II,:,.-II
. :. .,
:.,.-. -,--,-:i-..
,2-,
.,-I,.
I ."..-.
:-,,...
: .: --.-: `' , ; Z.I.
-..:- .,-I
.. ',-....-"I,-. :'
,.-'.
..`
, 1, '.-..II-7I',-.
-.-- ,-.
I I-,..:
II.
- .I.I:,
.
.-.II 4 -I-.I.1...,
,F.
, I-,I ..i ,.2
'. .. ...
,,..----,.-,,1-.
-. ,"'--.,.I.- -_,,
"-II
...
"I-,
-,
`,..
,I
I-",
.II....I-..I
,
"-,
,
",-,.
.,. -.
I-,.-'-I,,
"--.:-.!,,-..1.
,,,,--,,-,.I,-, Z. -I-,
I, -.. ,r
I -::-:.:,-'t,,,,,,-Ii-.
-.,.
.
,-, ,,:--.,-. ..-,-: 1-i..
-- --,I-.
..-,
, --,,
-i,,.- ----'.,..
,,-.-,:.,.
:,.-..
,
:-,I-.,--,--,-,-,,
.,.,,
,,----- - ..:",,,--,..,--:1:
,
-.
-,I--L ,,`,,
-II,..--,
,,-,..
,,-,-,--r-,
-- :1---.-..1
-..
,-:,.-...-,
,,,-,-,,--.:
--,-%.:-.
,.,----.
...-"'
,,
-;,,:.
-I- ,, .,7.-,-.-.--%
,,,,,,
--",I.;- ,-, ', ,-', ,.-.I"--I-:I
,,-,,,.,-:--.-:,
"II-..----Z:I::,.-I-,.,,,-.,
.-,, , --.
-, ... ",--.,-,
-",.,,
-,-,-,,.-.
I..-...
: -,.-...------,--,.-:.--. I.-...,,.-..,.,,
p,
-,
,----,.,-,---..-.-"," - :,t -.
,-1
: --,,,.
., -.",,,-,,,."I
-,--.I'-..-,,:-"
-; ---:.:-7,-;
:'.,.,.---a
,:..,-,,!,.; -,-I.--:.:,
,:I-.,,,:. ..-:,I--7.--.-.-.-I----.I.-..
:,'.,:.-.
I--",,--,-A.,,.
, -,,,,---.,--.-..,;r,
,,,.1.,.--,-2--,
':,:..!.--..---'
,.:,,-:
11-.-. -,--.-11-`.-,-,,.-., ;.-Ic
.,II:,,--: -: q,- ,
-,,,. -:,
",-.-I--..I-1
-I.Ib` ,;-,
:,-,
, I I'a;.:
.:1 .,, ,'-. i,-1,.-,
-,I-..t-.:-:,
-,,;Z
-.10: I- -I.,,,---.I
,4,.,-.,,.I
-'.
.,I T- -.-- .--1,
.-i,
-:,-:',,,,,, ':.I
.,",
.,.-,-11.I,.
.,
,..,.,..
-",
,...-.I-,.
,-..I.....III.,.---.
I..::-,,,. .,,,
-...-`,:
-.I.-..7:, ,I--.--,
,,-.
7 ,-.
-..
.
,1.,-,-,'.-,-.--".,,,
:II:-_-,.
,-m.-I.iI",---,.,-.,
.1II17
,- .,:, : -,.,_,
,,.I.::II,.
-.,-.
..
1.II.. -II.,?,..q--1
,- .---.,
-,,-.., -,-,'.
,-.-.,-- ---.",:11--,
, I--,-,.
-:---,,-,
:,-....
-;,-I
: Z-.,,,----.. -,,,I,..I-II,-I-. ...
;:,..I
-.-:..,
I .II.-.
.:I:.,- ,-Z,,.T ..,I.
,-.-,
:.--,.
II.
",,..,
,-.
,-..
.....I.-I.,I-..
,:r
:-' - ,,.11II,,-,---.":4,-..
-,..I
,I,-II.,
,.--.:.I...
',, .F......l
.,,.-,
-,---....-I.
I'.
".-.
-,-,--,
.-.
:-,..--.
,Z..---...
,,..
",..:..,
i, ;----1 ,,-,..,-,.--II
I-.,-I.
I -,--,,;
-,-,;: '.
;,I,-,-;.
..
-..1
;,
-.
-,%-,,,III,--.II
,.---,-::,-"-,-,
.".--I,-,--..
.I-:Z,..
I,-1-,,.-".,:
,
"----,.-.I
b:..1
:,
-,.,
.--,--II
-.
.,-I, ,r.,-1.:I
..
'-' ', ,--.
'.I.--:
";,-.
'I,,---:-:, :-, 4rj:,-,,
., I1,I.;I:.,..I1.IIt
-.,
-: ..-II.,I ,-;
7-,-..I ",
"i--::-,--"-----,-.
,.-.-.:..-..,
v-,
-1,.,,.I ,..,, ,:-- ,'.
, -,---,,I.,I..
-:,---...
-.-icI4 II"I-.-:,..I--.I1,
",---. ,,,',
'. --.
,.-:.,
,,.,-,---.,,-:,--.',--,.-, -------::
',-I. ,,-,Z
2. -Z-:
Z---,--..--,
.'.1.'.-:::-"-,.i,%:..-,II--..,:--I-,I
,-,"'.
-.
,II.-.
I.``-,,11:,,:--,,,
`-,,,;-....
..,I- ,.:,-.--.,,,-.I.,.I,-,:
1
..
..
-,,:-"
,---.,1
....:I,
%'..:---,,-:----:,
--.
I..I.--;
.,.1,i,,:
.- ,-,I.,,-,.--p:,.
:;,,',:
: -. ---I.-:I---1-.,...-,,,.-,,_4
.-;..
,"I-"I,.,.I.,i,---,7-I--I
-;,,, -,-i:,
,.-.,--.
:-,.,- :4:: T. --,-.1,q,I., ,, .,
.1.II,.
..".-"',
., -,---..
,-,t
,
- ---.
i". -"
-,,"--,-,
,I,--..,%..I
:-i-.-I,-.I1.:I..:,.- .,,-..
I .:.I',..
-,,.-,-I--,-,I
.:1
.,---.:..I.--,--,'I
-I,-.
---I-----,'.-,
-,.,
I..
..,-,...,--,!-,
.
"-,
,,,--.
,I
I,'
-...
",
..,.-:,-1
-I
I-,.
-,
,---:-,:,,'
"', ,..-I---I
,,
.,,.
.".:.
".---,I---%
ll' -I'-:,.
.,I
"
-,-,,--,
,':
-:f:,- ,,:, .-,.---I.:.-1
i-l. -.I..
.1.
,-,--!
-.
1.; -I,-.:
-:-,---r
,:-----,.,.-,, :,.-,
II1 ,, :"
... -.,, . ---- ":;
Z,,7"-,
-,., ,
.,:,,
-,,-,.
,
"
.---,
-.-II..-,,
., I-, !.-..,,..,
.11 -. I-,.,
-..,..I-II.
.,.-,,,;
-,-,...:..I.
,., -- ,:,-.-,-,-.---.,-:.,I
'.
::.,,
,.i:
II!-7,
-.
--,,:",,,,---;
t;
:
:-.
,.-.
.,I.,
,.-I1
-.
-,.---,
.. ,-.
,,:.,I,I-.-,,
...I,II--.
1-,
..
-, --...t-----.i-:,-,.,
,;,:"., -,.-1,.
. 1,
"
-:,,. - 4-;.--,
-,--.,-,'-,; oIi,,
',-,----!,
, -,
,---.
.
..,,'.. I-.,-F,;
,,-;I-,.-..--.-,-,:,---1
'.- -I.-.,.-....,'.--,:-, ,.-;,
--.
-.
:-,.-,-Ip--,-,I`II,;I..
,.- ,,. , ,
.. -,,....F.
t , -,; ,-,-7-2.,-.-1-I-.-.;
l'-,:,
---, -I--.,9:--,- I,...
,: :, ":-- .':,. "F.':1,
,,;.-:-,.-,-I
, 1.:.,I-IZI..
:..-....-,
..
-1 7 ",,.-%--;
;--,,,--1--,,,..,.II.!,
1,.--,I....I,:.-, -,--,,..I.-F--;--.- -,
,,-.-.,-I-I-,-,..
I--,,: ,
--.--:-:.-,-...
.-..,-,,.::
-III.
-,,..-- ,,'---..,:,,...
I,--.,
I:.'.,-,
-:,-,-Z
.-,-.
",,,-,.,,,,"'":, .'
-,
. :'.-.I--.
LI,--.II,i,--,,.-.,,--.i-..I--.-- I,,-..v--I1....
,-":_.,,--..F-,1,I..-,---- : 1,,--II.,,",t: .,,---. ,:.,,,I,-,,-- -,,.,
-,.,
-,-A-.:-,-,
.
.z---,I",
.,
,,.,:,I-.
,-I--,"I,---:-7z .
-,-...I-.I--,-.,
,,,.,...
1-. ,.I.I-.,--, "I.I-:
. . ,.41--,
,-I,-,---.-,.,:
-,--:
,.1!.,,I. I..
,I,--,.,,..:
..
:;-I-- -.
-I--,-II-I-.1.:1,-.-,,.-.,,
-,:,
,,
.
,-1,,
-,.II,I:
.-.'.
-!
1.
-:"I
1'.,-:-.
-j,,
r.,.,:,-;,.
.I-.,--,
,--"
-..
.-.
:-,,- :, ,.:.:-_.,.
."
z-,-,,
,..
:' .,-.:,.,.,,,I.I
-,
,--,--, -,,,I.;,,,-,
,,., II..,.,
:,
,, -_-,-. ,...
- ,',,.", -..--..,,,:..1 .-. ,--.':
..
1- I,.--...,.-i,--:
,.,I,
-, _-.I:
:
-.---,.Z.:-_
-L
;,--I
-,,
,.:,.,
..:.-.,
..,.
.: :, .I.I-.-.,,..
,..: :.I,--4.I ..I.
v.,
.I-I-,:tI.,
-;.-,7L,,-..
I.,
, :,3-I,
-':---:-...:
....-'-,:,",
..,:
;', -:-:- ------,.-- ",,.11--,-`..,
,-,1.-----....
