Page 289 - Special Topic Session (STS) - Volume 3

STS544 Jonathan W. et al.
(3) = + ,

where = [, ] =nx2n and = ( , −1 ) = 2nx1 and, then, in transposed

form for all t as

(4) = + ,

where = [ , … , ] =nxT, = [, … , ] = 2nxT, and = [ , … , ] =nxT

Consider column vectorization rule () = [ ]() ,

where, for the moment, A, B, and C denote any matrices conformable to the
matrix multiplication , vec(◦) denotes the columnwise vectorization of a
matrix (column one on top of column two, etc.) and  denotes the Kronecker
product. Applying the rule to equation (4), gives

(5) ̅ = (1  X)γ + ̅,

2
where ̅ = ()=nTx1, = ( ) = 2n x1, and ̅ = () = nTx1. The

n-variable generalization of the pattern of and in 4-variable equation (2)
implies that

, , 0, … ,0, , … ,
(6) = (0, … 0, , … , , , 0, … 0, 21, … , 2, 31 32 31 3
21
1
11

, … , ,−1 , , … , ) .
1
1

Because zeros in make corresponding columns of (  ) unnecessary, we

eliminate these unnecessary elements of and columns of (  ) and write

equation (7) as

(7) ̅ = ̅ + ̅,
̅
where and ̅, respectively, denote  and with the unnecessary
̅

columns and elements removed, so that ̅ = kx1, where k denotes the number
of non-identically zero elements of .
Theil-Goldberger mixed-estimation imposition of equality restrictions on
elements of ̅ goes as follows by adding pseudo data to the bottom of the
̅
data matrix [̅ ]. In exact form, parameter equality restrictions may be
expressed as 0 = ̅, where is a qxk matrix of 0, 1, and -1 elements and q
denotes the number of restrictions on the k elements of ̅, so that q < k.
According to the mixed estimation strategy, we impose restrictions with a
chosen degree of looseness or tightness regulated by positive scalar , such
that a higher value of indicates greater Bayesian tightness. Thus, we extend
0 = ̅ to

(8) 0 = ̅ +

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