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

SLIDE 1

26 26 Apr pril, l, 20 2016 16 EM EMLyon Semin inar ar

Asymptotic ic Nor

• rmali

lity of

• f R-estim

imators for

• r a

a si simple le linear regression model with Generalized Lehmann’s Alter ernativ ive Mod

• dels

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

• r Dalib

ibor

• r Volny

ny at Universi ersity ty of Rouen en and Professo fessor Sana a Louhich ichi at Univer ersit ity of Grenob

• ble,

e, in France ce.

SLIDE 2

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.

SLIDE 3

Abstract tract: We We look at at a simple linear ar regres ression

• n 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

• n 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

• n that

that the the obs

• bserv

ervati ations

• ns 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

• ne samp

ample locati

• cation
• n

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 based on

• n Rank

Rank stati tatisti tics cs ev even en in in the the cas cases es where where the the error error distributi tribution

• n is

is skewed

• ewed. The

The as asymp mptoti totic effi effici ciency ency of

• f our
• ur es

esti timators mators seem eem a lot

• t 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

• f mean

mean-vari ariance ance approach(CAPM) roach(CAPM) appears ears when when the the error

• r term

term distri tributi bution

• n is

is skewed wed. Time me seri eries es data ata in in finance nance are are very ery often

• ften seri

erial ally weak weakly dep epend endent ent (not (not iid iid). Some

• me res

resul ults ts, or

• r ex

extens tensions

• ns for

for a ser eries es of

• f weak

weakly dep epend endent ent obs

• bserv

ervati ation

• ns 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

• int

nt work work wi with th Profes Professor Dal Dalibor bor Vol

• lny

ny at at Uni nivers ersity ty

• f
• f Rouen

en and Profes fessor

• r Sana Louhi

hichi chi at at Univers ersity ty of

• f Grenobl

noble, e, in in France nce.

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[1] Jureck Jureckova, J.(1971 1971). “Nonparametric Es Esti timate mate of

• f Reg

Regres ression

• n Coeffi

Coeffici cients ents.” The The Annal Annals of

• f

Stati tisti tics cs. Vol.47

• 47. No

No.4. 1328 1328-1338 1338. [2] Koul Koul, H. L.(1977 1977). “Behavior of

• f Robus

Robust Es Esti timators mators in in the the Reg Regres ression

• n Mod

Model el wi with th Dep Depend endent ent Errors

• rs.” The

The Annal als of

• f 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

• lny

ny, D. (2015 2015).”On the the asym asymptoti ptotic norma normality ty of

• f the

the R-estim estimators ators of

• f

the the slop

• pe parameters

arameters of

• f simp

mple linear near reg regres ression

• n mod

model els wi with th pos

• siti

tivel ely dep epend endent ent errors errors.” Tohok

• hoku

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

• n 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

• f As

Associ

• ciated

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

• n 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

• f

depend endent ent sequences uences.” The The Annal als of

• f Probabi

bability ty Vol. 24 24, No

• No. 4, 2098

2098-2127 2127.

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Rank statistics： Score function is ”t-1/2”.

:1. Motivations with an old work Estimates of Beta and Residuals by LSE and R-estimator

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

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

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• 20

20 40

Normal Probability Paper

Observed Value Cumulative Percent 0.01 1 5 20 50 80 95 99 99.99

• 30
• 20
• 10

10 20 30 40

Normal Probability Paper

Observed Value Cumulative Percent 0.01 1 5 20 50 80 95 99 99.99

OKI 2010 Jan.-2014 Apr. residuals (LSE, Rank-Est)

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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 εi.

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2 2

2 2 2 2 2 2 2 2 1 2 2 2 2 2

1 ˆ ( ) ( ) , 1 1 ˆ ( ) , 12{ ( ) } where s =sample variance of explaining variable. Efficiency Comparison (Ratio) ˆ ( ) 3 1 ( 1), a function of

• nly.

ˆ ( )

LSE Rank LSE Rank

x f x dx ns ns f x dx e e

 

              

 

iid case. Comparison of Asymptotic Variances.

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

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In In Miura(1985), , err error term erms wer ere mod

• dele

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: Yi=βxi+εi , i=1,2,…,n. εi 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

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: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.

SLIDE 19

.

