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PROFESSOR: All right.

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Let's see.

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We're going to start
today with a wrap

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up of our discussion of
univariate time series

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analysis.

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And last time we went through
the Wold representation

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theorem, which
applies to covariance

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stationary processes, a
very powerful theorem.

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And implementations of
the covariance stationary

00:00:44.880 --> 00:00:47.700
processes with ARMA models.

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And we discussed
estimation of those models

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with maximum likelihood.

00:00:54.840 --> 00:00:57.010
And here in this
slide I just wanted

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to highlight how when
we estimate models

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with maximum likelihood
we need to have

00:01:03.340 --> 00:01:07.280
an assumption of a probability
distribution for what's random,

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and in the ARMA structure
we consider the simple case

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where the innovations,
the eta_t,

00:01:14.120 --> 00:01:17.437
are normally
distributed white noise.

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So they're independent and
identically distributed

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normal random variables.

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And the likelihood
function can be

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maximized at the maximum
likelihood parameters.

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And it's simple to implement
the limited information maximum

00:01:34.450 --> 00:01:38.200
likelihood where one conditions
on the first few observations

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in the time series.

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If you look at the likelihood
structure for ARMA models,

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the density of an outcome
at a given time point

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depends on lags of that
dependent variable.

00:01:53.700 --> 00:01:58.000
So if those are unavailable,
then that can be a problem.

00:01:58.000 --> 00:02:01.930
One can implement limited
information maximum likelihood

00:02:01.930 --> 00:02:04.680
where you're just conditioning
on those initial values,

00:02:04.680 --> 00:02:07.740
or there are full information
maximum likelihood methods

00:02:07.740 --> 00:02:09.750
that you can apply as well.

00:02:09.750 --> 00:02:13.170
Generally though the
limited information case

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is what's applied.

00:02:16.300 --> 00:02:18.800
Then the issue is
model selection.

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And with model
selection the issues

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that arise with time
series are issues

00:02:23.870 --> 00:02:27.480
that arise in fitting any
kind of statistical model.

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Ordinarily one will
have multiple candidates

00:02:30.040 --> 00:02:32.270
for the model you
want to fit to data.

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And the issue is how
do you judge which

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ones are better than others.

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Why would you prefer
one over the other?

00:02:38.700 --> 00:02:43.230
And if we're considering a
collection of different ARMA

00:02:43.230 --> 00:02:49.670
models then we could say, fit
all ARMA models of order p,q

00:02:49.670 --> 00:02:53.100
with p and q varying
over some range.

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p from 0 up to p_max,
q from q up to q_max.

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And evaluate those
p,q different models.

00:03:02.270 --> 00:03:05.570
And if we consider sigma
tilde squared of p,

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q being the MLE of
the error variance,

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then there are these
model selection criteria

00:03:12.490 --> 00:03:14.150
that are very popular.

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Akaike information criterion,
and Bayes information

00:03:16.450 --> 00:03:18.060
criterion, and Hannan-Quinn.

00:03:18.060 --> 00:03:22.850
Now these criteria all
have the same term,

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log of the MLE of
the error variance.

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So these criteria don't
vary at all with that.

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They just vary with
this second term,

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but let's focus first
on the AIC criterion.

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A given model is
going to be better

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if the log of the MLE for the
error variance is smaller.

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Now is that a good thing?

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Meaning, what is
the interpretation

00:03:50.186 --> 00:03:52.560
of that practically when you're
fitting different models?

00:03:55.720 --> 00:03:58.060
Well, the practical
interpretation

00:03:58.060 --> 00:04:02.980
is the variability of the
model about where you're

00:04:02.980 --> 00:04:05.380
predicting things, our
estimate of the error variance

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is smaller.

00:04:06.120 --> 00:04:09.940
So we have essentially a
model with a smaller error

00:04:09.940 --> 00:04:12.300
variance is better.

00:04:12.300 --> 00:04:15.720
So we're trying to minimize
the log of that variance.

00:04:15.720 --> 00:04:18.040
Minimizing that is a good thing.

00:04:18.040 --> 00:04:23.095
Now what happens when
you have many sort

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of independent variables
to include in a model?

00:04:26.120 --> 00:04:28.930
Well, if you were doing a
Taylor series approximation

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of a continuous function,
eventually you'd

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sort of get to probably
the smooth function

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with enough terms, but suppose
that the actual model, it does

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have a finite number
of parameters.

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And you're considering
new factors,

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new lags of
independent variables

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in the autoregressions.

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As you add more
and more variables,

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well, there really
should be a penalty

00:04:55.480 --> 00:05:00.140
for adding extra variables
that aren't adding

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real value to the model in terms
of reducing the error variance.

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So the Akaike
information criterion

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is penalizing different
models by a factor that

00:05:11.020 --> 00:05:14.490
depends on the size of the model
in terms of the dimensionality

00:05:14.490 --> 00:05:17.380
of the model parameters.

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So p plus q is
the dimensionality

00:05:19.340 --> 00:05:24.020
of the autoregression model.

00:05:24.020 --> 00:05:29.770
So let's see.

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With the BIC criterion the
difference between that

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and the AIC criterion is
that this factor two is

00:05:39.030 --> 00:05:43.090
replaced by log n.

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So rather than having a sort
of unit increment of penalty

00:05:49.120 --> 00:05:54.460
for adding an extra parameter,
the Bayes information criterion

00:05:54.460 --> 00:05:59.730
is adding a log n penalty
times the number of parameters.

00:05:59.730 --> 00:06:03.700
And so as the sample size
gets larger and larger,

00:06:03.700 --> 00:06:07.030
that penalty gets
higher and higher.

00:06:07.030 --> 00:06:12.610
Now the practical interpretation
of the Akaike information

00:06:12.610 --> 00:06:18.140
criterion is that it is
very similar to applying

00:06:18.140 --> 00:06:21.520
a rule which says, we're
going to include variables

00:06:21.520 --> 00:06:28.740
in our model if the square of
the t statistic for estimating

00:06:28.740 --> 00:06:34.850
the additional parameter in the
model is greater than 2 or not.

00:06:34.850 --> 00:06:41.710
So in terms of when does the
Akaike information criterion

00:06:41.710 --> 00:06:45.670
become lower from adding
additional terms to a model?

00:06:45.670 --> 00:06:49.150
If you're considering two models
that differ by just one factor,

00:06:49.150 --> 00:06:52.530
it's basically if the t
statistic for the model

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coefficient on that factor is a
squared value greater than two

00:06:56.890 --> 00:06:57.830
or not.

00:06:57.830 --> 00:07:03.310
Now many of you who have
seen regression models before

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and applied them, in
particular applications

00:07:05.560 --> 00:07:08.270
would probably
say, I really don't

00:07:08.270 --> 00:07:11.590
believe in the value
of an additional factor

00:07:11.590 --> 00:07:15.590
unless the t statistic
is greater than 1.96,

00:07:15.590 --> 00:07:18.410
or 2 or something.

00:07:18.410 --> 00:07:20.320
But the Akaike
information criterion

00:07:20.320 --> 00:07:22.410
says the t statistic
should be greater

00:07:22.410 --> 00:07:24.450
than the square root of 2.

00:07:24.450 --> 00:07:27.290
So it's sort of a weaker
constraint for adding variables

00:07:27.290 --> 00:07:28.510
into the model.

00:07:28.510 --> 00:07:31.551
And now why is it called
an information criterion?

00:07:31.551 --> 00:07:33.050
I won't go into
this in the lecture.

00:07:33.050 --> 00:07:34.940
I am happy to go into
it during office hours,

00:07:34.940 --> 00:07:38.085
but there's notions
of information theory

00:07:38.085 --> 00:07:41.480
and Kullback-Leibler
information of the model

00:07:41.480 --> 00:07:43.500
versus the true
model, and trying

00:07:43.500 --> 00:07:47.730
to basically maximize the
closeness of our fitted model

00:07:47.730 --> 00:07:48.960
to that.

00:07:48.960 --> 00:07:50.770
Now the Hannan-Quinn
criterion, let's

00:07:50.770 --> 00:07:52.270
just look at how that differs.

00:07:52.270 --> 00:07:57.480
Well, that basically has a
penalty midway between the log

00:07:57.480 --> 00:07:58.460
n and two.

00:07:58.460 --> 00:08:01.320
It's 2*log(log n).

00:08:01.320 --> 00:08:03.890
So this has a penalty that's
increasing with size n,

00:08:03.890 --> 00:08:07.680
but not as fast as log n.

00:08:07.680 --> 00:08:11.480
This becomes
relevant when we have

00:08:11.480 --> 00:08:16.240
models that get to be very large
because we have a lot of data.

00:08:16.240 --> 00:08:17.780
Basically the more
data you have,

00:08:17.780 --> 00:08:19.860
the more parameters
you should be

00:08:19.860 --> 00:08:22.220
able to incorporate in
the model if they're

00:08:22.220 --> 00:08:27.360
sort of statistically valid
factors, important factors.

00:08:27.360 --> 00:08:29.960
And the Hannan-Quinn
criterion basically

00:08:29.960 --> 00:08:36.179
allows for modeling processes
where really an infinite number

00:08:36.179 --> 00:08:38.870
of variables might
be appropriate,

00:08:38.870 --> 00:08:40.860
but you need larger
and larger sample sizes

00:08:40.860 --> 00:08:44.640
to effectively estimate those.

00:08:44.640 --> 00:08:51.530
So those are the criteria that
can be applied with time series

00:08:51.530 --> 00:08:52.330
models.

00:08:52.330 --> 00:08:55.210
And I should point
out that, let's see,

00:08:55.210 --> 00:08:59.720
if you took sort of
this factor 2 over n

00:08:59.720 --> 00:09:03.530
and inverted it to n over
two log sigma squared,

00:09:03.530 --> 00:09:07.560
that term is basically one of
the terms in the likelihood

00:09:07.560 --> 00:09:09.190
function of the fitted model.

