So, we did a bunch of stuff on estimation
of parameters in the constant expected
return model.
And, so estimation is one part of doing
statistical analysis.
We have a model, it has parameters.
You need to estimate those parameters from
data, and so went over a pretty extensive
review of the statistical theory of
estimation, how to evaluate estimates, how
to compute standard errors, how to use the
bootstrap to get standard errors, how to
compute confidence intervals.
Now, we're going to look at doing
statistical inference based the model.
So, and that, that's the, the name of
hypothesis testing.
And, again, to briefly review, when you do
hypothesis testing, there's some maintain
view of that, that you want to test.
And, in you know, with respected constant
expected return model, we might want to
test hypothesis about certain values of
the parameters.
We want to, we want to test hypothesis
that the, the expected returns for all the
assets are greater than zero.
We might want to test hypothesis that the
correlation between, you know, two pairs
of stocks is the same.
Or we might want to test hypothesis that
returns really are normally distributed.
Or we could be interested in the
hypothesis that the correlation between
two stocks is constant over the entire
data sample.
So, there are many different kinds of
hypothesis that we could be testing in a
given context.
And so, you know, the null hypothesis is
what's being maintained, you know?
The correlations are the same between two
pairs of assets, for example.
And then, the alternative hypothesis says,
you know, what, what's the alternative
view of the world?
For example, the correlations between two
pairs of, of returns are not the same.
And, when you do hypothesis testing, you
have to make a decision.
Do you reject the null hypothesis or do
you not reject the null hypothesis.
And, when we do testing, we specify the
level of the test which is the probability
that you reject the null hypothesis given
the null hypothesis is true.
And usually, this is some small number
like one percent or five%.
And it's the, again, it's the probability
of making a particular type of error.
Then you can sort the test statistic from
the data.
And, the test statistic, you know, again,
summarizes the data evidence in favor or
against the null hypothesis.
And typically, when you do hypothesis
testing, if your test statistic is big,
that's data evidence against the null
hypothesis and that's indicating that the
decision you should make is to reject it.
If the test statistic is small, then,
that's data evidence in favor of the null
hypothesis and, and you shouldn't reject
in, in that case.
And then, based on the magnitude of the
test statistic, you have to make a
decision.
And the formal theory of hypotheses
testing, we usually have values of the
test statistic that lie in the rejection
region.
So, if the test statistic isn't big
enough, then you should reject.
And so, the rejection region is a cut off
point typically of how big the test
statistic needs to be in order for you to
reject.
And this cut-off region is typically
called a critical value, and the critical
value has the property such that if say,
the absolute value of the test statistic
is bigger than the critical value you
reject, but if the absolute value of the
test statistic is less than the critical
value you don't reject.
Now, where this critical value comes from
depends upon the test statistic that you
use and the hypothesis that you're
testing.
And, so we'll go through several examples
to illustrate how this works.
Now, in any decision making context and a
hypothesis testing context is a decision
making context.
So, there are two decisions that you can
make.
You either reject the null hypothesis or
you don't reject the null hypothesis,
okay?
And, there, there is reality, what's
actually true in the world.
What's true is, the null hypothesis, it, I
mean, it could be the state of the world
that the null hypothesis that you're
testing is really true.
But, it could also be that the null
hypothesis you're testing is really false.
And again, I'd like to think of, you know,
the criminal justice system example.
You're sitting on a jury, you're trying to
decide, you know, the fate of, you know,
someone's on trial for murder.
You have to decide whether or not, and,
and then the maintain hypothesis that you
are innocent until proven guilty.
So, the null hypothesis is this, this
person is innocent.
And, the decision you're going to make are
you reject the null hypothesis.
So, you reject that the person is
innocent, you conclude that they're
guilty, or you decide not to reject the
null hypothesis.
That is, you, you conclude this person
really is innocent.
Now, in reality, this person is a killer
or he's not a killer, right?
So, in reality, you know, the person is
really innocent.
And then, if you reject the null
hypothesis when the person is really is,
then you make what's called a type one
error, okay?
So, that's sending the innocent person to
the gas chamber.
That's a bad thing to do.
On the other hand, if the person is
innocent and you don't reject the null
hypothesis, you conclude that the innocent
person is innocent and the person gets to
go free, and you've made the right
decision.
Now, the other state of the world could be
is that the null hypothesis is false, so
that the guy's really a killer.
And, you reject the null hypothesis, so
you found that the killer is guilty.
So, okay, you've, you've sent the killer
to the gas chamber.
And depending upon your politics that
could be no error.
And the other situation is for the, for
the guilty person.
You don't reject the null hypothesis, so
you found out that the guilty person is
really innocent.
So, you've decided this person innocent
when they're in fact, guilty, that's
called a type two error.
So, this is the error where you allow the
serial killer back out into the public.
And, that's a bad thing to do.
And you would like to make sure that you
don't make those kinds of errors.
Now, when you do hypothesis testing, the
level of the test is the probability of
type one error.
It's the probability that you reject an
hypothesis when it's true, okay?
So again, this is the, the probability of,
of find, finding that an innocent person
is, is guilty.
And, the goal of hypothesis testing is, is
to make the level of the test small.
So, you'd like to minimize this type of
level.
And so, we usually set our test such that
we have a pre-specified probability of
committing this kind of error, five%, one
percent are typical values.
Now, something called the power of the
test is one minus the probability of type
two error.
So, the power of the test is the
probability of rejecting the null
hypothesis, given the null hypothesis is
false.
So, you'd like to have a test to be able
to determine if the null hypothesis isn't
true.
So, in the courtroom example, you know, a
legal system that has good power is a
legal system that convicts guilty people,
right?
And, a good legal system is a system that
does not convict innocent people.
Now, there's a problem in the statistical
testing context is that it's impossible to
simultaneously have the level of test
approximately zero, and have power close
to 100%, okay?
Because if you make the level of test go
down to zero, that is, if you never
falsely, you know, convict somebody, then,
in, in some respects, you're making a
decision that you're never going to
convict anybody, which means that this
probability would get close to zero.
And similarly, if you want to have high
power, then you tend to push up the, the
probability of type one error.
If you want to, you know, increase, you
know, try to increase this probability of
rejecting null hypothesis when the null
hypothesis is false, you know, then, then
you tend to also commit higher probability
of type one error.
So, these, these, two properties of test
have conflicting views.
And the goal in, in hypothesis testing is
given a probability of type one error
construct a test that has the higher
power.
And so, the kinds of test statistics that
we're going to be looking at, have the
property that they tend to have you know,
high power in a decision context.