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ECONOMICS
DIFFERENT CONCEPTS OF
MATRIX CALCULUS
by
Darrell A Turkington
Business School
The University of Western Australia
DISCUSSION PAPER 11.03
DIFFERENT CONCEPTS OF
MATRIX CALCULUS
by
Darrell A Turkington
Business School
The University of Western Australia
DISCUSSION PAPER 11.03
Let Y be a p  q matrix whose elements y i j s are differentiable functions of the elements x r s s of a
m  n matrix X. We write Y  Y  X  and say Y is a matrix function of X. Given such a set up
we have mnpq partial derivatives we can consider:
i  1,
,m
 yi j
j  1,
,m
 xrs
r  1,
,p
s  1,
, q.
The question is how to arrange these derivatives. Different arrangements give rise to different
concepts of derivatives in matrix calculus.
Concept 1
The derivative of the p  q matrix Y with respect to the m  n matrix X is the
pq  mn matrix.
  y11

  x11


  y p1
 x
 11
 Y  vec Y 


 X  vec X 
  y1q
 x
 11


  yp q
 x
 11
 y11
 y11
 x m1
 x1n
 y p1
 y p1
 x m1
 x1n
 y1q
 y1q
 x m1
 x1n
 yp q
 yp q
 x m1
 x1n
 y11 

 xmn 


 y p1 
 x m n 
.

 y1q 
 xmn 



 yp q 
 x m n 
Notice that under this concept the mnpq derivatives are arranged in such a way that a row of
 vec Y
, gives the derivatives of a particular element of Y with respect to each element of X and a
 vec X
column gives the derivatives of all the elements of Y with respect to a particular element of X.
Notice also in talking about the derivatives of yi j we have to specify exactly where the ith row is
located in this matrix. Likewise when talking of the derivatives of all the elements of Y with
respect to particular element x r s of X again we have to specify exactly where the s th column is
located in this matrix.
1
This concept of a matrix derivative is strongly advocated by Magnus and Neudecker [see for
example Magnus and Neudecker (1985) and Magnus (2010)]. The feature they like about it is that
 vec Y
is a straight forward matrix generalization of the Jacobian Matrix for y  y  x  where y
 vec X
is a p  1 vector which is a real value differentiable function of a m  1 vector x. This Jacobian
matrix is defined as  y /  x.
Concept 2
The derivative of the p  q matrix Y with respect to the m  n matrix X is the mp  nq matrix
 Y
 x
11
 Y 

X 
 Y
  x m1

Y 
 x1n 



Y 
 x m n 
where  Y /  x r s is the p  q matrix given by
  y11

  xrs
Y 

x r s 
  y p1
x
 rs
for r  1,
, m, s  1,
 y1q 

 xrs 


 y pq 
 x r s 
, n.
This concept of a matrix derivative is discussed, for example, in Dwyer and MacPhail (1948),
Dwyer (1967), Roger (1980) and Graham (1981).
Concept 3
Suppose y is a scalar but a differentiable function of all the elements of a m  n matrix X. Then we
could conceive of the derivative of y with respect to X as the m  n matrix consisting of all the
partial derivatives of y with respect to the elements of X. Denote this m  n matrix as
 y
 x
11
 y 

X 
 y
  x m1

y 
 x1n 

.

y 
 x m n 
2
We could then conceive of the derivative of Y with respect to X as the matrix made up of the
 yi j /  X. Denote this mp  qn matrix  y / X. This leads to the third concept of the derivative of
Y with respect to X.
The derivative of the p  q matrix Y with respect to the m  n matrix X is the mp  nq matrix
  y11

X
 Y 

X 
  y p1
 X

 y1q 

X 
.

 yp q 
 X 
This is the concept of a matrix derivative studied in detail by MacRae (1974) and discussed by
Dwyer (1967), Roger (1980), Graham (1981) and others.
From a theoretical point of view Parring (1992) argues that all three concepts are permissible as
operators depending on which matrix or vector space we are operating in and how this space is
normed.
CASE WHERE Y IS A SCALAR
Suppose Y is a scalar, y say. This case is common in statistics and econometrics. Then concept 2
and concept 3 are the same and concept 1 is the transpose of the vec of either concept. That is for y
a scalar and X a m  n matrix

y
y
y
 y 

and
  vec
.
X
X
X

X


Examples where Y is a scalar
1.
Suppose y is the determinant of a non-singular matrix. That is y  X where X is a nonsingular matrix.
Then
y


 X  vec  X 1   .
X


3
(1)
From Eq.(1) it follows immediately that
y
y

 X  X 1  .
X X
2.
Consider y  Y where Y  XAX is non-singular.
Then


y
 Y A X Y 1  A X Y 1 .
X
It follows from Eq. (1) that



y
 Y  Y 1  A    Y 1  A   vec X
X
 Y  vec X   Y 1  A  Y 1  A  .



