Bilateral gamma distributions and processes in financial mathematics, Stochastic Processes and Their Applications, 2008, 118: 261-283. doi: 10.1016/j.spa.2007.04.006.

[16]

Lévy, P., Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1937.

[17]

Madan, D. B., Estimating parametric models of probability distributions, Methodology and Computing in Applied Probability, 2015, 17: 823-831. doi: 10.1007/s11009-014-9409-4.

[18]

Madan D., Carr, P. and Chang, E., The variance gamma process and option pricing, European Finance Review, 1998, 2: 79-105. doi: 10.1023/A:1009703431535.

[19]

Madan, D. B. and Schoutens, W., Self-similarity in long horizon returns, Mathematical Finance, 2020, 30: 1368-1391.

[20]

Madan, D. B. and Schoutens, W., Nonlinear Valuation and Non-Gaussian Risks, Cambridge University Press, Cambridge, UK forthcoming.

[21]

Madan, D. B., Schoutens, W. and Wang, K., Measuring and monitoring the efficiency of markets, International Journal and Theoretical and Applied Finance, 2017, 20.

[22]

Madan, D. B., Schoutens, W. and Wang, K., Bilateral multiple gamma returns: Their risks and rewards, International Journal of Financial Engineering, 2020, 07(01): 2050008, https://doi.org/10.1142/S2424786320500085.

[23]

Madan D. B. and Seneta, E., The variance gamma (VG) model for share market returns, Journal of Business, 1990, 63: 511-524. doi: 10.1086/296519.

[24]

Madan, D. B. and Wang, K., Asymmetries in financial returns, International Journal of Financial Engineering, 2017.

[25]

Merton, R. C. and Samuelson, P. A., Fallacy of the log-normal approximation to portfolio decision-making over many periods, Journal of Financial Economics, 1974, 1: 67-94. doi: 10.1016/0304-405X(74)90009-9.

[26]

Sato, K., Lévy processes and infinitely divisible distributions, Cambridge Uinversity Press, Cambridge UK, 1999.

[27]

Schoutens, W., Lévy processes in finance, John Wiley and Sons, Hoboken, New Jersey, 2003.

[28]

Shepard, N., Stochastic volatility models, In: Durlauf, S. N. and Blume, L. E. (Eds.), Macro econometrics and Time Series Analysis, The New Palgrave Economics Collection, Palgrave and Macmillan, London, 2010.

[29]

Singleton, K. J., Estimation of affine asset pricing models using the empirical characteristic function, Journal of Econometrics, 2001, 102: 111-141. doi: 10.1016/S0304-4076(00)00092-0.

[30]

Taylor, S. J., Modeling financial time series, Wiley, Chichester, 1986.

