Causal Linear Stochastic Dependencies: The Formal Theory

TRENDS IN MATHEMATICAL PSYCHOLOCY E. Dexreef and J. Van Bu e n h u t (edilors) 0 Ekevier Science Publisgrs B. V. (North-Holland), 1984

317

CAUSAL LINEAR STOCHASTIC DEPENDENCIES: THE FORMAL THEORY R o l f Steyer University o f Trier T r i e r , Federal R e n u b l i c o f Germany

The f o r m a l background o f t h e t h e o r y o f causal l i n e a r s t o c h a s t i c deoendence i s p r o v i d e d , which was i n t r o d u c e d by S t e y e r (1984).

The t h e o r y p r e s e n t e d i s

concerned w i t h those k i n d s o f deDendencies which can be d e s c r i b e d by s o e c i f y i n g t h e f u n c t i o n a l form of a c o n d i t i o n a l e x o e c t a t i o n E(Y1X).

This includes also

those s i t u a t i o n s i n which X i s a m u l t i d i m e n s i o n a l random v a r i a b l e .

The main concepts o f t h e t h e o r y

a r e causal and weak causal l i n e a r s t o c h a s t i c dependencies, t h e d e f i n i t i o n o f which i s based on t h e p r e - and equiorderedness r e l a t i o n s o f s i g m a - f i e l d s and s t o c h a s t i c v a r i a b l e s , on t h e n o t i o n o f p o t e n t i a l d i s t u r h i n q s i g m a - f i e l d s and v a r i a b l e s , as w e l l as o n t h e i n v a r i a n c e and on t h e average c o n d i t i o n s

.

These

concepts a r e f o r m a l l y d e f i n e d and t h e i r p r o o e r t i e s a r e s t u d i e d i n some d e t a i l . Causal l i n e a r stochast i c dependence i s d e f i n e d b y t h e preorderedness cond i t i o n t h a t t h e i n f l u e n c i n g v a r i a b l e i s antecedent t o t h e i n f l u e n c e d v a r i a b l e and by t h e i n v a r i a n c e cond i t i o n , whereas weak causal l i n e a r s t o c h a s t i c dependence i s d e f i n e d by t h e oreorderedness and average condi ti ons

.

Both, t h e in v a r i ance and t h e average

c o n d i t i o n s , and t h e r e f o r e b o t h k i n d s o f causal hynotheses, can e m i r i c a l l y be t e s t e d i n e x n e r i m e n t a l as we1 1 as i n nonexperimental o b s e r v a t i o n a l s t u d i e s . 1. INTRODUCTION

“ C o r r e l a t i o n does n o t Drove c a u s a l i t y ” , i s a s t a t e m e n t u n i v o c a l l y found in

318

R . Steyer

t e x t b o o k s on a p p l i e d s t a t i s t i c s .

However, whenever one l o o k s f o r a d e f i -

n i t i o n o f causal dependence, one e i t h e r f i n d s t r e a t m e n t s on n h i l o s o p h i c a l t h e o r i e s o n l y l o o s e l y r e l a t e d t o t h e concepts o f c o r r e l a t i o n , r e q r e s s i o n , and t h e e x o e r i m e n t a l c o n t r o l t e c h n i o u e s such as r a n d o m i z a t i o n (see e . g . Basozzi (1980) o r H e i s e ( 1 9 7 5 ) ) , o r t r e a t m e n t s o f e x n e r i m e n t a l and q u a s i e x o e r i m e n t a l c o n t r o l t e c h n i a u e s ( s e e e.g. Cook R Camobell ( 1 9 7 9 ) ) .

Althouah

these d i s c u s s i o n s a r e v e r y u s e f u l and i n s t r u c t i v e i n many r e s n e c t s , t h e r e

i s no f o r m a l t h e o r e t i c a l c o n n e c t i o n between t h e c o r r e l a t i o n , r e q r e s s i o n , and a n a l y s i s o f v a r i a n c e models a n n l i e d and t h e c o n t r o l t e c h n i o u e s d i s c u s sed.

Mhat i s , i n f o r m a l terms, t h e d i f f e r e n c e between an a n a l y s i s o f v a r i -

ance model in a n u r e l y o b s e r v a t i o n a l s t u d y a t one hand and i n a randomized exoeriment a t the other? S t e y e r (1983, 1984) has t a k e n some f i r s t s t e n s t o h r i d g e t h e pan between i d e a s o f causal denendence, s t o c h a s t i c models , and e x n e r i m e n t a l c o n t r o l technioues.

The b a s i c Q u e s t i o n r a i s e d i s , which a r e t h e f o r m a l o r o p e r t i e s

t h a t make a s t o c h a s t i c model a causal one?

R e s t r i c t i n g the discussion t o

l i n e a r s t o c h a s t i c deoendencies, i.e. t o t h o s e denendencies w h i c h can be d e s c r i b e d by c o n d i t i o n a l e x n e c t a t i o n s , S t e y e r (1984) nronosed such p r o p e r t i e s d e f i n i n a two tyoes o f nonsnurious o r causal l i n e a r s t o c h a s t i c dependencies, a weak and a s t r o n g one, each o f w h i c h a r e d i s t i n g u i s h e d f r o m noncausal l i n e a r s t o c h a s t i c deoendencies by two c o n d i t i o n s .

-

Beside t h e p r e -

orderedness c o n d i t i o n t h a t t h e i n f l u e n c i n g v a r i a b l e X i s a n t e c e d e n t t o t h e i n f l u e n c e d v a r i a b l e Y , t h e c r u c i a l c o n d i t i o n f o r causal l i n e a r s t o c h a s t i c dependence of a random v a r i a b l e Y on a, p o s s i b l y m u l t i d i m e n s i o n a l , v a r i a b l e X i s t h e i n v a r i a n c e c o n d i t i o n p o s t u l a t i n g t h a t E(YIX,IJ)

random

as=

E(YIX) + HOW h o l d s f o r a l l o o t e n t i a l d i s t u r b i n g v a r i a h l e s 14, where HOW i s a !\r-measurable c o m o o s i t i o n o f a f u n c t i o n H w i t h W .

The i n v a r i a n c e c o n d i -

t i o n s t a t e s t h a t , i f any n o t e n t i a l d i s t u r b i n g v a r i a b l e W i s i n c l u d e d as another c o n d i t i o n i n g v a r i a b l e E(Y!X,W),

,it

o n l y adds t o E(Y1X) i n t h e e q u a t i o n f o r

leaving E ( Y I X ) invariant.

The c r u c i a l c o n d i t i o n f o r weak causal l i n e a r s t o c h a s t i c deoendence, on t h e o t h e r hand, i s t h e average c o n d i t i o n D o s t u l a t i n q t h a t , f o r a l l D o t e n t i a l d i s t u r b i n g v a r i a b l e s GI, E(YIX=x) = /E(YIX=x, W=w) PW(dw) h o l d s , f o r Pxalmost a l l E ( Y I X = x ) .

T h i s c o n d i t i o n means t h a t E(YIX=x) i s t h e average o f

t h e c o n d i t i o n a l e x p e c t a t i o n s E(YIX=x, W=w) across PX-almost a l l E(YIX=x). t i o n can be w r i t t e n :

t h e v a l u e s w o f W, f o r

I f W i s a d i s c r e t e s t o c h a s t i c v a r i a b l e , t h i s equa-

Garsat linear stochactic dependencies

E(YIX=x) =

Iv, E ( Y I X = x , W=W)

319

P(I,l=w).

I t w i l l be shown t h a t t h i s average condition holds, f o r examnle, i f X and a l l p o t e n t i a l d i s t u r b i n a variahles GI a r e s t o c h a s t i c a l l y independent, a cond i t i o n which may be assumed t o hold i n randcmized exDeriments, where X i n dicates group membershin and the potential disturbina variahles 14 represent pronerties of the experimental units ( s u b j e c t s ) before o r a t the tine of the treatment. This nrovides the desired link t o the experimental control technioues such as randomization and matching.

In the formulation of the nreorderedness, invariance and average conditions above, formally undefined terms have been used t h a t have t o be eliminated i n order t o construct a formal theory consisting only of mathematically well-defined terms. So f a r , we have no formal c r i t e r i a t o decide whether o r not a s p e c i f i e d s t o c h a s t i c variable X i s antecedent t o a sDecified s t o c h a s t i c variable Y , and whether o r not a s p e c i f i e d s t o c h a s t i c variable W 2 i s a p o t e n t i a l disturbing v a r i a b l e . Can W1 = X be a notential disturbing variable w i t h respect t o the dependence of Y on X? Can 1J2 = X+Z be a pot e n t i a l disturbinq variable? Are variables mediating between X and Y pot e n t i a l disturbing v a r i a b l e s , o r variables which a r e influenced by Y? I n the following s e c t i o n s , we f i r s t summarize some b a s i c concents of probab i l i t y theory and introduce notational conventions. Then, we give a formal d e f i n i t i o n of the pre- and equiorderedness r e l a t i o n s and study t h e i r prooert i e s . Next, we define notential disturbing sigma-fields a n d variables, which w i l l make the averaae and invariance conditions discussed above well defined formal concepts, too. Then, we give formal d e f i n i t i o n s of causal and weak causal l i n e a r s t o c h a s t i c denendencies , and i n v e s t i g a t e the properties o f these concepts. 2 . SOME BASIC CONCEPTS OF PROBABILITY THEORY AND NOTATION

I n this s e c t i o n , a b r i e f summary and notational concepts o f p r o b a b i l i t y theory a r e given, which a proDer understanding of the theory proposed. the reader i s r e f e r r e d t o Bauer (1974) , Breiman (1977) , Halmos (1969) , o r Loeve (1977, 1978).

conventions of some basic seem t o be e s s e n t i a l f o r For d e t a i l e d introductions , (1968) , GBnsler and S t u t e

3 20

W. Steyer

The fundamental assumntion o f e v e r v s t o c h a s t i c s u b s t a n t i v e ( i .e. p s y c h o l o n i c a l , s o c i o l o g i c a l , e t c . ) model i s t h a t t h e e x n e r i m e n t , o r more g e n e r a l l y , t h e n a r t o f r e a l i t y t o be d e s c r i b e d , can be r e p r e s e n t e d by a n r o b a b i l i t y sDace, t h e d e f i n i t i o n o f w h i c h i s based on t h e f o l l o w i n g concents. Let

F:

=

i1,Z. . . . I be t h e s e t o f n a t u r a l numbers.

A s i g m a - f i e l d A on n i s

d e f i n e d t o be a s e t o f subsets o f I! with t h e f o l l o w i n g t h r e e p r o p e r t i e s . ( b ) I f A E A, t h e n AC E A , where AC := R - A denotes t h e cornC A. (a) Dlement o f A . ( c ) I f (Ai, i € N) i s a sequence o f elements Ai o f A , t h e n t h e i r u n i o n uia

Ai

i s an element o f A .

The i n t e r s e c t i o n o f a f a m i l y (Ai,

i c I ) o f s i a m a - f i e l d s Ai

on n i s a l s o a

I f E i s a s e t o f nonempty s u b s e t s o f R , t h e n tRe s i p s E i s d e f i n e d t o be t h e i n t e r s e c t i o n o f a l l those s i g m a - f i e l d s on R , w h i c h c o n t a i n E as a s u b s e t .

s i a m a - f i e l d on n.

f i e l d A ( E ) g e n e r a t e d by t h e s e t system

A measurable space i s d e f i n e d t o be a p a i r (Q,A)

o f a s e t Q and a sigma-

f i e l d A on ?. L e t ( : ? , A ) be a measurable soace.

