So the box plot gives you these kind of
robust measures of a distribution.
So here's an illustration of the box plot
for the monthly returns on Microsoft.
And so this black line in the middle
represents the, the median.
This is roughly the interquartile range.
It's not exactly it and then if you look
at the online help for a box plot, you'll
see why.
Notice that this is not quite symmetric.
So by definition, the median is in the
middle of the interquartile range, but
This is so, so that these, these fences
are, the, the, the edges of the box are
not quite the, the.
The first and the third quartile, and then
you have, You know, essentially these
fences that, illustrate, you know, when
you have extreme outliers in the data.
So these points out over here, essentially
are representing those values that are,
you know, what, what we characterize as an
extreme outlier.
And this would be in the right tail, and
this would be in the left tail.
So you have a, the bulk of the
distribution is roughly symmetric.
You have two big negative outliers and you
have a few big positive outliers.
And again, if you look at the online help
for the box plot, ya know, they'll tell
you exactly how this fence and this fence
is determined.
From the point of view of interpretation,
the big power of box plots comes when you
compare multiple series.
Because it allows you to, you know, very
quickly look at the distribution of
multiple series in the same plot.
So, here's a, a box plot where I have the
box plot of the Gaussian white noise.
This is the simulated data that comes from
a normal distribution that has the same
mean invariance as the Microsoft returns.
These are the, this is the box plot for
the actual Microsoft returns and this is
the box plot for the SMP 500 returns.
And so we see that they're on the same
scale which is percent.
So the average return for all of these
things is pretty close to zero and a
little bit positive.
So, that's the median return.
We see that you know, again, Microsoft and
Guassian White Noise, the middle of the
distribution is, is roughly the same.
And, what we see with the Microsoft data
is, we have some outliers here.
With the Gaussian white noise data we
don't have the outliers because normal
distribution doesn't really produce
outliers.
And with the S and P 500 data, we see the
smaller spread than Microsoft, and we see
the small two negative outliers.
So again, we can get sort of an idea of
the shape of the distribution.
So it's similar to the histogram, but
again it's based on robust measures, and
it's quite useful for doing quick
comparisons of many series at once.
Now, I've put together a little four graph
summary for looking at the distribution of
asset returns.
And this four graph summary is borrowed
from a very nice book by Rene Carmona
called The Statistical Analysis of
Financial Data.
And in this four graph summary there's
going to be a histogram, a box plot, a
smooth histogram, and then a QQ plot
relative to a normal.
So we have four pictures and it looks like
this.
So, say for Microsoft, here we have the
histogram, here we have the smooth density
down here so you can get a rough shape of
the distribution.
And then, we have the box plot over here,
and the QQ plot relative to the normal.
And all of these, these three plots are
giving you similar information.
We see kind of a long left tail, a
negative skewness, so we see a little long
left tail over here.
We see the negative outliers in the box
plot, which is corresponding to these
observations here.
And in the Q-Q plot relative to normal,
we're seeing this drop-down relative to
the normal graph, and this dipping up a
little bit on the right hand side.
So if you look at, you know, a, a p-, a
summary graph like this, you can get an
idea that Microsoft returns are.
Kind of approximately normally distributed
in terms of the bulk of the distribution
looks normal.
But there is some negative skewness and
fatter tales relative to the normal
distribution.
Okay.
Alright.
So when we have two or more random
variables, then, we have, we might want to
look at some descriptive statistics that
tell us the relationship between two or
more variables.
Now, when we studied probability theory,
when we were looking at the dependence
between two random variables, we looked at
covariance and correlation.
Now, from the point of view of descriptive
statistics, we can, we have, say we have
two random variables X and Y.
And then we observe the sample X1, Y1, X2,
Y2.
So, we can think of X as the return on
Microsoft, Y as the return on the S and P
500; and then our sample is the data that
we download from Yahoo.
Okay?
We assume that these returns, you know,
follow a multi-, a bi-variant normal
distribution.
For example, as, as being kind of the
benchmark.
And then, say we wanna measure the
dependence between the Microsoft returns,
and the S and P 500 returns.
We'd wanna compute the, sample covariance
and the sample correlation, to get, an
idea of what the data say the relationship
between these returns are, okay.
