So, the first kind of data descriptive
Statistics is going to be a graphical
statistics.
It's called the histogram.
In the histogram, the idea is we want to
describe the shape of the distribution of
the underlying data.
And so, how do we construct a histogram?
So, we order the data from largest to
smallest.
Then, we divide the range of the data
into, say, in equally spaced bins.
And then, we count the number of
observations in each of the bins and then
we create a bar graph of those counts.
And we might normalize the area so that
it's equal to one.
And so, we're going to get a, you know,
kind of a bar chart of the distribution of
data and we can think of this histogram as
a very crude estimate for the underlying
probability curve, alright?
So, in R, one of the reasons why R is very
nice is that it has all of these data
descriptive statistics built into the
language, right?
So, if we want to create a histogram, we
just use a function called hist and it
produces our histogram plot for us.
And we can change the number of bends and,
you know, do all sorts of things as an
option to the hist function.
And so, I encourage you, you know, just to
play around with it and, you know, and,
and see what you get.
So, let's look at some histograms of the
underlying data and try to get an idea of
what the distribution looks like, okay?
Now, before I do that I want to use
Gaussian white noise as a benchmark model
for the returns on Microsoft and the
returns on the S and P 500.
And we want to think about whether or not
the returns on Microsoft and the returns
on S and P 500 can be considered as a
realization from a Gaussian white noise
process, okay?
So, to help you do this thought experiment
I've plotted up here, the monthly
continuously compounded returns on
Microsoft.
This is the actual data.
Now here, what I've done is, I've created
a computer simulation of, of a Gaussian
white noise that has the same mean value
as Microsoft and it has the same
volatility, the same standard deviation as
Microsoft.
And I just created the 120 random draws
from that you know, Gaussian distribution,
that's calibrated to have the same mean
and standard deviation as Microsoft.
And so, here's the computer simulation and
here's Microsoft.
Now again, the differences between these
simulations are, you know, here it's the
volatility changing in the actual data is
not reflected in the Gaussian white noise
data because this always has the same
volatility all the way throughout the
process, okay?
So, that's one thing that's clearly
different between the simulated data and
the actual data.
And then, the question is well are the
things that are similar, the mean values
of these two series are the same by
construction?
And what about the empirical distribution?
Let's calculate the histogram of the
actual data and calculate the histogram of
the normal data and see if they look
similar.
So, here's a histogram plot of the monthly
returns on Microsoft, okay?
So this is done using the hist command in
R and I don't use any special options.
I use just all the defaults okay.
So, this is an estimate of the underlying
probability curve.
So, right away, one of things that you
should notice about this is that it looks
very similar to a bell-shaped
distribution, okay?
And so, this is one of the reasons why the
normal distribution is often used as a
benchmark because when you look at
histograms of returns they, they resemble
a normal distribution, okay?
Now, it's blocky because, you know, we're,
we're, we're dividing these things into
bins.
Now again, when we look at a histogram,
you can say, where is the histogram
centered, that's the mean return, the
volatility is the standard deviation in
the histogram, the skewness is the
asymmetry in the histogram, and the excess
kurtosis is telling us about how thick the
tails are relative to a normal
distribution, okay?
So, here, you know, if we think there
might be a slight long left tail.
So, if we're going to guess, it might have
a negative skewness.
And the tails, you know, kind of are, you
know, well it's hard to say.
Are they fatter than the normal
distribution?
It's hard to tell from the histogram by
itself.
But the fact that there is this kind of
big, outlying value, is a clue that
perhaps the kurtosis is bigger than three.
Okay.
Now, here is a histogram of the computer
simulated Gaussian data, okay?
This thing should look like a normal
curve, because it actually generated from
a normal distribution, okay?
It has the same mean as the returns on
Microsoft and the volatility is the same
volatility of Microsoft and notice the,
the range of the data.
So, it goes from, you know, -two to +two.
And, if we look at the actual Microsoft
data, this is going out to -0.4 to 0.4.
So, it's looking like the Microsoft
returns have fatter tails than, than the
Gaussian data, because of the, how far
they're going out in the actual data,
okay?
Now, if you look at the S and P 500.