I
",---,,,--.-!,,.- .,
",
,:,.-II-..,.-:, " ,";- 1 I-:
,.,.;,-I--`,-,,-,.-.,---,- -t-,,-----,....,,.-,--.
1,,-- -',-,
,.III...-T.
I--,:,,..-"
!I;i
--,.b
-.- I .1.,,","-,
.,--,, -%,,- 1:
",.".
.- II ;.I -;
,..I,.
, -1-.-.::--I;:.
,-.,, ...-.
':-.I-11,,.-,... -,,,..-...
,---.
,.-:-..I.,"n,,,:,,
11.-.-,-.-1
-::
-IA--a-1
-,-;.,
r.-ImoftI..-,-:'.-:,,:-,-:..::-1.:
-1
-,IIi
I, ..11rI
.IIi.
, I"
, "-.,
,
I,-.
-.
I,I!,11W II,-,2,..I
I..i.:I--II.
,,I.:
.I
,.
,,-,,---.
=--i.
,`.
...74,.-,
.,
I
:-.;,:--,-.,,.,II.I.,
- F,
,-,
-I-: -t,::;
.-.
:w,--..
:.,-.,'-%,
,-,
-::-.,.' '- :
-,--.- .-,-..
.-I:, 'o-.
-,
, :,".- ::t -I.I.
...
1,-"
,I---:.-I--,-VII..-I
.,.-.1
,
-,",-'
.:,l,,7.I--1
-,I......
-.
,.-I,,I--I,;
:,'.-.
-.
1.7.,- --"i..----.
I-.11-I,.-II-.
. -- , .I:-",
..
-I
-.- -lll--:
,-,
, ,,;-I
:-..-, I,
.4I--,--- .i-.
2.--.,,I.. ,".
,.-,-,
---, --..--:-I
I-,,;I",-.:,.. '. -.
-.
Z
...
.I'.,--,--.X,.,--;.
...-;-.- -1-- - r,,-,-;
:'--'
,,".,..,
--.---'.
.,,.-,,
,-.%:.-.
-: ,.:":."
.:--I-I--,;iI.,,:
I.-I.--.--,----,
II
,.--:--,.7,
,,:11-,:
.,,
,.,- IJ 1.
-:,.--II,-.--,,-I
,I.-I:I.j.
;.Ib-II I ,;;
rl,,-..-o..I.-tI
-.'.. -,:,-.I,.:I--, .,II-,,. ..I
': .-.
-,.;, .,
..
,.:i-I,- ..-.--.T
,.-,:..,,--.-,1--,
,,- ",--,-,
Ii-I.::i,-I
0.
, -:,"I.--I,
---,.,
, -, .II,
- . ,-,..
I;.!:--,
,.."7:.--,II-,
-,IZ
...-1.
"- .,,t. ,.-I.
,I.A;-.,..,,.I..- .p,.
,
,.,,:-,-.,-I-,,,I-:!.I.
-II-..--.I-11..II--I
-,..
-.
..
I
-"-I-,
--.
..
11,III...-1
,-.
I,-,:..I
,m-,,, . .,II
,i . --:
11-I p11-II2..:.:..:
, --..'.I,,
.I-I.,'.
: z',.., ,,
,,-,--.1i.'-: .I--.11T...
,,
: ..:---.-,-I.L.,I-.,.,.I.....-I-I-II.-.'--d..
-.--,II
-;-..-J-I-,,,
.1.. ,.::I.:
,:I-:,-,
- ,-,--..1,
II.-,:
,--i,I,
-Ii,--I-...
",:-I--: Z;-.
., .,'.
, ,.-:.--,-.-I.,I-I,-.-..., -,--,
,,- -,,--- -,.
,,.
,,
-".-I
: L-.
,,b7:, ..-.,
.,,,.o,-.;.:
, -. -,:
---.,.1..
.-..
:.I--9
1.:.,-7
,,-- -I.II-,',-i.
-,
,--:
,,:--I.,
;, I-,-.-1..-II-..,
-t2,.17I.-1,.,::
-..,,,..,:---,.I,,.- --.
.! i, 1 : .I..,.I'.'
,1.,;_r--I:"...I",.-...-.-.%
:I
.T,-I,.,-,.I
-.I--,
.---,---.-..-.. I--..
I:",,.-.iI.
,.--- ... ....-. ,,;,I"I.,' ..-I----'.
..
-II..., .,.-.-,..I..I
j.,..,.._.
'.
,,.I""..
:-::,.;
.--::.
I-,
I
,...-.....
0,.,
,..,.1
, -,--,,
:
I.-,;;,,I
11-.--,.I--',,.,'
.--- I.,I.-.I.Z;
11-------.
,,, -I.--.I--.
.,.,'...
...:-I-,--II.-t---...,f
I.II:.q-I,-.,.
..
I.:.
'.-.I-.
.-, %,l-..:%i--"-,:,
I-, L
:I:-:1 ::...
p1.
I-,F,.,..-II,
Ij.I-.--:.I-.-.,i--,.;",I--I:.,I,
,,,,II..
-,,,,.I_-.-_-I.I...
:--c-,.--171
-..
-.
,,I.,-;r:_%
-,Ii-.-I:1
:o
--,,-,-.-..I--:i:-7 ..,I,-II
,,,-"
-.-:,
I...1..,I-I..-,, .,-..--.II:-:;:-."
..
-,,--.-;.,.4.,II,'i -,%:
:,.,
..,I&I..:.'I
-,-,:I,,I--,:11:I.-:t::..
.:I-,,
1....-,,,
: ",
,.;a,-:
.
,,'',-.-.
il:,,I..-. =I,-I,.-,-,...,-.-.I,-,,-,:,,-.-1-,-::-,"-_,,,.
:I,,-,-.!
I,,
:,-..-.
,
,,t,-,,
-I,:.--.
,",,-II"
""
,---.,-I.,,-II..,I,.
,-.., -:
II
...-..
,.;,
,,
-ll--.. 1I,-.,.-tI-,
,,,.-..
,.I.-.
::---.-1-.
,
.::.-...:7.-II
.;-:....--.
-:,-..
.
I'.I,12,-,,%..I--II.I.
,
------...I-I,._I
:..
-,,
I
:-."--,,:,-:
.-..
-,,I.I..:d
I,-,
, "..- ,, :- '..,1:1,--,:, ,
:- :,::
; 1.--.-,,
- ,Z,
'.,-,,-,-:.-,,,I'
,..--.
;,.;iI.Iix
--.,.:,
- 7..
--.,
-,-:-,.:,.
.I,.
-I. .
.
.
-I...I.: --.
--.I.I.1-LII
-.
,
.
-."IZ,-,..L.
L
I-.III
.--,.,,-I,,,I
.
:
--,b-,
.
II
,-,.':-.I.I;,-:II
, -:,
.:,"`I
111,I
I. i .?.t..--I
-,--Id:,
1,.- ..
-,I--I-.
-,:
': , :
,
,----. ,,:-t.,:,-....-;--.I.II.-II.
I--.-:
I
.,I:,-III...,I,.:.-:I--I-`-I;,
--.-II.1--.:.:
III-Z-.
I-..I,:-:%:,-,..
-- I-....-I.,,,I .
.1,.-.---.
----.
-:.
---11
, ,..I.,-,-,.,.1.,.,,.;,.,,_,.
-.---...-I..,k1-.;.-------o..I1.
-,.--I.-.-I,-I--.::--,..-:..,II--I-,n,-:'..
,-,,,,
,,,
I .----...j.......-..,,I-L-.,
.I--:
, ...
3..,.
,.7..
.--
,,:-I
.. ,--
. . -- .-.
-. I - -- - I- -I.---..-...-
-1TF71
11-1.::t, -,----.- ,:.--I''
-I1.
-:;-!'.."',-...-,
E.-
k .I-'
-.
Z.."-,,
-,
I--
: 1. t'.-
:I., , --,,-
MASS~~~~~cHUSETT~~~~S i'~~i~INST :I'TUT
TECHNOLOY
OF
*
'
-
'@
; '
;
1
n
.X
A
t~~~~~~~~~~~~~~~~~~~~~~A
:
a.
<;
'
l
;-
'
:k:
POST ERIOR INFERENCE FOR ST RUCT URAL
PARAMETERS USING CROSS-SECTION
AND TIME SERIES DATA
by
G. M. Kaufman*
Sloan School of Management
M. I. T.
OR 007-71
December 1971
(Written for the Session on Bayesian Analysis of Simultaneous
Equations Models, World Congress of the Econometric Society,
Cambridge, England, September, 1970.)
I wish to express my thanks to Abba Krieger for doing the
numerical calculations in Section I. This research was supported
in part by a grant from the Cambride Project.
Introduction
Our purpose is to discuss features of an explicit Bayesian analysis
of a simultaneous equation system when both time series and crosssection data are available.
The process generating time series data
we shall assume to be the usual set of stochastic equations
(j)+ F z(
R
where
)
-
(j)
()
=
--
jl
,
B(mxm) non-singular and
*~(1)
...
,
(mxr) are coefficient matrices, fixed
for all
, z(j ) (rxl) is a vector of predetermined observable variables
and
and u() (mxl) are random vectors.
()
that {ui,
j=1,2,...}
In particular we assume
is a sequence of mutually independent identically
distributed Normal random vectors with mean O and variance matrices
, nor Z are known with certainty.
and that neither B,
,
The system
may be re-expressed in reduced form
(
y(j) =
Z
)j
+
j=1,2
,
when B is non-singular, where H_
N)
M =
I and E)
-B_
(1)
(2)
Yyztt
zt
Z Z
the likelihood function for B, f, and
Bt _2Ne
I~~t/li~~~e-?FtriZ_
C -
]
B_
Defining
(N)], and
FYI t
Y
BB_ X-½trZ _ [B1 ]M[B
=
t
-N
ZI ½X]
is
(3)
2
The identification problem is essentially one of determining
structural parameters from those of the reduced form.
Its Bayesian
counterpart is that time series observations generated according to ( 1 )
do not alter a posteriori the prior conditional density of B given reduced form parameters.
Cross-section data bearing directly on elements of
Here we shall explore properties of Bf and
(Bi ri)
'
The first is:
and ~ clearly will.
posterior to observing
cross-section data bearing on individual rows of (B
differing sets of assumptions.