A o r o b a b i l i t y m a s u r e P: A

+

[0,11 i s de-

f i n e d t o be a f u n c t i o n a s s i g n i n g each A E A a r e a l n o n n e g a t i v e number w i t h (a) P(0) = 0.

the following three o r o o e r t i e s . sequence o f elements o f A w i t h Ai P(Uia4

Ai)

P(Ai).

= Eia

n A

j ( c ) P(R) = 1.

=

0

( b ) I f (Ai,

iEN) i s a

f o r i # j , then

A o r o b a b i l i t y space can now be d e f i n e d t o be a t r i p l e (n,A,P)

o f a set

c a l l e d t h e s e t of e l e m e n t a r y e v e n t s , a s i g m a - f i e l d A o f s u b s e t s o f n, where t h e elements A

E

A a r e c a l l e d e v e n t s , and a D r o b a b i l i t y measure P: A

which a s s i g n s t h e n r o b a b i l i t y P(A) t o each e v e n t A L e t (C,A) and ( n ' , A ' ) (A,A')-measurable,

be measurable snaces.

E

+

[0,11

A.

P lraoping X: R

+

n' i s c a l l e d

iff, f o r a l l A' E A',

X-'(A')

:=

{U f

C:

X ( W ) E A ' ) E A.

1 1 The s e t X- ( A ' ) := { X - ( A ' ) : A ' E A ' ) i s c a l l e d t h e s i g m a - f i e l d g e n e r a t e d 1 by X and A ' . X- ( A ' ) may a l s o be denoted by A ( X , A ' ) . I f X i s real-valued 1 and N ( 1 ) - d i m e n s i o n a l , A(X,A') = X- ( A ' ) i s a l s o denoted by A ( X ) and A' i s onlRN('), which i s defined t o N(I) be t h e s i g m a - f i e l d g e n e r a t e d b y t h e s e t system o f a l l open i n t e r v a l l s o f understood t o b e t h e Bore1 s i a m a - f i e l d 6

RN('),

N(I) EN.

321

Causal linear stochastic dependencies

L e t (Ri,Ai),

i E I , be a f i n i t e o r i n f i n i t e sequence o f measurable spaces. i E I ) o f (A,Ai)-measurable

A s i g m a - f i e l d generated b y a sequence (Xi, p i n g s Xi:

+

Ri

i.e.

t h e s i g m a - f i e l d s XY1(Ai),

-1

i E I ] := AIUiEIXi

A[(Xi,Ai),

(A,)]

.

I f I = ~ l , . . , , N ( I ) ~ i s f i n i t e and ni

=IR,

a l t e r n a t i v e n o t a t i o n A(X1,...,XN(I))

i n s t e a d o f A[(Xi,Ai),

L e t @,A)

mao-

i s defir.ed t o be the s i g r a - f i e l d generated by t h e u n i o n c f

and ( n ' , A ' )

f o r a l l i E I,we a l s o use t h e

be measurable spaces.

i E I].

An ( n ' , A ' ) - s t o c h a s t i c

i s d e f i n e d t o be a mapDing X: R

X on t h e p r o b a b i l i t y space (R,A,P)

variable +

fi'

A N(1)-dimensional r e a l - v a l u e d s t o c h a s t i c v a r i -

t h a t i s (Aye')-measurable.

)-stochastic variable. N(I) The m e a s u r a b i l i t y c o n d i t i o n i n t h e d e f i n i t i o n o f a ( 2 ' , A ' ) - s t o c h a s t i c

, ) ' i (s Na I @ a b l e , f o r example,

R

vari-

able i m p l i e s t h a t

1 X- (A')

:=

{U

E R:

X(W) E A ' ) E A , f o r a l l A '

E

A'.

T h i s a l l o w s t o d e f i n e t h e d i s t r i b u t i o n Px: E , ' + t0.11 o f a ( R ' , A ' ) - s t o c h a s t i c v a r i a b l e X by

1 PX(A') = P[X- ( A ' ) ] , f o r a l l

A'EA'.

The d i s t r i b u t i o n P x o f X i s a D r o b a b i l i t y measure on A ' .

3. PRE- AND EQUIORDEREDNESS

A necessary c o n d i t i o n f o r a s t o c h a s t i c dependence o f Y on X t o be causal i s t h a t X i s antecedent o r , synonymously, p r e o r d e r e d t o Y . I n exoeriments, f o r example, t h e t r e a t m e n t v a r i a b l e s a r e m a n i p u l a t e d b e f o r e t h e e f f e c t s on t h e dependent v a r i a b l e s a r e assessed, and i n nonexperimental s t u d i e s , t o o , t h e i n f l u e n c i n g v a r i a b l e s have t o be antecedent t o t h e i n f l u e n c e d v a r i a b l e s , i f a causal s t a t e m e n t s h o u l d make sense a t a l l .

Even r e c i p r o c a l causal de-

pendence can much b e t t e r be t h o u g h t o f as a process o f mutual i n f l u e n c i n g o f t h e v a r i a b l e s i n v o l v e d (see e.g. S t e y e r ( 1 9 8 2 ) ) , where, a t each p o i n t o f t h e process, t h e c a u s i n g v a r i a b l e i s D r e o r d e r e d t o t h e caused one.

Thus,

r e p r e s e n t i n g r e c i p r o c a l causal r e l a t i o n s by dynamic models n o t o n l y a l l o w s t h e p r e s e r v a t i o n o f a s y m e t r y o f i n f l u e n c e (see e.g. Simon ( 1 9 5 2 ) ) f o r r e c i p r o c a l c a u s a l i t y , b u t i s a l s o much more congruent w i t h t h e dynamic n a t u r e

322

R . Steyer

Preorderedness o f a s t o c h a s t i c v a r i a b l e X t o a second one,

Y,

i s t h e f i r s t concent t o b e d e f i n e d w h i c h draws on t h e f a c t t k l a t (n,A,P)

is

of r e a l i t y .

assumed t o r e p r e s e n t a n r o c e s s .

A ( ~ ~ ' , A ' ) - s t o c h a s t i cnrocess on a n r o b a b i l i t v snace be a f a m i l y (Xt,

t C T) o f (n',A')-stochastic

(n,A,p)

i s defined t o

v a r i a h l e s on (n,A,P).

one m i o h t t h i n k o f d e f i n i n q Xs t o be n r e o r d e r e d t o Xt, However, such a concent w o u l d be t o o r e s t r i c t e d .

Hence,

i f s < t, s , t E T .

Oftentimes, variables

have t o be c o n s i d e r e d t h a t a r e d e f i n e d , f o r examnle, bv X := X

S

. Xt,

s#t.

O b v i o u s l y , t h e d e f i n i t i o n o f nreorderedness c o n s i d e r e d above c o u l d n o t be a n n l i e d t o X, because n o t o n l y one b u t two D o i n t s o f t i m e a r e i n v o l v e d i n t h e d e f i n i t i on o f X

.

These problems a r e n o t o n l y o f t h e o r e t i c a l , b u t a l s o o f much n r a c t i c a l i n terest.

I f , f o r examnle,

the e f f e c t o f a psycholooical V e s t r e n t (theraoy,

t r a i n i n g , e t c . ) i s t o be i n v e s t i q a t e d , such a t r e a t m e n t u s u a l l y extends over several sessions.

Hence, i f we l e t T be t h e i n d e x s e t o f t h e s e s s i o n s ,

t h e n t h e t r e a t m e n t v a r i a b l e , i n d i c a t i n g whether o r n o t a n e r s o n r e c e i v e s a t r c t a t w n t , cennot he a s s i a n e d t o one s i n g l e t E T . i f we l e t

T

The same n r o b l e m o c c u r s ,

denote t h e c o n t i n u o u s t i m e covered hy one s e s s i o n .

t i o n o f t h e D s y c h o l o o i s t cannot be a s s i g n e d t o a s i n a l e t o u t l i n e d above.

E

An i n t e r v e n T i n t h e way

S i m i l a r l y , reDeated e v e n t s , such as b e i n o cheated repeated-

l v h v one's mother, may he a cause o f someone b e i n g d i s t r u s t f u l .

Again,

t h e c a u s i n g v a r i a b l e cannot be a s s i a n e d t o a s i n q l e P o i n t of t i m e , a p r o blem which Suppes (1970) seems t o n e o l e c t ( c f . S t e y n i i l l e r (1983) p. 6 0 2 ) ) .

3! .J.

-

!

F i g u r e 1.

A m o n o t o n i c a l l y i n c r e a s i n g f a m i l y (At,t

E

T ) o f sigma-fields.

323

Grusal linear stochartic dependencies

I n o r d e r n o t t o r u n i n t o these d i f f i c u l t i e s ,

we b a s i c a l l y f o l l o w t h e ap-

nroach proposed by S t e y e r (1983). which i s , however, c o r r e c t e d i n some p o i n t s and s t u d i e d i n m r e d e t a i l . (Xt,

T h i s apnroach i s n o t based on a process

t E T ) , b u t on a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y (At,

fields,

t E T) o f siama-

(see F i g u r e 1). L e t T be a s u b s e t o f t h e s e t R o f r e a l numbers.

A monotonically increasina f a m i l y ( A t ,

t E T) o f s i p m a - f i e l d s i s d e f i n e d

t o be a f a m i l y o f s i g m a - f i e l d s At ofi a s e t R, t E T C X R , w i t h t h e p r o p e r t y that, i f s

t,

f

s,t E

T, then As

C At.

Such a m o n o t o n i c a l l y i n c r e a s i n g

f a m i l y o f s i g m a - f i e l d s i s o b t a i n e d , f o r examole, i f we d e f i n e At t o be t h e s i g m a - f i e l d generated by a l l Xs, s E T, s 6 t, where (Xt, t o be a r e a l - v a l u e d s t o c h a s t i c p r o c e s s . t i c nrocess (Xt, (At,

t

E

t

E

T ) i s assumed

However, we do n o t need a stochas-

T) t o construct a monotonically increasing f a m i l y

t E T) o f siama-fields,

as i s shown i n t h e f o l l o w i n g examDle.

3.1. EXAMPLE Consider an e x p e r i m e n t i n which two c o i n s , each h a v i n g one metal and one o l a s t i c s i d e , a r e t o s s e d o n t o a p l a t e h a v i n g t h e p r o p e r t i e s of an e l e c t r o maanet which i s on o r o f f w h i l e t h e two c o i n s a r e tossed. ( F o r a more det a i l e d d e s c r i p t i o n o f t h i s example, see S t e y e r , (1984)).

I n t h i s applica-

t i o n , we may choose R t o be t h e s e t o r o d u c t o f t h e s e t s R l = Ial,a21, Ibl,b2}, and n3 = {c1,c21. Hence, each element (ai,b.,c ) o f R denoJ k t e s one o f t h e e i g h t elementary events t h a t Coin 1 shows s i d e i, Coin 2

O2 =

shows s i d e j, and t h e e l e c t r o magnet i s o f f ( k = l ) o r on ( k = O ) .

Further-

t o be t h e s e t o f a l l subsets o f a .

more, we choose t h e s i g m a - f i e l d A on

I f X denotes t h e s t o c h a s t i c v a r i a b l e i n d i c a t i n a t h e s t a t e o f t h e e l e c t r o magnet and t h e v a r i a b l e s Yi,

i=1,2,

i n d i c a t e t h e outcome o f t o s s i n g c o i n i,

a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y (At, t E T ) o f s i c m a - f i e l d s i s e a s i l y obt a i n e d i f we d e f i n e T := {1,21, A1 := A(X), and A 2 := A(X,Y,,Y,), A(X,Y1,Y2)

where

denotes t h e s i g m a - f i e l d generated by t h e r e a l - v a l u e d s t o c h a s t i c

v a r i a b l e s X, Y1,

Y2.