Figure 1.  Fit of Variance Gamma (VG) and Bilateral Gamma (BG) to the stationary equilibrium return distribution of a GARCH(1,1) process. The observed tail probabilities are represented by circles. The VG and BG model tail probabilities are shown by red and black dots.
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Figure 2.  Sample paths for 63 days of returns from VGCTC and BGCTC. The VGCTC paths employ Cases 7 and 2 from the respective Tables of representative parameter values.
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Figure 3.  Bilateral Gamma parameter quantiles for the return at the five-, ten-, fifteen-, and twenty-day horizons. The parameters were fit to simulated returns for the period using a 512 point quantization of the full set of 3920 estimated parameter values. Presented are the quantiles for the 512 sets of long horizon return densities.
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Figure 4.  VGCTC second-period variance conditional on the first-period squared return. The eight cases refer to Table 2.
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Figure 5.  BGCTC second-period variance conditional on the first-period squared return. The eight cases refer to Table 4.
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Table 1.  Parameter quantiles for VGCTC. Displayed are the quantiles for VGCTC model parameters estimated for data on a thousand returns for 196 stocks taken at 20 randomly selected end dates between January 2008 and December 2019.
Quantile $\sigma$ $\nu$ $\theta$ $\alpha$
1 0.0215 0.2098 2.1693 0.02
5 0.1321 0.3169 6.2207 0.02
10 0.1481 0.3785 8.5184 0.02
25 0.1819 0.4843 11.867 0.3091
50 0.2316 0.6068 15.575 0.5431
75 0.3148 0.7485 20.244 0.7738
90 0.4367 0.9314 26.874 0.9795
95 0.5278 1.1636 36.265 0.98
99 0.6602 6.5334 255.21 0.98
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Quantile $\sigma$ $\nu$ $\theta$ $\alpha$
1 0.0215 0.2098 2.1693 0.02
5 0.1321 0.3169 6.2207 0.02
10 0.1481 0.3785 8.5184 0.02
25 0.1819 0.4843 11.867 0.3091
50 0.2316 0.6068 15.575 0.5431
75 0.3148 0.7485 20.244 0.7738
90 0.4367 0.9314 26.874 0.9795
95 0.5278 1.1636 36.265 0.98
99 0.6602 6.5334 255.21 0.98
Table 2.  Representative VGCTC parameters. The 3920 parameter sets for 196 stocks taken at 20 randomly selected end dates for data on a thousand returns were quantized into 8 representative centroids. The final column shows the proportion represented by each centroid.
Case $\sigma$ $\nu$ $\theta$ $\alpha$ Prop.
1 0.0153 0.5785 0.0015 0.9038 15.43
2 0.0144 0.5605 0.0016 0.6572 11.12
3 0.0175 0.7606 0.0017 0.6935 14.65
4 0.0194 0.7224 0.0017 0.3558 17.51
5 0.0126 0.3350 0.0014 0.9397 10.94
6 0.0285 1.1169 0.0040 0.4664 8.62
7 0.0146 0.4685 0.0015 0.5550 10.13
8 0.0156 0.4822 0.0016 0.3177 11.60
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Case $\sigma$ $\nu$ $\theta$ $\alpha$ Prop.
1 0.0153 0.5785 0.0015 0.9038 15.43
2 0.0144 0.5605 0.0016 0.6572 11.12
3 0.0175 0.7606 0.0017 0.6935 14.65
4 0.0194 0.7224 0.0017 0.3558 17.51
5 0.0126 0.3350 0.0014 0.9397 10.94
6 0.0285 1.1169 0.0040 0.4664 8.62
7 0.0146 0.4685 0.0015 0.5550 10.13
8 0.0156 0.4822 0.0016 0.3177 11.60
Table 3.  Parameter quantiles for BGCTC. Displayed are quantiles of estimated parameters for the BGCTC model. Estimates were obtained for 196 stocks taken at 20 random end dates with a thousand prior returns between January 2008 and December 2019.
Quantile $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $\alpha$
1 0 0.1782 0.0033 0.1615 0.02
5 0.0043 0.9129 0.0042 0.8294 0.02
10 0.0050 1.0638 0.0048 0.9918 0.02
25 0.0064 1.3026 0.0062 1.2678 0.2526
50 0.0084 1.5988 0.0082 1.5789 0.5010
75 0.0118 2.0148 0.0120 2.0002 0.7316
90 0.0171 2.5118 0.0186 2.5029 0.9339
95 0.0215 2.9618 0.0256 2.9599 0.98
99 0.0327 4.4274 0.5241 4.3772 0.98
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Quantile $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $\alpha$
1 0 0.1782 0.0033 0.1615 0.02
5 0.0043 0.9129 0.0042 0.8294 0.02
10 0.0050 1.0638 0.0048 0.9918 0.02
