Simple techniques for likelihood analysis of univariate and multivariate stable distributions: with extensions to multivariate stochastic volatility and dynamic factor models

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Munich Personal RePEc Archive

Simple techniques for likelihood analysis

of univariate and multivariate stable

distributions: with extensions to

multivariate stochastic volatility and

dynamic factor models

Tsionas, Mike

Athens University of Economics Business

10 May 2012

Online at

https://mpra.ub.uni-muenchen.de/40966/

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(19)

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Marginal posterior distributions of MCMC ABC

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0.7 0.8 0.9 1 1.1 1.2 1.3 0 2 4 6 d e n s it y

1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 1 2 3 4 5 d e n s it y

Marginal posterior distributions of

MCMC ABC

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Marginal posterior distributions of

MCMC ABC

1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 1 2 3 4 5 d e n s it y

Marginal posterior distributions of MCMC

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1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 1 2 3 4 5 d e n s it y

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2 4 6 8 10 12 14

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Exact Mixture (M=5)

3 0 2 '( 2 – 7G= G@

( = + . G( - + 2 .

+ . 7 2

2

0 2 0 2 0 2

0 2 '('( &'('( &&& 2222

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 -4 -2 0 2 m e a n s o f n o rm a l m ix tu re = -0.5 1 2 3 4 5

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 2 4 6 8 s .d . o f n o rm a l m ix tu re = -0.5

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.1 0.2 0.3 0.4 0.5 p ro b a b il it ie s o f n o rm a l m ix tu

re = -0.5

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 -4 -2 0 2 m e a n s o f n o rm a l m ix tu re = 0.2

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 2 4 6 8 s .d . o f n o rm a l m ix tu re = 0.2

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.2 0.4 0.6 0.8 p ro b a b il it ie s o f n o rm a l m ix tu

re = 0.2

% %%

% 4444 !#4!#4!#4!#4 2222

; =H '(H

α β 2 ? G=((

G'=(( 2 '( (((

2 -( .

F _α _β =H=H=H=H '(H'(H'(H'(H =(( ' '( @( <( ' /)- ' /$(

(27)

( -= ' <%' ' <%) ( =( ' <%' ' <%) ( <( ' <%' ' <%* '=(( ' '( @( <( ' /*/ ' /*% @( =( ' /*% ' /*) @( -= ' /** ' /** ( -= ' /** ' /)-( =/)-( ' /*' ' /*/ ( <( ' /*- ' /*/ ' $( @( <( ' <== ' <== @( =( ' <=/ ' <=% @( -= ' <=/ ' <=% ( -= ' <=$ ' <=$ ( =( ' <=/ ' <=$ ( <( ' <=/ ' <=$

/

//

/

!

6 2 +9 - -(($ 2

7 *' -('-. ' *%) 6 7474 2 00

+ G'% G( (('. !#4 2 !F0 7474 2 2

0 “ ” 2 7474 2 00 !#4

2 !F0 2 2

7G= 0

2

2 +μσπ.

+ .

3 +μσπ. + . ? 2 8:

2 2 ?

8: 0 '-( ((( -( (((

2 '( '( ((( ?

6 2 +' ((( .

2 @ @

+: 4 ? -((/ -((< : B -((<.

0 2 0 2 0 2

0 2 '''' 7 2'''' 7 27 27 2

1.250 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

5 10

d

e

n

s

it

y

Exact (FFT) ABC-ANF Mixture

-0.40 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

2 4 6 8

d

e

n

s

it

y

(28)

$ $$

$ &&&& 2222 !;4A!;4A!;4A!;4A BBBB 7474 74747474 7474 !;4A + . B . α

. ! 2 # + !#4 . 2 '-( (((

-( ((( '( !;4A –

!;4A – '/ _+7474. ₂ ₇ _D

2 -=H B – 2 2 8 B –

+7474. 2

7

!;4A– ( ($/=

+( ('(. +( ((<.( <')

!;4A– +!#4. ( ')*

+( (-=. +( (').( /=$ +( '--.' $=' +( ')'.@( ('* !;4A – + . ( ')( +( (-*. +( (').( /=/ +( '-(.' $=' +( ')*.@( ('' !;4A – +7474. ( ')* +( (-=. +( ('=.( /=$ +( '--.' $=' +( ')'.@( ('*

B– @( ''$

+( '-). +( ('=.( <=(

B– +!#4. @( (/<

+( (/=. +( (''.( /*- +( ''.' =$- +( ('=.@( '%

B– + . @( (<(

+( (/*. +( ('-.( /*( +( '-.' =$( +( ('%.@( '$

B– +7474. @( (<(

+( (/'. +( (''.( /*( +( ''.' ==( +( ('=.@( '$

F

F 2 !#4 2 !F0

0 2 '- 2

k

G= '= -= 0 2

0 2 0 20 2

0 2 '-'-'-'- &&&& !F0!F0!F0!F0 2222

0 5 10 15 20 25 30 35 40

0 5 10 15 20

number of points, k

p o s te ri o r, %

Marginal posterior distribution of k

-4 -3 -2 -1 0 1 2 3 4

0 20 40 60 80 100 k p o s te ri o r, %

Placement of points, k=5

-5 -4 -3 -2 -1 0 1 2 3 4 5

0 10 20 30 40 50 k p o s te ri o r, %

-4 -3 -2 -1 0 1 2 3 4

0 5 10 15 20 25 k p o s te ri o r, %

(29)

0 2 0 2 0 2

0 2 '*'*'*'* 7 27 27 27 2 2222 ????

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40

number of grid points, k

p

o

s

te

ri

o

r,

%

Marginal posterior, number of grid points

0 1 2 3 4 5 6 7

0 0.2 0.4 0.6 0.8 1

radians

c

u

m

u

la

ti

v

e

s

p

e

c

tr

a

l

m

e

a

s

u

re

Cumulative spectral measures

Projection I Projrction II Mult. Normal Sph. Harmonic

0 2 0 20 2

0 2 ')')')') 7777 7 27 27 27 2

1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2

0 5 10 15 20

d

e

n

s

it

y

Multivariate Stable: Marginal posteriors of

Projection I Projection II Mult. Normal Sph. Harmonic Copula

-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2

5 10

d

e

n

s

it

y

Multivariate Stable: Marginal posteriors of

(30)

<

<<

<

< '

3 < '

3

4

! +-$.

