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Components of Volatility and their Empirical Measures

Components of Volatility and their Empirical Measures. DIPANKOR COONDOO Economic Research Unit, Indian Statistical Institute, Kolkata PARAMITA MUKHERJEE Monetary Research Project, ICRA Limited, Kolkata. Notions of Volatility.

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Components of Volatility and their Empirical Measures

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  1. Components of Volatility and their Empirical Measures DIPANKOR COONDOO Economic Research Unit, Indian Statistical Institute, Kolkata PARAMITA MUKHERJEE Monetary Research Project, ICRA Limited, Kolkata

  2. Notions of Volatility Of Financial Analysts: Variability of a financial variable as measured by its Std. Dev. Of Econometricians: Conditional Heteroskedasticity

  3. Other Related Issues • Historical Volatility - Non Parametric Measure • Stochastic Volatility - GARCH-based Parametric Analysis • Changing Volatility - Rolling Sample Measure – Can be examined both in Historical & Stochastic Set up

  4. What led to what I talk about here? • Non-comparability of volatility of variables measured in different units • Basis for comparison of the Volatilities of FIIN to India and BSE return, say • Are there different aspects of volatility that need to be compared?

  5. Three Components of Volatility • Strength : Range of Amplitude of Fluctuation due to Volatility • Duration : Portion of Time the Variable is in Volatile State • Persistence: Inertia of large and small fluctuations

  6. Strength of Volatility Green has less strength than Blue

  7. Duration of Volatility Volatile State Normal State

  8. Persistence of Volatility Blue is more persistent than Black

  9. The Decomposition Methodology • Given Series has trend and volatility • An ARIMA with GARCH error will fit well Step 1: Fit ARIMA. Get the residuals e(t), T = 1, T. Standardise these residuals as w(t) = abs (e(t)/s), t = 1,T, where s = std. dev(e(t),t = 1,t)

  10. Note that w(t) is non-negative, by construction.Step 2: Estimate the PDF of w(t). We used non-parametric kernel method of density estimation of Silverman (1986). For every observed value of w, the ordinate of the estimated PDF is The Decomposition Methodology

  11. The Decomposition Methodology

  12. Table 1: Summary Descriptive Statistics

  13. Table 2: Results of Unit Root Tests BRET CMR FIIN -16.48 -7.73 -9.11 ADF-statistic -3.97 -2.87 -1.94 5% Critical Value Intercept No trend or Trend and Model Selected intercept intercept 2 3 4 lag order

  14. Table 3: Results of GARCH (1,1) Estimation

  15. FIIN Density

  16. BRET Density

  17. Table 4. Variable-specific Estimates of Volatility Components Based on Entire Sample

  18. Table 5: A Summary of Rolling Sample Estimation Returns Volatility Window- Mean/CV Variable Component width BRET CMR FIIN 15-day mean 0.66 0.67 0.63 cv 0.51 1.12 0.51 S 90-day mean 0.68 0.75 0.65 cv 0.22 0.54 0.33 Entire sample 0.692 0.714 0.673 15-day mean 0.6 0.72 0.63 cv 0.23 0.12 0.17 D 90-day mean 0.7 0.79 0.71 cv 0.09 0.05 0.06 Entire sample 0.773 0.807 0.769 15-day mean -0.01 0.24 -0.01 cv -47.31 0.91 -15.78 P 90-day mean 0.2 0.45 0.08 cv 0.67 0.33 1.54 Entire sample 0.25 0.51 0.22

  19. S Measure for 15-day Window Width 7 6 BRET CMR 5 FIIN 4 3 2 1 0 1 32 63 94 125 156 404 466 497 528 559 590 652 807 187 218 249 280 311 342 373 435 621 683 714 745 776 Note: Scales shifted for BRET and CMR

  20. D Measure for 15-day Window Width 2.25 2.05 1.85 1.65 1.45 1.25 1.05 0.85 0.65 0.45 0.25 1 33 65 97 257 321 353 385 417 449 481 513 545 577 129 161 193 225 289 609 641 673 705 737 769 801 Note: Scales shifted for BRET and CMR BRET CMR FIIN

  21. P Measure for 15-day Window Width 3 BRET CMR FIIN 2.5 2 1.5 1 0.5 0 -0.5 -1 1 27 53 79 105 157 235 287 339 365 417 469 495 547 599 651 677 729 781 807 131 183 209 261 313 391 443 521 573 625 703 755 Dotted lines are shifted scales for respective variables

