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

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 Kernel densities of

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 Summary descriptive statistics of returns* SERIE EuroBPCANYenSP500Nikkei FTSE100 IBEX35 Size Kurtosis r 1 (1) Q | Y | (50) r 1 (2) Q Y 2 (50) * F iltered by fitting MA(1) and/or correcting possible outliers >5  t/T Empirical application

18 Estimation results of ARSV(1) models SERIESEuroBPCANYenSP500Nikkei225FTSE100IBEX35     (0.005) (0.002) (0.003) (0.003) (0.002) (0.004) (0.005) (0.005) (0.001) (0.002) (0.003) (0.004) (0.004) (0.004) ∞ (0.003) (0.004) ∞   Q |  | (10) Q |  | (50) * 81.5** 22.2* 103** ** * 75.2* Q  2 (10) Q  2 (50) ** 117** ** 123** 23.9** 70.7* 43.3** 91.1** 38.9** 81.3**

19 Estimation results of LMSV models SERIESEuroBPCANYenSP500Nikkei225FTSE100IBEX35  d  ∞ ∞   Q |  | (10) Q |  | (50) ** 86.3** 28.8* 89.5** 22.4* 70.3* 45.3** 115** * 68.3* Q  2 (10) Q  2 (50) ** 71.8* * 18.9* ** 80.9** 21.1*

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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