00:09:09.190 --> 00:09:11.320
So you can see how this
criterion is basically

00:09:11.320 --> 00:09:16.280
manipulating the
maximum likelihood

00:09:16.280 --> 00:09:21.845
value by adjusting it for a
penalty for extra parameters.

00:09:28.080 --> 00:09:28.590
Let's see.

00:09:28.590 --> 00:09:29.360
OK.

00:09:29.360 --> 00:09:31.595
Next topic is just
test for stationarity

00:09:31.595 --> 00:09:32.470
and non-stationarity.

00:09:35.540 --> 00:09:40.870
There's a famous test called
the Dickey-Fuller test, which

00:09:40.870 --> 00:09:47.550
is essentially to evaluate
the time series to see if it's

00:09:47.550 --> 00:09:49.010
consistent with a random walk.

00:09:49.010 --> 00:09:51.490
We know that we've been
discussing sort of lecture

00:09:51.490 --> 00:09:56.880
after lecture how simple random
walks are non-stationary.

00:09:56.880 --> 00:10:02.960
And the simple random walk is
given by the model up here,

00:10:02.960 --> 00:10:05.920
x_t equals phi
x_(t-1) plus eta_t.

00:10:05.920 --> 00:10:09.345
If phi is equal
to 1, right, that

00:10:09.345 --> 00:10:11.740
is a non-stationary process.

00:10:11.740 --> 00:10:14.190
Well, in the
Dickey-Fuller test we

00:10:14.190 --> 00:10:17.640
want to test whether
phi equals 1 or not.

00:10:17.640 --> 00:10:23.090
And so we can fit the AR(1)
model by least squares

00:10:23.090 --> 00:10:29.110
and define the test statistic to
be the estimate of phi minus 1

00:10:29.110 --> 00:10:33.667
over its standard error where
phi is the least squares

00:10:33.667 --> 00:10:36.250
estimate and the standard error
is the least squares estimate,

00:10:36.250 --> 00:10:39.630
the standard error of that.

00:10:39.630 --> 00:10:44.970
If our coefficient phi is
less than 1 in modulus,

00:10:44.970 --> 00:10:47.430
so this really is a
stationary series,

00:10:47.430 --> 00:10:56.080
then the estimate phi converges
in distribution to a normal 0,

00:10:56.080 --> 00:10:58.950
1 minus phi squared.

00:10:58.950 --> 00:11:04.920
And let's see.

00:11:04.920 --> 00:11:09.990
But if phi is equal
to 1, OK, so just

00:11:09.990 --> 00:11:12.450
to recap that second
to last bullet point

00:11:12.450 --> 00:11:17.580
is basically the property that
when norm phi is less than 1,

00:11:17.580 --> 00:11:22.510
then our least squares
estimates are asymptotically

00:11:22.510 --> 00:11:26.420
normally distributed
with mean 0 if we

00:11:26.420 --> 00:11:29.740
normalize by the true value,
and 1 minus phi squared.

00:11:29.740 --> 00:11:34.220
If phi is equal to
1, then it turns out

00:11:34.220 --> 00:11:38.470
that phi hat is super-consistent
with rate 1 over t.

00:11:38.470 --> 00:11:44.520
Now this super-consistency
is related

00:11:44.520 --> 00:11:50.900
to statistics converging
to some value,

00:11:50.900 --> 00:11:55.590
and what is the rate of
convergence of those statistics

00:11:55.590 --> 00:11:56.490
to different values.

00:11:56.490 --> 00:12:04.840
So in normal samples we can
estimate sort of the mean

00:12:04.840 --> 00:12:06.440
by the sample mean.

00:12:06.440 --> 00:12:13.960
And that will converge to
the true mean at rate of 1

00:12:13.960 --> 00:12:14.570
over root n.

00:12:17.660 --> 00:12:23.070
When we have a
non-stationary random walk,

00:12:23.070 --> 00:12:28.890
the independent
variables matrix is such

00:12:28.890 --> 00:12:35.250
that X transpose X over
n grows without bound.

00:12:35.250 --> 00:12:42.270
So if we have y is equal
to X beta plus epsilon,

00:12:42.270 --> 00:12:46.750
and beta hat is equal to
X transpose X inverse X

00:12:46.750 --> 00:12:55.940
transpose y, the
problem is-- well,

00:12:55.940 --> 00:13:01.080
and beta hat is
distributed as ultimately

00:13:01.080 --> 00:13:03.500
normal with mean beta
and variance sigma

00:13:03.500 --> 00:13:07.430
squared, X transpose X inverse.

00:13:07.430 --> 00:13:10.160
This X transpose
X inverse matrix,

00:13:10.160 --> 00:13:14.750
when the process is
non-stationary, a random walk,

00:13:14.750 --> 00:13:15.650
it grows infinitely.

00:13:19.320 --> 00:13:23.980
X transpose X over
n actually grows

00:13:23.980 --> 00:13:31.280
to infinity in magnitude just
because it becomes unbounded.

00:13:31.280 --> 00:13:34.560
Whereas X transpose X over
n, when it's stationary

00:13:34.560 --> 00:13:36.130
is bounded.

00:13:36.130 --> 00:13:39.210
So anyway, so that leads
to the super-consistency,

00:13:39.210 --> 00:13:41.930
meaning that it converges
to the value much faster

00:13:41.930 --> 00:13:44.590
and so this normal
distribution isn't appropriate.

00:13:44.590 --> 00:13:47.730
And it turns out there's
Dickey-Fuller distribution

00:13:47.730 --> 00:13:50.910
for this test statistic,
which is based on integrals

00:13:50.910 --> 00:13:55.150
of diffusions and one
can read about that

00:13:55.150 --> 00:14:01.990
in the literature on unit roots
and test for non-stationarity.

00:14:01.990 --> 00:14:05.760
So there's a very rich
literature on this problem.

00:14:05.760 --> 00:14:12.470
If you're into econometrics,
basically a lot of time's

00:14:12.470 --> 00:14:15.870
been spent in that
field on this topic.

00:14:15.870 --> 00:14:22.370
And the mathematics gets
very, very involved,

00:14:22.370 --> 00:14:26.190
but good results are available.

00:14:26.190 --> 00:14:30.230
So let's see an application
of some of these time series

00:14:30.230 --> 00:14:30.730
methods.

00:14:37.550 --> 00:14:40.740
Let me go to the
desktop here if I can.

00:14:40.740 --> 00:14:46.690
In this supplemental material
that'll be on the website,

00:14:46.690 --> 00:14:49.540
I just wanted you
to be able to work

00:14:49.540 --> 00:14:51.150
with time series,
real time series

00:14:51.150 --> 00:14:53.250
and implement these
autoregressive moving

00:14:53.250 --> 00:14:58.220
average fits and understand
basically how things work.

00:14:58.220 --> 00:15:03.440
So in this, it introduces
loading the R libraries

00:15:03.440 --> 00:15:06.190
and Federal Reserve data into
R, basically collecting it

00:15:06.190 --> 00:15:07.370
off the web.

00:15:07.370 --> 00:15:11.300
Creating weekly and monthly
time series from a daily series,

00:15:11.300 --> 00:15:14.160
and it's a trivial thing to do,
but when you sit down and try

00:15:14.160 --> 00:15:16.460
to do it gets involved.

00:15:16.460 --> 00:15:20.210
So there's some nice
tools that are available.

00:15:20.210 --> 00:15:22.080
There's the ACF
and the PACF, which

00:15:22.080 --> 00:15:25.710
is the auto-correlation
function and the partial

00:15:25.710 --> 00:15:29.030
auto-correlation
function, which are

00:15:29.030 --> 00:15:30.690
used for interpreting series.

00:15:30.690 --> 00:15:35.350
Then we conduct Dickey-Fuller
test for unit roots

00:15:35.350 --> 00:15:39.920
and determine, evaluate
stationarity, non-stationarity

00:15:39.920 --> 00:15:42.170
of the 10-year yield.

00:15:42.170 --> 00:15:48.010
And then we evaluate
stationarity and cyclicality

00:15:48.010 --> 00:15:51.240
in the fitted autoregressive
model of order 2

00:15:51.240 --> 00:15:53.350
to monthly data.

00:15:53.350 --> 00:15:56.940
And actually 1.7 there,
that cyclicality issue,

00:15:56.940 --> 00:15:59.980
relates to one of the
problems on the problem set

00:15:59.980 --> 00:16:02.850
for time series,
which is looking at,

00:16:02.850 --> 00:16:06.030
with second order
autoregressive models,

00:16:06.030 --> 00:16:10.510
is there cyclicality
in the process?

00:16:10.510 --> 00:16:12.890
And then finally
looking at identifying

00:16:12.890 --> 00:16:16.960
the best autoregressive model
using the AIC criterion.

00:16:16.960 --> 00:16:21.500
So let me just page through
and show you a couple of plots

00:16:21.500 --> 00:16:22.470
here.

00:16:22.470 --> 00:16:22.970
OK.

00:16:22.970 --> 00:16:26.379
Well, there's the
original 10-year yield

00:16:26.379 --> 00:16:28.170
collected directly from
the Federal Reserve

00:16:28.170 --> 00:16:32.360
website over a 10 year period.

00:16:32.360 --> 00:16:34.600
And, oh, here we go.

00:16:34.600 --> 00:16:35.540
This is nice.

00:16:35.540 --> 00:16:36.040
OK.

00:16:42.580 --> 00:16:43.730
OK.

00:16:43.730 --> 00:16:46.870
Let's see, this
section 1.4 conducts

00:16:46.870 --> 00:16:49.930
the Dickey-Fuller test.

00:16:49.930 --> 00:17:03.080
And it basically
determines that the p-value

00:17:03.080 --> 00:17:06.420
for non-stationarity
is not rejected.

00:17:06.420 --> 00:17:12.819
And so, with the augmented
Dickey-Fuller test,

00:17:12.819 --> 00:17:15.089
the test statistic is computed.

