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Forecasting Financial Volatilities with Extreme Values: The Conditional AutoRegressive Range (CARR)...
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Transcript of Forecasting Financial Volatilities with Extreme Values: The Conditional AutoRegressive Range (CARR)...
Forecasting Financial Volatilities Forecasting Financial Volatilities with Extreme Values:with Extreme Values:
The Conditional AutoRegressive RThe Conditional AutoRegressive Range (CARR) Modelange (CARR) Model
- JMCB (2005)- JMCB (2005)
Ray Y. Chou 周雨田Academia Sinica, & National Chiao-Tung Universit
yPresented at
南開大學經濟學院4/11-12/2007
2
Motivation
Provide a dynamic model for range in resolving the puzzle of the fact that although theoretically sound, range has been a poor predictor of volatility empirically.
References of the “static range” models include Parkinson (1980), Garman and Klass (1980), Beckers (1983), Wiggins (1991), Rogers and Satchell(1991), Kunitomo (1992), and Yang and Zhang (2000).
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4
Main Results
CARR is ACD but with new interpretations and implications.
CARR has two properties: QMLE, and Encompassing.
Empirical results using daily S&P500 index are satisfactory.
5
Range as a measure of the “realized volatility”
Simpler and more natural than the sum-squared-returns (measuring the integrated volatility) of Anderson et.al. (2000)
6
Range vs. Integrated Volatility
Simple to obtain, e.g.,WSJ Unbiased estimator of the
standard deviation sampling frequency
determined by the data compiler, almost continuous
Known distribution –Feller (1951), Lo (1991), quality control values
Difficult to compute, N.A. for earlier time periods
Unbiased estimator of the variance
Sampling frequency is arbitrarily decided by the econometrician, see Chou (1988) for a critique
Distribution unknown, e.g., ln(IV) ~ Normal?
7
Range measured from a discrete price path
Let {P} be the logarithm of the price of a speculative asset. Normalize the range observation interval to be unity, e.g., a day, and further suppose the price level is only observed at every 1/n interval, the range can then be defined as
},{}{,1 PMinPMaxRntt
tn
tn
tt ,...,2
1,1
1,1
8
Range for a non-constant mean price process
If the sample mean of P over the interval t-1 to t,
is not a constant, then the range can be written in the following way:
)]()([0
110
11
nt
k
j
k
n
jtnk
nt
k
j
k
n
jtnk
nt PPMinPPMaxR
9
Range as an estimate of the standard deviation
Parkinson (1980) and others proved that under some regularity assumptions, then can be consistently estimated by the range with a scale adjustment. E(R) =
Lo (1991) proves that the limiting distribution of the rescaled range is a Brownian bridge on a unit interval. And the constant will be determined by the dependence structure of {P}
Hence a dynamic model of the range can be used as a model for the volatility.
10
The observation frequency parameter, n
The higher n is, the more frequently we observe the price between Pand P
If n* is the true frequency parameter then, Rn is a downward biased estimator of the true range if n<n*. Further, the bias is a decreasing function of n. Hence the case n=1, gives the least efficient estimator.
11
The Conditional AutoRegressive Range (CARR) model:
t=Rt/t , the normalized range, ~ iid f(.),
andt is the conditional mean of Rt , i, j > 0
.111
q
jj
p
ii For stationarity,
tttR
jt
q
jjit
p
iit R
11
12
A special case of CARR: Exponential CARR(1,1) or ECARR(1,1)
It’s useful to consider the exponential case for f(.), the distribution of the normalized range or the disturbance.
Like GARCH models, a simple (p=1, q=1) specification works for many empirical examples.
11 ttt R
13
ECARR(1,1) (continued)
The unconditional mean of range is given by .