3.
 

Consider y  Z where Z  X BX .
Then


y
 Z  vec X   B  Z1   B  Z1   .


X
It follows from Eq.(1) that


y y

 Z Z1X B  Z1X B .
X X
4.
Let y  trAX .
Then
y
 A .
X
It follows from Eq.(1) that
y
  vec A  .
X
5.
Let y  trXAX .
Then
y
  vec  AX  AX   .
X
It follows from Eq.(1) that
y y

 AX  AX.
X X
6.
Let y  trXAXB .
4
Then
y y

 BXA  BXA .
X X
It follows from Eq.(1) that
y
  vec  BXA  BXA   .
X
****
These examples suffice to show that it is a trivial matter moving between the different concepts of
matrix derivatives when Y is a scalar. In the next section we derive transformation principles that
allow us to move freely between the three different concepts of matrix derivatives in more
complicated cases. These principles can be regarded as a generalisation of the work done by Dwyer
and Macphail (1948) and by Graham (1980).
MATHEMATICAL PREREQUISITES
1.
Kronecker Products
Let A = {aij} be a m x n matrix and B be a p x q matrix. The Kronecker product of A and B,
denoted by A  B is the mp x nq matrix given by
 a11B

AB  
 a m1B

a1n B 

.
a m n B 
Let
 a1 


A
   a1
 m 
a  
an .
Then
 a1  B 


AB  
   a1  B
 m

 a   B
Moreover if x is a r x 1 vector then
5
a n  B .
 x   a1 


x  A  



m 
 x  a  
so the ith row of x  A is x   a i for i = 1,…,m.
Similarly
x  B  x  b1
x  bq
so the jth column of x  B is x  b j for j = 1,…,q.
Locating the ith row and the jth column of A  B
The ith row
If i is between 1 and p
a1  bi
If i is between p+1 and 2p
a 2  bi
If i is between (m-1)p and pm
a m  bi .
Write
i  (c  1)p  i
where c is between 1 and m, i is between 1 and p. Then ith row of A  B is
a c  bi .
eg. Let A be 2 x 3, B be 4 x 5 and suppose I want the 7th row of A  B . Write
7   2  1 4  3.
So c = 2, i  3 and
(A  B)7  a 2  b3.
Consider the n x n identity matrix In and write I n   e1n
e nn  . The ith column ein acts as a
selection matrix.
i.e. a c  ecmA , bi  eipB.
So
(A  B)i  (ecm  eip )(A  B) .
The jth column
6
Write
j  (d  1)q  j
with d between 1and n and j between 1 and q.
Then
(A  B) j  a d  b j  (A  B)(edn  eqj ) .
2.
Generalized Vecs and Rvecs
Let A be a m x n matrix and write
 a1 


A
   a1
 m 
a  
an  .
Then
 a1 
 
vecA    , rvecA  a1
 an 
 
Let A be a m x np matrix and write


A   A1
 mxn
a m .
Ap  .
mxn 
Then
 A1 
 
vecn A    .
 Ap 
 
Similarly if B is np x q and write
 B1 
 pxq 
B .
 
n 
B
 pxq 
Then
rvecp B   B1
Bn  .
Relationships
i)
If A is m x np then
(vecn A)  rvecn. A
ii)
A generalized vec can always be undone by taking an appropriate generalized rvec and
vice versa. For example, if A is m x n and vecjA and rveciA exist then
7
rvecm (vec jA)  A
vecn (rveci A)  A.
iii)
Suppose a and b are vectors, b being p x 1. Then
vec p (a   b)  ab
rvec p (a  b)  ba .
3.
Elementary Matrices
The elementary matrix E ijmn is a m x n zero-one matrix whose elements are all zero except in the
(i,j)th position which is 1. i.e.
Eijmn  eimenj  .
Recall for A and B m x n and p x q matrices respectively
(A  B)i  a c  bi .
Hence,
vecq (A  B)i  a c bi  Aecmeip B
(2)
 AEcimp B
Similarly,
rvecp (A  B) j  BE qn
A .
jd

