Συμπερασματολογίακαιεπιλογήμεταβλητών...

xi

xiii

1 11.1 . . . . . . . . . . . . . . 21.2 . . . . . . . . . . . 31.3 . . . 6

2 92.1 . . . . . . . . . 92.2 . . . . . . . . . . . . . . . . . . 102.3 . . . . . . . . . . . . . . . . . . 112.4 142.5 . . . . . . . . . . . 18

3 213.1 . . . . . . . . . . . . . . . . . . . . . 223.2 . . . . . . . . . . . . . . . . 233.3 . . . . . . . . 263.4 . . . . . 273.5 . . . . . . . 293.6 Bootstrap . . . 313.7 . . . . . . . . . . . . . . . . . . 333.8 . . . . . . . . . . . . . . . . . . . . . . 353.9 . . . . . . . . . . . . 363.10 . . . . . . . . . . . . . . 373.11 . . . . 383.12 Markov chain Monte Carlo . . . . . . . . . . . . . . 39

3.12.1 Gibbs . . . . . . . . . . . . . . . . . . . . 403.12.2 Metropolis-Hastings . . . . . . . . . . . . 41

iii

4 - 454.1 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 -

. . . . . . . . . . . . . . . . . . . . . . . 524.4 -

. . . . . . . . . . . . . . . . . . . . . . . . . 53

5 575.1 Akaike Information Criterion . . . . . . . . . . . . . . . . . . . 585.2 Bayesian Information Criterion . . . . . . . . . . . . . . . . . 595.3 Likelihood ratio test . . . . . . . . . . . . . . . . . . . . . . . 60

6 636.1 . . . . . . . . . . . . . . . . . . . . . . . 646.2 . . 676.3 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.4 -

. . . . . . . . . . . . . . . . . . . . . . . . 68

Matlab 75.1 Matlab . . . . . . 75.2 Matlab . . . . . . . . . . . 75.3 Matlab . . . . . . . 78

1.1 . . . . . . . . . . . . . . . 51.2 459 ( )

. . . . . . . . . . . . . . . . . . . 7

2.1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Engel , 235 - . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 . . . . . . 263.2 15% 4%

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

v

2.1 - . . . . . . . . . . . . . . . . . . . . . 12

3.1 - . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.1 . 666.2 . . 706.3 -

: . . . . . . . . . . . . . . . . . . . . . . . . . . 716.4 -

: . . . . . . . . . . . . . . . . . . . . . . . . . . 726.5 -

: . . . . . . . . . . . . . . . . . . . . . . . . . . 736.6 -

: . . . . . . . . . . . . . . . . . . . . . . . . . . 74

. - , - . , (-, , ).

. , ( ) : ( ) . , - . , , - , .,

. , - AIC BIC, - .

xi

- , . , , . ( ) . .,

. , . email . . . .,

, , , .

xiii

1

There are three kinds of lies:lies, damned lies, and statistics.

Benjamin Disraeli

(regression analysis) (Y ) (X) (Xi i = 1, . . . , n). H Y (response variable) Xi (cova-riates).

- , ; Y Xi (linear regression). . . - , -.

1

1.

.

(simple linear regression) , - (multiple linear regression). - .

1.1

n -. . {Yi, Xi1, . . . , Xik}, i = 1, . . . , n, n k . -

yi = 0 + 1xi1 + . . .+ kxik + i, i = 1, . . . , n,

0 - 0 1, . . . , n - Y , (.. 4 - Y X4 ). i ( ) . - . E(i) = 0, . - ( i i = 1, . . . , n).

E(ij) =

{

2 i = j0 i 6= j

-

E(Y |X) = 0 + 1X1 + . . .+ kXk.

2

1.2.

0, 1, . . . , k, (. ). . ,

(.. yi = 0 + 1xi1 + 2x2i2 + . . . + nxik + i yi = 0 + 1xi1 + 22xi2 + . . .+ nxik + i)., . , , 1 X1 Y - , . - .

yi = 0 + 1xi1 + . . .+ kxik + i i = 1, . . . , n.

.

Y =

y1y2...yn

, X =

1 x11 . . . x1k1 x21 . . . x2k......

......

1 xn1 . . . xnk

, =

01...k

, =

12...n

Y = X + ,

E() = 0 V () = 2I 2 I n n.

1.2 -

, - . (ordinary least squares) .1

1 -(weighted least squares) .

3

1.

- . 0, 1, . . . , k - Y ( ) . - i . ,

=

012...n

,

Y = X

i = Y X,

Yi (fitted value) Yi - (observed value).

1.1 - Y . (Yi, Xi)

, Y = X (. Yi ). - , i = Yi Yi. , 0 ( ), .

4

1.2.

Y

X

Yi

Yi

ii

Yi

Yi

1.1:

i2 = = (Y X)(Y X)= Y Y X Y Y X + X X= Y Y 2 X Y + X X.

(Y Y 2 X Y + X X)

= 0.

X X = X Y

X X ,

= (X X)1X Y.

E() = V () = 2(X X)1 .

.

5

1.

X X -, -. - Y . ( X X |(X X)| = 0) . - , , .

(...) - (quadrative loss function)r(u) = u2,

n

i=1

r(Y X) =n

i=1

(Y X)2.

E[Y |X = x], Y E[(Y Y )2|X = x].

1.3 -

, . - , , . , - , Y . - Y , .

6

1.3.

50

60

70

80

5 10 15 20 250 X

Y

1.2: 459 ( )

, 459 - [;]. 1.2 ( ). X ( ) - Y (). - y = 0 + 1x+ 2x

2 + .

(...). (-) Y (outliers) , - . , E(Y |X = 25) 65 - 25 65000 . - , 25 . -

7

1.

25 65000 . , , . 0.5 . ( ) 25 ( y) P (Y < y|X = 25) = 0.5. 80%

25 ; 90% 15 2 . ; .

(robust) .

8

2

Is it safe? Is it safe?

Marathon Man

Dr Christian Szell

2.1 -

- , ., , ... -

E[Y |X = x] , . ( ) - Y Xi. -. . . , Y Xi - (quantile regression).

9

2.

-.

{3, 6, 7, 8, 8, 10, 13, 15, 16, 20}

(10 ). - 1 3 1

4

34 -

. 7. () 1 1 . 8+10

2= 9

8 10 - 1. (10.6) . , p- p

100 1p

100 p (0, 100).

90 90% 10%. , 50 .

2.2

- . 1978 Koenker Bassett[?] - , . , - , - , Y . -

,

1 .. 2k+1 k k

10

2.3.

- . - F F1 . ,

, p100 , p (0, 1). , x

P (X x) = p.

p = 0.5 - (median regression). ()

F (x) = 1 ex.

F1(p, ) = ln{1 p}

0 p 1.

,

F1(0.25, ) =ln{4/3}

,

F1(0.5, ) =ln{2}

,

F1(0.75, ) =ln{4}

.

25% .

P (X ln{2}

) = 0.5 .

2.1 .

2.3

F1y (p|X) = Qy(p|X).