For stationarity, < 1 This model is identical to the EACD of En
gle and Russell (1998)
14
CARRX- Extension of CARR
ltl
L
ljt
q
jjit
p
iit XR ,1
111
15
CARR vs. ACD identical formula
CARR Range data, positive
valued, with fixed sample interval
QMLE with ECARR A new volatility
model
ACD Duration data,
positive valued, with non-fixed sample interval
QMLE with EACD Hazard rate
interpretation
16
CARR vs. GARCH
CARR Cond. mean model Range is measurable Asymptotic properties are simp
ler, less restrictions on moment conditions
Modeling variance of asset returns only
More efficient as more information is used
Include SD-GARCH as a special case with n=1
GARCH Cond. variance model Volatility unobservable Complicated asymptotic p
roperties, stringent moment conditions
Modeling mean/variance simultaneously
Not as efficient as CARR
17
Property 1: The QMLE property
Assuming any general density function f(.) for the disturbance term t, the parameters
in CARR can be estimated consistently by estimating an exponential-CARR model.
Proof: see Engle and Russell (1998),
p.1135
18
The Standard Deviation GARCH (SD-GARCH)
jt
q
jjit
p
iit r
11
Let rt (=Pt-Pt-1) be the return of the asset from
t-1 to t. The volatility equation of an SD-GARCH model is
19
Property 2: The Encompassing Property
Without specifying the conditional distribution, the CARR(p,q) model with n=1 is equivalent to a SD-GARCH(p,q) model of Schwert(1990) and others. Given the QMLE property, any SD-GARCH model can be consistently estimated by an Exponential CARR model.
Proof: It’s sufficient to show that with n=1, the range Rt is equal to the abs. value of the return, rt.
Rt = Max(P t-1, P t) – Min(P t-1, P t) = | P t – P t-1 | = | r t | .
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Empirical example: S&P500 daily index
Sample period: 1982/05/03 – 2003/10/20 Data source: Yahoo.com Models used: ECARRX, WCARRX Both daily and weekly data are used for
estimation but only weekly results are reported The weekly model is used to compare with a
weekly GARCH model, using four different measured volatilities: SSDR, WRSQ, RNG, and AWRET as benchmarks.
21
-16
-12
-8
-4
0
4
8
12
250 500 750 1000
W eekly Return
0
5
10
15
20
25
30
250 500 750 1000
W eekly Range
Figure 1: S&P500 Index W eekly Returns and Ranges, 5/3/1982-10/20/2003
Figure 1: S&P 500 Index Weekly Returns and Ranges 5/3/1982-10/20/2003
Weekly Range
Weekly Return
22
Table 1:Summary Statistics for the Returns and Ranges of Weekly S&P 500 Index
Return Absolute Return RangeMean 0.19544 1.675032 3.146566Median 0.36289 1.331176 2.661446Maximum 8.46172 13.00708 26.69805Minimum -13.007 0.002411 0.706926Std. Dev. 2.22171 1.471722 1.828565Skewness -0.5559 2.309058 3.284786Kurtosis 6.3816 12.38987 30.39723Jarque-Bera 591.323 5109.846 37042.49Probability 0 0 0Auto-Correlation Function (lag)
ACF (1) -0.062 0.207 0.53ACF (2) 0.068 0.101 0.426ACF (3) -0.031 0.147 0.386ACF (4) -0.037 0.087 0.356ACF (5) -0.011 0.064 0.311ACF (6) 0.082 0.13 0.348ACF (7) -0.024 0.142 0.326ACF (8) -0.029 0.101 0.285ACF (9) -0.012 0.101 0.233
ACF (10) -0.006 0.105 0.277ACF (11) 0.059 0.091 0.25ACF (12) -0.023 0.083 0.225
Q(12) 26.335 191.52 1564.7
23
1111
ttjt
q
jjit
p
iit rrR
tttR
t ~ iid f(.)