. . 0 2 8 4 +'<</.'<

! +-/.

3 2

6 2

2 2

2 2 3

2 ? 2

2 2

+-<.

8 4 +'<<* *.

! 2 7474

3 2 '((

K@( = ( =L : +7G- *. '(@%

2 7G'(

/ //

/ !!!! 2222 2222ε----_Μ_G'(_G'(_G'(_G'(

G@( < G@( = G@( -= G( -= G( = G( <

G' ' - )- ( $/( ( (/*=

- /( ( ='- ( (</) * %* ( (='= ( '(* * %' ( '$$ ( '(* - -< * <* ( '(-' =< ( /%( ( '(-'(( * %$ ( '%$ ( '(* * %- ( (<=) ( '(* * *$ ( '<' ( '(* * %/ ( '(/ ( '(*

' -< ( /%< ( '*% ' =) ( )<( ( (==* - =* ( (='< ( '*% - )= ( '<- ( (==< ' =% ) '% ( '*% ( %-= ( /%= ( ''/ * -- ( '/< ( '*$ - %% ( (<$% ( ('($ ' /= ( '<- ( (</< * )% ( '') ( ''%

( -<$ ( <)% ( -') ' (* ( )<* ( ()/' ' )) ( (='< ( ''% - '( ( '<* ( (''< ( $(/ ) '< ( -') @( )(% ' '* ( '*$ * (( ( -(- ( '$< - %( ( (<$% ( ((=)= ' (' ( '<* ( (%$) * () ( ''= ( (($=$

( -<* ( <)< ( -') ' (* ( )<* ( ()$< ' )) ( (='< ( ''% - '( ( '<* ( (''< ( %<) ) -( ( -') @( )'' ' '* ( '*$ - << ( -(- ( '$< - %( ( (<$% ( ((=)= ' (' ( '<* ( (%$' * () ( ''= ( (($=%

' *( ' '/ ( -') ' -( ( ='= ( (%*= - )( ( (=-( ( '<< - -- ( '<* ( (*-= ' == ) '% ( '*$ ( ('*/ ' -* ( ('*% * *) ( '/< ( -'= - %= ( (<$% ( ('*( ' *( ( '<= ( ($/$ * '( ( ''= ( (***

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G' * @( -%$ ) '' ( '='_{' '/ ( =/( ( ('=*} ( )/) ( (=-- ( '-' - -/ ( '<% ( (')= ' <( ' -' ( '$' @( )<( ' )( ( -*/ ' // ( '<< ( -=' - %$ ( (<$$ ( ('=/ ' *$ ( '</ ( ((<<% - )( ( '') ( ('-)

@( %%$ ) (* ( --$ ' '= ( =/< ( ('*% @( (%*' ( (=--( '==

- -$ ( '<% ( ((/-/ ' )% ' =% ( '(' @' '< ' (/ ( --) ' /) ( -($ ( -)/ - %$ ( (<$/ ( (($%$ ' *= ( '<< ( ((/'% - )( ( ''= ( (($=/

@( /$) ) (/ ( -=) ' '* ( =<- ( ('(' @( )(' ( (=-- ( '=-- '=--$ ( '<% ( ((%'' ' *= ' /$ ( (=%< @' %* ' (' ( -)< ' /' ( -'( ( -=) - %$ ( (<$/ ( ((=/-' *) ( ((=/-'<< ( ((%)-- )( ( ''= ( ((=$((%)--

((=$-@( /%% ) (/ ( -=% ' '* ( =<- ( ('(' @( */= ( (=-- ( ')< - -$ ( '<% ( ((%($ ' *= ' // ( (==/ @' %) ' (= ( -=( ' $/ ( -'( ( -== - %$ ( (<$/ ( ((=$% ' *) ( '<< ( ((%)) - )( ( ''= ( ((=%/

@( %'' * <- ( -)-' -)-') ( =<( ( (-)-')( @( ($(= ( (=--( '%'

- -$ ( '<% ( (($/= ' *$ ' %- ( (<-' @' '' ' (= ( -'/ ' /$ ( -($ ( -)-- %% ( (<$/ ( (($-)--= ' *= ( '<< ( ((/-$ - )( ( ''= (

(($--@( '/$ * </ ( '$% ' -- ( =<( ( (*-= ( =$' ( (=-* ( '=) - -$ ( '<= ( (-*( ' *- ( %=* ( '/$ @( $=$ ' -( ( '/% - )) ( '<* ( '/$ - %- ( (<$% ( ('/% ' )' ( '<< ( ('%( - *< ( '') ( ('</

G' = @' )- * /* ( '<=

' '- ( =<$ ( (-'- @' =) * <$ ( -((' (< ( %(( ( ('=' @' =$ ) (' ( -(*' (< ( %(( ( ('*' @' %= ) (= ( -(*' (< ( %(( ( ('*' @' == * << ( -((' (< ( =<< ( ('=( @' */ * /- ( '<=' '( ( =<$ ( (-'*

(31)

@( '*$ ( (=-) ( '%$ - -' ( '<= ( ((/=* ( %<- ( $*$ ( '<* @' %( ' (< ( '<= - (/ ( '<) ( '<= - =/ ( (<$% ( ((%%* ' *% ( '<< ( ('(< - *) ( '') ( (($%(

@( ))* ( (=-) ( '%= - -' ( '<% ( ((%=% ( ))< ( $<* ( '<= @- (' ' (/ ( -(( - (* ( '<= ( '<< - =/ ( (<$% ( ((=*-' *= ( ((=*-'<< ( ((/)( - *) ( '') ( ((=<=

@( =$- ( (=-) ( '%) - -' ( '<% ( ((=<-( *)- ((=<-( /'< ((=<-( '<= @- -( ' (< ( -(* - (' ( '<= ( '<< - =/ ( (<$% ( (()// ' *= ( '<< ( (($=) - *) ( '') ( ((=)'

@( =%) ( (=-) ( '%* - -' ( '<% ( ((=<= ( *)< ( /'$ ( '<= @- '< ' '( ( -(* - (' ( '<= ( '<< - =/ ( (<$% ( (()/< ' *= ( '<< ( (($=/ - *) ( '') ( ((=)*