11

2.

2.1: - .

FY (y) QY (p)

() 1 ex ln{1p}

(, ) 1 (y) (1 p) 1

(, ) y

p( ) + (, s) 1

1+e(y)/s s ln 1p

p

, p = 0.9 , Qy(0.9|X) 90 Y Xi 90% Y Xi P (Y y|X) = 0.9. -

, Y .

, . , . (low tail) -, (upper tail). - , . (..) . , ..., . , , Y .

... - ( i ) , -

12

2.3.

50

60

70

80

5 10 15 20 250 X

Y

75%

50%

25%

2.1: -

, Y . -

yi = 0,p + 1,pxi1 + . . .+ k,pxik + i,p i = 1, . . . , n.

Qy(p|X) = 0,p + 1,pX1 + . . .+ k,pXk.

Qy(i,p|X) = 0 ... E(i|X) = 0. . , -

1.2, 459 , - Y ( ). 2.1 1.2 -

(25%, 50%, 75%)

13

2.

Y . , - .

P (Y 60|X = 25) = 0.5

0.5 25 60000 . ( ). 2.1 25%

25 79000 . 75

P (Y 79|X = 25) = 0.75

P (Y > 79|X = 25) = 0.25 .

- 50 . - Y Xi.

2.4 -

- ( ) Y Xi , - .

, n

i=1 r(Y X) r(u) = u2 .

yi = 0,0.5 + 1,0.5xi1 + . . .+ k,0.5xik + i,p i = 1, . . . , n.

14

2.4.

Y Xi.

Y =

y1y2...yn

, X =

1 x11 . . . x1k1 x21 . . . x2k......

......

1 xn1 . . . xnk

, 0.5 =

0(0.5)1(0.5)...

k(0.5)

, 0.5 =

1,0.52,0.5...

n,0.5

Y = X0.5 + 0.5.

0.5 ( ) 0.5

n

i=1

0.5(Y X0.5)

0.5(u) = 0.5|u| (absolute lossfunction).

0.5(u) = 0.5|u|= 0.5uI[0,)(u) (1 0.5)uI(,0)(u).

IA(u) =

{

1 u A0 u / A

A. p,

p

n

i=1

p(Y Xp)

p(u) = p|u| .

p(u) = p|u|= puI[0,)(u) (1 p)uI(,0)(u).

(check function).

15

2.

p p

p = argminp

{n

i=1

p(Y Xp)}

0.5 = argmin0.5

{n

i=1

0.5(Y Xp)}.

, . . yi ,

yi =

K

k=1

k,pxik + i,p

=K

k=1

(1k,p 2k,p)xik + (i,p i,p).

1k,p 0, 2k,p 0 k = 1, . . . , K i,p 0, i,p 0 i = 1, . . . , n. ,

min1k,p,

2k,p,i,p,i,p

n

i=1

pi,p + (1 p)i,p

yi =K

k=1

(1k,p 2k,p)xik + (i,p i,p)

1k,p,

1k,p, i,p, i,p 0 (i, k).

A = (X,X, I,I), z = (1k,p, 2

k,p,

i,p,

i,p)

c = (0, 0, pi, (1 p)i)

minzcz

16

2.4.

Y

1000 30002000 4000 5000

500

1000

1500

2000

2.2: Engel , 235

Az = y (z 0)

maxw

wy

wA c

w (p1, p)n [;]. X (min cz = maxwy). -

. Engel , . -, Engel X Y . 2.2 1857 235 - X Y [;]. - p {0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95}

17

2.

( ).

. . ( ) ., -

Y -. - . 2.2 , . , k , k . , - , .

2.5 -

- (labour economics). Chamberlain[?] , , 10 28% 90 0.3%. -

18

2.5.

15.8% - .

(reference charts) - , , ., (survival analysis) -

(.. ) , .

, , - . - 50 , - . qp(x) , p0 100(1 p0)% qp0(x), .

, . - , , . Levin[?] - , - , , - . Tian[?] ( , , , -, , ..)

19

2.

5 50. . - . , .

20

3

You come at the king, you bestnot miss.

The Wire

Omar Devone Little

2 - . - . -, . ( ) . - -, . , 2 : ( ) . .

21

3.

3.1

(likelihood function) - X1, X2, . . . , Xn

L() = L(; x1, x2, . . . , xn) = f(x1, x2, . . . , xn; )

. . X1, X2, . . . , Xn f(x; )

L() = L(; x1, x2, . . . , xn) =n

i=1

f(xi; ).

X1, X2, . . . , Xn Bernoulli().

f(x; ) = x(1 )1x x = 0, 1 0 < < 1.

L() =n

i=1

f(xi; )

=

n

i=1

xi(1 )1xi

=

xi(1 )n

xi .

X1, X2, . . . , Xn N(, 2) 2 ). = (, 2)

f(x;, 2) =122

exp{ 122

(x )2} R, 2 0

L(, 2) =

n

i=1

f(xi;, 2)

=n

i=1

122

exp{ 122

(xi )2}

= (22)n/2exp{ 122

n

i=1

(xi )2}.

22

3.2.

- Xi, i = 1, . . . , n , .

3.2

,

L() = max

L().

, . . -

l() = logL()

.

X1, X2, . . . , Xn Bernoulli(). L() =

xi(1 )n

xi.

L(), l() = logL().

l() = logL()

=

xi ln + (n

xi) ln(1 ).

l()

= 0.

l()

=

xi

n

xi1 = 0

23

3.

xi

xi = n

xi

n =

xi =

xin

.

= x l()

2l()

2|= < 0

2l()

2|= =

xi

2 n

xi

(1 )2

=

xix2

n

xi(1 x)2

=

xi

(

xin

)2 n

xi

(1 (

xin

)2

= . . .

= n2( 1xi

+1

n xi) < 0.

(...) X1, . . . , Xn Bernoulli() = x.

-. -. l()

, -

. l() . ,

, I()

I() = 2

2l()

Fisher . - Fisher

I() = E[ 2

2l()]

.

24

3.2.

I(), .

. l(),

se() = I1/2(),

. , X1, X2, . . . , Xn

N(, 2) ,. =.

L() = (22)n/2exp{ 122

n

i=1

(xi )2}

,

l()

= 0,

. ,

2l()

2|= < 0

= x

2l()

2|= =

n

2< 0.

,

I() =n

2

se() =n.

25

3.

L()

3.1:

,

2l()2

|= < 0 - , 3.1. .

3.3 -

-.

I -

(Invariance). u() - , u() u().

26

3.4.

II ... , n .

III ... I1().

IV ...

3.4

- . Fisher .

L() L()L()

> C

(cutoff point) C. , D() ..

D() = 2(l() l())

D() X2

D() X21.

,

Pr(L()

L()> C) = Pr(log

L()

L()> logC)

= Pr(logL()

L()< logC)

= Pr(2 logL()

L()< 2logC)

= Pr(D() < 2c)

c = log(C).

27

3.