4.
Commutation Matrix K mn
If A is a m x n matrix then Kmn is the mn x mn zero-one matrix defined by
K mn vecA  vecA
Results about Kmn
nm
 E11


nm
 E1m

nm

E n1



E nm
nm 
i)
K mn
ii)
If A is m x n, B is p x q then
8
(3)
 a1  b1 




 m

1
a  b  
   B  A  Kqn
K p m (A  B)  


 a1  b p 




 m

p 
a  b  
and  B  A  Kqn   B  a1
iii)
B  a n .
ith row of Kpm(A  B)
By a similar analysis to that of above.
[K pm (A  B)]i  a i  bc
for i  (c  1)m  i and
vecq [Kpm (A  B)]i  AEimpc B
iv)
(4)
The jth column of  B  A  Kq n
By a similar analysis to that of above
 B  A  K 
qn j
 b j  ad
where j   d  1q  j and
rvecm   B  A  K q n   A E dn qj B.
j
v)
(5)
If X is a m  n matrix then
vec  X  IG    I m  vec m K mG  vecX.
****
5.
The Matrix Um n
U m n is the m2  n 2 matrix given by
Um n
mn
 E11


 E mm1n

mn

E1n

.
mn 
Emn 
Let A, B, C, D be r  m, s  m, n  u and n  v matrices respectively. Then
 A  B Umn  C  D   vecBA rvecCD.
9
(6)
RELATIONSHIPS BETWEEN THE DIFFERENT CONCEPTS
We can use our generalized vec and rvec operators to spell out the relationships that exist between
our three concepts of matrix derivatives. We consider two concepts in turn.
Concept 1 and Concept 2
The submatrices in Y / X are
  y11

  xrs
Y 

 xrs 
  y p1
x
 rs
for r  1,
, m and s  1,
 y1q 

 xrs 


 y pq 
 x r s 
, n. In forming the submatrix Y / x r s we need the partial derivatives of
the elements of Y with respect to x r s . When we turn to concept 1 we note that these partial
derivatives all appear in a column of Y / X. Just as we did in locating a column of a Kronecker
product we have to specify exactly where this column is located in the matrix Y / X. If s is 1 then
the partial derivatives appear in the rth column, if s is 2 then they appear in the m  r th column, if
s is 3 in the 2m  r th column and so on until s is n in which case the partial derivatives appear in
the  n  1 m  r th column. To cater for all these possibilities we say x r s appears in the th
column of  Y /  X where
  s 1 m  r
and s  1,
, n. The partial derivatives we seek appear in that column as the column vector
  y11

  xrs


  y p1
 x
rs



  y1q
 x
rs



  ypq
 x
rs

10








.









If we take the rvecp of this vector we get Y / x r s so
Y
Y / x rs  rvecp 

  X 
where
  s 1 m  r, for s  1,
, n and r  1,
(7)
, m.
Now this generalized rvec can be undone by taking the vec so
 Y 
Y


  vec 
  X 
  xrs 
(8)
If we are given  Y /  X and we can identify the th column of this matrix then Eq.(7) allows us to
move from concept 1 to concept 2. If, however, we have in hand Y / X we can identify the
submatrix Y / x r s and Eq.(8) will then allow us to move from concept 2 to concept 1.
Concept 1 and Concept 3
The submatrices in Y / X are
 yi j

x11
yi j 

X 
 yi j
 x
 m1
for i  1,
, p and j  1,
yi j 

x1n 


yi j 
x m n 
, q. In forming the submatrix yi j / X we need the partial derivative of
yi j with respect to the elements of X. When we examine Y / X we see that these derivatives
appear in a row of Y / X.
Again we have to specify exactly where this row is located in the matrix Y / X. If j is 1 then the
partial derivatives appear in the ith row, if j  2 then they appear in the p  i th row, if j  3 then
in the 2p  i th row and so on until j  q in which case the partial derivative appear in  q 1 p  i th
row. To cater for all possibilities we say the partial derivatives appear in the t th row of Y / X
where
t   j 1 p  i
and j  1,
, q. In this row they appear as the row vector
 yi j

 x11
yi j
yi j
x m1
x1n
11
yi j 
.
x m n 
If we take the vec m of this vector we obtain the matrix
 yi j

 x11


 yi j
 x
 1n
yi j 

x m1 


yi j 
x m n 
which is  yi j / X  . So we have


 Y  
  vec m 
 
X 
 X  t  
yi j
where t   j 1 p  i, for j  1,
, q and i  1,
(9)
, p.
As
 yi j 
 Y 
vec m 




 X  t   X 
and this generalized vec can be undone by taking the rvec we have
 yi j 
 Y 
.