@( ))( ( (=-) ( '%) - -' ( '<% ( ((%)/ ( ))$ ( $<( ( '<= @- (* ' '' ( -(( - (- ( '<= ( '<< - =/ ( (<$% ( ((=-% ' *= ( '<< ( ((/*$ - *) ( '') ( ((=/<

@( ')% ( (=-) ( '%/ - -' ( '<= ( ((/*( ( %<) ( $*% ( '<) @' %- ' (< ( '<= - ($ ( '<) ( '<= - =$ ( (<$% ( ((%)/ ' *= ( '<< ( ('(< - ** ( '') ( (($)' G' $ @- *( * /< ( '%(_{' ($ ( =*$ ( '*(}

@( $'$ ( (=-< ( ')( - (( ( '<* ( (-(-' )(-(-' ( $/% ( (-(-')$ @- $- ( '*= ( '%-( -*$ '%-( ')* '%-( '== - *- ( (<$* ( ((%)/ ' *= ( '<$ ( (%)) - '' ( '') ( (')(

@- -- * /% ( '%= ' (= ( =)' ( '-= @( $=% ( (=-< ( ')* - (( ( '<* ( ('/$ ' )' ( /'% ( ')% @- /= ( '*= ( '%% ( '<% ( ')* ( '=/ - *- ( (<$* ( ((%*= ' *) ( '<$ ( (=$' - '' ( '') (

('*-@- ** * $< ( '$-( </' '$-( ==- '$-( ''< @( =(( ( (=-< ( ')( - (- ( '<* ( ('%< ' )< ( </% ( ')/ @- /( ( '*$ ( '$* ( '%* ( ')- ( '%= - ** ( (<$* ( ((%)( ' *' ( '</ ( ()$= - '* ( '') ( ('-)

@- -/ * $/ ( '$* ( <=/ ( =)% ( '-( @( ))% ( (=-/ ( '*< - (- ( '<* ( ('%-' )/ ('%-' (( ( ('%-')$ @- /= ( '*$ ( '$* ( '/' ( ')- ( '%% - ** ( (<$* ( ((%-* ' -< ( '</ ( ()$' - '* ( '') ( (''<

@- )' * /( ( '$( ( <<% ( ==) ( '--@( =(% ( (=-< ( '*< - (- ( '<* ( ('$% ' =( ( <$< ( ')< @- $) ( '*$ ( '$' ( '%$ ( ')- ( '%* - *) ( (<$* ( ((%=) ' *- ( '</ ( ()<* - ') ( '') ( ('-/

@- *( * $< ( '%= ( </% ( =)) ( '-< @( )'/ ( (=-< ( '*/ - (- ( '<* ( ('$< ' )$ ( <)$ ( ')< @- $) ( '*$ ( '%% ( --/ ( ')- ( '%' - ** ( (<$* ( ((%)-' *( ( ((%)-'<$ ( (==((%)-' - '* ( '') ( ('-< G' < @- <- * %% ( ')'_{' (' ( =-' ( '*$}

@( /-< ( (=-< ( '=( ' <) ( '<- ( ('%/ ' '= ( %(= ( ')* @- <( ( '*) ( '%$ ( --< ( ')* ( '%-- '%--/ ( (<$* ( ((=)( ' ** ( '<% ( (%%-- (% ( '') ( (''%

@- << * $' ( '*/ ' (' ( =-( ( '*$ @( /=/ ( (=-< ( '=' ' <) ( '<- ( ('%/ ' ') ( %(( ( ')* @- <' ( '*) ( '%/ ( -*' ( ')) ( '%-- '%--/ ( (<$* ( ((=*$ ' ** ( '<% ( (%%' - (% ( '') ( (''=

@* (- * $$ ( '** ' (' ( ='$ ( '*/ @( /$< ( (=*( ( '=* ' <) ( '<- ( ('%$ ' ') ( =<= ( ')) @- <= ( '** ( '%< ( -*$ ( ')) ( '%) - -/ ( (<$- ( ((=*) ' ** ( '<% ( (%%) - (% ( '') ( (''=

@- <% * $( ( '*$ ' (' ( =-' ( '*$ @( /=* ( (=-< ( '=-' <) ( '=-'<- ( ('=-'%/ ' ') ( %(* ( ')* @- <= ( '*) ( '%/ ( --= ( ')) ( '%* - -/ ( (<$* ( ((=*< ' ** ( '<% ( (%=/ - (% ( '') ( (''%

@- << * /' ( '-% ' (( ( ='= ( '*< @( //* ( (=*( ( '=) ' <* ( '<- ( ('%% ' '* ( =<' ( ')= @* (* ( '** ( '$' ( -)' ( ')) ( '%= - -/ ( (<$- ( ((=** ' *- ( '<% ( (%%% - (% ( '') ( ('')

@- /% * %- ( ')) ( <<% ( =-= ( '*$ @( $/$ ( (=-< ( '=( ' <= ( '<- ( ('%< ' ') ( %') ( ')* @- <) ( '*) ( '%= ( -(/ ( ')* ( '%' - -< ( (<$* ( ((=)= ' *- ( '<% ( (%%' - ($ ( '') ( (''$

F ! @F 2

5 '(@%

'(@=

< <<

< !!!! 2222 2222ε----

ΜG-G-G-

G-G@( < G@( = G@( -= G( -= G( = G( <

G' ' - '/ * )% ( ')%

* -) ( %'- ( /=) ' )= * $* ( '/-- -- ( <*= ( /'/ ( =<' ) (* ( -*)' '/ ' *$ ( $%% ( =/( ) (- ( -*$' '$ ' *% ( $%* ' )- * $) ( '/)- -- ( <*) ( /'% - '= * =' ( '*<* -* ( %)$ ( /%'

G' * @( -/$ * =' ( -)*

' '< ' '= ( $=$ @( $=* * $= ( -%/( %-$ ' )% ( $*- @' (' * /% ( -$*( -=/ ' $* ( $-$ @( <<* * /- ( -/<( -%< ' $( ( $'' @( $') * $* ( -%$( %'- ' )$ ( $** @( -$- * )< ( -=-' -( ' '* ( $)/

G' = @' $- * )$ ( --'