F(.) symmetric around zero. Assume:Functional form of F is unknown. εi ,i=1,2,…,n , (iid or) weakly dependent(associated)

＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊ ＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊

Assume: Functional form of h(t: θ) is known. (1). X1,…,Xn distributed with G(x)=h(F(x): θ). Xi = εi ,i=1,2,…,n. Transformation parameter θ. (2). X1,…,Xn distributed with G(x)=h(F(x- μ): θ). Xi = μ +εi ,i=1,2,…,n. εi follows G(x)=h(F(x): θ) Simultaneous estimation of θ and location parameter μ. (3). Yi=α+βxi+εi =βxi+(α+ εi ), i=1,2,…,n. Yi :G(y)=h(F(y-α-βxi): θ). εi ,i=1,2,…,n are distributed with G(.)=h(F(.):θ). Estimation of θ, μ and regression parameter β.

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Generalized Lehmann’s Alternative models G(x-μ)=h(F(x- μ):θ), for −∞<x<∞ where F(.) is symmetric distribution function about zero.

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* * 2

. Lehmann(1953): h(t: )=t , 0< , 1. Proportional Hazard: h(t: )=1-(1-t) , 0< . 1. Contamination: h(t: )=(1- )t+ t , Lehmann(1953?). 1 (Maximum with Trancarted Poisson Inp E u xamples lse): h(t: )=

t

e

  

             , 1 Seen in T.Ferguson(1967.Locally most powerful Rank test is Wilcoxon). and others (See Miura&Tsukahara(1993)). e 

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Remark. Ge Generalized Lehmann’s Alternative model (GLAM) an and Sk Skew Symmetry ry: :

G(x:δ,F)=h(F(x): δ) g(x: δ,F)=h’(F(x): δ)f(x) This compares to Azzalini’s skew symmetry : f(x)2F(δx), or =f(x)2K(δx) …….K(.) is any symmetric d.f. Thus, 2F(δx) or f(x)2K(δx) in skew symmetry corresponds to h’(F(x): δ) in GLAM.

1 ( ) 1

( : , , ) 2 ( ( )) ( ( ): , , ) 2 ( ( ))

t F x

h t F K K F u du and h F x F K K F u du    

 

 

 

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Louhich, Volny and Miura (2015) Cov(X1, Xn) = o(n-b ) , b> 4 Shao and Yu (1996) Cov(X1, Xn) = o(n-b ) , b> (3+√33)/2

Weak Dependence and Associated sequence of random variables

SLIDE 25

A Sim Simpl ple Li Linear Reg egressio ion Mod

• del

Yi = βxi +εi, i=1,2,…, n. Where εi’s are independent and identically distributed with G(x-μ)=h(F(x-μ):θ) The expectation of εi is the following;

1 1

E[ ] E[Y x ] h(F(x): )= + ( ) h(t: ) +m( )

i i i

xd F t d        

  

    

 

SLIDE 26

Rem emark. A A Comment on a relation to Jensen’s α in in the the Th Theory ry of

• f

Fin inance: e: CAP CAPM μｐ-r= r= (α+) +)+βpM

pM(μM-r)

r).

Take Y for a excessive return of financial asset, and take x for the excessive return of market portfolio. Then, CAPM says that Intercept alpha in the regression is zero. However, the real market often shows that it is not zero. That corresponds in our model : μ+m(θ) is Jensen’ alpha. and CAPM says that the above expectation is zero. Note that m(θ)=0 for θ=θ*, since the transformation function h(t:θ*)=t, and F is symmetric about zero. m(θ) can be positive and negative according to the value of θ.

( ) , i=1,2,..., n.

i i i

e Y x    

E[ ( )] +m( )

i

e    

1 1

( ) ( ) h(t: ) m F t d  



 

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:3. Construction of Rank Statistics and the Derived R-Estimates:

with a brief proof for their Asymptotic Normality : R-Estimates can estimate Beta only while LSE has to estimate Intercept as well simultaneously. : R-Estimators do not care about the symmetry of error term distribution. : Asymptotic linearity of Rank Statistics as a function of a tentative variable. : Asymptotic Normality of R-estimators and their Asymptotic Variances.

