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

6A.5
Box-Cox Regression Model
Consider the following regression model
Y
λ
i
=
β
1
+
β
2
X
i
+
u
i
Y
>
0
(37)
where 
λ
(Greek lamda) is a parameter, which may be negative, zero, or positive. Since 
Y
is raised to
the power 
λ
, we will get several transformations of 
Y
, depending on the value of 
λ
.
Equation (37) is known as the Box-Cox regression model, named after the statisticians Box and
Cox.
1
Depending on the value of 
λ
, we have the following regression models, which are shown in
tabular form:
Value of 
λ
Regression Model
1
Y
i
=
β
1
+
β
2
X
i
+
u
i
2
Y
2
i
=
β
1
+
β
2
X
i
+
u
i
0.5
Y
i
=
β
1
+
β
2
X
i
+
u
i
0
ln
Y
i
=
β
1
+
β
2
X
i
+
u
i
0.5
1
√
Y
i
=
β
1
+
β
2
X
i
+
u
i
1.0
1
Y
i
=
β
1
+
β
2
X
i
+
u
i
As you can see, linear and log-linear models are special cases of the Box-Cox family of
transformations.
Of course, we can apply such transformations to the 
X
variable(s) also. It is interesting to note that
when 
λ
is zero, we get the log-transformation of 
Y
. The proof of this is slightly involved and is best
left for the references. (Calculus-minded readers will have to recall the l’Hopital Rule.)
But how do we actually determine the appropriate value of 
λ
in a given situation? We cannot
estimate Eq. (37) directly, for it involves not only the regression parameters 
β
1
and 
β
2
but also 
λ
,
which enters nonlinearly. But it can be shown that we can use the method of maximum likelihood to
estimate all these parameters. Regression packages exist to do just that.
We will not pursue this topic here because the procedure is somewhat involved.
However, we can proceed by trial and error. Choose several values of 
λ
, transform 
Y
accordingly,
run regression (37), and obtain the residual sum of squares (RSS) for each transformed regression.
Choose the value of 
λ
that gives the minimum RSS.
2
1
G.E.P. Box and D.R. Cox, “An Analysis of Transformations,” 
Journal of the Royal Statistical Society,
B26,
1964, pp. 211–243.
2
For an accessible discussion, refer to John Neter, Michael Kutner, Christopher Nachtsheim, and
William Wasserman, 
Applied Linear Regression Models,
3rd ed., Richard D. Irwin, Chicago, 1996.
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