L
i)
uider two
observations bearing on row i of (_,-,
are generated according to an independent Normal regression process
with residual error variance
structural parameters.
.,i'Ui a nuisance parameter unrelated to
This process is independent of the process
generating time series data; in addition, the processes generating
observations bearing on individual rows (
L) are mutually independent.
The second assumption varies from the first only in that the regression
process generating cross-section data bearing on (Bi
error variance proportional to
) has residual
ii
While we shall usually assume that the prior assigned to structural
parameters _, ,
and _ or alternately to reduced form parameters _ and _,
is non-informative, the functional form of the posterior density of B,
_ and
will be precisely the same as if we had assigned a prior natural
conjugate to the time series process, to the cross-section process, or to
both, the only difference appearing in the values of parameters of the
posterior.
reader.
We leave the modifications so induced as an exercise for the
3
Under the assumption that cross-section residual error variances
are not related to the time series variance matrix
density
, the marginal
of (B ) has components in the form of a product of
(multivariate) Student kernels--poly-t kernels.
Posterior poly-t
densities have been studied by Tiao and Zellner [7], Dickey [3,4], and
appear also in Chetty [2].
We show that under certain circumstances
the marginal posterior density of the reduced form parameter _ is
proportional to a ratio of matric poly-t densities.
This ratio bears
an interesting relationship to the posterior density for a single row
of
a
that Dreze [5] uses to derive a Bayesian version of a LIMLE estimator
when cross-section residual error variances are proportional to corresponding
time series error variances.
Using the same data as Dreze [5], we compute the posterior mode of
parameters in the demand equation of Tintner's model of supply and demand
for meat, poultry, and fish.
While purely illustrative in nature, the
computations indicate that the procedure is surprisingly robust given
that there is a substantive difference in specification of the model for
cross-section data used by Dreze and that used in our calculations.
The assumption that cross-section residual error variances are
proportional to corresponding aii's leads to a posterior density for
(E ) unconditional as regards ~ that is extremely complicated.
We
present some results for the special case m = 2, and give an asymptotic
expansion of this density whose leading term contains a Normal factor.
4
I - Time Series and Cross-Section Variances Unrelated
The first set of assumptions mentioned above leads to a likelihood
function for (Bi
_:-i
m
-n i
u
i)
and the uis
of the form
i -i
-½{[(B
e
.)-(B
i
)][(B.
)-(B.
+Si}/ui
(I.1)
i=l
Here ui is a nuisance parameter and {ni,si,P(B
i
sufficient for inference about (B
Provided that each n
integration over ui E
rFi)
and ui.
i=1,2,...,m
(0,),
F)} a set of statistics
1,
>
yields a term proportional to
1
m
-½n
{[(B i r.)-(B
i
H
)]P[(B.
i
.)-(B. r.)](B+s.}
(1.2)
i=l
m
when a prior of the form
-1
u.i du.dB.dr. is assigned to B, r,
cross-section data is observed.
of (Bi
and only
This is the kernel of the posterior density
-i) unconditional as regards the nuisance parameters ui
When in addition, time-series data generated according to ( 1 )
are observed, the kernel of the likelihood of
the form (I.1) times (3).
, A,
E
and the u's is of
5
If we assign a non-informative prior
reduced form parameters
IB BtI - (m
for g
r
and E.
+r )
L
BBtt
-_|
2½(m+l)dQdBdT
to B and the
and _, this induces a prior
IZl-1
(m+ l )
(I.3)
Assignment of a uniform prior d
to B in place of a
non-informative prior leads to a substantial computational simplification:
the determinant in B "disappears"
under certain conditions and we shall examine the
implications of this modification.
With no increase in computational
complexity, we can as asserted earlier assign independent Normal-gamma
priors to the (Bi
)
, ui ssiand/or a (degenerate) prior of the form
BBtl(m+r)jL -(m+l)
-½tr E 1[B
]M' [B it
(I.4)
Upon assigning a prior proportional to
IB B t I-½
( +r)
I l-½(mr+l)U1
1
ll
(I.5)
6
to B, Z,
r and the u.'s, and then integrating the product of (I.5)
1
> 0 and u.
and the likelihood (I.1) times ( 3 ) over
E
(0,),
i=1,2,...m yields a posterior density for B and F unconditional as regards
E
and the u
i's
proportional to:
I=
B t I(N-m-r) I[B [
L]-N
t I
(I.6)
m
-n.
{[(B
r
r
From a Bayesian
(Bi
r )-(B.
r >]+s i
standpoint, a "full information" analysis bearing
on, say, the first row of (_ )
requires use of all of (I.6); i.e. we must
compute the marginal density of (1 i
) by integrating (I.6) over the
range set of all parameters in (B ) save (B
rl).
The resulting posterior
density takes into account all information generated by the prior as well
as all time series and cross-section data.
This integration appears
difficult, and a useful representation of the resulting density remains
to be found.
I[B r]M[B
However,
]t
partitioning [B] =
= (B1 rF) (B
>
1 )t
where E = M-M(B we1
){(B
F){(B )(B(B
density
that
the1see
is 1
we see that the density is
[B
1
4 ]E[B
]t
F )M. Upon substituting,
7
)
r-l) M(B
{ (B
}
(I.7a)
t
{[(B
)-(B
r)]P1[(B
(B
r) -(
1
r
-½2n
+s)
times
IB BtlI(N-m-r)
2[
_-2]N
__]
E[_2
(I.7b)
-in.
m
{[(B
H
)-(Bl
r.)]P[(B
r.)-(B
r.)] +Si}
i=2
Thus the posterior density of (2
L2) given (B1 '1) = (B1 F)
is, aside
from the complication introduced by the first row of B being fixed, is of
the same functional form as that of (B F), while (I.7a) is the kernel of
a (degenerate) poly-t density (cf. Dickey [3]), Tiao and Zellner
[7]).
Replication of this decomposition m-2 times allows us to write the kernel
c1.7)
as
B
(N-m-r)
times a marginal poly-t kernel in (B
)
(m-l) conditional poly-t kernels in rows (Bi Fi , i=2,...,m.
1F )
and
H
If we transform from (B, ) to (,
D),
1) has a
we find that (,
posterior kernel
(I.8)
m
iH { [
i=l
i
(-
(-Bi -i
]-
[Bi (I-
-n.
-
)
-(B
-i)]
+Si
B and write terms in them as
We may complete the square in the -1
+~
--
{[Bi-_
1i ()]i(_)[Bi-i(
- -i
C)
-½n.
+i S (n)}
=
(I.9)
I
where
()
Si(ww
- [I-i Zi
] [I-H t
[-i
x
i]Li[-1
(D
and
0
S
W
Si
(S+
i
i)
(
Aside from the term
i -i)t -
i
(
i
t
o
B Btl
½m arising as the non-informative prior
assigned to i, (1.9) shows that the distribution of L given
= _ is a
product of (conditionally) independent Student densities, centered at the
_i()'s with variances [S
-1
s with
variances [S 2.(D/(n-m-2)]_i
1
1i ()
provided that ni- m > 2.
9
A slight alteration of the prior (I.5) will in fact lead to
given
= f having rows that are conditionally independent and Student:
assign a uniform prior dB to B in place of a non-informative prior
or alternately assign a prior
Z½(+l).
IZI- (2m+l) to
in place of
(This latter assignment changes N in (1.5) to N-m.)
Adopting the former modification of (I.5),
10
2
an
integration over
density of =.
cE
m
gives the kernel of the marginal posterior
To this end observe that the reciprocal of the normalizing
constant of (I.9) is proportional to
(
_)/S(_)-
_
* (S ())
Letting ci = S i + (Bi i )P(B- r.)t and using the well known fact that
(v-lt+c) =
(xV xt+c) =
l-llc-lx+V,
we may write
VI1c1xtx+VjI, we may write
1_i(
-½ (ni-m)
-sO(~~
li
(I.10)
½(n -m-l)
-l
;-
-1
'* [I-]ci
[;-n[c. P.1(
rl)
(By rl)_i]
t
l -1_\(
B) F
-½2
(in m)
[
lp.+p.
]
It
-1+ = I [I-
Partitioning
-(i)
P
11-1
P
=i
(i)
=-12
, (i) (mxm),
,~:=
1l
=
i- p (i)
i =21
(i)
=-22
I
and
1 (i)
-1
-
-
t(B
H - c
i
rP(B
) (t
=I
i - -i1
-i
H (i'
-
_ (i)
r )P.+Pi
=
==
I (i)
H (i)
H--2
, H1
(mxm),
and completing the square in (I.10), it becomes
The range set of B is in fact the ordinary matrix representation of the real
linear group of order m. Since the set of values of B such that I = 0 is
of measure zero in
, we can integrate over
m
- m
- Euclidean space.
11
[_)_(i) =22(i)
t12
_
M
-=12 (i)
-22
2
[ (i) (i)
·
2
Mpi-
t(ni-m-1)
(i)
--1.2
(i)i)(i)
t '
-2
k2 [ L12
]22[H2i
2-22]t +
[_pi
(
2 _
i=1
l[_n12
kl
n -H
2-22
[
22 (i)
]
H2
(ni-m )
2= 1.
is then:
The complete kernel of the posterior density of
[-4222
i
1.21-
-
i+(i)(i
)
-
(i)
(ni- m
-
)
22+
)__~(
H(i)
(i) ii)
'i
[ -=-2) 22
namely, a ratio of matric-poly-t kernels.
-
]+
]
)(
it
+
The denominator is a matric
version of the poly-t kernels that have appeared thus far.
When cross-section data is available only on the first row of (
Ax,
and no time series data is available, the resulting kernel suggests an
interesting rough analogy:
Dreze [5] has invented a Bayesian analogue
of the usual LIMLE; he finds a posterior density of a single row of (B I)
consisting of a ratio of quadratic forms and chooses its mode as a point
estimator.
Here we have a matric-version of this ratio for the reduced
form...a reversal of roles!
12
If we continue to assume that cross-section data is available
only on the first row, and assign a prior of the form (I.5),
the posterior kernel of (
) is expressible as
- m- r
1 ½(N
B B
then
) times
a poly-t kernel:
1Btl
(N -m - r )
1[2
2
2]E[_
= ]
N
(I.12)
_(B1) M(B
{
where (
2
4
rl) t}½N{[(B 1 rl)-(B
)]P
[(B
r
)-(B
)
+
1
) are rows 2 through m of (B F).