  rvec 
 X  t 
 X 
(10)
If we have in hand Y / X and if we can identify the t th row of this matrix the Eq.(9) allows us to
move from concept 1 to concept 3. If, however, we have obtained Y / X so we can identify the
submatrix yi j / X of this matrix then Eq.(10) allows us to move from concept 3 to concept 1.
Concept 2 and Concept 3
Returning to concept 3, the submatrices of Y / X are
 yi j

x11
yi j 

X 
 yi j
 x
 m1
and the partial derivative
yi j
x r s
yi j 

x1n 


yi j 
x m n 
is given by the  r,s  th element of this submatrix. That is
yi j
 yi j 

 .
x r s  X r s
12
It follows that
  y11 


  X  r s
Y 

x r s 
  y p1 

  X  r s

 y1q  

 
 X  r s 
.

 y p q  

 
 X  r s 
(11)
Starting now with concept 2, the submatrices of Y / X are
 y11

 x r s
Y 

x r s 
 y p1
 x
 rs
y1q 

x r s 


y pq 
x r s 
and the partial derivative yi j / x r s is the  i, j th element of this submatrix. That is
 Y 

 .
x r s  x r s 
ij
yi j
It follows that
  Y 


  x11 i j
yi j 

X 
  Y 
  x m1 
i j

 
 
i j 

.

 Y  

 
 x m n i j 
 Y

 x1n
(12)
If we have in hand y / X then Eq.(11) allows us to build up the submatrices we need for Y / X.
If however, we have a result for Y / X then Eq.(12) allows us to obtain the submatrices we need
for Y / X.
Tranformation Principles One
Several matrix calculus results when we use concept 1 involve Kronecker products whereas the
equivalent results, using concepts 2 and 3 involve elementary matrices. In this section we see that
this is no coincidence.
We have just seen that
13
where
Y
 Y 
 rvec p 

x r s
 X 
(13)


 Y  
  vec m 
 
X 
 X  t  
(14)
  s 1 m  r and that
yi j
where t   j  1 p  i. Suppose now that Y / X  A  B where A is a q  n matrix and B is a
p  m matrix.
Then from Eq.(3) we have

rvecp  A  B   BEmn
rs A ,
so using Eq.(13) we have that
Y
 BE mr s n A.
x r s
From Eq.(2) we have
vecm  A  B t   A Eqjip B
so from Eq.(14)
yi j
X
  A Eqjip B  B E pq
ji A.
This leads us to our first transformation principle.
The First Transformation Principle
Let A be a q  n matrix and B be a p  m matrix. Whenever
Y
 AB
X
regardless of whether A and B are matrix functions of X or not
Y
 BE mr s n A
x r s
and
yi j
X
 B Eipqj A
and the converse statements are true also.
****
14
For this case
 B E11m n A
Y 

X 
mn
 B E m1 A
mn
B E1n
A 

   I m  B  U m n  I n  A  ,
mn
B E m n A 
where U m n is the m2  n 2 matrix, given by
Um n
 E11m n


 E mm1n

mn

E1n

.
mn 
Em n 
From Eq.(6)
 A  B Umn  C  D   vec BA rvecCD ,
so
Y
  vec B  rvec A  .
X
In terms of concept 3 for this case
 B E11p q A
Y 

X 
pq
 B E p1 A
pq
B E1q
A

   I p  B  U p q  Iq  A    vec B  rvec A  .
pq
B E p q A 
In terms of the entire matrices we can express the First Transformation Principle by saying that the
following statements are equivalent:
Y
 AB
X
Y
  vec B  rvec A 
X
Y
  vec B  rvec A  .
X
Examples of the Use of the First Transformation Principle
1.
Y  A  B for A p  m and B n  q.
Then it is know that
AXB
 B  A.
X
It follows that
AXB
 A E mr s n B
x r s
15
and
  AXB i j
X
 A E ipjq B
Moreover
AXB
  vec A  rvec B 
X
AXB
  vec A  rvec B  .
X
2.
Y  XAX where X is a n  n symmetric matrix.
Then it is know that
XAX
 E nr sn AX  XAE nr sn .
x r s
It follows that
  XAX i j
X
 E injn XA  AXE injn
and that
XAX
  XA  I n    I n  AX  .
X
Moreover
XAX
  vec I n  rvec AX    vec AX  rvec I n 
X
Y
  vec I n  rvec XA    vec XA  rvec I n  .
X
3.
Y  X  IG where X is a m  n matrix.
We have seen that vec  X  IG    I n  vec m K mG  vec X so
  X  IG 
X
 In  vecm K mG .
It follows that
  X  IG 
x rs
  vecm K G m  E rsmn
and
  X  IG 
  vecm K G m  Eikjn where k  G 2 n.
X
16
Moreover
  X  IG 
 vec  vec m K m G   rvec I n    vec I m G   rvec I n 
X
  X  IG 
 vec  vec m K m G   rvec I n    vec I m G   rvec I n  .
X
4.
Y  AX1B where A is p  n and B is n  q. Then it is known that
  AX 1B 
ij
X
 X 1A E ipqj BX 1.
It follows straight away that
AX 1B
 AX1E nrsn X 1B,
x rs
and that
AX 1B
 BX 1  AX 1.
X
Moreover
AX 1B
   vec AX 1  rvec X 1B 
X
and