Table 2: Estimation of the CARR Model Using Weekly S&P500 Index with Exponential Distribution,
5/3/1982~10/20/2003
ECARR(1,1) ECARR(2,2) ECARRX(1,1)-a ECARRX(1,1)-b
LLF -2204.888 -2204.824 -2199.039 -2199.062 0.1435 (4.123) 0.1616 (1.500) 0.2128 (5.776) 0.2066 (5.750)
0.2434 (7.828) 0.2635 (5.772) 0.2570 (7.724) 0.2361 (8.805)
0.0218 (0.114)
0.7112 (20.857) 0.4259 (0.596) 0.6954 (22.465) 0.7045 (22.818)
0.2377 (0.463) -0.0960 (-5.110) -0.0967 (-5.620) -0.0255 (-0.674)
Q(12) 14.600 (0.264) 14.594 (0.264) 12.579 (0.400) 12.308 (0.421)
W2 40.355 (0.000) 40.414 (0.000) 41.529 (0.000) 41.479 (0.000)
24
1111
ttjt
q
jjit
p
iit rrR
tttR
t ~ iid f(.)
Table 3:Estimation of the CARR Model Using Weekly S&P500 Index with Weibull Distribution 5/3/1982~10/20/2003
WCARR(1,1) WCARR(2,2) WCARRX(1,1)-a WCARRX(1,1)-b
LLF -1810.485 -1810.363 -1781.963 -1782.092 0.1803 (4.467) 0.1733 (1.183) 0.2509 (6.041) 0.2559 (6.173)
0.3086 (18.191) 0.3142 (18.554) 0.2535 (8.228) 0.2678 (14.528)
-0.0108(-0.042
9)
0.6362 (28.672) 0.6152 (0.734) 0.6656 (25.440) 0.6590 (29.108)
0.0284 (0.052) -0.1150
(-11.172)
-0.1147(-11.19
8) 0.0173 (0.641) θ 2.4025 (51.575) 2.4017 (49.818) 2.4742 (51.501) 2.4727 (52.746)
Q(12) 16.889 (0.154) 16.218 (0.181) 14.943 (0.245) 15.196 (0.231)
W2 6.152 (0.000) 6.179 (0.000) 6.208 (0.000) 6.238 (0.000)
25
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2
1 .4
1 2 3 4
F ig u r e 2 : R e s id u a l D e n s ity : E C A R R ( 1 ,1 )
26
.0
.1
.2
.3
.4
.5
.6
.7
0 5 10 15 20 25 30
Figure 3: Transformed Residual Density: W CARR(1,1)
27
Table 4: Forecast Comparison Using RMSE and MAE 5.02
1
1 ])(ˆ[),( mVMMVThmRMSE htht
T
t
)(ˆ),(1
1 mVMMVThmMAE htht
T
t
ssdr wrsq wrng awret
horizon carr Garch carr Garch carr Garch carr Garch1 9.270 11.328 18.990 19.310 1.956 2.263 2.018 2.0582 9.960 11.820 19.239 19.653 2.055 2.358 2.043 2.0914 11.230 12.596 19.565 19.791 2.238 2.452 2.074 2.1068 11.240 12.396 19.527 19.799 2.395 2.561 2.086 2.12513 11.604 12.674 19.604 19.760 2.482 2.595 2.104 2.13126 11.212 12.041 19.262 19.380 2.402 2.427 2.028 2.05350 10.531 10.187 11.483 11.716 2.135 2.033 1.667 1.704
RMSE
ssdr wrsq wrng awret
horizon carr Garch carr Garch carr Garch carr Garch1 6.768 8.015 9.610 9.878 1.376 1.640 1.492 1.4852 7.339 8.440 9.703 9.995 1.441 1.687 1.499 1.4924 7.846 8.652 9.492 10.072 1.609 1.724 1.474 1.4838 7.761 8.715 9.031 9.984 1.691 1.865 1.435 1.495