0 < < 1 c = 12X21,(1) X

21,(1)

100(1 ) X21 ,

Pr(L()

L()> C) = Pr(X21 < X

21,(1)) = 1

c

{, L()L()

> C}

100(1 )% . c

Fisher 6.7% , = 0.05 0.01 c 0.15 0.04 . 95% 99% 15% 4%. .. X1, X2, . . . , X10 10 -

, Xi = 1 i Xi = 0 . , - X =

10i=1Xi = 8, ,

, . .. X B(n, ) n = 10 ,

f(x; ) =

(

n

x

)

x(1 )nx, x = 1, 2, . . . , 10, 0 < < 1

x = 8 .

l() = x log + (n x) log(1 )

=x

n= 0.8 .

3.2 . C = 0.04 C = 0.15. = 0.8 C = 0.04 (0.41, 0.98) c = 0.15 (0.50, 0.96). 99% 95%.

28

3.5.

L()

0.0

0.2

0.4

0.6

0.8

1.0

0.20.0 0.8 1.00.4 0.6

3.2: 15% 4%

- . 95% 95% - , 95% .

3.5

-, . = (1, 2, . . . , s) i 2. . = (1, 2, . . . , s) s- .

Fisher -

29

3.

.

Iij() = 2

ijl()|= i, j = 1, 2, . . . , s.

Fisher

Iij() = E[ 2

ijl()]

i, j = 1, 2, . . . , s.

, l() ... I(). X1, X2, . . . , Xn -

N(, 2) ( ). = (, 2)

f(x;, 2) =122

exp{ 122

(x )2}, R, 2 0.

L() = L(, 2) = (22)n/2exp{ 122

n

i=1

(xi )2}.

1, 2 = 1 2 = 2

l(1, 2)

1= 0

l(1, 2)

2= 0

,

2l(1, 2)

2i|i=i < 0 i = 1, 2.

1 = x, 2 =1

(xi x)2.

30

3.6. BOOTSTRAP

- 2 , . Fisher

I() =

(

n/2 00 n/(24)

)

, - . , PL(i), = (i, ),

PL(i) = max

L(i, ) = L(i, )

PL(2) = (22)n/2exp{ 122

n

i=1

(xi x)2}

PL() =(2

n

n

i=1

(xi )2)n/2

exp{n2}

.

3.6 Bootstrap

Bootstrap Effron (1979) , . Bootstrap . X1, X2, . . . , Xn F

X1 = x1, X2 = x2, . . . , Xn = xn. .

31

3.

( ) F . - Bootstrap , B, n F , - Fn . Fn (x1, x2, . . . , xn) - 1/n x1, x2, . . . , xn

Fn =

ni=1 I(xi x)

n

I(A) . n Fn (x1, x2, . . . , xn). B

i, i = 1, 2, . . . , B. B ,

B , .

s =

1

B

B

i=1

(i )2

=1

B

B

i=1

i.

F . n = 2 X1 = c < X2 = d

. X1 , X2

Pr(Xi = c) = Pr(Xi = d) =

1

2, i = 1, 2.

n = 2 22

(c, c), (c, d), (d, c), (d, d)

14.

i=

X1 +X2

2=

c p = 14

c+d2

p = 12

d p = 14

i = 1, 2, . . . , B

32

3.7.

i i

B 2

. = 1B

Bi=1 i

. -

Bootstrap .

1 2

(

)

CovB(1, 2

) =

1

B

B

i=1

(1i 1

)(i2

2)

(1i, i2

) Bootstrap i

. Bootstrap . , Bootstrap - . ( n nn

) . Bootstrap

[;], [;], HungChen[?].

3.7

- . , , ()

- () 0, .

-

()d = 1

() = 1.

() (prior distribution) ( )

33

3.

. , x =(x1, x2, . . . , xn) L(x|) = L(). , (posterior distribution) p(|x) Bayes

p(|x) = L(x|)()L(x|)()d , .

Bayes -

p(|x) L(x|)().

x = (x1, x2, . . . , xn) X Poisson() ,

f(x|) = xe

x!, 0.

E(X) = V (X) = .

L(x|) =n

i=1

exi

xi!

en

xi

Gamma(r, q),

() =qr

(r)r1e{q}, > 0,

E() =r

qV () =

r

q2.

Bayes

p(|x) r1exp{q} exp{n}

xi

= (r+

xi1)exp{(q + n)}= R1exp{Q}

34

3.8.

R = r +

xi Q = q + n. ,

|x Gamma(R,Q)

, - . p(|x)

Ep(|x) =

L(x|)()d

L(x|)()d

.

3.8

(Credibility intervals ) - . , - C(x) 100(1 )%

C(x)

p(|x)d = 1 .

1 - C(x). . ( ) . - 1 100(1)% . 1 . ( )

C(x) = { : p(|x) }

C(x)

p(|x)d = 1 .

35

3.

( ) - (highest posterior density intervals/regions). - (a, b). .

3.9 -

, L(x|) (). . , , .

. - , - . (uninformative prior ) , - prior . , - .

I prior .

II prior ( posterior prior).

III prior (.. ), . () .

IV , prior .

36

3.10.

3.10

, () . Bayes

L(x|)()d

. x = (x1, x2, . . . , xn)

X Poisson() . [0, 1], , () = 1, 0 1

L(|x) exp(n)

xi .

1

0

exp(n)

xid.

, . - . Gamma(r, q), |x Gamma(R,Q). ()

p(|x) (), () (conjugate prior).

, . -

(exponential family).

L(x|) = h(x)g()exp{t(x)c()}

37

3.

3.1: - .

() p()x (n, ) Beta(r, q) Beta(r + x, q + n x)x1, . . . , xn () Beta(r, q) Beta(r + n,

xi n)x1, . . . , xn Poisson() Gamma(r, q) Gamma(r +

xi, q + n)x1, . . . , xn Normal(, 1), ( ) Normal(b, c1) Normal( cb+nxc+n , 1c+n )

h, g, t, c

L(x|)dx = g()

h(x)exp{t(x)c()}dx = 1.

, Poisson , Gamma, . 3.1

. () .

3.11

. . = (1, 2, . . . , s)

s -. () , L(x|)

38

3.12. MARKOV CHAIN MONTE CARLO

Bayes

p(|x) = L(x|)()L(x|)()d

. . (conditional posterior distribu-

tion) pi(i|y, i) i i

pi(i|x, i) p(|x)

i . - p(|x) i, .

(marginal posterior distribution) p(i|x) - i

p(i|x) =

p(|x)di,

p(|x) i.

3.12 Markov chain Monte Carlo

- Markov chainMonte Carlo. - . , . Markov chain Monte Carlo (MCMC) -

, , . MCMC , Gibbs Metropolis-Hastings.

39

3.

3.12.1 Gibbs

Xi|, N(, 1 ) i =1, 2, . . . , n , .

f(x|, ) =n

i=1

(

2)1/2exp{

2(xi )2} n/2exp{

2

n

i=1

(xi )2}

,

N(0,1

0) , Gamma(0, 0)

, . - ,

f(, |x) e2n

i=1(xi)2

e02(0)2

n2+01e0

, . - , ,

(|, x) N(00 + n

i=1 xi0 + n

,1

0 + n),

(|, x) Gamma(0 +n

2, 0 +

ni=1(xi )2

2).