00:17:15.089 --> 00:17:19.849
Its significance is
evaluated by the distribution

00:17:19.849 --> 00:17:21.760
for that statistic.

00:17:21.760 --> 00:17:24.790
And the p-value tells
you how extreme the value

00:17:24.790 --> 00:17:28.910
of the statistic is,
meaning how unusual is it.

00:17:28.910 --> 00:17:33.950
The smaller the p-value, the
more unlikely the value is.

00:17:33.950 --> 00:17:35.910
The p-value is what's
the likelihood of getting

00:17:35.910 --> 00:17:39.690
as extreme or more extreme a
value of the test statistic,

00:17:39.690 --> 00:17:41.150
and the test
statistic is evidence

00:17:41.150 --> 00:17:43.075
against the null hypothesis.

00:17:43.075 --> 00:17:48.850
So in this case the p-values
range basically 0.2726

00:17:48.850 --> 00:18:00.760
for the monthly data, which
says that basically there

00:18:00.760 --> 00:18:03.345
is evidence of a unit
root in the process.

00:18:06.530 --> 00:18:08.980
Let's see.

00:18:08.980 --> 00:18:09.480
OK.

00:18:09.480 --> 00:18:10.896
There's a section
on understanding

00:18:10.896 --> 00:18:12.815
partial auto-correlation
coefficients.

00:18:16.740 --> 00:18:20.180
And let me just state what
the partial correlation

00:18:20.180 --> 00:18:21.010
coefficients are.

00:18:21.010 --> 00:18:22.676
You have the
auto-correlation functions,

00:18:22.676 --> 00:18:25.850
which are simply the
correlations of the time

00:18:25.850 --> 00:18:28.190
series with lags of its values.

00:18:28.190 --> 00:18:30.020
The partial
auto-correlation coefficient

00:18:30.020 --> 00:18:36.640
is the correlation that's
between the time series

00:18:36.640 --> 00:18:42.180
and say, it's p-th lag that is
not explained by all lags lower

00:18:42.180 --> 00:18:42.720
than p.

00:18:42.720 --> 00:18:45.690
So it's basically the
incremental correlation

00:18:45.690 --> 00:18:50.460
of the time series variable with
the p-th lag after controlling

00:18:50.460 --> 00:18:51.540
for the others.

00:18:55.650 --> 00:18:57.420
And then let's see.

00:18:57.420 --> 00:19:01.480
With this, in section
eight here there's

00:19:01.480 --> 00:19:07.220
a function in R called ar, for
autoregressive, which basically

00:19:07.220 --> 00:19:11.170
will fit all autoregressive
models up to a given order

00:19:11.170 --> 00:19:14.230
and provide diagnostic
statistics for that.

00:19:14.230 --> 00:19:18.110
And here is a plot of the
relative AIC statistic

00:19:18.110 --> 00:19:20.640
for models of the monthly data.

00:19:20.640 --> 00:19:25.100
And you can see that basically
it takes all the AIC statistics

00:19:25.100 --> 00:19:28.950
and subtracts the smallest
one from all the others.

00:19:28.950 --> 00:19:33.495
So one can see that according
to the AIC statistic

00:19:33.495 --> 00:19:40.110
a model of order seven is
suggested for this treasury

00:19:40.110 --> 00:19:40.892
yield data.

00:19:43.670 --> 00:19:46.140
OK.

00:19:46.140 --> 00:19:49.500
Then finally because these
autoregressive models

00:19:49.500 --> 00:19:52.920
are implemented with
regression models,

00:19:52.920 --> 00:19:56.780
one can apply
regression diagnostics

00:19:56.780 --> 00:20:02.180
that we had introduced earlier
to look at those data as well.

00:20:02.180 --> 00:20:04.140
All right.

00:20:04.140 --> 00:20:07.495
So let's go down now.

00:20:14.978 --> 00:20:16.125
[INAUDIBLE]

00:20:16.125 --> 00:20:16.625
OK.

00:20:25.770 --> 00:20:27.970
[INAUDIBLE]

00:20:27.970 --> 00:20:28.660
Full screen.

00:20:28.660 --> 00:20:31.170
Here we go.

00:20:31.170 --> 00:20:31.670
All right.

00:20:36.700 --> 00:20:41.070
So let's move on to the
topic of volatility modeling.

00:20:44.350 --> 00:20:50.290
The discussion in
this section is

00:20:50.290 --> 00:20:53.640
going to begin with just
defining volatility.

00:20:53.640 --> 00:20:56.450
So we know what
we're talking about.

00:20:56.450 --> 00:21:01.740
And then measuring volatility
with historical data

00:21:01.740 --> 00:21:05.190
where we don't really apply sort
of statistical models so much,

00:21:05.190 --> 00:21:07.810
but we're concerned with
just historical measures

00:21:07.810 --> 00:21:10.180
of volatility and
their prediction.

00:21:10.180 --> 00:21:11.450
Then there are formal models.

00:21:11.450 --> 00:21:14.230
We'll introduce Geometric
Brownian Motion, of course.

00:21:14.230 --> 00:21:17.080
That's one of the standard
models in finance.

00:21:17.080 --> 00:21:18.710
But also Poisson
jump-diffusions,

00:21:18.710 --> 00:21:22.240
which is an extension of
Geometric Brownian Motion

00:21:22.240 --> 00:21:24.300
to allow for discontinuities.

00:21:24.300 --> 00:21:28.410
And then there's a property
of these Brownian motion

00:21:28.410 --> 00:21:30.860
and jump-diffusion
models which is models

00:21:30.860 --> 00:21:33.400
with independent increments.

00:21:33.400 --> 00:21:43.620
Basically you have disjoint
increments of the process,

00:21:43.620 --> 00:21:45.750
basically are independent
of each other, which

00:21:45.750 --> 00:21:51.270
is a key property when there's
time dependence in the models.

00:21:51.270 --> 00:21:54.040
There can be time dependence
actually in the volatility.

00:21:54.040 --> 00:21:55.980
And ARCH models were
introduced initially

00:21:55.980 --> 00:21:57.084
to try and capture that.

00:21:57.084 --> 00:21:58.500
And were extended
to GARCH models,

00:21:58.500 --> 00:22:00.910
and these are the
sort of simplest cases

00:22:00.910 --> 00:22:03.530
of time-dependent
volatility models

00:22:03.530 --> 00:22:06.680
that we can work
with and introduce.

00:22:06.680 --> 00:22:11.630
And in all of these the sort
of mathematical framework

00:22:11.630 --> 00:22:14.820
for defining these models
and the statistical framework

00:22:14.820 --> 00:22:18.050
for estimating their parameters
is going to be highlighted.

00:22:18.050 --> 00:22:22.100
And while it's a
very simple setting

00:22:22.100 --> 00:22:24.710
in terms of what
these models are,

00:22:24.710 --> 00:22:28.090
these issues that
we'll be covering

00:22:28.090 --> 00:22:33.200
relate to virtually all
statistical modeling as well.

00:22:33.200 --> 00:22:36.120
So let's define volatility.

00:22:36.120 --> 00:22:36.620
OK.

00:22:36.620 --> 00:22:40.480
In finance it's defined as the
annualized standard deviation

00:22:40.480 --> 00:22:43.380
of the change in price or
value of a financial security,

00:22:43.380 --> 00:22:45.280
or an index.

00:22:45.280 --> 00:22:49.630
So we're interested
in the variability

00:22:49.630 --> 00:22:55.220
of this process, a price
process or a value process.

00:22:55.220 --> 00:22:59.240
And we consider it on an
annualized time scale.

00:22:59.240 --> 00:23:03.910
Now because of that, when
you talk about volatility

00:23:03.910 --> 00:23:10.550
it really is meaningful to
communicate, levels of 10%.

00:23:10.550 --> 00:23:17.500
If you think of, at what level
do sort of absolute bond yields

00:23:17.500 --> 00:23:19.480
vary over a year?

00:23:22.440 --> 00:23:25.120
It's probably less than 5%.

00:23:25.120 --> 00:23:26.242
Bond yields don't--

00:23:26.242 --> 00:23:27.950
When you think of
currencies, how much do

00:23:27.950 --> 00:23:30.860
those vary over a year.

00:23:30.860 --> 00:23:32.790
Maybe 10%.

00:23:32.790 --> 00:23:35.480
With equity markets,
how do those vary?

00:23:35.480 --> 00:23:39.700
Well, maybe 30%, 40% or more.

00:23:39.700 --> 00:23:43.170
With the estimation and
prediction approaches,

00:23:43.170 --> 00:23:46.030
OK, these are what
we'll be discussing.

00:23:46.030 --> 00:23:47.930
There's different cases.

00:23:47.930 --> 00:23:52.830
So let's go on to
historical volatility.

00:23:52.830 --> 00:23:56.270
In terms of computing
the historical volatility

00:23:56.270 --> 00:23:59.350
we'll be considering
basically a price

00:23:59.350 --> 00:24:02.080
series of T plus 1 points.

00:24:02.080 --> 00:24:06.811
And then we can get
T period returns

00:24:06.811 --> 00:24:08.310
corresponding to
those prices, which

00:24:08.310 --> 00:24:12.450
is the difference in
the logs of the prices,

00:24:12.450 --> 00:24:14.300
or the log of the
price relatives.

00:24:14.300 --> 00:24:18.370
So R_t is going to be
the return for the asset.

00:24:18.370 --> 00:24:22.710
And one could use
other definitions,

00:24:22.710 --> 00:24:26.340
like sort of the absolute
return, not take logs.

00:24:26.340 --> 00:24:30.160
It's convenient in much
empirical analysis,

00:24:30.160 --> 00:24:34.250
I guess, to work with the
logs because if you sum

00:24:34.250 --> 00:24:37.990
logs you get sort of
log of the product.

00:24:37.990 --> 00:24:41.830
And so total cumulative
returns can be computed easily

00:24:41.830 --> 00:24:43.670
with sums of logs.

00:24:43.670 --> 00:24:47.140
But anyway, we'll work
with that scale for now.