The example above, i n v o l v i n c j o n l y t h r e e v a r i a b l e s X, Y1 and Y 2 and t h e i r generated s i g m a - f i e l d s ,

i s very simple.

However, i t w i l l h e l p t o i l l u s -

t r a t e t h e concept o f oreorderedness t h a t i s d e f i n e d as f o l l o w s . 3.2.

DEFINITION

L e t (n,A,P)

be a p r o b a b i l i t y space, l e t T be a s u b s e t o f t h e s e t l R of r e a l

313

R . Steyer

numbers, l e t (At, f i e l d s w i t h A(uET b l e s on (c?,A,P), (i)

Lfe say t h a t

t F T) be a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y o f sigma-

At) C A , l e t X,

'$1

be ( P ' , A ' ) - ,

(n",A")-stochastic

varia-

r e s n e c t i v e l y , and f i n a l l y , l e t C , 0 C A be two s i g m a - f i e l d s . C i s preordered t o

P w i t h r e s p e c t t o (A t '- t

E T), i f f

( ' i f f ' i s an a b r e v i a t i o n f o r ' i f and o n l y i f ' ) (a) there i s a

s

E

A s , fl

T, w i t h C

( b ) t h e r e i s an element t

E

T, t >

p

AS and

s , w i t h 1) C At,

As, for a l l

0

S E T , s < t.

( i i ) 'Je say t h a t X i s n r e o r d e r e d t o W w i t h r e s p e c t t o (At,

t

T),

iff

C o n d i t i o n s ( a ) and ( b ) h o l d w i t h 1 ( C ) C = X- ( A ' ) , 1 = U - I ' A ' ).

I f no c o n f u s i o n i s p o s s i b l e , t h e e x n l i c i t r e f e r e n c e t o (At, t t T) may be A c c o r d i n g t o C o n d i t i o n s ( a ) and ( b ) of E e f i n i t i o n 3.2,

omitted.

a smallest t E T f o r which

s

F

T, s

<

D is

there i s

a s u b s e t o f A t and t h a t t h e r e i s an element

t, such t h a t C i s a s u b s e t o f A,.

I n t h e examnle d i s c u s s e d

! and above, where a l l v a r i a b l e s i n v o l v e d a r e r e a l - v a l u e d ( i . e . R' = 9'' = R A' = A " = 8 , where B denotes t h e Bore1 s i g m a - f i e l d on R ) , X i s o r e o r d e r e d 1 t o Y 1 and Y2, because A ( X ) : = X- ( R ) A1, whereas A(Y1) : = Y,-l(R) ! .$ A1 -1 I n t h i s example, Y1 and A ( Y 2 ) : = Y 2 ( B ) A1 , b u t A(Y1), A(Y2) C A2. and Y 2 a r e e q u i o r d e r e d i n t h e f o l l o w i n g sense. 3.3. DEFINITION L e t t h e Dresumptions and n o t a t i o n s o f 3.2 be v a l i d . (i)

We say t h a t C and 0 a r e e q u i o r d e r e d w i t h r e s p e c t t o (At, ( a ) t h e r e i s an element t E T w i t h C, ( b ) t h e r e i s no element s E T,

s

D c At

and

< t, w i t h C C

t E T), i f f --

AS o r 1) C A,.

( i i ) We say t h a t X and I d a r e e q u i o r d e r e d w i t h r e s p e c t t o (Att,

t E T), i f f

C o n d i t i o n s ( a ) and ( b ) h o l d w i t h ( c ) C = X- 1( A ' ) and D = k J - ' ( A ' ' ) . A c c o r d i n g t o C o n d i t i o n s ( a ) and ( b ) o f D e f i n i t i o n 3.3, t h e r e i s a s m a l l e s t element t E T w i t h C and P b e i n g b o t h subsets o f At, and t h e r e i s no e l e ment s E T, s < t, such t h a t C o r 0 a r e subsets o f A s . I t i s e a s i l y seen t h a t t h e diagram o f any r e c u r s i v e o a t h a n a l y s i s model can

be t r a n s l a t e d i n t o a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y o f s i g m a - f i e l d s so t h a t t h e concepts o f p r e - and equiorderedness can be a p p l i e d . nram o f F i g u r e 2, f o r examole, we may d e f i n e T := {1,2,31,

For the path diaA 1 := A(Z

1' Z 2 ) ,

Causal linear stochattic dependencies

325

.

A2 := A(Z1,Z2,Z3) , a n d A3 := A(Z,,. . ,Z4). Obviously, Z1 and Z2 a r e equiordered, Z1 and Z2 a r e both preordered t o Z3 and Z4, and Z3 i s nreordered t o Z4 with respect t o ( A t , t E T) ( s e e Definitions 3.2 and 3 . 3 ) .

Fipure 2.

Path diagram f o r a recursive model w i t h four v a r i a b l e s .

We now t r e a t some formal Drooerties of the pre- and equiorderedness relat i o n s . The f i r s t one i s t h a t preorderedness of sigma-fields, as well as o f s t o c h a s t i c v a r i a b l e s , i s a s t r i c t order r e l a t i o n . Note t h a t a l l propos i t i o n s in the followincl theorems a r e made w i t h respect t o a given family ( A t , t E T ) of sigma-fields. 3.4. THEOREM

Let the presumptions and notations of 3.2 be v a l i d , l e t F C A be a sigmaf i e l d a n d l e t X, W, U be ( n ' , A ' ) - , (n",A")-, and (n"' ,A"')-stochastic variables on the probability space ( n , A , P ) , respectively. ( i ) I f C i s preordered t o 77, then D i s not preordered t o C (asymmetry). ( i i ) I f C i s preordered t o t, and Q i s preordered t o F, then C i s a l s o preordered t o F ( t r a n s i t i v i t y ) ( i i i ) Preorderedness of sigma-fields i s a s t r i c t order r e l a t i o n on the s e t of a l l sigma-fields being subsets of A. ( i v ) Propositions ( i ) t o ( i i i ) a l s o hold f o r s t o c h a s t i c variables X, C' and U taking the r o l e of C, Q, and F, resoectively.

.

Proof. -

We only have t o show t h a t the properties ( i ) and ( i i ) hold f o r sigmaf i e l d s , because a s t r i c t order r e l a t i o n i s defined by the assymetry and t r a n s i t i v i t y p r o p e r t i e s , and preorderedness of s t o c h a s t i c variables i s defined through t h e i r generated sigma-fields. I f C i s preordered t o D w i t h respect t o (At, t E T ) , then there (i)

326

R . Stcyer

i s an element s

E

T with C

C

AS, 0

p

A S . A s (At,

c a l l y i n c r e a s i n o , t h e r e i s n o element t C

E

t E T ) i s monotoni-

T, t > s , such t h a t D

p At, which i m n l i e s t h a t P i s n o t p r e o r d e r e d t o If C i s D r e o r d e r e d t o D, t h e n l a ) there i s a s F T w i t h C c A S , D As.

C.

st

At,

C

At,

(ii)

i s p r e o r d e r e d t o F , then

If

P

T, t > s, w i t h

(b) there i s a t

c At,

f

F c

( c ) t h e r e i s an element u F T, u > t, u i t b

and

AL, F p A s , f o r a l l

s F T , s < u .

Hence, C o n d i t i o n s ( a ) t c ( c ) i m p l y t h a t t h e r e i s an element s F p/ith C C A c ,

F

c A U'

F

iA

F

$T A , ,

and t h e r e i s an e l e m e n t u

~ f, o r a l l s

T, s

<

C

T, u

T

> s, with

u.

Theorem 3.4 does n o t i m o l y t h a t a l l s i g m a - f i e l d s b e i n q subsets o f A a r e i n t h i s r e l a t i o n , b u t t h a t t h e asymmetry and t r a n s i t i v i t y c o n d i t i o n s a r e f u l fil led.

3.5.

THEOREM

L e t t h e presumptions and n o t a t i o n s o f 3 . 2 and 3.4 be v a l i d . ordered t o

D, then

C i s a l s o preordered t o F := A ( C

Proof. -

I f C i s preordered t o

D,

U

I f C i s pre-

U).

then

( a ) t h e r e i s an element s E T w i t h C c A s , U $ A s , and ( b ) t h e r e i s an element t E T, t > s , w i t h P C A D ' (At, t E T ) b e i n q m o n o t o n i c a l l y i n c r e a s i n g , i m p l i e s

t

( c ) C c At,

F c At,

because C ,

D c At imDlies F

C

p As, d

S E T . s
At.

( d ) F i s n o t a s u b s e t o f A S , s E T, s i t, because o f ( b ) . Hence, t n e r e i s an e l e m e n t s t T w i t h L' C A S , F $T AS (see ( a ) and ( a ) ) , and t h e r e i s a t

T, t > s , w i t h F C At,

F

AS, V s t

T, s

< t (see

( c ) and ( d ) ) , which proves t n e theorem A c c o r d i n q t o t h e f o l l o w i n g theorem, equiorderedness i s an e q u i v a l e n c e r e l a tion, 3 . 6 . THEOREM

L e t t h e oresumptions and n o t a t i o n s o f 3 . 2 and 3 . 4 be v a l i d . ( i ) I f t h e r e i s a t 6 T w i t h C C A t and no s E T, s < t, w i t h C C i s equiordered t o i t s e l f ( r e f l e x i v i t y ) .

C

A s , then

&sol

327

linear stochatic dependencies

( i i ) I f C and D a r e e q u i o r d e r e d , t h e n D and C a r e e q u i o r d e r e d (symmetry). ( i i i ) I f C and P as w e l l as D and F a r e e q u i o r d e r e d , t h e n C and F a r e e q u i ordered ( t r a n s i t i v i t y ) . ( i v ) Equiorderedness o f s i g m a - f i e l d s i s an e q u i v a l e n c e r e l a t i o n on t h e s e t o f a l l s i g m a - f i e l d s Ci Ci

c At and no s

C

A , f o r which t h e r e i s an element t

E

T with

T, s < t, w i t h Ci c A S .

E

( v ) P r o n o s i t i o n s ( i ) t o ( i v ) a l s o h o l d f o r equiorderedness o f s t o c h a s t i c v a r i a b l e s X, W, 11 t a k i n g t h e r o l e s o f C , D, and F, r e m e c t i v e i y . Proof. (i) (ii)

C o n d i t i o n s 3.3a and 3.3b a r e o b v i o u s l v t r u e f o r C = P . I s obvious.

( i i i ) I f C and U a r e e q u i o r d e r e d , then ( a ) t h e r e i s a s m a l l e s t t E T such t h a t C , P

C

At and no s E T,

C As o r D c: As. F are equiordered, t h i s imDlies t h a t also

s < t, w i t h C

If

1)

anL

( b ) F c At and t h a t ( c ) t h e w i s no s E T, s < t, w i t h P C As o r F

(1

As.

( a ) t o ( c ) i m p l y t h a t t h e r e i s a t E T w i t h C, F C At,

whereas ( a )

F C A,. F a r e e q u i o r d e w d , which i m p l i e s t h e t r a n s i t i v i t y

and ( c ) i m p l y t h a t t h e r e i s no s E T, s < t, w i t h C C AS o r Hence, C and omperty

.

(iv)

An e q u i v a l e n c e r e l a t i o n i s d e f i n e d b y r e f l e x i v i t y , s y m t r y , and

(v)

T h i s o r o n o s i t i o n i s i m o l i e d by 3.6i t o 3 . 6 i v Y i f we d e f i n e -1 C := X ( A ' ) , P := l!J-'(A"), and F := U-'(A''I ) .

transitivity.