(conditional conjuga-cy ) Gibbs . d (1, 2, . . . , d) f(1, 2, . . . , d|x). Gibbs

I = ((0)1 , (0)2 , . . . ,

(0)d ).

II (1)1 f(1|x, (0)2 ,

(0)3 , . . . ,

(0)d ).

III (1)2 f(2|x, (1)1 ,

(0)3 , . . . ,

(0)d ).

40

3.

, , ,

N(0,1

0) , Gamma(0, 0)

, . - ,

f(, |x) {n

i=1

1

1 + (xi )2}e

02(0)2

n2+01e0I[ > 0].

,

f(|, x) {n

i=1

1

1 + (xi )2}e

02(0)2

f(|, x) {n

i=1

1

1 + (xi )2} n2 +01e0I[ > 0].

Gibbs. Gibbs , Metropolis-Hastings .

I d 1, . . . , d 1.

II ((0)1 , (0)2 , . . . ,

(0)d ).

III (0)1 (1)1

f(1|x, (0)2 , (0)3 , . . . ,

(0)d ).

IV (0)2 (1)2

f(2|x, (1)1 , (0)3 , . . . ,

(0)d ).

V . . .

VI (0)d (1)d

f(d|x, (1)1 , (1)2 , . . . ,

(1)d1).

42


VII III V I.

, , -. Metropolis-Hastings i ( ). MCMC -

j (j)1 , (j)2 , . . . ,

(j)d

(j+1)1 , 1. - Metropolis-Hastings .

1. can1 -

q(can1 |(j)1 ,

(j)2 , . . . ,

(j)d ).

2. 1 (j+1)1

(j+1)1 =

{

can1 p

(j)1 1 p

p = min{1, f(can1 |x,

(j)2 , . . . ,

(j)d )

f(j1|x, (j)2 , . . . ,

(j)d )

q(j1|can1 , (j)2 , . . . ,

(j)d )

q(can1 |(j)1 ,

(j)2 , . . . ,

(j)d )

}

f(can1 |x, (j)2 , . . . ,

(j)d ) -

1 1 = can1

f(j1|x, (j)2 , . . . ,

(j)d ).

,

q(can1 |(j)1 ,

(j)2 , . . . ,

(j)d )

. Metropolis-Hastings - Gibbs. Gibbs Metropolis-

Hastings q(can1 |(j)1 ,

(j)2 , . . . ,

(j)d ) =

f(can1 |(j)1 ,

(j)2 , . . . ,

(j)d )

(j+1)1 =

can1

1[;].

43

3.

44

4

And Id join the movementIf there was oneI could believe inYeah Id break bread and wineIf there was a church I couldreceive in.

Acrobat

U2

, ( ).

45

4.

4.1

-

yi = 0 + 1xi + i i = 1, 2, . . . , n

i N(0, 2).

=[

0, 1, 2]

L(0, 1, 2|y) (2)n2 exp{ 1

22

n

i=1

(yi 0 1xi)2}

l(0, 1, 2|y) = n

2ln(2) 1

22

n

i=1

(yi 0 1xi)2.

l0l1l2

=

12

ni=1(yi 0 1xi)

12

ni=1 xi(yi 0 1xi)

n22

+ 124

ni=1(yi 0 1xi)2

0 =

[

0, 1, 2]

2 =

ni=1(yi 0 1xi)2

n

0 1

[

nn

i=1 xin

i=1 xin

i=1 x2i

]

[

0 1

]

=

[

ni=1 yi

ni=1 xiyi

]

.

0 = y 1x

46

4.1.

1 =

ni=1 xiyi 1n

ni=1 xi

ni=1 yi

ni=1 x

2i 1n(

ni=1 xi)

2

., Fisher

I() = 2

nn

i=1 xi 0n

i=1 xin

i=1 x2i 0

0 0 (22)1n

yi = 0 + 1xi1 + . . .+ kxik + i, i = 1, . . . , n.

i N(0, 2I).

Y =

y1y2...yn

, X =

1 x11 . . . x1k1 x21 . . . x2k......

......

1 xn1 . . . xnk

, =

01...k

, =

12...n

Y = X + ,

E() = 0 V () = 2I 2 - I n n.

=[

, 2]

.

yi N(i, 2) i = xi xi =

1xi1...xik

i = 1, . . . , n.

L() =( 1

22)n/2

exp{ 122

n

i=1

(yi xi)2}

47

4.

l(, 2|y) = n2ln(22) 1

22

n

i=1

(yi xi)2

= n2ln(22) 1

22[(y X)(y X)]

= n2ln(22) 1

22[yy 2 X y + X X]

l(, 2|y)

= 122

[[yy 2 X y + X X]

]

= 122

[2X y + 2X X]

=1

22[X y X X].

,

l(, 2|y)

= 0

1

22[X y X X] = 0

X X = X y

= (X X)1X y.

2

l(, 2|y)2

= n22

+1

24[y X)(y X)]

l(, 2|y)2

= 0

n22

+1

24[y X)(y X)] = 0

1

24[y X)(y X)] = n

22

1

2[y X)(y X)] = n.

48

4.2.

2 =1

n

n

i=1

(y X )2.

[;] Fisher

= (X X)1X Y,

2 =1

n

n

i=1

(yi xi)2

I() = 2(X X).

I1() = 2(X X)1.

... - Franklin[?]Gonzalez[?].

4.2

.

yi = 0 + 1xi1 + . . .+ kxik + i, i = 1, . . . , n.

i N(0, 2I)

Y = X + ,

E() = 0 V () = 2I 2 I n n. , ,

=[

, 2]

49

4.

L(, 2|y) (2)n/2exp[(y X)(y X)

22].

(y X)(y X) = (y X X +X)(y X X +X)= [y X X( )][y X X( )]= (y X)(y X) + ( )X X( )= S + ( )X X( ).

S = (y X)(y X)

= (X X)1X y

... .

L(, 2|y) (2)n/2exp[S + ( )X X( )

22].

= [, 2] - (NIG(a, d,m, V )) (Normal Inverse Gam-ma) a, d,m, V

() =(a/2)d/2

(2)p/2|V |1/2(d/2)(2)(d+p+2)/2exp[{(m)V 1(m)+a}/(22)],

() (2)(d+p+2)/2exp[{( m)V 1( m) + a}/(22)]

p(|y) ()L(|y)

p(|y) (2)(d+n+p+2)/2exp[Q/(22)]

Q = (y X)(y X) + ( m)V 1( m) + a= (V 1 +X X) (V 1m+X y) (mV 1 + yX) + (mV 1m+ yy + a)= ( m)(V )1( m) + a,

50

4.2.

V = (V 1 +X X)1,

m = (V 1 +X X)1(V 1m+X y),

a = a+mV 1m+ yy (m)(V )1m.