00:24:47.140 --> 00:24:47.640
OK.

00:24:47.640 --> 00:24:52.080
Now the process R_t, the
return series process,

00:24:52.080 --> 00:24:55.400
is going to be assumed to
be covariance stationary,

00:24:55.400 --> 00:24:59.820
meaning that it does
have a finite variance.

00:24:59.820 --> 00:25:04.900
And the sample estimate
of that is just

00:25:04.900 --> 00:25:10.730
given by the square root
of the sample variance.

00:25:10.730 --> 00:25:13.445
And we're also considering
an unbiased estimate of that.

00:25:16.360 --> 00:25:20.770
And if we want to
basically convert these

00:25:20.770 --> 00:25:22.570
to annualized
values so that we're

00:25:22.570 --> 00:25:24.410
dealing with a
volatility, then if we

00:25:24.410 --> 00:25:28.672
have daily prices of
which in financial markets

00:25:28.672 --> 00:25:30.130
they're usually--
in the US they're

00:25:30.130 --> 00:25:33.550
open roughly 252 days
a year on average.

00:25:33.550 --> 00:25:37.580
We multiply that sigma
hat by 252 square root.

00:25:37.580 --> 00:25:44.110
And for weekly, root 52, and
root 12 for monthly data.

00:25:44.110 --> 00:25:48.870
So regardless of the
periodicity of our original data

00:25:48.870 --> 00:25:51.700
we can get them onto
that volatility scale.

00:25:56.410 --> 00:26:00.960
Now in terms of
prediction methods

00:26:00.960 --> 00:26:05.980
that one can make with
historical volatility,

00:26:05.980 --> 00:26:12.230
and there's a lot of work
done in finance by people

00:26:12.230 --> 00:26:15.060
who aren't sort of
trained as econometricians

00:26:15.060 --> 00:26:18.570
or statisticians, they basically
just work with the data.

00:26:18.570 --> 00:26:23.840
And there's a standard for
risk analysis called the risk

00:26:23.840 --> 00:26:30.780
metrics approach, where the
approach defines volatility

00:26:30.780 --> 00:26:33.470
and volatility estimates,
historical estimates, just

00:26:33.470 --> 00:26:35.750
using simple methodologies.

00:26:35.750 --> 00:26:39.870
And so that's just go
through what those are here.

00:26:39.870 --> 00:26:46.940
One can-- basically
for any period t,

00:26:46.940 --> 00:26:49.710
one can define the
sample volatility,

00:26:49.710 --> 00:26:53.670
just to be the sample standard
deviation of the period t

00:26:53.670 --> 00:26:55.100
returns.

00:26:55.100 --> 00:26:58.430
And so with daily
data that might just

00:26:58.430 --> 00:27:00.800
be the square of
that daily return.

00:27:00.800 --> 00:27:05.150
With monthly data it could be
the sample standard deviation

00:27:05.150 --> 00:27:08.240
of the returns over the
month and with yearly it

00:27:08.240 --> 00:27:10.860
would be the sample
over the year.

00:27:10.860 --> 00:27:15.150
Also with intraday data, it
could be the sample standard

00:27:15.150 --> 00:27:22.900
deviation over intraday periods
of say, half hours or hours.

00:27:22.900 --> 00:27:26.810
And the historical
average is simply

00:27:26.810 --> 00:27:30.320
the mean of those
estimates, which

00:27:30.320 --> 00:27:32.490
uses all the available data.

00:27:32.490 --> 00:27:34.830
One can consider the
simple moving average

00:27:34.830 --> 00:27:38.280
of these realized volatilities.

00:27:38.280 --> 00:27:44.330
And so that basically is using
the last m, for some finite m,

00:27:44.330 --> 00:27:46.260
values to average.

00:27:46.260 --> 00:27:53.296
And one could also consider
an exponential moving average

00:27:53.296 --> 00:27:57.170
of these sample
volatilities where

00:27:57.170 --> 00:28:02.060
we have-- our estimate of the
volatility is 1 minus beta

00:28:02.060 --> 00:28:05.600
times the current
period volatility

00:28:05.600 --> 00:28:08.550
plus beta times the
previous estimate.

00:28:08.550 --> 00:28:10.780
And these exponential
moving averages

00:28:10.780 --> 00:28:15.990
are really very nice
ways to estimate

00:28:15.990 --> 00:28:19.740
processes that change over time.

00:28:19.740 --> 00:28:23.440
And they're able to track
the changes quite well

00:28:23.440 --> 00:28:27.214
and they will tend to
come up again and again.

00:28:27.214 --> 00:28:28.880
This exponential
moving average actually

00:28:28.880 --> 00:28:31.990
uses all available data.

00:28:31.990 --> 00:28:34.770
And there can be discrete
versions of those where

00:28:34.770 --> 00:28:37.615
you say, well let's use not
an equal weighted average

00:28:37.615 --> 00:28:39.490
like the simple moving
average, but let's use

00:28:39.490 --> 00:28:44.220
a geometric average of the last
m values in an exponential way.

00:28:44.220 --> 00:28:46.790
And that's the exponential
weighted moving average

00:28:46.790 --> 00:28:47.780
that uses the last m.

00:28:54.191 --> 00:28:54.690
OK.

00:28:54.690 --> 00:28:55.190
There we go.

00:29:03.109 --> 00:29:03.609
OK.

00:29:06.610 --> 00:29:11.870
Well, with these different
measures of sample volatility,

00:29:11.870 --> 00:29:17.610
one can basically build
models to estimate them

00:29:17.610 --> 00:29:23.990
with regression
models and evaluate.

00:29:23.990 --> 00:29:26.650
And in terms of the
risk metrics benchmark,

00:29:26.650 --> 00:29:30.140
they consider a variety
of different methodologies

00:29:30.140 --> 00:29:32.080
for estimating volatility.

00:29:32.080 --> 00:29:35.000
And sort of determine
what methods are best

00:29:35.000 --> 00:29:38.320
for different kinds of
financial instruments.

00:29:38.320 --> 00:29:42.030
And different financial indexes.

00:29:42.030 --> 00:29:44.140
And there are different
performance measures

00:29:44.140 --> 00:29:45.000
one can apply.

00:29:45.000 --> 00:29:47.740
Sort of mean squared
error of prediction,

00:29:47.740 --> 00:29:51.020
mean absolute error
of prediction,

00:29:51.020 --> 00:29:53.360
mean absolute prediction
error, and so forth

00:29:53.360 --> 00:29:55.680
to evaluate different
methodologies.

00:29:55.680 --> 00:30:00.640
And on the web you can actually
look at the technical documents

00:30:00.640 --> 00:30:03.530
for risk metrics and they
go through these analyses

00:30:03.530 --> 00:30:06.700
and if your interest is in a
particular area of finance,

00:30:06.700 --> 00:30:09.810
whether it's fixed income
or equities, commodities,

00:30:09.810 --> 00:30:13.220
or currencies,
reviewing their work

00:30:13.220 --> 00:30:15.160
there is very
interesting because it

00:30:15.160 --> 00:30:20.740
does highlight different
aspects of those markets.

00:30:20.740 --> 00:30:25.690
And it turns out that basically
the exponential moving average

00:30:25.690 --> 00:30:30.040
is generally a very good
method for many instruments.

00:30:30.040 --> 00:30:38.050
And the sort of discounting
of the values over time

00:30:38.050 --> 00:30:41.340
corresponds to having roughly
between, I guess, a 45

00:30:41.340 --> 00:30:45.910
and a 90 day period in
estimating your volatility.

00:30:45.910 --> 00:30:50.690
And in these approaches
which are, I guess,

00:30:50.690 --> 00:30:52.930
they're a bit ad hoc.

00:30:52.930 --> 00:30:54.250
There's the formalism.

00:30:54.250 --> 00:30:57.530
And defining them is
basically just empirically

00:30:57.530 --> 00:30:58.750
what has worked in the past.

00:31:03.760 --> 00:31:04.260
Let's see.

00:31:08.610 --> 00:31:12.170
While these things are
ad hoc, they actually

00:31:12.170 --> 00:31:13.840
have been very, very effective.

00:31:13.840 --> 00:31:23.970
So let's move on to
formal statistical models

00:31:23.970 --> 00:31:25.940
of volatility.

00:31:25.940 --> 00:31:30.740
And the first class is-- model
is the Geometric Brownian

00:31:30.740 --> 00:31:31.240
Motion.

00:31:31.240 --> 00:31:37.700
So here we have basically
a stochastic differential

00:31:37.700 --> 00:31:41.960
equation defining the model
for Geometric Brownian Motion.

00:31:41.960 --> 00:31:44.950
And Choongbum will be
going in some detail

00:31:44.950 --> 00:31:49.360
about stochastic
differential equations,

00:31:49.360 --> 00:31:52.300
and stochastic calculus
for representing

00:31:52.300 --> 00:31:55.590
different processes,
continuous processes.

00:31:55.590 --> 00:32:00.910
And the formulation
is basically looking

00:32:00.910 --> 00:32:08.470
at increments of the price
process S is equal to basically

00:32:08.470 --> 00:32:14.910
a mu S of t, sort of a drift
term, plus a sigma S of t,

00:32:14.910 --> 00:32:18.930
a multiple of d W
of t, where sigma

00:32:18.930 --> 00:32:21.130
is the volatility of
the security price,

00:32:21.130 --> 00:32:25.810
mu is the mean return
per unit time, d W of t

00:32:25.810 --> 00:32:29.830
is the increment of a standard
Brownian motion processor,

00:32:29.830 --> 00:32:31.350
Wiener process.

00:32:31.350 --> 00:32:38.210
And this W process is
such that it's increments,

00:32:38.210 --> 00:32:42.160
basically the change in value
of the process between two time

00:32:42.160 --> 00:32:46.410
points is normally
distributed, with mean 0

00:32:46.410 --> 00:32:51.720
and variance equal to the
length of the interval.

