Mann, Jitendar S.

English

Research Papers in Economics

Staff Papers Series
P 69-1 January 1969
TESTING HYPOTHESIS ON THE RELATIVE
SIZE OF THE COEFFICIENTS IN
REGRESSION MODELS
J. S. Mann
Department of Agricultural Economics
University of Minnesota
Institute of Agriculture
St. Paul, Minnesota 55101P 69-1  January 1969
TESTING HYPOTHESIS ON THE  RELATIVE SIZE  OF  THE
COEFFICIENTS  IN  REGRESSION  MODELS
J.  S, MannTesting Hypothesis  on the Relative Size  of the
Coefficients  in  Regression Models
J. S. Mann
In the traditional multiple regression  analysis a hypothesis
about a set  of regression coefficients  is tested.  This hypothesis
is  generally stated  in terms  of  one or more  coefficients being equal
to  zero  or  some other given value.  (See, for example, Johnston 5,
Section 4.3 7,  Goldberger L,  Section 7, Chapter 47  and Mood  and
Graybill /,  Theorem 13.6_7).  However,  often the  researcher is faced
with the problem of testing a hypothesis about the relative  size  of
various  coefficients;  for example, two coefficients  are  equal.  This
can be handled with the help  of a matrix of  restrictions  (r in  the
following discussion).  Johnston  3, pp.  131-133_7 has developed  a
test  for a simple  case where r has rank  1.  In  this paper a readily
usable technique  for the  general  case  is  presented.  First,  the tra-
ditional regression model  is  set up.  Second, the general method  for
testing hypothesis  on  the relative size  of the coefficients  is
developed.  Finally,  the method  is applied to a few  leading cases.
The  Regression Model
Let  y be a linear  function of k independent  variables x1, x2 ---
Xk  and a disturbance term u.  If  a sample of T observations on  y and
x's  is  taken, the relationship  can be written as-2-
y  =  B  +  a  x  +  6  x  +  ---  +  B  x  +  u
t  o  1  It  2  2t  k  kt  t
for  t  =  1,  2,  -- ,  T
These  T equations  can  be  set
Y  =
where
=
X  =
a =
and
out  in  matrix  notation  as
X  8+  u
Y1 y1
y2
yt
1x  x  ---  x
11  21  1 kl
1  X  X  ---  x
12  22  k2
1 x  x
1T  2T
u
1
u
2
L  T
---  XkT
r-  - I
.-3-
If we  make the  following assumptions:
2
(1)  u is  normally distributed with Eu =  0 and  Euu' = a  I,
(2)  X has  rank k+l
then,  the maximum  likelihood  estimator  of  B is[ 2,  p.111]
-1
=  (X'  X)  X'  y
General Hypothesis
The  general hypothesis  can be  stated  as
H  :  r  6= R
o
where r is  a G x k + 1  matrix  of G independent restrictions  on  6  (that
is,  r has rank  G).  For example, consider a regression problem with three
independent  variables
Yt  + B1  Xlt  + 2  x2t + 83 x3t + ut
We  want  to  test  82  =  3.  The  null hypothesis  can be  stated  as
H  :  =
o  2  3
or  8  - 8  =
2  J
or  (O O 1 -1)
r80
a  1
6
2
_ 3
=  0
Here r = (O 0 1 -1) and  R = O.  (In this case G = 1.)
As another example,  in  a three  independent variables  case,  let
the  null hypothesis be  8  = 0 and  =  8  . In  other words,
0  '  2  3-4-
H  :  6  =0
o  0
8  =0
82-  =0
0
0
Here 1  0
r  =
0  0
0
R  =
0
0
1
O 07
-1I
0
1
and G =  2.
The  general hypothesis
If  the estimated value  of
obtain
Subtracting  (1)  from  (2)
r
a
,A =  R can  also be written as
6  =  (r$  - R)  =  0
that  is  0  ,  is  substituted,  we
6  =  (r  - R)
6-  6  =r (  - )
If  the  null  hypothesis  is  true,  that  is  6  =  0,