13 7.442 8.853 8.917 10.061 1.752 1.919 1.420 1.49626 6.801 8.315 8.107 9.305 1.635 1.748 1.344 1.44950 6.239 7.175 5.973 7.368 1.506 1.525 1.194 1.298
MAE
28
Table 5: CARR versus GARCH, in forecasting SSDR SSDRt+h = a + b FVt+h(CARR) + ut+h
SSDRt+h = a + c FVt+h(GARCH) + ut+h
SSDRt+h = a + b FVt+h(CARR) + c FVt+h(GARCH) + ut+h
Forecast horizon Explanatory Variables
1 -0.117 (-0.067) 0.540 (5.579) 0.317 1 6.271 (2.606) 0.540 (1.826) 0.043 1 2.799 (1.396) 0.734 (6.384) -0.789 (-3.337) 0.368 2 1.640 (0.965) 0.470 (4.421) 0.214 2 8.650 (4.093) 0.272 (1.131) 0.011 2 5.070 (2.831) 0.721 (5.022) -0.963 (-3.985) 0.290 4 6.184 (3.266) 0.257 (2.766) 0.052 4 12.787 (5.228) -0.200 (-0.968) 0.006 4 9.709 (4.303) 0.570 (4.213) -1.077 (-3.798) 0.146 8 8.829 (3.109) 0.117 (0.771) 0.007 8 13.661 (6.181) -0.324 (-1.665) 0.016 8 11.066 (4.044) 0.424 (2.404) -0.853 (-4.490) 0.071
13 12.740 (4.167) -0.109 (-0.624) 0.004 13 15.758 (6.715) -0.536 (-2.743) 0.045 13 14.031 (4.565) 0.254 (1.222) -0.783 (-3.679) 0.056
h intercept FV(CARR) FV(GARCH) Adj. R-sq.
29
Table 6: CARR versus GARCH, in forecasting WRSQ WRSQt+h = a + b FVt+h(CARR) + ut+h
WRSQt+h = a + c FVt+h(GARCH) + ut+h
WRSQt+h = a + b FVt+h(CARR) + c FVt+h(GARCH) + ut+h
Forecast horizon Explanatory Variables
h intercept FV(CARR) FV(GARCH) Adj. R-sq.1 5.727 (1.605) 0.168 (1.171) 0.011 1 9.118 (2.116) 0.008 (0.020) 0.000 1 7.702 (1.743) 0.300 (1.726) -0.534 (-1.103) 0.019 2 7.781 (2.853) 0.062 (0.707) 0.001 2 12.899 (3.420) -0.441 (-1.802) 0.010 2 11.307 (3.188) 0.320 (2.193) -0.990 (-2.495) 0.029 4 11.496 (2.516) -0.128 (-0.807) 0.004 4 14.593 (2.936) -0.630 (-1.682) 0.020 4 14.057 (2.625) 0.099 (0.731) -0.783 (-2.330) 0.022 8 12.870 (2.536) -0.217 (-1.027) 0.008 8 15.045 (3.206) -0.671 (-1.954) 0.023 8 14.774 (2.718) 0.044 (0.214) -0.727 (-2.386) 0.023
13 16.061 (2.815) -0.438 (-1.598) 0.020 13 14.988 (3.646) -0.643 (-2.187) 0.020 13 16.715 (2.903) -0.254 (-0.869) -0.396 (-1.789) 0.024
30
Table 7: CARR versus GARCH, in forecasting WRNG WRNGt+h = a + b FVt+h(CARR) + ut+h
WRNGt+h = a + c FVt+h(GARCH) + ut+h
WRNGt+h = a + b FVt+h(CARR) + c FVt+h(GARCH) + ut+h
Forecast horizon Explanatory Variables h intercept FV(CARR) FV(GARCH) Adj. R-sq.