00:32:54.354 --> 00:32:56.770
And increments on disjoint
time intervals are independent.

00:33:01.690 --> 00:33:10.810
And well, if you
divide both sides

00:33:10.810 --> 00:33:16.535
of that equation by S of t then
you have d S of t over S of t

00:33:16.535 --> 00:33:20.120
is equal to mu dt
plus sigma d W of t.

00:33:20.120 --> 00:33:25.495
And so the increments d S
of t normalized by S of t

00:33:25.495 --> 00:33:29.600
are a standard Brownian motion
with drift mu and volatility

00:33:29.600 --> 00:33:30.100
sigma.

00:33:36.200 --> 00:33:44.570
Now with sample data
from this process,

00:33:44.570 --> 00:33:46.890
now suppose we have
prices observed

00:33:46.890 --> 00:33:50.820
at times t_0 up to t_n.

00:33:50.820 --> 00:33:53.960
And for now we're not going
to make any assumptions

00:33:53.960 --> 00:33:57.950
about what those time increments
are, what those times are.

00:33:57.950 --> 00:33:59.724
They could be equally spaced.

00:33:59.724 --> 00:34:01.015
They could be unequally spaced.

00:34:03.550 --> 00:34:10.420
The returns, the log of the
relative price change from time

00:34:10.420 --> 00:34:15.880
t_(j-1) to t_j are
independent random variables.

00:34:15.880 --> 00:34:19.610
And they are independent.

00:34:19.610 --> 00:34:21.800
Their distribution is
normally distributed

00:34:21.800 --> 00:34:27.330
with mean given by mu times the
length of the time increment,

00:34:27.330 --> 00:34:31.580
and variance sigma squared times
the length of the increment.

00:34:31.580 --> 00:34:35.909
And these properties will
be covered by Choongbum

00:34:35.909 --> 00:34:38.139
in some later lectures.

00:34:38.139 --> 00:34:41.750
So for now what we can
just know that this is true

00:34:41.750 --> 00:34:46.420
and apply this result.
If we fix various time

00:34:46.420 --> 00:34:49.130
points for the observation
and compute returns this way.

00:34:49.130 --> 00:34:51.260
If it's a Geometric
Brownian Motion

00:34:51.260 --> 00:34:55.610
we know that this is the
distribution of the returns.

00:34:55.610 --> 00:34:58.190
Now knowing that
distribution we can now

00:34:58.190 --> 00:35:01.620
engage in maximum
likelihood estimation.

00:35:01.620 --> 00:35:02.120
OK.

00:35:02.120 --> 00:35:06.030
If the increments are
all just equal to 1,

00:35:06.030 --> 00:35:09.140
so we're thinking
of daily data, say.

00:35:09.140 --> 00:35:13.600
Then the maximum likelihood
estimates are simple.

00:35:13.600 --> 00:35:17.570
It's basically the sample mean
and the sample variance with 1

00:35:17.570 --> 00:35:20.340
over n instead of 1 over
n minus 1 in the MLE's.

00:35:20.340 --> 00:35:26.520
If delta_j varies
then, well, that's

00:35:26.520 --> 00:35:30.810
actually a case
in the exercises.

00:35:30.810 --> 00:35:39.100
Now does anyone,
in terms of, well,

00:35:39.100 --> 00:35:46.730
in the class exercise the issue
that is important to think

00:35:46.730 --> 00:35:53.640
about is if you consider a given
interval of time over which

00:35:53.640 --> 00:35:57.660
we're observing this Geometric
Brownian Motion process,

00:35:57.660 --> 00:36:03.440
if we increase the sampling
rate of prices over a given

00:36:03.440 --> 00:36:06.990
interval, how does that
change the properties

00:36:06.990 --> 00:36:09.400
of our estimates?

00:36:09.400 --> 00:36:11.840
Basically, do we obtain
more accurate estimates

00:36:11.840 --> 00:36:14.450
of the underlying parameters?

00:36:14.450 --> 00:36:19.420
And as you increase
the sampling frequency,

00:36:19.420 --> 00:36:21.830
it turns out that some
parameters are estimated much,

00:36:21.830 --> 00:36:26.190
much better and you
get basically much

00:36:26.190 --> 00:36:28.730
lower standard errors
on those estimates.

00:36:28.730 --> 00:36:31.900
With other parameters
you don't necessarily.

00:36:31.900 --> 00:36:35.140
And the exercise is
to evaluate that.

00:36:35.140 --> 00:36:37.350
Now another issue
that's important

00:36:37.350 --> 00:36:42.550
is the issue of sort of what
is the appropriate time scale

00:36:42.550 --> 00:36:46.910
for Geometric Brownian Motion.

00:36:46.910 --> 00:36:48.750
Right now we're
thinking of, you collect

00:36:48.750 --> 00:36:52.055
data, whatever the
periodicity is of the data

00:36:52.055 --> 00:36:54.430
is you think that's your period
for your Brownian Motion.

00:36:54.430 --> 00:36:56.630
Let's evaluate that.

00:36:56.630 --> 00:37:01.655
Let me go to another example.

00:37:08.200 --> 00:37:09.060
Let's see here.

00:37:13.515 --> 00:37:15.350
Yep.

00:37:15.350 --> 00:37:15.850
OK.

00:37:15.850 --> 00:37:17.360
Let's go control-minus here.

00:37:24.830 --> 00:37:25.424
OK.

00:37:25.424 --> 00:37:25.924
All right.

00:37:31.026 --> 00:37:32.060
Let's see.

00:37:32.060 --> 00:37:33.640
With this second
case study there

00:37:33.640 --> 00:37:41.200
was data on exchange rates,
looking for regime changes

00:37:41.200 --> 00:37:43.920
in exchange rate relationships.

00:37:43.920 --> 00:37:46.560
And so we have data
from that case study

00:37:46.560 --> 00:37:49.880
on different foreign
exchange rates.

00:37:49.880 --> 00:37:57.390
And here in the top panel
I've graphed the euro/dollar

00:37:57.390 --> 00:38:01.460
exchange rate from
the beginning of 1999

00:38:01.460 --> 00:38:05.370
through just a few months ago.

00:38:05.370 --> 00:38:12.830
And the second panel is a
plot of the daily returns

00:38:12.830 --> 00:38:14.730
for that series.

00:38:14.730 --> 00:38:21.860
And here is a histogram
of those daily returns.

00:38:21.860 --> 00:38:28.990
And a fit of the Gaussian
distribution for the daily

00:38:28.990 --> 00:38:33.270
returns if our sort of
time scale is correct.

00:38:33.270 --> 00:38:37.350
Basically daily returns
are normally distributed.

00:38:37.350 --> 00:38:41.630
Days are disjoint in
terms of the price change.

00:38:41.630 --> 00:38:45.160
And so they're independent
and identically distributed

00:38:45.160 --> 00:38:46.960
under the model.

00:38:46.960 --> 00:38:49.480
And they all have the
same normal distribution

00:38:49.480 --> 00:38:52.330
with mean mu and
variance sigma squared.

00:38:55.220 --> 00:38:55.870
OK.

00:38:55.870 --> 00:39:00.000
This analysis assumes
basically that we're

00:39:00.000 --> 00:39:03.340
dealing with trading days for
the appropriate time scale,

00:39:03.340 --> 00:39:04.630
the Geometric Brownian Motion.

00:39:09.640 --> 00:39:10.440
Let's see.

00:39:10.440 --> 00:39:15.300
One can ask, well, what
if trading dates really

00:39:15.300 --> 00:39:19.240
isn't the right time scale,
but it's more calendar time.

00:39:19.240 --> 00:39:22.060
The change in value
over the weekends

00:39:22.060 --> 00:39:26.050
maybe correspond to price
changes, or value changes

00:39:26.050 --> 00:39:28.150
over a longer period of time.

00:39:28.150 --> 00:39:30.980
And so this model
really needs to be

00:39:30.980 --> 00:39:35.270
adjusted for that time scale.

00:39:35.270 --> 00:39:41.190
The exercise that
allows you to consider

00:39:41.190 --> 00:39:45.660
different delta t's shows you
what the maximum likelihood

00:39:45.660 --> 00:39:47.429
estimates-- you'll
be deriving maximum

00:39:47.429 --> 00:39:49.470
likely estimates if we
have different definitions

00:39:49.470 --> 00:39:52.180
of time scale there.

00:39:52.180 --> 00:40:02.992
But if you apply the calendar
time scale to this euro,

00:40:02.992 --> 00:40:05.200
let me just show you what
the different estimates are

00:40:05.200 --> 00:40:09.590
of the annualized mean return
and the annualized volatility.

00:40:09.590 --> 00:40:16.030
So if we consider trading days
for euro it's 10.25% or 0.1025.

00:40:16.030 --> 00:40:22.390
If you consider clock time, it
actually turns out to be 12.2%.

00:40:22.390 --> 00:40:25.070
So depending on how
you specify the model

00:40:25.070 --> 00:40:28.640
you get a different
definition of volatility here.

00:40:28.640 --> 00:40:36.170
And it's important to
basically understand

00:40:36.170 --> 00:40:40.650
sort of what the assumptions
are of your model

00:40:40.650 --> 00:40:47.480
and whether perhaps things
ought to be different.

00:40:47.480 --> 00:40:53.700
In stochastic modeling,
there's an area

00:40:53.700 --> 00:40:57.030
called subordinated
stochastic processes.

00:40:57.030 --> 00:41:04.220
And basically the idea is, if
you have a stochastic process

00:41:04.220 --> 00:41:08.770
like Geometric Brownian Motion
of simple Brownian motion,

00:41:08.770 --> 00:41:14.005
maybe you're observing that
on the wrong time scale.

00:41:14.005 --> 00:41:15.963
You may fit the Geometric
Brownian Motion model

00:41:15.963 --> 00:41:17.560
and it doesn't look right.

00:41:17.560 --> 00:41:19.740
But it could be that
there's a different time

00:41:19.740 --> 00:41:21.180
scale that's appropriate.