6  =  r  (  a  - 6  ).
6  is  an  indication  of  how much the estimated value  r8  deviates
or
and
and
_
or 1
Lo
80
1
2
83
(1)
(2)
0
0-5-
from  the  true  value  r  B.
Now  we  proceed  to  derive  a  test  statistic  for  6  . It  is  known
that  B  - B  has  normal  distribution  with  mean  0  and  variance
a2  (X'  X) 1 A,  Theorem 13.3].  Since  a linear  function  of  a
normally distributed random variable has  a normal distribution,
6  =  r  (  i  - I)  is  normally distributed with
E  6 = r E  (  6  - B )  =  0, and
A  A
E  6  d  '  =  r  E  (S-B)  (S-  )'  r
2-  1
=  a  r  (X'  X)  r'
Using  the  fact  that  if  a.  be  independent  normally  distributed
2
variables  (i  =  1, 2, --, k) with mean =  0 and  variance  =  a , then
2
£  ai
z 
= 1
a2
2
has a X  distribution with k degrees of  freedom,  in  the present case
,  -1  . -1
6  [r  (xx)  r'  6
2
has  2  distribution  with  G degrees  of  freedom.
A~~~~  2
Further,  '  u  is  also  distributed  as  X  with  T-k  -1
a2
A  A  A
degrees  of  freedom  and  u'  u  is  independent  of  6  Z,  Theorem  13.3_.
The  ratio  of  two  independent  Chi-square  variables  each  divided  by  its
degrees  of  freedom  has  the  Snedecor  F  distribution.  Therefore,  the
ratio  1  1  6  [  (X'  X)1  r'  6
ratio  - -
a2  G
1  1  u'  u
a2  T-k-l-6-
^  -1  -1
(T-k-l)lr  (8-8)1'  [r  (X'X)  r]  [r  (  - )
G  u'  u
has  F  distribution.
G, T-k-l
Applications
The discussion  in this section  is  limited to  a case  of three  in-
dependent variables:
Y  t  1  X  +  2 xt  +  t  3 X3t,
#  _
where AO
8
Al  =  (X'  X)- 1 X'y.
2
3
Let the  i  jth element  of  (X'  X) 1 be  designated as  c.
ij
Case  1.  H :  8  =
o  2  3
An  economic  example would be  the coefficients to two related  goods
in  the demand function for  a commodity.  It  may be  conjectured that
those two goods  have  identical  influence on the demand  for the given
commodity and  may be  combined  into one  good.  In this  case the hypothesis
of  equality of two coefficients  is tested.
The  null hypothesis  can  also be  stated as
H  :8  - =0,
o  2  3-7-
_--I
or
(O  O  1  -1)
Thus, r =  (0  0 1 -1) and R =  0.
Then
r  (X')  (  o 1 r'  =  1 -1)
=  (c20-c30  C21-c31  C22-C32
B
3
a
0
1
2
3
Hence
J
00
c10
c20
30
6  =
11
c21
C31
C23-C33)
=  c  -c  - c  +c
22  32  23  33
The test  statistic  is
0
A  A
(  B2  3).
C02  C03
C12  C13
c22  C23
C32  c33
0
1
-1
F  = (T-4)
1,T-4
-~~~~~~~~~~~~~~~~~~~ 82  - .3  (22
u
U'
- 32-  23  +  33]
Case  2:  We may want to test the  hypothesis
are  in a given ratio.  For example,
H  :  =k  0
o  2  3
or  8  - k 8  =0
2  3
or  (O  0 1  -k)
Here, r = (O 0 1 - k) and R = 0.
that two  coefficients
BO
8
1
32
83
_I-  _
=  0
0
0
1
-1
A-  A
- 3 (2  3g
I.-..
I
IThen
r  (X'  X)1  r'  =  (0  0  1  - k)
=  (c  - kc
20  30
c  - kc
21  31
c22  - kc32  c  - kc  )
22  32  23  33
2
=  (c  - kc  - kc  + k  )
22  32  23  33
and
F  T  =  (T-4)  (  2  - k  3)
U'  U
u'  u
[k  2 7  -1  3
[c22-kc32-kc23+  k  c33]  (  2-k  3 )
is  the test  statistic.
Case 3:  Consider a Cobb - Douglas production  function
1  2  x3  u Yt  = t  X2t  3t  t
Taking  log of both  sides
log Yt =1  g  + 8  log  x  +  log  x  +  l  +log  u t  0  1  it  2  2t  3  3t  t
where  3  =  log(
We  want  to  test
H  :  8  +
o  1  2
or
the  hypothesis  of  constant  returns  to  scale,  that  is