, d = d + n

p(|y) = p(, 2|y) NIG(a, d, m, V )

, 2 p(, 2|y) 2 .

p(|y) =

p(, 2|y)d2

(2)(d+n+p+2)/2exp[Q/(22)]d2

{1 + ( m)(aV )1( m)}(d+p)/2,

t d m

aV .

2

p(2|y) =

p(, 2|y)d

(2)(d+n+p+2)/2exp[Q/(22)]d

(2)(d+2)/2exp[a/(22)],

2 - (IG(a, d)) (Inverse Gamma) a d. -

Sorensen,Gianola[?] OHagan[?].

51

4.

4.3

-

- - . i. -

i N(0, 2).

Laplace (assymetric Laplace). - U Laplace

gp(u) = p(1 p)exp{p(u)}, 0 < p < 1, u [0,)

p(u) = puI[0,)(u) (1 p)uI(,0)(u)

(check function). p = 1/2 gp(u) =14exp{ |u|

2} Laplace

. p, . 12p

p(1p) p > 1/2,

p < 1/2 0 p = 1/2 12p+2p2

p2(1p)2

p .

yi = 0,p + 1,pxi1 + . . .+ k,pxik + i,p i = 1, . . . , n.

Y Xi.

Y =

y1y2...yn

, X =

1 x11 . . . x1k1 x21 . . . x2k......

......

1 xn1 . . . xnk

, p =

0(p)1(p)...

k(p)

, p =

1,p2,p...

n,p

52

4.4.

Y = Xp + p.

, p, p

n

i=1

p(Y Xp)

p(u) = p|u| . p

L(yi|p, p, x1, . . . , xn) = (p(1 p)

p)nexp{ 1

p

n

i=1

p(yi pxi)}

0 < p < 1, p > 0.

Y , ... . [;]

p = argminp

[n

i=1

p(yi pxi)]

p =1

n

n

i=1

p(yi pxi).

- Bootstrap [;]., Laplace , - tick-exponential . - Komunjer [;].

4.4

-

- , MCMC

53

4.

(Markov chain Monte Carlo) - . MCMC . , , - . y = (y1, y2, . . . , yn),

p, p

(p, p|y) L(y|p, p)p(p)p(p), p(p), p(p) p, p L(y|p, p)

L(yi|p, p, x1, . . . , xn) = (p(1 p)

p)nexp{ 1

p

n

i=1

p(yi pxi)}

0 < p < 1, p > 0.

p Yu Moyeed[?] (improper) p , (proper).1

p 0 2p p ( )

p(p) exp{2p22p

}

p(p) 1.

(p, p|y) (p(1 p)

p)nexp{ 1

p

n

i=1

p(yi pxi)}exp{2p22p

}

1 p()

0 c H0.

(x) < c H0.

(x) = c H0 q,

c, q

qPr((x) = c|H0) +Pr((x) < c|H0) =

.

1 (x), H0. Likelihood ratio test

61

5.

62

6

There is no such thing as animpartial jury because there areno impartial people. There arepeople that argue on the web forhours about who their favoritecharacter on Friends is.

The Daily Show

Jon Stewart

(hedgefunds) . hedgefunds - . . 40% Euronext, - . - / . -

(risk factors) . - hedge fund, - .

63

6.

- .

stepwise regression AIC BIC .

, , , , . .

- . , , -. - . . 4

14 (risk factors) - - (0.1, 0.25, 0.5, 0.75, 0.9). AIC BIC. - - . - MATLAB .

6.1

4 : Convertible Arbitrage (CA), Equity Non-Hedge (ENH), Distressed Secu-

64

6.1.

rities (DS), Merger Arbitrage (MA). CA (- , ) (short selling). - ENH (leverage). DS MA . - 1990 2005. - . 6.1

- . 6.1 . (ENH) (CA, MA). . ENH CA 4 . , 4 - . DS, MA . - . -

1.RUS Russell 3000 (Rus-sell 3000 equity index excess return).

2.RUS(1) Russell 3000 (Rus-sell 3000 equity index excess return lagged once).

3.MXUS ( ) Morgan Stanley(Morgan Stanley Capital Idternational world excluding USA index e-xcess return).

4.MEM MorganStanley (Morgan Stanley Capital Idternational emerging markets indexexcess return).

65

6.

6.1: -

.

.. . . . 25 v. 50 v. 75 v.CA 0.48 0.99 5.37 1.18 0.03 0.68 1.14ENH 1.00 4.05 3.63 0.53 1.60 1.70 3.53DS 0.84 1.75 8.50 0.68 0.05 0.81 1.72MA 0.50 1.08 13.38 2.39 0.08 0.64 1.12

5.SMB Fama and French (Fama and Frenchs size).

6.HML Fama and French(Fama and Frenchsbook-to-market).

7.MOM Carhart (Carharts momentum factor).

8.SBGC - Salomon Brothers (Salomon Brothers world government andcorporate bond index excess return).

9.SBWG Salomon Brothers (SalomonBrothers world government bond index excess return).

10.LHY Lehman (Lehman high yield indexexcess return).

11.DEFSPR BA-A 10 (Difference between the yield on the BAA-rated corporate bonds and the 10-year bonds).

12.FRBI Federal ReserveBank (Federal Reserve Bank competitiveness weighted dollar-index e-xcess return).

13.GSCI Goldman Sachs (Goldman Sachs com-modity index excess return).

14.VIX S&P 500 (Change in S&P500 implied volatility index).

66

6.2.

6.2

- 6.2. , ( 0) - (standarderrors).

. , BIC AIC BIC .

6.3 -

- 6.3, 6.4, 6.5, 6.6. -, ( 0) (standard errors) Bo-otstrap. - . - . . (10 90 ) - .

67

6.

. - MA 90 10 14 - 50 3. 7 13 10 90 , MA ( ). 9 11 CA (AIC) 11 ENH ( AIC).

6.4

-

6.2 6.3, 6.4, 6.5, 6.6 , - , - DS, ENH BIC. - . hedge funds. -

, , . - 6.3 hedge funds. - , . , , - , . , .

68

6.4.

- , BIC. , BIC .

69

6.

6.2: .. .

