Transcript Components of Volatility and their Empirical Measures: A Note

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
Of Financial Analysts: Variability of a financial
variable as measured by
its Std. Dev.
Of Econometricians:
Conditional
Heteroskedasticity
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
What led to what I talk about here?
1. Non-comparability of volatility of variables
measured in different units
2. Basis for comparison of the Volatilities of FIIN
to India and BSE return, say
3. Are there different aspects of volatility that
need to be compared?
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
Strength of Volatility
Green has less strength than Blue
Duration of Volatility
Volatile State
Normal State
Persistence of Volatility
Blue is more persistent than Black
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)
The Decomposition Methodology
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
1 T
ˆ
fT ( w) 
 K [( w  wt ) / h]
T .h t 1

K [.] : kernel function,  K (u ) du  1

h : bandwidth or smoothing parameter
The Decomposition Methodology
Step 3. Find mode of w. Call it w. 
ˆ.
Also find mean of w  w and call it w
w: average amplitude in nonvolatile state
ˆ : average amplitude in volatile state
w
ˆ w
Strength : S  w
Duration: D  1  F ( w)
where F ( w): a measure of portion of sample
period the variable is in volatile state
Persistence: P  a measure of autocorrn of w
Table 1: Summary Descriptive Statistics
BRET
CMR
FIIN
Mean
-0.00004
8.38
34.07
Median
0.00092
8.03
23.10
Max
0.09
22.50
983.20
Min
-0.07
0.50
-509.50
Std. Dev.
0.02
2.10
120.04
Skewness
-0.07
2.58
0.76
Kurtosis
5.33
13.99
9.66
Jarque-Bera
190.15
5160.66
1632.13
Sample Size
840
840
840
Table 2: Results of Unit Root Tests
ADF-statistic
5% Critical Value
Model Selected
lag order
BRET
-16.48
-1.94
No trend or
intercept
2
CMR
-7.73
-3.97
Trend and
intercept
3
FIIN
-9.11
-2.87
Intercept
4
Table 3: Results of GARCH (1,1) Estimation
item
BRET
Coefficient
CMR
Std. Error
Coefficient
FIIN
Std. Error
Coefficient
Std. Error
0.025773
24.63928
2.921969
mean equation
intercept
0.000723
0.000592
7.990946
variance equation
intercept
4.56E-05
9.87E-06
0.196689
0.012257
274.2189
61.64922
ARCH(1)
0.161038
0.036078
1.08646
0.059423
0.124378
0.013505
GARCH(1)
0.713714
0.052307
0.144736
0.019215
0.863543
0.014828
Adjusted R2
-0.005257
-0.03849
-0.00979
FIIN Density
1.00
0.80
0.60
0.40
0.20
0.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
BRET Density
1.00
0.80
0.60
0.40
0.20
0.00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
Table 4. Variable-specific Estimates of Volatility
Components Based on Entire Sample
BRET
Amplitude of Fluctuation
w
Average Amplitude Normal Phase(
w* )
Average Amplitude Volatile Phase (
Strength of Volatility (S)
CMR
0.295
0.133
) 0.987
0.847
FIIN
0.256
0.929
0.692
0.714
0.673
0.773
0.807
0.769
1st Order Autocorrelation of w
0.25
0.51
0.22
2ndOrder Autocorrelation of w
0.18
0.36
0.20
3rdOrderAutocorrelation of w
0.11
0.25
0.20
Duration of Volatility
Proportion of Volatile Days (D)
Persistence of Volatility (P)
Table 5: A Summary of Rolling Sample
Estimation Returns
Volatility
Component
Windowwidth
15-day
S
D
P
Mean/CV
mean
cv
90-day
mean
cv
Entire sample
15-day
mean
cv
90-day
mean
cv
Entire sample
15-day
mean
cv
90-day
mean
cv
Entire sample
Variable
BRET
0.66
0.51
0.68
0.22
0.692
0.6
0.23
0.7
0.09
0.773
-0.01
-47.31
0.2
0.67
0.25
CMR
0.67
1.12
0.75
0.54
0.714
0.72
0.12
0.79
0.05
0.807
0.24
0.91
0.45
0.33
0.51
FIIN
0.63
0.51
0.65
0.33
0.673
0.63
0.17
0.71
0.06
0.769
-0.01
-15.78
0.08
1.54
0.22
S Measure for 15-day Window Width
7
6
BRET
CMR
5
FIIN
4
3
2
1
Note: Scales shifted for BRET and CMR
807
776
745
714
683
652
621
590
559
528
497
466
435
404
373
342
311
280
249
218
187
156
125
94
63
32
1
0
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
Note: Scales shifted for BRET and CMR
