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A g e n e r a l i z a t i o n o f g e n e r a l i z e d b e t a d i s t r i b u t i o n s
M i c h a e l B . G o r d y
B o a r d o f G o v e r n o r s o f t h e F e d e r a l R e s e r v e S y s t e m
A p r i l 8 , 1 9 9 8
A b s t r a c t
T h i s p a p e r i n t r o d u c e s t h e \ c o m p o u n d c o n u e n t h y p e r g e o m e t r i c " ( C C H ) d i s t r i b u t i o n . T h e C C H
u n i e s a n d g e n e r a l i z e s t h r e e r e c e n t l y i n t r o d u c e d g e n e r a l i z a t i o n s o f t h e b e t a d i s t r i b u t i o n : t h e G a u s s
h y p e r g e o m e t r i c ( G H ) d i s t r i b u t i o n o f A r m e r o a n d B a y a r r i ( 1 9 9 4 ) , t h e g e n e r a l i z e d b e t a ( G B ) d i s t r i -
b u t i o n o f M c D o n a l d a n d X u ( 1 9 9 5 ) , a n d t h e c o n u e n t h y p e r g e o m e t r i c ( C H ) d i s t r i b u t i o n o f G o r d y
( f o r t h c o m i n g ) . I n a d d i t i o n t o g r e a t e r e x i b i l i t y i n t t i n g d a t a , t h e C C H o e r s t w o u s e f u l p r o p e r t i e s .
U n l i k e t h e b e t a , G B a n d G H , t h e C C H a l l o w s f o r c o n d i t i o n i n g o n e x p l a n a t o r y v a r i a b l e s i n a n a t u r a l
a n d c o n v e n i e n t w a y . T h e C C H f a m i l y i s c o n j u g a t e f o r g a m m a d i s t r i b u t e d s i g n a l s , a n d s o m a y a l s o
p r o v e u s e f u l i n B a y e s i a n a n a l y s i s . A p p l i c a t i o n o f t h e C C H i s d e m o n s t r a t e d w i t h t w o m e a s u r e s o f
h o u s e h o l d l i q u i d a s s e t s . I n e a c h c a s e , t h e C C H y i e l d s a s t a t i s t i c a l l y s i g n i c a n t i m p r o v e m e n t i n t
o v e r t h e m o r e r e s t r i c t i v e a l t e r n a t i v e s .
J E L C o d e s : C 1 0 , D 9 1 , G 1 1
T h e v i e w s e x p r e s s e d h e r e i n a r e m y o w n a n d d o n o t n e c e s s a r i l y r e e c t t h o s e o f t h e B o a r d o f
G o v e r n o r s o r i t s s t a . I a m g r a t e f u l f o r t h e h e l p f u l c o m m e n t s o f K a r e n D y n a n , A r t h u r K e n n i c k e l l ,
W h i t n e y N e w e y , a n d M a r t h a S t a r r - M c C l u e r . P l e a s e a d d r e s s c o r r e s p o n d e n c e t o t h e a u t h o r a t
D i v i s i o n o f R e s e a r c h a n d S t a t i s t i c s , M a i l S t o p 1 5 3 , F e d e r a l R e s e r v e B o a r d , W a s h i n g t o n , D C 2 0 5 5 1 ,
U S A . P h o n e : ( 2 0 2 ) 4 5 2 - 3 7 0 5 . F a x : ( 2 0 2 ) 4 5 2 - 5 2 9 5 . E m a i l : h m g o r d y @ f r b . g o v i .
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T h e b e t a d i s t r i b u t i o n i s w i d e l y u s e d i n s t a t i s t i c a l m o d e l i n g o f b o u n d e d r a n d o m v a r i a b l e s . I t i s
e a s i l y c a l c u l a t e d , c a n t a k e o n a v a r i e t y o f s h a p e s , a n d , p e r h a p s a s i m p o r t a n t l y , n o n e o f t h e o t h e r
c o m m o n l y u s e d d i s t r i b u t i o n f u n c t i o n s h a v e c o m p a c t s u p p o r t .
1
H o w e v e r , i t s a p p l i c a t i o n i s l i m i t e d
i n i m p o r t a n t w a y s . F i r s t , a s a t w o p a r a m e t e r d i s t r i b u t i o n , i t c a n p r o v i d e o n l y l i m i t e d p r e c i s i o n i n
t t i n g d a t a . I t i s d e s i r a b l e t o h a v e m o r e p a r a m e t r i c a l l y e x i b l e v e r s i o n s o f t h e b e t a t o a l l o w a r i c h e r
e m p i r i c a l d e s c r i p t i o n o f d a t a w h i l e s t i l l o e r i n g m o r e s t r u c t u r e t h a n a n o n p a r a m e t r i c e s t i m a t o r .
S e c o n d , t h e b e t a d o e s n o t o e r a n a t u r a l a n d c o n v e n i e n t m e a n s o f i n t r o d u c i n g e x p l a n a t o r y v a r i a b l e s .
I n a B e t a ( p ; q ) d i s t r i b u t i o n , t h e p a r a m e t e r s ( p ; q ) j o i n t l y d e t e r m i n e b o t h t h e s h a p e a n d m o m e n t s
o f t h e d i s t r i b u t i o n . T h e r e i s n o s a t i s f a c t o r y w a y o f c o n d i t i o n i n g t h e m e a n b y s p e c i f y i n g p a n d q a s
f u n c t i o n s o f e x p l a n a t o r y v a r i a b l e s a n d r e g r e s s i o n c o e c i e n t s . T h i r d , t h e b e t a i s i n c o n v e n i e n t f o r
u s e i n B a y e s i a n a n a l y s i s . I t i s c o n j u g a t e f o r b i n o m i a l s i g n a l s , b u t n o t f o r s i g n a l s o f a n y c o n t i n u o u s
d i s t r i b u t i o n .
R e c e n t r e s e a r c h h a s c o n t r i b u t e d t h r e e g e n e r a l i z a t i o n s o f t h e b e t a w h i c h a d d r e s s o n e o r m o r e o f
t h e s e l i m i t a t i o n s . A r m e r o a n d B a y a r r i ( 1 9 9 4 ) d e n e t h e G a u s s h y p e r g e o m e t r i c ( G H ) d i s t r i b u t i o n
b y t h e d e n s i t y f u n c t i o n
G H ( x ; p ; q ; r ; ) =
x
p 1
( 1 x )
q 1
( 1 + x )
r
B ( p ; q )
2
F
1
( r ; p ; p + q ; )
f o r 0 < x < 1 ( 1 )
w i t h p > 0 ; q > 0 a n d w h e r e
2
F
1
d e n o t e s t h e G a u s s h y p e r g e o m e t r i c f u n c t i o n . T h e G H c o l l a p s e s t o
t h e o r d i n a r y b e t a i f e i t h e r r = 0 o r = 0 , a n d t o t h e b e t a - p r i m e i f q = = 1 . A r m e r o a n d B a y a r r i
a p p l y t h e G H t o a B a y e s i a n q u e u i n g t h e o r y p r o b l e m .
A r e l a t e d d i s t r i b u t i o n i s i n t r o d u c e d b y M c D o n a l d a n d X u ( 1 9 9 5 ) a s t h e \ g e n e r a l i z e d b e t a "
1
E x c e p t , o f c o u r s e , t h e u n i f o r m d i s t r i b u t i o n , w h i c h i s a s p e c i a l c a s e o f t h e b e t a .
1
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( G B ) d i s t r i b u t i o n . T h e G B i s d e n e d b y t h e p d f
G B ( x ; a ; b ; c ; p ; q ) =
j a j x
a p 1
( 1 ( 1 c ) ( x = b )
a
)
q 1
b
a p
B ( p ; q ) ( 1 + c ( x = b )
a
)
p + q
f o r 0 < x
a
< b
a
= ( 1 c ) ( 2 )
a n d z e r o o t h e r w i s e w i t h 0 c 1 a n d b ; p a n d q p o s i t i v e . A s i n t h e o r d i n a r y b e t a d i s t r i b u t i o n ,
t h e p a r a m e t e r s p a n d q c o n t r o l s h a p e a n d s k e w n e s s . P a r a m e t e r s a a n d b c o n t r o l \ p e a k e d n e s s "
a n d s c a l e , r e s p e c t i v e l y . G i v e n a = b = 1 , t h e p a r a m e t e r c s h i f t s t h e G B f r o m t h e o r d i n a r y b e t a
d i s t r i b u t i o n ( c = 0 ) t o t h e b e t a - p r i m e d i s t r i b u t i o n ( c = 1 ) .
G o r d y ( f o r t h c o m i n g ) g e n e r a l i z e s t h e b e t a i n a n u n r e l a t e d d i r e c t i o n . T h e \ c o n u e n t h y p e r g e o -
m e t r i c " d i s t r i b u t i o n C H ( p ; q ; s ) i s d e n e d b y t h e p d f
C H ( x ; p ; q ; s ) =
x
p 1
( 1 x )
q 1
e x p ( s x )
B ( p ; q )
1
F
1
( p ; p + q ; s )
f o r 0 < x < 1 : ( 3 )
w h e r e
1
F
1
i s t h e c o n u e n t h y p e r g e o m e t r i c f u n c t i o n d e n e d i n A b r a m o w i t z a n d S t e g u n , e d s ( 1 9 6 8 ,
1 3 . 1 . 2 ) ( h e r e a f t e r c i t e d a s \ A S " ) .
