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Chapter 4. Multiple Regression Analysis: Inference
We can summarize what we have done so far as follows.
Specification of multiple linear regression model
OLS (Ordinary Least Squares) estimator
minimizes the SSR
Once the OLS estimators are computed, we can compute
Predicted values of
:
Regression residuals:
Estimator of the variance of error term
We have seen that parameter estimators are random and vary from sample to sample. The variances and
covariances can be expressed as
where
depends on the data of explanatory variables. The estimated variance and covariance are computed
by replacing the unknown
with its estimator
In this chapter we will consider the test of hypothesis.
Test of Hypotheses
Hypothesis, Test Statistic, and Rejection Region
Specification of Hypotheses
A hypothesis about the parameters is an assertion (or a theory) about the true value of the parameter. For
example,
which says that the explanatory variable
has no effect on the dependent variable, or
(
)which says that the marginal effect of
and
are the same. The assertion we wish to test is
called a
null hypothesis
or a
maintained hypothesis
, and it is typically denoted by
H
0
. The null hypothesis
is tested against an
alternative hypothesis
which is typically denoted by
and can take different forms
depending on what we wish to test. Examples are
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Some authors use the term
onetailed
and
twotailed
test. We will use these terms interchangeably.
(a)
(b)
(c)
When a hypothesis specifies a specific value such as
, it is called a
simple hypothesis
. When a
hypothesis specifies multiple values such as
, it is called a
composite hypothesis
. Thus, (a) tests a simple
null against a simple alternative hypothesis, and (b) and (c) test a simple null against composite alternative
hypothesis. When a hypothesis contains more than one statement, it is called a
joint hypothesis
. An example
of a joint null hypothesis
.
Test Statistic
Once the null and alternative hypotheses are specified, we need to choose a test statistic. The choice of the
test statistic determines how we use the sample information for the test. This choice can be arbitrary, but a
function of the estimator of parameter under consideration is commonly used as a test statistic. For example,
we may use the OLS estimator
as a test statistic in testing the null hypotheses specified in the examples
above.
Decision Rule  Choice of Rejection Region
Given the test statistic, the decision rule specifies when to reject and when to accept
. For example, we
may decide to reject
in example (a) if the test statistic
is closer to 0 than to 1. More specifically, we
reject
if
and accept
if
for some constant
c
in an interval [0, 1]. For example (b), we may
reject
if
or if
for some constant value
c
. The set of values for which
will be rejected is
called a
rejection region
, or a
critical region
, of the test. The rejection region usually takes the form of a
one
sided
or a
twosided
rejection region:
Onesided rejection region
:
Twosided rejection region
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 Three '11
 Su
 Econometrics, Normal Distribution, Null hypothesis, Statistical hypothesis testing, Joint Hypotheses, Test of equality

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