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All right.
So, let's talk a little bit about

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hypothesis testing.
Now, again, this is part of the, you know,

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pro, you know, probability and statistics
review.

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So, everyone in their previous statistics
class has had some exposure to hypothesis

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testing, I assume doing T tests and stuff
like that.

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And so, I just want to remind you a little
bit about the hypothesis testing

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methodology.
And then, mention, you know, what are some

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of the interesting hypotheses that we
could conduct in our constant expected

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return model.
So, so this is the basic idea with, with

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hypothesis testing.
So, with hypothesis testing, there's a

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decision to be made.
So, there is some hypothesis.

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Now, a hypothesis is some assumption about
your model.

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So, hypothesis could be that true mean is
equal to zero, okay?

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And then, so that, so you, you specify
something that's being maintained and

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that's called the null hypothesis.
So, 80 is some, maintain hypothesis like

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the, the true mean is equal to zero.
And then, there's an alternative

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hypothesis which is, you know, if, if, if
this isn't true, then, what else could be

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true.
So, the alternative could be, well, the

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mean is not equal to zero, okay?
And so, what you want to do is you want to

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test, you want to gather data, and you
want to see if the data is in favor of the

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null hypothesis, or if it's more in favor
of the alternative hypothesis.

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So, hypothesis testing is really about
making decisions, right?

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And, you know, in our case, in our model,
you know, what are the kinds of things

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that are interesting?
For example, we might want to test the

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hypothesis that our returns are normally
distributed, right?

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And, cuz returns are normally distributed
then, you know, it gives us an easy way to

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do certain calculations.
But if our data's not normally distributed

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then, you know, we should be searching for
better distributions.

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So, in finance, and particularly in
looking at return data, you know, that is

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an interesting hypothesis to, to look at.
Another interesting hypothesis is, you

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know, Are the parameters of our model
constant over time, right?

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Is the mean return constant over a five
year period, or is it changing?

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Is the correlation between two returns
constant over our observed data, or is it

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changing over time?
So, those are kind of interesting

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hypothesis that I think that we want to
look at and, and we would like to try to

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evaluate with data.
Alright.

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So, we're going to use the methodology of
hypothesis testing to do this.

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So, anytime, you know, you start with a
null and alternative hypothesis and then

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you're going to do a test.
You're going to gather data, and with your

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data you're going to construct a test
statistic, and the test statistic is going

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to give you data evidence in favor or
against the hypothesis.

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Now, because hypothesis involves,
hypothesis testing involves a decision,

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the decision is you either reject the null
hypothesis or you do not reject the

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hypothesis, okay?
So, think of, you know, in the context of

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like, drug trials, right?
You're going to either conclude that the

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drug cures cancer or it doesn't, right?
And, that's a very important decision.

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Because if you have a, found a drug that
cures cancer, right?

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You know?
You've you know, made a ton of money for

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your company and, and improve the world
and so on and so forth.

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But, if the drug doesn't cure cancer or do
anything, then you know, it should be

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thrown out and you should, you know, look
elsewhere.

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So, very often in hypothesis testing, the
decision is very important.

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I like to, to talk about hypothesis
testing often in a, a courtroom setting.

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So, think of yourself sitting on a jury,
and you have to decide whether or not

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someone's guilty or innocent.
And, and to make it more interesting,

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suppose you're on a murder trial.
And so, the idea is, you know, you're

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innocent until proven guilty.
So, the null hypothesis is the person on

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trial is not a serial killer.
But the alternative hypothesis is, the guy

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is a serial killer.
And you, as the jury, have to decide

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whether or not this guy's going to, to go
to the gas chamber or not, right?

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So, your decision has a real implication
to what's going on.

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So, you don't want to make an error.
So, in any decision, you know, there could

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an error that could be made.
It could be the case that this person is

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really innocent, right?
And then, an error would be is you convict

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the innocent guy, right?
That's rejecting the null hypothesis when

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the null hypothesis is true, right?
So, you're concluding that the guilt, the

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innocent person is guilty when in fact,
they're not guilty.

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That's a bad thing to do.
You don't want to send an innocent person

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to the gas chamber.
So, you want the probability that you

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reject the null hypothesis when it's in
fact true, to be very, very, very small.

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You, and in the, in the context of a court
room you want the evidence to be beyond

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the reasonable doubt, right?
And so, that's like saying you want this,

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this probability of rejecting the null
hypothesis when null hypothesis is true to

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be very small.
So, in, in statistical testing, what we

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call the level of the test is the
probability that we reject the null when

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it is in fact true.
And, and we would like that to be a small

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value, like one percent or five%.
Five percent can often be very big.

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Particularly, if you're on a murder trial,
beyond a reasonable doubt should be

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smaller than five percent probability.
[laugh] One should think that should be

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like 0.001 or something like that.
So, you, any test that you perform,

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there's a pre-specified tolerance for
committing this kind of error.

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And again, that's called the level of the
test.

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The next step in hypothesis testing is you
gather data, and you summarize the data

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evidence in the form of a test statistic.
And very often in statistics, a test

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statistic is something called a
t-statistic.

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And, and then, the test statistic
generally has the property that if the

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test statistic is big, then, that's data
evidence against the null hypothesis, and

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you should reject it, okay?
And if the test statistic is small, then

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that's data evidence in favor of the null
hypothesis and you shouldn't reject it.

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So again, it's like t is summarizing all
of the arguments from the defense and the

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prosecuting attorney.
And then, you know, it's synthesizing that

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evidence into some numerical value, and if
that numerical value is big, then you

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reject the null.
And if it's small, you don't reject.

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Now, also in the language of hypothesis
testing, you shouldn't use phrases like, I

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accept the no hypothesis, you know?
And, and from the logical point of view,

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accepting is sort of establishing the fact
that it is true.

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Well, all you can really do is evaluate
the data evidence against the null

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hypothesis.
It's sort of like this, trying to evaluate

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this statement, all swans are white,
right?

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So, you can't, in order to prove that all
swans are white, you have to find every

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swan that ever existed and show that is
white.

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In order to disprove the hypothesis, you
just need to find one black swan, right?

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So, at best you cannot reject the null
hypothesis if all you find are white

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swans, that doesn't mean that every swan
is white.

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It just means that you haven't find a
black one yet.

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And, and according to your data evidence,
you can't reject the statement that all

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swans are white.
So, never say, I accept the Null

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hypothesis.
Say, I can't reject it.

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Okay.
So, hypothesis tests, test statistics

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reject the null hypothesis when the test
statistic is big.

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And, how do you determine if the test
statistics is big?

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In statistics, we usually have something
called a rejection region.

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Which are a range of values such that if
the test statistic is in the rejection

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region, you determine that the test
statistic is big enough to reject the null

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hypothesis.
And usually, a rejection region is defined

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by a critical value such that if the
absolute value of the test statistic is

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bigger than the critical value, then you,
you have data evidence to reject.

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And if the test statistic is less than a
critical value, then, then you don't