( ='* ' )% ( $$< @' /< * =$ ( --)( -=$ ' %* ( $$% @' <* * =/ ( -*(( ')< ' $( ( $$( @- (' * %* ( --<( '=* ' %< ( $$' @' /% * =% ( -*)( -%= ' %' ( $%% @' %= * )) ( --%( =(% ' )= ( $$)

G' $ @* *( * (% ( '=<

( -'( ' =% ( /)' @* -< * (( ( '%)( ')= ' %' ( /*% @* -< * (= ( '%/( '-' ' %- ( /*- @* -< - << ( '$'( '** ' %- ( /-< @* -< * '' ( '%%( ')- ' %' ( /*) @* -< - << ( '%)( --* ' == ( /*%

G' < @* -< - $$ (

'<-( ')/ ' )' '<-( /'<-(/ @* *< - /( ( '/=( '*( ' )- ( /'= @* *< - /- ( '/=( '-) ' )* ( /'= @* *< - $% ( '/$( '*= ' )- ( /'* @* *' - $% ( '<'( '*< ' )- ( /(< @* *' - $' ( '<)( '%- ' )( ( /(%

F ! @F 2

(32)

0 2 0 2 0 2

0 2 '='= 4'='= 444 ++++ .... 2222 ----_7G-_7G-_7G-

7G-“ ” 2

-20 -15 -10 -5 0 5 10 15

0 0.05 0.1 0.15 0.2 0.25

ordinate, log2

d e n s it y

= 1.30, = -0.25

Exact Approximate

-20 -15 -10 -5 0 5 10 15

0 0.05 0.1 0.15 0.2

ordinate, log2

d e n s it y

= 1.30, = 0.25

Exact Approximate

-20 -15 -10 -5 0 5 10

0 0.05 0.1 0.15 0.2 0.25

ordinate, log2

d e n s it y

= 1.70, = -0.25

Exact Approximate

-25 -20 -15 -10 -5 0 5 10

0 0.05 0.1 0.15 0.2 0.25

ordinate, log2

d e n s it y

= 1.70, = 0.25

Exact Approximate

<

-B +'<<'. !;4A

7 +-('-. – –

2 # 2

7: 7474

0 8 2 +-((-.

I +-((). 2 2

!

+*(.

. . + . .

5 2 2 + I -(().

$ " " " " " " ( (

&

+*'. " " " " " $ " / " / " /

2 3

2 ) 2

7

2 + .

<

<<

<

7

7 < '

#

< '

#

< '

#

< '

#

7 B +7 B. 2

7 B +4 C

(33)

+*-.

(&

0 ?

2 " @ 2

? (& !

4 C ! +-((<. ? 2 2

C

& +-((%. 7 – A 2 4

2 +'<<) '<<=. 6 2 4 8 +-((*. ?

7 –A 2 2 ?

+4 F -((%

A ; ? '<<) 2 . 7 B

3 ?

2

7 ! F 0 +7!F0.

77 !! FF 00 +7!F0.+7!F0.

7 ! F 0 +7!F0.

For a fixed value of , consider a fixed set of 5 matrices ) .

1. Given the data compute the empirical characteristic function:

&

& , where , Let

& &

&

.

2. Denote the parameter vector by . Draw a parameter vector .

3. Simulate artificial data 6 , and compute the characteristic

function & & , where , and is

5 a matrix.

4. Define

&

& &

& , & ).

5. Compute & & , & & ), and cov denotes the empirical covariance of

&

, an approximation to the optimal covariance matrix of the MANF.

6. Define the likelihood function of the MANF:

(34)

% 7474

! !#4

– 2 -( ₂

=(H

!#4 7!F0 2

5 7!F0 5

)

2

d

3

2 0

&

?

L

2 ?

&

" " 5 "

"

+ ? 2 5

? 2

S

–

7!F0 3 !

G' =( G @( =( +. . 0

3

2 7 2 2 2

2 0

" "

&

! _" " " +**.

7 ? 2 _"

5 2

<

<<

< -

--

-

7

7 B

B

7

0

2

2 7 B

2 2

( +*).

( +*=.

- 2 2 3

( +*%.

ds 2 - 2

3

+*$.

-( _<(H _? _{2 2} _G'(( _{= (((}

K' '( ' <(L K@( <( ( <(L 2

(35)

5 5 3

3 * * ) 6 2 !#4

Propose draws for the parameters and in the discrete case or 3 * in the

normal approximation.

Simulate artificial data for or using parameters or 3 * .

Compute the spectral measure ( and in the discrete

case, and ₍ , where _d_s denotes expectation taken with respect to

, given the time-varying parameters in the normal case.

Compute and the empirical

equivalent or the moving-blocks approximation.

Accept the draw if ! , for some constant ) .

0

2

!#4 =(H

-We partition the sample into 0 subsamples of approximately equal size, say . Denote each

partition by , 0.

For each subsample obtain ( using either the normal or the discrete approximation. As a

result we have estimates of the stable distribution parameters , along with

in the discrete case or in the normal approximation case. The estimates are obtained using ABC-ANF for each subsample when or are fixed.

Use least squares to fit 3 or ,

0, where and are obtained from ABC-ANF for the entire sample, and *,

, where denotes the empirical (co)variance, .

Set

0

0 ,

0

0 , and for 3 or denote by the least

squares quantities.

The proposal is , where for the discrete measure, and

for the normal approximation.

0

* 0 6

! 0 ) 3

2 0) -( '(

5 “ ”

6 “ ” 6

(36)

'( '( '(

-'( - !!!! &&&&

3 7 +'<<<.

! “K L 2 2

2 2 ”

+A '<$= 7 4 '<<$.

7 +'<<<. 2

"

' ! " _" '

"

) _" ' ' _" 0 ' ' '

) "

" 6 “ - 2 2”

2 : 2

@ ! 5

( ( ! +7 -(('. F !

-# ! ! ! 1

-! 1 !

-- 0 7

+-((*.

: 5 @ 5

2

" " " 4 2 ' +

. 2

7 +'<<<.

2 2

2 @ 2 2

2 2 2

2 4 2

+ .