BRET
CMR
FIIN
801
769
737
705
673
641
609
577
545
513
481
449
417
385
353
321
289
257
225
193
161
129
97
65
33
1
0.25
P Measure for 15-day Window Width
3
BRET
CMR
FIIN
2.5
2
1.5
1
0.5
0
-0.5
Dotted lines are shifted scales for respective variables
807
781
755
729
703
677
651
625
599
573
547
521
495
469
443
417
391
365
339
313
287
261
235
209
183
157
131
105
79
53
27
1
-1
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
737
705
673
641
609
577
545
513
481
449
417
385
353
321
289
257
225
193
161
129
97
65
33
1
0.25
D Measure for 90-day Window Width
1.1
1
0.9
0.8
0.7
0.6
BRET
CMR
FIIN
Note:Scale shifted for BRET
737
705
673
641
609
577
545
513
481
449
417
385
353
321
289
257
225
193
161
129
97
65
33
1
0.5
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
Dotted lines are shifted scales for respective variables
729
701
673
645
617
589
561
533
505
477
449
421
393
365
337
309
281
253
225
197
169
141
113
85
57
29
1
-0.2
Table 6:
Correlation between day to day variations of estimated
volatility components for different pairs of variables
Volatility
component
S
D
P
Window-width
15-day
90-day
15-day
90-day
15-day
90-day
Correlation for the variable-pair
BRET-CMR
-0.02
0.43
-0.34
-0.38
-0.12
-0.23
BRET-FIIN
0.23
0.51*
0.05
0.23
0.07
-0.16
CMR-FIIN
0.06
0.25
0.06
0.19
0.05
0.42
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)
Adjusted R-squared
0.025349
0.86514
0.020244
1.252201
0.2109
Durbin-Watson stat
1.985785
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)
Adjusted R-squared
-0.010605
0.621627
0.007934
-1.336662
0.1817
Durbin-Watson stat
2.007996
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)
Adjusted R-squared
0.018263
0.824973
0.010017
1.823169
0.0686
Durbin-Watson stat
2.005559
Table 8A:
Non-parametric Volatility explained by three components
BRET_SD15 Regressed on
Coefficient Std. Error
Constant
BRET_S15
BRET_D15
BRET_P15
BRET_SD15Lag1
BRET_SD15Lag2
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
t-Statistic
Prob.
0.182372
0.017221
10.58988
0
0.290415
0.016038
18.10794
0
-0.252909
0.024861
-10.17302
0
0.011946
0.011538
1.035377
0.3008
0.812226
0.033928
23.9396
0
-0.051401
0.029838
-1.722664
0.960513
0.960271
0.074802
4.571429
969.1867
1.548439
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.0853
0.925065
0.375287
-2.340672
-2.306313
3974.679
0
Forecast Model
2
0.030
1.8
0.028
1.6
0.026
0.024
1.4
0.022
1.2
0.020
1
0.018
0.8
0.016
0.6
0.014
0.4
0.012
0.2
0.010
BRET_SD15
BRET_SD15Forecast
GarchVolBRET_MA15
Forecast Model:
BRET_SD15=C(1)+C(2)*BRET_S15+C(3)*BRET_D15+C(4)*BRET_P15+C(5)*BRET_SD15LAG1+C(6)*BRET_SD15LAG2
Table 8B:
Non-parametric Volatility explained by three components
CMR_SD15 Regressed on
Coefficient
Std. Error t-Statistic
Constant
CMR_S15
CMR_D15
CMR_P15
CMR_SD15Lag1
CMR_SD15Lag2
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.212444
0.453337
-0.265469
0.001537
0.664036
-0.133352
0.976942
0.976801
0.107611
9.460969
669.8826
1.023739
0.032426
0.014953
0.046262
0.0182
0.030368
0.023436
6.551635
30.31661
-5.738427
0.084428
21.86596
-5.690007
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0
0
0
0.9327
0
0
0.68998
0.706518
-1.613324
-1.578964
6923.154
0
Table 8C:
Non-parametric Volatility explained by three
components
FIIN_SD15 Regressed on
Coefficient Std. Error t-Statistic
Constant
FIIN_S15
FIIN_D15
FIIN_P15
FIIN_SD15Lag1
FIIN_SD15Lag2
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.183537
0.398881
-0.283178
0.003833
0.721912
-0.021108
0.976067
0.975921
0.065868
3.540332
1072.563
1.443571
0.017932
0.020162
0.026796
0.011566
0.032756
0.028928
10.23524
19.78417
-10.56775
0.331404
22.03913
-0.729696
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0
0
0
0.7404
0
0.4658
0.85531
0.424477
-2.595044
-2.560652
6655.9
0
Table 9A:
Non-parametric Volatility Forecasting Model
BRET_SD15 Regressed on
Coefficient Std. Error
t-Statistic
Prob.
