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Chapter 3. Multiple Regression Analysis: Estimation
Simple linear regression model: an intercept and one explanatory variable (regressor)
Multiple linear regression model: an intercept and many explanatory variables (regressors)
OLS estimation method is exactly same as in the simple linear regression model. OLS estimators are derived
from minimizing the residual sum of squares (
SSR
) with respect to the parameters
The predicted values of
and the regression residuals are computed by
and the unbiased estimator of the variance of error term is
Note that the number of coefficients
is subtracted from the sample size
n
in the denominator.
Interpretation of the marginal effect
If all explanatory variables are different, then
represents the marginal effect of
, holding all other
explanatory variables constant.
Example 3.2. Hourly wage in loglinear model
An increase in education by one year increases
by 0.092, which is equivalent to an increase in wage
by 9.2%. An increase in labor market experience by one year increases the average wage by 0.41%, etc.
Note:
. This is based on the first order Taylor expansion
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A Measure of Goodnessoffit
We introduced the squared correlation coefficient (
) between
and
as a measure of how well our
model explains the dependent variable. A higher
means a better model fitting. We showed
We can use this in the multiple regression equation also.
Statistical Properties of OLS Estimator in Multiple Regression Model
OLS estimators are unbiased if
(1)
and they are BLUE if
(1)
(2)
(3)
Assumption (1) implies
,
for all
i
and
j
,
for all
i
and
j
The formulas of variance of
and covariance between
and
are quite complicated to write out, though
it has a simple form in matrix form. We only need to know that the estimated variances and covariances use
the unbiased estimator
.
What are the new issues in the multiple regression model?
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 Three '11
 Su
 Econometrics

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