Linear Regression – General
Linear Tests
Cobb-Douglas Production Function
(Multiplicative Form)
Source: C.W. Cobb and P.H. Douglas (1928). “A Theory of Production”, American
Economic Review Vol. 18 (Supplement) pp. 139-165.
Linear Regression Model with Constraints
Y  X 
subject to K T   m
K
T
  m  0
where: Y  n  1 X  n  p '   p ' 1 K T  k  p '
minimize L*  Y  X  * Y  X  *  2 T  K T  *  m 
T
 Y T Y  2Y T X  *   *T X T X  * 2 T K T  * 2 T m
with respect to  *,  where   k 1:
set
L
T
T
 2 X Y  2 X X  * 2 K   0 
 *
set
^
L
T
T
 2  K  * m   0  K  *  m

^
T

 X X K   *  X T Y 
 T
   

0      m 
 K
^
X X  * K   X TY
T
Solving for Constrained Regression Coefficients
A
Note (Assuming A is full rank): A   11
 A21
A12 
A22 
 A111  I  A12 F2 A21 A111   A111 A12 F2 
1
A 
where: F2   A22  A21 A111 A12 

 F2 A21 A111
F2


1
X X
 T
 K
T


1
1
1 
 T 1 
T
T
X
X
I

K
K
X
X
K
KT  X T X  






K


 
1
0

1
1
T
T
T
T
K
X
X
K
K
X
X






1

Note: F2  0  K T  X T X  K

1

  KT  X T X  K
1




^
1 
1
  *   X T X   I  K KT  X T X  K



1
1
1

1
1

K K

 K
T
T
X X 
X X 
T
T
1
K

1
K

1
1

1 
1
1
K T  X T X   X TY   X T X  K K T  X T X  K

  X T X  X TY   X T X  K K T  X T X  K
1

K KT  X T X  K
 KT  X T X  K
X X 
T


1

1


1
1






1
1
1
1 
 T 1 
T
T
T
T
^
X
X
I

K
K
X
X
K
K
X
X








  


   *  

1
1
1
   
KT  X T X  K KT  X T X 


1
T


1
X X 



  X TY 


 m 



1
m
K T  X T X  X TY  m     X T X  K K T  X T X  K
1
^
1
1

1
 T ^

K m


Error Sum of Squares for Constrained Model
^
^
^
^
^

^

e*  Y  X  *  Y  X   X   *     e  X   *   




where e  Y  X  and SS  Residual Complete   eT e
^
T
^
^

^
 
^

 SS  ResidualReduced   e * e*   e  X   *      e  X   *    

 



T
T
T
^
^
^
^
 T ^
 T ^

 e e  2 *   X e   *   X X  *  






Note: X T e  X T  I  P  Y  0
T
T
^
^
^
 T ^

 SS  ResidualReduced   e e    *    X X   *   




T

^
1
1
^

with   *       X T X  K K T  X T X  K



1
 SS  ResidualReduced   SS  ResidualComplete  
K  X X  K  K  X X  X X  X X  K K  X X  K 
  m   K  X X  K  K  X X  K  K  X X  K   K   m 



  m   K  X X  K   K   m   Q



T
 T ^

 K   m



  KT


  KT

 T ^

 K   m


T
^
T
^
T
1
1
T
T
T
1
1
T
T
1
1
T
1
T
T
T
T
^
T
1
1
T
T
T
T
1
1
T
1
T
^
1
 T ^

K m


Application to (Multiplicative) Cobb-Douglas
Production Function
• Annual Data 1899-1922 (Indexed to 1899)
• Dependent Variable: Q ≡ Quantity Produced
• Independent Variables: K ≡ Capital
L ≡ Labor
Model: Q   K  K L L 
0   K ,  L  1 E    1
 ln  Q   ln      K ln  K    L ln  L   ln   
 Q*   *   K K *   L L *  *
Note (Ignoring Error Term):
Elasticity of Q wrt K :  K 
Q Q  Q   K 
K
 K 1  L 



K
L
K
 

K L
K K  K   Q 
K L
Q Q  Q   L 
L
K
 L 1 



K

L
 

K L
L L  L   Q 
K L
Elasticity of Scale:  = K   L   K   L
Elasticity of Q wrt L:  L 

