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Lectures 31 & 32  Functional form
1) The next topic we will discuss concerns the functional form of the regression equation.
That is, do the
variables enter in linear form, as squares, logs, etc..
Once again, we should allow theory to determine this
as far as possible.
Your text gives you the impression that this can almost always be accomplished.
Unfortunately, this is not so!
Most economic theories tell you only the sign that should be on the
coefficient.
(For instance, in macro you learned that consumption is a positive function of income, but the
theory said nothing about whether income should be squared, or enter as a log, etc.).
Thus, here is a place
where art plays an important role.
It would be better if theory did indicate the form of the variables,
unfortunately this is not always the case.
We will begin with a discussion of the interpretation and reliability of the constant term
.
We have seen that
there are 2 reasons for not placing much confidence in the estimate of the constant term.
These are:
a) We do not have sample values close to the origin
, and hence cannot get an accurate estimate for
the intercept.
b) The constant term adjusts to correct for the fact that the expected value of the error term may
not be equal to 0
, as is required by the classical assumptions.
Thus the estimate we get for the
constant term may not be the true constant, reflecting this adjustment.
The interpretation of the constant is that it tells us the value of the dependent variable when the independent
variables take on the value = 0.
Some economic theories may imply that the constant should be equal to 0.
For instance, in a production function we would normally assume that if the amount of labor used is equal
to 0, then output will also be equal to 0, implying that the intercept is equal to 0.
The question then arises
as to whether we should force the constant to be equal to 0 when doing the regression.
This is called
‘suppressing the constant’.
The answer is that we should not do this.
The following picture helps to
understand why:
$
Y
$
Y
with constant
0
≠
Y
X
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If we force the constant to be equal to 0 then we are estimating the regression line that goes through the
origin.
Given the sample we actually have, this clearly increases the size of the slope estimate we would
have obtained had we not suppressed the constant.
Thus, our tvalues will increase.
The problem here is
really just a version of (a) above.
Although theory tells us that the function should go through the origin,
the sample we have is not close enough to the origin to let us know what the shape of the function appears
like in that region. In the sample we actually have we obtain the regression equation with the positive
intercept.
If the theory is true, then we must have a nonlinear relationship.
But we do not have sample
values that enable us to estimate what the nonlinear relation looks like. Hence, we will get a more accurate
representation of the true relationship if we estimate within the sample we have and don't suppress the
constant, even though theory says the constant = 0.
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 Spring '11
 YongJinPark
 Econometrics, Regression Analysis, Inverse function, Logarithm, Rsquared

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