Applying least absolute deviation regression to regressiontype estimation of the index of a stable distribution using the characteristic function

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1 Applyig least absolute deviatio regressio to regressiotype estimatio of the idex of a stable distributio usig the characteristic fuctio J. MARTIN VAN ZYL Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State, PO Box 339, Bloemfotei, South Africa Abstract Least absolute deviatio regressio is applied usig a fixed umber of poits for all values of the idex to estimate the idex ad scale parameter of the stable distributio usig regressio methods based o the empirical characteristic fuctio. The recogized fixed umber of poits estimatio procedure uses te poits i the iterval zero to oe, ad least squares estimatio. It is show that usig the more robust least absolute regressio based o iteratively re-weighted least squares outperforms the least squares procedure with respect to bias ad also mea square error i smaller samples. Keywords Stable Distributio; Idex; Characteristic fuctio; Estimatio. Mathematical Subject Classificatio 62F10; 60E07; 62E10; 1. Itroductio A procedure usig robust LAD regressio based o the empirical characteristic fuctio (c.f.) evaluated at a fixed umber of poits to estimate parameters of the symmetric stable distributio is proposed. Deote the c.f. by φ ( t) Page 1. Let x,..., 1 x deote a sample of size i.i.d. observatios. The sample c.f. is estimated for a 1 give value of t as ˆ itx ( ) j φ t = e. Suppose the c.f. is estimated at K poits 1 j= 1 t,..., t, = 1,..., K. (1980) showed that the trasformatio K ˆ ca be used to costruct liear regressio equatios to 2 log( log( φ ( t ) )) estimate the parameters. The resultig regressio equatio is highly heteroscedastic ad there is also a much more complicated autocorrelatio

2 structure tha a simple autoregressive process of low order. (1980) foud that the optimal values for calculatig the empirical c.f. are t,..., t, t = π / 25, = 1,..., K. A complicatio is that K depeds o the 1 K uow value of the idex of the stable distributio ad the estimatio results are very sesitive to the umber of poits. Thus the two problems whe usig the empirical c.f. regressio approach is to fid the optimal value of K ad which poits must be chose. Methods were derived usig a fixed umber of poits ad other usig a umber which is a fuctio of the uow parameters. This wor will focus o comparig the LAD regressio estimatio procedure usig a fixed umber of poits for all values of the uow parameters with that of Kogo ad Williams (1998), which also mae use of a fixed umber of poits. The followig aspects will be tae ito accout i this wor: The iterval where the residual variace of the regressio reaches a miimum ad is most costat. It was foud that this iterval is approximately for t [0.5,1.0] ad usig poits chose i this iterval leads to excellet results with respect to the bias of the estimated parameters but performs reasoable with respect to MSE. The more robust least absolute regressio (LAD) maig use of iteratively reweighted least squares (IRLS) is tested. IRLS for LAD estimatio use weigths with are iversely proportioal to the absolute value of the residuals ad may perform good i regressio problems where heteroscedasticity is preset. It was foud that if a fixed umber of poits K=20, i the iterval [0.1,1.0] is used, the excellet estimatio results with respect to both MSE ad bias were foud over the whole rage of parameters. The sesitivity of the various procedures with respect to the umber of poits ad which poits are chose. It was foud that the LAD procedure with poits chose o the iterval [0.1,1.0], the results is robust with Page 2

3 respect to the choice of K, but for the poits t,..., t, t = π / 25, = 1,..., K, the wrog choice of K ca lead to very 1 K biased estimators of the idex whe least squares estimatio is performed. The more robust LAD regressio usig the IRLS method ad a fixed umber of poits outperforms the Kogo ad Williams (1998) procedure i samples with up to a few hudred observatios. Kogo ad Williams (1998) suggested te poits [0.1, 0.2,,1.0] usig least squares regressio. Some sewess i the data does have a small ifluece o the estimatio results. LAD estimatio outperforms the (1980) estimatio method where the umber ad choice of poits where the c.f. is calculated is chose usig iitial estimatio of the parameters. For a give sample, the c.f., φ ( t), regressio equatios are formed based o calculatig the empirical characteristic fuctio at poits t 1,..., t, t = π / 25, = 1,..., K. I practice the parameters ad specifically K the idex is uow ad K is a fuctio of the uow idex of the stable distributio. Whe the c.f. is calculated at the optimal poits this method performs excellet, but the method of (1980) is very biased whe choosig K icorrectly. This wor will focus o the estimatio of the idex of symmetrically stable distributed data. Such data are ofte used i maret ris aalysis ad especially whe worig with log returs of assets traded i a maret. Some of this is reviewed i the boos by Cize, Härdle ad Wero, eds. (2011), Getle, Härdle, Mori, eds, Deote the c.f. of the stable distributio by φ ( t) where log ( t) α α φ = σ t {1 iβsig( t) ta( πα / 2)} + iµ t, α 1, ad log φ( t) = σ t {1 + iβsig( t)(2 / π )log( t )} + iµ t, α = 1. Page 3