Constant
BRET_S15 Lag1
BRET_D15 Lag1
BRET_P15 Lag1
BRET_SD15Lag1
BRET_P15 Lag2
0.057006
0.065216
-0.055691
0.06902
0.927875
-0.103754
0.021796
0.021957
0.031278
0.030397
0.018618
0.030054
2.615513
2.970124
-1.780506
2.270634
49.83784
-3.452256
0.0091
0.0031
0.0754
0.0234
0
0.0006
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Residual Series
Jarque-Bera
Probability
0.944372
0.944032
0.088784
6.440066
828.1604
1.793371
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
6770.647
0.0000
0.925065
0.375287
-1.99796
-1.9636
2773.982
0
Table 9B:
Non-parametric Volatility Forecasting Model
CMR_SD15 Regressed on
Coefficient
Std. Error
t-Statistic
Prob.
Constant
CMR_S15Lag1
CMR_D15Lag1
CMR_S15Lag2
CMR_D15Lag3
0.539725
0.81667
-0.305795
0.094666
-0.329692
0.077949
0.035396
0.142694
0.035166
0.141923
6.924074
23.07214
-2.143016
2.691998
-2.323039
0
0
0.0324
0.0072
0.0204
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Residual Series
Jarque-Bera
Probability
0.878311
Mean dependent var
0.877716
S.D. dependent var
0.245462
Akaike info criterion
49.2256
Schwarz criterion
-9.268205
F-statistic
0.992321 Prob(F-statistic)
11381
0.0000
0.687053
0.701938
0.034716
0.063376
1474.213
0
Table 9C:
Non-parametric Volatility Forecasting Model
FIIN_SD15 Regressed on
Coefficient Std. Error
Constant
FIIN_S15 Lag3
FIIN_S15 Lag4
FIIN_D15 Lag6
FIIN_SD15 Lag1
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Jarque-Bera
Probability
0.066596
0.080775
-0.103597
-0.072937
0.993259
0.965969
0.965801
0.078577
5.01968
922.5482
1.978857
2653.505
0.0000
0.019491
0.031703
0.0295
0.028755
0.015763
t-Statistic
3.416721
2.547885
-3.511783
-2.536479
63.01025
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0007
0.011
0.0005
0.0114
0
0.856792
0.4249
-2.243394
-2.214624
5769.176
0
BRET
2.10
2.2
BRET
2
1.80
1.8
1.4
1.20
1.2
0.90
1
0.8
0.60
0.6
0.30
0.4
BRET_SD15Observed
BRET_SD15Forecast
807
776
745
714
683
652
621
590
559
528
497
466
435
404
373
342
311
280
249
218
187
156
125
94
63
0.2
32
0.00
Forecast
1.6
1
Observed
1.50
CMR
CMR
3.00
2.50
CMR_SD15Observed
CMR_SD15Forecast
2.00
1.50
1.00
0.50
0.00
1
32 63 94 125 156 187 218 249 280 311 342 373 404 435 466 497 528 559 590 621 652 683 714 745 776 807
811
781
751
721
691
661
631
601
571
541
511
481
451
421
391
361
331
301
271
241
211
181
151
121
91
61
31
1
Observed
FIIN
FIIN_SD15Forecast
1.15
1
0.65
0.5
0.15
0
Forecast
FIIN
2.5
2.15
FIIN_SD15Observed
2
1.65
1.5