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Lecture 23 – Hypothesis Tests
We will now begin to discuss hypothesis tests.
As we will see, hypothesis tests and confidence intervals
are closely inter-related.
Recall from introductory macroeconomics that economic theory states that the demand for money is
inversely related to the rate of interest.
Let M = money demand and R = the rate of interest.
Then the
regression model we would seek to estimate is given by:
MBB
R
tt
=+
+
01
ε
t
,
and the estimated regression equation would be given by:
$$$
R
.
According to the theory, the sign of the slope coefficient should be negative.
Now this is what we mean by
a hypothesis.
A hypothesis is simply some theoretical assertion concerning the value of the parameters.
Suppose when we did the above regression, the estimated value of the slope turned out to be equal to 400.
This is much above 0, i.e., it is a large positive number.
We would take this as evidence that the hypothesis
is probably not true.
Note, I have said probably
not true.
We cannot ever be certain, because we know that
our estimate is just that, an estimate.
It is quite possible that the true parameter value indeed is negative
and we just happened to select an odd sample giving us the value of 400.
But note that if the true parameter
value really is negative, then the probability of getting an estimate equal to 400, while not = 0, is
nevertheless very small.
And this is the basic intuition behind the idea of a hypothesis test.
We hypothesize that our parameter has a value (or range of values) based on theory.
We then look at the
value of the estimate that we obtain from our random sample.
If the estimated value is very far from the
hypothesized value, we take this as an indication that the hypothesis is not true.
We can never be certain
that it is not true because of the possibility of having selected an odd sample, but we can be quite confident,
i.e., we can know with a high probability, that the hypothesis is rejected by the empirical evidence.
What
we will do now is to formalize this intuitive idea.
The hypothesis which is being tested (and which is determined by economic theory) is called the
null
hypothesis
and is denoted by
.
H
o
The usual practice is to state the null hypothesis in a form we would like
to reject
.
So for instance, in the money example above, theory says that the coefficient on the interest rate
variable should have a negative sign, i.e., the coefficient is less than 0.
Our null hypothesis would then be
stated as: the coefficient is positive, i.e., is > 0.
If we reject this hypothesis then we in fact are stating that
the empirical evidence confirms the theory, and since we would like our theory to be confirmed by the
evidence, we hope to reject the null hypothesis
.
We will see later on why we set up the null hypothesis as
something we would like to reject. Rejecting the null hypothesis means that we are accepting what is
called the alternative hypothesis
.
In the above example with money demand, the alternative hypothesis is
that the coefficient on the interest rate variable has a negative sign.