ACTAUNIVERSITATISLODZ IENSIS FOLIA OECONOMICA 90, 1989

A C T A
U N I V E R S I T A T I S
FOLIA OECONOMICA
L O D Z I E N S I S
90, 1989
Zbigniew Wasilewski
REMARKS ON BLUS RESIDUALS
1. INTRODUCTION
The BLUS r e s i d u a l s t h e o r y
( B e s t L i n e a r U n b ia s e d
w ith S c a la r c o v a r ia n c e m a t r i x )
[ü ]
in
is
g en er a ly r e la t e d
c o r r e la tio n stru ctu re
be
t o p r o b le m s
l i n e a r r e q r e s s i o n m od els*
im p ortan t
stru ctu re
for
r e sid u a ls
d e v e l o p e d by T h e i l [ 7 ]
and K o e r t s
o f e s t i m a t i n g t h e error t e r m
T h is e s t im a t o r p o s s e s s e s
a s t h e unknown d i s t u r b a n c e s .
sta tistica l
in fe re n c e
about
the
same
T h i s se e m s t o
the
sto c h a stic
o f t h e r e g r e s s i o n m o d e l.
We h a v e a r e g r e s s i o n m od e l o f t h e fo r m
у * X0 + u
,
(1)
w h e r e u n d er common a s s u m p t i o n s :
( a ) X i s an
t a in s the v a lu e s
n:k
n o n s t o c h a s t ic m a trix
ta k e n by th e
p erio d s:
(b ) lim n” 1f X'X)
П - аз
is
E(uu')
к w h ic h c o n -
in d epend en t v a r i a b le s
in
n
a f i n i t e n o n s in g u la r m a trix ;
( c ) t h e v e c t o r o f random
m en ts w i t h z e r o
к
o f rank
d is tu r b a n c e s , has u n c o rre la ted e l e -
mean and c o n s t a n t
v a ria n ce,
i.e.
E(u)
=
0,
*= a 2 1;
( d) in a d d itio n i t
is
o f t e n assum ed,
th at
the
d istu r b a n c e s
are n orm ally d i s t r i b u t e d .
*
Lecturer at the
ty od Łódź.
I n s t i t u t e of Econometrics
and S t a t i s t i c s ,
U niversi-
Due t o l a c k
o f k n o w le d g e a b o u t
in f e r e n c e s about
b a s e d on some
stra ig h tfo r w a r d
u
h a v in g r e l a t e d p r o p e r t ie s *
t o t r e a t as such
a p p rox im a tion
errors
a p p r o x im a tio n c r i t e r i o n ,
is
Ц | У1 - x ^
so fa r
to it s
th is
of
b|.
у - Xb
an e s t i m a t e t h e
o f th e chosen
or
w h e re
b'
The m i n i m i z a t i o n
a ttr a c tiv e a n a ly tic a l
is
m ust be
I t se em s
vector
o f tho
g i v e n by
m in im iz a tio n
for in sta n c e
ZKy^- x'^ b )2
o f the error
t h e m o st p o p u l a r c r i t e r i o n
cr ite r io n
o f d istu r b a n ce s,
a s s u m p t i o n s ( c ) and ( d )
the s t o c h a s t ic
estim a te o f
r e a l v a lu es
sum o f s q u a r e s
of estim a tio n ,
in p a r t due
and n u m e r i c a l p r o p e r t i e s .
we o b t a i n t h e w e l l - k n o w n
U s in g
le a s t squares
estim a to r
ß
b * ( X ' X ) " 1X' y
(2)
and t h e c o r r e s p o n d i n g v e c t o r
o f le a st-sq u a r e s r e sid u a ls
e = у - Xb = My =* Mu
w here,
(3)
M = ( I - X ( X' X) 1 X' ) X
m a t r i x o f rank
is
th e id em poten t
n :n p r o j e c t io n
n - k.
The e s t i m a t o r o f
u
(a) i t
is
in th e dependent v a r ia b le ;
(b) i t
i s u n b ia sed ;
lin ea r
has th e fo llo w in g d e s ir a b le p r o p e r t ie s :
( c ) i t has the s m a lle s t
tim a tio n e r r o r s ,
e x p e c t e d sum
w ith in th e s e t
o f squares
of a l l lin e a r
o f the
es-
and u n b i a s e d
es-
tim a to rs.
