Forecasting Financial Volatilities with Extreme Values: The Conditional AutoRegressive Range (CARR) Model - JMCB (2005) Ray Y. Chou 周雨田 Academia Sinica, & National Chiao-Tung University Presented 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

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:

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 P  and P   If n* is the true frequency parameter then, R n 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 =R t / t, the normalized range, ~ iid f(.), and  t is the conditional mean of R t ,  i,  j > 0 For stationarity,

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.

13 ECARR(1,1) (continued)  The unconditional mean of range is given by .  For stationarity,  < 1  This model is identical to the EACD of Engle and Russell (1998)

14 CARRX- Extension of CARR

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 simpler, 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 properties, 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) Let r t (=P t -P t-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 R t is equal to the abs. value of the return, r t. R t = Max(P t-1, P t ) – Min(P t-1, P t ) = | P t – P t-1 | = | r t |.

20 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 Figure 1: S&P 500 Index Weekly Returns and Ranges 5/3/ /20/2003 Weekly Range Weekly Return

22 Table 1:Summary Statistics for the Returns and Ranges of Weekly S&P 500 Index ReturnAbsolute ReturnRange Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability000 Auto-Correlation Function (lag) ACF (1) ACF (2) ACF (3) ACF (4) ACF (5) ACF (6) ACF (7) ACF (8) ACF (9) ACF (10) ACF (11) ACF (12) Q(12)

23  t ~ iid f(.) Table 2 : Estimation of the CARR Model Using Weekly S&P500 Index with Exponential Distribution, 5/3/1982~10/20/2003

24  t ~ iid f(.) Table 3:Estimation of the CARR Model Using Weekly S&P500 Index with Weibull Distribution 5/3/1982~10/20/2003

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27 Table 4: Forecast Comparison Using RMSE and MAE ssdrwrsqwrngawret horizoncarrGarchcarrGarchcarrGarchcarrGarch RMSE ssdrwrsqwrngawret horizoncarrGarchcarrGarchcarrGarchcarrGarch MAE

28 Table 5: CARR versus GARCH, in forecasting SSDR SSDR t+h = a + b FV t+h (CARR) + u t+h SSDR t+h = a + c FV t+h (GARCH) + u t+h SSDR t+h = a + b FV t+h (CARR) + c FV t+h (GARCH) + u t+h Forecast horizon Explanatory Variables hinterceptFV(CARR)FV(GARCH)Adj. R-sq.

29 Table 6: CARR versus GARCH, in forecasting WRSQ WRSQ t+h = a + b FV t+h (CARR) + u t+h WRSQ t+h = a + c FV t+h (GARCH) + u t+h WRSQ t+h = a + b FV t+h (CARR) + c FV t+h (GARCH) + u t+h Forecast horizon Explanatory Variables

30 Table 7: CARR versus GARCH, in forecasting WRNG WRNG t+h = a + b FV t+h (CARR) + u t+h WRNG t+h = a + c FV t+h (GARCH) + u t+h WRNG t+h = a + b FV t+h (CARR) + c FV t+h (GARCH) + u t+h Forecast horizon Explanatory Variables hinterceptFV(CARR)FV(GARCH)Adj. R-sq.

31 Table 8: CARR versus GARCH, in forecasting AWRET AWRET t+h = a + b FV t+h (CARR) + u t+h AWRET t+h = a + c FV t+h (GARCH) + u t+h AWRET t+h = a + b FV t+h (CARR) + c FV t+h (GARCH) + u t+h Forecast horizon Explanatory Variables hinterceptFV(CARR)FV(GARCH)Adj. R-sq.

32 Table 9: Encompassing Tests using West’s (2001) V-Procedure

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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