Now, in, looking at pairwise
relationships, there's a graphical,
diagnost-, a graphical descriptive
statistic called a scatter plot.
And a scatter plot is just an x-y plot of
your bi-variant data.
So you can plot the returns on the SMP500
on one axis and returns of Microsoft on
the other.
And then you can see what the data looks
like.
So for example.
Here's a scatter plot of the monthly
returns on Microsoft versus the S and P
500.
So here, I, I put Microsoft on the X axis.
The S and P 500 on the Y axis.
And the black lines here represent.
This is the mean for Microsoft, this is
the mean for the S and P 500.
Okay?
So remember when we, we studied
co-variance and correlation, we, we looked
at these probability scatter plots.
And you know?
Essentially, when you look at this data,
you know?
What would you say, are, is there a
positive or negative relationship between
Microsoft and the S and P 500?
Positive, right?
Cuz as Microsoft returns go up, S and P
500 returns tend to go up as well.
As the Microsoft returns go down, the S
and P 500 returns are going down.
Okay?
So here we see, we see a negative
relationship.
And later on, we'll compute the sample
co-variance and correlation.
And we, there's a positive sample
co-variance and the sample correlation
is.6.
So again, there's a reasonably strong
positive linear association between these
two returns.
Okay?
Now a.
When you have more than two returns,
there's a nifty function in R called
pairs, P, A, I, R, S.
And what the pairs function does, it
creates all pair y scatter plots.
So here I have three data series, my
computer-simulated Gaussian white noise,
the returns on Microsoft, and the returns
in S and P 500.
And so now I have what's called a scatter
plot matrix.
And, so this graph right here, this is the
scatter plot with Gaussian white noise on
this axis and Microsoft on this axis.
And we see the scatter plot looks like a
shock and blast.
There appears to be no linear association
between them.
So we would assume that the covariance is
close to zero and the correlation is close
to zero, based on this plot.
This plot represents Gaussian White Noise
on this axis, and the S and P 500 on this
axis.
Okay?
And, again, this plot also kinda looks
like a s-, a shotgun blast, where it
doesn't appear to be any systematic
positive or negative relationship in the
data.
If anything, you know, there might be what
might, what looks, maybe, to be a slight
negative relationship in, in the data.
But it certainly isn't very strong.
So one might expect this sample covariance
to be slightly negative, and the
correlation to be a negative number
that's, but pretty close to zero.
This plot over here is just, we put
Microsoft on this axis and Gaussian white
noise on this axis.
So this plot is a, just slipping the axis
from this plot, okay?
And similarly, this, this plot down here
is flipping this plot.
And then finally our last plot shows
Microsoft on this axis, SMP 500 on this
axis.
So that's what I showed you before.
And we see positive linear relationships.
So, these two series are positively
correlated positive covariance and a
positive correlation.
So again, this is nice if you have ten
assets.
You can do this and then you can get a
very quick summary of what appears to be
the linear association between the
variables.
And you can also I mean again, cause these
are plots.
Even if there's a non linear association
you know, you would see, you could
possibly see that in, in the plot itself.
So, We often summarize the sample
variances and covariances in a matrix.
And actually before I get there let me
define the sample statistics.
So if you wanna compute the sample
covariance, then you would take your
sample of your first data series, the
sample of your second data series.
You compute this sample average of x minus
its mean, times y minus its mean.
And that's sometimes this is called at,
little s with a subscript xy, or sigma hat
xy.
So this is the sample covariance and then
the sample correlation is the sample
covariance divided by the product of the
sample standard deviations.
So if we're working in R, the bar function
computes sample covariance matrix.
So if you so if we have three assets, and
we have a three by three matrix, then have
the variances along the diagonals and the
covariances on the off-diagonals.
That's what the bar function computes.
Notice that there isn't a, actually, there
is a co function, but since with the co
function, does it same thing as, as a bar.
The core function COR, gives you the
sample correlation matrix.
So, here.
So if I take my three data series, my
Gaussian white noise, my Microsoft, and my
S and P 500.
I use the bar command.
And then this is the variance/covariance
matrix.
The sample variances are along the
diagonals.
And the covariances are on the off
diagonals.
