文档介绍：Ch. 7 Violations of the Ideal Conditions
1 Specification
Selection of Variables
Consider a initial model, which we assume that
Y = X1β1 + ε,
It is not unusual to begin with some formulation and then contemplate adding
more variable (regressors) to the model:
Y = X1β1 + X2β2 + ε.
2 2
Let R1 be the R-square of the model with fewer regressor, and R12 be the
R-square of the model with more regressors. It is apparent as we have shown
2 2 2
earlier that R12 > R1. Clearly, it would be possible to push R as high as desired
by adding regressors. This problem motivates the use of the adjusted R-square,
T 1
R¯2 = 1 −(1 R2)
− T k −
−
It has been suggested that the adjusted R-square does not penalize the loss of
degree of freedom heavily, two alternative have been proposed paring
models are
T + k
R˜2 = j (1 R2)
j T K − j
− j
and Akaike’s information criterion:
0
e ej 2k 2k
AIC = ln j + j = ln σˆ2 + j .
j T T j T
Although intuitively appealing, these measures are a bit unorthodox in that
they have no firm basis in theory (unless that are used in time series analysis
model). Perhaps a somewhat more palatable alternative is the method of step-
wise regression; However, economists have tends to avoid stepwise regression
method for the break down of inference procedures.
1
Omission of Relevant variables
Suppose that a correctly specified regression model would be
Y = X1β1 + X2β2 + ε,
where the two parts of X have k1 and k2 columns, respectively. If we regress Y
on X1 without including X2, that is you have estimate the model
Y = X1β1 + ε,
and obtain the estimator as
ˆ 0 −1 0 0 −1 0
β1 = (X1X1) X1Y = (X1X1) X1(X1β1 + X2β2 + ε)
0 −1 0 0 −1 0
= β1 + (X1X1) X1β2 + (X1X1) X1ε.
0 ˆ
Taking the expectation, we see that unless X1X2 = 0 or β2 = 0, β1 is biased:
ˆ 0 −1 0
E(β1) = β1 + (X1X1) X1X2β2.
ˆ
The variance of β1 is
ˆ 2 0 −1
V ar(β1) = σ(X1X1) .
If we puted the correct regression, including X2