AX1B
  vec X1A rvec BX 1 .
X
5.
Y  AXBXC where X is m  n, A is p  m, B is n  m and C is n  q.
Then it is well known that
AXBXC
 A E mrs n BXC  AXBE rsm n C.
x r s
It follows that
  AXBXC i j
X
 A E ipjq CXB  BXA E ipjq C
and
AXBXC
  CXB  A    C  AXB  .
X
17
Moreover
AXBXC
  vec A  rvec BXC    vec AXB  rvec C  .
X
and
AXBXC
  vec A  rvec CXB    vec BXA  rvec C  .
X
As I hope these examples make clear this transformation principle ensure is a very easy matter to
move from a result involving one of the concepts of matrix derivatives to the corresponding results
for the other two concepts. Although this principle covers a lot of cases, it does not cover them all.
Several matrix calculus results for concept 1 involve multiplying a Kronecker product by a
commutation matrix. The following transformation principal covers this case.
Transformation Principle Two
Suppose then that
Y
 Kq p C  D   D  C Kmn
X
where C is a p  n matrix and D is a q  m matrix. Forming Y / x r s from this matrix requires that
we first obtain the th column of this matrix where
  s 1 m  r and we take the rvecp of this
column. From Eq.(5) we get
Y
 C E snrm D
x r s
In forming yi j / X from Y / X we first have to obtain the t th row of this matrix, for
t   j 1 p  i and then we take the vec m of this row. The required matrix yi j / X is the
transpose of the matrix thus obtained. From Eq.(4) we get
yi j
X
  C Eipqj D   D Eqjip C.
This leads us to our second transformation principle.
The Second Transformation Principle
Let C be a p  n matrix and D be a q  m matrix. Whenever
Y
 Kq p C  D
X
18
regardless of whether C and D are matrix functions of X or not
Y
 C Esnrm D
x r s
yi j
X
 D E qjip C
and the converse statements are true also.
****
For this case
 C E11n m D
Y 

X 
nm
 C E1m D
nm
 E11n m
C E n1
D 


   Im  C  
nm
 E1m
C E nn mm D 

nm

E n1

  I n  D    I m  C  K m n  I n  D  .
E nn mm 
In terms of Y / X we have
 DE11q p C
Y 