On t h e o t h e r h a n d ,
estim a to r,
E (ee ')
however,
a s s u m in g t h a t
=
e
Thus t h e l e a s t
t e s t in g purposes.
squares
r e s id u a ls are
d e p e n d s on t h e
C lea rly
e
fe r en t c o r r e la tio n str u c tu r e ,
m a trix
M
sin g u lá ry
is
sid u a ls th a t
n o t u n iq u e
t i m a t e d can
of
n - к )
c o r re la ted
p a rticu la r
b u t due t o
th e fa c t th a t
of
n - к
the
a
for
dif-
1 - s rep ro je c tio n
tran sform ed
re-
M or eove r t h e o b t a i n e d s o l u t i o n i s
o f the
be v er y Im portan t.
th e ir
u seful
can be tran sform ed t o have
we can o b t a i n o n l y
are u n c o r r e la t e d .
and
X m a t r i x . T h i s mak-
o f d istu rb a n ces l e s s
d i s t r i b u t e d ( t h e rank
and t h e c h o i c e
th is
(4)
squares estim a to r
s id u a ls are
th e co v a r ia n ce m a trix
o
= a I,
i s g iv e n by
( m u u 'M) = a 2M
c o v a r ia n c e m a tr ix
e s the l e a s t
E(ee')
к
re sid u a ls th a t
In t h e p a p e r we t r y
are n o t e s t o make some
ev id en ce,
a b o u t t h e .m u tu al
t r a n s f o r n a t i o n and t h e
r e l a t i o n s betw een th e
accuracy o f the
mean s q u a r e e r r o r ( MSE) ,
in the c a se
o f o u tlie r s
p r o p o s e t o u s e r e s i d u a l s m i n i m i z i n g t h e sum
t o o b t a in n on -b iiased r e s i d u a l s
such a c a s e .
b a sis
e s iir .a tio n ,
of
the
neasured
by
among d a t a .
We
o f a b s o l u t e d e v ia t io n s
w ith s c a l a r c o v a r ia n c e m a tr ix ,
in
2. NOTES ON THE CONSTRUCTION OF THE BLUS RESIDUALS
S u p p o se t h a t an
t r a n s fo rn a tio n
(n - k )
ô = C'y.
b ia s e d r e s i d u a l s w itli
(i)
E( é ) = 0
The c o n d i t i o n s
s n
m a trix
We w i l l c a l l
С
é
d e fin e s
S c a la r c o v a r ia n ce m a trix
and
(ii)
(i),
(ii)
E(éé')
a
lin ea r
a v e c to r o f L inear
( LUS)
Un-
if,
= o2I
re q u ir e o n ly t h a t
C'X = 0 and C'C =
- I.
T h er e e x i s t
w h ic h f u l f i l l
of
n - к
m ethods o f
th ese c o n d itio n s.
d eriv a tio n
more o r l e s s
a rb itra r y ,
o f the
A l l o f them
r e s i d u a l s t o be e s t i m a t e d .
d u a ls i s
LUS
a few
m a trix
req u ire the
The c h o i c e o f
С
ch o ic e
th ese
so th a t the d e f in it io n
r e siof
the
r e s i d u a l s i s n o t u n iq u e.
S u p p o se we p a r t i t i o n
- ( С'oC
)
such t h a t th e
co m p o n e n ts o f
u
su b scrip t
0
X' =
resp ond s t o th e re m a in in g
n - к
o f t h e rows
We a d d i t i o n a l y a s s u m e ,
The c o n d i t i o n s
cases.
C'X » 0
w r it t e n u sin g th e p a r t i t i o n o f th e
T h is i s
к : к
and
to
and s u b s c r i p t
o f th e m a trix
th a t the
(x' X ^ ) and
corresponds
w h ic h a r e n o t e s t i m a t e d
by s im p le r e o r d e r in g
sin g u la r.
u' = ( u ' Q u'.,),
a lw a y s
m a tr ix
С
к
cor-
p o ssib le
X.
C'C = I n_j,
m a tr ice s
the
1
C' =
and
X
can
X
is
now
as
n o ilbe
fo l-
lo w s ,
(5)
CQ
is
t h u s d e t e r m i n e d u n i q u a l y from ( 5 ) and e q u a l s ,
C' o = - C' l X1Xo ' 1
(5a)
w here t h e a s s u m p t io n o f n o n s i n g u l a r i t y o f
S u b stitu tin g
C'C -
is
used.