  K


  L

  1  Constant Returns to Scale (1% Change in all inputs = 1% Change in output)
Data, Hypothesis, and OLS Estimator
year
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
Q
100
101
112
122
124
122
143
152
151
126
155
159
153
177
184
169
189
225
227
223
218
231
179
240
K
100
107
114
122
131
138
149
163
176
185
198
208
216
226
236
244
266
298
335
366
387
407
417
431
L
100
105
110
118
123
116
125
133
138
121
140
144
145
152
154
149
154
182
196
200
193
193
147
161
ln(Q)
4.6052
4.6151
4.7185
4.8040
4.8203
4.8040
4.9628
5.0239
5.0173
4.8363
5.0434
5.0689
5.0304
5.1761
5.2149
5.1299
5.2417
5.4161
5.4250
5.4072
5.3845
5.4424
5.1874
5.4806
ln(K)
4.6052
4.6728
4.7362
4.8040
4.8752
4.9273
5.0039
5.0938
5.1705
5.2204
5.2883
5.3375
5.3753
5.4205
5.4638
5.4972
5.5835
5.6971
5.8141
5.9026
5.9584
6.0088
6.0331
6.0661
ln(L)
4.6052
4.6540
4.7005
4.7707
4.8122
4.7536
4.8283
4.8903
4.9273
4.7958
4.9416
4.9698
4.9767
5.0239
5.0370
5.0039
5.0370
5.2040
5.2781
5.2983
5.2627
5.2627
4.9904
5.0814
 *
    K  H 0 :  K   L  1
  L 
 K T   0 1 1 m  1
X'X
24
128.5556 119.1054
128.5556 693.4555 639.9174
119.1054 639.9174 592.0168
X'Y
121.8561
655.4095
605.9387
INV(X'X)
55.80062 5.912343 -17.617
5.912343 1.194064 -2.48016
-17.617 -2.48016 6.226807
Beta-hat
-0.17731
0.233053
0.807278
Y'Y
SS(Model) SS(Res)
S^2
620.3713 620.3003 0.070982 0.003549
Constrained Estimator and F-test
INV(X'X)K
-11.7047
-1.28609
3.746651
INV(K'XXIK)
0.406412
B*-B
0.191854
0.021081
-0.06141
Beta-hat*
0.014545
0.254134
0.745866
K'Beta-m
0.040332
H0 : K T   m  0 H A : K T   m  0
TS : Fobs
INV(X'X)K
-11.7047
-1.28609
3.746651
INV(K'XXIK)
0.406411632
B*-B
0.191854
0.021081
-0.06141
Beta-hat*
0.014544619
0.254134156
0.745865844
H0
Q k

~ Fk ,n  p '
MS  ResidualComplete 
 k  1, n  p '  24  3  21
K'Beta-m
0.040332
Note: SS  Residual Reduced 
T
Y'Y
B*'X'Y
B*'X'XB
SS(Res*)
Q
620.3713 620.283291 620.2668998 0.071643
0.000661
Y'Y
SS(Model)
SS(Res)
F_obs
F(.05)
P-value
620.3713 620.300343 0.070981638
0.195583 4.324794 0.662831
^
^

 

  Y  X  *  Y  X  *

 

^
^
^
 Y T Y  2  *T X T Y   *T X T X  *
1
^

  *   X ' X  X 'Y 


t-test/Confidence Interval (k=1 Hypothesis)
Parameter: K 
^
Estimator: K 
T
T
1
 T ^
^
T
V  K    K V    K   2K T  X T X  K


 
^
1
 T ^
^
T
V  K    K V    K  s2K T  X T X  K


 
H0 : K T   m H A : K T   m
^
^
TS : tobs 
KT   m


V  KT  


^
^
^

KT   m
H0
s2 K T  X T X  K
1
~
tn  p '
1   100% Confidence Interval: K   t /2,n p ' s 2 K T  X T X  K
T
1
^
K'Beta-m
S^2
K'*INV(X'X)K V(KB)
SE(KB)
0.040332 0.00338008 2.460559494 0.008317 0.091197
t_obs
t(.025,21)
P-value
0.442248 2.07961384 0.662830702
K'Beta
K'Beta
Lower
Upper
0.850677 1.229986