I t s h o u l d be n o t e d t h a t n o t e v e r y s i g m a - f i e l d C

c A i s equiordered t o i t -

I t may hapaen t h a t C i s n o t i n t h e equiorderedness r e l a t i o n , because

self.

t h e r e i s no s m a l l e s t t E T w i t h C c At.

I f T = R , C = IB,n}, f o r example,

t h e r e i s no s m a l l e s t t E T w i t h I @ s ~ C} At.

The e q u i v a l e n c e r e l a t i o n o f

equiorderedness i s n o t d e f i n e d on t h e s e t o f a l l s i g m a - f i e l d s

CiC

A, b u t

o n l y on t h e s e t o f a l l Ci C A , f o r which t h e r e i s a s m a l l e s t t E T w i t h Ci c A t .

? .7. THEOREM L e t t h e n w s u m p t i o n s and n o t a t i o n s o f 3.2 and 3.4 be v a l i d . I f C and

D a r e e q u i o r d e r e d , t h e n C and F

: = A(C

u P ) , as w e l l as D and

328

R. Steyer

T-' a r e e n u i o r d e r e d . Proof. -

I f C and ? a r e e q u i o r d e r e d , then t h e r e i s a s m a l l e s t t

F

T such

tlltlt

( a ) C,

P c A t , and

( b ) t h e r e i s no s F T, s < t, w i t h C C AS o r f i

C o n d i t i o n ( a ) i m p l i e s F C At,

C

because F i s t h e s m a l l e s t s i g m a - f i e l d

c o n t a i n i n q b o t h r and P as s u b s e t s .

Condition (h) i m o l i e s t h a t

t h e r e i s no s F- T, s < t, w i t h C, ?, o r F C A 5 . C aid

A5.

This implies t h a t

F as w e l l as D and F a r e e q u i o r d e r e d .

3.P,. THF09Fb1 L e t t h e n w s u m o t i o n s and n o t a t i o n s o f 3.2 and 3.4 be v a l i d . ( i ) I f C i s preordered t o dered t o F .

D, and

i, and

F a r e e q u i o r d e r e d , then C i s n r e o r -

( i i ) I f C , P a r e e q u i o r d e r e d and P i s n r e o r d e r e d C.c F, t h e n C i s D r e o r d e r e d

t o F. ( i i i ) P r o o o s i t i o n s ( i ) and ( i i ) a l s o h o l d f o r s t o c h a s t i c v a r i a b l e s X,

01

and

I1 t a k i n o t h e r o l e s o f C , D and F , r e s o e c t i v e l y . Proof. -

(i)

I n t h i s case, ( a ) t h e r e i s an element s F

t E T, t > s, w i t h P

C

T with At,

C

C

0 $ At,

F

As and an element f o r a l l s E T, s < t .

As, I,

Furthermore, (b) F

C

A t and t h e r e i s no s E T, s < t, w i t h ?

for all s

F

C

As o r F C AS,

T, s < t .

( a ) and ( b ) i m p l y t h a t t h e r e i s a s E T w i t h C C A?.

r

an c i e f i e r t t E T, t > s, w i t h F E At,

F p AS, f o r a l l s

which i m p l i e s t h a t C i s o r e o r d e r e d t o

F (see 3 . 2 ) .

(ii)

A s , atiu E

T, s < t,

Is o b v i o u s .

( i i i ) T h i s o r o o o s i t i o n i m m e d i a t e l y f o l l o w s f r o m 3 . 8 i and 3 . 8 i i , define

r

:= Y - l ( A ' ) ,

? := W - ' ( A " ) ,

an(!

F := \I-'(A'''

i f we

).

Another conceot needed i n t h e f o l l o w i n g s e c t i o n s i s t h a t o f e o u i - o r p r e o r deredness, where ' o r ' denotes t h e l o c i c a l d i s j u n c t i o n . edness i s a weak o r d e r r e l a t i o n .

Equi- o r nreorder-

Aoain, t h e o r o o o s i t i o n s a r e always made

w i t h r e s m c t t o an u n d e r l y i n q f a m i l y (At,

t E

T) o f s i g m a - f i e l d s , t h e

Grusal linear stochastic dependencies

329

exnlicit reference t o which i s omitted only for reasons of convenience. 3.9. DEFINITION

Let the nresumptions and notations of 3.2 be valid. ( i ) Isle say t h a t C i s equi- or nreordered t o 0 , i f f ( a ) C and 0 are enuiordered, or ( b ) C i s preordered t o D. ( i i ) Ire say t h a t X i s enui- or oreordered t o 'J, i f f C i s equi- or preordered t o P a n d ( c ) C = X- 1(A') and P = l , f - l ( A " ) . 3.10. THEOREPI

Let the nresumptions and notations of 3.2 and 3.4 be valid. ( i ) I f there i s a t E T with C C: At such t h a t there i s no s E T , s < t. with C C A,, then C i s equi- or prcordered t o i t s e l f (reflerivi t y ) . ( i f ) If C i s equi- or oreordered t o 0 , and D i s equi- or nreordered t o C , then C and U are equiordered (antisymnetrv). ( i i i ) I f C i s equi- or oreordered t o P , and P i s equi- or oreordered t o F, then C i s equi- or preordered to F ( t r a n s i t i v i t y ) . ( i v ) Equi- or preorderedness o f s i y a - f i e l d s i s a weak order relation on the s e t of all sicma-fields Ci C A , for which there i s a t E T with Ci C A

t such t h a t there i s no s T, s < t, with C C A,. ( v ) Pronositions 3.10i t o 3.10iv also hold for equiorderedness of stochast i c variables X, W , U taking the roles of C, P, and F, resnectively. Proof. -

ProDerty 3.10i follows from 3.6i. In order t o prove 3.1Oiiy we consider four cases. First, b o t h C and U are preordered t o one another, w h i c h can be excluded, because of 3.4i. Second, C and D are equiordered. In this case, the pronosition 3.10ii i s imnlied by 3.6ii. Third, C i s preordered t o P and D i s equiordered t o C . This contradicts Condition ( b ) of Definition 3.3. Fourth, U i s oreordered t o C , and C i s equiordered t o U . Again, this case cannot occur, because of 3.3b. The proof of prooerty ( i i i ) follows the same line of arguments as t h a t o f property 3.10ii. For the f i r s t case t h a t C , 0 and 0, F are equiordered, the property i s imolied by3.6iii. For the second case t h a t C i s preordered t o C and U i s nreordered t o F, ProDerty 3.lOiii i s

330

R. Steyer

i m n l i e d by 3 . 4 i i i .

F o r t h e t h i r d case t h a t C i s n r e o r d e r e d t o

U , and

0 and F a r e e q u i o r d e r e d , i t i s i m o l i e d hy 3 . 8 i , and f o r t h e l a s t case t h a t C and I! a r e e q u i o r d e r e d , whereas D i s o r e o r d e r e d t o F , i t i s i m ( i v ) A weak o r d e r r e l a t i o n i s r'efined by r e f l e x i v i t y ,

o l i e d by 3 . 8 i i .

antisymmetry, and t r a n s i t i v i t y . t o ( i v ) , i f we d e f i n e f : = X - ' ( A ' ) , 3.11.

( v ) This n r c n o s i t i o n f o l l o w s from ( i )

D

: = ~ ~ ~ " ( A " and ) , F := U-'(A''').

THEOREM

L e t t h e nresumptions and n o t a t i o n s o f 3.2 and 3.4 be v a l i d .

o r o r e o r d e r e d t o P , then C i s a l s o e q u i - o r o r e o r d e r e d t o F ?roof. -

( a ) I f C i s p r e o r d e r e d t o P , then, accordin!

I f C i s equi:= A ( C

u 27).

t o 3.5i, C i s also preor-

dered t o F . ( b ) I f C and

D

a r e e q u i o r d e r e d , then, a c c o r d i n g t o 3 . 7 i , C and F a r e

also equiordered.

( a ) and ( b ) i m l y t h e theorem.

4 . POTENTIAL DISTURRING SIGVA-FIELDS AND VPRIABLES

Ey i n t u i t i o n , i t i s obvious t h a t nreorderedness o f X t o

Y i s a necessary

b u t n o t s u f f i c i e n t c o n d i t i o n f o r t h e s t o c h a s t i c denendence o f Y on X t o he causal.

Two a l t e r n a t i v e second necessary c o n d i t i o n s , t h e i n v a r i a n c e and

a v e r a w c o n d i t i o n s have been d i s c u s s e d by S t e y e r (1984).

To s t a t e e i t h e r

of them o r e c i s e l y , we need a f o r m a l d e f i n i t i o n o f a n o t e n t i a l d i s t u r b i n g v a r i a b l e and a p o t e n t i a l d i s t u r b i n g s i q m a - f i e l d .

The aim i s t o d e f i n e v a r i -

a b l e s which a r e p o s s i b l y confounded w i t h X and thus make t h e denendence o f Y on X a s p u r i o u s one. !n t h e f o l l o w i n g d e f i n i t i o n we need t h e conceDt o f p r o b a b i l i t y measure t h a t i s n o n t r i v i a l w i t h r e s p e c t t o a s i g m a - f i e l d C. w h i c h i s d e f i n e d t o b e a Drob a b i l i t y measure Q on f such t h a t t h e r e i s a t l e a s t one C

0

q(c)

<

E

C with

1.

4.1. DEFINITION L e t (n,A,P)

be a p r o b a b i l i t y sDace, l e t (Aty t E T ) , T C l R , be a monotoni-

c a l l y increasing family o f sicma-fields w i t h A denote s i g m a - f i e l d s , variables

on (n,A,P),

(

~ At)J c ~ A,~ l e t C y D , F c A

and f i n a l l y l e t X, W be (n' , A ' ) - , respectively.

(n",A")-stochastic

331

Cuusal linear stochatic dependencies

D i s c a l l e d p o t e n t i a l d i s t u r b i n g sigma-field w i t h respect t o -(Aty t E T) and C l o r , ifC = X-l(A')y w i t h r e s D e c t t o (Aty t E T) and X I , (i)

i f f t h e f o l l o w i n g two c o n d i t i o n s h o l d : (a)

D

i s e q u i - o r n r e o r d e r e d t o C , and

( b ) t h e r e i s a p r o b a b i l i t y measure r) on A t h a t i s n o n t r i v i a l w i t h r e s c e c t t o b o t h C and D such t h a t C and 0 a r e s t o c h a s t i c a l l y independent. (ii)

-(At,

181

i s c a l l e d a D o t e n t i a l d i s t u r b i n g variable w i t h resDect t o

t

E

T) and C [ o r , i f C = X-l(A')y

w i t h r e s n e c t t o (At,

t

E

T)

and 8 ,

i f f C o n d i t i o n s ( a ) and ( b ) h o l d , as w e l l as ( c ) U = GI- 1( A " ) . C o n d i t i o n 4 . l a i s chosen i n o r d e r t o e x c l u d e m e d i a t o r v a r i a b l e s f r o m t h e s e t o f potential disturbing variables.

M e d i a t o r v a r i a b l e s , say U, a r e

v a r i a b l e s f o r which X i s n r e o r d e r e d t o U and U i s Dreordered t o Y .