AIC 2 4 5 8 10

0.0042 0.0494 0.0347 0.0331 0.1370 0.0493s.e. 0.0007 0.0160 0.0106 0.0213 0.0531 0.0226

CABIC 2 4 8 10

0.0043 0.0539 0.0387 0.1232 0.0575s.e. 0.0007 0.0158 0.0103 0.0525 0.0220

AIC 1 2 4 5 6 9 10 12 13 14

0.0063 0.6920 0.0448 0.0704 0.4370 -0.1045 0.0812 0.0640 0.1613 0.0271 0.0780s.e. 0.0008 0.0332 0.0212 0.0175 0.0294 0.0234 0.0494 0.0288 0.0796 0.0142 0.0316

ENHBIC 1 4 5 6 10 14

0.0066 0.6969 0.0673 0.4548 -0.0950 0.0720 0.0948s.e. 0.0008 0.0335 0.0174 0.0284 0.0230 0.0289 0.0307

AIC 2 4 5 8 10 11 12 14

0.0074 0.0924 0.0898 0.0956 0.2301 0.0823 -2.7857 0.1513 -0.0556s.e. 0.0009 0.0229 0.0168 0.0287 0.0748 0.0306 0.8219 0.0770 0.0292

DSBIC 2 4 5 8 10 11

0.0073 0.0787 0.0988 0.0952 0.2161 0.0918 -3.1576s.e. 0.0009 0.0225 0.0145 0.0292 0.0759 0.0309 0.8192

AIC 1 2 4 5 6 7 11

0.0037 0.0996 0.0653 0.0405 0.0732 0.0516 0.0298 0.8227s.e. 0.0006 0.0221 0.0164 0.0136 0.0233 0.0221 0.0135 0.5584

MABIC 1 2 4

0.0043 0.0787 0.0681 0.0460s.e. 0.0006 0.0210 0.0152 0.0130

70

6.4.

6.3: : .. . .

0.10 AIC 2 3 4 5 6 8 9 10 11 -0.0083 0.0960 -0.0732 0.0981 -0.0703 -0.0609 0.2603 0.1015 0.0716 -2.0879s.e. 0.0018 0.0480 0.0546 0.0382 0.0727 0.0682 0.1557 0.1181 0.0542 2.0486

0.25 1 2 3 4 8 10 11 13 14 -0.0001 0.0673 0.0759 -0.0405 0.0268 0.1199 0.0881 1.1586 0.0220 0.0315s.e. 0.0013 0.0384 0.0279 0.0406 0.0253 0.0848 0.0444 1.5808 0.0163 0.0388

0.50 1 2 5 10 11 0.0059 0.0470 0.0439 0.0282 0.0972 2.3189s.e. 0.0006 0.0130 0.0141 0.0182 0.0364 0.6483

0.75 2 4 5 6 7 8 11 13 0.0099 0.0297 0.0244 0.0293 -0.0325 -0.0431 0.1298 1.7262 0.0147s.e. 0.0006 0.0179 0.0100 0.0223 0.02221 0.0159 0.0399 0.5544 0.0130

0.90 2 4 5 6 7 8 9 10 11 12 13 0.0125 0.0470 0.0231 0.0257 -0.0184 -0.0169 0.1072 -0.0507 0.0144 1.3574 -0.0619 0.0224s.e. 0.0011 0.0306 0.0202 0.0374 0.0315 0.0157 0.0777 0.0612 0.0756 0.8819 0.0888 0.0207

CA0.10 BIC 2 3 4 8 11

-0.0090 0.0763 -0.0515 0.1004 0.3214 -1.5298s.e. 0.0017 0.0502 0.0417 0.0272 0.1074 1.7396

0.25 2 8 10 0.0001 0.0801 0.1398 0.1368s.e. 0.0009 0.0237 0.0516 0.0453

0.50 1 2 5 10 11 0.0059 0.0470 0.0439 0.0282 0.0972 2.3189s.e. 0.0005 0.0119 0.0130 0.0174 0.0353 0.7283

0.75 2 4 5 6 7 8 11 0.0095 0.0360 0.0246 0.0346 -0.0343 -0.1454 0.1354 1.7523s.e. 0.0006 0.0165 0.0112 0.0191 0.0215 0.0171 0.0372 0.5899

0.90 2 4 5 10 11 0.0135 0.0469 0.0254 0.0434 0.0225 1.4495s.e. 0.0007 0.0281 0.0183 0.0185 0.0654 0.6323

71

6.

6.4: : .. . .

0.10 AIC 1 2 4 5 6 7 9 10 11 13 -0.0065 0.6239 0.0668 0.0988 0.3765 -0.1062 0.0137 0.1657 0.1366 -1.0523 0.0291s.e. 0.0014 0.0337 0.0260 0.0254 0.0357 0.0339 0.0183 0.0568 0.0544 1.1602 0.0143

0.25 1 2 4 5 6 9 10 13 14 -0.0019 0.6191 0.0493 0.0896 0.3838 -0.0997 0.1176 0.0803 -0.2368 0.0257s.e. 0.0012 0.0546 0.0348 0.0360 0.0553 0.0394 0.0809 0.0360 0.0177 0.0503

0.50 1 3 4 5 6 7 8 14 0.0061 0.6358 0.0405 0.0670 0.4986 -0.1042 -0.0030 0.1146 0.0855s.e. 0.0012 0.0530 0.0399 0.0272 0.0477 0.0376 0.0194 0.0982 0.0424

0.75 1 3 4 5 6 7 11 12 14 0.0130 0.6677 0.0266 0.0841 0.4943 -0.1164 -0.0218 1.0484 0.2198 0.0768s.e. 0.0011 0.0532 0.0395 0.0314 0.0401 0.0404 0.0271 1.0466 0.0943 0.0644

0.90 1 2 3 4 5 6 7 11 12 13 14 0.0194 0.7972 0.0404 0.0289 0.0333 0.4596 -0.0906 -0.0313 0.9426 0.2875 0.0559 0.1717s.e. 0.0019 0.1183 0.0507 0.0443 0.0358 0.0725 0.0616 0.0381 1.3739 0.1597 0.0375 0.1284

ENH0.10 BIC 1 2 4 5 6 9 10 11 13

-0.0066 0.6241 0.0667 0.0988 0.3670 -0.1183 0.1828 0.1097 -1.4624 0.0337s.e. 0.0014 0.0290 0.0268 0.0231 0.0350 0.0256 0.0597 0.0469 1.2791 0.0143

0.25 1 4 5 6 9 10 14 -0.0012 0.6604 0.0849 0.4333 -0.0803 0.1154 0.0647 0.0622s.e. 0.0012 0.0475 0.0330 0.0480 0.0405 0.0579 0.0392 0.0342

0.50 1 4 5 6 14 0.0066 0.7335 0.0653 0.4626 -0.0779 0.1339s.e. 0.0009 0.0460 0.0220 0.0497 0.0369 0.0388

0.75 1 4 5 6 14 0.0126 0.7033 0.0744 0.5043 -0.1011 0.1060s.e. 0.0010 0.0456 0.0191 0.0426 0.0380 0.0760

0.90 1 4 5 6 13 14 0.0191 0.8209 0.0194 0.5412 -0.0399 0.0511 0.1668s.e. 0.0014 0.0852 0.0360 0.0562 0.0626 0.0346 0.1172

72

6.4.

6.5: : .. . .