00:41:21.180 --> 00:41:24.990
And it's really Brownian
motion on that time scale.

00:41:24.990 --> 00:41:29.830
And so formally it's called
a subordinated stochastic

00:41:29.830 --> 00:41:30.330
process.

00:41:30.330 --> 00:41:32.160
You have a different
time function

00:41:32.160 --> 00:41:35.970
for how to model the
stochastic process.

00:41:35.970 --> 00:41:40.530
And the evaluation of
subordinated stochastic

00:41:40.530 --> 00:41:43.750
processes leads to consideration
of different time scales.

00:41:43.750 --> 00:41:48.320
With, say, equity markets,
and futures markets,

00:41:48.320 --> 00:41:50.987
sort of the volume of trading,
sort of cumulative volume

00:41:50.987 --> 00:41:53.070
of training might be really
an appropriate measure

00:41:53.070 --> 00:41:54.880
of the real time scale.

00:41:54.880 --> 00:41:56.820
Because that's a
measure of, in a sense,

00:41:56.820 --> 00:41:59.000
information flow
coming into the market

00:41:59.000 --> 00:42:01.870
through the level of activity.

00:42:01.870 --> 00:42:06.720
So anyway I wanted to highlight
how with different time scales

00:42:06.720 --> 00:42:08.320
you can get different results.

00:42:08.320 --> 00:42:11.660
And so that's something
to be evaluated.

00:42:11.660 --> 00:42:13.620
In looking at these
different models,

00:42:13.620 --> 00:42:15.420
OK, these first few
graphs here show

00:42:15.420 --> 00:42:18.880
the fit of the normal model
with the trading day time scale.

00:42:22.400 --> 00:42:22.966
Let's see.

00:42:22.966 --> 00:42:25.340
Those of you who've ever taken
a statistics class before,

00:42:25.340 --> 00:42:29.780
or an applied statistics, may
know about normal q-q plots.

00:42:29.780 --> 00:42:33.830
Basically if you
want to evaluate

00:42:33.830 --> 00:42:37.960
the consistency of
the returns here

00:42:37.960 --> 00:42:41.620
with a Gaussian
distribution, what we can do

00:42:41.620 --> 00:42:49.410
is plot the observed
ordered, sorted returns

00:42:49.410 --> 00:42:52.790
against what we would
expect the sorted returns

00:42:52.790 --> 00:42:56.200
to be if it were from
a Gaussian sample.

00:42:56.200 --> 00:42:58.940
So under the Geometric
Brownian Motion model

00:42:58.940 --> 00:43:04.530
the daily returns are a sample,
independent and identically

00:43:04.530 --> 00:43:06.740
distributed random variable
sampled from a Gaussian

00:43:06.740 --> 00:43:07.750
distribution.

00:43:07.750 --> 00:43:11.510
So the smallest return should
be consistent with the smallest

00:43:11.510 --> 00:43:14.080
of the sample size n.

00:43:14.080 --> 00:43:18.870
And what's being plotted here
is the theoretical quantiles

00:43:18.870 --> 00:43:21.980
or percentiles versus
the actual ones.

00:43:21.980 --> 00:43:24.930
And one would expect that
to lie along a straight line

00:43:24.930 --> 00:43:30.100
if the theoretical quantiles
were well-predicting

00:43:30.100 --> 00:43:32.670
the actual extreme values.

00:43:32.670 --> 00:43:37.760
What we see here is that as the
theoretical quantiles get high,

00:43:37.760 --> 00:43:40.990
and it's in units of
standard deviation units,

00:43:40.990 --> 00:43:45.080
the realized sample
returns are in fact

00:43:45.080 --> 00:43:47.540
much higher than would be
predicted by the Gaussian

00:43:47.540 --> 00:43:49.210
distribution.

00:43:49.210 --> 00:43:52.400
And similarly, on
the low end side.

00:43:52.400 --> 00:43:54.550
So there's a normal
q-q plot that's

00:43:54.550 --> 00:43:57.800
used often in the
diagnostics of these models.

00:43:57.800 --> 00:44:04.910
Then down here I've actually
plotted a fitted percentile

00:44:04.910 --> 00:44:06.160
distribution.

00:44:06.160 --> 00:44:12.470
Now what's been done here
is if we modeled the series

00:44:12.470 --> 00:44:16.790
as a series of Gaussian
random variables

00:44:16.790 --> 00:44:24.960
then we can evaluate
the percentile

00:44:24.960 --> 00:44:27.130
of the fitted Gaussian
distribution that

00:44:27.130 --> 00:44:29.430
was realized by every point.

00:44:29.430 --> 00:44:38.780
So if we have a return of say
negative 2%, what percentile

00:44:38.780 --> 00:44:40.540
is the normal fit of that?

00:44:45.720 --> 00:44:50.410
And you can evaluate the
cumulative distribution

00:44:50.410 --> 00:44:54.750
function of the fitted model at
that value to get that point.

00:44:54.750 --> 00:44:59.100
And what should the
distribution of percentiles

00:44:59.100 --> 00:45:04.410
be for fitted percentiles if
we have a really good model?

00:45:04.410 --> 00:45:07.370
OK.

00:45:07.370 --> 00:45:08.180
Well, OK.

00:45:08.180 --> 00:45:09.860
Let's think.

00:45:09.860 --> 00:45:14.890
If you consider the 50th
percentile you would expect,

00:45:14.890 --> 00:45:18.800
I guess, 50% of the data to
lie above the 50th percentile

00:45:18.800 --> 00:45:21.930
and 50% to lie below the
50th percentile, right?

00:45:21.930 --> 00:45:22.530
OK.

00:45:22.530 --> 00:45:24.160
Let's consider,
here I divided up

00:45:24.160 --> 00:45:27.840
into 100 bins
between zero and one

00:45:27.840 --> 00:45:31.955
so this bin is the
99th percentile.

00:45:38.630 --> 00:45:40.460
How many observations
would you expect

00:45:40.460 --> 00:45:45.590
to find in between the
99th and 100 percentile?

00:45:49.800 --> 00:45:51.170
This is an easy question.

00:45:51.170 --> 00:45:52.150
AUDIENCE: 1%.

00:45:52.150 --> 00:45:53.070
PROFESSOR: 1%.

00:45:53.070 --> 00:45:53.790
Right.

00:45:53.790 --> 00:45:55.290
And so in any of
these bins we would

00:45:55.290 --> 00:46:01.450
expect to see 1% if the
Gaussian model were fitting.

00:46:01.450 --> 00:46:06.690
And what we see is that,
well, at the extremes

00:46:06.690 --> 00:46:08.600
they're more extreme values.

00:46:08.600 --> 00:46:13.720
And actually inside there
are some fewer values.

00:46:13.720 --> 00:46:17.660
And actually this is exhibiting
a leptokurtic distribution

00:46:17.660 --> 00:46:20.070
for the actually
realized samples;

00:46:20.070 --> 00:46:22.080
basically the middle
of the distribution

00:46:22.080 --> 00:46:24.280
is a little thinner
and it's compensated

00:46:24.280 --> 00:46:26.660
for by fatter tails.

00:46:26.660 --> 00:46:29.440
But with this
particular model we

00:46:29.440 --> 00:46:33.900
can basically expect to
see a uniform distribution

00:46:33.900 --> 00:46:39.690
of percentiles in this graph.

00:46:39.690 --> 00:46:46.990
If we compare this with
a fit of the clock time

00:46:46.990 --> 00:46:51.770
we actually see
that clock time does

00:46:51.770 --> 00:46:59.490
a bit of a better job at getting
the extreme values closer

00:46:59.490 --> 00:47:01.110
to what we would
expect them to be.

00:47:01.110 --> 00:47:07.506
So in terms of being a better
model for the returns process,

00:47:07.506 --> 00:47:09.380
if we're concerned with
these extreme values,

00:47:09.380 --> 00:47:12.320
we're actually getting
a slightly better value

00:47:12.320 --> 00:47:13.720
with those.

00:47:13.720 --> 00:47:16.590
So all right.

00:47:16.590 --> 00:47:20.890
Let's move on back to the notes.

00:47:20.890 --> 00:47:28.410
And talk about the
Garman-Klass Estimator.

00:47:28.410 --> 00:47:30.905
So let me do this.

00:47:34.625 --> 00:47:36.080
All right.

00:47:36.080 --> 00:47:37.123
View full screen.

00:47:43.040 --> 00:47:45.334
OK.

00:47:45.334 --> 00:47:46.120
All right.

00:47:46.120 --> 00:47:48.090
So, OK.

00:47:48.090 --> 00:47:50.990
The Garman-Klass
Estimator is one

00:47:50.990 --> 00:47:55.410
where we consider the
situation where we actually

00:47:55.410 --> 00:47:59.410
have much more information
than simply sort of closing

00:47:59.410 --> 00:48:01.980
prices at different intervals.

00:48:01.980 --> 00:48:05.374
Basically all transaction
data's collected

00:48:05.374 --> 00:48:06.290
in a financial market.

00:48:06.290 --> 00:48:08.150
So really we have
virtually all of the data

00:48:08.150 --> 00:48:11.280
available if we want
it, or can pay for it.

00:48:11.280 --> 00:48:14.270
But let's consider
a case where we

00:48:14.270 --> 00:48:18.300
expand upon just having
closing prices to having

00:48:18.300 --> 00:48:22.190
additional information over
increments of time that

00:48:22.190 --> 00:48:27.230
include the open,
high, and low price

00:48:27.230 --> 00:48:28.355
over the different periods.

00:48:33.500 --> 00:48:35.940
So those of you who are
familiar with bar data

00:48:35.940 --> 00:48:41.000
graphs that you see whenever you
plot stock prices over periods

00:48:41.000 --> 00:48:46.870
of weeks or months you'll
be familiar with having

00:48:46.870 --> 00:48:48.610
seen those.