SLIDE 29

1 1 1 , 1

where, c (x x). '( ( )) J(.) can be taken to be J (t,g) 1 S (b) = J (R (b)/(n+1))(x x) n 1 = J (R (b)/(n+ =- ( 1)) c , n ( ))

i n n i i i n i i i i

g G t g G t

       

  

 

( ) , i=1,2,..., n.

i i i

e b Y b x  

1

R (b) = rank of e (b) among {e (b), j=1,2,...,n} = I{e (b) (b)}

i i j n i j j

e







This is a monotone step function of b. Steps at b=(yi – yj)/ (xi – xj )

SLIDE 30

1 , ,

S (b) = J ( ) ( )

n n b

t dV t

 



( 1) , ( ) 1 2 1 , 1 , 1 1 2 1

1 where ( ) and ( ) can be written by ( ) ( ) where is a piecewise linear version of 1 G ( ) { ( ) } and 1 ( ) [ { ( ) } ] : Here th

k

n t n b D b k n i i n b n b n n n i i n n i i n i i

V t c c V t V t W G G x I e b x n W x c I e b x x c Note

        

      

    

1 2 1

e mathematical essence is the convergence of 1 [ { } ] that has been proved for iid s Louhichi, Miura and Voly (201 equence of 5) has proved this for a bou , 1,2,..., . nded score funct

n i i i n i i i

c I x x c i n  

 

  

 

in and for weakly dependent sequences.

SLIDE 31

1 1 , ,

S ( ) = J ( ) ( ) converges to J ( ) ( )

n n b

t dV t t dV t

  



 

1 2 , , a 1

a Asymptotic Linearity: b= + , a sup S (b)-{S ( )+a ( ) ( , ) ( /n)}

n n n i A i

A n J t J t g dt c

  

 

 



 

1 1 2 1

n(β -β) ( , , ) J ( ) ( ) / ( ) ( , ) (lim /n)

n i n i n

T J h g t dV t J t J t g dt c

    

 

  

This has been proved by Louhichi , Miura and Volny(2015) in a slightely different form, for a bounded score function J(.), both for iid and weakly dependent ( α-mixing, and associated )sequence of εi , i=1,2,…, n. But it is not in the above from: Statistics is an integral integrated by dVn,b (t) For that, we need a further result of weak convergence of empirical distribution function in a form with q(t) function that will take care of order of convergence near the both end of the interval [0,1]. We are on our way for it.

SLIDE 32

Th This was as do done by y Jur Jureckova (19 (1969,1 ,1971) for

• r i.

i.i.d i.d.case.

We would like to treat weakly dependent cases. Koul (1977) did for strongly mixing sequence of random variables. Louhichi, Miura and Volny (2015) worked for associated sequence of random variables. Both only for bounded score functions. We should like to extend this to unbounded score functions, But, not yet done in weakly dependent cases.

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Once we have estimates of

• f β,

take residuals to

• proceed

for

• r estimation of
• f μ an

and θ.

Note: Estimation method for β can be Least squares estimates, Rank-statistic-based Estimates or any

• ther as long as its estimation error has

√n - order asymptotic distribution.

SLIDE 37

Miura and d Tsukaha hara ra (1993) ) defin fined d and d prove ved d asympt mptotic Norma mali lity of rank-est stim imates s

• f θ and

d µ. Iid case se. This can work well also so for weakl kly depe penden dent cases s when we use the results

• f Shao and Yu (1996).

).

What we do here is to combine the two results where we have to use “residuals = estimated random errors” for estimation of θ and µ. Iid case and weakly dependent case as well. Koul (1977) and Louhichi et’al (2015).

SLIDE 38

Shao & Yu (1996) THEOREM 2.4. Let { Un, n > 1} be a stationary associated sequence of uniform [0, 1] random variables. If Cov(U１, Un) = O(nーνーε) for some ν ≥ (3 +√‾ 33)/2 and ɛ > 0, then we have En (.)/q(.) → B*(.)/q(.) in D[0, 1] for any weight function q satisfying q(t) > C(t(l - t))(l-3/v)/2 for some C > 0.

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ˆ ˆ ( )

i n i n i

X Y x    

1

1 ˆ ˆ ( : ) { ( ) }, ( , )

n n n i n i

G x I X x x n  



    



( ) ( ) ( 1) ( 1) ( ) (0) (1) ( 1) ( )

ˆ ( ) ˆ ˆ ˆ ˆ ( : ) , [ ( ), ( )], i=1,2,...,n. ˆ ˆ 1 ( 1){ ( ) ( )} 1 1 ˆ ˆ ˆ ˆ ( ) ( ) , ( ) ( )

i n n n i n i n i n i n n n n n n n

x X i G x x X X n n X X X X X X n n          

  

          

ˆ ( : ): ) ( : ): )

n n

G x r G x r   

1

ˆ ˆ ˆ ( : ) ( ( : ): ) 1

i n n n

i Z r G h r n  



 