( 5a ) in ( 6 ) o b ta in ,
C ' l X l X0 - 1X ' 0 - 1X ' l Cl + C ' l Cl - I n _k
S u b stitu tin g
X ' QXo
c '^ C x 'x
-
XQ
-
x;
( x ' X ) ~ 1X ’ 1 ( I
= C' ,
by
X'X - X ' ^
we o b t a i n ' ,
x l ) “ l x ' 1c 1 + C ' 1C1 = c ^ x , [ ( x ' x ) - 1
L e t us p u t now
+
- X1 ( x ' x ) " 1x ' 1 ) “ 1 x 1 ( x ' x ) “ 1 ] X ' 1C1 + C ' 1 C 1
[ Х ^ Х ' Х Г ’ Х'., + X ^ X ' X ) “ ^ ' ,
• х ^ х ' х Г ’х ' ,
(7)
(8)
(I - Xl(X'X)- 1 X ' t ) “ 1 .
+ I ] С, * I n_ k
A - X^X'X)- ^',,
th e n
(8 )
can b e w r i t t e n as
C ', ( A + A ( I - A ) ~ 1A + I ) C 1 =
= C ' , ( A ( I + ( I - A ) ” 1A ) + I ) C r =
c',1
A ( ( I - A ) " 1 ( I - A) + ( I - A0) _ 1 A
A)) +
(8a)
+ (I
- A )(I
- A )“ 1]C ,
= C ^ C A ( I - A) “ 1( I - A + A) + ( I - A ) ( i - A ) “ IJC1
= c '^ i
- a ) " 1c 1 = C ^ ' I
- X ^ X 'X ) - ^ ' , ) “ ^ ,
I n th e d e r i v a t i o n the i n v e r s i o n of th e p ro d u c t X’X - XjX. was
obt a i n e d u s i n g th e f o l lo w in g u p d a ti n g form ulae due t o Gauss.
Lema 1. L e t A be p : p rank p symmetrix m a t r i x , and suppose t h a t X and
V a r e q : p rank q m a t r i c e s . Then, p ro v id ed t h a t th e i n v e r s e s e x i s t ,
(A + X’Y ) " 1 - a " ' - а Л
’ ( I + YA " ' х ’ Г ' у а " 1.
It
f o l l o w s from
c ',( i
-
Thus
C1
trix (i
from
8a
th at
C1
m ust s a t i s f y ,
х , ( х ' х Г 1 x ' 1) - 1 c 1 = i n _k
must b e c h o s e n
- X^( X' X)
1x ' , )
1
(9)
t o b e any
and
CQ
is
fa c to riz a tio n
then
o f t h e ma-
d eterm in ed
u n iq u e ly
C '0 =
T h e i
1
[ 7 J , Г8J
showed t h a t t h e
use o f s p e c t r a l
p o s i t i o n o f th e m a tr ix
(I - X^X'X)
LUS r e s i d u a l s w i t h t h e
sm a lle st exp ected re sid u a l
Dug t o t h i s
a d d itio n a l p rop erty th e se
b e s t L i n e a r U n b ia s e d
It
1Х ^ )
f o l l o w s fro m
in v e r s e o f the
1
to find
decom-
Ct l e a d s t o
sum o f s q u a r e s .
r e s id u a l s are
ca lle d
the
S c a l a r c o v a r i a n c e m a t r i x ( BLUS) r e s i d u a l s .
d ir e c t m u ltip lic a tio n
m a tr ix
(T h eil
( i - X1 ( X ' X ) - 1 X ' 1 )
[7]),
ex ists
that
and i s
the
o f the
form ,
(.1 - х ^ х ' х Г ’ х ' , Г 1 = i + x 1 ( x ' o x o ) ~ 1 x ' 1 =
w h e re
(10)
ZZ'
Z = X.X - 1 .
I
о
The m a t r i x
I - x 1 ( X ' X )” 1X ^
o f the p o s it iv e
tiv e
+
I
d e fin ite .
is,
s e m i - d e f i n i t e m a trix
Thus, th e r e e x i s t s
as a
n o n s in g u la r su b m a tr ix
M = I - X( X' X)
' X' ,
p o si-
a sq uare o r th o g o n a l m a trix
P,
such t h a t
P'(I
- X ^ 'X ^ x ^ P
= D
P'(I
- Xt ( X ' X ) “ 1X1 ) ' ' 1P = D~1
( 11
)
_1
w h e re
P 'P = I
ent roots of
and
D,D
are d ia g o n a l
I - X1( X ' X ) _1 X1
and
(i
m a trices
w ith th e
- X1( X ' X ) " 1X1 ) ~ 1
la t-
on
the
eq u a tio n in
(11)
main d i a g o n a l 2 .