These

v a r i a b l e s do n o t t h r e a t e n t h e v a l i d i t y o f a causal p r o p o s i t i o n b u t may e l a b o r a t e i t . They a r e i m p o r t a n t t o d i s t i n g u i s h between d i r e c t and t o t a l e f 2 f e c t s . C o n d i t i o n 4 . l b i s added t o e x c l u d e s t o c h a s t i c v a r i a b l e s l i k e X, X , ax, e t c . f r o m t h e s e t o f p o t e n t i a l d i s t u r b i n g v a r i a b l e s .

I t i s n o t pos-

s i b l e t o c o n s t r u c t a p r o b a b i l i t y measure (1 on A t h a t i s n o n t r i v i a l w i t h 2 2 r e s p e c t t o A ( X ) and A ( X ) f o r examole, such t h a t X and X a r e s t o c h a s t i c a l l y independent w i t h r e s p e c t t o t u r b i n g variable,

4.2.

then X+U,

X-U,

Q. Furthermore, i f U i s a D o t e n t i a l d i s f o r example, a r e n o t D o t e n t i a l v a r i a b l e s .

EXAMPLE

Consider a s t u d y on t h e e f f e c t i v e n e s s o f a r e t r a i n i n g propram on t h e r a t e o f r e c i d i v i s m ( f o r a more d e t a i l e d d e s c r i p t i o n o f t h e example see S t e y e r (1984)).

We m i g h t choose 0

where (a. ,bj,ck) 1

{(a.,b.,ck): 1 J

i,j,k

= Oyl},

r e p r e s e n t s one o f t h e e i g h t p o s s i b l e e l e m e n t a r y e v e n t s

t h a t a p a r o l e e i s male o r female (ai),

takes p a r t a t t h e r e t r a i n i n g program

o r n o t ( b . ) , and comnits a c r i m e o r n o t a f t e r t h e r e t r a i n i n g ( c k ) . I4e J choose t h e s i g m a - f i e l d A on n t o be t h e s e t o f a l l subsets o f n, which con-

@, the

e i g h t s e t s {(ai,b.,c ) } and a l l unions o f these s e t s . J k t h e r o n , we m i g h t choose t h e s t o c h a s t i c v a r i a b l e s

sists o f

Fur-

331

R. Steyer

1, i f t h e o a r o l e e i s female

II =

0, o t h e r w i s e

1, i f t h e a a r o l e e takes n a r t a t t h e r e t r a i n i n g X = I

0, o t h e r w i s e y = '

1, i f t h e r e i s no r e c i d i v i s m f o r t h e a a r o l e e

0, o t b e w i s e , If

t o r e p r e s e n t t h e e v e n t s t h a t m i o h t be observed. A2 := A(1d.X).

and A3 : = A('l,X,Y),

X i s nreordcred t c Y.

T

: = {1,2,3'*, A1 : = A ( I J ) ,

then '4 i s o r e o r d e r e d t o b o t h X and Y , and

Furthermore, !.I t h e n i s a n o t e n t i a l d i s t u r b i n g v a r i -

i + l e w i t h r e s p e c t t o X and (At,

t

E

T ) , hecause

(1

i s enui- o r preordered t o

X and 4 . l b i s f u l f i l l e d , too, as i s e a s i l y seen, i fwe choose ')(X=x) = l ( l l = w ) = 0.5 and O(X=x,LI=w)

= 0.25,

f o r a l l combinations o f values o f X and

1.1 .

O b v i o u s l y , t h e f o r m a l i z a t i o n o f t h i s example, c o n s i s t i n a o f o n l y t h r e e d i chotomous v a r i a b l e s , i s v e r v r e s t r i c t e d . n a r t o f the r e a l i t y i n v e s t i g a t e d .

I t reoresents o n l v a very small

I t does n o t r e o r e s e n t a l l v a r i a b l e s t h a t

m i g h t be i m p o r t a n t f o r t h e Dhenomena o f r e c i d i v i s m o f o a r o l e e s .

I t does

n o t contain Variables l i k e the parolee's m a r i t a l s t a t u s , the k i n d o f crime committed, b e i n g r a i s e d o r n o t i n a complete f a m i l y , a l backaround, e t c .

the parolees education-

I t would be n o s s i b l e , o f course, t o f o r m u l a t e a new mo-

d e l i n c l u d i n g a l l these v a r i a b l e s .

However, such an e x p l i c i t enumeration

of v a r i a b l e s o c c u r i n o i n t h e D a r t o f r e a l i t y s t u d i e d , w o u l d n e v e r be complete.

T h e r e f o r e , an a l t e r n a t i v e k i n d o f a p o l i c a t i o n o f t h e f o r m a l t h e o r y

s h o u l d be c o n s i d e r e d . F i r s t , one may i n t e r n r e t t h e n r o b a b i l i t y space ( R , A , P )

t o renresent a speci-

f i e d e x n e r i m e n t , o r more a e n e r a l l y , t h e p a r t o f r e a l i t y t o be d e s c r i b e d . Second, one m i q h t d e f i n e T

:=lR and i n t e r p r e t each s i g w a - f i e l d At,

t E T,

t o r e p r e s e n t a l l e v e n t s w h i c h mav o c c u r up t o p o i n t t o f t i m e , i n c l u s i v e l y . F o r t h i s k i n d o f i n t e r p r e t a t i o n s o f t h e formal s t r u c t u r e s (n,A,P) (At,

t

5

T),

and

i t i s n o t necessary t o know a l l t h e e v e n t s i n each A t o r even

enumerate them e x p l i c i t e l y .

I f (n,A,P)

represents a s D e c i f i e d D a r t o f rea-

l i t y , t h e r e i s no a n b i g u i t y about t h e v a r i a b l e s and e v e n t s i n v o l v e d o r about

t h e i r temooral o r d e r i n g so l o n g as i t i s made c l e a r enough w h i c h p a r t o f r e a l i t y i s considered,

333

h r a l linear stochastic dependencies

I f we f o l l o w t h i s second g a t h o f i n t e r n r e t i n s t h e f o r m a l t h e o r y i n t h e s t u -

dy on t h e e f f e c t i v e n e s s o f t h e r e t r a i n i n g w i t h n a r o l e e s , a l l o r o p e r t i e s o f t h e n a r o l e e b e f o r e t h e r e t r a i n i n s a r e o o t e n t i a l d i s t u r h i n a v a r i a b l e s , e.a. t h e k i n d o f t h e p a r o l e e ' s crime, t h e k i n d o f exneriences i n j a i l , e t c . Obviously, a causal a r o n o s i t i o n i s i n f u l l accordance w i t h o u r i n t u i t i o n o n l y f o r t h i s second k i n d o f i n t e r n r e t a t i o n o f t h e s i g m a - f i e l d s At,

allo-

w i n g f o r a l l p o t e n t i a l d i s t u r b i n g v a r i a b l e s , d i s t u r b i n a v a r i a b l e s which a r e t h e r e , even though t h e y m i q h t n o t b e observed. However, so l o n g as one mentions

w i t h r e s p e c t t o which f a m i l y (At,

t E T)

a causal o r o p o s i t i o n i s made, causal o r o p o s i t i o n s as d e f i n e d i n t h e n e x t s e c t i o n a r e m e a n i n g f u l and n o t t r i v i a l , so l o n g as a t l e a s t one p o t e n t i a l d i s t u r b i n q variable i s allowed f o r .

The more o o t e n t i a l d i s t u r b i n g v a r i a b l e s

a r e i m n l i c i t l y o r e x p l i c i t l y a l l o w e d f o r by t h e i n t e r o r e t a t i o n o f t h e siGma f i e l d s At,

t

E

T, t h e more meaninoful a causal p r o n o s i t i o n i s .

At a r e i n t e r p r e t e d t o r e o r e s e n t a l l events t h a t mav o c c u r

UP

Only i f t h e

t o point t o f

t i m e i n c l u s i v e l y , i n t h e r e a l nrocess considered, w i l l causal o r o p o s i t i o n s f u l l y c o i n c i d e w i t h t h e i n t u i t i v e meaning o f t h e term ' c a u s a l ' .

I t should

b e n o t e d , however, t h a t t h e formal t h e o r y prooosed i s n e u t r a l w i t h r e s p e c t t o these q u e s t i o n s o f adequate a o n l i c a t i o n .

5. CAUS4L LINEAR STnCHASTIC DEPENDENCE U s i n g t h e concents o f p r e - and eouiorderedness and o f p o t e n t i a l d i s t u r b i n o v a r i a b l e s t r e a t e d i n t h e n r e v i o u s s e c t i o n s , as w e l l as t h a t o f a c o n d i t i o n a l e x o e c t a t i o n , we mqv now C o r m a l l y d e f i n e causal l i n e a r s t o c h a s t i c dependence and z t u d y i t s f o r m a l p r o p e r t i e s .

As n o t o n l y causal l i n e a r q t n c h a s t i c

dependence, h i i t a l s o independence i s o f i n t e r e s t , we f i r s t d e f i n e causal lin e a r s t o c h a s t i c in/deDendence,

which means dependence o r independence.

Tn-

dependence a d d i t i o n a l l y r e q u i r e s E(Y IC)as=E(Y), and deoendence t h e n e g a t i o n o f it.

5 . 1 . DEFINITION L e t Y be a r e a l - v a l u e d s t o c h a s t i c v a r i a b l e on t h e p r o b a b i l i t y sDace (n,A,P) w i t h f i n i t e e x p e c t a t i o n E(Y), l e t X be a ( n ' , A ' ) - s t o c h a s t i c be a ( n " , A " ) - s t o c h a s t i c l e t (At, A(uET

v a r i a b l e on (n,A,P),

v a r i a b l e and 1.1

l e t C C A be a s i g m a - f i e l d ,

t E T) be a monotonically increasing f a m i l y o f sigma-fields w i t h

At) c A , and l e t E(YIC,D)

e x p e c t a t i o n o f Y g i v e n A(C

U

D).

:= E[YIA(C

U

D)]

denote t h e c o n d i t i o n a l

We say t h a t Y i s c a u s a l l y l i n e a r l y

334

R. Sfeyer t F: T ) , i f f t h e f o l -

s t o c h a s t i c a l l y in/denendent on C w i t h r e s p e c t t o (At, lowing conditions hold: (a)

C i s p r e o r d e r e d t o A(Y) w i t h r e s D e c t t o (At,

t E T) (nreorderedness).

(b)

For a l l o o t e n t i a l d i s t u r b i n ? sigma-fields 0 w i t h resnect t o (At, a r d c,

t E T)

h o l d s , where Ho1.1 denotes t h e comnosi t i o n o f an A"-measurable r e a l - V a l ued f u n c t i o n tJ w i t h a (V,A")-measurable

s t o c h a s t i c v a r i a b l e ',I ( i n v a r i a n c e ) .

IffC o n d i t i o n s ( a ) and ( b ) , as w e l l as ( c ) E ( Y / T ) as= E(Y) ( l i n e a r s t o c h a s t i c indeoendence) h o l d , we say t h a t Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l v independent from D w i t h r e s p e c t t o

(At,

t F T ) , and denendent on C, o t h e r w i s e .

1

If C o n d i t i o n s ( a ) and ( b ) h o l d , as we11 as ( d ) C = X- ( A ' ) , we a l s o say that

Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y i n / d e p e n d e n t on X w i t h r e s o e c t

-

t o (At,

t t T).

I f no c o n f u s i o n i s n o s s i b l e , we mav o m i t t h e e x p l i c i t r e f e r e n c e t o (At,

t E

T).

There a r e a numher o f s i t u a t i o n s i n w h i c h E q u a t i o n (1) h o l d s .