3

& 2 + -('-. “ ”

+ . 2

2

Use the method of Meerschaert and Scheffler (2003) to estimate principal directions in the direction of heaviest tails, . As in Tsionas (2012) this can be applied21 in 0 subsamples to

obtain an estimate .

For each subsample the method of Meerschaert and Scheffler (2003) also yields estimates ,

0, from which the empirical covariance can be obtained.

In the case of general (non-symmetric) multivariate stable distributions the samples

, so estimates , 0 can be obtained along with and

.

0 0

2 !#4 !#4@!F0 2

-' _– ₂

(37)

A ( (

0 2 F

2 !2

!#4 !#4@!F0

3 *

6 “&&&& @@@@ @@@@#### ” +&&&& . 2

!#4 !#4 !F0 7

# 2

3 * 2

“ ”

7 4 3 *

-2 2 - 2

Given parameters compute the discrete approximation to the spectral measure for all

subsamples through , 0.

Compute , , for each 0.

Use LS fit to obtain 3 , *.

Use 3 * and 3 * to formulate a multivariate normal proposal for ABC or ANF.

& 7 +'<<<.

7474 #

“ ”

2 7 4

3 *

)

3 * ? 7

2 - 2

6 ? !

0 7 4

' ((( ? 7474 '-( (((

'( ((( '( '( (((

'( '( '(

'( ;;;; 7777 4444 G' $= G' $= G' $= G' $= G(G(G(G(

3 _*

2 d

- @( ''

+( ('%. +( (-(.( <' +( ('(.( '- +( ('-.' $% +( ($(.@( ('=

-F @( ''

+( ('). +( ('=.( <* +( ('%.( '- +( ('*.' $= +( (%'.@( ('*

& @!#4- @( '*

+( ('<. +( (--.( <) +( ('=.( '' +( ('*.' $$ +( ($-.@( ('/

& @!#4- @( ''

+( (''. +( ('<.( <) +( ('*.( ') +( ('*.' $) +( ($-.@( ('=

& @!F0- @( ''

(38)

& @!F0- @(

'-+( ('=. +( (''.( <- +( ('*.( '* +( (-*.' $% +( (/=.@(

('-& @!F0- @( ')

+( (-'.

( /= +( (*-.

( '% +( (--.

' /' +( ()*.

@@@

& @!F0- @( ')

+( ('=. +( (-'.( <- +( ('-.( '' +( ('*.' $$ @@@ !

- @( '=

+( (*-. +( ()).( <= +( (*=.( '= +( ()'.' $< +( (<'.(

'--F @( '=

+( (''. +( ('-.( <= +( (''.( '= +( ('-.' $< +( (*'.@( ($

& @!#4- @( '*

+( ()'. +( ()*.( <- +( (=*.( '- +( (-=.' $- +( (-/.( (*

& @!#4- @( ''

+( ('*. +( (''.( <( +( ('(.( '' +( (''.' $) +( (-).(

(-& @!F0- @(

'-+( ()*. +( (*'.( <' +( (*(.( '- +( (-'.' $= +( (--.(

(-& @!F0- @( ''

+( (''. +( ('-.( <( +( ('-.( '( +( ('(.' $% +( (--.( ('=

& @!F0- @(

'-+( ()'. +( (*=.( <' +( (--.( '- +( (*-.' $% @@@

& @!F0- @( ''

+( ('-. +( (').( <- +( (($.( '' +( ((/.' $= @@@

- @( -)

+( (%=. +( ($%.( $$ +( (%$.( -- +( ($$.' %* +( (*).( -)

-F @( ''

+( (($. +( ((/.( <* +( ((*.( '( +( ((%.' $= +( (''.@(

('-& @!#4- @( ''

+( (%%.

( <* +( ($/.

( '( +( (/-.

' $= +( ($/.

@( ('-+( ()).

& @!#4- @(

'-+( ((). +( ((*.( <' +( (('.( '' +( ((-.' $* +( (''.(

('-& @!F0- @( '(

+( ($'. +( (%%.( <( +( (*=.( '- +( (%$.' $* +( ()%.(

('-& @!F0- @( '(

+( ((*. +( ((=.( <( +( (('.( '' +( ((-.' $= +( (''.(

('-& @!F0- @( ''

+( ($$. +( (%$.( <( +( (/-.( '- +( ($'.' $) @@@

& @!F0- @( ''

+( ((). +( ((*.( <( +( ((-.( '( +( ((-.' $= @@@

F & & 7

2 2 2 !

3

+0 . ? + ! . 2

0 7 4 2

-- '(

-F &

6 2 !#4 !F0

2 !F0 2

C

0 2 '% 2 +

'( ((( .

+'( '(( =(( '((( . 0

(39)

0 2 0 2 0 2

0 2 '%'%'%'%

7 B 7

F & @!F0 !F0 5 &

@ '(

0 1 2 3 4 5 6 7

0 0.2 0.4 0.6 0.8 1 radians s p e c tr a l m e a s u re Period 10

0 1 2 3 4 5 6 7

0 0.2 0.4 0.6 0.8 1 radians s p e c tr a l m e a s u re Period 1000 True SD PD-ANF True SD PD-ANF True SD PD-ANF True SD PD-ANF ! &

+ 2 .

“ ”

2 “ ” ; + .

2 0 2 2

<

<<

< *

**

*

7

7 B

B

7

3 ? 2 6 7474 F - + . ! 2

1,...,

t

n

(&

" @ 2 ?

(& 7

( 2 2

( F

! _" _" _"

"

j

1,...,

N

7 F +'<<). α@

α@ 2 +

. 0 _" _" _" _"

(40)

(

2 "

2 5 2 7474 7 B

2 C - 2

!#4

-Draw parameters from a proposal distribution.

Simulate artificial data , fix the length, -, of moving blocks, and let

- - , -.

Compute the simulated log characteristic function and the empirical characteristic

function , where

- - , -, is the moving blocks in the data.

Accept the draw using the ABC or ANF criteria.

7474

& 3

&

! (& (& 2@ @

2 "

2

0 0 2

α@

5 5

7 @A 2 2 !#4 !F0 0 @ 2 +

.

2@ @

! 2@ @

2 " 2

7474 5

& 0

2

!#4 !F0 2

2 +<(H '(H . 6 5

%(H $=H

<

<<

< )

))

)

6 '(( O & ’ + -*@-<D'(D-((<.