Re-assembling the quadratic forms in (I.12) generated by the time series
data and writing them as
1.2
+ [_L+
U2]
--
22
M]t
1
[
N
,
if we use a fundamental decomposition from the general linear hypothesis,
and write this determinant as a product of a Student kernel for
B = B and a matric Student kernel for L2 given
integrating out I% the kernel of B and
= B and
l is found to be
1
given
= rl' upon
13
|{(BM
{(B
1M 12
2B1 ) + [+B
2
1Mllt
B
(N-m-r+l)
-1 t }-(N-m+l)
Bt
-1
]2[ +BlM122 ]
2
(I.13)
I
S
- t
()
11-11
41.2(-ll_-BY +S
+.. 1
l-l+(B-B
(1) (1) 1
2 ==-22 2
-1
- 1+(~..
r P(1)
Here we have partitioned P
that
1 given
=
-
[-+
lL7--
) (1)2
()
l)
p1)
l=12
P(1) (mxm) so as to show
(1)
p
=21
(1)
i
=22
1
' 11
is poly-t.
Again, a slight modification of the prior assigned to B dramatically
simplifies things.
A uniform prior in place of a non-informative prior
drops the determinant I
Lt1-m
from (1.13). Upon adopting the Normalization
rule, setting the first column of B equal to a column of l's, Bll = 1,
and defining
'Ml 2
= 11.2
=(1)
l
'
11 scalar,
'
11 scalar,
22
'V'll
-12
--11.2
l -2l
IP.
-K2
@22\
14
=*11.2l.
(B
B111.2-1 t
1
-1
22-412)t
Q1(B12)
and
S
+ (B
= S
(1)
)11.2(B
-
_
-1
t
P2
+ 1 %22Bld
(B121222 (-2 B12 12 ) t
+
and
22 >
-"22
i.e. provided
Q2(B12);
r-m+l > 0,
1Ql(-k2)]- '-2r
is the kernel
-in1
of a proper Student density and similarly for [Q2 (B1 2 )]
22
>
Consequently if we let
and nl-m+l > 0.
Vl = N-m-r+l, v
-22
=
L2/v
=
-22
-
r
**
-B
1Ql (-k2)
2
*
LB
= nN-r,
22(12
-1
=-B11222'
-
=T
the kernel (.13)
-1
-
(B -B
)222'r~ 1
12 =12I A
the
becomes
kernel
(I.13)
(N-m-l) v2n1 times
times
becomes
when
(1.14)
15
[Q2(B2)]
[Q (B2)-½r
+
[Lt-4 t-½(N-m+l)
* {1 +{l [*-4]"B
[l-rB]-22[rl-rB]}
*{1 + t[rl-r
Here rl given
**
*
*
(1.15)
(I.15)
**-t Gno
322rl--B
1 = B
and unconditional as regards Bi, i=2,...,m is
poly-t.
A rough cut at a Bayesian version of a maximum likelihood estimator
of (B1
l) would be to determine the mode of the posterior kernel (1.15)
When the only cross-section information available bears solely on the
first row of (BI),
given that the prior we assigned doesn't do too
much violence to a priori beliefs, we can label this a full information
estimator, since we are using all available information.
If (I.15)
ignores additional prior information or cross-section data bearing on the
second row of
).thvn
it is in a loose sense a limited information
procedure.
Finding a maximizer (B02, F0) of (I.15)is in general quite
involved.
When both N and n1 become large, however, a reasonable
approximation to (
2 ,P1 )
can be found by using an expansion of two-
factor poly-t kernels given by Tiao and Zellner [7].
The leading term
in their expansion is a Normal kernel, which for the poly-t kernel of
a
[ *
A*
* *
rl given
= B
in (I.15),is centered at r
-B
-B -2][-22
-- 22approximation,
an (approximate)
then we may compute
If we adopt this-1
If we adopt this approximation, then we may compute an (approximate)
-1
16
first find a maximizer B12
maximizer of (I.15) in two steps:
[Q1-12)]
[Q2(B12)1
.
(B
'r
IQ1-2Ma
Then for B
fixed atB
of
a
compute F.
An alternate approximation that (we conjecture) is more accurate,
but requires more calculation is:
choose as an approximation to B12
-12
a
which maximizes (I.15) subject to F fixed at F as
that value of B
-12
to -1ject
r
a
1fixed
as
defined above; i.e. for a generic B12 compute the value of (I.15) at
a
h
t
a
l
12or maximizer
a
of (I.12
for a maximizer of (.15).
nB
B
valuesof12
Search over
17
An Example
To illustrate the application of the preceding calculations, we
shall analyze the example treated by Dreze in [5] under a different
specification as to how cross-section data comes into play.
He examines
Tinter's model of supply and demand for meat, poultry, and fish, using
as "cross-section" data estimates of price and income elasticities of
demand for meat products in Sweden computed by Wold and Jureen [8 ].
we can adopt it as a vehicle for
Use of this example has an advantage:
studying the effect of differing assumptions about the role of crosssection data in a Bayesian context.
Since this is intended solely as a numerical example to illustrate
the methodology, we do not attempt to justify the particular numbers we
use.
(See Dreze [5] for a discussion.)
Dreze uses the cross-section
data to generate Normal-gamma prior for (
12
iL)
(B12 is (lxl) in this
example) of the form
r11 e -[(B
th
r )(B
where all is the (1,1)
r
[(B
element of
/a
(i.16)
, thus assuming that the (residual error)
variance of the cross-section data is (proportional to) that of the residuals
in the first time series equation.
This prior is improper, but informative.
He then computes a maximizer of
{(1 B1 2 )M
(12B1 ))t(Nm-r+l)
{(
12t}- (12 -12
%
{(1 =-12
_)w (1 B_12 ) t--j(+v-2)
18
where in our notation W =
2
+
and v is a parameter Dreze
manipulates in his numerical calculations.
rationale.)
(See [5 ] p. 28 for the
Upon finding a maximizer of this ratio, he computes an
estimator of rl, by observing that since ri given B-1 = B
a maximizer of the density of rf given B
= B
is Student,
is the mean of r
given B
= B.
From (I.13), we see that the factor {(1 B2)Mll2(I B-12)
also appears under the assumption that cross-section residual error
variances and time series error variances are unrelated.
of a single Student kernel for (2
However, in place
1l) whose parameter set blends time
series data {M,N} and cross-section data { ,
B12
'
-1' nl}, we have a
product of Student kernels in (B12 Fl).
In terms of deviations from means, the demand and supply equations
in Tintner's model are of the form
Y1
+
B1 2 y2
+
rllZl
+
r12 z2
+
r13 3
=
1
Y1
+
B22Y2
+
rllZ
+
r122
+
r13z3
=
U2
and
We do not distinguish between specifications but rather leave all exogenous
variables in the model and assume that, in a classical sense, it is underidentified in each equation.
Tintner examined two models with different
a priori (exact) specifications.
-
-
Under-identification in this particular
-~~~~~_
See [5 ] for a detailed description of the model.
19
instance corresponds reasonably well with a priori judgments, since there
is some question as to which exogenous variables really belong in the
model.
Furthermore, under-identification is more natural in a Bayesian
context since it avoids the necessity of assuming that we know the
value of a particular parameter with certainty when in fact we don't.
An exact comparison of results is not possible, but by some
jiggling of the numbers, we can match up the prior in [5 ] and a Student
prior for (
12
of (I. 13).
F1 ), then use the latter as the cross-section "component"
(In particular, the prior (I.16 ) unconditional as regards
°11 is improper and has infinite variance.)
Dr'ze [5] sets
O
83.000
3.333
U
333.300
Lt
333.300
and
(B1 2 rL) = (.54, -.17, 0, 0).
12
-
The matrix P is expressed in units of the computed variance of the
i-l
indirect least squares residuals (in the just-identified case):
33.33.
Consequently, we scale the raw time series data M to correspond with P1
and use {N,M}, N = 23 and
1.36954
M =
-.35244
3.67191
-.53648
.98386
1.58149
8.35459
.85033
1.23576
83.43365
3.61172
12.20477
2.53480
.73078
2.62699
20
Under the specifications for cross-section data we have been exploring,
we must (see (I.13)) assign values to elements of the parameter set
{Pi'
Sl
nl, (Bil'
is not possible
l)}.
As previously stated, an exact match with (I.16)
because of the difference in models; but we can set P1
and (B1 , r ) equal to the values displayed above and then manipulate
s1 and n
to explore features of (I.13).
To this end we used the second method for approximating (B12 r)
suggested earlier and calculated Table la
and (s
= 2000, n
= 7,8,10).
for (s1
While in practice n
=
14, n 1 = 3,5,6,10,20)
and s
will usually be
determined by the process generating cross-section data, here our choice
of values is ad hoc.
However, when a prior (I.16) is assigned to (B12 i1)
12-
the conditional variance of (B12 L 1 ) given ll
= a 11 is (
.
Under
our specification the cross-section component of (I.13) is a Student kernel;
the density generated by this kernel has variance sl(P
that n1 > 6.
Hence at s
=
14 and n
/(nl-6) provided
= 20, this variance is also (o )
Of course the same is true for all s1 and n 1 such that sl/(nl-6) = 1.
For fixed P'
the ratio
sl/(nl-6) determines the "peakedness"
of this
component of (I.13); hence we expect that increasing sl with n 1 held fixed
affects the estimates in the same way as decreasing n 1 with s
held fixed.
The "cross-section" factor in (I.13) may be written as {[(B ril)ts 1
e
-1
(B1 l)]Pi [ ] +}
1; setting sl = 0 duplicates the exponent of (I.16)
in the synthetic representation
1
_
F(½
1 e)
(B1 2
12 -1
1)
**For n 1
> 6.
For nl > 6.
{
·
I
}
n-1
h
dh, but then (1.13) "blows up" at
21
When n
< 5, a prior of the formC{[(B1 2
t -n
(B1 2 r )]t+sl}
1
is improper even when L
1 )(- -1
1)
> O and s
1
(B12
> 0.
1 )-
This
corresponds for our specification to the improper prior (1.16) in the
specification of [5].
