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The Classical Linear Regression Model (CLRM)
The CLRM makes the following assumptions:
A-1: The regression model is linear in the parameters as in
๐๐
๐๐=๐ต๐ต1+๐ต๐ต2๐๐2๐๐ +๐ต๐ต3๐๐3๐๐ +โฏ+๐ต๐ต๐๐๐๐๐๐๐๐ +๐ข๐ข๐๐
it may or may not be linear in the variables Y and the Xs.
A-2: The regressors are assumed to be fixed or nonstochastic in the sense that their
values are fixed in repeated sampling. This assumption may not be appropriate
for all economic data, but as we will show later, if X and u are independently
distributed the results based on the classical assumption discussed below hold true
provided our analysis is conditional on the particular X values drawn in the
sample. However, if X and u are uncorrelated, the classical results hold true
asymptotically (i.e.in large samples.)1
A-3: Given the values of the X variables, the expected, or mean, value of the error term
is zero. That is,2
๐ธ๐ธ(๐ข๐ข๐๐|X)= 0 (1.8)
where, for brevity of expression, X (the bold X) stands for all X variables in the
model. In words, the conditional expectation of the error term, given the values
of the X variables, is zero. Since the error term represents the influence of factors
that may be essentially random, it makes sense to assume that their mean or
average value is zero.
As a result of this critical assumption, we can write (1.2) as:
๐ธ๐ธ(๐๐
๐๐|X)=๐ฉ๐ฉ๐ฉ๐ฉ +๐ธ๐ธ(๐ข๐ข๐๐|X)
=๐ฉ๐ฉ๐ฉ๐ฉ (1.9)
which can be interpreted as the model for mean or average value of ๐๐
๐๐ conditional
on the X values. This is the population (mean) regression function (PRF)
mentioned earlier. In regression analysis our main objective is to estimate this
function. If there is only one X variable, you can visualize it as the (population)
regression line. If there is more than one X variable, you will have to imagine it
to be a curve in a multi-dimensional graph. The estimated PRF, the sample
counterpart of Eq. (1.9), is denoted by ๐๐
๏ฟฝ๐๐=๐๐๐๐. That is, ๐๐
๏ฟฝ๐๐=๐๐๐๐ is an estimator of
๐ธ๐ธ(๐๐
๐๐|๐๐).
1 Note that independence implies no correlation, but no correlation does not necessarily imply
independence.
2 The vertical bar after ๐ข๐ข๐๐ is to remind that the analysis is conditional on the given values of X.