2
G o r d y ( f o r t h c o m i n g ) s h o w s t h a t a b e t a p r i o r a n d g a m m a s i g n a l
g i v e s r i s e t o a C H p o s t e r i o r a n d a p p l i e s t h i s p r o p e r t y t o a u c t i o n t h e o r y .
I n t h i s p a p e r , I u n i f y a n d f u r t h e r g e n e r a l i z e t h e s e t h r e e d i s t r i b u t i o n s . T h e c o n s t a n t o f p r o p o r -
t i o n a l i t y i n t h e n e w p d f i s t h e p r o d u c t o f a b e t a f u n c t i o n a n d a c o m p o u n d c o n u e n t h y p e r g e o m e t r i c
f u n c t i o n . T h e r e f o r e , I d e n o t e t h i s d i s t r i b u t i o n t h e \ c o m p o u n d c o n u e n t h y p e r g e o m e t r i c " ( C C H ) .
T h e C C H i s d e n e d a n d d e s c r i b e d i n S e c t i o n 1 . S p e c i a l c a s e s a r e d i s c u s s e d i n S e c t i o n 2 . I n p a r t i c -
u l a r , I s h o w t h a t t h e b e t a , G B , G H , C H a n d g a m m a d i s t r i b u t i o n s a r e a l l s p e c i a l c a s e s o f t h e C C H .
E m p i r i c a l a p p l i c a t i o n s t o m e a s u r e s o f h o u s e h o l d l i q u i d a s s e t s a r e p r o v i d e d i n S e c t i o n 3 .
2
T h i s f u n c t i o n i s d e n o t e d t h e r e a s M a n d r e f e r r e d t o a s t h e \ d e g e n e r a t e " h y p e r g e o m e t r i c f u n c t i o n i n G r a d s h t e y n
a n d R y z h i k ( 1 9 6 5 , 9 . 2 1 0 . 1 ) . I n M a t h e m a t i c a , i t i s t h e H y p e r g e o m e t r i c 1 F 1 .
2
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1 D e n i t i o n o f t h e C C H
I d e n e t h e C C H b y t h e d e n s i t y f u n c t i o n
C C H ( x ; p ; q ; r ; s ; ; ) =
x
p 1
( 1 x )
q 1
( + ( 1 ) x )
r
e x p ( s x )
B ( p ; q ) H ( p ; q ; r ; s ; ; )
f o r 0 < x < 1 = ( 4 )
f o r 0 < p , 0 < q , r 2 < , s 2 < , 0 1 , a n d 0 < .
3
T h e f u n c t i o n H i s g i v e n b y
H ( p ; q ; r ; s ; ; ) =
p
e x p ( s = )
1
( q ; r ; p + q ; s = ; 1 ) ( 5 )
w h e r e
1
i s t h e c o n u e n t h y p e r g e o m e t r i c f u n c t i o n o f t w o v a r i a b l e s d e n e d i n G r a d s h t e y n a n d
R y z h i k ( 1 9 6 5 , 9 . 2 6 1 . 1 ) b y
1
( ; ; ; x ; y ) =
1
X
m = 0
1
X
n = 0
( )
m + n
( )
n
( )
m + n
m ! n !
x
m
y
n
( 6 )
a n d w h e r e ( a )
k
i s P o c h h a m m e r ' s n o t a t i o n , i . e . , ( a )
0
= 1 ; ( a )
1
= a ; ( a )
k
= ( a )
k 1
( a + k 1 ) . F o r
c o n v e n i e n c e i n e x p o s i t i o n , I r e f e r t o H a s a \ c o m p o u n d c o n u e n t h y p e r g e o m e t r i c " f u n c t i o n r a t h e r
t h a n a s a \ c o n u e n t h y p e r g e o m e t r i c f u n c t i o n o f t w o v a r i a b l e s . "
I n A p p e n d i x A , I s h o w t h a t e q u a t i o n ( 4 ) i n t e g r a t e s t o o n e . T o g u a r a n t e e t h a t t h e C C H d i s t r i -
b u t i o n i s w e l l - d e n e d e v e r y w h e r e o n t h e p a r a m e t e r s p a c e , I p r o v e i n A p p e n d i x C t h e t h e o r e m
T h e o r e m 1
F o r a l l ( p ; q ; r ; s ; ; ) s u c h t h a t p > 0 , q > 0 , r 2 < , s 2 < , 0 < 1 a n d > 0 , H ( p ; q ; r ; s ; ; ) i s
a n i t e p o s i t i v e r e a l n u m b e r .
I t i s s t r a i g h t f o r w a r d t o c h e c k t h a t t h e m o m e n t g e n e r a t i n g f u n c t i o n f o r t h e C C H i s g i v e n b y
M ( t ) =
H ( p ; q ; r ; s t ; ; )
H ( p ; q ; r ; s ; ; )
3
= 0 i s h a n d l e d a s a s p e c i a l c a s e . S e e t h e U H d i s t r i b u t i o n i n S e c t i o n 2 .
3
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a n d t h e k t h o r d e r m o m e n t s a r e g i v e n b y
E ( X
k
) =
( p )
k
( p + q )
k
H ( p + k ; q ; r ; s ; ; )
H ( p ; q ; r ; s ; ; )
: ( 7 )
T h e o r e m 1 i s s u c i e n t t o g u a r a n t e e t h a t a l l m o m e n t s o f X e x i s t .
G i v e n t h e r e s t r i c t i o n s o n t h e p a r a m e t e r s , t h e
1
f u n c t i o n i n e q u a t i o n ( 5 ) c a n a l w a y s b e e x -
p r e s s e d a s a n i n n i t e s e r i e s i n w h i c h a l l t e r m s a r e n o n - n e g a t i v e ( s e e A p p e n d i x B ) . T h e r e f o r e , t h e
H f u n c t i o n c a n b e c a l c u l a t e d w i t h o u t n u m e r i c a l r o u n d - o p r o b l e m s . D e s p i t e i t s a p p a r e n t c o m -
p l e x i t y , i t i s q u i c k l y c a l c u l a t e d o v e r m o s t o f t h e r e l e v a n t p a r a m e t e r s p a c e .
4
C o m p u t a t i o n t i m e
d e c r e a s e s w i t h p ; a n d a n d i n c r e a s e s w i t h q ; j r j a n d j s j . I t a p p e a r s t h a t a n d j o i n t l y h a v e
t h e l a r g e s t e e c t . T o t a k e a n e a s y e x a m p l e , c o m p u t a t i o n o f H ( p = 2 0 ; q = 2 ; r = 5 ; s = 0 ; = 1 ; = 1 )
t o 1 4 p l a c e s a c c u r a c y t a k e s 0 . 0 0 0 2 s e c o n d s o n a S p a r c S t a t i o n 1 0 . T o t a k e a m o r e d i c u l t e x a m p l e ,
H ( 2 ; 2 0 ; 1 5 ; 1 0 ; 0 : 0 1 ; 0 : 0 1 ) r e q u i r e s 0 . 6 s e c o n d s . T o t h e e x t e n t t h a t t h e C C H i s e m p l o y e d t o g e n e r a l -
i z e t h e b e t a d i s t r i b u t i o n , r a t h e r t h a n t h e b e t a - p r i m e d i s t r i b u t i o n , c o m p u t a t i o n a l l y e a s y c a s e s w i l l
p r e d o m i n a t e i n e m p i r i c a l a p p l i c a t i o n s .
F i g u r e s 1 a , 1 b a n d 1 c p l o t t h e C C H p d f f o r a v a r i e t y o f p a r a m e t e r v a l u e s . T h e g u r e s s h o w t h a t
t h e r o l e o f p a r a m e t e r s p a n d q i n t h e C C H i s m u c h t h e s a m e a s i n t h e o r d i n a r y b e t a d i s t r i b u t i o n .
P a r a m e t e r r e s c a l e s t h e d i s t r i b u t i o n f o r l o n g e r o r s h o r t e r s u p p o r t . T h e r e m a i n i n g p a r a m e t e r s r ,
s a n d \ s q u e e z e " t h e d e n s i t y f u n c t i o n t o t h e l e f t o r r i g h t . W h i l e t h e s h a p e s p o r t r a y e d i n t h e s e
g u r e s a r e q u a l i t a t i v e l y f a m i l i a r f r o m t h e b e t a d i s t r i b u t i o n , t h e C C H c a n a l s o t a k e o n a w i d e r a n g e
o f m u l t i - m o d a l o r l o n g - t a i l e d s h a p e s w h i c h t h e b e t a c a n n o t . E x a m p l e s a r e p r e s e n t e d i n F i g u r e 2 .
T h e p a r a m e t e r s a l l o w s f o r a c o n v e n i e n t m e t h o d o f c o n d i t i o n i n g t h e C C H d i s t r i b u t i o n o n
e x o g e n o u s v a r i a b l e s . T h e b o t t o m p a n e l o f F i g u r e 1 b s h o w s t h a t i n c r e a s i n g ( d e c r e a s i n g ) s s q u e e z e s
t h e d i s t r i b u t i o n t o t h e l e f t ( r i g h t ) . I n A p p e n d i x D i t i s p r o v e d t h a t t h e m e a n c h a n g e s m o n o t o n i c a l l y
4
S o f t w a r e i n C a n d M A T L A B i s a v a i l a b l e u p o n r e q u e s t .