So, so notice that we see a negative
sample covariance between Microsoft and
Gaussian white noise and a negative
covariance between Microsoft, sorry,
between the S and P 500 and the Gaussian
white noise.
And we have a positive covariance between
the S and P 500 and Microsoft.
Now, covariance is just direction of
association, correlation of strength.
So when you look at the correlation
matrix, you have ones along the diagonal
because the correlation with each series
with itself by definition is one.
And so here's the correlation between the
Gaussian white noise and the Microsoft,
that's -.19 Not very strong.
Correlation between Gaussian white noise
and S and P 500 is -.24.
Now this is a completely spurious
correlation, because the Gaussian white
noise was computer generated.
Right?
It has nothing to do with the actual S and
P 500 data, but we still had a number
that's, you know, kinda large and
negative, and that's just by chance.
Right?
If I simulated another Gaussian white
noise, then this correlation, probably
close to zero or could be a little
positive, or something like that.
The Microsoft and S and P 500 data has a
reasonably strong correlation, that's .6.
The last descriptive statistic I want to
talk about is, has to do with time series
dependence.
So again we were covering probability
theory, we went through the time series
concepts section, and we were trying to
think about time dependence in data.
And so we defined what are called
autocorrelations.
We defined the covariance between y(t) and
y(t-1), the correlation between y(t) and
y(t-1) as well.
Well, we can compute sample versions of
these quantities.
So given, the time series of data.
If, and, and if we're interested in
determining if there's linear dependence
over time.
Then we can compute the sample auto
covariance and the sample auto
correlation, and see if these things are
different from zero.
So the sample autocovariance is just at
lag J, is just the sample covariance
between XT and XT minus J.
And the sample auto correlation is just
this sample auto co-variance divided by
the sample variance, and these are
measures of linear dependence between a
variable and its lags, and then we can do
a graphical plot called the sample auto
correlation function, where we just plot
this sample auto correlation against the
lag.
So, notice that when you calculate this
sum.
We start the summation at J plus one,
right, because we're looking at the
summation.
So, the first index in this sum is J plus
one, so we cut, we look at here.
That's going to be X, J plus one minus the
mean.
And then we have XJ + one - J.
So that's X1.
Right?
Minus the mean.
So we're looking at the relationship
between XJ and J lags from XJ.
So from the first observation to the Jth
observation.
Then the next term in the sum is XJ + one.
And then this would be X2, and so on.
So we're looking at the relationship
between, you know, X at time T and its
lag.
So if I guess if you draw a picture.
Right?
So we have you know, one, two, three,
four, you know, up to J, and so this would
be XJ, and then we have X1, and then we
have XJ plus one X2.
So when we're calculating this sum, so
we're looking at the relationship between
this variable and this variable, this
variable and that variable, and in terms
of the computation of this.
If we started this at zero.
Then this would be XO.
Or say, if we started this at one, then
this would be X1.
But this would be X1-J.
And we don't have data, for that.
So, this, this notation is to emphasize
that we have to do this computation based
on the data sample that we actually
observed.
Okay, so here's an example of sample auto
correlations.
So this is, these are plots that are
measuring the estimated time dependence in
the data.
And the top graph is for the Gaussian
white noise.
And the middle graph is for Microsoft.
The bottom graph is for the S and P 500.
And this vehicular is being plotted here.
So, this is I guess, lag one, lag two, lag
three, lag four.
So for Gaussian white noise, the estimated
correlation between xt and xt minus one
is, is actually zero.
It seems, you're just, because you're not
seeing here.
The estimated correlation between xt and
xt minus two is a small negative number.
So the scale here this is minus 0.15.
So this value is you know like minus 0.05.
So this is a very small number.
Now on these graphs are blue dotted lines,
okay.
These blue dotted lines are thresholds to
determine whether or not these values are
statistically different from zero.
Alright.
After the midterm I'll explain where these
blue dotted lines come from.
They're essentially based upon 95 percent
confidence interval for these estimates.
So when you look at this graph, how you're
supposed to read it, is if any of the
estimated auto correlations extend beyond
the blue line, then those auto
correlations are thought to be
statistically different from zero at the
at with, with 95 percent confidence.
And so for the Gaussian white noise, we
see none of these auto- correlations lie
outside of the blue dotted lines.
