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)

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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

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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

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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.

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-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

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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)

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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)

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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 )

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.0

.1

.2

.3

.4

.5

.6

.7

0 5 10 15 20 25 30

Figure 3: Transformed Residual Density: W CARR(1,1)

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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

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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.

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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  

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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  

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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  

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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