Suppose f o r

example, t h a t t h e c o n d i t i o n a l c x n e c t a t i o n o f t h e Y q i v e n C and V i s t h e

sum o f a c o m p o s i t i o n FOX and a c o m n o s i t i o n HoU, E(Y[C,D) as= FOX + Hob1 where F i s an A'-measurable and H an A"-measurable r e a l - v a l u e d f u n c t i o n , whereas X,

111

s r e ( C , A ' ) - , (D.A")-measurable,

respectively.

T h i s w i l l be

r c f e r r e d t o as the a c ' d i t i v i t y c o n d i t i o n , a s p e c i a l case o f which i s t h e mu1 t i p l e r e g r e s s i o n e q u a t i o n E(Y[ X,IsI)

as= by" + byXX + byMW.

The a d d i t i v i t y c o n d i t i o n a l s o h o l d s , f o r example, i f

whereas i t does n o t h o l d , f o r example, i f

W a f finear stochastic dependencies

335

T h i s l a s t e q u a t i o n c o n t r a d i c t s t h e a d d i t i v i t y c o n d i t i o n , because o f t h e m u l t i n l i c a t i v e t e r m X-11, which i s n e i t h e r C- n o r D-measurable.

5 . 2 . TEEOREM L e t t h e Dresumptions and n o t a t i o n s o f 5 . 1 be v a l i d . (a)

If

f o r a l l p o t e n t i a l d i s t u r h i n g s i g m a - f i e l d s P, E(YIC,P)

as= FOX t Ho\J,

(2)

where F i s an A ' - and H an A"-measurable r e a l - v a l u e d f u n c t i o n , X i s (C,A')and If i s (PYA")-measurable ( a d d i t i v i t y ) ; i f (b)

!' and

111

a r e s t o c h a s t i c a l l y indenendent, and i f

( c ) C i s p r e o r d e r e d t o A(Y) w i t h r e s p e c t t o (At, t E T ) , then Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/denendent on C . Proof. -

I n o r d e r t o prove t h i s o r o p o s i t i o n , we suDpose t h a t t h e e x n e c t a t i o n

E(HoI.1) i s equal t o z e r o .

Note t h a t t h i s does n o t r e s t r i c t g e n e r a l i t y ,

because, b y s u b t r a c t i n g i t s e x p e c t a t i o n , any c o m n o s i t i o n H'oN w i t h an e x p e c t a t i o n unequal zero can e a s i l y be t r a n s f o r m e d i n t o a c c m p o s i t i o n

No:: w i t h e x p e c t a t i o n z e r o .

I t i s e a s i l y seen t h a t t h e c o n d i t i o n s ( a ) and ( b ) o f theorem 5 . 2 i m n l y F q u a t i o n ( l ) , because

(see A.17)

E(Y1X) as= E[E(YIC,D)IXl as= E( FoXtHoYI X)

(see 2)

as= ~ ( ~ 0 x t1 ~ E ()H O W / X ) as= FOX

t

E(Hold) = FOX

( s e e A.15)

,

(see A.9)

where E(Ho191) = 0 and t h e theorem i s used t h a t s t o c h a s t i c independence nf

and X i m p l i e s E(H0lJIX) as= E(HoY).

k'e may now i n s e r t FOX as=

E ( Y ] X ) i n t o Equation ( 2 ) , which y i e l d s E q u a t i o n ( l ) ,which proves t h e theorem. We now t u r n t o a n o t h e r s i t u a t i o n i n which Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/dependent on C.

Sul)pose, E(YlX,W)

t o o , E q u a t i o n (1) i s f u l f i l l e d . w h i c h E(YIX,V)

as= E(Y1X).

I n this situation,

14 b e i n g a c o n s t a n t i s a s n e c i a l case i n

as= E(Y1X) i s t r u e .

Hence, i f t h e r e i s t h e p o s s i b i l i t y t h a t

a n o t e n t i a l d i s t u r b i n g v a r i a h l e i s an a c t u a l one, i t m i g h t be h e l d c o n s t a n t a t one o f i t s values.

Thus, t h e e x p e r i m e n t a l c o n t r o l t e c h n i q u e o f h o l d i n g

o o t e n t i a l d i s t u r b i n g v a r i a b l e s c o n s t a n t , can be based on t h e i n v a r i a n c e condition.

I t s h o u l d be noted, however, t h a t t h e v a l i d i t y o f a causal

336

R. Sreyrr

o r o n o s i t i o n i s then r e s t r i c t e d t o t h e case t h a t LI i s c o n s t a n t a t i t s v a l u e L1.

.

5 . 3 . THEOREr4 L e t t h e oresumntions and n o t a t i o n s o f 5 . 1 b e v a l i d . ( a ! C i s o r e o r d e r e d t o A ( v ) w i t h r e s n e c t t o (At, t

If C

T ) , and i f

( b ) f o r a l l n o t e n t i a l d i s t u r h i n q si?ma-fields P , E(V!C,D) (C-conditional

as= E ( Y ! C ) ,

(3)

l i n e a r s t o c h a s t i c indeoendence o f Y f r o m

D),

t h e n Y i s cau-

s a l l y l i n e a r l y s t o c h a s t i c a l l v i n / d e n e n d e n t on C. Proof. -

E n u a t i o n ( 3 ) i m n l i e s E n u a t i o n ( l ) , because we may d e f i n e H and I.1 such t h a t Hob1 = 0 .

5 . 4 . COROLL73Y L c t tk,e oresumDtions and n o t a t i o n s o f 5 . 1 be v a l i d . A ( Y ) w i t h r e s p e c t t o (At,

I f C i s preordered t o

t E T ) and

(4)

E(Y1C) as= Y

(comnlete dependence of Y on C), then Y i s c a u s a l l y l i n e a r l v s t o c h a s t i c a l l y in/deaendent on C. Proof. -

E(YIC) as= Y i m n l i e s F(vlC,?)

as= E I E ( v l C ) l C , f l

as= E ( Y I C ) ,

f o r a l l sivma-fields

DcA

,

(see A . 1 2 )

which i m D l i e s 5.3b. Hence, causal l i n e a r s t o c h a s t i c deoendence does n e i t h e r r e q u i r e a l l o f t h e v a r i a t i o n o f t h e dependent v a r i a b l e Y t o be determined h y C ( s e e E q u a t i o n (4)),

n o r i s i t necessary t h a t a l l i n f l u e n c i n g v a r i a b l e s t h a t a r e e q u i - o r

n r e o r d e r e d t o C a r e known ( s e e E n u a t i o n ( 3 ) ) . notential disturbing siama-fields

D

It i s only required that a l l

behsve as d e s c r i b e d by E n u a t i o n ( 1 ) .

This equation i s f u l f i l l e d already i f Y i s C - c o n d i t i o n a l l y l i n e a r l y stochast i c a l l y independent f r o m

D, i . e .

i f E q u a t i o n ( 3 ) h o l d s , o r i f X and

s t o c h a s t i c a l l y independent and a d d i t i v i t y can be assumed.

W are

Causal linear stochactic dependencies

337

G. SII.fP1.F: CPVSAL REG-LINEAR DEPEVCENCE I n t h e case o f a s i m p l e r e p - l i n e a r denendence o f Y on X which i s c h a r a c t e r i zed b y t h e s i m p l e r e g r e s s i o n e q u a t i o n E(Y1X) as= ayo

+

ayXX,

(5)

the invariance condition implies t h a t , f o r a l l Dotential d i s t u r b i n g variables

1.1,

Idhatever t h e tyDe o f t h e f u n c t i o n H01.1,

the regression c o e f f i c i e n t

aYX i s l e f t unchanged when t i i r n i n o from E n u a t i o n ( 5 ) t o ( 6 ) , o r o v i d e d t h e i n v a r i a n -

ce condi ti on h o l ds

.

Another remarkable p r o n e r t y o f t h e s i m o l e r e g r e s s i o n c o e f f i c i e n t a Y X i s t h a t i t i s equal t o t h e CorresDonding r e g r e s s i o n c o e f f i c i e n t ayX1,lq,l

of

t h e c o n d i t i o n a l s i m p l e r e g r e s s i o n o f Y on X g i v e n I.!=w:

i f t h e invariance condition holds.

T h i s i s e a s i l y seen f r o m E q u a t i o n ( 6 ) ,

because, f o r #=w, E(vlX,I.l=w) as= aYO + ayXX + H(w), where H(w) i s a c o n s t a n t so t h a t we may d e f i n e ayoIl,,=w

.- aYO + H(w)

and

._

aYXI,,,=w .- ayX. Thus, E q u a t i o n ( 5 ) and t h e i n v a r i a n c e c o n d i t i o n i m p l y , t h a t aYX i s a l s o t h e r e a r e s s i o n c o e f f i c i e n t o f t h e c o n d i t i o n a l s i m p l e r e p r e s s i o n o f Y on X g i v e n I J = w ,

where I d i s any p o t e n t i a l d i s t u r b i n g v a r i a b l e

This motivates the f o l l o w i n g 6 . 1 . DEFINITION

L e t t h e presumptions and n o t a t i o n s o f 5 . 1 be v a l i d .

We say t h a t Y i s s i m p l y

c a u s a l l y r e g - l i n e a r l y in/deDendent on X and t h a t aYX i s t h e s i m p l e causal r e g - l i n e a r e f f e c t o f X on Y , i f f Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y i n / dependent on X and E q u a t i o n ( 5 ) h o l d s .

338

R . Stryer

7 , IVEAK CAUSAL LINEAR STOCHASTIC DEPENDENCE I t may be a r y e d t h a t t h e k i n d o f causal l i n e a r s t o c h a s t i c deoendence d i s cussed above i s u n r e a l i s t i c a l l y r e s t r i c t i v e .

L e t us c o n s i d e r once again

t h e i n v a r i a n c e c o n d i t i o n s a v i n q t h a t , f o r a l l p o t e n t i a l d i s t u r b i n g sigmaf i e l d s 0. E(YlC,P)

as= F ( Y 1 C )

+ HolJ,

where tio1.l denotes t h e c o m p o s i t i o n o f an A"-measurable r e a l - v a l u e d f u n c t i o n H w i t h a (D,/.")-measurable

stochastic variable

1.1.

I f , f o r examnle, C i s

generated b y t h e r e a l - v a l u e d N ( J ) - d i m e n s i o n a l s t o c h a s t i c v a r i a b l e

x

= (X1,*.

.,YN(J)),

i t i s e a s i l y seen t h a t t h e r e a r e no m u l t i o l i c a t i v e

terms i n v o l v i n g a P-measurable f u n c t i o n and one o r more o f t h e random v a r i a -

.

Therefore, the invariance c o n d i t i o n exclub l e s X . , j E J = (1,. . , N ( J ) l . J des i n t e r a c t i o n s i n t h e a n a l y s i s o f v a r i a n c e sense o f any o r d e r between t h e v a r i a b l e s X . and any o o t e n t i a l d i s t u r b i n g v a r i a b l e 1 J . This m n c ' i t i o n i s a J r a t h e r s t r i c t one t h a t cannot be guaranteed t o h o l d even i n randomized exq e r i m e n t s , where t h e t e r m ' c a u s a l ' seems t o be a D n r o n r i a t e , t o o , t o charact e r i z e t h e dependence o f Y on t h e t r e a t m e n t v a r i a b l e s X . i n d i c a t i n g memberJ s h i p t o t h e e x p e r i m e n t a l grouos. Therefore, a weak tync? o f causal l i n e a r s t o c h a s t i c denendence i s now i n t r o duced.