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F i g u r e 1 a : C o m p o u n d C o n u e n t H y p e r g e o m e t r i c P D F s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7Varying p
x
f ( x )
q=4, r=10, s=0, ν=1, θ=0.75
p=0.5
p=1
p=6
p=12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7Varying q
x
f ( x )
p=6, r=10, s=0, ν=1, θ=0.75
q=0.5
q=1q=6
q=12
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F i g u r e 1 b : C o m p o u n d C o n u e n t H y p e r g e o m e t r i c P D F s ( c o n t i n u e d )
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5Varying r
x
f ( x )
p=6, q=4, s=0, ν=1, θ=0.75
r=40
r=20r=0
r=−20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3Varying s
x
f ( x )
p=6, q=4, r=p+q, ν=1, θ=0.75
s=6 s=−6s=0
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F i g u r e 1 c : C o m p o u n d C o n u e n t H y p e r g e o m e t r i c P D F s ( c o n t i n u e d )
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
Varying ν
x
f ( x )
p=6, q=4, r=p+q, s=0, θ=0.75
ν=1/3
ν=2/3
ν=1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
Varying θ
x
f ( x )
θ=0.2
θ=0.5θ=1
θ=2
θ=5
p=6, q=4, r=p+q, s=0, ν=1
7
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F i g u r e 2 : M u l t i - m o d a l a n d L o n g - T a i l e d C C H P D F s
0 0.2 0.4 0.6 0.8 10
1
2
3
4Example 1
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2Example 2
0 0.2 0.4 0.6 0.8 10
1
2
3
4Example 3
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5Example 4
0 0.2 0.4 0.6 0.8 10
1
2
3
4Example 5
0 0.2 0.4 0.6 0.8 10
1
2
3Example 6
P a r a m e t e r v a l u e s :
E x a m p l e 1 : p = 2 ; q = 3 ; r = 1 5 ; s = 2 0 ; = 1 ; = 0 : 2 5 .
E x a m p l e 2 : p = 5 ; q = 1 : 2 ; r = 3 7 ; s = 2 6 ; = 1 ; = 0 : 3 6 .
E x a m p l e 3 : p = 1 2 ; q = 0 : 4 ; r = 2 5 ; s = 1 0 ; = 1 ; = 0 : 1 4 .
E x a m p l e 4 : p = 0 : 8 ; q = 0 : 7 ; r = 1 5 ; s = 1 0 ; = 1 ; = 2 .
E x a m p l e 5 : p = 0 : 2 ; q = 2 5 ; r = 4 0 ; s = 1 0 ; = 1 ; = 0 : 4 .
E x a m p l e 6 : p = 4 ; q = 0 : 2 ; r = 0 ; s = 1 0 ; = 1 ; = 2 .
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w i t h s , i . e . ,
P r o p o s i t i o n 2
d E ( X )
d s
= V a r ( X ) < 0 :
T h e r e f o r e , i f z
t
i s a v e c t o r o f e x o g e n o u s v a r i a b l e s , i t m i g h t b e a s s u m e d t h a t X
t
C C H ( p ; q ; r ; s
t
; ; )
f o r s
t
= z
t
. T h e r e g r e s s i o n c o e c i e n t s a r e e a s i l y e s t i m a t e d a l o n g w i t h t h e x e d p a r a m e t e r s
( p ; q ; r ; ; ) b y m a x i m u m l i k e l i h o o d . T h e d e r i v a t i v e o f t h e f u n c t i o n H w i t h r e s p e c t t o s h a s a
s i m p l e a n a l y t i c e x p r e s s i o n , s o t h e c o m p u t a t i o n a l b u r d e n o f a d d i n g v a r i a b l e s t o z
t
i s n o t e s p e c i a l l y
l a r g e .
L e m m a 3
d
d s
H ( p ; q ; r ; s ; ; ) =
p
p + q
H ( p + 1 ; q ; r ; s ; ; ) :
T h e p r o o f i s i n A p p e n d i x D .
L i k e t h e b e t a , t h e C C H d i s p l a y s a s i m p l e r e e c t i o n p r o p e r t y . I f X C C H ( p ; q ; r ; s ; ; ) a n d
i f Y = 1 = X , t h e n Y C C H ( q ; p ; r ; s ; ; 1 = ) . L i k e t h e C H , t h e C C H m a y b e u s e f u l i n
B a y e s i a n d e c i s i o n p r o b l e m s , b e c a u s e i t i s c o n j u g a t e f o r g a m m a d i s t r i b u t e d s i g n a l s . I f t h e p r i o r f o r
a n u n k n o w n r a n d o m v a r i a b l e X i s t h e C C H ( p ; q ; r ; s ; ; ) a n d s i g n a l W j ( X = x ) G a ( ; x ) , t h e n
t h e p o s t e r i o r d i s t r i b u t i o n o f X j ( W = w ) i s C C H ( p + ; q ; r ; s + w ; ; ) .
2 S p e c i a l c a s e s
G e n e r a l i z e d B e t a : W h e n s = 0 , r = p + q , = ( 1 c ) = b , a n d = 1 c , t h e C C H s i m p l i e s t o
t h e G B w i t h a = 1 . T h e C C H d i s t r i b u t i o n c a n e a s i l y b e e x t e n d e d t o a c c o m m o d a t e a p e a k e d n e s s
p a r a m e t e r a a s w e l l , w h i c h i s i g n o r e d h e r e t o s i m p l i f y e x p o s i t i o n . T h e r e s t r i c t i o n i s r e l a t i v e l y m i n o r ,
b e c a u s e i f X G B ( a ; b ; c ; p ; q ) t h e n X c a n b e r e s c a l e d a s
~
X X
a
C C H ( p ; q ; r = p + q ; s = 0 ; =
( 1 c ) = b ; = 1 c ) .
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G a u s s H y p e r g e o m e t r i c : W h e n s = 0 a n d = 1 , t h e C C H ( p ; q ; r ; s = 0 ; = 1 ; ) s i m p l i e s t o
t h e G H ( p ; q ; r ; ) f o r = ( 1 ) = . T h e f u n c t i o n H s i m p l i e s t o
H ( p ; q ; r ; s = 0 ; = 1 ; ) =
r
2
F
1
( r ; q ; p + q ; 1 ) =
2
F
1
( r ; p ; p + q ; ( 1 ) = ) ( 8 )
w h e r e t h e l a s t t r a n s f o r m a t i o n f o l l o w s f r o m A S 1 5 . 3 . 4 .
C o n u e n t H y p e r g e o m e t r i c : W h e n = 1 a n d = 1 , C C H ( p ; q ; r ; s ; = 1 ; = 1 ) s i m p l i e s
t o t h e c o n u e n t h y p e r g e o m e t r i c d i s t r i b u t i o n C H ( p ; q ; s ) . I t i s c l e a r f r o m t r a n s f o r m a t i o n ( T 4 ) i n
A p p e n d i x B t h a t H ( p ; q ; r ; s ; = 1 ; = 1 ) =
1
F
1
( p ; p + q ; s ) .
G a m m a : W h e n = 0 a n d s > 0 , C C H ( p ; q ; r ; s ; = 0 ; ) s i m p l i e s t o t h e g a m m a d i s t r i b u t i o n
G a ( p ; s ) . W h e r e a s t h e G B d i s t r i b u t i o n i n c l u d e s t h e g a m m a d i s t r i b u t i o n a s a l i m i t i n g c a s e , t h e
C C H i n c l u d e s t h e g a m m a a s a s p e c i a l c a s e .
C o n u e n t H y p e r g e o m e t r i c U : A n i n t e r e s t i n g l i m i t i n g c a s e o f t h e C C H a r i s e s a s ! 0 a n d
! 0 s u c h t h a t ( 1 ) = ! f o r a c o n s t a n t v a l u e s u c h t h a t s = 0 . T h e l i m i t i n g d i s t r i b u -
t i o n i s b a s e d o n t h e \ U " c o n u e n t h y p e r g e o m e t r i c f u n c t i o n d e n e d i n A S 1 3 . 1 . 3 .
5
L e t t h e \ U H "
d i s t r i b u t i o n b e d e n e d b y t h e p d f
U H ( x ; p ; r ; s ; ) =
x
p 1
( 1 + x )
r
e x p ( s x )
( p ) U ( p ; p + 1 r ; s = )
f o r 0 < x < 1 : ( 9 )
I n A p p e n d i x A i t i s s h o w n t h a t t h i s d e n s i t y i n t e g r a t e s t o o n e . M o m e n t s a r e g i v e n b y
E ( X
k
) = ( p )
k
U ( k + p ; k + p + 1 r ; s = ) )
U ( p ; p + 1 r ; s = ) )
:
5
T h i s f u n c t i o n i s d e n o t e d a s t h e \ d e g e n e r a t e " h y p e r g e o m e t r i c f u n c t i o n i n G r a d s h t e y n a n d R y z h i k ( 1 9 6 5 ,
9 . 2 1 0 . 2 ) . I n M a t h e m a t i c a , i t i s t h e H y p e r g e o m e t r i c U .
1 0
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W h e n s = 0 , = 1 a n d r = p + q , t h e U H ( p ; r = p + q ; s = 0 ; = 1 ) s i m p l i e s t o t h e b e t a - p r i m e .
W h e n = 0 o r r = 0 , t h e U H ( p ; r ; s ; ) f o r s > 0 s i m p l i e s t o t h e g a m m a d i s t r i b u t i o n G a ( p ; s ) .
