linear regression model with Generalized Lehmanns Alter ernativ - PowerPoint PPT Presentation

26 26 Apr pril, l, 20 2016 16 EM EMLyon Semin inar ar Asymptotic ic Nor ormali lity of of R-estim imators for or a a si simple le linear regression model with Generalized Lehmann’s Alter ernativ ive Mod odels ls ( Jensen’s alpha ). ). Ryozo Miura Hitotsubashi University (Professor Emeritus) Tohoku University (Visiting Professor) Japan The working ing pape per r (referr ferred ed in the end of my talk) ) is a joint int work with h Profes fessor or Dalib ibor or Volny ny at Universi ersity ty of Rouen en and Professo fessor Sana a Louhich ichi at Univer ersit ity of Grenob oble, e, in France ce.

Outline of today’s Talk :1. Motivations with an old work Estimates of Beta and Residuals (Skewed distribution) by LSE and R-estimator Comparison of Asymptotic Variances. Differences of Estimated Values using real data. :2. Statistical Model with Skewed Error Terms. A Simple Linear Regression Model with a Generalized Lehmann’s Alternative model. Possible applications in Finance: Market risk of Stock Portfolio . Jensen’s Alpha :3. Rank Statistics and the derived R-Estimators. R-Estimates estimate Beta only while LSE has to estimate Intercept as well simultaneously. LSE may be modified by M-estimator with a score function suitable to a skewed error terms. :and Asymptotic Normality of R-Estimators (weakly dependent cases). Asymptotic linearity of Rank Statistics as a function of a tentative variable. Asymptotic Normality of R-estimators.

Abstract tract: We We look at at a simple linear ar regres ression on model el. Mi Miura& ura&Tsuk ukahara ahara (1993 1993) defi efined ned R-es esti timators mators for for General Generalized ed Lehmann’s Al Alternati ternative Mod Model els (or (or Trans ransformati formation on mod model els) and and prov roved ed thei their as asymp mptoti totic normal normality ty und under er the the as assump umpti tion on that that the the obs observ ervati ations ons are are ind ndep epend endent nt and and identi entical cally distri tributed buted. Thi This mod model el incl nclud udes es a us usual ual one one samp ample locati ocation on model els. Combi Combini ning ng thi this res resul ult wi with th Jureck Jureckova(1971 1971), ), we we can can es esti timate mate beta beta and and al alpha ha based on based on Rank Rank stati tatisti tics cs ev even en in in the the cas cases es where where the the error error distributi tribution on is is skewed ewed. The The as asymp mptoti totic effi effici ciency ency of of our our es esti timators mators seem eem a lot ot better better than than the the us usual ual leas east square quare es esti timates mates in in iid iid cas case. The The mod model el can can be be us used ed to to des escri cribe be so so that that the the so so-cal called ed Jensen’s al alpha ha in in the the worl world of of mean mean-vari ariance ance approach(CAPM) roach(CAPM) appears ears when when the the error or term term distri tributi bution on is is skewed wed. Time me seri eries es data ata in in finance nance are are very ery often often seri erial ally weak weakly dep epend endent ent (not (not iid iid). Some ome res resul ults ts, or or ex extens tensions ons for for a ser eries es of of weak weakly dep epend endent ent obs observ ervati ation ons are are also under er way way (Koul ul(1977 1977), ), Louhi hichi chi and et’ al al (2015 15)) )). The The wor worki king ng pap aper er is is a joi oint nt work work wi with th Profes Professor Dal Dalibor bor Vol olny ny at at Uni nivers ersity ty of of Rouen en and Profes fessor or Sana Louhi hichi chi at at Univers ersity ty of of Grenobl noble, e, in in France nce.