X 
qp
 DE1p C
DE qq1p C 

   I p  D  K p q  Iq  C  .
DE qq pp C 
In terms of the full matrices we can express the Second Transformation Principle as saying that the
following statements are equivalent:
Y
 Kqp C  D
X
Y
  I m  C  K m n  I n  D 
X
Y
  I p  D  K p q  Iq  C  .
X
As an example of the use of this second transformation principle let Y  AXB where A is p  n
and B is m  q. Then it is known that
AXB
 K p q  B  A  .
X
It follows that
AXB
 BE smr n A
x r s
19
and that
  AXB i j
X
 AE pjiq B.
In terms of the entire matrices we
Y
  I n  B  K n m  I m  A 
X
Y
  I q  A  K q p  I p  B  .
X
****
Principle 2 comes into its own when it is used in conjunction with principle 1. Many matrix
derivatives come in two parts: one where principle 1 is applicable and the other where principle 2 is
applicable.
For example we often have
Y
 A  B  Kq p C  D ,
X
so we would apply principle 1 to the A  B part and principle 2 to the Kq p  C  D part.
Examples of the Combined Use of Principles One and Two
1.
Let Y  XAX where X is m  n, A is m m. Then it is well known that
XAX
 K n n  In  XA    In  XA  .
X
It follows that
XAX
 Esnrm AX  XA E mr s n
x r s
and that
  XAX i j
X
 A X E nj in  A X E injn .
Moreover
XAX
 K m n  I n  AX    I m  XA  U m n  K m n  I n  AX    vec XA  rvec I n  .
X
XAX
  I n  AX  K n n   I n  A X  U n n   I n AX  K n n   vec A X  rvec I n  .
X
20
2.
Let Y  XAX where X is m  n and A is n  n. Then it is known that
XAX
 XAEsnrm  E rms n AX.
x r s
It follows that
  XAX i j
X
 E mj i m XA  Eimj m XA
and
XAX
 K m m  XA  I m    XA  I m  .
X
Moreover
XAX
  I m  XA  K m n  U m n  I n  AX    I m  XA  K m n   vec I m  rvec AX  .
X
and
XAX
 K m m  I m  XA   U m m  I m  XA   K m m  I m  XA    vec I m  rvec AX  .
X
3.
Let Y  AXBXC where A is p  n, B is m  m and C is n  q. Then it is known that
  AXBXC i j
X
 BXCE qjip A  BXA E ipjq C.
It follows using our principles that
AXBXC
 CEsnrm BXC  AXBE mn
rs C
x r s
and that
AXBXC
 K q p  A  CXB    C  AXB  .
X
In terms of the entire matrices we have
AXBXC
  I m  A  K m n  I n BXC    I m  AXB  U m n  I n  C 
X
  I m  A  K m n  I n  BXC    vec AXB  rvec C  .
AXBXC
  I p  BXC  K p q  Iq  A    I p  BXA  U p q  Iq  C 
X
  I p  BXC  K p q  Iq  A    vec BXA  rvec C  .
21
4.
Let Y  AXBXC where A is p  m, B is n  n and C is m  q. Then it is well known that
AXBXC
 K q p  AXB  C    CXB  A  .
X
Using our principles we obtain
AXBXC