Poly-t densities need not be unimodal.
BtI
-l
deleted) for rLfor
1 fixed at
at
(N1 = 23, s
= 14, n
Graphs
of (1.13) (with
]
LA2 +F[M22+ri-2
B =2Jl
2
2J
and
and
= 3,5,6) show that in fact it is bimodal.
As can
be seeno for n1 = 3, the maximizer of (1.13) is the larger of the two
modal values of (1.13), while for n
= 5,6, it is the smaller.
The
smaller modal values of B12 are close to B12 while the larger ones are in
the range of estimates derived in [5] by Dreze using his analogue of
LIMLE estimation.
For n > 10, the larger modal value disappears and the
graph becomes unimodal, with the modal value of B12 not far from B12.
We conclude that as n1 increases with sl and N held fixed, the crosssection component of (1.13) quickly begins to play a dominant role.
The
behavior for sl = 2000 is somewhat different, as can be seen from the graphs.
Tables 2a and 2b display variance matrices for
estimates in Tables la and lb.
corresponding to the
Those in Table 2a were computed using
a large (time series) sample approximation to the exact variance of
0
*
given B12 = B12; i.e. [ 2
*
+ -2
]
.
-1
While having substantially the same
flavor as those computed by Dreze, the conditional variances of the
are much larger, and increase as n1 increases relative to s1.
The kernels displayed in these graphs have NOT been normalized.
li's
22
Table la
Posterior Modal Values for (I.13)
N
S1
-
n1
B1 2
_
-
23
14
3
Fll
_.__
1.611
.370
6
-.10205
.0002533
-.0001052
-.1948
-.00003659
-.0001684
.006892'
-.006214
-.084180
1.483
-.1884
1.23
.449
1
23
2000
r1 3
_
.380
5
f12
..
._
-.001285
-.02747
-.15910
.008316
-.03899
-.08845
-.009888
-.009568
_
_
- - -
I
10
.4'4
-. UlZJ
-. UZi48
-. UZZMb
20
.514
-.08955
-.04546
-.05446
7
.874
-.06674
.1443
-. 4856
8
.874
-.06430
.1477
-.5026
10
.890
.0640
.1462
-.5226
Table lb
Modal Values from Cases B.1 and B.2 of [5]
B12-
--
Fll
r1 2
r13
CASE B.1:
(v=2)
1.22
-. 166
.0003
-. 0014
CASE B.2:
(v=r+m+l)
1.32
-.174
.00018
-.0014
l-4
Ii
i-r
j1I
l
'
i
, ,
' 1 1~
i.
1---:
-i---:
I
' ·
-+
i
t
-t
fL- i j
-
T j /t
I
iii-i!~i
.,
II
hii *C
z
C
*a
II I i
i i ,_
, 4 1
ii
i
:
!--
cl3
t
1- · · :
i
1 -·
I
t
I
.
.... 1~
!- 1- !,
;
· C -^ ?
.
tf·1·:
.?.I__.
i
i
,...L
z
:-1
i.
C14
0I:,
'
'
i I _ X Il--i ii
- .---t- -
I-
',
i
I
3L
N
I
w
fI
-4
0
I
:1~
(0
CP
i,
1 i~
i
z
0
'-4
I
t 'r
t~
!
, I
l,
'
I
tlc
I 1 I4
i-
1
. i,
. .
1
1
I
l
i
f.om
ft7i
I
i
1
I
_
!
i
It
1
,
, I .
i
I
I
I
...!t
I
i
_i
I! 1
_
i !
i i
_, -
; '
1
1
;
i
i
I
j
I
i ;
!
1 ]
. I
-, I . '
I
I
t
!
:T1
1
i
I
.
,,
I
t
.
,
1
;
:
i
I
iI
i
:1
I 1
,
..
t
, ;
I
i
I.
C -
1
Ii
I
1
!II1
1
'
jP
L
II
Iii
i
I;
i
!
I
-
ce
,
:,
I
;
I
;
jI
, :,I
i !
I
I
I
'
i
j
I
i
:
. t
.
I
__
-H
iii
lI
!I--;
ii
i
;i
11
1,
i
I
I
I
; ' I
....
.
i
ti
t --
i
i
i i
.
,
i
,I
.;
I
ii
.i
i -
i
i!
Ii
ii
i b I ! i
I .i
ti-t
11
,
Ii
i
i
!
t
i
i
i
.
11
i
!I
I
i
;i
:
iI
I
--- t-PCCt--C---i
]i
i
IC
,
l
,
.
.
!
.
4
.
7::
.
t
I
I
:
:
i
;
;
!
i
:
I
I
;
:
1
1
,
! ! ;
,
I
!
;
i :
I
!
i
I
1
,
,
i
.;
.I
i
! 1. )
I
!
;
:
i
.1
!
.
i
'
cI'
!
0
-C-·' ' ----Cf-----
i.
.
j .i I
. i.
! ;.1
I
I
i
I
i
'111
1i
I
t
-
I
!i
i,
1
I
· ·
Ii
I
I
:
i
I . I i i : I
I
: i;
, I11
i
-
I
I
I
I
-.
:I
;
I
t
i
;1
-11i
I,
It
i
i
i
:1
Ii
I
i;
i!
11
i
.
i
I
;i
_
_
II
i
I
' I--c-
i
, -. ·
i
i
1;ii
,i
..
. .:
I
I1
iI
|
I
I
i i
i
t
i
I
I
I I :,
! i !
,
i
1
.
I
i .. i
i I I
,
J-
-i
i11
i-I
,
!,
I ' : ' - -j Itl- t
-
rI,
r
I
i -I! -,
.
t
to
I
,
II;I
~ I_ ~~
.
-i-Lc+--cCc-cL.
Il
ji
1
----
i
II
I --
.
. ! I :i
ii !,
II , ; I
,
I ', I !
i
,i
~
.
II
I
!
1
I I
i
I
i
i!
IIi
I i
i I
iii
Ii
T. i
4 i
: ,
! i
I
.
Ii
!
1i
It
.1r
-t
j
I
Ii
i ti
i
!
!!
iI
,
]
1
Ij
'
--I:: t-1
i
i,
L t mmi i
11~11
1!
,
I
Cl
ii
!
tl
..
0--
I'i
,i
t
-
.,
i
-!f--
I It
~--·---
--------
·-- i
i-
i-- ·-- i--i
, .I,,
.
t
t
I
H
>!
I----JI
r
; ' '
!T I:.
r It t- -i i1 !- '
---
0
PL4
O
i
i~~i
· · · :
1
..1
-r-1-:-- j-
11
'i
-' -i--t---
t
1-
'--t
I-·
·. I.
.
I
0
0
o
H-
_
I, I I'i I
ii i
-
,I
1
0
c'q
1! '
i.S
'
,
I '
-IiiI
II I i,
_
! I
!
--_
.~~
0
co
a)
O
c'
.
__.
-- -
:1
.
.
c J I
·
. .'i I
i
I
I
I.'
i ,
i
I
'i
1T
--IT
. I
I
14~4
1
tV. ; t
}.......
i
'
': '!
i
i
iI
H
I
[--,
II
II
II
:.
I
N
I
I
:
'
!
!I
,
.
i
I.
f
. I
e
e-
;
.
tIt-
,
ff---L-iI
-L---L-
''~_ .ii
-i... !i!
-- -~--
l
-t_,
li
C
i' j
'
t i-:
,- 4 i.
:
F-?
II
EI
','
:
'
.
U1
C4
11
m
H
.i.,
F H
I ,-
-L
t/:
o
P4
'I' - -' f: i--i
4.
t
!···
!- i
i
1
j
t
- -- ,
i
.
I
I-t
i
'- '
-j-t -t - . :
* ,t
-i
ft
l
.
J
-i
,
T !-
,
. . ..i
.
1!t
i ji ,
l
-- L-c-cc--
. : ·i . ;
! L,,.,1
I
.
F
iI I
I
I
I
!
I
i
,
---cc---
------ cccL
4-i~~~~~~~
I
I
C
!jFI'
j
.
.
" ,,
I
0
r
i
i .i
t-
I
i
;
----
I1
I1
-i
i-
~
i-
-1
NV
-
Irt:
I
cc-
, .
floods,
i
----
I
I
"
i
J.r
!
p7-t-t-i-i '
i- c-.i
-1--r-- t- ;...i.. .
.i_
,_, I ..1 ,...
-i i--:--i t-'r-i ii rt
44.
;- -i--l-- I
L I. 1 :
i
I f, '
l
- I .I i
I-, - -i
-t;
-tt
i-f--
iJ:
' i,
t
i
+
-- '---tr-:
i--L..-i- i.. .i--i
: :
1, j,
I
Eq
¢4
-~~
I
i
i i..
z0
O
:
i i-:! i irl
/i.1i
II
i , I i : iii ,
H
,--7 : :' :'
-
i
!
0
'
,--j : i ' t r
, i
H
U
-L
-::-i
-
'
;
. 'I
i
:..I_
; -
i- '
=
I
IIC
1C14
..
=
r
i.
.,
i tvCN4
i
·
I
*
It
I
___
_
I:i
i -t
::: ::::: :kf
_
I
*o
i iT-.
_
1
I
I;
1
..
7
1
i.
i
'
!I
i
~-t-~---
7--
i-
-- C11-·-*_
I
r
ii
'
-
I
I
- - - -
I
ill r
--I
I
0~~~-
r)
.
0
1
,iiii
o
''
In
:
0
I ... '
In
i
.
__
4
_._
I
1
- - - - - - - -: -
. 1
. 1
.
T_ :
I
:i
_
,
;_
i
.
I
,
·
_
;
iI
Ii
I1;
*
II-C
-ii
i
Ii
j;
'-4
I
·- -
r--+-·
-·1-
I
. ..
r -· --r·-·
tt-
t
-'-
PSI
·S
'
·
Ii
;I
!i
G
..
i
It
I
t ]4
+eN
CN¢
I
.
I
I
ii-
t I -I - -
r
C..,
-
t
r-
II I i_ i -
IIq
C4 A11
O
U
tIr
1-4
1i
W
I
0
I
i ii
i
; i 11 ·
H
4
4-
4;
s
C
i--I
U
iii i
rl
O
CS
pq
0
P:
O
V3
O
o
O
H
·-L
Ilit
;
/3
f
i
-
-I
. . . t-t'
J
Eq1
C)r
:i
__-
. .