--5 & +-('-. 2

2 " 9 * '<<% 7 -' -('- 4

9 # ! A 2@8 2 F E

8 7

3 0 2 '$ 2 3 * 2

–

(41)

0 2 0 20 2

0 2 '$'$'$'$ AAAA 2222 2222 –

-9

-8.5

-8

-7.5

-7

-6.5

-6

0

20

40 Posterior means of

0.05

0

0.1

0.15

0.2

0.25

0.3

10

20 Posterior means of

1.95

0

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

20

40 1/2

Posterior means of

1/2

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75

0

20

40 Posterior means of

d

e

n

s

it

y

2 2 0 2 '/ -(

(42)

0 2 0 2 0 2

0 2 '/'/ 7 2'/'/ 7 27 27 2 Δ

-0.10 0 0.1 0.2 0.3 0.4 0.5 0.6

5 10 15

d

e

n

s

it

y

3 0 2 '< 2 !F0

& &'(( + . 5 2 + 2 . 6

&'(( -( =(H '(( + 2

@ 2 . 0 2 *(H

2 0 &'((

2 0 2 '< 6

0 + '(( '(( .

--“ ” + . < '(( “ ” +

(43)

0 2 0 20 2

0 2 '<'<'<'< &&&&

0 2 0 20 2

0 2 '<'<'<'< &&&& α !F0!F0!F0!F0 &&&& &'(( &'(( &'(( &'((

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

posterior means, ANF

p

o

s

te

ri

o

r

m

e

a

n

s

,

P

(44)

0 2 0 2 0 2

0 2 -(-( 7 2-(-( 7 27 27 2 α 2 ;2 ;2 ;2 ;

0.8 1 1.2 1.4 1.6 1.8 2

posterior means, ANF

p

o

s

te

ri

o

r

m

e

a

n

s

,

P

D

2 0 2 -'

0 2 0 20 2

0 2 -'-'-'-' 7 27 27 27 2 &'(( &'(( &'(( &'((

1.58 1.59

1.6 1.61

1.62 1.63 1.64

1.65 1.66 1.67

-0.3 -0.28 -0.26 -0.24 -0.22 -0.2 -0.18 -0.16 -0.140

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.4 0.4

0.5

0.5 0.5

0.6

0.6 0.6

0.7

0.7 0.8 0 0.8

.9

1.59 1.6 1.61 1.62 1.63 1.64 1.65

(45)

'' '' ''

'' 4444 2222 &'(( &'(( &'(( &'((

;F

& !F0 & 4 !F0 7 2 '(

& !F0

α ( ** ( '- ( )* ' *- ( =' (

'-β ( -' ( *= ' '- ' -( ( *= ( '$

τ ( )'@' *= ( )=@( $- ( /%@' $' ( %'@' -/ @( '-@( %' @( '$@(

%-δ ( *$ ( )( ' ** ( <* ( )( ( -'

Δ ( )= ( )= ( '% ' '= ( *- ( ''

Φ ( *- ( == ' '' ( *- ( )= ( -'

F ;F 4 2 2 0 τ

& “& ” !F0 “! F 0 ”

4 @ =(H -=H

0 2 0 20 2

0 2 --- 777 27 222 α β

1.540 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7

10 20 30 40

d

e

n

s

it

y

ANF PD ABC

-0.50 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05

5 10 15

d

e

n

s

it

y

(46)

0 2 0 2 0 2

0 2 -*-* 7-*-* 777 τ

0 2 4 6 8 10 12 14 16 18 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

lag

a

c

f

ANF PD ABC

0 2 0 20 2

0 2 -)-)-)-) ;;;; 4444 2222

-0.040 -0.02 0 0.02 0.04 0.06 0.08

5 10

rank correlation

0 10 20 30 40 50 60 70 80 90 100

0.8 0.85 0.9 0.95 1

lag

a

c

(47)

0 2 0 2 0 2

0 2 -=-= 7 2-=-= 7 27 27 2 4 4 4 4 2 ;2 ;2 ;2 ;

; 2 + . + 2 . 2 !F0

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 0

20 40

-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 5

10

1.5 1.55 1.6 1.65 1.7 1.75 1.8 0

10 20

-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 5

10

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0

20 40

-0.40 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 10

20

1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 0

10 20

-0.150 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 5

10

1.4 1.5 1.6 1.7 1.8 1.9 2

0 20 40

-0.70 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 5

10

1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 0

20 40

-0.350 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 5

(48)

'

''

'(

((

(

0

7

7 '

''

'(

((

( '

''

'

-#

4

+*/.

5 5 !

5 2 5 5 3

+ . (&

2 ? 2

3

,

i i

3

:

3 0 5

6 2

@ -* _A

2 2 2 6 ?

@ 2 ? ?

“ 2 ” + E '<<% !2 6 -((( : 6 -((*.

2 +'</(. 3

+*<.

+)(.

%

&

&&

+)'.

5

! 6

% # ’ 0 2 E +'<<% =%=@=%%.

2 2 /_" +

" / + / 0 -) ₊ ₂ _{'( 0}

-* _{: J} ₄ _+-((). _B _+-((<.

-) ₆ _@

2 @

(49)

) 0

2 5 0

%

k

1, 2,...

& 2 3 2 2

+! ). ! 2

@

6 2 !#4

Fit the normal k-factor model using MCMC25 to obtain posterior draws for

, for , where is an a priori known bound on . For

estimate univariate stable distributions for each time series

. Take as proposal a multivariate Student-t (see below) based on maximum likelihood

estimation26 with mean , , and covariance27

6 , where denote the ML quantities. For take the same

proposal independently of .

Use a multivariate Student-t proposal for , with the stated parameters. The degrees of freedom,

n

p

, where

p

is the number of parameters.

Draw a candidate parameter vector from the proposal and simulate model (39)-(40) to obtain

artificial data . Compute the simulated characteristic function

.

If ! , for some positive constant , accept the draw with probability.

Use (41) and the Laplace approximation to obtain the log-marginal likelihood for , and

draw a value for . If the log-marginal likelihood is denoted by / , the probability of model

is28 / / , .

3 -<

/ +)-.

+*<.

2 +)'.