3 E m p i r i c a l a p p l i c a t i o n s
T o d e m o n s t r a t e t h e u s e o f t h e C C H d i s t r i b u t i o n , I m o d e l t w o m e a s u r e s o f h o u s e h o l d l i q u i d a s s e t s :
t h e r a t i o o f h o u s e h o l d l i q u i d a s s e t s t o t o t a l h o u s e h o l d n a n c i a l a s s e t s , a n d t h e l o g o f h o u s e h o l d
l i q u i d a s s e t s . T h e d i s t r i b u t i o n s o f t h e s e v a r i a b l e s a c r o s s h o u s e h o l d s a r e i m p o r t a n t i n u n d e r s t a n d i n g
m o t i v e s f o r s a v i n g a n d t h e d e t e r m i n a n t s o f p o r f o l i o a l l o c a t i o n . F o r e x a m p l e , t h e b u e r - s t o c k m o d e l
o f s a v i n g ( C a r r o l l 1 9 9 2 ) a n d t h e l i f e - c y c l e m o d e l ( s e e , e . g . , P o t e r b a a n d S a m w i c k 1 9 9 7 ) d i e r i n
i m p l i c a t i o n s f o r t h e c r o s s - s e c t i o n a l d i s t r i b u t i o n o f l i q u i d a s s e t s . C r o s s - s e c t i o n a l p a t t e r n s i n p o r t f o l i o
s h a r e a s s i g n e d t o l i q u i d , l o w r i s k a s s e t s m a y p e r m i t t e s t s o r c a l i b r a t i o n s o f m o d e l s o f p o r t f o l i o c h o i c e ,
s u c h a s L a i b s o n ( 1 9 9 7 ) a n d B o d i e , M e r t o n a n d S a m u e l s o n ( 1 9 9 2 ) . M o r e d i r e c t l y , t h e d i s t r i b u t i o n s
o f t h e s e v a r i a b l e s m a y s h e d l i g h t o n t h e e e c t s o f i n c o m e d i s t r i b u t i o n a n d d e m o g r a p h i c t r a n s i t i o n s
o n s a v i n g s a n d o n m a r k e t r i s k - p r e m i a , a n d o n t h e i m p o r t a n c e o f l i q u i d i t y c o n s t r a i n t s i n a g g r e g a t e
c o n s u m p t i o n .
A l l d a t a c o m e f r o m t h e 1 9 9 5 S u r v e y o f C o n s u m e r F i n a n c e s . T h e S C F p r o v i d e s d a t a o n t h e
a s s e t s a n d l i a b i l i t i e s o f U . S . h o u s e h o l d s ( s e e K e n n i c k e l l , S t a r r - M c C l u e r a n d S u n d e n ( 1 9 9 7 ) f o r a
g e n e r a l d e s c r i p t i o n ) . I n o r d e r t o p r o v i d e a d e q u a t e r e p r e s e n t a t i o n o f t h e h i g h l y s k e w e d d i s t r i b u t i o n
o f a s s e t s , t h e S C F o v e r s a m p l e s w e a l t h y h o u s e h o l d s , s o i n a l l c a l c u l a t i o n s b e l o w I a p p l y S C F s a m p l e
w e i g h t s t o o b t a i n u n b i a s e d e s t i m a t e s o f t h e p o p u l a t i o n d i s t r i b u t i o n . W e i g h t s a r e n o r m a l i z e d t o
m e a n o n e t o a v o i d b i a s i n g t h e s i z e o f t e s t s t a t i s t i c s .
6
I d e n e l i q u i d a s s e t s , L I Q , a s t h e s u m o f c h e c k i n g d e p o s i t s , s a v i n g s d e p o s i t s , m o n e y m a r k e t a n d
6
P a r a m e t e r s e s t i m a t e d o n t h e u n w e i g h t e d s a m p l e s d o d i e r f r o m t h e r e s u l t s r e p o r t e d b e l o w , b u t a r e q u a l i t a t i v e l y
s i m i l a r . I n p a r t i c u l a r , L R t e s t s t a t i s t i c s y i e l d t h e s a m e c o n c l u s i o n s .
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m o n e y m a r k e t m u t u a l f u n d d e p o s i t s , a n d c a l l a c c o u n t s a t b r o k e r a g e s . T o t a l n a n c i a l a s s e t s , F I N ,
i n c l u d e s L I Q , s t o c k s , b o n d s , t r u s t s , c a s h v a l u e o f w h o l e l i f e i n s u r a n c e , a n d o t h e r n a n c i a l a s s e t s .
7
W e i g h t e d m e d i a n a n d m e a n s a m p l e v a l u e s a r e $ 1 , 6 0 0 a n d $ 1 3 , 2 6 1 f o r L I Q a n d $ 1 0 , 0 0 0 a n d $ 8 8 , 6 2 9
f o r F I N . R o u g h l y 9 % o f h o u s e h o l d s h a v e n o n a n c i a l a s s e t s a n d r o u g h l y 1 3 % h a v e n o l i q u i d a s s e t s .
R o u g h l y 2 0 % h a v e p o s i t i v e l i q u i d a s s e t s b u t n o o t h e r n a n c i a l a s s e t s . T o b o u n d t h e l i q u i d i t y r a t i o
i n ( 0 ; 1 ) , I a s s u m e h o u s e h o l d i h a s a n a d d i t i o n a l c
i
d o l l a r s i n \ p o c k e t c a s h , " a n d a n o t h e r f
i
d o l l a r s
i n s m a l l l o a n s t o f r i e n d s , w h e r e c
i
a n d f
i
a r e i i d d r a w s f r o m a U n i f o r m ( 0 ; 1 5 0 ) d i s t r i b u t i o n . I t h e n
d e n e L I Q R A T
i
( L I Q
i
+ c
i
) = ( F I N + c
i
+ f
i
) .
8
T h e t o p p a n e l o f F i g u r e 3 s h o w s a U - s h a p e d
h i s t o g r a m f o r L I Q R A T . R o u g h l y h a l f o f U . S . h o u s e h o l d s m a i n t a i n e i t h e r a v e r y l o w o r v e r y h i g h
p e r c e n t a g e o f h o u s e h o l d n a n c i a l a s s e t s i n l i q u i d d e p o s i t s .
T a b l e 1 p r e s e n t s t h e r e s u l t s o f m a x i m u m l i k e l i h o o d e s t i m a t i o n o f d i s t r i b u t i o n p a r a m e t e r s f o r
L I Q R A T u s i n g t h e b e t a , c o n u e n t h y p e r g e o m e t r i c , G a u s s h y p e r g e o m e t r i c a n d C C H d i s t r i b u t i o n s .
9
B e c a u s e 0 a n d 1 a r e n a t u r a l b o u n d s f o r L I Q R A T , i s x e d t o 1 .
1 0
T h e e s t i m a t e d d i s t r i b u t i o n s a r e
p l o t t e d i n t h e b o t t o m p a n e l o f F i g u r e 3 . A l l o f t h e e s t i m a t e d d i s t r i b u t i o n s a r e U - s h a p e d , b u t t h e y
d o d i e r v i s i b l y . F u r t h e r m o r e , L R t e s t s s t r o n g l y r e j e c t ( a t w e l l b e y o n d 1 % l e v e l s ) t h e e q u i v a l e n c e
o f t h e r e s t r i c t e d d i s t r i b u t i o n s t o t h e C C H . T h e C C H t h u s o e r s a s i g n i c a n t l y b e t t e r t t o t h e
d a t a t h a n b e t a , C H o r G H .
I t m i g h t b e e x p e c t e d t h a t L I Q R A T w i l l d e p e n d o n h o u s e h o l d c h a r a c t e r i s t i c s . H i g h e r i n c o m e
h o u s e h o l d s m a y h a v e l e s s n e e d t o k e e p s a v i n g s i n a h i g h l y l i q u i d f o r m . H i g h e r l e v e l s o f e d u c a t i o n
7
T h e s e a r e s t a n d a r d d e n i t i o n s u s e d i n a n a l y s i s o f t h e S C F d a t a . S e e t h e U R L
< h t t p : / / w w w . b o g . f r b . f e d . u s / p u b s / o s s / o s s 2 / 9 5 / c o d e b k 9 5 p t 5 . h t m l > a t t h e F e d e r a l R e s e r v e B o a r d ' s w e b
s i t e .
8
T h i s p r o c e d u r e c a u s e s e s t i m a t e d p a r a m e t e r s t o v a r y w i t h e a c h s e t o f r a n d o m d r a w s . I h a v e r u n t h e p r o g r a m
m a n y t i m e s , h o w e v e r , a n d n d t h a t r e s u l t s a p p e a r t o b e s t a b l e .
9
S t a n d a r d e r r o r s a r e o b t a i n e d u s i n g t h e m e t h o d o f B e r n d t , H a l l , H a l l a n d H a u s m a n ( 1 9 7 4 ) . I a b s t r a c t f r o m
v a r i a t i o n a s s o c i a t e d w i t h t h e i n c l u s i o n o f i m p u t e d v a l u e s i n t h e S C F b y u s i n g o n l y t h e r s t s e t o f i m p l i c a t e s . R u b i n
( 1 9 8 7 ) d i s c u s s e s c o r r e c t i o n t o t h e s t a n d a r d e r r o r s f o r m u l t i p l y i m p u t e d d a t a . I n t h i s a p p l i c a t i o n , t h e c o r r e c t i o n s h o u l d
b e q u i t e s m a l l a n d i s i g n o r e d .
