1. 1 A NEW SPECIFICATION TOOL FOR STOCHASTIC VOLATILITY MODELS BASED ON THE TAYLOR EFFECT A. Pérez 1 and E. Ruiz 2 1 Dpto. Economía Aplicada Universidad de Valladolid (Spain) 2 Dpto. Estadística Universidad Carlos III de Madrid (Spain) XXXIII SIMPOSIO DE ANÁLISIS ECONÓMICO XXXIII SIMPOSIO DE ANÁLISIS ECONÓMICO Zaragoza, December 12, 2008

2. 2 Outline  Motivation and background  The Taylor property in ARSV and LMSV models  A new tool for model adequacy based on Taylor effect Asymptotic and finite sample properties  Empirical application  Conclusions

3. 3 Motivation and background Sample autocorrelations of absolute returns are larger than those of squares; Taylor (1986) Autocorrelations of powered-absolute returns, |y t | , are highest for  1, i.e. absolute returns ; Ding et al (1993)  Taylor effect, Granger and Ding (1995) Such autocorrelations tend to persist for long lags ==> possible long-memory in volatility; Ding et al (1993)

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6. 6 Reasons for interest in autocorrelations of |y t |  Model adequacy: any theoretical model should be able to replicate these sample correlation patterns; Ding et al (1993), Baillie and Chung (2001), Karanasos et al (2004) Model selection: correlations of squares are not enough to discriminate models; Franq & Zakoian (2008) The power transformation that maximizes correlations related to predictability; Higgins & Bera (1992) Improved estimators of conditional heteroskedastic models; Deo et al (2006)

7. 7 Taylor effect in LMSV models LMSV(1, d,0) model: y t : series of returns  t : the volatility  t ~ IID(0,1) symmetric  t ~ NID(0, ) independent of  t d <0.5, |  |<1 for stationarity Note: d =0 => ~ AR(1) => ARSV(1) model Note:  =Corr(  t,  t+1 ) + Gaussian => A-LMSV model

8. 8 Kurtosis of y t => k y = k  exp( ) Moments and dynamic structure: ACF of | y t |  in LMSV model; Harvey (1998)  k (  )=  (| y t | ,| y t-k |  ) = where   ={E( |  t | 2  )}/{E( |  t |  )} 2   k (  ) depends on:, , d, , distribution of  t is the acf of

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10. 10 Let  We focus on  1 (  ) and  max (1)  1 (  )= d =0: ARSV(1) =>  h (1)=   =0: LMSV(0, d,0) =>  h (1)= d /(1- d ) Persistence

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12. 12 A new tool for model adequacy Asymptotic properties: Gourieroux & Jasiak (2002) Consistent estimator of  max ( k ) Asymptotically Normal:

13. 13 Finite samples: Monte Carlo experiment 1000 series of sizes T={500, 1000, 5000}  ={0.8,0.98}, ={0.01,0.05,0.1}, d ={0.3,0.45}  t ~N(0,1) and Student t-7 A grid of values of  (0,3)   j  {0.01,0.02,…,3} For each replicate i  compute r 1i (  j ), j=1,…,300 For each  j  sample mean of r 1i (  j ) and 90%,95% For replicate i  Pick up, i =1,..,1000

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15. 15 Kernel densities of

16. 16 New tool proposed Given a data set and its fitted SV model with its estimated parameters: (  =   N(0,1)) Define: = Test for model adequacy  H 0 :  max (k)=  Reject, at  %, when outside the 100(1-  )% confidence region of the asymptotic distribution

17. 17 Summary descriptive statistics of returns* SERIE EuroBPCANYenSP500Nikkei FTSE100 IBEX35 Size Kurtosis 2512 4.384 6047 5.760 8053 6.499 6041 6.509 10778 8.161 4676 7.192 4735 6.503 3991 6.639 r 1 (1) Q | Y | (50) 0.073 492 0.134 3229 0.202 6965 0.142 1770 0.214 16676 0.205 5192 0.195 5600 0.231 6938 r 1 (2) Q Y 2 (50) 0.054 269 0.097 2437 0.163 3033 0.198 1409 0.246 9294 0.139 1727 0.239 5890 0.181 3765 * F iltered by fitting MA(1) and/or correcting possible outliers >5  t/T Empirical application 0.72 0.82 0.98 1.14 0.70 0.63 3.26 1.07 2.31 1.03 0.95 0.69 3.45 1.51 1.01 1.03

18. 18 Estimation results of ARSV(1) models SERIESEuroBPCANYenSP500Nikkei225FTSE100IBEX35     0.992 (0.005) 0.003 (0.002) 7.35 0.988 (0.003) 0.011 (0.003) 4.45 0.988 (0.002) 0.022 (0.004) 9.75 0.979 (0.005) 0.017 (0.005) 4.65 0.994 (0.001) 0.009 (0.002) 10.1 0.991 (0.003) 0.015 (0.004) 9.45 0.986 (0.004) 0.013 (0.004) ∞ 0.991 (0.003) 0.016 (0.004) ∞   3.8903.9944.3164.5603.8104.2143.3003.576 Q |  | (10) Q |  | (50) 8.8 54.2 20.6* 81.5** 22.2* 103** 9.4 64.8 15.7 86.4** 12.4 64.7 21.6* 75.2* 10.4 63.5 Q  2 (10) Q  2 (50) 8.5 54.1 17.4 61.2 72.0** 117** 8.4 46.5 73** 123** 23.9** 70.7* 43.3** 91.1** 38.9** 81.3** 1.01 1 0.69 0.68 0.90 0.87 0.72 0.70 0.95 0.94 0.91 0.89 1.38 1.33 1.14 1.10

19. 19 Estimation results of LMSV models SERIESEuroBPCANYenSP500Nikkei225FTSE100IBEX35  d  --- 0.930 0.003 7.38 --- 0.647 0.092 5.21 0.756 0.495 0.028 15.7 --- 0.488 0.283 8.41 --- 0.876 0.038 12.3 0.931 0.494 0.003 9.25 0.913 0.440 0.004 ∞ --- 0.663 0.122 ∞   3.8503.8844.1014.1023.6304.1993.2843.394 Q |  | (10) Q |  | (50) 8.4 55.4 28.3** 86.3** 28.8* 89.5** 22.4* 70.3* 45.3** 115** 13.4 61.5 22.2* 68.3* 15.4 62.3 Q  2 (10) Q  2 (50) 6.8 53.7 18.064.918.064.9 26.8** 71.8* 7.0 48.1 12.5 75.7* 18.9* 63.4 34.4** 80.9** 21.1* 59.8 --- 0.26 0.25 0.50 0.48 --- 0.24 1.08 1.05 ---

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21. 21 Conclusions 1.ARSV, LMSV and A-LMSV models are able to generate Taylor effect for the most realistic parameter sets 2.Sample and theoretical autocorrelations of | y t |  peak at similar values of  3.Use as an additional tool for model adequacy of a fitted SV model 4. consistent and asymptotically Normal