2 “ 2” F 2 2

2 2 +'</(.

-= _E _+'<<%. _+'=.@+'$. _!

-% ₇ ₂ ₀₀

-$ ₂ ₂ _7: _@

-/ ₃ ₂ ₇₄₇₄

2 “ ” 2 2

-< ₄ ₈ _; ₆ _{+'<<$. !}

C

+! =. ( (= ( '( A =H 2 00

(50)

3 2 !' <(H

?

n

0

G' ((( 7 4 2 2 +*<.

+)(. <(H

2

G' ((( ' (((

2

3 ! ' + . <(H + . <(H

<(H

2

5 2 =(H*(

!' 4 !' 4 !' 4 !' 4

3 33 333 3B

G' $( G@( -( G' -( G@( /( G' -( G( /( G' $( G( -(

G= G= G= G=

G-G'(( (

*$-( -)% –( %'$ ( *)= ( -'$ –( %'( ( *)* ( --( –( %(% ( *$* ( -)( –( %'*

G=(( ( *=<

( -(< –( =/'

( *(< ( '/) –( ==$

( *(%

( '/( @( ==/ _{( -'(}( *=/–( =/)

G' ((( ( *=$

( -'* –( %('

( *--( '$/ –( =<*

( *'< ( '$) –( %(%

( *=$ ( -'' –( %('

G- ((( (

*=-( -*=-($ –( =$<

( *(* ( '$' –( ==% ( *(* ( '$' –( ==% ( *=) ( -(% –( =$/

G'( G) G'( G) G'( G) G'( G)

G'(( ( --)

( '%$ –(

*=-( '<) ( '=' –( -<'

( '<= ( ')$ –( -/*

( --$ ( '%% –( *=*

G=(( ( '%)

( (<< –( *'(

( ''-( ''-($< –( '/<

( ''' ( (/( –( '<*

( '%= ( (</ –( *(/

G' ((( ( '=(

( (/* –( *($

( (<= ( (%* –( '$<

( (<= ( (%) –( '/'

( '=) ( (/) –( *(=

G- ((( ( '=)

( (/( –( *--( *--(/$ ( (=* –( '$$ ( (/$ ( (=) –( '$% ( '=) ( ($< –( *-'

G-( G/ G-( G/ G-( G/ G-( G/

G'(( ( '/(

( '*% –( -%= ( '$% ( '*$ –( -=-( '$-( '*= –( -=-( '/' ( '*/ –(

-%-G=(( ( (/'

( (%) –( '-( ( ($$ ( (%( –( ''* ( (/( ( (%- –( '') ( (/-( (/-(%* –( ''/ G' ((( ( (=<

( ()% –( (/=

( (== ( ()* –( ($<

( (=% ( ()* –( (/'

( (=< ( ()% –( (//

G- ((( ( ())

( (** –( (%= ( ()( ( (*' –( (=/ ( ()' ( (*' –( (=/ ( ()) ( (** –( (%$ G'((

G-G'(( G-G'(( G'((

G-G' ((( ( *''

( -(' –( ))( ( (/% ( (%* –( '-) ( (/* ( (%- –( '-* ( *(< ( -(' –( )*< G* ((( ( *(<

( -(* –( )*$ ( (%< ( (=( –( ''' ( ($( ( (=( –( ''( ( *(/ ( -(% –( )*% F <(H + . ' (((

<(H ' (((

0 ! &'(( 2 7474

2 '-( ((( -( ((( 2

'( ; 2 !'

*( _! ₂

! 5 2

(51)

2 “ ” 2 =(H

*(H 3 “ ” *

2 ? &

0 2 !' 2 !* 0 2 !'

2 3 2 7474

0 2 !- 0 !

2 2 0 !

0 2 0 2 !*

0 2 0 ! 2

0 2 0 2 0 2

0 2 -%-%-%-% &&&&

1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

number of factors, k

p

ro

b

a

b

ili

ty

(52)

0 2 0 20 2

0 2 -$-$-$-$ 4444 2222

2 )=

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

Posterior means, normality

P

o

s

te

ri

o

r

m

e

a

n

s

,

s

ta

b

le

Posterior means of normal versus stable factor analysis

0 2 0 2 0 2

0 2 -/-/ 7 2-/-/ 7 27 27 2 0000 !!!! 2

2 “4 ” 2

+ . + . “ ”

2

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9

0 5 10 15 20

d

e

n

s

it

y

Common , Symmetric stable

,

-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2

5 10 15

d

e

n

s

it

y

Common ,

(53)

'

''

'(

((

( -

--

-

! 7

0

7 2 ?

2 - 2 A 0 7 2 2

2

5 2 7

"

" " +)*.

2 7

"

" " +)).

3 7

3 - 7

- 2

7

+)=.

+)%.

+)*. +)). 2 + . + . +)*.

0 +)'.

"

% #

$

&

&&

+)$. 6

! !

!- & 7

!- &!- & 77

!- & 7

&

+ . ;F 4 F

( +( ('=.( (('- ( *=* @' -'- ( ((-$

' @( ((-$+( (--. ( )=' @' === ( (()'

( @( ((-'+( ('$. ( /'( ' *=% ( (()=

( @( ((*)_{+( (-$.} ( $%% ' -'- ( (('*

( +( (('$.( ('= ( =%' ( /<$ ( ((()=

' +( (==.( '$' ( -'- ' -)= ( ((%'

( +( ('%.( (-$ ( *=* @' )=' ( ((*)

( +( ()).( '// ( )$$ ( =%' ( ((=)

(54)

+( (--. +( (**.

' ' $'-_{+( ('=.} _{+( ()'.}@( ('- ( =%% @' *<' ( ((*) ( ((*$

(

' </< ( ((*)

+( (''. +( ((-=. ( *)= ( ==% ( (()= ( ((*$

'

' =%( @( -'

+( ('). +( (--. ( ))) ( /'- ( ((-- ( ((*<

( -$*

+( ('=. ( /'< @' -'( (

(('-(

%'-+( ('$. ( $'( ' **$ ( ((-=

( ''= ( (==

+( (*=. +( ('-. ( %=( ' -'= ' '-- ( ((*' ( (('

( '-- ( (*'

+( (-'. +( (''. ( $'( ( $'$ ' *)' @' -- ( $ (

((-( ((-(*= ( $'- ( (-'

+( ('$. +( (--. +( (-%. ( %'( @' *-( ( ((-'

F ;F 4 ’ +'<<). 2 2 F

'

''

'(

((

( *

**

*

0

3 2 ? 2 0 2 ? 2

2 3

2

2 ? +0 : -((' 0 A : ; -((( -((=.