1 0
T h e G B i s n o t i n c l u d e d i n t h i s c o m p a r i s o n b e c a u s e i t i s e q u i v a l e n t t o t h e b e t a w h e n r e s t r i c t i o n s a = 1 a n d
c = b = 1 = 0 a r e i m p o s e d .
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F i g u r e 3 : D i s t r i b u t i o n o f L i q u i d A s s e t R a t i o
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35Histogram
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5Estimated PDFs
LIQRAT
Beta
CH
GH
CCH
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T a b l e 1 : E s t i m a t e d D i s t r i b u t i o n a l P a r a m e t e r s f o r L i q u i d i t y R a t i o
y
B e t a C H G H
z
C C H
p 0 : 5 0 1 0 : 6 4 6 1 : 0 6 2 0 : 9 3 9
( 0 : 0 1 1 ) ( 0 : 0 1 7 ) ( 0 : 0 5 1 ) ( 0 : 0 3 4 )
q 0 : 6 8 7 0 : 4 7 7 0 : 5 2 6 0 : 6 4 9
( 0 : 0 1 1 ) ( 0 : 0 1 8 ) ( 0 : 0 1 5 ) ( 0 : 0 2 6 )
r 1 : 0 5 2 6 : 2 9 6
( 0 : 0 5 0 ) ( 2 : 8 1 9 )
s 1 : 5 8 5 5 : 7 3 2
( 0 : 1 0 9 ) ( 1 : 8 6 7 )
0 : 0 3 2 0 : 2 8 5
( 0 : 0 0 7 ) ( 0 : 0 8 6 )
L o g L i k : 8 3 8 : 0 8 9 3 9 : 8 7 1 0 2 2 : 1 1 0 4 2 : 5
y : S t a n d a r d e r r o r s i n p a r e n t h e s e s . 4 2 9 9 o b s e r v a t i o n s i n s a m p l e .
S a m p l e w e i g h t s a r e n o r m a l i z e d t o h a v e m e a n o n e . i s x e d a t
1
z : T h e p a r a m e t e r i n t h e s t a n d a r d f o r m f o r t h e G H i s g i v e n b y
= ( 1 ) =
m a y b e a s s o c i a t e d w i t h g r e a t e r s o p h i s t i c a t i o n i n n a n c i a l m a t t e r s , a n d s o a g r e a t e r p r o p e n s i t y
t o h o l d a s s e t s o t h e r t h a n l i q u i d d e p o s i t s . C o m p a r e d t o h o u s e h o l d s i n t h e 3 5 - 6 5 a g e g r o u p , o l d e r
a n d y o u n g e r h o u s e h o l d s a r e l i k e l y t o h a v e h i g h e r L I Q R A T . O l d e r h o u s e h o l d s m a y b e m o r e r i s k
a v e r s e , a n d s o p r e f e r s a f e r a s s e t a l l o c a t i o n s . Y o u n g e r h o u s e h o l d s m a y b e m o r e l i k e l y t o f a c e i n c o m e
u n c e r t a i n t y a n d t o h a v e n e a r - t e r m g o a l s f o r s a v i n g ( e . g . , d o w n p a y m e n t f o r h o m e p u r c h a s e ) , a n d
t h u s t o h a v e g r e a t e r d e m a n d f o r b o t h s a f e t y a n d l i q u i d i t y . I n T a b l e 2 , I p r e s e n t r e s u l t s f r o m t h e
e s t i m a t i o n o f t h e c o n d i t i o n a l d i s t r i b u t i o n L I Q R A T
i
C C H ( p ; q ; r ; s
i
; 1 ; ) . I a s s u m e s
i
= z
i
,
w h e r e t h e z
i
i n c l u d e h o u s e h o l d i n c o m e , a n d d u m m y v a r i a b l e s f o r t h e a g e a n d l e v e l o f e d u c a t i o n o f
t h e h e a d o f h o u s e h o l d , a n d i s a v e c t o r o f c o e c i e n t s .
R e s u l t s a r e p r e s e n t e d i n T a b l e 2 . E s t i m a t e s f o r p ; q ; r a n d a r e c l o s e t o t h o s e f o r t h e u n c o n d i -
t i o n a l C C H ( l a s t c o l u m n o f T a b l e 1 ) . T h e c o e c i e n t s a r e i n d i v i d u a l l y a n d j o i n t l y d i e r e n t f r o m
z e r o a t t h e 1 % c o n d e n c e l e v e l . A s p r e d i c t e d , h i g h e r i n c o m e a n d h i g h e r l e v e l o f e d u c a t i o n s h i f t t h e
d i s t r i b u t i o n t o t h e l e f t . T h e y o u n g ( A G E < 3 5 ) a n d t h e o l d ( A G E > 6 5 ) m a i n t a i n h i g h e r l i q u i d i t y
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r a t i o s t h a n t h o s e i n b e t w e e n .
T a b l e 2 : E s t i m a t e d C o n d i t i o n a l D i s t r i b u t i o n f o r L i q u i d i t y R a t i o
y
C o e S . E .
p 0 : 9 4 5 0 0 : 0 3 4 7
q 0 : 6 4 9 7 0 : 0 2 6 2
r 4 : 8 9 5 3 2 : 0 6 5 6
0 : 2 4 8 5 0 : 0 7 9 6
C O N S T A N T 7 : 6 7 7 3 1 : 5 4 3 0
L O G ( 1 + I N C O M E ) 0 : 3 7 7 9 0 : 0 2 4 3
A G E < 3 5 0 : 5 6 7 5 0 : 0 9 7 3
A G E > 6 5 0 : 2 4 4 1 0 : 0 9 2 0
E D U 1 ( N O H S D I P L O M A )
z
1 : 1 6 8 6 0 : 1 1 4 6
E D U 2 ( H S D I P L O M A )
z
0 : 6 6 5 9 0 : 0 9 8 7
E D U 3 ( S O M E C O L L E G E )
z
0 : 4 7 0 2 0 : 1 1 3 1
L o g L i k : 1 1 9 4 : 7 5
y : 4 2 9 9 o b s e r v a t i o n s i n s a m p l e . S a m p l e w e i g h t s a r e n o r m a l i z e d
t o b e m e a n o n e . i s x e d a t 1 .
z : C O L L E G E D I P L O M A i s t h e r e f e r e n c e c a t e g o r y f o r t h e e d -
u c a t i o n d u m m i e s .
F i g u r e 4 d e m o n s t r a t e s t h e e e c t o f c o n d i t i o n i n g t h r o u g h s o n t h e C C H d i s t r i b u t i o n . S u b j e c t
1 i s a 2 5 y e a r o l d h i g h s c h o o l d r o p - o u t e a r n i n g $ 2 0 , 0 0 0 p e r y e a r ; s u b j e c t 2 i s 7 0 y e a r s o l d , h a d
s o m e c o l l e g e , a n d e a r n s $ 4 0 , 0 0 0 p e r y e a r ; a n d s u b j e c t 3 i s a 4 5 y e a r o l d c o l l e g e g r a d u a t e e a r n i n g
$ 1 0 0 , 0 0 0 p e r y e a r . P a r a m e t e r s a r e t a k e n f r o m T a b l e 2 . A l l t h r e e d i s t r i b u t i o n s a r e U - s h a p e d , b u t
t h a t o f S u b j e c t 1 i s m o s t t i l t e d t o w a r d s h i g h L I Q R A T a n d t h a t o f S u b j e c t 3 i s m o s t t i l t e d t o w a r d s
l o w L I Q R A T . T h e m e a n s o f t h e s e t h r e e d i s t r i b u t i o n s a r e 0 . 3 1 , 0 . 2 2 , a n d 0 . 1 6 , r e s p e c t i v e l y .
T h e s e c o n d m e a s u r e o f h o u s e h o l d l i q u i d i t y i s t h e l o g o f t o t a l h o u s e h o l d l i q u i d a s s e t s ( i n c l u d i n g
t h e r a n d o m l y d r a w n a m o u n t o f p o c k e t c a s h ) . T h i s i s d e n e d a s L O G L I Q
i
l o g ( 1 + L I Q
i
+ c
i
) .
T h e t o p p a n e l o f F i g u r e 5 s h o w s a s i n g l e - p e a k e d h i s t o g r a m f o r L O G L I Q .
T a b l e 3 p r e s e n t s t h e r e s u l t s o f m a x i m u m l i k e l i h o o d e s t i m a t i o n o f d i s t r i b u t i o n p a r a m e t e r s f o r
L O G L I Q u s i n g t h e g a m m a , g e n e r a l i z e d b e t a a n d C C H d i s t r i b u t i o n s . I n p r i n c i p l e , L O G L I Q i s
u n b o u n d e d ( s o = 0 ) , b u t f o r t h e p u r p o s e s o f t h i s e x a m p l e I a l l o w t o b e e s t i m a t e d f r e e l y . I a l s o
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F i g u r e 4 : C o n d i t i o n a l C C H P D F s f o r L i q u i d A s s e t R a t i o
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
LIQRAT
Subject 1Subject 2Subject 3
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i n c l u d e i n T a b l e 3 e s t i m a t e s f o r t w o s p e c i a l c a s e s o f t h e C C H . T h e r s t , l a b e l l e d G B ( s ) , i s t h e G B
e x t e n d e d t o i n c l u d e a n e x p ( s x ) t e r m . T h e s e c o n d , l a b e l l e d C H ( ) , i s t h e C H w i t h t h e ( 1 x )
q 1
t e r m g e n e r a l i z e d t o ( 1 x )
q 1
.
