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Multiple Regression Analysis
The
three-variable linear model:
Multiple regression analysis is used for testing hypothesis about the
relationship between a dependent variable, Y, and two or more independent
variables, Xs, and for prediction. The three-variable linear regression
model can be written as
Yi=
b0 + b1X1i + b2X2i
+ ui
For the case of k independent or explanatory variables, we have
Yi=
b0 + b1X1i + b2X2i
+ .......+ bkXki + ui
where
X2i represents, for
example, the ith observation on independent variable X2.
The additional assumption (to those of the simple regression model) is that the there is no exact linear relationship between the X's.
The
first five assumptions of the multiple regression linear model are
exactly the same as those of the simple OLS regression model. That is, the
first three assumptions can be summarized as ui~N(0,σu2).
The fourth assumption is E(uiuj)=0 for i=j; and the
fifth assumption is E(Xiui) =0 . The only additional
assumption required for the multiple OLS regression linear model is that
there is no exact linear relationship between the axis. If two or more
explanatory variables are perfectly linearly correlated, it will be
impossible to calculate OLS estimates of the parameters because the system
of normal equations will contain two or more equations that are not
independent. If two or more explanatory variables are highly but not perfectly
linearly correlated, then OLS parameter estimates can be calculated, but
the effect of each of the highly linearly correlated variables on the
explanatory variable cannot be isolated.
In
case of multiple regression analysis with two independent or
explanatory variables:
Parameter
b0 is the constant term or intercept of the regression and
gives the estimated value of Yi when X1i = X2i =
0.
Parameter
b1 measures the change in Y for each one-unit change in X1
while holding X2 constant. Slope parameter b1 is a partial
regression coefficient because it corresponds to the partial derivative of
Y with respect to X1, or
.
Parameter
b2 measures the change in Y for each one-unit change in X2
while holding X1 constant. Slope parameter b2 is the
second partial regression coefficient because it corresponds to the
partial derivative of Y with respect to X2, or
.
Since,
,
and
are obtained by the OLS
method, they are also best linear estimators. That is E(
)=
, E(
)=
, and E(
)=
and
,
,
are lower than for any other
unbiased linear estimator.
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© 2002
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Evgenia Vogiatzi <<Previous Next>>
