The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

Part One
Single-Equation Regression Models
2. The Chow test will tell us only if the two regressions (8.7.1) and (8.7.2) are different,
without telling us whether the difference is on account of the intercepts, or the slopes, or
both. But in Chapter 9, on dummy variables, we will see how we can answer this question.
3. The Chow test assumes that we know the point(s) of structural break. In our exam-
ple, we assumed it to be in 1982. However, if it is not possible to determine when the struc-
tural change actually took place, we may have to use other methods.
16
Before we leave the Chow test and our savings–income regression, let us examine one
of the assumptions underlying the Chow test, namely, that the error variances in the two
periods are the same. Since we cannot observe the true error variances, we can obtain their
estimates from the RSS given in the regressions (8.7.1a) and (8.7.2a), namely,
ˆ
σ
2
1
=
RSS
1
n
1
−
2
=
1785
.
032
10
=
178
.
5032
(8.7.6)
ˆ
σ
2
2
=
RSS
2
n
2
−
2
=
10,005
.
22
14
−
2
=
833
.
7683
(8.7.7)
Notice that, since there are two parameters estimated in each equation, we deduct 2 from
the number of observations to obtain the df. Given the assumptions underlying the Chow
test, 
ˆ
σ
2
1
and 
ˆ
σ
2
2
are unbiased estimators of the true variances in the two subperiods. As a
result, if 
σ
2
1
=
σ
2
2
, that is, the variances in the two subpopulations are the same (as assumed
by the Chow test), then it can be shown that
ˆ
σ
2
1
σ
2
1
ˆ
σ
2
2
σ
2
2
∼
F
(
n
1
−
k
),(
n
2
−
k
)
(8.7.8)
follows the 
F
distribution with (
n
1
−
k
) and (
n
2
−
k
) df in the numerator and the denomi-
nator, respectively, in our example 
k
=
2, since there are only two parameters in each sub-
regression.
Of course, if 
σ
2
1
=
σ
2
2
, the preceding 
F
test reduces to computing
F
=
ˆ
σ
2
1
ˆ
σ
2
2