The Chow test, proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known a priori (for instance, a major historical event such as a war). In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.
Suppose that we model our data as
y
t
=
a
+
b
x
1
t
+
c
x
2
t
+
ε
.
If we split our data into two groups, then we have
y
t
=
a
1
+
b
1
x
1
t
+
c
1
x
2
t
+
ε
and
y
t
=
a
2
+
b
2
x
1
t
+
c
2
x
2
t
+
ε
.
The null hypothesis of the Chow test asserts that
a
1
=
a
2
,
b
1
=
b
2
, and
c
1
=
c
2
, and there is the assumption that the model errors
ε
are independent and identically distributed from a normal distribution with unknown variance.
Let
S
C
be the sum of squared residuals from the combined data,
S
1
be the sum of squared residuals from the first group, and
S
2
be the sum of squared residuals from the second group.
N
1
and
N
2
are the number of observations in each group and
k
is the total number of parameters (in this case, 3). Then the Chow test statistic is
(
S
C
−
(
S
1
+
S
2
)
)
/
k
(
S
1
+
S
2
)
/
(
N
1
+
N
2
−
2
k
)
.
The test statistic follows the F distribution with
k
and
N
1
+
N
2
−
2
k
degrees of freedom.
Remarks
The global sum of squares (SSE) is often called the Restricted Sum of Squares (RSSM) as we basically test a constrained model where we have 2K assumptions (with K the number of regressors).
Some software like SAS will use a predictive Chow test when the size of a subsample is less than the number of regressors.