-- -
--i
I
-·-CL---3-i
,
,
.
- . ,
i
L
' -. I
-
-
F
--i
-
i- ·-
'
-
,
' ,. I ;
;
4
,
'
i t,
J
4 +|iS
i
t
i
.
- -
I
.
.
.
.
U)
W
'-i---r
I
i- '
!-/.
t
O P,
! O
l
D
-r-
t;
t
<
-i ---
.'
.
=
-
'
, .1'
.
-:
C
....
t
t
-.-
-I
1
!
5
.....
I
;
I
i
i
i
I
---rc--------t--c--cc
I
,
i
. . .i-. .. ?4
. .. . . _.....-* ...
I- . j ?w
. .....
i
----...
it .. '.' ! ' ...
x'_ ... '+...
. T.
.
,' ...t:t !'-- ~"r-t--!'
|
--
|
,
r
r
--
i
-- ' -c- ., -1
i
i 1 --- : - i--i
i--l i -i-t---;--l
-:--i
'
~ .
-.
'
...
... r
i
'
t
' ! ?4 I'
: -i
._
. 4'-r
' ! - t '',1I t1;4l!,, , t_0_
I~~~~~~~~~~~~
r-t-
_ t I l-
, J! ,
,t I ;
iI
t iI-i 1 1 i-.
, ,
I1-l1
'
'
i
i
: f : :
t -t
1 t ' .- i
I-ril
t
r--ii
I
·--
i
:
i
'
1
i
I
0
·
i
1:11 i I - i
f
t
1
-:-- i
ii '
i
·
;
.. i
i :
IH
I
1
, , l
jlillil'j
--
-
C14
0
?
_D
I
!
_/
.
i
I
__ I
;
.
T
i
4t
1
- t-i.
J
L _.
_i i _i
i- i-I i- i
rl
I
4C e
II
··f-··C-
.-1 1-
-
..
;.
i!
-
-
i
·
t .
~iI
-
-
. . .
I
I - - .
. I
--- . _ .
-- .,
4
14--i-~
.
I
I
I
.
i
i
Ii
I I
.
r
J~
-N
-
r
I
1'
1--·
'-L'-r-`'-*"''
__
-U-
--·-+-
i! !r
11 :
-- - .-.--I
----- r - --.
I
I j
I
i
Ii
ici--w----r
i
I
+l
C4
C4
4C
1- *
*C kp
I 1
.
I11
+ I
ICI----·-----Y--L
i
I
: 1
(
_
CN
,
Ir
O
.
:
C f
t-l( *
O
; I
-,
--i---' -t
i-- -: --
.
.
:
I
-
I
.'
-·--i----:C-: L1
-i-l . -i-
;
-
.
: -
.i .
N
IIt
.
:
:
I
.
.- -
r
I-C---C-I-----
.
i3
H
H
C0
t
t
-.
1 1
I
I---
.
t
I
i
t
- i- i 4 j - 1- I
'
:0
i
i
I t
.
I
I
iiltliir
o
t
: ---
: -t
i
z
0
i
4 0! ji8
I I
;
.
i
.
-- -.---
--·- ·- ·--- ·-----------
.----- . - ·-i.i , I
c4
H
z
iIi
Ii!
¢z
I
1 t;
.I
I
i!
-"--Ti-!
.
i ".
--
7 -,
..
I
i
I
I
.
I i
i4
-,
..
-
*
t!
! I
' I
·-
.
_r
..
...
..
-
;.
.
i '4'.I
.
.4
.Ii
I . Ij
..
I
'
,
i
I
I
. I
I
.
.
I
i
;
-C
i.
I
.!
.
!,
. I
.
II
I1
:
j
.
.
--.
.
4
i
I'
i
... .A:,~-·--
!
.
I.
'
I
j
i
t1 I
4
ii
I
i!
i
i
I:
I;
: :
i I ,
i!
i
ii
il l
--
,
1
,
I
I
.
I.i
?
-4
I
L
!
iI! Ii:
i
.
1
T I
I I1
ii
..
I
i
1
:
1:
; 11
i
!,
i
I
1
.i;
I
,
,
.
1,
,
; I I
i ,
I I I
1
1
II,,
:
i 1 1
!1
11
I,
II I .;
I;
iI
: ,
II
_
:
I.
, ;
1
iI
. i.
'
: II
1
iI
;
ii,
.,
j
;1
,-
i
-·
.
,,
;i
,I
i!
i:
I
.
'
i
-
i
I
I
i
o
I
1
I
I
-
I
I
. .
II
-I -
\ I _
.>
-. --.- 9 -
,[
, *, j q
i 1i
I! I1 .
i
! ,
! : . I
!
. i i
i- i
i. T i
.
..
1
.
t i
,
i-
!i1
.
-
-- - -,
...
.'
.',
li
I~~~~
.
'.'
..
.
.
......
I
I -
1
.i
t
4
It
r·-·
..
1
,
i... c
r- ;
i
1 ·
i
_
'
t-4- - t--,
'
r---
-
i
1
I
.
. .
,
---
it
II .;
i
!I
...
:
. . . .I-
·
iI
_.
_
ti
I ... i
I-
|
F
F
j ':
s ,
i
I- tT
I..
_.
. . ..
i:4
i i
.
I
;;.
.
. .
' .
_
i
i
.
I
!Ii
~~~~~
I
~
.
..
: ,
i
"O"
i
.
I
4
,
.
,
'i i
; iI
!
I
I
I
I
P
04 I ~I
. 1.
,
s
* I
..
I
Il- i
__
t
i
U
i i!
!. I I
I
..
. I
_;:
11
J.
.
_
ii II +
i!
: I
.
..
,
f
*
(
'__-~~~~ -f
I
, I
~~ ~
I
-- -7
i
I
-
i
!
I I
. '
i i
. .
I
I 4I
_____~~~~~~~~~~~~~~~~~~~~~i._~~~~~~~~~~~~~~..__~~~~~~~~
t
i
o
·
· i
C
o
Ho
C
Cl4
H--
CD
\O
......
._
,,,,
· ,
_/ I i~~~~~~~~l
1
/
·- f
-' -t--i-i-t
-·
_I
.
I
_ _
- |
|
|
|s
w
1~~~~~~~~~~
I
I
|
_
i
-i
!
i
^
I
^
__I4
r
w
T
_T
i
I
1
s
r-
1.
! I
I~~~~~~~~~~~~~~~~~~
II
II
--·
.
En
,
I
1
I iI
1
i
~i
i !
I
II
i
!I!
f
II
r-
CN
-
-c-·--cc--t-----c-I
.,
.I:
!i ; I
-
I.
I
I
I
I1-,II
I:
. I
I
I I
;
,
II
iI
1I
. I
1
.
! i. i
,.
I
I
I
I.
,
t
!
I _
,
i -?----i !
I
.
I
I
I
I
I.I
IIiI
I
II
II
I
! I
I:i
Ii!
I
I:
I
I
I
I
i
'
I
I----
,-4)I
I
I
,
I
I
I
.iI11
;
I
III-I
1
I
:
FZ4
0O
H
*zT
r-i
N
II
i
',
;
! -!i i
Z
' '
'
'
' -i-iJ
-1
*0.
o
,.: i.
1
'
.1
-;
I ,,
C)
t
O
-'.'
0,
'-4
C14
,
01
.
.
·- 7
---
I
4
|
.
,
Ii
! I
i
II
'
It
I
. .
j
,
.i
;
t I
,j,!I
I
,!1I
I
i
..
. .j
[
:I I i i,
i
I
!I
-
J
-A....
I
- I
I 1
i
, !
I
I
-I
; : !; 1
H
I
;
i
I -i
I
ii
t
j
I
I
i ,
A,I
I.I 1
! II i
i] t
1
, i.!11!
C-)
1
iI
I
:
i;
I
.I
i
I11
i
'-4
'* le
I
.
I
-4
111
i t i I
I
i il
I_i i
: I i r iT
i
-e
·--
:
iI
__1
-I
I
·- t-,--!-·t
-tc+
I I
---14--I-t -i-----
r.
t
..
-·
i --·---·
.
-7-.-
H
0
Err
C1
:
I
.
.
.
.
.
.
.
-
i
i
: 1
I i
ii
,
1
11.
I
iII
I
,i
.I.
I
I
. :I
I
, 11
--·
-·
; I
; I
!
i
I
.
,
L
I
ii
11
;
,
1i
_
i·.
!
4
i ;
, !I!' j; , i j.!
-I-I
I
-t... I~~~~~~
!
- rF . ,, .
1--
.-- l
.. .I
I
1
1
I1
. . 1
1
. -
't
.
t
;
I
t
1 .
. i
I
!
i i7
~
111
111
ii r
!,
II1
IIi
i
!
!,
! ...
ill
1 ,I
.
!iI
3 ! Ii i;i--~
I!-:
I
IJ
I
J
'
I
i1
1
11
:
I
i
_-
,illllI
I
.I
i i
---
t
S
0
co
,--I
:
s
/
'I
i
i
I
I
t-
!
!
:7
.,
:I
;I
i I
I
II
II
I
I
!
.
,
,
j:
,
.
!
i Ij
I
i
I
I.!
41
I
.
i
t
II
I
i
i
I
Ii
t
I
!
t
I
i
I
I
I
;
.
I
|
, ,
|
4
!
I
i
I
.
I
I
I
I
.
.
.
I
I
ii
. .
I
i
Io
I J
i
I
;
:!
I
II
i
:;
I
.II
1
,
,I
|
i
i :I , I I
Ii
-~-~cc--t--~I
;I 4
I
?i
i
I
,:
e+-c
: ; , i
i
i
.
.
I 111. ' ij
.
j
I 1
j
i
; iI
I
i
,
I
.
I
.. ·
I
i
._
.
.I
I
I.
I
.
I
-
.
I
I
.;I
i-i
ii-1
.
!I
I
.,,
j
-i-1
¢
i
II
I
.
I
I
II
;
,
I 1: II
1 I
i :,
. .- ^-
..