  22. S Measure for 90-day Window Width 2.75 2.5 2.25 BRET CMR FIIN 2 1.75 1.5 1.25 1 0.75 0.5 0.25 1 33 65 97 129 161 193 225 257 449 481 513 545 577 609 289 321 353 385 417 641 673 705 737

  23. D Measure for 90-day Window Width 1.1 1 0.9 0.8 0.7 0.6 BRET CMR FIIN 0.5 1 33 65 97 129 161 193 225 257 289 321 353 385 417 449 481 513 545 577 609 641 673 705 737 Note: Scale shifted for BRET

  24. P Measure for 90-day Window Width 1.2 BRET_90 CMR_90 FIIN 90 1 0.8 0.6 0.4 0.2 0 -0.2 1 29 57 85 113 141 169 197 225 253 281 309 337 365 393 421 449 477 505 533 561 589 617 645 673 701 729 Dotted lines are shifted scales for respective variables

  25. Table 6: Correlation between day to day variations of estimated volatility components for different pairs of variables Volatility Window-width Correlation for the variable-pair component BRET-CMR BRET-FIIN CMR-FIIN 15-day -0.02 0.23 0.06 S 90-day 0.43 0.51* 0.25 15-day -0.34 0.05 0.06 D 90-day -0.38 0.23 0.19 15-day -0.12 0.07 0.05 P 90-day -0.23 -0.16 0.42

  26. Table 7A: Component-wise Forecast : Strength BRET_S15=C(1)+C(2)*BRET_S15LAG1+C(3)*BRET_S15LAG2 +C(4)*BRET_SD15LAG1 Coefficient Std. Error t-Statistic Prob. C(1) 0.040277 0.011506 3.500639 0.0005 C(2) 0.949454 0.036686 25.8806 0 C(3) -0.045665 0.035586 -1.283214 0.1998 C(4) 0.025349 0.020244 1.252201 0.2109 Adjusted R-squared 0.86514 Durbin-Watson stat 1.985785

  27. Table 7B: Component-wise Forecast :Duration BRET_D15=C(1)+C(2)*BRET_D15LAG1+C(3)*BRET_D15LAG2 +C(4)*BRET_SD15LAG1 Coefficient Std. Error t-Statistic Prob. C(1) 0.124057 0.016135 7.688657 0 C(2) 0.677629 0.034416 19.68965 0 C(3) 0.133242 0.034411 3.872083 0.0001 C(4) -0.010605 0.007934 -1.336662 0.1817 Adjusted R-squared 0.621627 Durbin-Watson stat 2.007996

  28. Table 7C: Component-wise Forecast : Persistence BRET_P15=C(1)+C(2)*BRET_P15LAG1+C(3)*BRET_P15LAG2 +C(4)*BRET_SD15LAG1 Coefficient Std. Error t-Statistic Prob. C(1) -0.017607 0.009949 -1.769719 0.0771 C(2) 1.039735 0.034724 29.94281 0 C(3) -0.157909 0.034455 -4.583119 0 C(4) 0.018263 0.010017 1.823169 0.0686 Adjusted R-squared 0.824973 Durbin-Watson stat 2.005559

  29. Table 8A: Non-parametric Volatility explained by three components BRET_SD15 Regressed on Coefficient Std. Error t-Statistic Prob. Constant 0.182372 0.017221 10.58988 0 BRET_S15 0.290415 0.016038 18.10794 0 BRET_D15 -0.252909 0.024861 -10.17302 0 BRET_P15 0.011946 0.011538 1.035377 0.3008 BRET_SD15Lag1 0.812226 0.033928 23.9396 0 BRET_SD15Lag2 -0.051401 0.029838 -1.722664 0.0853 R-squared Mean dependent var 0.960513 0.925065 Adjusted R-squared S.D. dependent var 0.960271 0.375287 S.E. of regression Akaike info criterion 0.074802 -2.340672 Sum squared resid Schwarz criterion 4.571429 -2.306313 Log likelihood F-statistic 969.1867 3974.679 Durbin-Watson stat Prob(F-statistic) 1.548439 0

  30. Forecast Model

  31. Table 8B: Non-parametric Volatility explained by three components CMR_SD15 Regressed on Coefficient Std. Error t-Statistic Prob. Constant 0.212444 0.032426 6.551635 0 CMR_S15 0.453337 0.014953 30.31661 0 CMR_D15 -0.265469 0.046262 -5.738427 0 CMR_P15 0.001537 0.0182 0.084428 0.9327 CMR_SD15Lag1 0.664036 0.030368 21.86596 0 CMR_SD15Lag2 -0.133352 0.023436 -5.690007 0 R-squared 0.976942 Mean dependent var 0.68998 Adjusted R-squared 0.976801 S.D. dependent var 0.706518 S.E. of regression 0.107611 Akaike info criterion -1.613324 Sum squared resid 9.460969 Schwarz criterion -1.578964 Log likelihood 669.8826 F-statistic 6923.154 Durbin-Watson stat 1.023739 Prob(F-statistic) 0