25 0.0064 1.3026 0.0062 1.2678 0.2526
50 0.0084 1.5988 0.0082 1.5789 0.5010
75 0.0118 2.0148 0.0120 2.0002 0.7316
90 0.0171 2.5118 0.0186 2.5029 0.9339
95 0.0215 2.9618 0.0256 2.9599 0.98
99 0.0327 4.4274 0.5241 4.3772 0.98
Table 4.  Representative parameter sets for BGCTC. The 3920 parameter estimates for 196 stocks taken at 20 random end dates between January 2008 and December 2019 were quantized into eight centroids. The last column shows the proportions of points represented by each centroid.
Case $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $\alpha$ Prop.
1 0.0091 1.5746 0.0102 1.3424 0.9284 10.67
2 0.0076 1.9910 0.0080 1.7468 0.9532 18.29
3 0.0050 3.5498 0.0058 2.8806 0.8854 8.20
4 0.0087 1.7630 0.0102 1.3803 0.2893 9.52
5 0.0091 1.7250 0.0098 1.5000 0.9541 10.58
6 0.0115 1.3579 0.0125 1.1849 0.7729 17.00
7 0.0064 2.4888 0.0073 2.0861 0.9140 13.14
8 0.0158 1.0939 0.0178 0.9166 0.3755 12.60
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Case $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $\alpha$ Prop.
1 0.0091 1.5746 0.0102 1.3424 0.9284 10.67
2 0.0076 1.9910 0.0080 1.7468 0.9532 18.29
3 0.0050 3.5498 0.0058 2.8806 0.8854 8.20
4 0.0087 1.7630 0.0102 1.3803 0.2893 9.52
5 0.0091 1.7250 0.0098 1.5000 0.9541 10.58
6 0.0115 1.3579 0.0125 1.1849 0.7729 17.00
7 0.0064 2.4888 0.0073 2.0861 0.9140 13.14
8 0.0158 1.0939 0.0178 0.9166 0.3755 12.60
Table 5.  Squared return correlations. For 196 stocks taken at 20 random end dates squared return correlations were estimated from paths of 252 days. For the data column, they were prior returns. For the models they were from a single simulated path using estimated parameter values based on a thousand days of data.
Quantile Data VGCTC BGCTC
1 0.0265 0.1034 0.1457
5 0.0605 0.1659 0.1928
10 0.0913 0.1906 0.2279
25 0.1639 0.2352 0.2983
50 0.2459 0.2850 0.3474
75 0.3187 0.3281 0.4076
90 0.3820 0.3649 0.4646
95 0.4241 0.3936 0.4912
99 0.4972 0.4786 0.5449
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Quantile Data VGCTC BGCTC
1 0.0265 0.1034 0.1457
5 0.0605 0.1659 0.1928
10 0.0913 0.1906 0.2279
25 0.1639 0.2352 0.2983
50 0.2459 0.2850 0.3474
75 0.3187 0.3281 0.4076
90 0.3820 0.3649 0.4646
95 0.4241 0.3936 0.4912
99 0.4972 0.4786 0.5449
Table 6.  Decay rate quantiles. For 196 stocks and 20 randomly selected end dates between January 2008 and December 2019 from a sequence of 252 squared returns an autocorrelation function was computed. The log autocorrelation function was regressed on the logarithm of the time lag. Displayed are quantiles for slope coefficients. The data column employed return sequences. The models used paths simulated from parameter estimates.
Quantile Data VGCTC BGCTC
1 −0.0718 −0.0580 −0.0210
5 −0.0175 −0.0249 0.0436
10 0.0055 −0.0075 0.0726
25 0.0482 0.0238 0.1158
50 0.1100 0.0610 0.1943
75 0.1888 0.1105 0.3393
90 0.2843 0.1714 0.6415
95 0.3520 0.2206 0.9243
99 0.5271 0.2760 1.6103
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Quantile Data VGCTC BGCTC
1 −0.0718 −0.0580 −0.0210
5 −0.0175 −0.0249 0.0436
10 0.0055 −0.0075 0.0726
25 0.0482 0.0238 0.1158
50 0.1100 0.0610 0.1943
75 0.1888 0.1105 0.3393
90 0.2843 0.1714 0.6415
95 0.3520 0.2206 0.9243
99 0.5271 0.2760 1.6103
Table 7.  VGCTC scaling coefficient quantiles. Displayed are quantiles of scaling coefficients. The coefficients are obtained by regressing the empirical cumulant at the longer horizon on the same for the shorter horizon. The empirical cumulant is based on 10000 simulated VGCTC paths.
Quantile $c_{5,10}$ $c_{5,15}$ $c_{5,20}$
1 0.6178 0.6160 0.6271
5 0.6774 0.6808 0.6821
10 0.7203 0.7177 0.7187
25 0.7776 0.7768 0.7695
50 0.8545 0.8548 0.8453
75 0.9601 0.9625 0.9423
90 1.0890 1.0886 1.0803
95 1.1828 1.2213 1.2029
99 2.0039 3.0077 4.0080
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Quantile $c_{5,10}$ $c_{5,15}$ $c_{5,20}$
1 0.6178 0.6160 0.6271
5 0.6774 0.6808 0.6821
10 0.7203 0.7177 0.7187
25 0.7776 0.7768 0.7695
50 0.8545 0.8548 0.8453
75 0.9601 0.9625 0.9423
90 1.0890 1.0886 1.0803
95 1.1828 1.2213 1.2029