! ,
;
i
I
i
I
;
- :
. . .
i
-
II
I-I
- ..
s
I
111
"i
l
i
I
j
i i
I
I
I I
I114
i
i
i!I i I It
_-
'
i
'
4
'Ii I
i I t
I
. .
::I
.
. . . :
I
1,
,
!
,,
L
.
-1i,,11
I
. I-I
i
-
I 1 11
' I!
I I
11 i
I
4.,
j
. .
- i~i
I i
.I _
.. .I.,
I
-i .;...i----l
I
- i
'
i
i
i
1
;I
i
1.
I11
i I
I
I
I .
t
,~
·
.
; I
I
i, ....
1
I .
II.j
4
ij j
_
ii
J1
i
Ii
i
,
1
t
i
i
1i
i.
,
1
i
i
i
I
.i
1II
. ! !i-
- ,
1
..
i i'
1:1
I
1i;
i
_ __
1
.
III
I 1
:IiI 1
1:
,1
:
,-I
i
4
I
-
-1 -1
i
!,
.
, : .
.
ij
.
i 111 -
-_ I
I I
I
__
I
,
.
;.. i
_.i
__i
-- i
I-,
iiI
'i-- .: I-
-4
.
I
-11-F
-
"i
:
.
i
i .,,
-
i
!I
iI
. [
:i
;1 .,
I I i , .
I
it
I I
I i! I!
i
I
.
I
I,
I
i
.
I
. . .
. .,
,1- I '
,
! 1,, 1 i
I
;
I
I I
. . I
-
I
.
:
,
I
I
. . . . . . .
i
ii ii 1t
I
C14
.-
1
I
-
I
.I
.
,
-1--' :--
O
i i---
i_
-- 4 -!. ----I
--; i
-
0
..
.
O
cO
0
.
.A
I
23
Table 2a
0
Large Sample Estimate of Variance Matrix of F1 given B12 = B12
N = 23, s
= 14, n
= 3
F
r1 3
11
rll
.08971
-.00009
.00299
.11335
F12
r13
-.01904
-. 07287
.45937
-.00549
.50078
r1 3
N = 23, sl = 14, n
0
= 5
l11
rll
-
.00299
r13
N = 23, s1 = 14, n
-.00030
= 6
Fll
r12
ll
.07565
.00533
-.01886
F12
-.00533
.12571
-.00054
rF13
-. 0186
-.00054
.12980
24
Table 2b
Variance Matrix, Case B.2 of Dreze [5]
rll
rll
r 12
r 13
(.019)2
r±2
-(.002)2
(.010)2
F13
-.00001
~ O
(.010)2
25
II - Cross-Section and Time Series Variances Related
In the sequel we shall assume that the variance of the error term
in each cross-section equation is proportional to the marginal variance
of the error term in the corresponding structural equation.
with (B. r.),
1
1
Namely,
S i , and ni defined as before, the likelihood function
1
generated by cross-section data
i),
(Bi
Si,
, n} bearing on
individual rows of (B I) is*
m
-½n
I a..
i=l ll
-½{[(Bi r.)(B
i)--
e
th
where ail is the i
-
F )P[(B
=1
component of
are known with certainty.
.)-(B.
.)
1
+ S}/k1
1
(11)
and the scale factors ki, i=1,2,...,m
Provided that the k i are known with certainty
we may set them each equal to one with no loss in generality and do so.
If we assign a non-informative prior to _, _, and _, posterior to
observing cross-section data leading to (II.1) and n time series
observations generated according to ( 1 ) and summarized by the sufficient
set {M,n}, the kernel of (
I) unconditional
as regards
is, in an
obvious notation proportional to
_-l
-½tr-1&
1tl(n-m-r)
_
-½(n+m+l)
f=i
1
-S~q-½n
e-½qn ii
i
a ii
_
_
= 0, P. = 0.
By definition, when n
m
m
=
-
d
i~l
~,
(11.2)
26
where qi
0 when ni
=
0 and qi > 0 when n i > 0.
The integral has not to the writer's knowledge been explored before,
We show below that when
and appears difficult to evaluate in general.
m = 2, the posterior kernel of (B
(1)
D
may be expressed as a product of
BAt ½(n-m-r) times independent Student kernels in (B1 r1)
and (B2 r2)
and
(2)
an infinite sum with generic term expressible in terms of
a hypergeometric function F1 of two arguments (cf. Appell and
Kampe de Feriet [1]), the arguments being ratios (ql/All+ql)
and (q2 /A22 +q2 ) of quadratic forms.
Evaluation of (. 2) When m
Rewriting N
n for notational clarity, we show that
e-½tr
e
E>0
equals
2
I-e
i 1A
3
e
i=l
-
i1
d
(11.3)
27
N+ (n 1+n 2 )
r(½(N+nln2- 1)) (All+ql)
2
- (N+n1 )
2)
- (Nn
(A 2 2 +q 2 )
{([A12 /A1 1 A 2 2 ](-R 1 ) (l-R2 ))
½
'Fi(J+
where R
i
1 +,
(II .4a)
½(N+n,)+j, ½(N+n +n 2 )+j; R ,R) '
- qi/Aii+qi, i1,2,
22 r(½(N+n,)+j )r((N+n2 )+ior
(i+-)
:. 4b)
..
( . .
r(½ (N+nl+n2)+j)
¢; Pr
and
F 1 (j+,
('+nl)+j,
1
(N+nlf+n
2)-,.
(N.e'
~ ~~~~~~~
2)
2 )+j; ~ R 1 ,R~~~~~~~~~~~
2%.-2
(II.4c)
I'
£o
-
o
EE.
m-O n=O
[(j~)
)+j ]j[(N+n
lm+ n [L(N+n
2 )+j)] n
1
Tdt ll.tN-n
- --
L A \"
-
i-L 11
4-n
2r
·
-m
Fn
m n
2
28
The function F
is absolutely convergent on the (open) unit disc;
we have written (II.4)
so as to show how it depends
on the square
.
of the "correlation" coefficient A 1 /,AlA22
2
Proof:
-to
-1
to H =
and then from H
To this end transform first from
ion is I~12
-H12HlH22.
transforma
the
first
of
Jacoban
=
The
to p = H1 2 /AH1 1 H22 .
The Jacobian of the first transformation is
Hll-- , so that J(
and that of the second is
(12)-3
=(p
-1
(HllH22) .
)
(H1 1,H 22
1122
-1
As ail
=
))
=
_|
H
(H
2
Hii(l-p ), i=1,2, (II.3) may be written
as
+1
-½H
(2
e
11
f
f
2
11 q(lP )
- ½H2 2 {A2 2 +q2 (-p
)} - pA1 2 H1 1 2
0 0 -1
½ (N+nl)-1i (N+n )-1
11H
22
2 ½(N+nl+n2-1)-i
(l-p )
dpdH 1 1 dH2 2 '
N+j + (nl+n2 )
If we define c
= 2
((½
(N+n+j
2
rewrite A
+ q1p
))
2
) = (Aii+qi)(1-Ri
)
R i - qi/Aii+qi,
exp {-PA12/HllH22 and integrating over Hii E (0,c),
expressed as
r(½(N+n 2 +j)) and
i=1,2,
upon expanding
(II.3) may be
H
1122
)
29
Z
[-Aj /(A +ql)
j .
12
11
½(N+nl+j)
(N
(A22 q2)
222
+j)
2 +n
(11.5)
ipJ'
+1
+1
-1
2,½(N
(1-RP1
J
2)½
2 (Nl+nn
1 2-1)-i
)½
+n j
(
N
) 2,
(1-R2 )
-2+-)do
Aside from pJ, the integrand in the above integral is symmetric about
0 in p, so for j odd, this integral is 0.
u =p
, J(p-u) = ½u
, a typical integral in (II.5) is
1~n
El'-1--1~1-----(N+nl+
u
j
f
0
When j is even, upon defining
(1-R U)
+n-1)-i
) (N+n + j
(l-R2 u)
du
(11.6)
(II.6)
This last integral is an integral representation of a hypergeometric
function of two variables (R1 and R2 ).
r(½(j+l) ) r ((N+nl+n 2-1) ) /r(½(N+nl+n2 +j))
Substitution yields
-1
-H
E to H
-1
H H
Z
12 22 21' J(
I-3
=
1.32
1.2
times F 1 as defined in (II.4c).
(II.4a).
In the special case, say, q2
from
Namely, (II.6), equals
=
0 and n2 = 0, if we transform in (II.3)
and then from H to (Hl.2H12,H22),
H
(Hl.2',H12 ,H22))
(H
H-322
22 ,so
=
_
HJJ(H
H(H
(II.3) may be written as
(111,1))
(Hll.2,H12
H.2
22
-
30
c
-½11.2 (All+ql) -½122A22.1
c
f f f
00 -o
1
-1 2
(H
+I
A)
22
21
22 2 1
Integratine
sdt12dH22
-H
11
d
(N+nl-1)-l ½(N-i)-l
11.2
H22
ll
Hl11.2
®
Integrating over H21
21 e (--,+-) first, then H 22' H.2
11.2 £ (0, ) yields
N+
-
{2
v7
rF(½N)r(½(N+nl-))
+
- (N+n1 -l) -½N
11 +q)1(A
A22.1
We can arrive at (II.4d) directly from (II.4a) as follows:
n2 = q2
=
(II.4d)
when
0, F1 as expressed in (II.4c) reduces to
1m
m!
-½
(1-R
(l-R1)
m=O
cj becomes 22j r(j+)r(N+j)
(l-R1 )
+),
and R
=
so (II.4a) is proportional to
= A½ /(All+q) ½ times
All-
c
j=0
22
j
r(j+½)r(PzN+j)
r(2j+l)
2
[A12 /A11 A 22]
Using the relation r(j+!)/r(2j+l) = //
2 2r(j+l),
the above sum reduces to
30a
A2
oo
rN
v=0
E
j=o
2
j
jI
-½N
A12
[A12/AllA22]
=
r(½N) [1
-
A2 AllA22
Algebraic rearrangment yields (II.4d).
A feature of F1 worth exploiting for computational purposes has
been recorded in [ 1]:
F1 (ct,B,',y;x,y)
F1 can be expressed as an infinite sum
=
Z
m=0
[Y]m m!
2 F 1 (-m,';
B+
-)xm
';1- )x
where 2F1 is a Gaussian hypergeometric function--in fact a polynomial
of degree m in 1-(y/x).