  32. Table 8C: Non-parametric Volatility explained by three components FIIN_SD15 Regressed on Coefficient Std. Error t-Statistic Prob. Constant 0.183537 0.017932 10.23524 0 FIIN_S15 0.398881 0.020162 19.78417 0 FIIN_D15 -0.283178 0.026796 -10.56775 0 FIIN_P15 0.003833 0.011566 0.331404 0.7404 FIIN_SD15Lag1 0.721912 0.032756 22.03913 0 FIIN_SD15Lag2 -0.021108 0.028928 -0.729696 0.4658 R-squared 0.976067 Mean dependent var 0.85531 Adjusted R-squared 0.975921 S.D. dependent var 0.424477 S.E. of regression 0.065868 Akaike info criterion -2.595044 Sum squared resid 3.540332 Schwarz criterion -2.560652 Log likelihood 1072.563 F-statistic 6655.9 Durbin-Watson stat 1.443571 Prob(F-statistic) 0

  33. Table 9A: Non-parametric Volatility Forecasting Model BRET_SD15 Regressed on Coefficient Std. Error t-Statistic Prob. Constant 0.057006 0.021796 2.615513 0.0091 BRET_S15 Lag1 0.065216 0.021957 2.970124 0.0031 BRET_D15 Lag1 -0.055691 0.031278 -1.780506 0.0754 BRET_P15 Lag1 0.06902 0.030397 2.270634 0.0234 BRET_SD15Lag1 0.927875 0.018618 49.83784 0 BRET_P15 Lag2 -0.103754 0.030054 -3.452256 0.0006 R-squared 0.944372 Mean dependent var 0.925065 Adjusted R-squared 0.944032 S.D. dependent var 0.375287 S.E. of regression 0.088784 Akaike info criterion -1.99796 Sum squared resid 6.440066 Schwarz criterion -1.9636 Log likelihood 828.1604 F-statistic 2773.982 Durbin-Watson stat 1.793371 Prob(F-statistic) 0 Residual Series Jarque-Bera 6770.647 Probability 0.0000

  34. Table 9B: Non-parametric Volatility Forecasting Model CMR_SD15 Regressed on Coefficient Std. Error t-Statistic Prob. Constant 0.539725 0.077949 6.924074 0 CMR_S15Lag1 0.81667 0.035396 23.07214 0 CMR_D15Lag1 -0.305795 0.142694 -2.143016 0.0324 CMR_S15Lag2 0.094666 0.035166 2.691998 0.0072 CMR_D15Lag3 -0.329692 0.141923 -2.323039 0.0204 R-squared 0.878311 Mean dependent var 0.687053 Adjusted R-squared 0.877716 S.D. dependent var 0.701938 S.E. of regression 0.245462 Akaike info criterion 0.034716 Sum squared resid 49.2256 Schwarz criterion 0.063376 Log likelihood -9.268205 F-statistic 1474.213 Durbin-Watson stat 0.992321 Prob(F-statistic) 0 Residual Series Jarque-Bera 11381 Probability 0.0000

  35. Table 9C: Non-parametric Volatility Forecasting Model FIIN_SD15 Regressed on Coefficient Std. Error t-Statistic Prob. Constant 0.066596 0.019491 3.416721 0.0007 FIIN_S15 Lag3 0.080775 0.031703 2.547885 0.011 FIIN_S15 Lag4 -0.103597 0.0295 -3.511783 0.0005 FIIN_D15 Lag6 -0.072937 0.028755 -2.536479 0.0114 FIIN_SD15 Lag1 0.993259 0.015763 63.01025 0 R-squared 0.965969 Mean dependent var 0.856792 Adjusted R-squared 0.965801 S.D. dependent var 0.4249 S.E. of regression 0.078577 Akaike info criterion -2.243394 Sum squared resid 5.01968 Schwarz criterion -2.214624 Log likelihood 922.5482 F-statistic 5769.176 Durbin-Watson stat 1.978857 Prob(F-statistic) 0 Jarque-Bera 2653.505 Probability 0.0000

  36. BRET

  37. CMR

  38. FIIN

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