INTRODUCTION TO ECONOMETRICS II
ECO 306
NOUN
89 For the purpose of linear regression analysis, only the second type of linearity is important.
Nonlinearity in the variables can always be sidestepped by using appropriate definitions. For example, suppose that the relationship was of the form
√
…[2.77] By defining Z
2
=
, Z
3
=
√
, Z
4
=
etc, the relationship can be rewritten
…[2.78] and it is now linear invariables as well as in parameters. This type of transformation is only beautifying, and you will usually seethe regression equation presented with the variables written in their nonlinear form. This avoids the need for explanation and extra notation. But [2.76] is nonlinear in both parameters and variables and cannot be handled by a mere redefinition. That is, even if attempted, the equation cannot be made linear by defining Z =
and replacing
with Z; since you do not know
, you have noway of calculating sample data for Z.
However, you could define
1
Z
X
, the equation now becomes
…[2.79] and this is linear, which is the regress of Y onZ. The constant term in the regression will bean estimate of
and the coefficient of Z will bean estimate of
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