T h e e s t i m a t e d d i s t r i b u t i o n s a r e p l o t t e d i n t h e b o t t o m p a n e l o f F i g u r e 5 . A l l a r e s i n g l e - p e a k e d ,
b u t d i e r e n c e s a r e c l e a r l y v i s i b l e . O n c e a g a i n , L R t e s t s s t r o n g l y r e j e c t ( a t w e l l b e y o n d 1 % l e v e l s )
t h e e q u i v a l e n c e o f t h e r e s t r i c t e d d i s t r i b u t i o n s t o t h e C C H . T h e C H ( ) i s c l o s e s t t o t h e C C H , b o t h i n
l i k e l i h o o d a n d t o t h e e y e . T h e G B a n d G B ( s ) a p p e a r t o u n d e r p e r f o r m e v e n t h e g a m m a d i s t r i b u t i o n .
I n t h e e s t i m a t i o n o f t h e f u l l C C H , r h a s a n e s p e c i a l l y l a r g e s t a n d a r d d e v i a t i o n . T h i s c a n b e s e e n
i n t h e e s t i m a t i o n o f t h e C C H d i s t r i b u t i o n f o r L I Q R A T i n T a b l e 1 a s w e l l , b u t i s m o r e p r o n o u n c e d
h e r e . O n e r e a s o n i s s i m p l y t h a t f a i r l y l a r g e c h a n g e s i n r a r e n e e d e d t o p r o d u c e m o d e r a t e c h a n g e s
i n t h e d i s t r i b u t i o n ( s e e F i g u r e 1 b ) . H o w e v e r , i t a p p e a r s t h a t j o i n t i d e n t i c a t i o n o f r ; a n d s m a y
s o m e t i m e s b e w e a k .
T a b l e 3 : E s t i m a t e d D i s t r i b u t i o n a l P a r a m e t e r s f o r L o g ( L i q u i d i t y )
y
G a m m a G B
z
G B ( s )
z
C H ( )
z
C C H
p 1 0 : 2 8 4 3 7 : 5 6 2 4 9 : 4 1 7 2 : 4 8 0 0 : 4 4 6
( 0 : 0 3 8 ) ( 0 : 3 0 9 ) ( 0 : 1 6 9 ) ( 0 : 1 5 0 ) ( 0 : 3 8 7 )
q 3 : 5 8 9 1 7 : 8 3 6 2 0 : 7 4 1 1 4 : 3 2 6
( 0 : 1 6 8 ) ( 0 : 7 5 0 ) ( 0 : 7 9 1 ) ( 2 : 8 4 8 )
r p + q ] 1 ] 1 5 : 7 9 7
( 1 5 : 4 7 6 )
s 1 : 4 0 5 2 : 9 5 4 1 : 6 8 6 1 : 0 7 9
( 0 : 0 1 2 ) ( 0 : 0 8 3 ) ( 0 : 0 8 6 ) ( 1 : 0 6 4 )
0 : 0 5 5 0 : 0 5 5 0 : 0 5 5 0 : 0 5 5
( 0 : 0 0 2 ) ( 0 : 0 0 1 ) ( 0 : 0 0 1 ) ( 0 : 0 0 1 )
] 1 ] 0 : 2 1 4
( 0 : 1 6 1 )
L o g L i k : 9 5 0 6 : 4 9 7 7 9 : 0 9 6 7 4 : 7 9 2 6 0 : 5 9 2 4 7 : 9
y : S t a n d a r d e r r o r s i n p a r e n t h e s e s . 4 2 9 9 o b s e r v a t i o n s i n s a m p l e . S a m p l e w e i g h t s a r e n o r m a l i z e d
t o h a v e m e a n o n e .
z : T h e G B i s e s t i m a t e d w i t h r e s t r i c t i o n a = 1 ; t h e p a r a m e t e r i s e q u i v a l e n t t o c = b i n t h e s t a n d a r d
f o r m f o r t h e G B . T h e G B ( s ) d i s t r i b u t i o n i s t h e G B e x t e n d e d t o i n c l u d e a n e x p ( s x ) t e r m .
T h e C H ( ) d i s t r i b u t i o n i s t h e C H w i t h t h e ( 1 x )
q 1
t e r m g e n e r a l i z e d t o ( 1 x )
q 1
. I m p o s e d
p a r a m e t e r r e s t r i c t i o n s a r e s h o w n i n h a r d b r a c k e t s .
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F i g u r e 5 : D i s t r i b u t i o n o f L o g L i q u i d A s s e t s
0 2 4 6 8 10 12 14 16 180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4Histogram
0 2 4 6 8 10 12 14 16 180
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2Estimated PDFs
LOGLIQ
Gamma
GB
CH( ν)
CCH
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C o n c l u s i o n
R e c e n t w o r k o n e x t e n d i n g t h e b e t a d i s t r i b u t i o n h a s g i v e n r i s e t o t h r e e n o n - n e s t e d g e n e r a l i z a t i o n s :
t h e G a u s s h y p e r g e o m e t r i c d i s t r i b u t i o n o f A r m e r o a n d B a y a r r i ( 1 9 9 4 ) , t h e g e n e r a l i z e d b e t a d i s -
t r i b u t i o n o f M c D o n a l d a n d X u ( 1 9 9 5 ) , a n d t h e c o n u e n t h y p e r g e o m e t r i c d i s t r i b u t i o n o f G o r d y
( f o r t h c o m i n g ) . I n t h i s p a p e r , t h e s e d i v e r g e n t s t r a n d s a r e u n i e d i n t o a n e w s i x p a r a m e t e r d i s t r i b u -
t i o n , w h i c h I d e n o t e t h e \ c o m p o u n d c o n u e n t h y p e r g e o m e t r i c " d i s t r i b u t i o n . P l o t s o f t h e C C H f o r
a v a r i e t y o f p a r a m e t e r v a l u e s s h o w t h a t t h e C C H d e n s i t y c a n t a k e n o t o n l y t h e f a m i l i a r v a r i e t y o f
U - s h a p e d a n d s i n g l e - p e a k e d f o r m s o f t h e b e t a d i s t r i b u t i o n , b u t a l s o a v a r i e t y o f m u l t i - m o d a l a n d
l o n g - t a i l e d f o r m s .
U s e o f t h e C C H i s d e m o n s t r a t e d w i t h t w o m e a s u r e s o f h o u s e h o l d l i q u i d a s s e t s . I n e a c h e x a m p l e ,
e s t i m a t e d C C H d i s t r i b u t i o n s a r e q u a l i t a t i v e l y s i m i l a r t o e s t i m a t e s u s i n g m o r e r e s t r i c t i v e d i s t r i b u -
t i o n s , b u t t h e a d d i t i o n a l p r e c i s i o n i n t i s h i g h l y s i g n i c a n t s t a t i s t i c a l l y a n d c l e a r l y v i s i b l e i n p l o t s
o f t h e t t e d d e n s i t i e s .
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A I n t e g r a t i n g t h e C C H d e n s i t y
L e t A b e t h e i n t e g r a l
A
Z
1 =
0
x
p 1
( 1 x )
q 1
( + ( 1 ) x )
r
e x p ( s x ) d x ( 1 0 )
w h e r e 0 < p , 0 < q , r 2 < , s 2 < , 0 < 1 , a n d 0 < . M a k e t h e c h a n g e o f v a r i a b l e u = 1 x
t o g e t
A =
p
e x p ( s = )
Z
1
0
( 1 u )
p 1
u
q 1
( 1 ( 1 ) u )
r
e x p ( ( s = ) u ) d u : ( 1 1 )
T a k e t h e s e r i e s e x p a n s i o n o f t h e e x p o n e n t i a l f u n c t i o n t o g e t
A =
p
e x p ( s = )
1
X
m = 0
( s = )
m
m !
Z
1
0
u
q + m 1
( 1 u )
p 1
( 1 ( 1 ) u )
r
d u
=
p
e x p ( s = )
1
X
m = 0
( s = )
m
m !
( q + m ) ( p )
( p + q + m )
2
F
1
( r ; q + m ; p + q + m ; 1 )
= B ( p ; q )
p
e x p ( s = )
1
X
m = 0
( q )
m
( p + q )
m
m !
s
m
1
X
n = 0
( r )
n
( q + m )
n
( p + q + m )
n
n !
( 1 )
n
= B ( p ; q )
p
e x p ( s = )
1
X
m = 0
1
X
n = 0
( q )
m + n
( r )
n
( p + q )
m + n
m ! n !
s
m
( 1 )
n
w h e r e ( a )
i
i s P o c h h a m m e r ' s n o t a t i o n , i . e . , ( a )
0
= 1 ; ( a )
1
= a ; ( a )
i
= ( a )
i 1
( a + i 1 ) . T h e s e c o n d
l i n e f o l l o w s f r o m t h e r s t b y A S 1 5 . 3 . 1 , a n d t h e t h i r d l i n e f r o m t h e s e c o n d b y A S 1 5 . 1 . 1 . B y t h e
d e n i t i o n o f H ( p ; q ; r ; s ; ; ) i n e q u a t i o n ( 5 ) , t h e l a s t l i n e g i v e s A = B ( p ; q ) H ( p ; q ; r ; s ; ; ) , w h i c h
g u a r a n t e e s t h a t e q u a t i o n ( 4 ) i n t e g r a t e s t o u n i t y .