4 The parameters are the idex α (0,2], scale parameter σ > 0, coefficiet of sewess β [ 1,1] ad mode µ. The symmetric case with µ = 0, β = 0 will be cosidered i this wor. (1980) made use of the properties of the c.f. 2 α α ad usig the fact that φ( t) = exp( 2 σ t ) derived the model which does ot ivolve β ad µ whe estimatig the idex α ad σ : φ t = σ + α t, (1.1) 2 α log( log( ( ) )) log(2 ) log( ) a simple liear regressio model ca be formed y = m + αω + ε. (1.2) The c.f. is estimated for a give value of t, for a sample of size i.i.d. observatios,..., 1 x1 x, as ˆ itx ( ) j φ t = e, ad j= 1 y = ˆ φ, 2 log( log( ( t ) )) α m = log(2 σ ), ω = log( t ), ε a error term. (1980) derived the optimal poits t = π / 25, = 1,..., K, ad optimal values of K was suggested for various sample sizes ad α ' s. I practice for a specific sample size α is uow ad choosig it icorrectly leads to icorrect estimatio results. A expressio for the covariace ˆ φ ˆ φ ad thus the variace of 2 2 cov( ( t j ), ( t ) ) ˆ ( ) 2 φ t j is give by (1980). This expressio depeds o the 2 uow parameters, ad thus also Var(log( log( ˆ φ ( t ) ))) maig weighted regressio problematic. j Paulso et al. (1997) showed that by usig stadardized data estimatio results ca be improved ad all estimatio i this wor will be performed o stadardized data. (1980) foud that this regressio equatio does ot deped o the locatio parameter. (1980) suggested usig a trucated mea of 25% ad the Fama ad Roll ( 1971) estimator of the scale parameter σ : ˆ σ = ( x x ) /1.654, Page 4

5 where the.72 ad.28 deote percetiles of the data, to stadardize the data. The same stadardizatio will be used before applyig the LAD regressio. Kogo ad Williams (1998) used iitial estimators usig the method of McCullogh (1986) to perform stadardizatio. The LAD results will be compared o stadardized data as described above agaist the method assumig K ow, K estimated ad also the Kogo-Williams procedure (Bora ad Wero (2010c)). The optimal K for the procedure of (1980) is chose usig a iitial estimate of the idex usig the Mcullogh (1986) estimate as applied by Bora ad Wero (1910a). Much research was doe o usig a approximate covariace matrix ad geeralized least squares to estimate the parameters. A excellet overview of this approach is give by Besbeas ad Morga (2008). They suggested usig arithmetic spacig of t s which performs very well but the optimal umber of poits chose is also ot idepedet of the uow parameters. The wor of Feuerverger ad McDuough (1981a, 1981b,1981c), ad Bauer (1982) is also of importace where weighted least squares ad geeralised least squares were applied. But also o defiite umber of samplig poits which will perform good over the whole rage of the idex. It is show i sectio 2 that the residual variace is highly heteroscedastistic with respect to t. This might lead to a decrease i the efficiecy, ad also icorrect estimates of the variaces of the estimated parameters. The variace of the residuals, ε 's for a give t ad the true parameters, is estimated usig simulated samples. Residuals of a sample was calculated usig the true parameters as ε = y m αω, ad from these residuals the variace, Var( ε ) = Var( ε t, σ, α), was estimated. The sesitivity of the procedure with respect to K is also ivestigated. I sectio 3 simulatio studies were coducted to compare the various estimators. Page 5