So we see no evidence of time dependence
in the data.
And again, that should be the case.
Because the Gaussian white noise is
computer simulated with no time dependence
in the data.
On the other hand, if we look at the
Microsoft returns, we see that the first
return is negative, and it's about -.2.
So this is the correlation between the
return in month T and the return in month
T -one.
And notice that it's negative.
And it extends beyond, beyond the blue
dotted line.
So we would view this as being
statistically significant.
So there appears to be a negative
correlation between the return this month
and the return last month.
Okay?
Now, the other auto-correlations.
So this is the, return, the
autocorrelation between the return at
month T and the return at month T minus
two.
So this is the lag two autocorrelation,
the lag three autocorrelation and so on.
Here, notice that the lag two is not
beyond the blue dotted line but the lag
three is slightly positive, 'kay.
So if we run through the Microsoft data
and we view this graph we would say there
appears to be some evidence for time
dependence in the data, you know.
But if we, then, if you look at the S and
P 500 data.
We see none of the sample
auto-correlations are outside of the blue
dotted line.
So for the S and P 500 data, there appears
to be no evidence for time-dependence in,
in the date.
In statistical analysis you should never
say there is, right, because you're, in
statistics you're never certain, you're
never 100 percent certain of anything.
Right?
All you can say, and you want to think of
yourself as, like, sitting on the jury
evaluating evidence, right?
Is the evidence in favor of, or is the
evidence against, something, right?
So you would interpret this as, there's
data evidence in favor of some time
dependence in the Microsoft returns.
When we look at the Microsoft data, just
how do you interpret this negative
correlation?
And it's literally saying that, if the
return this month is positive.
Then, there's a tendency for the return
next month to be negative.
Right, so there's kinda of a reversal
that's going on here.
>> And vice versa.
>> And vice versa, and if the return is
negative this month there is a tendency
for the return to be positive next month.
Right, so, one could ask the question,
what could be causing such a reversal?
Now, in finance there is some explanation
for the reversal effect in asset returns
that's known as the bid-ass bounce.
I'll come back to that a little bit later
on.
But that's, that's one explanation.
It's, it's not a very good explanation for
monthly data.
It's, it's a better explanation if, if you
have say, asset returns computed every
minute.
But for monthly data, it's a bit of a
mystery.
You know, why is this negative correlation
here?
Okay, so to summarize, I like to call
these stylized facts.
So in, in this section we looked at three
assets.
We looked at Microsoft, the S and P five,
actually we only looked at two assets.
We looked at Microsoft and we looked at
the S and P 500.
And of course, one is always tempted to
try to make sweeping generalizations based
upon, or, maybe I should rephrase this.
You should be careful not to make sweeping
generalizations based on the analysis of
two assets.
But, you know, I've, I've analyzed a lot
more than two assets, and others have
analyzed a lot more than two assets.
And, so when you look at monthly
continuously compounded returns, these are
the results that you tend to find when you
look at, you know, lots and lots and lots
of different assets.
At the monthly basis, returns appear to be
approximately normally distributed, okay?
They don't follow the normal distribution
exactly, But the normal distribution is
not a horrible mistake.
All right?
There's some noticeable negative skewness
and excess kurtosis in typical assets,
right?
If you look at bivariate relationships,
many assets are contemporaneously
correlated, that is, and they tend to be
positively correlated.
And so, and that's one of the things as
well, if you look at temporal dependence
in the data.
At the monthly level there is not a whole
lot of evidence for strong temporal
dependence, so assets are approximately
uncorrelated over time.
Then again when you look at many assets,
the Microsoft data showed there was some
negative dependence on one lag, but when
you looked at, you know, other lags there
was not much that was there.
So these are broad, what I call stylized
facts and you want to keep these sort of
in the back of your mind.
Stylized facts are useful at the model
building phase.
So if you want to build a probability
model for asset returns, we want our
probability model to capture the basic
stylized facts of the data, okay.
If our model doesn't capture the basic
stylized facts, it's not a good model.
And we're always gonna keep that in mind
as we look at particular models in this
class.
We always wanna ask ourselves, you know,
if we you know, simulate data from our
model, does it look like real data?