31
Posterior Density of (B(2),
The posterior density of
directly from (.4)
Si + [( B
) unconditional as regards
upon substituting [
r )-(Bi ri)]~ [(B
i
)(-
_ )i
_jM[B LJ
for qi.
)t
Normalization rule, set the first column of
follows
for A and
If we adopt the
equal to a column of ones
and partition
M=
[::-21
21
, mll scalar,
then All + ql
1 may be conveniently rewritten as
S + [(B
r )-(B" -))],[(B " L
)-(B- -,,
1
12 -1)
12 1
1
12
212
where
&l = -22 +
by defining
el
_=
22nl
l
l(12 22 4l)
(B
+ = [[ml2+
r)
]
1 2+(B 112
2 -1)-1
;
, we have
S1
m112 + S1 + [(12
)+m1
2 22 ] 1
S ,,-
[(B12 -)
-1t
m12-22
The sum A 2 2 + q2 may be rewritten in precisely the same way with the
obvious notational modifications.
See Raiffa and Schlaifer [6], p. 3 1 3 .
32
Then (II.4) implies that the posterior
-(2)
density of B
to IB
and L unconditional as regards Z has a kernel proportional
t(n-m-r) times
(1)
a product of independent Student kernels, one involving (B12 rl),
the other (B22 r ), and
(2)
a complicated function expressible in terms of powers of (a)
ratios of quadratic forms appearing in the cross-section and
time series likelihoods and (b) a cross-product term
(1 B12
)M(1 B22 r)t;
12 :-1
22 -2
namely,
/B
1
[(B1 2
trI-½(N+n
1/(n-m-r)
~s 1 +E(B
1 )l[(B 12 l
-( 12
1 ) (B 12
1
)
1)]}
(II.7)
)
- (N+n
{
+
[(B2
L2 )(B
(N,nln 2; A,qlq)
22
)][(B
22
2
2)-(B22
-)]}
2
where the function ~ is implicitly defined by (II.4).
In the special case where cross-section data is available only on,
say, the first row of (
and (2.24)
L), (II.4d)leads directly to the densities (2.23)
erived by Drbze in [5].
2
33
Approximation for Large N
It is natural to inquire what happens to the posterior density of
(B, I) as N
+
m but n1 and n2 remain fixed and finite.
We shall show
) simplifies considerably by use of an asymptotic
that the kernel (II.7
approximation to (EI.7) whose leading term contains a Normal factor in
(B(2),
(The proof is given in the Appendix.)
).
In order to simplify exposition we shall write
q1
n(1
q2
(1
+
Al
-),
1
n11
All
1
l(N+nl) ( 1
1
=
+
)'
+
A2
(
2n 2
+
+
2n2A22
A 22 =
(N+n 2)(1 + N2
and
A 12 =
/[(1
+
)
where
nl = $1' 62n 2
=
S2 '
=
mll.2 '
=
Si',
1(N+nl)
(N+n 2 )
and for i=1,2,
Q
[(B
r i)-(Bi 2
ii
-2)(i/i)[(Bi
i)2
and
Ai
=
[(Bi2
4
)-(Bi
2 -i)](
2/Bi)[(Bi2
i) (Bi2 4)]t
=
S 2,
34
The cross product term
- [(B 1
with
= all
.2 2
r)
+ m1 2
] (2
22
/)[(B
2
rl) +
122
In addition, we define
-vT N(1 +
-
g
12
)
_
t
=
[
Q1
j2 (N+nl)(N+n2 )(1 + Nn )(1
1
I
+ -)
N+n
2
With these definitions, an approximation to (II.7) that is o(N
)
is, up to a constant of proportionality, 1B DtI(n-m-r) times
-½Q
1e
2+g
elnl(l +
1
{1 + N-l
1)
(P1 + [
][1 + ge
1
]
1
25(1 + N1
1~(II.8)
A2
%1nl(l + {)
1
+ N+n
N(P
2
[
+
[.
Q2
2(1 + N+n2)
+o (N-1) } .
]1
+ ge-g]
35
Here, as in (3.2) and (3.3) of Tiao and Zellner [7],
'= 1
PP0
p
=
2
2
2(r+l)Q], and
[Q
¼[Q
p
1 - 2(r+l)Q 1 ], a
For large N, g is approximately (-,/
r)
are quadratic forms in (B
[Q2
[Q
2
].
2
Since Q1 and Q2
i/ r--)(1 + 0).
and in (B22
(2
2(r+)
) respectively and 0 is a
bilinear form in these variables, we can if we choose, complete the
square and approximate the leading factor in (II.8 ) by a Normal kernel
whose argument is a 1 x 2(m+r) vector [(B1
1 )(B2
2 )].
Additional information can be squeezed from (II.8) about the
behavior of ((2)
) as n1
-*+
, n 2 + , or both for fixed but large N.
Holding N fixed, as n 1 + A, since g is proportional to
+ n)
1
, it approaches zero, whereupon for very large N
(Nnl)
(1
and n,
l1 Btl½(n-m-r) times
A2
-½Q1 -½Q2
e
{1 +
1
2
+
1
1
'
+(P
N+n 2 1
2
2 2(1 +
)}
(II.9)
Q2 )
N~n2
is a rough approximation of (II.7 ) up to a constant of proportionality.
36
APPENDIX
Proof of (II.8)
A typical term in (II.4a) is aside from IB t 1½(n
(-2) [ (J+1)
]m+nr ( (N+n+m+j))r((N+n
2 +n+j)
- m-
r) proportional to
)
j!2mnl r (½(N+nl+n2 +m+n+j))
( A.1)
j
m n
A12 ql q2
(N+nl+j )+
(A11+q 1 )
(N+n2 +j )+n
(A2 2 +q 2 )
Using the Stirling approximation to the gamma function, the ratio of the
first term in (A.1 ) to
- {(2 7)½(½N)(N
approaches one as
1)e
N +
N}
x
J [(j+1) m
(-/)
+n N j
Jlm!nI
( A.2)
.
The second part of ( A.1) is
N½J mn
n1l2
(I
+
x
) (1 +
n.
;(N
(l+N+n
1
)m(l
+
x {[ 1 (Nn+nl)] ½(N
n2
½(N+
(A.3)
l
--
1-j)+m
(1 +
Q2
(N+n2 +j)+n
)
)
) n(N+) 2
37
The terms in curly brackets in ( A.2) and (A.3 )do not involve m, n,
so we delete them.
or j,
Upon doing so, a generic
term of
(II. 4a),can be written up to a factor of proportionality as
(-/)j[(j+l)
N nln2
]m+n
(
'i+m
i+n
(N+n1 ) ½
(N+n2 )
j!m!n!
(A.4)
( m n
C 1 e2
½j+m 2j +n
1~ ~2
(1 +
''
_Q1
J(1+11
l)m(
+
½(N+nl+j)+m
2n
Q2
Whiletheabsolute
error of the
(N+n 2 +j)+n
N+n
While the absolute error of the Stirling approximation to Fix)
x -+
increases beyond all bounds, (A.4 ) shows nevertheless that terms
in (II.4a) with
zero as N
m
m
greater than zero
or n
with j, m, and n fixed.
Now write
Q1 -2 (N+nl)
)
N+n1
(1+
and
as
1)
1
-½Q1
a
Z pk / (N+nl-r-l)
k=O
k
approach
38
Q2 -½(N+n 2)
(1 + N+n2)
2~
as in (3.2) of [7
-N2
pl/ (N+n 2 -r-1)
Q
Q=O
with Pk and p
defined as in (3.3) of [7 ]. Then as N
(II.4a) is proportional to
e-½Q 1-Q
2
PkP
k-O Z-O (N+nl )k (N + n 2)
(A.g)
00
co
00
(-
x { Z
Z
z
j=0 m=O n=O
Since p
Q'1
)
[
(j+l
) ]m+nNJ
j[m!nl(N+ni
!I
m+n
12
(N+n)
j+n
(1 +
CJ 1p2
m nln2
X.
= PO = 1, and Pk and pI for k,
1 2
½2i+m ½Ij+n
I
·/J
A
x
)j (1 + -)m(1
nl
+
2)n
n2
J
(1 + N+n)
N+n 1
1
½i2j+m
(1 +
~
Q2 ' and r, as N +- , the only values of m, n, k, and Q, for which (A.5)
does not vanish are k = Q = m = n = 0.
) is
½i+n
2
> 0 are functions only of
Using the definition of g given earlier (A.5
I-- -- i
)
2
39
-½QQ -Q
e
2
{1
2 + o(N - 1 )
1
½(j+1) 1n(1 + n
x
{
+
j=0
(N+nl )(
+
Q1
+n
(A.6)
)
½(j+l) 2n 2 (1 + n)
2
+
+ o(N-1
)
}
Q2
(
Since
X
j=0
4-
= ge
,
2)
N +
(1
+n 2 )
upon multiplying out terms in curly brackets
!
in (A.6) and dropping those of order N-2 , a rearrangement gives (11.8 ).
40
References
[1]
Appell, P. and de Feriet, J. (1926), Functions Hypergeometriques,
(Gauthiers-Villars, Paris).
[2]
Chetty. (1969), On Pooling of Time Series and Cross-Section Data,
(Econometrica).
[3]
Dickey, J. M. (1967), Matricvariate Generalizations of the
Multivariate t Distribution and Inverted Multivariate t
Distribution,(Annals of Math. Stat., Vol. 38, No. 2, pp. 503511).
[4]
. (1967), Expansions of t Densities and Related Complete
Integrals, (Annals of Math. Stat., Vol. 38, No. 2, pp. 511-518.
[5]
Drze, J. (1968), Limited Information Estimation from a Bayesian
Viewpoint, (CORE Discussion Paper No. 6816, University of
Louvain).
[6]
Raiffa, H. and Schlaifer, R. (1961), Applied Statistical Decision
Theory, (Division of Research, Harvard Business School, Boston).
[7]
Tiao, G. and Zellner, A. (1964), Bayes Theorem and the Use of Prior
Knowledge in Regression Analysis, (Biometrika, Vol. 51, No. 1 and
2, pp. 219-230).
[8]
Wold H. and Jureen L. (1953), Demand Analysis, (J. Wiley and Sons,
Inc., New York).