by
On p r e m u l t i p l y i n g b o t h S i d e s o f t h e s e c o n d
P and p o s t m u l t i p l y i n g by P ' we o b t a i n ,
(I
2
- Х ^ Х ' Х Г ^ Г 1 = I + ZZ' = PD- 1 P'
Given Che p o s i t i v e
r o o t s are p o s i t i v e .
d e fin itn e ss of
I - X,(X*X)
(12)
-1
Xj
a ll
these
laten t
The c o n d i t i o n ( 9 ) can b e t h u s
w ritten
in th e
form ,
C ^ P D ľ ’ p ' C, » I
w h ic h i s
fulfilled
fo r the
m a tr ix
C,
g iv e n b y ,
C, = PD1 / 2 P'
L e t us now c o n s i d e r
tr ix
Mn
th e c h a r a c t e r i s t i c eq u a tio n
fo r the
ma-
= ( I + Z Z ' ) -1 ,
[ (X + Z Z ' ) " 1 - d i l ]
(15)
PjL
On p r e m u l t i p l y i n g ( 1 5 ) by
I + ZZ'
we h a v e ,
\
[ I - d i I - ZZ' di ] p 1 = 0
and a f t e r d i v i d i n g by
(-d ^
<1 6 )
and s u b s t i t u t i n g
X1XQ-1
for
Z
we
f in a l ly o b ta in ,
CX1X0 "1( X 1X0 ' 1 ) ' - ( 1 / d i -
l)]pi - 0
From ( 1 7 ) we f i n d t h a t t h e
- X ^ X ' X V ’X,
lu es" o f
ch a ra cteristic
are th e p r in c ip a l
T h i s m a t r i x can be t r e a t e d
(17)
as a m a tr ix o f
ex p la n a to r y v a r ia b le s
vectors
co m p o n e n ts o f t h e
of
I +
m a t r i x X., XQ •
" in d ex tran sform ed v a -
w ith m a tr ix
XQ
as a
b a sis
of
t h i s tra n sfo r m a tio n .
S in c e th e p o s i t i v e
ord er (n - k)
n - 2k
: (n - k )
and o f rank
zero la te n t ro o ts
H ence, a t l e a s t
к
se m i d e f i n i t e m a t r i x
n - 2k
and a t m o st
o f the
d 's
к
X1XQ
or l e s s
к
( ^ i x0
i t has
p o sitiv e
are e q u a l t o
/ i s
at
of
le a st
la te n t ro o ts.
1
and a t m o st
o f them eure l e s s t h a n 1 .
T a k in g t h i s
С
1
=
i n t o a c c o u n t we ca n
*
1 /9
£. d
P i 15/
i=l
= и„
now r e w r i t e
i n t h e fo rm ,
n"k
+
Z 1 d i 1/ 2 О
1-1
pi p , i *
- di 1 /2 ) P i P ' i
(18)
Thus t h e t r a n s f o r m a t i o n m a t r i x
t a i n e d , from s u b m a t r i x
by a d d in g
к
M11
3.
o f the
Р^Р'*.
m a tr ice s
a2 ,
tr a n s fo r m a t io n m a tr ix
BLUS r e s i d u a l s i s ,
(T h e i
1
d e p e n d s on t h e c h o i c e
[4]
re sid u a ls
w here
e) ( e -
e= (O',
t h e sum o f t h e
O
- d 1/ 2 )
1
(19)
XQ.
c o v a r ia n c e m a trix o f
co v a r ia n ce m a tr ice s o f
and t h e 1 - s - BLUS d i f f e r e n c e s ,
E [( é - u ) ( é - u ) ' ]
+ E[( é -
o f the b a sis
showed t h a t t h e
r e s id u a ls eq u a ls
1 - s
appart
- 2 ( n - k) - 2 trC 1 =
N e u d e c k e r
the
M
factor
([7])
Jc
- 2 ( n - k ) - 2 ( n - 2k + V
d 1/2) = 2 £
f=*1
1
i=1
%
t h e BLUS
by t h e
OF THE TRANSFORMATION
squares o f th e
equal to
E [(é - u , ) ' ( ž - u ^ ]
Thus i t
i s ob-
THE PRICE OF THE SCALAR COVARIANCE CONDITION
The e x p e c t e d sum o f
factor
1-s
f o r BLUS r e s i d u a l s
o f u n i t rank s c a l e d
AND THE CHOICE OF THE BASIS
from t h e
C1
* E [(e - u )(e - u)' ] +
e)' ]
é')',
(20)
e ■ ( e ' 0 » e>i ^ f
t h e v e c t o r s o f BLUS,
and
1 - s r e s i d u a l s and
u = ^и ' о '
u’i^ ’
are
th e e r r o r term , r e s p e c t i -
v ely .
