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

(9.8.1)
Frisch–Waugh theorem.
16
) So to speak, we kill (deseasonalize) two birds (two series)
with one stone (the dummy technique).
If you want an informal proof of the preceding statement, just follow these steps:
(1) Run the regression of 
Y
on the dummies as in Eq. (9.7.2) or Eq. (9.7.3) and save the
residuals, say, 
S
1
; these residuals represent deseasonalized 
Y.
(2) Run a similar regression
for
X 
and obtain the residuals from this regression, say, 
S
2
; these residuals represent
deseasonalized 
X
. (3) Regress 
S
1
on 
S
2
. You will find that the slope coefficient in this
regression is precisely the coefficient of 
X
in the regression (9.7.4).
16
For proof, see Adrian C. Darnell, 
A Dictionary of Econometrics,
Edward Elgar, Lyme, U.K., 1995,
pp. 150–152.
I
II
Sales commission
X
*
Y
X 
(sales)
FIGURE 9.5
Hypothetical
relationship between
sales commission and
sales volume.
(
Note:
The intercept on
the 
Y
axis denotes
minimum guaranteed
commission.)
EXAMPLE 9.6
(
Continued
)
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296
Part One
Single-Equation Regression Models
where 
Y
i
=
sales commission
X
i
= 
volume of sales generated by the sales person
X
∗
= 
threshold value of sales also known as a 
knot
(known in advance)
17
D
= 
1
if 
X
i
>
X
∗
= 
0
if 
X
i
<
X
∗
Assuming 
E
(
u
i
)
=
0, we see at once that
E
(
Y
i
|
D
i
=
0,
X
i
,
X
∗
)
=
α
1
+
β
1
X
i
(9.8.2)
which gives the mean sales commission up to the target level 
X
∗
and
E
(
Y
i
|
D
i
=
1,
X
i
,
X
∗
)
=
α
1
−
β
2
X
∗
+
(
β
1
+
β
2
)
X
i
(9.8.3)
which gives the mean sales commission beyond the target level 
X
∗
.
Thus, 
β
1
gives the slope of the regression line in segment I, and 
β
1
+
β
2
gives the slope
of the regression line in segment II of the piecewise linear regression shown in Figure 9.5.
A test of the hypothesis that there is no break in the regression at the threshold value 
X
* can
be conducted easily by noting the statistical significance of the estimated differential slope
coefficient 
ˆ
β
2
(see Figure 9.6).
Incidentally, the piecewise linear regression we have just discussed is an example of a
more general class of functions known as