F o r t h e U H l i m i t i n g c a s e , s o l v e
A =
Z
1
0
x
p 1
( 1 + x )
r
e x p ( s x ) d x
2 0
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b y t h e c h a n g e o f v a r i a b l e u = x . T h e s o l u t i o n i s
A =
r
Z
1
0
u
p 1
( 1 + u )
r
e x p ( ( s = ) u ) d u
=
r
( p ) U ( p ; p + 1 r ; s = ) ( 1 2 )
w h e r e U i s t h e \ U " c o n u e n t h y p e r g e o m e t r i c f u n c t i o n ( s e e A S 1 3 . 1 . 3 a n d A S 1 3 . 2 . 5 ) .
B T r a n s f o r m a t i o n r u l e s f o r t h e
1
f u n c t i o n
I n t h e r e m a i n i n g a p p e n d i x e s , i t w i l l b e c o n v e n i e n t t o m a k e u s e o f a l t e r n a t i v e e x p r e s s i o n s f o r t h e
n e s t e d i n n i t e s e r i e s i n t h e d e n i t i o n e q u a t i o n ( 6 ) f o r
1
. F o r 0 < < a n d 0 y < 1 ,
1
( ; ; ; x ; y ) =
1
X
m = 0
( )
m
( )
m
x
m
m !
2
F
1
( ; + m ; + m ; y ) ( T 1 )
= e x p ( x )
1
X
m = 0
( )
m
( )
m
( x )
m
m !
2
F
1
( ; ; + m ; y ) ( T 2 )
=
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( + n ; + n ; x ) ( T 3 )
= e x p ( x )
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( ; + n ; x ) : ( T 4 )
T h e s e t r a n s f o r m a t i o n s a r e d e r i v e d u s i n g s t r a i g h t f o r w a r d m a n i p u l a t i o n s o f t h e n e s t e d i n n i t e s e r i e s .
A l l t h a t i s n e e d e d i s t h e r u l e ( a )
m + n
= ( a )
m
( a + m )
n
a n d t h e K u m m e r t r a n s f o r m a t i o n
1
F
1
( a ; b ; x ) =
e x p ( x )
1
F
1
( b a ; b ; x ) ( s e e A S 1 3 . 1 . 2 7 ) .
W h e n y < 0 , a p p l y t h e t r a n s f o r m a t i o n r u l e ( A S 1 5 . 3 . 4 )
2
F
1
( a ; b ; c ; y ) = ( 1 y )
a
2
F
1
( a ; c b ; c ; y = ( y 1 ) )
2 1
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t o t h e f o r m ( T 2 ) t o g e t
1
( ; ; ; x ; y ) = e x p ( x )
1
X
m = 0
( )
m
( )
m
( x )
m
m !
( 1 y )
2
F
1
( ; + m ; + m ; y = ( y 1 ) )
= e x p ( x ) ( 1 y )
1
X
m = 0
1
X
n = 0
( )
m
( )
m
m !
( )
n
( + m )
n
( + m )
n
n !
( x )
m
y
y 1
n
= e x p ( x ) ( 1 y )
1
X
m = 0
1
X
n = 0
( )
m + n
( )
n
( )
m + n
m ! n !
( x )
m
y
y 1
n
= e x p ( x ) ( 1 y )
1
( ; ; ; x ; y = ( y 1 ) ) : ( T 5 )
I f y < 0 , t h e n 0 < y = ( y 1 ) < 1 , s o f o r m s ( T 1 ) { ( T 4 ) c a n b e a p p l i e d t o t h e
1
t e r m i n t h e l a s t
l i n e .
C B o u n d i n g t h e f u n c t i o n H
I n t h i s a p p e n d i x , I p r o v e T h e o r e m 1 . F r o m t h e d e n i t i o n o f H i n e q u a t i o n ( 5 ) , i t i s c l e a r t h a t H i s
n i t e a n d p o s i t i v e e v e r y w h e r e o n t h e p a r a m e t e r s p a c e i f a n d o n l y i f t h e n e s t e d i n n i t e s e r i e s g i v e n
b y
1
( q ; r ; p + q ; s = ; 1 ) c o n v e r g e s t o a p o s i t i v e r e a l n u m b e r f o r a l l p > 0 , q > 0 , r 2 < , s 2 < ,
0 < 1 a n d 0 < . T h e r e f o r e , i t i s s u c i e n t t o s h o w t h a t
1
( ; ; ; x ; y ) c o n v e r g e s t o a p o s i t i v e
r e a l n u m b e r f o r a l l 0 < < , 2 < , x 2 < a n d e i t h e r 0 y < 1 ( i . e . , f o r 1 ) o r y < 0 ( i . e . , f o r
> 1 ) .
T h e t e c h n i q u e s n e e d e d t o b o u n d
1
d e p e n d o n t h e s i g n s o f , x a n d y . A s s u m e r s t t h a t > 0 ,
0 y < 1 , a n d x 0 . F o r t h i s c a s e , w e n e e d t h e l e m m a
L e m m a 4 F o r a l l x 0 a n d 0 < < , 1
1
F
1
( ; ; x ) e x p ( x ) .
P r o o f : E x p a n d t h e
1
F
1
s e r i e s a s
1
F
1
( ; ; x ) =
1
X
m = 0
( )
m
( )
m
x
m
m !
:
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T h e r s t t e r m ( m = 0 ) e q u a l s 1 a n d r e m a i n i n g t e r m s a r e n o n - n e g a t i v e , s o t h e s u m m a t i o n m u s t b e
g r e a t e r t h a n o r e q u a l t o 1 . S i n c e < , e a c h t e r m m u s t b e l e s s t h a n x
m
= m ! , s o
1
F
1
( ; ; x ) <
1
X
m = 0
x
m
m !
= e x p ( x ) :
C o n s i d e r t h e f o r m f o r
1
g i v e n b y ( T 3 ) . G i v e n 0 y < 1 , e a c h t e r m i n t h e e x p a n s i o n i s
n o n - n e g a t i v e , s o
1
( ; ; ; x ; y ) =
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( + n ; + n ; x )
<
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
e x p ( x ) = e x p ( x )
2
F
1
( ; ; ; y )
w h e r e t h e i n e q u a l i t y f o l l o w s f r o m L e m m a 4 . T h e
2
F
1
s e r i e s c o n v e r g e s f o r a l l y i n s i d e t h e u n i t c i r c l e
( A S 1 5 . 1 . 1 ) , s o t h i s e x p r e s s i o n i s n i t e . T a k i n g t h e l o w e r b o u n d ,
1
( ; ; ; x ; y ) =
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( + n ; + n ; x )
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
=
2
F
1
( ; ; ; y ) 1
w h e r e t h e r s t i n e q u a l i t y f o l l o w s f r o m t h e l o w e r b o u n d i n L e m m a 4 . T h e r s t t e r m ( n = 0 ) i n t h e
2
F
1
s e r i e s e x p a n s i o n e q u a l s 1 a n d r e m a i n i n g t e r m s a r e n o n - n e g a t i v e , s o t h e s u m m a t i o n m u s t b e
g r e a t e r t h a n o r e q u a l t o 1 .
N e x t , a s s u m e x < 0 . W e n e e d t h e l e m m a :
L e m m a 5 F o r x < 0 , 0 < < a n d n 0 ,
1
1
F
1
( ; + n ; x )
1
F
1
( ; ; x ) < e x p ( x ) :
T h e p r o o f i s s i m i l a r t o t h a t o f L e m m a 4 .
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C o n s i d e r t h e f o r m f o r
1
g i v e n b y ( T 4 ) . G i v e n 0 y < 1 , e a c h t e r m i n t h e e x p a n s i o n i s
n o n - n e g a t i v e , s o
1
( ; ; ; x ; y ) = e x p ( x )
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( ; + n ; x )
e x p ( x )
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( ; ; x )
= e x p ( x )
1
F
1
( ; ; x )
2
F
1
( ; ; ; y ) <
2
F
1
( ; ; ; y )
w h e r e b o t h i n e q u a l i t i e s f o l l o w f r o m L e m m a 5 . T o g e t a l o w e r b o u n d ,
1
( ; ; ; x ; y ) = e x p ( x )
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
1
F
1
( ; + n ; x )
e x p ( x )
1
X
n = 0
( )
n
( )
n
( )
n
y
n
n !
= e x p ( x )
2
F
1
( ; ; ; y ) :
T h u s , f o r > 0 a n d 0 y < 1 ,
1
c o n v e r g e s t o a p o s i t i v e n i t e n u m b e r f o r a l l x 2 < .
W h e n 0 , u s e f o r m s ( T 1 ) a n d ( T 2 ) f o r
1
a n d t h e t r a n s f o r m a t i o n f o r t h e
2
F
1
g i v e n i n
A S 1 5 . 3 . 3 :
2
F
1
( ; ; ; y ) = ( 1 y )
2
F
1
( ; ; ; y ) : ( 1 3 )
T h e f o l l o w i n g l e m m a b o u n d s t h e v a l u e s o f t h e
2
F
1
t e r m s i n
1
.