So, here's the again, we use the hist
command on the S and P 500 monthly returns
and so it's, it's pretty, I mean, the
histogram is pretty blocky here, you know,
the default I could have probably used a
few more bins.
But I think the striking feature of the
histogram is the fact that it has a pretty
clear, long, left tail.
So, there's a pretty clear negative
skewness in this distribution and so, you
know, if you think, is the normal
distribution a good model for the S and P
500 returns, probably not, you know,
because of this, this big negative
skewness.
Now if you want to compare Microsoft to
the S and P 500, we would like to use the
same x-axis scale for the two return
series.
And so, what I did was I first computed
the histogram for Microsoft, and then I
used the bin ranges for Microsoft to
compute the histogram for the S and P 500
and the reason for doing that is you can
see now that the S and P 500, the data or
much more concentrated around zero than
for Microsoft.
Microsoft has a much larger spread above
the average than the S and P 500.
And so, that's just another way of saying
that the volatility of Microsoft is higher
than the volatility than, than the S and P
500.
I made the comment that I thought the
kurtosis of this was, was bigger than
three, that the kurtosis is bigger than
the standard normal.
And the reason is because the, in the
actual data, notice that this is, the left
tail is -40%, the right tail is +40%.
When I did the computer simulated Gaussian
data, the right tail is only going to
-twenty percent and +twenty%.
So, the Microsoft returns are out here in
the left, left tail, and are out here in
the right tail.
So, the actual data produce more extreme
values than the computer simulation from a
normal distribution.
So, that's why I'm saying it's probably
the case that the kurtosis of the actual
Microsoft data is going to be bigger than
three.
Alright.
So, when we look at histograms, his
histograms are blocky, okay?
Now, if we want to eliminate the
blockiness of a histogram, we can compute
what's called a smoothed histogram.
And so, this is a graph, of a smoothed, of
the smoothed histogram and it's created
with the R function called, R function
called density.
And another name for a smoothed histogram
is a kernel density estimate, okay?
And literally, what you're doing, you
know, if you think about it, think about
drawing a smooth curve over the data,
that's what the kernel density algorithm
does.
And how do you get a smooth curve
throughout the data?
Well, at a given point, what you do is you
take an kind of a weighted average of
values around a particular point where the
weights you know, kind of decline going
out in this direction and decline going
out in that direction.
And then essentially, what you do is you,
you take that kind of smooth and you pass
it all the way across the data and that
allows you to draw this kind of smooth
curve over the histogram.
The, the algorithm for computing the
smooth histogram, this kernel density
estimate, this is taken from Rupert is of
the following form.
So, if you want to estimate the, the
probability curve at a single point, then
what we do is we take a weighted average
where the weights are determined by what's
called a kernel function.
And a kernel function is very often just a
symmetric probability distribution.
And usually, the kernel function is the
standard normal curve.
And essentially, what you're doing is
you're, you're taking a weighted average
of, of the values in a histogram where
you're weighting the points with a normal
distribution before and after the given
point.
And that allows you to smooth the
blockiness of, of, of the histogram, okay?
The, when you use this kind of function
there's a, a parameter B that's called a
bandwidth parameter and that controls the
degree of smoothing.
So, if B is big, then you get a lot of
smoothing and if B is small then you don't
get very much smoothing, okay?
So, anyway, for our purposes, I just want
you to know that sometimes, it's, it's
easier to look at the smooth version of
the histogram to get a, a general view of
the shape, instead of the histogram by
itself.
And so, notice you get, when you do the
smooth, you get some sort of wiggles out
here, you get like little wiggle over
here.
And so, you know, does this look like a
normal curve?
Well, not exactly, you know, it's a, a, a
nice normal curve would be perfect
symmetric and bell-shaped, whereas you
know, this is a little bit less, less
perfect, okay?
So, we can overlay, for example, the
smooth histogram on top of the, the
histogram from Microsoft.
And so, we see how the smooth, you know,
sort of picks up this little bump, picks
up this little bump here, you know,
captures the basic shape of the
distribution in the middle, gets that
little bump here from these two things,
and so on.
So, we often, you know, will report a
histogram with the little smooth line on
top of it.
And again, you just want to get an idea of
what the shape of the distribution looks
like.