6 2. The residual variace whe usig regressio type methods based o the empirical characteristic fuctio The residual variace, autocorrelatio structure whe usig least squares ad the sesitivity whe choosig K icorrectly with respect to the optimal K will be ivestigated i this sectio. The programs of Bora ad Wero (2010a), (2010b), (2010c), Bora et al (2011) were used whe applyig the method of Kogo ad Williams (1998) ad also whe applyig the method of (1980) usig iitial estimated parameters. Their iitial estimatio ad also stadardizatio was accordig to the wor of McCullogh (1986). I figure 1 the estimated residual variace based o the true parameters usig (1.2) ad o-stadardized data is plotted for various values of t. The variace of the residuals of M = 1000 samples of size = 200 each, with respect to t is show. The data was simulated with α = 1.5, σ = 0.1, β = 0, µ = 0. Similar patters was observed for other values of α, σ. The error variace is smallest i the iterval with t betwee 0.5 ad 1.0. The true parameters are used i the calculatio of the error variace, for example 2 2 ˆ ˆ t t t,, var( 1,..., M t,, ), σ = σ σ α = ε ε α σ, where ε α, m, t = y m α log( t), y = ˆ φ t j = M. j 2 log( log( ( ) )), 1,..., j j residual variace for a give t t Figure 1 Estimated residual variaces for give values of t, α = 1.5, σ = 0.1, = 200. Page 6

7 I figure 2 the estimated residual variace based o the estimated parameters usig (1.2) ad stadardized data is plotted for various values of t. The data stadardized usig the program by Bora ad Wero (2010a). Similar patters of the error variaces were observed for differet values of the idex. The autocorrelatio fuctio plot of the residuals for a specific sample is show i figure 3, which shows that a simple autoregressive type model will ot fit the residuals. Experimetatio showed that the patter is a complicated ARMA type model with terms of high order which is ot easily idetifiable, maig the use of regressio models with autocorrelated residuals very difficult residual variace t Figure 2 Estimated residual variaces for K=28 values of t, α = 0.9, σ = 1.0, = 200. Stadardized data ad variaces calculated from 500 regressios. Page 7

8 1 0.8 Sample Autocorrelatio Lag Figure 3 ACF of residuals, α = 0.9, σ = 1.0, = 200. Stadardized data. The bias whe usig the icorrect umber of poits whe usig the (1980) procedure o stadardized data is show. The estimated parameters was calculated based o 500 estimated values of the idex α = 1.3 for which the optimal K=22. The LAD estimated idex for various values of K is also show, ad it ca be see that the result is ot very sesitive with respect to K estimated idex K Figure 4 Estimated idex, various values of K, α = 1.3, σ = 1.0, β = 0.0, = 200. The solid lie whe usig π / 25, = 1,..., K. Average of 500 estimated idexes at each K. ad the dashed lie K LAD regressio usig K poits i the iterval [0.1:1.0]. Page 8

9 This small error variace together with the research of Kogo ad Williams (1998) motivated the use of iterval [0.1, 1.0] whe usig LAD regressio. Experimetatio showed that 20 uiformly distributed poits yielded good estimatio results whe usig LAD, with respect to bias ad MSE. 3. Compariso betwee estimatio procedures I this sectio a simulatio study was coducted. Stadardizatio was performed o all the data before estimatio ad the locatio parameter µ = 0, scale parameter σ = 1.0 was used for all the simulatios. Samples from symmetric distributios with locatio parameter zero were cosidered. For the LAD method, data was stadardized usig the Fama ad Roll (1971) estimator ad a 25% trimmed mea. K=20 poits were used for all values of the idex, ad the poits used were t = ( 1), = 1,..., 20. Iteratively reweighted least squares (IRLS) ca be used to miimize with respect to the absolute value orm if the weight matrix is a diagoal matrix with diagoal elemets equal to the iverse of the absolute value of the residuals. This techique was applied to fid the LAD estimators. The method ivolves a weighted least squares multiple liear regressio form, calculated iteratively, usig a diagoal weight matrix ( j) W with diagoal elemets with w = 1/(1 + e ), i = 1,..., 20, where e i is the i-th residual at iteratio j. This form is chose to avoid divisio by zero ad it ca be see that it is weighted regressio with weights iversely related to the size of the absolute value of the residuals. The results are based o simulated samples each time. No adjustmet was made if the LAD estimate was larger tha 2. ( j) i i For the Kogo ad Williams (1998) method the poits t = ( 1), = 1,...,10 were used. The results of the (1980) procedure choosig K usig the true idex is icluded as a referece. It should be oted this is ot comparable to the other three methods with respect to practical problems where there is o prior owledge of the parameters ad all are Page 9