The p r o b le m o f t h e
e v e r , m a in ly r e l a t e d
best
t o .th e pow er
e d on BLUS r e s i d u a l s .
b a s i s by
ch o ic e
T h e i 1
th e s e le c t io n o f ,
of th e b a sis
proposed t o ch oose
a so c a l l e d ,
g iv e n t e s t i n g
p e c t e d r e s i d u a l sum
o f squares c r it e r io n .
t io n s ought t o
a lte r n a tiv e
case o f te s t in g
how-
bases
mińimum
The " p e r m i t t e d s e t
in such a w ay, t h a t t h e
exof
b a sis observa-
w ith r e s p e c t t o th e
than th e rem a in in g o b s e r v a t io n s .
for s e r i a l c o r r e la t io n ,
such a
p e rm itted s e t o f
p rob lem , u s in g a
have " l e s s in fo r m a tio n va lu e"
h y p o th e sis,
is ,
o f th e co r re sp o n d in g t e s t b a s [8]
w ith r e s p e c t to th e
bases" ou ght t o be ch osen
Xq
for in s ta n c e ,
In t h e
such a s e t
o f "p erm itted b ases"
fir st
and
ш
and l a s t
H a r v e y
i s t u n ifo rm ely
p o ssib le
c o n sists
к - m
[5]
o f the b ases
cases,
w h e re
i n d i c a t e , how ever,
b est bases
c o r r e la tio n
a c o m p a r is o n
are a n o th er
b e o b t a i n e d by u s i n g t h e
1
( I -'X^(X'X)
1X ^ )
t o be s p e c i a l l y
to fin d
th e r e c u r s iv e com p u tation s
The t e s t s
a b it
re sid u a ls is
in (8a).
a ttr a c tiv e
of
T h is
The
that
the
type
and
c o r r e la t io n based
le ss
made.
can
m a tr ix
of
r e si-
due t o t h e s i m p l i c i t y o f
( P h i l i p s
for s e r ia l
a re o n l y
a-
for s e r ia l
t y p e o f LUS r e s i d u a l s
C1
It is
in
lo w p o w e r s .
C holesk y d eco m p o sitio n
d u a l s se e m s
sid u a ls,
t o succeed
o f th e ex a ct t e s t s
b a s e d on BLUS and r e c u r s i v e
re cu rsiv e r e s id u a ls
the
P h i l i p s
h ypotheses.
o n l y , by p r o p e r c h o i c e o f t h e b a s i s ,
In t h e same work
co n ta in
t h a t t h e r e do n o t e x -
for a ll a lte r n a tiv e
v o i d i n g t e s t s w h ic h h a v e r e l a t i v e l y
Q5]!).
that
0 < m < k.
H a r v e y
on r e c u r s i v e
re-
p o w e r f u l t h a n t h e BLUS t e s t s .
A. CHOICE OF THE BASIS IN THE PRESENCE OF OUTLIERS
The f a c t t h a t t h e LUS r e s i d u a l s
ponents o f th e v e c to r
and c a s e s
d u a ls.
u
much more t h a n
it
In c e r t a i n c a s e s ,
mong s a m p le d a t a ,
is
In t h e c a s e
is
s p r e a d , by
d u a ls.
t h i s can h a v e
of 1 - s
by
In s u c h
h^j
com-
1 - s
a-
on t h e i n -
e f f e c t o f the
elem en t o f th e
i - th
r e si-
ca ses a ls o the c h o ic e
p r o j e c t io n m a trix
- th
the
o f the
th e r e are o u t l i e r s
of
im p o rta n t.
estim a tio n the
means o f t h e
ij
n - к
se rio u s im p lic a tio n s
se em s t o b e more
D en oting th e
= X( X' X) _ 1 X'
in th e ca s e
e s p e c i a l l y when
f e r e n c e b a s e d on LUS r e s i d u a l s .
th e proper b a s is
e stim a te d o n ly
b l u r s t h e r e l a t i o n s h i p betw een r e s i d u a l s
1 - s
M,
to
o u tlier s
a ll
m a trix
r e siH
=
r e s i d u a l can b e w r i t t e n
i n t h e form ,
•j -
о
- i^ iy , -
E
hl j y j
(го
n
w here due t o t h e
£ 1 /n
in d e m p o t e n c y o f
p ro v id ed th e
H ence i f
h^
H,
h^
=
2
h i
and
h^
>
m odel c o n t a i n s a c o n s t a n t t e r m .