1 , 1

1 ˆ ˆ ˆ ˆ ( : ) { ( : ) } ( ( ( : ): ))

n n r n i n n n n i

L x I Z r x u h G x r n   

 

  



1

1 1 ( ) { }, t [0,1] 1

n n i

u t I t n n



   



1 ˆ ˆ , :0 ( : ) : ( : ) 0

ˆ ˆ ˆ where the rank: R ( : ) { ( : ) ˆ ˆ ( : ) ( : ) 1 1 1 1 ˆ S (r: ) = ( ) ( ) n 2 2( 1) n 2 2( 1) ( : )}.

i n i n

i n i n n n i Z r i n Z i n j n i n j r

R r R r J J n n r I Z r Z r

    

     

     

      

  

G-1(u)=F-1(h-1(u:θ)) G-1(h(t:θ))=F-1(t)

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Let , 0<s<C< . s r n    

ˆ for r and

n



, , , ,

ˆ ˆ S (r: )= {S (r: )-S (r: )}+ S (r: )

n n n n n n

n n n

   

   

1 , 1 1

1 ( ( : ) ( (1 : ) S (r: ) { ( ) [ ] ( )} 2 ( : ) (1 : )

n

U h t U h t n s dJ t h t h t



           



とするとき、rと に対して、n→∞のとき、 The e first term has asympt ptot

• tic

ic linearit earity unif ifor

• rmly

ly in s from Theorem eorem1. 1. The e second

• nd term was proved

ed in Theor eorem em2. 2.1 1 of Miura ra and Tsukah ahara ara （pag age. e.94 94,（2. 2.14 14））, to have e its limit it as follow lows. Ther eref efore,

• re, we hav

ave e the follow lowing ing.

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Let , 0<s<C< s r n    

ˆ

n



, , 1 1/2 1 ' 1 ' 1 2

ˆ {S (r: )-S (r: )} ( , , : ) ( ( )) ( ) ( , , ) ( ( )) (1 )

n n n

n T J h g xf F t J t dt T J h g xf F t J t d

     

  

 

  

 

Theor

• rem

em 1.

Then, n, as n→∞, it holds for r and unifor iforml mly y in s.

SLIDE 44

Sum ummary ry

Yi=βxi+εi , , i=1,2,…,n. (1) (1) εi ‘s ar are e iid iid dis istr tributed wit ith G(x (x-μ)=h(F(x-μ): ):θ), ), (2) (2) εi ‘s ar are e str stric ictly sta stati tionary and and weak eakly dep ependent with ith G(x (x-μ)=h(F(x-μ): ):θ). ).

(a) β and (θ, , μ) can an be e est estim imated sep eparately. (b) Once nce β is estimated with √n-order asymptotic normality for its estimation error, following Miura&Tsukahara(1993 and its extension), we can use the residuals to estimate (θ, , μ) ) with √n-order asymptotic normality in either cases that εi ‘s are iid iid or r that εi ‘s are wea eakly depen endent (str strongly mi mixin xing by Koul

• ul(1977), an

and as asso sociated Shao and Yu (1996): b>(3+√33)/2). (c) β can be estimated in many ways with √n-order asymptotic normality in iid case, e.g. by LSE, M-estimators and R-estimators. Then, for weakly dependent cases, we have R- estimators by Koul(1977) for bounded score function and stationary and εi ‘s bei eing str trongly mixing, and and by Louh Louhich ch, Miur iura and and Voln

• lny (2015)

) for bounded score functions and εi ‘s being stationary and as asso socia iated :b>4. (*) Remar

• emark. R-est

stim imators wor

• rk fo

for Multi ultiple le regr egressi sion mod model l as as well ell (Jur Jureck ckova 1969,1971)

SLIDE 45

*

More works remaining to be done.

Mathematical Theory : make sure that score can be unbounded in weakly dependent cases. : Construct estimators of asymptotic variances in weakly dependent cases. Applications in Finance : Market Risk Measurement: decomposition of stock portfolio market risk into systematic and idosincatic : Portfolio selection and portfolio performance. Try high-beta portfolio (low-beta portfolio) and see their performance with changes of classifications, in comparison with LSE-Beta portfolio selection. Estimates of θ and μ may be used as well to describe “Skew of Market”.

SLIDE 46

SLIDE 47

債券を加えた効果 途上国を加えた効果 ヘッジファンドファクタ を加えた効果 ① ② ③ ④ ⑤ ⑥ ④ ④ ③ ⑤ ⑥

SLIDE 48

Thank you for your attention.

。。。