+83  =  1
0 0
(0111)  0  =1
83
A"
-8-
C
00
C10
c 2 0
C30
__
C
01
C11
C2 1
C31
c0 2
C12
c22
C32
C
03
13
C
23
C 3 3
0
0
1
-k
0
0
1
-k-9-
Here, r =  (0 1 1 1) and R =  1
Then
-1
r (X'X) -r'  =  (0 1 1 1)
00  01  02  03
C1 0 Cll  C12  C13
C20  21  22  23
30  31  32  33
0
( 10 +  20 +  30  11 +  21 c31 C12 + c22 +  32 13 +  2 +c  33)
1
=  +c  c  +c  +  +c  +  +c  +c  +c  )
11  21  31  12  22  32  13  23  33
(Here  X'X  is  the  matrix  of  sums  of  squares  and  cross  products  of  the  logs
of  x  variables).
The  test  statistic  is
F  =
, T-4
"-1
(T-4)  (  +  2+-l)[c  +c+c+c 2 c  +c  +c  + c 1 +c2 3+c 3 (]1)
1  3  21  31  12  22  32  13  23  33  1  2+3
u'  u
Case  4:  In  order  to  demonstrate  the  method  for  more  than  one  restriction,
consider  a  demand  curve  l
y  a  =  x  8 1 x  u
t  It  2t  3t  t
where,
Y  =  per  capita quantity demanded,
x  =  price  of the  commodity,
it
x  =  price of the  competing good,  and
2t
x 3t  =  per  capita  income.
0
1
1
1
I I
I  r  11 ----  II  I-10-
The  function can  be  fitted by taking  log  of  both  sides and  following
the usual procedure  of  solving the normal equations,  X'X being  in
terms  of  logs  of the x variables.
We may want to test that the  commodity has unit price  elasticity
of demand  and  that the  demand function satisfies the homogeneity con-
dition, that  is  the  sum of own  price elasticity, cross elasticity and
income elasticity  is  zero.
H :  S  =  -1 and
o  1
1  +2  +  3 
0
8
or  0  1  -1
011  12
3
where  8  =  log  a
0
Here  0 1  0  o
r=
Lo  1 1  1
R =  -1
0 - ~
and  G  =  2.-11-
Now,
-1
r  (X'X)  r  =  0100
01  11 iL
CO  C01  C02  C03
c10  11  12  13
C20  C21  C22  C23
30  31  32  33
c
11
c  +c  +c
11  21  31
c
012
c  +  c  +c
12  22  32
00
c  1  1
13  0  1
c  +c  +c  0  1
13  23  33
c
11
=c  +c  +c
11  21  31
c  +  c  +  c
11  12  13
c+  +c  +c  +  c  +C  +  +c  +  +
11  21  31  12  22  32  13  23  3
And the test statistic  is
T-4 =  (T-4)  [6 + 1 l  1+  32+83  ]
2,'?-4
c  c  +  +  c
11  11  12  13
c+  +C  C  +C  +C  +C  +c  +C  +c  +c  +c
11  21  31  11  21  31  12  22  32  13  23  3
-1  Bl+l
1  2  a
A  A 2  u'  U
Finally, it  can be  shown  that the traditional test on  a set  of
coefficients  is  a special  case  of  the generalized  hypothesis.  For
example,
H  2 o  2
00
1  1
0  1
0  1
c
10is  equivalent  to
(0  O  1  0)
t0
2
3 3
that  is,  r  =  (0  0 1  0).
Now
-1
r  (X')  r'  =  (O O  1  0)
And
F  =  (T-4)  2  - 1
1,T-4  2  22  2
U'  U
is  the test  statistic
This may also be
^2
F  =  B
1,T-4  2
A  A
u'  U
c  ...-
22  --T-4
written  as
a
2
c22u  u  /  T-4
_\~~~
-12-
=  0
COO
c C
10
20
C30
C01
C c
11
c2 1
C 3 1
C0 2 03
C  C
12  13
C  C
22  23
C32  33
O 0
0
1
0
=22
2
2
=  t
T-4-13-
Literature  Cited
/_7  Goldberger, Authur S.  Econometric Theory, New York,
John Wiley & Sons,  1964.
ZJ7  Graybill, Franklin A.  An  Introduction to Linear Statistical
Models, Vol  I, New York, McGraw-Hill Book Co.,  1961.
3_]7  Johnston, J.  Econometric Methods, New York,  McGraw-Hill
Book Co.,  1963.
/4_7  Mood, Alexander M. and  Franklin A. Graybill, Introduction
to the Theory of Statistics, New York, McGraw-Hill Book
Co.,  1963.

TESTING HYPOTHESIS ON THE RELATIVE SIZE OF THE COEFFICIENTS IN REGRESSION MODELS

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