0.10 AIC 1 2 4 7 8 10 11 12 13 -0.0049 0.0565 0.1012 0.0865 0.0284 0.1460 0.1738 -2.4498 0.1634 0.0548s.e. 0.0019 0.0410 0.0386 0.0466 0.0331 0.1148 0.0737 1.3073 0.1219 0.0291

0.25 1 2 4 5 7 8 10 11 12 0.0013 0.0517 0.0885 0.0422 0.0742 0.0172 0.1879 0.1361 -4.1552 0.0601s.e. 0.0011 0.0298 0.0269 0.0255 0.0331 0.0186 0.1023 0.0681 1.1223 0.0923

0.50 2 3 4 5 8 10 11 13 0.0066 0.0671 0.0392 0.0725 0.0459 0.2820 0.0763 -4.0166 -0.0375s.e. 0.0012 0.0356 0.0332 0.0261 0.0636 0.1316 0.0770 1.3440 0.0289

0.75 1 2 4 5 6 7 8 9 10 11 12 0.0132 0.0580 0.0663 0.0907 0.2061 0.0664 0.0494 0.1487 0.1209 0.0567 -3.0243 0.2432s.e. 0.0014 0.0372 0.0414 0.0213 0.0524 0.0512 0.0369 0.1273 0.0858 0.0508 1.4736 0.1377

0.90 2 3 4 5 6 8 9 10 11 12 14 0.0209 0.0227 -0.0687 0.1291 0.2505 0.1051 0.2439 0.0795 0.0914 -2.0042 0.3356 -0.0300s.e. 0.0021 0.0575 0.0684 0.0374 0.0710 0.0450 0.1893 0.1125 0.0714 2.3445 0.1495 0.0563

DS0.10 BIC 1 2 4 8 10 11 12 13

-0.0045 0.0721 0.0912 0.0687 0.1782 0.1561 -2.5255 0.1923 0.0371s.e. 0.0018 0.0384 0.0391 0.0456 0.0902 0.0641 1.3520 0.1249 0.0261

0.25 1 2 4 5 8 10 11 0.0014 0.0336 0.0851 0.0479 0.0687 0.2319 0.1242 -4.4157s.e. 0.0010 0.0388 0.0274 0.0222 0.0330 0.0902 0.0648 1.1975

0.50 2 4 8 10 11 0.0059 0.0903 0.0847 0.2298 0.1082 -4.3613s.e. 0.0014 0.0298 0.0174 0.1233 0.0753 1.3483

0.75 2 4 5 8 10 11 0.0142 0.0539 0.0861 0.1690 0.2461 0.0731 -3.0941s.e. 0.0011 0.0366 0.0156 0.0492 0.0891 0.0632 1.2087

0.90 4 5 6 8 10 12 0.0202 0.1209 0.2548 0.1302 0.2649 0.0703 0.3399s.e. 0.0021 0.0282 0.0413 0.0419 0.1564 0.0580 0.1753

73

6.

6.6: : .. . .

0.10 AIC 1 2 3 4 5 6 7 11 12 13 -0.0070 0.1146 0.0915 0.0426 0.0604 0.0574 0.0378 0.0344 2.1157 0.1076 0.0369s.e. 0.0016 0.0453 0.0276 0.0501 0.0382 0.0322 0.0481 0.0245 1.3575 0.0944 0.0289

0.25 1 2 4 5 7 8 12 14 -0.0007 0.0916 0.0453 0.0319 0.0648 0.0182 0.1471 0.1189 -0.0411s.e. 0.0009 0.0306 0.0329 0.0222 0.0319 0.0148 0.0654 0.0892 0.0417

0.50 1 2 3 4 5 7 9 11 12 0.0051 0.0761 0.0671 -0.0379 0.0491 0.0544 0.0222 0.1074 1.3755 0.1353s.e. 0.0007 0.0320 0.0148 0.0231 0.0183 0.0206 0.0119 0.0452 1.0334 0.0539

0.75 1 2 4 5 6 9 11 12 13 14 0.0096 0.0562 0.0435 0.0214 0.0738 0.0182 0.0689 0.8799 0.1802 -0.0225 -0.0230s.e. 0.0007 0.0235 0.0223 0.0150 0.0257 0.0256 0.0383 0.7499 0.0621 0.0138 0.0286

0.90 1 2 5 7 8 10 11 12 13 14 0.0128 0.0449 0.0327 0.0713 -0.0303 0.1028 -0.0539 1.0319 0.1585 -0.0290 -0.0396s.e. 0.0007 0.0177 0.0234 0.0272 0.0179 0.0634 0.0351 0.6407 0.0572 0.0132 0.0288

MA0.10 BIC 1 2 4 7 13

-0.0059 0.1187 0.0890 0.0595 0.0275 0.0365s.e. 0.0019 0.0450 0.0369 0.0390 0.0238 0.0281

0.25 1 2 4 -0.0004 0.0696 0.0499 0.0507s.e. 0.0008 0.0276 0.0307 0.0198

0.50 1 2 5 0.0054 0.0837 0.0533 0.0690s.e. 0.0007 0.0246 0.0178 0.0139

0.75 1 2 5 12 13 0.0100 0.0699 0.0254 0.0772 0.0790 -0.0231s.e. 0.0006 0.0172 0.0156 0.0259 0.0334 0.0089

0.90 1 2 5 7 8 10 11 12 13 14 0.0128 0.0449 0.0327 0.0713 -0.0303 0.1028 -0.0539 1.0319 0.1585 -0.0290 -0.0396s.e. 0.0007 0.0199 0.0212 0.0242 0.0155 0.0549 0.0336 0.7342 0.0615 0.0145 0.0286

74

Matlab

Ignorance is bliss. Oedipusruined a great sex life by askingtoo many questions.

The Colbert Report

Stephen Colbert

.1 Matlab -

y - 6.1

mean(y)

std(y)

prctile(y,[25 50 75])

skewness(y)

kurtosis(y)

.2 Matlab -

- AIC, BIC . 14

75

. MATLAB

214 = 16384. 2 , . MATLAB ( ).

N=14;

all_models=zeros(1,N);

for i=1:N

models_i=nchoosek(1:14,i);

[a,b]=size(models_i);

models_i=[models_i zeros(a,N-b)];

all_models=[all_models; models_i];

end

save all_models all_models

all models =

1 0 0 0 . . . 02 0 0 0 . . . 0..................

14 0 0 0 . . . 01 2 0 0 . . . 01 3 0 0 . . . 0..................

13 14 0 0 . . . 0..................

1 2 3 4 . . . 14

16384 14 - . MATLAB AIC, BIC

.

function [minaic,minbic,B]=minaibi(y,X)

load all_models

load datafactors_hfrci.txt % Matrix that contains the risk factors

for i=1:size(all_models14,1)

model=all_models14(i,:);

model(model==0)=[];

76

.2. MATLAB

X=datafactors_hfrci(:,model);

[n,k]=size(X);

s=regstats(y,X,linear); % Linear regression

AIC=n*log(s.r*s.r)+2*(k+1); % Computing AIC, BIC

BIC=n*log(s.r*s.r)+log(n)*(k+1);

B(i,:)=[AIC BIC];

end

minaic=min(B(:,1)); % Computing the min AIC, BIC

minbic=min(B(:,2));

end

AIC - .

L() =( 1

22)n/2

exp{ 122

n

i=1

(yi xi)2}

l() = n2log(2) n

2log(2) 1

22

n

i=1

(yi xi)2}

= n2log(2) n

2log(2) 1

22n2

= n2log(2) + c.