[1] Jureck Jureckova, J.(1971 1971). “Nonparametric Es Esti timate mate of of Reg Regres ression on Coeffi Coeffici cients ents. ” The The Annal Annals of of Stati tisti tics cs. Vol.47 47. No No.4. 1328 1328-1338 1338. [2] Koul Koul, H. L.(1977 1977). “Behavior of of Robus Robust Es Esti timators mators in in the the Reg Regres ression on Mod Model el wi with th Dep Depend endent ent Errors ors. ” The The Annal als of of Stati tisti tics cs. Vol. 5, No No. 4, 681 681-699 699. [3] Louh Louhich chi,S ,S., Mi Miura, ura, R. and and Vol olny ny, D. (2015 2015). ”On the the asym asymptoti ptotic norma normality ty of of the the R-estim estimators ators of of the the slop ope parameters arameters of of simp mple linear near reg regres ression on mod model els wi with th pos ositi tivel ely dep epend endent ent errors errors. ” Tohok ohoku Uni nivers ersity ty Center Center for for Data ata Sci cience ence and and Data Data Serv ervice ce Res Research earch. Di Discus cussion on Pap Paper er No No. 49 49. Octo October ber 23 23, 2015 2015 http://www.econ.tohoku.ac.jp/econ/datascience/newpage7.html [4] Sana ana Louhi Louhich chi (2000 2000). “Weak Conv Converg ergence ence for for Emp Empiri rical cal Proces Processes es of of As Associ ociated ated Sequence equences. ” Ann Ann. Inst. Henri ri Poincaré ncaré, Probab babilités tés et et Stati tisti tiques ques Vol. 36 36, No No. 5, 547 547 – 567 567 [5] Mi Miura, ura, R. and and Tsukahar ukahara, H.(1993 1993). “One samp ample es esti timati mation on for for general eneralized ed Lehmann’s alternati ernative model els. ” Stati atisti tica ca Sinica ca. Vol. 3. 83 83-101 101. [6] Miura, R. (2014). “ Ippanka sareta Lehmann tairitsukasetsu moderu wo motiita tanjunsenkeikaiki moderu to juni suitei (A simple linear regression model with error terms represented by generalized Lehmann’s alternative models and Rank-based estimators). ” Syougaku Kenkyuu, Kwansei Gakuin Daigaku, March 2014. pp.89-108. (in Japanese) [7] Qi Qi-Man Man Shao hao and and Hao Hao Yu Yu (1993 1993). “Weak Conv Conver ergence ence for for Wei eighted hted Emp Empiri rical cal Proces Processes es of of depend endent ent sequences uences. ” The The Annal als of of Probabi bability ty Vol. 24 24, No No. 4, 2098 2098-2127 2127.

:1. Motivations with an old work Estimates of Beta and Residuals by LSE and R-estimator Rank statistics ： Score function is ” t-1/2 ” .

Differences of Estimated Values using real data. Monthly data Tokyo Stock Exchange From January 1952 to December 1981 (352 names) This 30years decomposed to sixe 5years(60 months)

Monthly data Tokyo Stock Exchange From January 1952 to December 1981 (352 names) This 30years decomposed to sixe 5years(60 months)

OKI 2010 Jan.-2014 Apr. residuals (LSE, Rank-Est) Normal Probability Paper Normal Probability Paper 99.99 99.99 99 99 95 95 Cumulative Percent Cumulative Percent 80 80 50 50 20 20 5 5 1 1 0.01 0.01 -30 -20 -10 0 10 20 30 40 -20 0 20 40 Observed Value Observed Value

In n Miu iura(1985) Lo Log-Normal( ξ , , τ ) sh shif ifted by y +1 +1. G(x G(x)= )= Ф (( ((lo log(x+1) ) - ξ )/ )/ τ ) The shift by +1 is a maximum shift to cover all the possible return rate. So it is rather ad-hoc since so much empty space=no observation area. Instead of +1, one may set a shift parameter s to adjust the shift to a data. Generalized Lehmann’s Alternative Model with a location parameter might be able to describe the residuals . It should be remarked that the advantage of R-estimators is its smaller asymptotic variance of estimation error. It is ¼ at tau=0.1 and 1/25 at tau=0.15 . (iid case) Tau is a scale parameter. Remark. Least square estimates can be improved by taking a more suitable Loss function among The family of M-estimates. LSE corresponds to the Normality of the distribution of ε i .

Comparison of Asymptotic Variances. 1 ˆ     2 2 ( ) x f x dx ( ) , LSE 2 ns 1 1 iid case. ˆ    2 ( ) ,  Rank 2 2 2 ns 12{ f x ( ) dx } 2 where s =sample variance of explaining variable. Efficiency Comparison (Ratio) ˆ   2 1  ( ) 3 1 2     2 2 LSE e 2 ( e 1), a function of only. ˆ     2 2 ( ) Rank

Fig. A rolling AR structure of a hedge fund's returns

In In Miura(1985), , err error term erms wer ere mod odele led with th Lo Log-Norm rmal( l( ξ , , τ ) ) shi shifted by +1 +1. G(x G(x)= Ф ((lo log(x+1) - ξ )/ )/ τ ) Then, in 2014 Miura (in Japanese) made a note: Y i = β x i + ε i , i =1,2,…,n. ε i s are iid distributed with G(x- μ )=h(F(x- μ ): θ ). After this iid cases, we have been working on weakl dependent cases. Remark. The computing procedure for R — estimates of β is the same for both cases: iid and weakly dependent cases. Hence the above graghs for R-estimates of β are valid in weakly dependent cases

:2 :2. Statistical Model : Generalized Lehmann's Alternative Models can capture the skewed (asymmetric) distribution with a degree of asymmetry. : Monotonicity assumption for the model is set so to accept R- estimators.