is
clo se to
1a
g ro ss e r ro r in
y^
w ill
not
n e c e s s a r i l y show
say in
e^,
r e sid u a ls.
it
is
tr ix
h^^
b u t i t m ig h t
h a p p e n s t o be l a r g e
The same e f f e c t
stro n g ly
X
up i n
if
and c h a r a c t e r o f
d u a ls.
i s p r e s e n t i n LOS
d ep e n d e n d n o t
o f the b a s i s .
n ife st it s e lf
co n ta m in a tio n
n.
t io n in th e
o b serv a tio n
o f on e o f b a s i s
On t h e o t h e r hand
cedure w i l l in flu e n c e
used a t th e
sum p tions t h a t
o u t l i e r s among
ma-
ch o ic e
r e siw i l l ma-
T h is i n f l u e n c e
of
de-
co n ta m in a -
recurrent
From
pro-
th is
p o in t
a p p r o p r ia te f o r exam in in g a s -
order o f th e c a s e s .
there
are
e v i d e n t b u t a l s o se em s
t h e norm al c a s e .
e i s o f th e L U S -tra n sfo rm a tio n sh o u ld
In t h e c a s e o f
b a s i s when
d ata i s n o t so
more I m p o r t a n t t h a n i n
the
on t h e
end o f t h e
proper ch o ic e o f th e
t h e sam ple
of
o b serv a tio n s
o n ly th e l a s t r e s id u a l s .
d e p e n d on t h e
BLUS r e s i d u a l s t h e
a l l the
o f re cu rsiv e
th e occurrence
o f view th e r e c u r s iv e r e s i d u a l s are
t o be
but a lso
i n tjhe c a s e
elsew h er e,
r e s i d u a l s , b u t now
by I n c r e a s i n g r e s i d u a l e r r o r s .
c r e s e s w ith
up
a ffects
o n l y on t h e s t r u c t u r e
T h is i s e v i d e n t
The c o n t a m i n a t i o n
show
and i t
not
G e n e ra lly th e ba-
co n ta in
any
o u tly in g
o b ser v a tio n s.
I n o r d e r t o g i v e some
o f o u t l i e r s on BLUS
b a s i s we make
ev id en ce
1 5 - e l e m e n t s s a m p le
from t h e "mean s h i f t
Y ' XP + ♦ .
w h e re
4^ СotfcS)
con stant
aa
o f th e
Each e x p e r i m e n t
In e a c h r e p l i c a t i o n
m a g n it u d e and c o n f i g u r a t i o n
in flu e n c e
r e s p e c t to th e ch o ic e
some s i m p l e n u m e r i c a l e x p e r i m e n t s .
c o n s i s t e d o f 500 r e p l i c a t i o n s .
en
about th e p o s s ib le
r e s i d u a l s w ith
we
gen erated
o u t l i e r model" w i t h g i v -
of a sin g le o u tlie r ,
(ac) + u
d e n o t e s t h e dummy
in i - t h
v a r ia b le
p o s i t i o n and z e r o s
w ith
co n ta m in a tio n
e l s e w h e r e and
u ~ N(0,
5a
and
о ).
The c o n t a m i n a t i o n c o n s t a n t
added t o t h e f i r s t ,
sp e c tiv e ly .
c e n t r a l and l a s t
F or e a c h s a m p le
four d i f f e r e n t b a se s
square e s tim a t io n
com p u tation s
and
w er e c a l c u l a t e d
la st
к
(BLUSe) o b s e r v a t i o n s
co r re sp o n d in g
v i a t i o n r e s i d u a l s ( B L U S l) .
( BLUSb) ,
to zero v a lu es
was
s a m p le
re sid u a ls
and t h e v a l u e s
e r r o r s ( MSE) w e r e c o m p a r e d .
к
10a
o b se r v a tio n in
t h e v a l u e s o f BLUS
c o n s is t o f the f i r s
( BLUSc) and t h e
o b ser v a tio n s
was e q u a l
rew ith
o f t h e mean
The b a s e s f o r BLUS
th e ce n tr a l
as w e l l as
к
th e
к
o f le a s t a b so lu te de-
The f i r s t t h r e e c r i t e r i o n s
\
w er e o f t e n
(see eg.
used in l i t e r a t u r e
proposed h ere
The mean v a l u e s
t y p e o f BLUS
of its
o f t h e MSE
r e s i d u a l s are
ta in s a ls o the
resp o n d in g
T h e i
m a in ly b eca u se
1
[в]),
o v e r 500 r e p l i c a t i o n s
g i v e n i n T a b le
fr a c tio n s o f the
th e la s t
one
is
rob u stn ess to o u t l i e r s .