( +)/.

+)<.

( ( @

5 6 2 (&

2 2 (& !

+'<$$. 2 +'<$$. 2 6 +'</'. !

- ! 2 +)/. – +)<.

2 2 2 6 +-((=.

# # ? +-((=.

+)/. – +)<.

2 4 +)/.

2 2 2 !

2

@ ;

“ ”

2@ 2 ! 2

" + ₅ .

+=(.

+ . +='.

3 “ ”

@ 2 + .

& - +& . 2 ! 2

+! '-. 0

(55)

!2 6 +-((( ! #. 6

-2 C

& +! '=. –

=( =( & +! '=. 2 “ ”

=(H $* -$ 2

&

! 0 2 !) 2 !% 3 0 2 !)

2 2 ? + .

2 ? + 2 . &'(( 3

2 2 3

0 2 != ? 2 ?

+ . 2 ? + 2 .

+ .

@ @

22 2

2 3 0 2 !% 2

2 6

0

' $= @( '

3 @

0 2 0 2 0 2

0 2 -<-<-< &-< &&& 2222 ???? &'(( &'(( &'(( &'((

1 2 3 4 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

p

ro

b

a

b

il

it

y

Generalized Stable Factor Model

1 2 3 4 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p

ro

b

a

b

il

it

y

(56)

0 2 0 2 0 2

0 2 *(*( &*(*( &&& ???? &'(( &'(( &'(( &'((

0 1 2 3 4 5 6 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

radians

s

p

e

c

tr

a

l

m

e

a

s

u

re

Generalized Stable Factor Model

Discrete Normal

0 1 2 3 4 5 6 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

radians

s

p

e

c

tr

a

l

m

e

a

s

u

re

Generalized Dynamic Stable Factor Model

Discrete Normal

0 2 0 20 2

0 2 *'*'*' 7 2*' 7 27 27 2 ???? 0000 7777

&'(( &'(( &'(( &'((

1.650 1.7 1.75 1.8 1.85 1.9 1.95 2 2

4 6 8 10 12 14

d

e

n

s

it

y

,

-0.20 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 5

10 15 20 25

d

e

n

s

it

y

,

3 0 2 !$ 2 2

5 3 2

(57)

0 2 0 2 0 2

0 2 *-*- 0*-*- 000 2222 ???? 0000 7777 &'(( &'(( &'(( &'((

-0.60 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20 30 40 50 60 70 80

loadings Posterior means of factor loadings,

-0.60 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1

2 3 4 5 6 7x 10

5

loadings Draws of factor loadings,

0 2 0 2 0 2

0 2 **** 7**** 777

1 2 3 4 5 6 7 8 9 10

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

lag

a

c

f

Max. ACF of draws for Static and Dynamic Generalized Stable Factor Model

Static Dynamic

4

2

4

2

4

2

4

2

3 !#4

& 2

3 !#4 2

2 2 0

! 2

3

(58)

2

2 5 3

2 2

! !#4 2

2 + 5

&'((. –

2 2 &

7 &

2 ?

2 2

2 2 @

6 2 !#4

7 4

# 2@ 3

2

3 2

!#4 “ ”7 4

;

! ? 7 3 ! 2 '<%= A 7 0 0

7 F I

!2 C 7 6 -((( # 2

9 # '/ **/@*=$

# ? E & # A C '</% 2 E ü

! 2 7 7 ' *$@)<

# 4 A ; 5 ; ; '<<* A 2 2 2

7 C ; '/ -==@-%%

# # 9 # & ? -((= 7 2 ! @ 2

2 +0!B!;. > 9 '-( '-( '-( '-(

*/$@)--# ! A 7 8 9 & 7 4 -('- !;4A

9

# 9 '<<= # 9 !

! <( %(=@%'*

# ? 9 & F # ; 5 '<<* !

9 7 ! )% '*@*'

4 7 '</' : 3!7 9

! 7 )' )*@%<

4 7 9 0 -((( ? 77

'% $<$–/*)

4 7 9 0 9 -((- 77 2

6 2 & M " ;

4 9 7 4 : 7 # 6 '<$% ! 2

9 ! ! $' *)(@*))

4 2 # F ; '<<= 7 7 0 = '**@'=*

4 2 +'<<). # 2 !;7!+ . 9

%) '/*–-(%

4 F 0 F +-((%. ! 2

9 '*) *)'–*$'

4 I C 7 ! -((< 7 B ! + .

A 0 2 B 2 A 2

! 9 7 F -((= @ 3 ; $*

'''@'-<

4 9 ; 8 ! ; : 6 '<<$ 4 2 # 2

9 ! ! <- <(*@<'=

I B -('- 9

9 9 8 -((( @

# + . 9 ; #

%- *@=%

2 ; ; 6 '</' ! - 2

9 ! ! $% $$)@$/'

0 2 ! & 7 2 '</' C

7 4 2 ; F I F @A '(<–

'--0 2 ! & 7 2 '</' C 0 9

! ! $% *$<–*/$

0 2 ! ; ! 7 '<$$ !

= //@<$

0 7 7 : -((' 2 ? ;

(59)

0 7 7 A 7 : : ; -((= 2 ? 3

; /- =)(@==)

0 7 7 A 7 : : ; -((= 2 ? C @

2 9 ! ! '(( /*(@/)(

0 6 ! B ? '<</. " 2 ; " 2 4 F ! !

9 - '@ -=

; ; B -(''

9 '%' *-=@**$

I 8 -('' F

DD 2D D(<($ -)='

2 ! # 7 2 -((' ! # @

2 & 2 /' '%(*@'%-*

9 0 8 9 2 '</( 3 2

? 9 ! ! $= '**@'*$

9 0 '<$$ 0 ! 7 P : B

7 ! 2 ! 2 *%=@*/* F @A

9 0 E '<<% 7 2 2 2 2

; 0 < ==$@=/$

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