00:48:48.610 --> 00:48:52.550
Now the Garman-Klass
paper addressed

00:48:52.550 --> 00:48:54.730
how can we exploit this
additional information

00:48:54.730 --> 00:49:00.290
to improve upon our estimates
of the close-to-close.

00:49:00.290 --> 00:49:03.600
And so we'll just
use this notation.

00:49:03.600 --> 00:49:06.200
Well, let's make some
assumptions and notation.

00:49:06.200 --> 00:49:09.470
So we'll assume that mu is equal
to 0 in our Geometric Brownian

00:49:09.470 --> 00:49:10.482
Motion model.

00:49:10.482 --> 00:49:12.190
So we don't have to
worry about the mean.

00:49:12.190 --> 00:49:14.260
We're just concerned
with volatility.

00:49:14.260 --> 00:49:18.530
We'll assume that the
increments are one for daily,

00:49:18.530 --> 00:49:20.760
corresponding to daily.

00:49:20.760 --> 00:49:23.640
And we'll let little f,
between zero and one,

00:49:23.640 --> 00:49:35.130
correspond to the time of day
at which the market opens.

00:49:35.130 --> 00:49:40.130
So over a day, from
day zero to day one at

00:49:40.130 --> 00:49:45.420
f we assume that
the market opens

00:49:45.420 --> 00:49:50.780
and basically the Geometric
Brownian Motion process

00:49:50.780 --> 00:49:53.020
might have closed
on day zero here.

00:49:53.020 --> 00:50:00.790
So this would be C_0 and it
may have opened on day one

00:50:00.790 --> 00:50:01.880
at this value.

00:50:01.880 --> 00:50:05.990
So this would be O_1.

00:50:05.990 --> 00:50:09.000
Might have gone up
and down and then

00:50:09.000 --> 00:50:14.801
closed here with the
Brownian Motion process.

00:50:14.801 --> 00:50:15.300
OK.

00:50:15.300 --> 00:50:22.240
This value here would
correspond to the high value.

00:50:22.240 --> 00:50:25.710
This value here would correspond
to the low value on day one.

00:50:25.710 --> 00:50:30.840
And then the closing
value here would be C_1.

00:50:30.840 --> 00:50:35.310
So the model is we have this
underlying Brownian Motion

00:50:35.310 --> 00:50:41.220
process is actually working
over continuous time,

00:50:41.220 --> 00:50:45.950
but we just observe it over
the time when the markets open.

00:50:45.950 --> 00:50:48.840
And so it can move between when
the market closes and opens

00:50:48.840 --> 00:50:52.640
on any given day and we have
the additional information.

00:50:52.640 --> 00:50:56.380
Instead of just the close, we
also have the high and low.

00:50:56.380 --> 00:51:00.450
So let's look at how we might
exploit that information

00:51:00.450 --> 00:51:03.070
to estimate volatility.

00:51:03.070 --> 00:51:03.570
OK.

00:51:03.570 --> 00:51:07.920
Using data from the first period
as we've graphed here, let's

00:51:07.920 --> 00:51:16.370
first just highlight what
the close-to-close return is.

00:51:16.370 --> 00:51:17.900
And that basically
is an estimate

00:51:17.900 --> 00:51:20.950
of the one-period variance.

00:51:20.950 --> 00:51:25.750
And so sigma hat 0 squared is
a single period squared return.

00:51:28.300 --> 00:51:33.130
C_1 minus C_0 has a distribution
which is normal with mean 0,

00:51:33.130 --> 00:51:36.930
and variance sigma squared.

00:51:36.930 --> 00:51:45.429
And if we consider
squaring that, what's

00:51:45.429 --> 00:51:46.470
the distribution of that?

00:51:46.470 --> 00:51:50.482
That's the square of a
normal random variable, which

00:51:50.482 --> 00:51:52.690
is chi squared, but it's a
multiple of a chi squared.

00:51:52.690 --> 00:51:57.250
It's sigma squared times a chi
squared one random variable.

00:51:57.250 --> 00:52:00.160
And with a chi squared
random variable

00:52:00.160 --> 00:52:03.150
the expected value is 1.

00:52:03.150 --> 00:52:09.140
The variance of a chi squared
random variable is equal to 2.

00:52:09.140 --> 00:52:11.430
So just knowing
those facts gives us

00:52:11.430 --> 00:52:17.120
the fact we have an unbiased
estimate of the volatility

00:52:17.120 --> 00:52:25.420
parameter sigma and the variance
is 2 sigma to the fourth.

00:52:25.420 --> 00:52:29.716
So that's basically
the precision

00:52:29.716 --> 00:52:31.510
of close-to-close returns.

00:52:34.140 --> 00:52:36.275
Let's look at two
other estimates.

00:52:40.170 --> 00:52:42.940
Basically the
open-to-close return,

00:52:42.940 --> 00:52:46.950
sigma_1 squared,
normalized by f,

00:52:46.950 --> 00:52:48.410
the length of the interval f.

00:52:48.410 --> 00:53:02.821
So we have sigma_1 is equal
to O_1 minus C_0 squared.

00:53:02.821 --> 00:53:03.320
OK.

00:53:06.500 --> 00:53:07.917
Actually why don't
I just do this?

00:53:07.917 --> 00:53:09.624
I'll just write down
a few facts and then

00:53:09.624 --> 00:53:11.270
you can see that the
results are clear.

00:53:11.270 --> 00:53:17.400
Basically O_1 minus C_0 is
distributed normal with mean 0

00:53:17.400 --> 00:53:19.860
and variance f sigma squared.

00:53:19.860 --> 00:53:25.100
And C_1 minus O_1 is
distributed normal with mean 0

00:53:25.100 --> 00:53:28.490
in variance 1 minus
f sigma squared.

00:53:28.490 --> 00:53:28.990
OK.

00:53:28.990 --> 00:53:31.410
This is simply using the
properties of the diffusion

00:53:31.410 --> 00:53:33.840
process over different
periods of time.

00:53:33.840 --> 00:53:37.070
So if we normalize
the squared values

00:53:37.070 --> 00:53:39.140
by the length of
the interval we get

00:53:39.140 --> 00:53:42.150
estimates of the volatility.

00:53:42.150 --> 00:53:46.670
And what's particularly
significant

00:53:46.670 --> 00:53:49.250
about these
estimates one and two

00:53:49.250 --> 00:53:52.360
is that they're independent.

00:53:52.360 --> 00:53:54.660
So we actually
have two estimates

00:53:54.660 --> 00:53:58.130
of the same
underlying parameter,

00:53:58.130 --> 00:54:01.090
which are independent.

00:54:01.090 --> 00:54:05.305
And actually they both
have the same mean

00:54:05.305 --> 00:54:08.360
and they both have
the same variance.

00:54:08.360 --> 00:54:11.650
So if we consider
a new estimate,

00:54:11.650 --> 00:54:15.790
which is basically
averaging those two.

00:54:15.790 --> 00:54:21.050
Then this new estimate has the
same-- is unbiased as well,

00:54:21.050 --> 00:54:26.340
but it's variance is basically
the variance of this sum.

00:54:26.340 --> 00:54:29.330
So it's 1/2 squared times
this variance plus 1/2 squared

00:54:29.330 --> 00:54:33.640
times this variance, which is
a half of the variance of each

00:54:33.640 --> 00:54:34.350
of them.

00:54:34.350 --> 00:54:38.000
So this estimate
has lower variance

00:54:38.000 --> 00:54:40.200
than our close-to-close.

00:54:40.200 --> 00:54:46.560
And we can define the efficiency
of this particular estimate

00:54:46.560 --> 00:54:50.870
relative to the
close-to-close estimate as 2.

00:54:50.870 --> 00:54:54.175
Basically we get
double the precision.

00:54:58.510 --> 00:55:01.940
Suppose you had the open,
high, close for one day.

00:55:01.940 --> 00:55:04.220
How many days of
close-to-close data

00:55:04.220 --> 00:55:08.343
would you need to have the
same variance as this estimate?

00:55:13.860 --> 00:55:14.729
No.

00:55:14.729 --> 00:55:15.645
AUDIENCE: [INAUDIBLE].

00:55:15.645 --> 00:55:17.550
Because of the three
data points [INAUDIBLE].

00:55:17.550 --> 00:55:18.360
PROFESSOR: No.

00:55:18.360 --> 00:55:20.590
No.

00:55:20.590 --> 00:55:22.565
Anyone else?

00:55:22.565 --> 00:55:23.760
One more.

00:55:23.760 --> 00:55:25.150
Four.

00:55:25.150 --> 00:55:25.760
OK.

00:55:25.760 --> 00:55:30.070
Basically if the
variance is 1/2,

00:55:30.070 --> 00:55:36.040
basically to get the standard
deviation, or the variance

00:55:36.040 --> 00:55:39.205
to be-- I'm sorry.

00:55:39.205 --> 00:55:40.770
The ratio of the
variance is two.

00:55:40.770 --> 00:55:41.430
So no.

00:55:41.430 --> 00:55:43.140
So it actually is close to two.

00:55:46.571 --> 00:55:47.070
Let's see.

00:55:47.070 --> 00:55:49.640
Our 1/n is-- so it
actually is two.

00:55:49.640 --> 00:55:50.140
OK.

00:55:50.140 --> 00:55:51.420
I was thinking standard
deviation units

00:55:51.420 --> 00:55:52.540
instead of squared units.

00:55:52.540 --> 00:55:55.900
So I was trying to
be clever there.

00:55:55.900 --> 00:55:59.270
So it actually is
basically two days.

00:55:59.270 --> 00:56:03.150
So sampling this
with this information

00:56:03.150 --> 00:56:05.842
gives you as much as two
days worth of information.

00:56:05.842 --> 00:56:06.800
So what does that mean?

00:56:06.800 --> 00:56:08.216
Well, if you want
something that's

00:56:08.216 --> 00:56:09.970
as efficient as daily
estimates you'll

00:56:09.970 --> 00:56:14.390
need to look back one
day instead of two days

00:56:14.390 --> 00:56:16.440
to get the same efficiency
with the estimate.