1.
re p lica tio n s
T h is
for
each
ta b le
con-
i n w h ic h
the c o r -
BLUS r e s i d u a l had'1' t h e s m a l l e s t MSE v a l u e .
T a b l e
,
I
Mean v a l u e s of th e MSE
Index o f c ontam inated
o b s e r v a t i o n and the
v a lu e of c o n ta m in atio n
Type o f BLUS r e s i d u a l s
BLUSb
BLUSc
BLUSe
BLUS1
1
5a
1.0179
(0 .0 8 )
0.5001
(0 .4 1 2 )
0.5929
(0 .1 7 2 )
0.5124
(0 .3 36 )
8
5о
— '3О"- <r
O'
Г
о о
0 .9 55 7
(0 .0 54 )
0.7 66 0
(0 .2 88 )
0.7783
(0 .1 6 4 ) .
15
50
4. 1716
(0 .5 5 2 )
5.0073
(0 .0 0 2 )
18.1478
(0 . 0 0 0 )
4.3454
(.0.446)
1
lOo
1.8990
(0 .0 3 6 )
0.6486
(0 . 4 9 0 )
0.7509
( 0 . 180)
0.7217
(0 .2 9 4 )
8
10c
1.1726
(0.518)
1,7468
(0.022)
1.2292
( 0 .3 4 4 )
1.3146
( 0 . 116)
No t e :
Mean v a lu e s of th e MSE o ver 500 r e p l i c a t i o n s o f th e e x p erim en t
f o r f o u r ty p e s of BLUS r e s i d u a l s and f i v e v a r i a n t s o f c o n t a m i n a t i o n . The v a lu e s in b r a c k e t s d e n o te s t h e f r a c t i o n of th e r e p l i c a t i o n s in which th e c o r re s p o n d in g MSE v a lu e was th e s m a l l e s t one.
The r e s u l t s o f
ex p er im en ts g a th e re d in T a b le 1
r e l a t i v e l y h ig h i n c r e a s e
o u tlie r s.
i n MSE i n t h e c a s e o f
T h is I n c r e a s e depends
o u t l i e r i n c o m p a r is o n
w ith o th e r o b s e r v a tio n s .
c h o i c e o f t h e b a s i s f o r LUS
in th e d a ta
on t h e r e l a t i v e
c o u n t, th e c h o ic e
b a s e - c h o ic e w ith
j
m a g n it u d e o f the
Thus
the
c o m p u t a t i o n s when t h e r e
i n some c a s e s .
o f th e b a s is
proper
are o u tlie r s
t y p e o f BLUS r e s i d u a l s
r e c t l y o b t a i n e d from
( i n T a b le 1
n - к
tra n sfo r m a tio n .
T a k in g t h i s i n t o a c -
co rresp o n d in g t o ze r o
s o l u t e d e v i a t i o n s ( LAD) r e s i d u a l s seem s
C 'i
a
se em s t o be o f s p e c i a l i m p o r t a n c e . The i d e n t i f i c a t i o n
o f o u t l i e r s can be d i f f i c u l t
the
in d ic a te s
n o t e d a s BLUS1)
n o n z e r o LAD
le a st
ab-
t o b e a good c h o i c e . T h i s
r e sid u a ls
can
by
be
means
d iof
Thus t h e r e e x i s t s m u t u a l c o r r e s p o n d e n c e
b e t w e e n LAD
and BLUSl r e s i d u a l s
assu m ing
к
n o n -estim a te d
BLUS
r e s id u a ls are equal zero.
The s y s t e m o f e q u a t i o n s
(22)
e LAD
l j
and
0
w h e re t h e s u b s c r i p t s m
z e r o and n o n z e r o
1
r e sid u a ls r e s p e c tiv e ly .
T h is co r re sp o n d s t o r e o r d e r in g th e
first
к
in g ly
XQ
tr ic e s.
are th o se ly in g
e T, _
LAD
(2 4 )
so th a t
h y p erp la n e.
and
(n - к )
n - к
th e v e c to r
у,
the
Accord: к
nonzero r e sid u a ls.