10 parameters are calculated usig the sample. The Bora ad Wero (2010a) method base the umber of poits o estimated iitial values. 3.1 Results for the estimatio of the idex The assumptio is ot made that the mea is zero ad stadardizatio is carried out also with respect to the mea i the simulatios. For the idex estimatio, the LAD methods outperforms the other procedures with respect to MSE i smaller samples ad much the same whe it was tested o sewed data where β = 0.5. It seems that i larger samples fidig iitial estimates ad the usig the optimal poits yields the best results. It ca be oted that the Fama ad Roll (1971) estimatio method which is used for stadardizatio, is strictly speaig valid for α 1, which may explai the wea estimatio results whe usig these estimates to stadardize the data whe α < 1, ad thus leads to poor results especially whe α = 0.7 ad usig the method of (1980). This poit was ot icluded i the figures. I table 1 the performace of usig least squares estimatio usig te poits uiformly chose i the iterval [0.5,1] is show. It ca be see that it yields excellet results with respect to bias, but the MSE is weaer tha the Kogo- Williams procedure. Overall LAD regressio is best to use whe both bias ad MSE is tae ito accout with a MSE cosidered as least or more importat tha bias. α LAD Least Squares [0.5,1.0] Kogo-Williams [0.1,1.0] ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE Table 1 Compariso of estimatio procedures of α with respect to bias ad MSE, Page 10

11 σ = 1.0, µ = 0, β = 0, = 100. LAD usig 20 poits o the iterval [0.1,1.0], least squares 10 poits o the iterval [0.5,1], ad the Kogo-Williams procedure 10 poits o the iterval [0.1,1]. α LAD (K optimal true α ) (Bora ad Wero) Kogo-Williams ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE Table 2 Compariso of estimatio procedures of α with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0, = MSE estimated idex idex α Figure 5 MSE of three procedures for various values of the idex α. Symmetric data ad estimatio performed o stadardized data, =100, estimated samples. Solid lie - LAD, dash dot - Kogo-Williams method ad dashed the method of with the umber of poits chose usig the method of McCullogh. Page 11

12 α LAD (K optimal true α ) (Bora ad Wero) Kogo-Williams ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE Table 3 Compariso of estimatio procedures of α with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0, = 200. The MSE of the various procedures is show i figure MSE estimated idex idex α Figure 6 MSE of three procedures for various values of the idex α. Symmetric data ad estimatio performed o stadardized data, =200, estimated samples. Solid lie - LAD, dash dot - Kogo-Williams method ad dashed the method of with the umber of poits chose usig the method of McCullogh. Page 12

13 α LAD (K optimal true α ) Kogo-Williams ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE Table 4 Compariso of estimatio procedures of α with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0, = x MSE estimated idex idex α Figure 7 MSE of three procedures for various values of the idex α. Symmetric data ad estimatio performed o stadardized data, =800, estimated samples. Solid lie - LAD, dash dot - Kogo-Williams method ad dashed the method of with the umber of poits chose usig the method of McCullogh. Page 13

14 α LAD (K optimal true α ) Kogo-Williams ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE ˆα Bias MSE Table 5 Compariso of estimatio procedures of α with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0.5, = MSE estimated idex idex α Figure 8 MSE of three procedures for various values of the idex α. β = 0.5 ad estimatio performed o stadardized data, =200, estimated samples. Solid lie - LAD, dash dot - Kogo-Williams method ad dashed the method of with the umber of poits chose usig the method of McCullogh. 3.2 Results for the estimatio of the scale parameter There is ot much differece betwee the estimators of the scale parameters. Agai it seems that LAD will be best to apply whe the samples are smaller ad i large samples the other procedures performs well. Page 14

15 LAD (K optimal) Kogo-Williams ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE Table 6 Compariso of estimatio procedures of σ with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0, = MSE estimated scale parameter idex α Figure 9 MSE of three procedures for various values of the scale parameterσ. Symmetric data ad estimatio performed o stadardized data, =100, samples. Solid lie - LAD, dash dot - Kogo-Williams method ad dashed the method of with the umber of poits chose usig the method of McCullogh. Page 15