99 2.0039 3.0077 4.0080
Table 8.  BGCTC scaling coefficient quantiles. Displayed are quantiles of scaling coefficients. The coefficients are obtained by regressing the empirical cumulant at the longer horizon on the same for the shorter horizon. The empirical cumulant is based on 10000 simulated BGCTC paths.
Quantile $c_{5,10}$ $c_{5,15} $ $c_{5,20}$
1 0.6673 0.6851 0.6513
5 0.7673 0.7741 0.7568
10 0.8164 0.8346 0.8125
25 0.9364 0.9462 0.9453
50 1.2345 1.2530 1.2425
75 1.9509 2.3806 2.4516
90 2.0126 3.0054 4.0017
95 2.0346 3.0318 4.0375
99 2.1364 3.1371 4.1237
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Quantile $c_{5,10}$ $c_{5,15} $ $c_{5,20}$
1 0.6673 0.6851 0.6513
5 0.7673 0.7741 0.7568
10 0.8164 0.8346 0.8125
25 0.9364 0.9462 0.9453
50 1.2345 1.2530 1.2425
75 1.9509 2.3806 2.4516
90 2.0126 3.0054 4.0017
95 2.0346 3.0318 4.0375
99 2.1364 3.1371 4.1237
Table 9.  Regression of time scaling coefficients on squared return correlations (SQRC) dependent. Presented are the results of regressing time scaling coefficients at the ten-, fifteen-, and twenty-day horizon on the associated squared return correlations. t-statistics in parenthesis.
Variable Constant SQRC
VGCTC
$c_{5,10}$ 0.9391 −0.2335
(62.48) −(4.66)
$c_{5,15}$ 0.9715 −0.3410
(66.91) (−6.71)
$c_{5,20}$ 0.9398 −0.2758
(62.38) −(5.49)
BGCTC
$c_{5,10}$ 1.9408 −2.6867
(87.57) (−30.06)
$c_{5,15}$ 2.6457 −4.8397
(66.91) (−30.35)
$c_{5,20}$ 3.2558 −6.8205
(58.23) (−30.25)
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Variable Constant SQRC
VGCTC
$c_{5,10}$ 0.9391 −0.2335
(62.48) −(4.66)
$c_{5,15}$ 0.9715 −0.3410
(66.91) (−6.71)
$c_{5,20}$ 0.9398 −0.2758
(62.38) −(5.49)
BGCTC
$c_{5,10}$ 1.9408 −2.6867
(87.57) (−30.06)
$c_{5,15}$ 2.6457 −4.8397
(66.91) (−30.35)
$c_{5,20}$ 3.2558 −6.8205
(58.23) (−30.25)
Table 10.  p-value quantiles. Presented are selected quantiles of p-values for distributional equality between 2019 and 2020 for 824 stocks.
Quantile p-value
1 2.92e−9
5 2.8e−7
10 5.92e−6
25 0.00022
50 0.0061
75 0.0466
90 0.1862
95 0.3316
99 0.6267
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Quantile p-value
1 2.92e−9
5 2.8e−7
10 5.92e−6
25 0.00022
50 0.0061
75 0.0466
90 0.1862
95 0.3316
99 0.6267
Table 11.  Bilateral gamma parameter quantiles.
Quantile 2019 2020
$b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$
1 0.0001 1.1635 0.0007 0.7347 0.0069 0.5897 0.0084 0.4720
5 0.0003 1.5256 0.0012 1.0004 0.0095 0.7328 0.0117 0.6220
10 0.0006 1.7956 0.0021 1.1641 0.0109 0.8460 0.0136 0.6921
25 0.0019 2.3483 0.0042 1.5314 0.0138 0.9958 0.0169 0.8474
50 0.0048 3.4532 0.0068 2.3362 0.0182 1.2260 0.0215 1.1009
75 0.0067 16.343 0.0095 6.7286 0.0250 1.5685 0.0272 1.4138
90 0.0092 113.26 0.0129 18.250 0.0347 2.1524 0.0331 1.8616
95 0.0110 184.27 0.0148 93.635 0.0413 2.6990 0.0370 2.2755
99 0.0171 312.31 0.0198 123.74 0.0583 5.4866 0.0503 4.6835
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Quantile 2019 2020
$b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$ $b_{p}$ $c_{p}$ $b_{n}$ $c_{n}$
1 0.0001 1.1635 0.0007 0.7347 0.0069 0.5897 0.0084 0.4720
5 0.0003 1.5256 0.0012 1.0004 0.0095 0.7328 0.0117 0.6220
10 0.0006 1.7956 0.0021 1.1641 0.0109 0.8460 0.0136 0.6921
25 0.0019 2.3483 0.0042 1.5314 0.0138 0.9958 0.0169 0.8474
50 0.0048 3.4532 0.0068 2.3362 0.0182 1.2260 0.0215 1.1009
75 0.0067 16.343 0.0095 6.7286 0.0250 1.5685 0.0272 1.4138
90 0.0092 113.26 0.0129 18.250 0.0347 2.1524 0.0331 1.8616
95 0.0110 184.27 0.0148 93.635 0.0413 2.6990 0.0370 2.2755
99 0.0171 312.31 0.0198 123.74 0.0583 5.4866 0.0503 4.6835
Table 12.  Squared return correlation quantiles.
Quantile 2019 2020
1 0.0095 0.0650
5 0.0341 0.1212
10 0.0553 0.1593
25 0.1155 0.2390
50 0.2095 0.3463
75 0.2854 0.4370
90 0.3500 0.5212
95 0.3766 0.5743
99 0.4454 0.6511
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Quantile 2019 2020
1 0.0095 0.0650
5 0.0341 0.1212
10 0.0553 0.1593
25 0.1155 0.2390
50 0.2095 0.3463
75 0.2854 0.4370
90 0.3500 0.5212
95 0.3766 0.5743
99 0.4454 0.6511
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