TESTING FOR RESTRICTIONS ON THE COEFFICIENTS OF THE
VARIABLES
Data 4-2
Consumption, ct= real consumption
GDP, yt=real gross domestic product
Wages, wages=total nominal wages
Price deflator, prdefl= implicit price deflator (price index), 1992=100
Type generate float rwages=wages/prdefl*100 to convert nominal wages to real
wages or use Data/Create or change variables/Create new variables and type
rwages and then the formula. A new series of rwages will be stored in the data set.
Also, generate float rprofit=yt-rwages to create Real Profits as another variable.
Then issue the command reg ct rwages rprofit. This is your unrestricted model.
ct ˆ  ˆ1rwages  ˆ2rprofit
(UNRESTRICTED) MODEL
Source
SS
df
MS
Model
Residual
31252859.4
38976.5104
2
33
15626429.7
1181.10637
Total
31291836
35
894052.456
ct
Coef.
Std. Err.
rwages
rprofit
_cons
.6932614
.735916
-222.1584
.0326064 21.26
.0488218 15.07
19.55271 -11.36
t
Suppose you want to test the restriction that
Number of obs =
36
F( 2, 33)
=13230.33
Prob > F
= 0.0000
R-squared
= 0.9988
Adj R-squared = 0.9987
Root MSE
= 34.367
P>t
0.000
0.000
0.000
[95% Conf.
.6269232
.6365872
-261.9387
Interval]
.7595996
.8352448
-182.3781
H 0 : ˆ1  ˆ2 such that marginal
propensity to consume out of real wages is equal to the marginal propensity to
consume out of real profits (meaning the MP out of savings are also equal across
these groups). You can do so using again the Wald test. Just type the following

test rwages=rprofit
( 1) rwages - rprofit = 0
F( 1, 33) =
Prob > F =
0.28
0.6017
Notice that F-test has a p-value of 0.6017 greater than any alpha. DO NOT
REJECT the null hypothesis,
H 0 : ˆ1  ˆ2
So, the MPC out of wages and MPC out of profits are equal to each other, refuting
an important misconception that saving propensity out of profits are higher than
that of wages!
Various
types
of
restrictions
can be tested using
this
simple method.
METHOD 1: WALD TEST ON RESTRICTIONS
Alternatively, you can generate the F-test using the ESSR and ESSUR in the formula
described in class. You will get the same value as F=0.277722 in the above table.
To get the restricted model, regress ct on yt. This means that restricted model
imposes the equality for the coefficients of rwages and rprofit combining them as
yt. The output of the restricted model is given below.
(RESTRICTED) MODEL
reg ct yt
Source
SS
df
MS
Model
Residual
31252531.4
39304.5352
1
34
31252531.4
1156.01574
Total
31291836
35
894052.456
ct
Coef.
Std. Err.
t
yt
_cons
.7102901
-221.4251
.0043199 164.42
19.29488 -11.48
Number of obs =
36
F( 1, 34)
=27034.69
Prob > F = 0.0000
R-squared
= 0.9987
Adj R-squared = 0.9987
Root MSE
=
34
P>t
[95% Conf.
Interval]
0.000
0.000
.701511
-260.637
7190692
-182.2132
Using the ESSR and ESSUR from these restricted and unrestricted models, you can
get the same F statistic and carry out the test as in done in the Wald test output.
METHOD 2: Indirect t-test
See class notes on the derivation of the model. The output needed for this test is
as follows. Regress ct on yt and rprofit.
Source SS
df
Model 31252859.4
Resid 38976.5104
Total
31291836
ct
Coef.
MS
2
33
15626429.7
1181.10637
35 894052.456
Std. Err.
t
yt
.6932614 .0326064 21.26
rprofit .0426546 .0809389 0.53
_cons -222.1584 19.55271 -11.36
P>t
0.000
0.602
0.000
Number of obs =
36
F( 2, 33)
=13230.33
Prob > F
= 0.0000
R-squared
= 0.9988
Adj R-squared = 0.9987
Root MSE
= 34.367
[95% Conf.
.6269232
-.1220167
-261.9387
Interval]
.7595996
.207326
-182.3781
This method, as explained in class, relies on a t-test on rprofit variable and is based
on p-value. We can not reject the null of the equality of coefficients as p-value for
rprofit=0.602>0.05 alpha. Same result!
METHOD 3: Direct t-test
No need for a separate estimation. Use the UR model above and calculate the ttest statistic as will be discussed in class.
t
ˆ1  ˆ2
s 2 ˆ  s 2 ˆ  2Cov( ˆ1, ˆ2 )
1
2
=(0.693262-0.735916)/sqroot(0.0326062+0.0488222-2(-0.001552))=-0.53
Cov(ˆ1, ˆ2 )
can be calculated as –0.001552 from Covariance Matrix on
coefficients using correlate rwages rprofit, covariance _coef option. Based on this tvalue of –0.53, you can not again reject the null! Same result!