A l l t h a t has t o be done i s t o r e p l a c e t h e i n v a r i a n c e by t h e averaoe

c o n d i t i c n , which can he f o r m u l a t e d as f o l l o w s : f o r a l l o o t e n t i a l d i s t u r b i n g v a r i a b l e s 1.1, t h e f o l l o w i n c j e q u a t i o n h o l d s f o r PX-almost a l l E(YIX=x): E(YIX=x) =

E(Y/X=X,IJ=W) P , ~ ( ~ w ) ,

where Pb, denotes t h e d i s t r i b u t i o n o f

14.

I f 1.1 i s a d i s c r e t e random v a r i a b l e ,

t h i s e q u a t i o n can be w r i t t e n E(Y [ X=X) = I w E ( Y I:(=x,W=w) where t h e summation i s o v e r a l l values w of W .

P(I,rqf), T h i s average c o n d i t i o n sim-

p l y means t h a t PX-almost a l l c o n d i t i o n a l e x p e c t e d v a l u e s E(YIX=x) a r e t h e average o f t h e c o n d i t i o n a l e x o e c t a t i o n s E(YIX=x,IJ=w) o f Y g i v e n X=x and W=w across t h e values w of 14.

Together w i t h preorderedness, t h e averace c o n d i -

t i o n d e f i n e s weak c a u s a l l i n e a r s t o c h a s t i c deDendeqce o f Y on X .

339

Causal linear stochastic dependencies

7.1.

DEFIYITION

L e t t h e oresumptions and n o t a t i o n s o f 5 . 1 be v a l i d .

Ire say t h a t Y i s weak-

l y c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/dependent on X, i f f

( a ) X i s o r e o r d e r e d t o Y w i t h r e s o e c t t o (At, ( b ) Fc-r a l l p o t e n t i a l d i s t u r b i n g v a r i a b l e s (At,

t

E

1.1

t

T) ( p r e o r d e r e d n e s q \ .

w i t h r e s o e c t t o X and

T) E( Y I X=X) =

f o r PX-almost a l l x

E

1 E (Y I X=X,IJ=w)

n' (averace c o n d i t i o n ) .

PI,,( &)

,

(7)

I f additionallv

( c ) E(Y1X) as= E(Y) ( l i n e a r s t o c h a s t i c independence), we say t h a t Y i s weakly c a u s a l l y ; i n c a r l y s t o c h a s t i c a l l y indenendent f r o m

X and, dependent on X, ot+em!ise. P s p e c i a l s i t u a t i o n i n which E q u a t i o n (7) i s f u l f i l l e d i s c h a r a c t e r i z e d by E(Y IX,L')

as= E(Y I X ) , which d e f i n e s X - c o n d i t i o n a l l i n e a r s t o c h a s t i c indepen-

dence o f Y from 14. 7 . 2 . THEOREM L e t t h e oresumptions and n o t a t i o n s o f 5.1 be v a l i d . ( a ) X i s p r e o r d e r e d t o Y w i t h r e s p e c t t o (At,

If

t E T) and

( b ) f o r a l l p o t e n t i a l d i s t u r b i n g v a r i a b l e s IJ,

then Y i s weakly c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/deoendent on X. Proof. -

E q u a t i o n (8) i m p l i e s E ( Y )X=x,I\r=w) = E ( Y IX=x), f o r P(X,l,,)-almost (x,w) E n ' x 0 " .

1 E(YIX=x,lJ=w)

all

Hence,

P1,,(dw) = / E ( Y I X = x ) PY(dw) = E(YIX=x).

The importance o f t h e average c o n d i t i o n , however, stems f r o m t h e f a c t t h a t i t i s i m p l i e d by t h e s t o c h a s t i c indeoendence o f a l l o o t e n t i a l d i s t u r b i n g

v a r i a b l e s W a t one s i d e and t h e N(J)-dimensional v a r i a b l e X = (Xl,...,XN(J)) on t h e o t h e r s i d e . 7 . 3 . THEOREM

L e t t h e oresumptions and n o t a t i o n s of 5 . 1 be v a l i d .

If

R . Steyer

340

( a ) X i s n r e o r d e r e d t o Y and ( b ) X and a l l p o t e n t i a l d i s t u r b i n ? v a r i a h l e s a r e s t o c h a s t i c a l l y indeoendent, then Y i s weakly c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y i n / d e n e n d e n t on X . Proof. -

A . 1 7 y i e l d s E[E(Y IX,l.l) 1x1 as= E ( Y I X ) , w h i c h i m n l i e s ( a ) € ( V / X = x ) = E[E(YIX,lJ)IX=x], f o r PX-almost a l l x 'toc':ast.ic indenendence o f X and ',I then i m p l i e s E(Y;X=x) = F:[E(YIX=x,l*!)]

=

E

9'.

E(Y!X=x,LJ=w) P,,(dP/)

\4iere t h e l a s t l i n e f o l l o w s f r o m ( a ) and 5.3.22 o f C l n s s l e r and S t u t e ((19771, P. 1 ' 9 ) .

I n a randomized e x p e r i m e n t , X = (Xl,

... ,XN(J))

may r e n r e s e n t t h e v e c t o r o f

t r e a t m e n t v a r i a b l e s i n d i c a t i n g t h e e x o e r i m e n t a l groun t o w h i c h an e x p e r i mental u n i t ( s u b j e c t ) b e l o n g s . disturbing variables

1.1

Randomization i m n l i e s t h a t a l l o o t e n t i a l

( r e n r e s e n t i n g any p r o p e r t i e s o f t h e e x n e r i m e n t a l

u n i t s before o r a t t h e t r e a t m e n t ) and t h e t r e a t m e n t v a r i a b l e

x

= (X

. ., X N ( J ) )

a r e s t o c h a s t i c a l l v independent.

c o n d i t i o n h o l d s i n randomized e x o e r i m e n t s .

Therefore, the a v e r a w

Thus, weak causal l i n e a r s t o -

c h a s t i c dependence ( o r independence, i f E(Y1X) as= E ( Y ) ) i s guaranteed by r a n d o m i z a t i o n ( f o r an examole see S t e y e r (1984)).

Matching, o f course,

serves the same nurnose, h u t o n l v f o r those o o t e n t i a l d i s t u r b i n g v a r i a b l e s L.1 w i t h r e s w c t t o which t h e e x n e r i m e n t a l groups a r e matched.

Thus, t h e

c o n t r o l techniques o f r a n d o m i z a t i o n and m a t c h i n g can he based on t h e t h e o r y nrooosed, t o o . However, and maybe more i m p o r t a n t l y , t h e t h e o r y o f c a u s a l l i n e a r s t o c h a s t i c deoendence i s a l s o r e l e v a n t f o r n o n e x ? e r i m e n t a l s t u d i e s , because bGth t h e i n v a r i a n c e and t h e averaoe c o n d i t i o n may be t e s t e d , once a n o t e n t i a l d i s t u r b i n a v a r i a b l e 1.1 i s observed.

I f n e i t h e r h o l d s , any causal h y p o t h e s i s

s h o u l d he r e j e c t e d f o r t h e model c c v s i d e r e d .

I f one o r b o t h o f them h o l d ,

one may m a i n t a i n t h e c a u s a l h y p o t h e s i s ( i n i t s weak o r i t s s t r i c t form) as l o n o as no p o t e n t i a l d i s t u r b i n g v a r i a b l e i s f o u n d f o r which t h e i n v a r i a n c e and t h e average c o n d i t i o n s do n o t h o l d .

T e s t i n g i f t h e averaqe c o n d i t i o n

( s e e E q u a t i o n ( 7 ) ) h o l d s i s a t e s t o f weak causal l i n e a r s t o c h a s t i c dependence and t e s t i n g i f t h e i n v a r i a n c e c o n d i t i o n (see E q u a t i o n ( 2 ) ) h o l d s i s a t e s t o f causal l i n e a r s t o c h a s t i c deoendence ( i n t h e s t r i c t e r sense).

O f

course, such t e s t s make sense o n l y if X i s assumed t o be n r e o r d e r e d t o Y .

I n t h e l a s t theorem, i t i s shown t h a t causal l i n e a r s t o c h a s t i c in/dependence

34 1

Grusal linear stochastic dependencies

i m n l i e s weak causal l i n e a r s t o c h a s t i c in/denendence.

7.4. THEOREFI I f V i s causally l i n e a r

L e t t h e presumptions and n o t a t i o n s o f 5.1 be v a l i d .

l y s t o c h a s t i c a l l y denendent on X, then Y i s a l s o weakly c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y dependent on X. Proof. -

'Je have t o show t h a t Eouation ( 1 ) i n p l i e s E n u a t i o n ( 7 ) .

imnl i e s : E(Y IX=X,H=W) = E[E(Y IX,sI) I X=X,M=W] ) = E [E(Y IX) !X=x,'J=w] t E ( H o ~ ~ I X = X , ~ J ==WE(YIX=x) Hence, E(Y1X.x)

= E(YlX=x,l.I=w)

E(YIX=x) =

=

because

/

/

E(YIX=x)

E(YIX=x,l.'=w) P,,,(dw)

1 H(w)

PI,,(&)

E[E(YIX,I*l)]

-

Equation ( 1 )

H(w).

t

H(w) and

p l , l ( ~= )

-I

/

[E(YIX=x,'J=w)

H(w) P,.l(cbr) =

-

H(w)I Pl,l(dw)

1 E(YIX=x,',l=w)

= E(HoIJ) = 0, which i s e a s i l y seen f r o m = E[E(V!X)] t E(HolJ),

which i m n l i e s E(HoIJ) = 0, because E[E(YIX,IJ)l

= E[E(YIX)l

= E(Y).

F o r examvles i l l u s t r a t i n g t h e a p p l i c a t i o n o f t h e the0r.y nroposed, as w e l l as f o r a d i s c u s s i o n r e 1 a t i r . g t h i s t h e o r y t o simultaneous e q u a t i o n modeling, the reader i s r e f e r r e d t o Steyer (1984). APPENDIX I n t h i s anpendix a r e o r e s e n t e d t h e d e f i n i t i o n o f , and some theorems p e r t i n e n t t o t h e c o n d i t i o n a l e x n e c t a t i o n E(YIC) o f Y n i v e n C t h a t a r e f r e q u e n t l y c i t e d i n t h i s chanter.

F o r an i n t r o d u c t i o n i n t o t h e conceot and i t s back-

around see e.g. Sauer (1974) o r LoPve (1977, 1978).

, Rreiman

(1968), Cannsler and S t u t e (1977),

A s h o r t e r and t h e r e f o r e more c o n v a n i e n t came f o r

E(Y I C ) i s C - c o n d i t i o n a l e x n e c t a t i o n of Y.

Y i s a v e r y general and u s e f u l concent.

The C - c o n d i t i o n a l e x n e c t a t i o n o f I t i s used, f o r examole, t o d e f i n e

t h e C - c o n d i t i o n a l n r o h a b i l i t y P(AlC) o f an e v e n t A , as w e l l as t h e C-condit i o n a l v a r i a n c e V(Y I C ) and c o v a r i a n c e C(Y,ZlC)

o f stochastic variables.

S n e c i a l C - c o n d i t i o n a l e x p e c t a t i o n s a r e a l s o o b t a i n e d , i f C i s t h e sigmaf i e l d generated by a s t o c h a s t i c v a r i a b l e X o r by a f a m i l y (X stochastic variables. e t c . may be used.

j'

j E J) o f

I n these cases, t h e n o t a t i o n s E ( y I X ) , E(YIX., j E J ) , J

342

R . Steyer

The mathematical d e f i n i t i o n g i v e n helow does n o t anneal v e r y much t o i n t u i tive insight.