L e m m a 6 F o r 0 y < 1 , 0 < < , a n d 0 ,
0 <
2
F
1
( ; ; ; y ) 1 :
P r o o f : A p p l y e q u a t i o n ( 1 3 ) . T h e r s t t e r m i n t h e e x p a n s i o n o f
2
F
1
( ; ; ; y ) i s 1 a n d t h e
r e s t a r e n o n - n e g a t i v e , s o
2
F
1
( ; ; ; y ) 1 . S i n c e > 0 a n d 0 < ( 1 y ) 1 ,
t h e r i g h t h a n d s i d e o f e q u a t i o n ( 1 3 ) m u s t b e s t r i c t l y p o s i t i v e , s o 0 <
2
F
1
( ; ; ; y ) . T o s h o w t h e
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u p p e r b o u n d , I s h o w r s t t h a t t h e d e r i v a t i v e o f
2
F
1
( ; ; ; y ) w i t h r e s p e c t t o y i s n o n - p o s i t i v e f o r
0 . A S 1 5 . 2 . 1 g i v e s :
d
d y
2
F
1
( ; ; ; y ) =
2
F
1
( + 1 ; + 1 ; + 1 ; y ) :
I f + 1 > 0 , t h e n t h e r s t t e r m i n t h e e x p a n s i o n o f
2
F
1
( + 1 ; + 1 ; + 1 ; y ) i s 1 a n d t h e r e s t a r e n o n -
n e g a t i v e . I f + 1 0 , t h e n t h e a r g u m e n t j u s t u s e d t o s h o w 0 <
2
F
1
( ; ; ; y ) a p p l i e s h e r e a s w e l l .
I n e i t h e r c a s e ,
2
F
1
( + 1 ; + 1 ; + 1 ; y ) > 0 . S i n c e = 0 , t h e d e r i v a t i v e
d
d y
2
F
1
( ; ; ; y ) 0 f o r
a l l 0 y < 1 . T h e r e f o r e ,
2
F
1
( ; ; ; 0 ) = 1 i s t h e u p p e r b o u n d o n
2
F
1
( ; ; ; y ) f o r 0 y < 1 :
T h e r e m a i n i n g a r g u m e n t s p a r a l l e l t h o s e u s e d e a r l i e r . F o r x 0 , u s e ( T 1 ) a n d L e m m a 6 t o s h o w
0 <
1
( ; ; ; x ; y )
1
F
1
( ; ; x ) :
F o r x < 0 , u s e ( T 2 ) , L e m m a 6 a n d t h e K u m m e r t r a n s f o r m a t i o n t o d e m o n s t r a t e t h e s a m e b o u n d s .
T h u s f a r , i t h a s b e e n a s s u m e d t h a t 0 y < 1 . F o r t h e c a s e y < 0 , r u l e ( T 5 ) g i v e s
1
( ; ; ; x ; y ) = e x p ( x ) ( 1 y )
1
( ~ ; ; ; x ; ~ y )
w h e r e ~ > 0 , ~ < a n d 0 ~ y y = ( y 1 ) < 1 . I t h a s a l r e a d y b e e n e s t a b l i s h e d t h a t t h e
r i g h t h a n d s i d e
1
c o n v e r g e s t o a n i t e p o s i t i v e n u m b e r , s o
1
( ; ; ; x ; y ) m u s t a s w e l l .
D P r o p o s i t i o n 2 a n d L e m m a 3
T o s h o w L e m m a 3 , s u b s t i t u t e t h e e x p r e s s i o n f o r
1
i n r u l e ( T 2 ) i n t o e q u a t i o n ( 5 ) t o g e t
H ( p ; q ; r ; s ; ; ) =
p
1
X
i = 0
( p )
i
( p + q )
i
i !
s
i
2
F
1
( r ; q ; p + q + i ; 1 ) :
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T a k e t h e d e r i v a t i v e w i t h r e s p e c t t o s :
d
d s
H ( p ; q ; r ; s ; ; ) =
p
1
X
i = 1
( p )
i
( p + q )
i
i !
i
s
i 1
2
F
1
( r ; q ; p + q + i ; 1 )
=
( p + 1 )
1
X
i = 0
( p )
i + 1
( p + q )
i + 1
i !
s
i
2
F
1
( r ; q ; p + q + i + 1 ; 1 )
=
( p + 1 )
1
X
i = 0
p ( p + 1 )
i
( p + q ) ( p + q + 1 )
i
i !
s
i
2
F
1
( r ; q ; p + q + i + 1 ; 1 )
=
p
p + q
H ( p + 1 ; q ; r ; s ; ; )
w h e r e t h e s e c o n d l i n e f o l l o w s f r o m a s h i f t o f i n d e x i n t h e r s t l i n e .
T o s h o w P r o p o s i t i o n 2 , t a k e t h e d e r i v a t i v e o f e q u a t i o n ( 7 ) w i t h r e s p e c t t o s a n d u s e L e m m a 3
t o s u b s t i t u t e f o r d e r i v a t i v e s o f H . T h i s g i v e s
d E ( X
k
)
d s
=
( p )
k
( p + q )
k
p
p + q
H ( p + k ; q ; r ; s ; ; )
H ( p ; q ; r ; s ; ; )
H ( p + 1 ; q ; r ; s ; ; )
H ( p ; q ; r ; s ; ; )
p + k
p + q + k
H ( p + k + 1 ; q ; r ; s ; ; )
H ( p ; q ; r ; s ; ; )
= E ( X
k
) E ( X ) E ( X
k + 1
) :
P r o p o s i t i o n 2 i s a s p e c i a l c a s e o f t h i s r e s u l t .
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R e f e r e n c e s
A b r a m o w i t z , M i l t o n a n d I r e n e A . S t e g u n , e d s , H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s n u m -
b e r 5 5 . I n ` A p p l i e d M a t h e m a t i c s S e r i e s . ' , N a t i o n a l B u r e a u o f S t a n d a r d s , 1 9 6 8 .
A r m e r o , C . a n d M . J . B a y a r r i , \ P r i o r a s s e s s m e n t s f o r p r e d i c t i o n i n q u e u e s , " T h e S t a t i s t i c i a n ,
1 9 9 4 , 4 3 ( 1 ) , 1 3 9 { 1 5 3 .
B e r n d t , E . , B . H a l l , R . H a l l , a n d J . H a u s m a n , \ E s t i m a t i o n a n d I n f e r e n c e i n N o n l i n e a r
S t r u c t u r a l M o d e l s , " A n n a l s o f E c o n o m i c a n d S o c i a l M e a s u r e m e n t , 1 9 7 4 , 3 / 4 , 6 5 3 { 6 6 5 .
B o d i e , Z v i , R o b e r t C . M e r t o n , a n d W i l l i a m F . S a m u e l s o n , \ L a b o r S u p p l y F l e x i b i l i t y a n d
P o r t f o l i o C h o i c e i n a L i f e C y c l e M o d e l , " J o u r n a l o f E c n o m i c D y n a m i c s a n d C o n t r o l , J u l y {
O c t o b e r 1 9 9 2 , 1 6 ( 3 { 4 ) , 4 2 7 { 4 4 9 .
C a r r o l l , C h r i s t o p h e r D . , \ T h e B u e r - S t o c k T h e o r y o f S a v i n g : S o m e M a c r o e c o n o m i c E v i d e n c e , "
B r o o k i n g s P a p e r s o n E c o n o m i c A c t i v i t y , 1 9 9 2 , ( 2 ) , 6 1 { 1 3 5 .
G o r d y , M i c h a e l B . , \ C o m p u t a t i o n a l l y C o n v e n i e n t D i s t r i b u t i o n a l A s s u m p t i o n s f o r C o m m o n -
V a l u e A u c t i o n s , " C o m p u t a t i o n a l E c o n o m i c s , f o r t h c o m i n g .
G r a d s h t e y n , I . S . a n d I . M . R y z h i k , T a b l e o f I n t e g r a l s , S e r i e s , a n d P r o d u c t s , N e w Y o r k : A c a -
d e m i c P r e s s , 1 9 6 5 .
K e n n i c k e l l , A r t h u r B . , M a r t h a S t a r r - M c C l u e r , a n d A n n i k a E . S u n d e n , \ F a m i l y F i n a n c e s
i n t h e U . S . : R e c e n t E v i d e n c e f r o m t h e S u r v e y o f C o n s u m e r F i n a n c e s , " F e d e r a l R e s e r v e B u l -
l e t i n , 1 9 9 7 , 8 3 ( 1 ) , 1 { 2 4 .
L a i b s o n , D a v i d , \ G o l d e n E g g s a n d H y p e r b o l i c D i s c o u n t i n g , " Q u a r t e r l y J o u r n a l o f E c o n o m i c s ,
1 9 9 7 , 1 1 2 ( 2 ) , 4 4 3 { 4 7 7 .
M c D o n a l d , J a m e s B . a n d Y e x i a o J . X u , \ A g e n e r a l i z a t i o n o f t h e b e t a d i s t r i b u t i o n w i t h
a p p l i c a t i o n s , " J o u r n a l o f E c o n o m e t r i c s , 1 9 9 5 , 6 6 ( 1 , 2 ) , 1 3 3 { 1 5 2 .
P o t e r b a , J a m e s M . a n d A n d r e w A . S a m w i c k , \ H o u s e h o l d p o r t f o l i o a l l o c a t i o n o v e r t h e l i f e
c y c l e , " W o r k i n g P a p e r 6 1 8 5 , N B E R S e p t e m b e r 1 9 9 7 .
R u b i n , D o n a l d B . , M u l t i p l e I m p u t a t i o n f o r N o n r e s p o n s e i n S u r v e y s W i l e y S e r i e s i n P r o b a b i l i t y
a n d M a t h e m a t i c a l S t a t i s t i c s , J o h n W i l e y & S o n s , 1 9 8 7 .