1 0.877 (1.362) 0.855 (5.927) 0.224 1 2.718 (3.167) 0.658 (2.322) 0.039 1 1.902 (2.463) 1.161 (6.098) -0.818 (-2.671) 0.256 2 1.414 (2.275) 0.738 (5.080) 0.154 2 3.836 (4.625) 0.264 (1.018) 0.006 2 2.882 (3.994) 1.210 (5.482) -1.214 (-3.529) 0.224 4 2.730 (3.493) 0.434 (2.583) 0.046 4 5.100 (5.440) -0.188 (-0.666) 0.003 4 4.172 (4.495) 0.977 (4.323) -1.291 (-3.597) 0.124 8 4.203 (3.971) 0.087 (0.355) 0.001 8 6.421 (7.013) -0.646 (-2.264) 0.037 8 5.468 (5.238) 0.769 (2.613) -1.390 (-4.346) 0.090
13 6.354 (5.510) -0.476 (-1.710) 0.027 13 7.386 (8.677) -0.990 (-3.867) 0.087 13 7.055 (6.266) 0.215 (0.616) -1.162 (-3.596) 0.090
31
Table 8: CARR versus GARCH, in forecasting AWRET AWRETt+h = a + b FVt+h(CARR) + ut+h
AWRETt+h = a + c FVt+h(GARCH) + ut+h
AWRETt+h = a + b FVt+h(CARR) + c FVt+h(GARCH) + ut+h
Forecast horizon Explanatory Variables h intercept FV(CARR) FV(GARCH) Adj. R-sq.
1 1.163 (1.689) 0.252 (1.593) 0.023 1 1.949 (2.173) 0.110 (0.372) 0.001 1 1.666 (1.826) 0.403 (1.946) -0.402 (-1.032) 0.032 2 1.581 (2.739) 0.153 (1.183) 0.008 2 2.713 (3.572) -0.164 (-0.739) 0.003 2 2.388 (3.244) 0.412 (1.989) -0.667 (-1.847) 0.033 4 2.206 (2.772) 0.011 (0.067) 0.000 4 3.091 (3.419) -0.290 (-1.088) 0.009 4 2.850 (2.962) 0.253 (1.358) -0.577 (-1.893) 0.019 8 2.751 (2.843) -0.118 (-0.543) 0.003 8 3.524 (4.098) -0.433 (-1.699) 0.020 8 3.303 (3.289) 0.179 (0.674) -0.605 (-2.016) 0.023
13 3.673 (3.315) -0.352 (-1.339) 0.017 13 3.760 (4.489) -0.499 (-1.966) 0.025 13 3.923 (3.475) -0.106 (-0.337) -0.415 (-1.498) 0.026
32
Table 9: Encompassing Tests using West’s (2001) V-Procedure
hthththt uGARCHVMCARRVMMV ˆˆ horizon t-ratio t-ratio
SSDR 1 1.180 (5.820) -0.219 (-1.220)2 1.140 (4.400) -0.161 (-0.744)4 0.786 (2.780) 0.228 (0.791)8 0.390 (0.792) 0.619 (1.240)
13 -0.167 (-0.484) 1.160 (3.090) WRSQ 1 1.010 (1.180) -0.012 (-0.013)2 0.241 (0.752) 0.771 (1.770)4 -0.169 (-0.389) 1.160 (1.960)8 -0.077 (-0.223) 1.070 (2.350)
13 0.484 (1.010) 0.528 (1.420) WRNG 1 1.190 (5.530) -0.198 (-0.934)2 1.130 (4.900) -0.133 (-0.596)4 0.825 (2.590) 0.177 (0.531)8 0.129 (0.327) 0.872 (2.150)
13 -0.137 (-0.456) 1.140 (3.650) AWRET 1 1.150 (1.600) -0.149 (-0.212)2 0.645 (1.220) 0.358 (0.660)4 0.043 (0.052) 0.957 (1.090)8 -0.192 (-0.317) 1.190 (1.840)
13 0.195 (0.353) 0.806 (1.460)
33
0
4
8
12
16
20
400 410 420 430 440 450
GARCH_FOR CARR_FOR SSDR
Figure 4: Volatility Forecasts: CARR vs GARCH
34
Conclusion with extensions
Robust CARR – Interquartile range Asymmetric CARR – Chou (2005b) Modeling return and range simultaneously MLE: Does Lo’s result apply to CARR? Aggregations