 AXBEsnrm C  AE mn
r s BX C
x r s
and
  AXBXC i j
X
 CE qjip AXB  A E ipjq CXB.
Moreover, we have
AXBXC
  I m  AXB  K m n  I n  D    I m  A  U m n  I n  BXC 
X
  I m  AXB  K m n  I n  D    vec A  rvec BXC  .
AXBXC
  I p  C  K p q  Iq  BXA    I p  A  U p q  I q  CXB 
X
  I m  AXB  K m n  I n  D    vec A  rvec CXB  .
The following results are not as well known:
5.
Let Y  DD where D  A  BXC with A p  q, B p  m and C n  q.
Then from Lutkepohl (1996) p.191 we have
DD
 K q q  C  DB   C  DB.
X
Using our principles we obtain
DD
 CEsnrm BD  BDE mn
rs C
x r s
and
  DD i j
X
 BDE qj iq C  BDE iqjq C.
In terms of the complete matrices we have
22
DD
  I m  C  K m n  I n  BD    I m  DB  U m n  I n  C 
X
  I m  C  K m n  I n  BD    vec DB  rvec C  .
DD
  Iq  BD  K q q  Iq  C    I q  BD  U q q  I q  C 
X
  Iq  BD  K q q  Iq  C    vec BD  rvec C  .
6.
Let Y  DD where D is as in 5.
Then from Lutkepohl (1996) p.191 again we have
DD
 K p p  DC  B    DC  B  .
X
It follows that
DD
 DCE snrm B  BE rms n CD
x r s
  DD i j
X
 BE pjip DC  BE ipjp DC
or in terms of complete matrices
DD
  I m  DC  K m n  I n  B    I m  B  U m n  I n  CD 
X
  I m  DC  K m n  I n  B    vec B  rvec CD 
DD
  I p  B  K p p  I p  DC    I p  B  U p p  I p  DC 
X
  I p  B  K p p  I p  DC    vec B  rvec DC  .
****
23
ECONOMICS DISCUSSION PAPERS
2009
DP
NUMBER
AUTHORS
TITLE
09.01
Le, A.T.
ENTRY INTO UNIVERSITY: ARE THE CHILDREN OF
IMMIGRANTS DISADVANTAGED?
09.02
Wu, Y.
CHINA’S CAPITAL STOCK SERIES BY REGION AND SECTOR
09.03
Chen, M.H.
UNDERSTANDING WORLD COMMODITY PRICES RETURNS,
VOLATILITY AND DIVERSIFACATION
09.04
Velagic, R.
UWA DISCUSSION PAPERS IN ECONOMICS: THE FIRST 650
09.05
McLure, M.
ROYALTIES FOR REGIONS: ACCOUNTABILITY AND
SUSTAINABILITY
09.06
Chen, A. and Groenewold, N.
REDUCING REGIONAL DISPARITIES IN CHINA: AN
EVALUATION OF ALTERNATIVE POLICIES
09.07
Groenewold, N. and Hagger, A.
THE REGIONAL ECONOMIC EFFECTS OF IMMIGRATION:
SIMULATION RESULTS FROM A SMALL CGE MODEL.
09.08
Clements, K. and Chen, D.
AFFLUENCE AND FOOD: SIMPLE WAY TO INFER INCOMES
09.09
Clements, K. and Maesepp, M.
A SELF-REFLECTIVE INVERSE DEMAND SYSTEM
09.10
Jones, C.
MEASURING WESTERN AUSTRALIAN HOUSE PRICES:
METHODS AND IMPLICATIONS
09.11
Siddique, M.A.B.
WESTERN AUSTRALIA-JAPAN MINING CO-OPERATION: AN
HISTORICAL OVERVIEW
09.12
Weber, E.J.
PRE-INDUSTRIAL BIMETALLISM: THE INDEX COIN
HYPTHESIS
09.13
McLure, M.
PARETO AND PIGOU ON OPHELIMITY, UTILITY AND
WELFARE: IMPLICATIONS FOR PUBLIC FINANCE
09.14
Weber, E.J.
WILFRED EDWARD GRAHAM SALTER: THE MERITS OF A
CLASSICAL ECONOMIC EDUCATION
09.15
Tyers, R. and Huang, L.
COMBATING CHINA’S EXPORT CONTRACTION: FISCAL
EXPANSION OR ACCELERATED INDUSTRIAL REFORM
09.16
Zweifel, P., Plaff, D. and
Kühn, J.
IS REGULATING THE SOLVENCY OF BANKS COUNTERPRODUCTIVE?
09.17
Clements, K.
THE PHD CONFERENCE REACHES ADULTHOOD
09.18
McLure, M.
THIRTY YEARS OF ECONOMICS: UWA AND THE WA
BRANCH OF THE ECONOMIC SOCIETY FROM 1963 TO 1992
09.19
Harris, R.G. and Robertson, P.
TRADE, WAGES AND SKILL ACCUMULATION IN THE
EMERGING GIANTS
09.20
Peng, J., Cui, J., Qin, F. and
Groenewold, N.
STOCK PRICES AND THE MACRO ECONOMY IN CHINA
09.21
Chen, A. and Groenewold, N.
REGIONAL EQUALITY AND NATIONAL DEVELOPMENT IN
CHINA: IS THERE A TRADE-OFF?
24
ECONOMICS DISCUSSION PAPERS
2010
DP
NUMBER
AUTHORS
TITLE
10.01
Hendry, D.F.
RESEARCH AND THE ACADEMIC: A TALE OF
TWO CULTURES
10.02
McLure, M., Turkington, D. and Weber, E.J.
A CONVERSATION WITH ARNOLD ZELLNER
10.03
Butler, D.J., Burbank, V.K. and