T h e r e f o r e , i t m i g h t be h e l p f u l t o r e c a l l t h a t E(YIX) i s a

s t o c h a s t i c v a r i a b l e , t h e values o f which a r e i d e n t i c a l w i t h t h e c o n d i t i o n a l e x n e c t a t i o n s E(Y1X-x) g i v e n X=x.

Another way t o t h i n k a b o u t E(Y1X) i s t h a t

i t i s a s t o c h a s t i c v a r i a b l e c o n s i s t i n g o f the b e s t n r e d i c t i o n s o f Y piven

a v a l u e x o f X.

I n many t e x t h o o k s on a p p l i e d s t a t i s t i c s , t h e v a r i a b l e con-

s i s t i n g o f t h e b e s t o r e d i c t i o n s o f Y i s denoted by f . o f as t h e mathematical e q u i v a l e n t o f

f.

E ( y I X ) may be t h o u g h t

S i m i l a r l y , E(Y1C) c o n s i s t s o f t h e

b e s t p r e d i c t i o n s o f Y based on t h e s i g m a - f i e l d C. Definition 1 L e t Y: 1

(r,A,P)

+lR

be a r e a l - v a l u e d s t o c h a s t i c v a r i a b l e on t h e n r o h a b i l i t y space

w i t h f i n i t e e x p e c t a t i o n E(Y) and l e t C

s t o c h a s t i c v a r i a b l e E(Y1C): R

+

A be a s i g m a - f i e l d .

C

The

!R i s c a l l e d t h e C - c o n d i t i o n a l e x p e c t a t i o n

o f Y , i f f E(Y1C) i s C-measurahle and for all C

EIlCE(Y/C)] = E(lCV), Let A

6

E

A be an e v e n t and lA i t s indicator function.

(1)

C.

P ( A 1 C ) : R +!R i s c a l -

l e d t h e C - c o n d i t i o n a l n r o b a b i l i t y o f A, i f f

P ( A I C ) : = E( 1,lC). L e t X be a ( n ' , A ' ) - s t o c h a s t i c

(2)

v a r i a b l e on (n,n,P).

X - c o n d i t i o n a l e x p e c t a t i o n o f Y, i f f C = A(X,A')

E(YIX) i s c a l l e d t h e

i s t h e s i g m a - f i e l d genera-

t e d by X and A',and E(Y1X) := E(Y1C). L e t (Xj,

j E J ) be a sequence o f ( n . , A . ) - s t o c h a s t i c

E(YIXj, j E J ) i s c a l l e d t h e ( X j , C = APjEJ

A(X.,A.)I J J

J

J

(3) v a r i a b l e s on (n,A,P).

j E J ) - c o n d i t i o n a l e x p e c t a t i o n o f Y, i f f

i s t h e s i g m a - f i e l d generated by [(X.,A.), J J

j

E

J1, and

E ( Y ( X j , j E J ) := E(Y1C).

) i s c a l l e d t h e (X1,.,.,XN(J))-conditional 4(Xj,Aj)]

(4) e x p e c t a t i o n o f Y,

i s t h e s i g m a - f i e l d generated by [(X.,A.), J J ,N(J) 1, PI(J) E N = (1,2,. . I , and

.

E(Y IX1,.

...XN(J))

:= E(Y I C ) .

j

E

J],

(5)

Garsal linear stochastic dependencies

Note t h a t E(Y1C) i s the general concent.

343

Hence, if n r o o o s i t i o n s are t r u e

f o r E ( Y ( C ) , then they a l s o h o l d for P(PlC), E(YIX), E(Y\X., j E J ) , and J ,XN(J)). Through the d e f i n i t i o n above, the C-condi t i o n a l expecE(Y lX1,.. t a t i o n E( Y C) i s cniauely determined o n l y w i t h nrobabi lit y one. Therefore ,

.

there are g e n e r a l l y c ' i f f e r e n t versions o f E ( Y IC) , which are, however, equi-

v a l e n t , w i t h o r o b a h i l i t y 1. almost s u r e l y i n general

Equations on E(Y[C) are t h e r e f o r e t r u e o n l y

, which

w i l l be abbreviated by the symbol ' a s = ' .

N o t i c e t h a t t h e r e i s a d i f f e r e n c e between the C-conditional expectation E(Y1C) o f Y, i . e . the c o n d i t i o n a l expectation o f Y w i t h resaect t o the s i q m a - f i e l d C and the c o n d i t i o n a l e x o e c t a t i o n E ( Y 1 C ) o f Y given the event C, and correspondinqly, a d i f f e r e n c e between the c o n d i t i o n a l nrohabi 1it y P ( A ( C ) o f the event A w i t h r e s n e c t t o the s i g m a - f i e l d C and the c o n d i t i o n a l

p r o b a b i l i t y P(A1C) o f the event A qiven the event C.

E(YIC) and P(A1C) are

s t o c h a s t i c variables, whereas E ( Y IC) and P(A(C) are r e a l - v a l u e d constants. I n the f o l l o w i n g theorem, a number o f p r o p o s i t i o n s on E(Y1C) are gathered.

Some o f them are on c o n d i t i o n s under which E(Y\C) i s equal t o a constant function.

I n these and r e l a t e d contexts, the symbol f o r the f u n c t i o n and

the constant ( i t s values) w i l l be t h e same. does n o t o n l y denote a r e a l cnnstan:,

T k syrrkol ' a ' , f o r example,

b u t a l s o a f u n c t i o n a: n

t a k i n g the value a f o r a l l w E 8. The n o t a t i o n E(Y(C)

:0,

+R

almost sure, i n

P r o o o s i t i o n (11) i s an a b b r e v i a t i o n f o r E(YIC)(w) 2 O(W), f o r almost a l l w E

n, where 0: R + R i s a f u n c t i o n d e f i n e d by O(W)

= 0, f o r a l l w E R.

THEORCtI 1

L e t Y be a real-valued s t o c h a s t i c v a r i a b l e on the p r o b a b i l i t y space (n,A,P)

w i t h f i n i t e e x p e c t a t i o n E(Y).

I f E(Y1C) i s the C-conditional ex-

p e c t a t i o n o f Y, then the f o l l a w i n g p r o p o s i t i o n s are true: EK(Y1C)l = E(Y) E(Y1C) = E(Y), i f C = InJI, where E(Y) : n E ( Y ) ( w ) = E(Y), f o r a l l w E R E[Y-E(YIC)l

+lR i s

d e f i n e d by

= 0.

E ( Y ( C ) as= Y,

i f Y i s C-measurable

E(Y1C) as= a, i f Y as= a, where a: n +lR i s defined by a(w) = a, f o r a l l w E R E(Y1C) 2 0, almost sure, i f Y 5 0, almost sure, where

0: n +lR i s d e f i n e d by O(W) = 0, f o r a l l w 6 R

.

344

R . Stuyrr

Accordin? t o E q u a t i o n ( 6 ) , t h e e x i e c t a t i o n o f t h e C-condi t i o n a l e x p e c t a t i o n o f Y i s equal t o t h e e x o e c t a t i o n of Y .

conditional exnectatinn o f if C =

P r o n o s i t i o n ( 7 ) shows t h a t t h e C-

i s equal t o the constant f u n c t i o n [ ( Y ) :

V

fi + R ,

in,@). A c c o r d i n ? t o E q u a t i o n ( 8 ) . t h e e x n e c t a t i o n o f t h e d i f f e r e n c e

Y-C(Y!C) o f Y and i t s C - c o n d i t i o n a l e x n e c t a t i o n i s z e r o .

c a l l e d the error or r esidual o f Y w i t h resaect t o

difference V-E(Y!C),

n l a v s an i m n o r t a n t r o l e i n a n n l i c a t i o n s ,

E(’!!C),

analied.

Note t h a t t h e

Pronosition ( 9 ) i s often

-

i s C-measurable, f o r examale, i f C i s t h e s i m a - f i e l d Generated

Y

hv the ( ? . A , ) - s t o c h a s t i c v a r i a h l e s X j E !I, and V = Xi, V = X X k , or 1’ J i’ j v = x t Xk, where .i,k C (1. Hence, s o e c i a l cases o f P r n n o s i t i o n ( 9 ) a r e ,

i

f o r examnle, E ( X ! X ) as= X ,

xi ’ i f i E J , J ) as= X..X i f i,k F J , 1 k’ ,!) as= Xi+Xk, i f i , k E ,1.

c ( x . ~ x . , j c ? ) as= 1

J

E(Xi.%k\Xj,

j

E(Xi+XklXj,

,j F

F

The f o l l o w i n g theorem c o n t a i n s n r o n o s i t i o n s on E(VIC), where Y i s t h e weiqht e d sum o r t h e n r o a u c t c f o t h e r s t o c h a s t i c v a r i a b l e s . V

2

Again, t h e n o t a t i o n

Z, almost sure, which occurs i n P r o o o s i t i o n ( 1 6 ) , i s an a h b r e v i a t i o n f o r Z(u;,

Y(.J)

f c r almost a l l w E R.

THEOREK 2 Let

and Z he r e a l - v a l u e d s t o c h a s t i c v a r i a h l e s on t h e n r o b a b i 1it y space

V

(?,f!,P)

w t h f i n i t e e x n e c t a t i o n s E ( Y ) and E ( Z ) , r e s o e c t i v e l v , and l e t C,

r 0 C 1. be two s i a m a - f i e l d s . I f E ( Y \ C e x o e c t a t i o n s o f Y and Z, r e s o e c t i v e l v

and E( Z ! C) a r e t h e C-condi t i o n a l t h e n t h e f o l 1owing Dronosi t i ons a r e

true : E ( Y 1 C ) as= E ( Z / C ) , i f Y as= Z .

E(Y.ZlC)

as= Y.E(ZlC),

finite EIE(VICO).Z\CI as= E(VIC,)*E(ZIC), E(avt>ZIC)

RS=

?E(YIC)

(12)

i f V i s C-measurable and i f E ( Y * Z ) i s

+

bE(ZIC),

(13)

i f Co c C .

(14)

i f a, b ElR.

(15)

€ ( v I c ) i E ( Z / C ) , a l m o s t s u r e , i -Y 6 Z, a l w c s t s u r e .

(16)

THEOREU 3 L e t Y he a r e a l - v a l u e d s t o c h a s t i c v a r i a b l e on t h e o r o b a b i l i t y soace ( O , A , P ) w i t h f i n i t e exoectation E(Y).

I f E(Y1C) i s t h e (‘-conditional

exoectation

Grusal linear stochastic dependencies

o f Y and Co

c C i s a sirpa-field,

345

then the f o l l o w i n g equations are t r u e :

EIE(YIC)ICol as= E(YICO) as= EIE(YICO)ICl , EIY-E(YIC)ICoI as= 0. The s i q m a - f i e l d Co w i l l he a subset o f the s i q m a - f i e l d C, f o r example, i f C i s generated by the (n.,A.)-stochastic

J J v a r i a b l e s Xk,

(nk,Ak)-stochastic

variables X.,

k E K, where K C J .

,7

and Co by t h e

.j E

Hence, s n e c i a l cases

o f the Equations (17) are E[E(YIX1,X2) I X1l as= E(YIX1) as= E[E(YIX1)IX1,X21 , E[E(YIX., ,j E J ) I X k , k E K1 as= E ( Y I X k , k E K ) J as= E I E ( Y I X k , k E K)IXj, j E 51. Equation (18) reveals t h a t the Co-conditional e x n e c t a t i o n o f t h e e r r o r o r

-

residual F = Y

This n r o ? o s i t i o n on F i s much

E(Y1C) i s zero, i f Co C C.

s t r i c t e r than t h a t o f Equation (8)

, sccording

t o which the ( u n c o n d i t i o n a l )

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