,

AIC = n log(2) + 2k.

BIC. - B 16384 2 AIC - BIC . . , B, ,

[val1 ind1]=min(B(:,1)); % Value of minaic and index of that model

[val2 ind2]=min(B(:,2)); % Value of minbic and index of that model

allmodels(ind1,:) %Model with minimum AIC

allmodels(ind2,:) %Model with minimum BIC

77

. MATLAB

, - , - CA AIC

b1=[2 4 5 8 10]; %Best model factors

X=datafactors_hfrci(:,b1);

s1=regstats(y,X,linear);

s1.beta %Regression coefficients

sqrt(diag(s1.covb)) %Standard errors

.3 Matlab

- AIC, BIC . (check function) - .

function sum = check(y,X,betaq,p)

sum=0;

qq=y-X*betaq;

for i=1:size(qq,1)

if (qq(i)>=0)

sum=sum+p*qq(i);

elseif (qq(i)

.3. MATLAB

[m n] = size(X);

u = ones(m, 1);

a = (1 - p) .* u;

b = -lp_fnm(X, -y, X * a, u, a);

function y = lp_fnm(A, c, b, u, x)

% Solve a linear program by the interior point method:

% min(c * u), s.t. A * x = b and 0 < x < u

% An initial feasible solution has to be provided as x

%

% Function lp_fnm of Daniel Morillo & Roger Koenker

% Translated from Ox to Matlab by Paul Eilers 1999

% Modified by Roger Koenker 2000--

% More changes by Paul Eilers 2004

% Set some constants

beta = 0.9995;

small = 1e-5;

max_it = 50;

[m n] = size(A);

% Generate inital feasible point

s = u - x;

y = (A \ c);

r = c - y * A;

r = r + 0.001 * (r == 0); % PE 2004

z = r .* (r > 0);

w = z - r;

gap = c * x - y * b + w * u;

% Start iterations

it = 0;

while (gap) > small & it < max_it

it = it + 1;

% Compute affine step

q = 1 ./ (z ./ x + w ./ s);

r = z - w;

Q = spdiags(sqrt(q), 0, n, n);

AQ = A * Q; % PE 2004

79

. MATLAB

rhs = Q * r; % "

dy = (AQ \ rhs); % "

dx = q .* (dy * A - r);

ds = -dx;

dz = -z .* (1 + dx ./ x);

dw = -w .* (1 + ds ./ s);

% Compute maximum allowable step lengths

fx = bound(x, dx);

fs = bound(s, ds);

fw = bound(w, dw);

fz = bound(z, dz);

fp = min(fx, fs);

fd = min(fw, fz);

fp = min(min(beta * fp), 1);

fd = min(min(beta * fd), 1);

% If full step is feasible, take it. Otherwise modify it

if min(fp, fd) < 1

% Update mu

mu = z * x + w * s;

g = (z + fd * dz) * (x + fp * dx) + (w + fd * dw) * (s + fp * ds);

mu = mu * (g / mu) ^3 / ( 2* n);

% Compute modified step

dxdz = dx .* dz;

dsdw = ds .* dw;

xinv = 1 ./ x;

sinv = 1 ./ s;

xi = mu * (xinv - sinv);

rhs = rhs + Q * (dxdz - dsdw - xi);

dy = (AQ \ rhs);

dx = q .* (A * dy + xi - r -dxdz + dsdw);

ds = -dx;

dz = mu * xinv - z - xinv .* z .* dx - dxdz;

dw = mu * sinv - w - sinv .* w .* ds - dsdw;

% Compute maximum allowable step lengths

fx = bound(x, dx);

fs = bound(s, ds);

80

.3. MATLAB

fw = bound(w, dw);

fz = bound(z, dz);

fp = min(fx, fs);

fd = min(fw, fz);

fp = min(min(beta * fp), 1);

fd = min(min(beta * fd), 1);

end

% Take the step

x = x + fp * dx;

s = s + fp * ds;

y = y + fd * dy;

w = w + fd * dw;

z = z + fd * dz;

gap = c * x - y * b + w * u;

%disp(gap);

end

function b = bound(x, dx)

% Fill vector with allowed step lengths

% Support function for lp_fnm

b = 1e20 + 0 * x;

f = find(dx < 0);

b(f) = -x(f) ./ dx(f);

AIC, BIC .

function [minaicq,minbicq,Q]=minaibiq(y,X,p)

load all_models

load datafactors_hfrci.txt % Matrix that contains the risk factors

for i=1:16384

model=all_models(i,:);

model(model==0)=[];

X=datafactors_hfrci(:,model);

[n,k]=size(X);

X=[ones(n,1) X];

betaq=rq(X, y, p); %Quantile regression function

ch=check(y,X,betaq,p);

AIC=2*n*log(ch)-2*n*log(p*(1-p))+2*n-2*n*log(n)+2*(k+1);

BIC=2*n*log(ch)-2*n*log(p*(1-p))+2*n-2*n*log(n)+log(n)*(k+1);

81

. MATLAB

Q(i,:)=[AIC BIC];

end

minaicq=min(Q(:,1)); % Computing the min AIC, BIC

minbicq=min(Q(:,2));

end

AIC

L(yi|p, p, x1, . . . , xn) = (p(1 p)

p)nexp{ 1

p

n

i=1

p(yi pxi)}

l() = n log(p(1 p)) n log(p)1

p

n

i=1

p(yi pxi)

= . . .

= n log(p(1 p)) n log(n

i=1

p(yi pxi)) + n log(n) n.

,

AIC = 2n log(n

i=1

p(yi pxi)) 2n log(p(1 p)) + 2n 2n log(n) + 2k.

BIC. - Q 16384 2 AIC - BIC . . , Q, ,

[val1 ind1]=min(Q(:,1)); % Value of minaic and index of that model

[val2 ind2]=min(Q(:,2)); % Value of minbic and index of that model

allmodels(ind1,:) %Model with minimum AIC

allmodels(ind2,:) %Model with minimum BIC

Bootstrap -

82

.3. MATLAB

function [bootestimates,covmat_boot]=b_bootstrap(F,data,p,b_lsq,B,m);

T=length(data);

bootestimates=zeros(length(b_lsq),B);

for bb=1:B

bootsample=unidrnd(T,1,m);

boot_factors=F(bootsample,:);

boot_data=data(bootsample);

b_boot=rq(boot_factors,boot_data,p);

bootestimates(:,bb)=[b_boot];

end

mles=repmat([b_lsq],[1,B]);

covmat_boot=(m/T)*(bootestimates-mles)*(bootestimates-mles)/B;

- , y , p, p, B m ., -

, - CA AIC

b1q=[2 3 4 5 6 8 9 10 11]

X=datafactors_hfrci(:,b1q);

[n k]=size(X);

X=[ones(n,1) X];

beta1q=rq(X,y,0.1);

[bootestimates,covmat_boot]=b_bootstrap(X,y,0.1,beta1q,100,189);

sqrt(diag(covmat_boot))

83

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