b
o f estim a ted
LAD
and i s e q u a l t o ,
(23)
X " 1y
o Jo
r e sid u a ls i s
- X ,b
1 LAD
G iv e n t h e o r d e r i n g
tio n
X0
can b e o b t a i n e d from ( 2 2 )
The v e c t o r o f LAD
'LAD
к : к
r e fe r s to the
A ssu m in g n o n s i n g u l a r i t y o f
b LAD
T
o b ser v a tio n s
in th e r e g r e s s io n
are r e s p e c t iv e ly
and
The v e c t o r
param eters
abso-
bLAD
0
I х’
*’ I
o f th e le a s t
can be w r i t t e n a s ,
о
X
l u t e d e v i a t i o n p r o b le m
fo r the s o lu t io n
then g iv e n b y,
-1
(24 )
У1 - x i x
o f the c a s e s
t h e r e s p e c t i v e BLUS
co rresp o n d in g
estim a to r
to the
of re sid u a ls
can
so lu be
w ritten a s,
= C 'y = C c ^ c ' , )
= C ' o y o + С ' , У1
(25)
-c ' i x i x o ’ 4
+ cv
i
Thus c h o o s i n g t h e b a s i s
we can o b t a i n t h e
к
y i - x i x o ' 1v
co rresp o n d in g
v e c t o r o f BLUS
n o n z e r o LAD r e s i d u a l s .
c o n ta in e d in
= cV
= c , 1<e LAD
t o z e r o LAD
r e sid u a ls d ir e c tly
N o te t h a t i n
th is
n o n -estim a te d r e s id u a ls
re m a in in g n - к r e s i d u a l s ,
w h ic h i s
o b t a i n e d from
1 - s
re sid u a ls.
case,
th e
i s n o t spread
r e sid u a ls
from
n - к
in fo r m a tio n
in to
the
t h e c a e s f o r BLUS r e s i d u a l s
f
REFERENCES
CO
Co o k
D. R. ,
[2]
K o e r t s
W e i s b e r g
J.
(1 9 6 7 ) ,
R e g re ssio n A n a ly s i s ,
Ľ3 ]
K o e r t s
J.,
and A p p lic a tio n
[4]
Some F u rth e r N o tes
H.
P h i l i p s
(1 9 6 9 ) ,
R e s id u a ls and
I n f lu e n c e
on D istu rb a n c e E s tim a te s in
J . Amer. S t a t i s t . A s s o c . ,
o f th e G eneral
N e u d e č k e r
(1982 ) ,
New York.
A b r a h a m s e
S t a t i s t . A sso c.,
[5]
S .,
Chapman and H a l l ,
in R e g r e s s io n ,
A. P.
I.
L in e a r M odel,
1067-1079.
(1 9 6 9 ) ,
On
th e
Theory
R otterdam U n i v e r s i t y P re s s.
A N ote on BLUS E s tim a tio n ,
J,
Amer.
949-952.
G. D. A. ,
S e r i a l C o r r e la tio n
H a r v e y
i n R e g re ss io n
A. C.
A n a ly s i s ,
(1974),
A S im p le T e s t f o r
J . Amer. S t a t i s t .
A s s o c .,
935-939.
[6]
T a y l o r
L.
[in :]
E rrors,
(1 9 7 4 ) ,
E s tim a tio n by
F r o n tie r s i n
M in im izin g th e
E c o n o m e tric s ,
ed .
Sum o f
P. Zarembka,
A b s o lu te
New
York
1974.
[7]
T h e i 1
n a ly s is ,
[8]
H.
(1 9 6 5 ) ,
The A n a ly s is
o f D istu rb a n c e s
in R e g re ss io n
A-
J . Amer. S t a t i s t . Assoc.
T h e i 1
H.
b lis h in g C o .,
(1 9 7 1 ) ,
p r in c ip le s o f
N orth
E c o n o m e tric s,
H o lland
Pu-
Amsterdam.
Z b ig n iew W a silew ski
UWAGI 0 RESZTACH BLUS
W p rac y rozważa s i e problemy
wadzącego do
o trz y m a n ia e s t y m a t o r a w e k to ra r e s z t m .n .k .
w aria n cji ko w ariancji,
proponcwano
w przypadku w ystępow ania
w y k o rz y s ta n ie zerowych r e s z t
n i m a l i z u j ą c e j sumę
cji.
d o ty c z ą c e wyboru bazy p r z e k s z t a ł c a n i a ,
estym acji
o k r e ś l a n i a bazy t e j
Umożliwia t o u n i k n i ę c i e wyboru o b s e r w a c j i ,
m ac ierzy
n ietyp o w y ch.
otrzymanych w wyniku
o d ch y le ń bezwzględnych do
c z n ie popraw ia j a k o ś ć e s t y m a c j i .
o skalarnej
obserw acji
pro-
Zami-
transform a-
nietypow ych d l a b a z y , co zna-