Chisholm, J.S.
THE FRAMES BEHIND THE GAMES: PLAYER’S
PERCEPTIONS OF PRISONER’S DILEMMA,
CHICKEN, DICTATOR, AND ULTIMATUM GAMES
10.04
Harris, R.G., Robertson, P.E. and Xu, J.Y.
THE INTERNATIONAL EFFECTS OF CHINA’S
GROWTH, TRADE AND EDUCATION BOOMS
10.05
Clements, K.W., Mongey, S. and Si, J.
THE DYNAMICS OF NEW RESOURCE PROJECTS
A PROGRESS REPORT
10.06
Costello, G., Fraser, P. and Groenewold, N.
HOUSE PRICES, NON-FUNDAMENTAL
COMPONENTS AND INTERSTATE SPILLOVERS:
THE AUSTRALIAN EXPERIENCE
10.07
Clements, K.
REPORT OF THE 2009 PHD CONFERENCE IN
ECONOMICS AND BUSINESS
10.08
Robertson, P.E.
INVESTMENT LED GROWTH IN INDIA: HINDU
FACT OR MYTHOLOGY?
10.09
Fu, D., Wu, Y. and Tang, Y.
THE EFFECTS OF OWNERSHIP STRUCTURE AND
INDUSTRY CHARACTERISTICS ON EXPORT
PERFORMANCE
10.10
Wu, Y.
INNOVATION AND ECONOMIC GROWTH IN
CHINA
10.11
Stephens, B.J.
THE DETERMINANTS OF LABOUR FORCE
STATUS AMONG INDIGENOUS AUSTRALIANS
10.12
Davies, M.
FINANCING THE BURRA BURRA MINES, SOUTH
AUSTRALIA: LIQUIDITY PROBLEMS AND
RESOLUTIONS
10.13
Tyers, R. and Zhang, Y.
APPRECIATING THE RENMINBI
10.14
Clements, K.W., Lan, Y. and Seah, S.P.
THE BIG MAC INDEX TWO DECADES ON
AN EVALUATION OF BURGERNOMICS
10.15
Robertson, P.E. and Xu, J.Y.
IN CHINA’S WAKE:
HAS ASIA GAINED FROM CHINA’S GROWTH?
10.16
Clements, K.W. and Izan, H.Y.
THE PAY PARITY MATRIX: A TOOL FOR
ANALYSING THE STRUCTURE OF PAY
10.17
Gao, G.
WORLD FOOD DEMAND
10.18
Wu, Y.
INDIGENOUS INNOVATION IN CHINA:
IMPLICATIONS FOR SUSTAINABLE GROWTH
10.19
Robertson, P.E.
DECIPHERING THE HINDU GROWTH EPIC
10.20
Stevens, G.
RESERVE BANK OF AUSTRALIA-THE ROLE OF
FINANCE
10.21
Widmer, P.K., Zweifel, P. and Farsi, M.
ACCOUNTING FOR HETEROGENEITY IN THE
MEASUREMENT OF HOSPITAL PERFORMANCE
25
10.22
McLure, M.
ASSESSMENTS OF A. C. PIGOU’S FELLOWSHIP
THESES
10.23
Poon, A.R.
THE ECONOMICS OF NONLINEAR PRICING:
EVIDENCE FROM AIRFARES AND GROCERY
PRICES
10.24
Halperin, D.
FORECASTING METALS RETURNS: A BAYESIAN
DECISION THEORETIC APPROACH
10.25
Clements, K.W. and Si. J.
THE INVESTMENT PROJECT PIPELINE: COST
ESCALATION, LEAD-TIME, SUCCESS, FAILURE
AND SPEED
10.26
Chen, A., Groenewold, N. and Hagger, A.J.
THE REGIONAL ECONOMIC EFFECTS OF A
REDUCTION IN CARBON EMISSIONS
10.27
Siddique, A., Selvanathan, E.A. and
Selvanathan, S.
REMITTANCES AND ECONOMIC GROWTH:
EMPIRICAL EVIDENCE FROM BANGLADESH,
INDIA AND SRI LANKA
26
ECONOMICS DISCUSSION PAPERS
2011
DP
NUMBER
AUTHORS
TITLE
11.01
Robertson, P.E.
DEEP IMPACT: CHINA AND THE WORLD
ECONOMY
11.02
Kang, C. and Lee, S.H.
BEING KNOWLEDGEABLE OR SOCIABLE?
DIFFERENCES IN RELATIVE IMPORTANCE OF
COGNITIVE AND NON-COGNITIVE SKILLS
11.03
Turkington, D.
DIFFERENT CONCEPTS OF MATRIX CALCULUS
11.04
Golley, J. and Tyers, R.
CONTRASTING GIANTS: DEMOGRAPHIC CHANGE
AND ECONOMIC PERFORMANCE IN CHINA AND
INDIA
11.05
Collins, J., Baer, B. and Weber, E.J.
ECONOMIC GROWTH AND EVOLUTION:
PARENTAL PREFERENCE FOR QUALITY AND
QUANTITY OF OFFSPRING
11.06
Turkington, D.
ON THE DIFFERENTIATION OF THE LOG
LIKELIHOOD FUNCTION USING MATRIX
CALCULUS
11.07
Groenewold, N. and Paterson, J.E.H.
STOCK PRICES AND EXCHANGE RATES IN
AUSTRALIA: ARE COMMODITY PRICES THE
MISSING LINK?
11.08
Chen, A. and Groenewold, N.
REDUCING REGIONAL DISPARITIES IN CHINA: IS
INVESTMENT ALLOCATION POLICY EFFECTIVE?
11.09
Williams, A., Birch, E. and Hancock , P.
THE IMPACT OF ON-LINE LECTURE RECORDINGS
ON STUDENT PERFORMANCE
11.10
Pawley, J. and Weber, E.J.
INVESTMENT AND TECHNICAL PROGRESS IN THE
G7 COUNTRIES AND AUSTRALIA
27