16 LAD (K optimal) Kogo-Williams ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE Table 7 Compariso of estimatio procedures of σ with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0, = 200. A plot of the bias of the various procedures is show i figure 6. It ca be see that overall all the methods performs well with respect to the estimatio of the scale parameter ad there is little differece betwee the methods. I the followig tables the procedures were compared usig sewed data with β = 0.5. It ca be see that usig LAD yielded good results. LAD (K optimal) Kogo-Williams ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE Table 8 Compariso of estimatio procedures of σ with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0.5, = 200. I table 9 results are give for sample size =800, simulated samples. Page 16

17 LAD (K optimal) Kogo-Williams ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE ˆ σ Bias MSE Table 9 Compariso of estimatio procedures of σ with respect to bias ad MSE, σ = 1.0, µ = 0, β = 0, = 800. It ca be see that the regressio usig LAD ad a fixed umber of poits to evaluate the empirical characteristic fuctio performs almost everywhere better that the Kogo-Williams ad the procedure where the umber of poits is based o iitial estimated parameters. 4. Coclusios LAD regressio performs better tha usig least squares Kogo ad Williams (1998). The estimated ˆα ca also be used to guess the best value of K, ad the performig estimatio based o the poits t = π / 25, = 1,..., K. Because the bias is very small, also whe usig least squares, usig the iterval [0.5,1.0], might be a very easy to calculate good estimate of K before usig the optimal poits derived by (1980). Refiemets ca be made to this procedure, with respect to the umber of t s used i the regressio. The simulatio study shows that the method performs well with respect to bias ad MSE over the whole rage of parameters commoly ecoutered i practical problems. Page 17

18 Refereces Besbeas, P, Morga, BJT (2008). Improved estimatio of the stable laws. Stat Comput 18: Bora, S., Misiore, A. ad Wero, R. (2011). Models for Heavy-tailed Asset Returs, I Statistical Tools for Fiace ad Isurace, P.Cize, W.Härdle, R.Wero (Eds.), 2d editio, Heidelberg : Spriger-Verlag. Bora, S. ad Wero, R. (2010a). STABLEREG: MATLAB fuctio to estimate stable distributio parameters usig the regressio method of, Statistical Software Compoets M429005, Bosto College Departmet of Ecoomics. Bora, S. ad Wero, R. (2010b). STABLECULL: MATLAB fuctio to estimate stable distributio parameters usig the quatile method of McCulloch, Statistical Software Compoets M429004, Bosto College Departmet of Ecoomics. Bora, S. ad Wero, R. (2010c). STABLEREGKW: MATLAB fuctio to estimate stable distributio parameters usig the regressio method of Kogo ad Williams, Statistical Software Compoets M429004, Bosto College Departmet of Ecoomics. Cize,P., Härdle, W., Wero, R., Eds. (2011). Statistical Tools for Fiace ad Isurace, 2d editio, Spriger-Verlag: Heidelberg Fama, E.F., Roll, R. (1971). Parameter Estimatio for Symmetric Stable Distributios, J. of the Am. Stat Assoc., 66, Feuerverger, A. ad McDuough, P. (1981a). O the efficiecy of empirical characteristic fuctio procedures. J. Roy. Stat. Soc. Ser B, 43, Feuerverger, A. ad McDuough, P. (1981b). O some Fourier methods for iferece. J. Am. Stat. Assoc., 76, Feuerverger, A. ad McDuough, P. (1981c). O efficiet iferece i symmetric stable laws ad processes. I Csörgö, M., Dawso, D., Rao, J., Saleh, A. (eds.). Statistics ad Related Topics, vol. 99, pp North-Hollad, Amsterdam. Getle, J.E., Härdle, W., Mori, Y., Eds. (2004). Hadboo of Computatioal Statistics: Cocepts ad Methods. New Yor: Spriger. Kogo, S.M. ad Williams, D.B. (1998). Characteristic fuctio based estimatio of stable distributios parameters. I: Adler, R.J, Feldma, R.E, Taqqu, M.S. (Eds.). A Practical Guide to Heavy Tails, pp Bosto: Birhäuser., I.A. (1980). Regressio-type Estimatio of the Parameters of Stable Laws. J Am Stat Ass 75: , I.A. ad Bauer, D.F. (1982). Asymptotic distributio of regressio-type estimators of stable laws. Commu. Stat. - Theory ad Methods, 11, McCulloch, J.H. (1986). Simple Cosistet Estimators of Stable Distributios, Comm. i Statistics Simulatio ad Computatio, 15, Paulso, A.S., Holcomb, W.E., Leitch, R.A. (1975). The Estimatio of the Parameters of the Stable Laws, Biometria, 62, Page 18

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