All right.
So, let's talk a little bit about
hypothesis testing.
Now, again, this is part of the, you know,
pro, you know, probability and statistics
review.
So, everyone in their previous statistics
class has had some exposure to hypothesis
testing, I assume doing T tests and stuff
like that.
And so, I just want to remind you a little
bit about the hypothesis testing
methodology.
And then, mention, you know, what are some
of the interesting hypotheses that we
could conduct in our constant expected
return model.
So, so this is the basic idea with, with
hypothesis testing.
So, with hypothesis testing, there's a
decision to be made.
So, there is some hypothesis.
Now, a hypothesis is some assumption about
your model.
So, hypothesis could be that true mean is
equal to zero, okay?
And then, so that, so you, you specify
something that's being maintained and
that's called the null hypothesis.
So, 80 is some, maintain hypothesis like
the, the true mean is equal to zero.
And then, there's an alternative
hypothesis which is, you know, if, if, if
this isn't true, then, what else could be
true.
So, the alternative could be, well, the
mean is not equal to zero, okay?
And so, what you want to do is you want to
test, you want to gather data, and you
want to see if the data is in favor of the
null hypothesis, or if it's more in favor
of the alternative hypothesis.
So, hypothesis testing is really about
making decisions, right?
And, you know, in our case, in our model,
you know, what are the kinds of things
that are interesting?
For example, we might want to test the
hypothesis that our returns are normally
distributed, right?
And, cuz returns are normally distributed
then, you know, it gives us an easy way to
do certain calculations.
But if our data's not normally distributed
then, you know, we should be searching for
better distributions.
So, in finance, and particularly in
looking at return data, you know, that is
an interesting hypothesis to, to look at.
Another interesting hypothesis is, you
know, Are the parameters of our model
constant over time, right?
Is the mean return constant over a five
year period, or is it changing?
Is the correlation between two returns
constant over our observed data, or is it
changing over time?
So, those are kind of interesting
hypothesis that I think that we want to
look at and, and we would like to try to
evaluate with data.
Alright.
So, we're going to use the methodology of
hypothesis testing to do this.
So, anytime, you know, you start with a
null and alternative hypothesis and then
you're going to do a test.
You're going to gather data, and with your
data you're going to construct a test
statistic, and the test statistic is going
to give you data evidence in favor or
against the hypothesis.
Now, because hypothesis involves,
hypothesis testing involves a decision,
the decision is you either reject the null
hypothesis or you do not reject the
hypothesis, okay?
So, think of, you know, in the context of
like, drug trials, right?
You're going to either conclude that the
drug cures cancer or it doesn't, right?
And, that's a very important decision.
Because if you have a, found a drug that
cures cancer, right?
You know?
You've you know, made a ton of money for
your company and, and improve the world
and so on and so forth.
But, if the drug doesn't cure cancer or do
anything, then you know, it should be
thrown out and you should, you know, look
elsewhere.
So, very often in hypothesis testing, the
decision is very important.
I like to, to talk about hypothesis
testing often in a, a courtroom setting.
So, think of yourself sitting on a jury,
and you have to decide whether or not
someone's guilty or innocent.
And, and to make it more interesting,
suppose you're on a murder trial.
And so, the idea is, you know, you're
innocent until proven guilty.
So, the null hypothesis is the person on
trial is not a serial killer.
But the alternative hypothesis is, the guy
is a serial killer.
And you, as the jury, have to decide
whether or not this guy's going to, to go
to the gas chamber or not, right?
So, your decision has a real implication
to what's going on.
So, you don't want to make an error.
So, in any decision, you know, there could
an error that could be made.
It could be the case that this person is
really innocent, right?
And then, an error would be is you convict
the innocent guy, right?
That's rejecting the null hypothesis when
the null hypothesis is true, right?
So, you're concluding that the guilt, the
innocent person is guilty when in fact,
they're not guilty.
That's a bad thing to do.
You don't want to send an innocent person
to the gas chamber.
So, you want the probability that you
reject the null hypothesis when it's in
fact true, to be very, very, very small.
You, and in the, in the context of a court
room you want the evidence to be beyond
the reasonable doubt, right?
And so, that's like saying you want this,
this probability of rejecting the null
hypothesis when null hypothesis is true to
be very small.
So, in, in statistical testing, what we
call the level of the test is the
probability that we reject the null when
it is in fact true.
And, and we would like that to be a small
value, like one percent or five%.
Five percent can often be very big.
Particularly, if you're on a murder trial,
beyond a reasonable doubt should be
smaller than five percent probability.
[laugh] One should think that should be
like 0.001 or something like that.
So, you, any test that you perform,
there's a pre-specified tolerance for
committing this kind of error.
And again, that's called the level of the
test.
The next step in hypothesis testing is you
gather data, and you summarize the data
evidence in the form of a test statistic.
And very often in statistics, a test
statistic is something called a
t-statistic.
And, and then, the test statistic
generally has the property that if the
test statistic is big, then, that's data
evidence against the null hypothesis, and
you should reject it, okay?
And if the test statistic is small, then
that's data evidence in favor of the null
hypothesis and you shouldn't reject it.
So again, it's like t is summarizing all
of the arguments from the defense and the
prosecuting attorney.
And then, you know, it's synthesizing that
evidence into some numerical value, and if
that numerical value is big, then you
reject the null.
And if it's small, you don't reject.
Now, also in the language of hypothesis
testing, you shouldn't use phrases like, I
accept the no hypothesis, you know?
And, and from the logical point of view,
accepting is sort of establishing the fact
that it is true.
Well, all you can really do is evaluate
the data evidence against the null
hypothesis.
It's sort of like this, trying to evaluate
this statement, all swans are white,
right?
So, you can't, in order to prove that all
swans are white, you have to find every
swan that ever existed and show that is
white.
In order to disprove the hypothesis, you
just need to find one black swan, right?
So, at best you cannot reject the null
hypothesis if all you find are white
swans, that doesn't mean that every swan
is white.
It just means that you haven't find a
black one yet.
And, and according to your data evidence,
you can't reject the statement that all
swans are white.
So, never say, I accept the Null
hypothesis.
Say, I can't reject it.
Okay.
So, hypothesis tests, test statistics
reject the null hypothesis when the test
statistic is big.
And, how do you determine if the test
statistics is big?
In statistics, we usually have something
called a rejection region.
Which are a range of values such that if
the test statistic is in the rejection
region, you determine that the test
statistic is big enough to reject the null
hypothesis.
And usually, a rejection region is defined
by a critical value such that if the
absolute value of the test statistic is
bigger than the critical value, then you,
you have data evidence to reject.
And if the test statistic is less than a
critical value, then, then you don't