00:56:16.440 --> 00:56:16.940
All right.

00:56:19.870 --> 00:56:21.760
The motivation for
the Garman-Klass paper

00:56:21.760 --> 00:56:26.035
was actually a paper
written by Parkinson

00:56:26.035 --> 00:56:31.210
in 1976, which dealt with using
the extremes of a Brownian

00:56:31.210 --> 00:56:34.730
Motion to estimate the
underlying parameters.

00:56:34.730 --> 00:56:39.590
And when Choongbum talks about
Brownian Motion a bit later,

00:56:39.590 --> 00:56:42.080
I don't know if you'll
derive this result,

00:56:42.080 --> 00:56:43.810
but in courses on
stochastic processes

00:56:43.810 --> 00:56:47.340
one does derive properties
of the maximum of a Brownian

00:56:47.340 --> 00:56:50.200
Motion over a given
interval and the minimum.

00:56:50.200 --> 00:56:53.350
And it turns out
that if you look

00:56:53.350 --> 00:56:57.070
at the difference between the
high and low squared divided

00:56:57.070 --> 00:57:02.140
by 4 log 2, this is an
estimate of the volatility

00:57:02.140 --> 00:57:03.460
of the process.

00:57:03.460 --> 00:57:06.300
And the efficiency
of this estimate

00:57:06.300 --> 00:57:12.589
turns out to be 5.2,
which is better yet.

00:57:12.589 --> 00:57:14.380
Well, Garman and Klass
were excited by that

00:57:14.380 --> 00:57:16.710
and wanted to find
even better ones.

00:57:16.710 --> 00:57:22.350
So they wrote a paper that
evaluated all different kinds

00:57:22.350 --> 00:57:23.000
of estimates.

00:57:23.000 --> 00:57:25.772
And I encourage you
to Google that paper

00:57:25.772 --> 00:57:27.480
and read it because
it's very accessible.

00:57:27.480 --> 00:57:31.060
And it sort of highlights the
statistical and probability

00:57:31.060 --> 00:57:33.150
issues associated
with these problems.

00:57:33.150 --> 00:57:34.800
But what they did
was they derived

00:57:34.800 --> 00:57:38.130
the best analytic
scale-invariant estimator,

00:57:38.130 --> 00:57:46.130
which has this sort of bizarre
combination of different terms,

00:57:46.130 --> 00:57:48.840
but basically we're
using normalized values

00:57:48.840 --> 00:57:52.270
of the high, low, close
normalized by the open.

00:57:52.270 --> 00:57:55.840
And they're able to get
an efficiency of 7.4

00:57:55.840 --> 00:57:59.390
with this combination.

00:57:59.390 --> 00:58:06.700
Now scale-invariant estimates,
in statistical theory,

00:58:06.700 --> 00:58:08.130
they're different
principles that

00:58:08.130 --> 00:58:11.610
guide the development of
different methodologies.

00:58:11.610 --> 00:58:16.420
And one kind of principle is
issues of scale invariance.

00:58:16.420 --> 00:58:20.600
If you're estimating a scale
parameter, and in this case

00:58:20.600 --> 00:58:22.210
volatility is telling
you essentially

00:58:22.210 --> 00:58:25.680
how large is the
variability of this process,

00:58:25.680 --> 00:58:29.750
if you were to say multiply your
original data all by a given

00:58:29.750 --> 00:58:33.950
constant, then a
scale-invariant estimator

00:58:33.950 --> 00:58:37.560
should be such that your
estimator changes in that case

00:58:37.560 --> 00:58:39.710
only by that same scale factor.

00:58:39.710 --> 00:58:42.710
So sort of the
estimator doesn't depend

00:58:42.710 --> 00:58:45.700
on how you scale the data.

00:58:45.700 --> 00:58:50.230
So that's the notion
of scale invariance.

00:58:50.230 --> 00:58:55.092
The Garman-Klass paper
actually goes to the nth degree

00:58:55.092 --> 00:58:56.800
and actually finds a
particular estimator

00:58:56.800 --> 00:59:00.120
that has an efficiency
of 8.4, which

00:59:00.120 --> 00:59:02.470
is really highly significant.

00:59:02.470 --> 00:59:08.350
So if you are working
with a modeling process

00:59:08.350 --> 00:59:11.680
where you believe that the
underlying parameters may

00:59:11.680 --> 00:59:16.080
be reasonably assumed
to be constant

00:59:16.080 --> 00:59:19.590
over short periods
of time, well,

00:59:19.590 --> 00:59:21.870
over those short periods
of time if you use

00:59:21.870 --> 00:59:24.360
these extended
estimators like this,

00:59:24.360 --> 00:59:26.320
you'll get much more
precise measures

00:59:26.320 --> 00:59:28.970
of the underlying parameters
than from just using

00:59:28.970 --> 00:59:33.160
simple close-to-close data.

00:59:33.160 --> 00:59:33.730
All right.

00:59:33.730 --> 00:59:38.790
Let's introduce Poisson
Jump Diffusions.

00:59:38.790 --> 00:59:44.480
With Poisson Jump
Diffusions we have

00:59:44.480 --> 00:59:50.560
basically a stochastic
differential equation

00:59:50.560 --> 00:59:52.560
for representing this model.

00:59:52.560 --> 00:59:55.750
And it's just like the
Geometric Brownian Motion model,

00:59:55.750 --> 01:00:00.930
except we have this additional
term, gamma sigma Z d pi of t.

01:00:00.930 --> 01:00:04.390
Now that's a lot of
different variables,

01:00:04.390 --> 01:00:07.370
but essentially what
we're thinking about

01:00:07.370 --> 01:00:20.300
is over time a Brownian Motion
process is fully continuous.

01:00:20.300 --> 01:00:24.390
There are basically no jumps in
this Brownian Motion process.

01:00:24.390 --> 01:00:27.610
In order to allow
for jumps, we assume

01:00:27.610 --> 01:00:33.960
that there's some process pi of
t, which is a Poisson process.

01:00:33.960 --> 01:00:38.480
It's counting process that
counts when jumps occur,

01:00:38.480 --> 01:00:40.160
how many jumps have occurred.

01:00:40.160 --> 01:00:44.020
So that might start
at 0 at the value 0.

01:00:44.020 --> 01:00:50.680
Then if there's a jump
here it goes up by one.

01:00:50.680 --> 01:00:54.280
And then if there's another
jump here, it goes up by one,

01:00:54.280 --> 01:00:55.890
and so forth.

01:00:55.890 --> 01:01:02.000
And so the Poisson Jump
Diffusion model says,

01:01:02.000 --> 01:01:04.450
this diffusion
process is actually

01:01:04.450 --> 01:01:06.630
going to experience
some shocks to it.

01:01:06.630 --> 01:01:08.940
Those shocks are
going to be arriving

01:01:08.940 --> 01:01:11.010
according to a Poisson process.

01:01:11.010 --> 01:01:12.740
If you've taken
stochastic modeling

01:01:12.740 --> 01:01:17.925
you know that that's a sort
of a purely random process.

01:01:17.925 --> 01:01:21.630
Basically exponential
arrival rate of shocks occur.

01:01:21.630 --> 01:01:23.740
You can't predict them.

01:01:23.740 --> 01:01:28.630
And when those
occur, d pi of t is

01:01:28.630 --> 01:01:31.690
going to change from 0
up to the unit increment.

01:01:31.690 --> 01:01:33.740
So d pi of t is 1.

01:01:33.740 --> 01:01:36.540
And then we'll realize
gamma sigma Z of t.

01:01:36.540 --> 01:01:40.975
So at this point we're
going to have shocks.

01:01:43.580 --> 01:01:50.900
Here this is going to be gamma
sigma Z_1 And at this point,

01:01:50.900 --> 01:01:57.960
maybe it's a negative
shock, gamma sigma Z_2.

01:01:57.960 --> 01:01:59.440
This is 0.

01:01:59.440 --> 01:02:03.400
And so with this overall
process we basically

01:02:03.400 --> 01:02:05.720
have a shift in the
diffusion, up or down,

01:02:05.720 --> 01:02:08.070
according to these values.

01:02:08.070 --> 01:02:11.746
And so this model allows for
the arrival of these processes

01:02:11.746 --> 01:02:13.870
to be random according to
the Poisson distribution,

01:02:13.870 --> 01:02:17.770
and for the magnitude of the
shocks to be random as well.

01:02:21.350 --> 01:02:24.740
Now like the Geometric
Brownian Motion model

01:02:24.740 --> 01:02:31.350
this process sort of has
independent increments, which

01:02:31.350 --> 01:02:35.630
helps with this estimation.

01:02:35.630 --> 01:02:39.550
One could estimate this
model by maximum likelihood,

01:02:39.550 --> 01:02:43.020
but it does get tricky
in that basically

01:02:43.020 --> 01:02:45.970
over different increments
of time the change

01:02:45.970 --> 01:02:50.200
in the process corresponds
to the diffusion increment,

01:02:50.200 --> 01:02:53.550
plus the sum of the
jumps that have occurred

01:02:53.550 --> 01:02:55.820
over that same increment.

01:02:55.820 --> 01:03:01.170
And so the model ultimately
is a Poisson mixture

01:03:01.170 --> 01:03:05.410
of Gaussian distributions.

01:03:05.410 --> 01:03:11.040
And in order to evaluate this
model, model's properties,

01:03:11.040 --> 01:03:14.370
moment generating functions
can be computed rather directly

01:03:14.370 --> 01:03:15.380
with that.

01:03:15.380 --> 01:03:19.650
And so one can understand how
the moments of the process

01:03:19.650 --> 01:03:22.410
vary with these different
model parameters.

01:03:22.410 --> 01:03:28.410
The likelihood function is
a product of Poisson sums.

01:03:28.410 --> 01:03:32.270
And there's a closed form
for the EM algorithm, which

01:03:32.270 --> 01:03:34.280
can be used to
implement the estimation

01:03:34.280 --> 01:03:37.280
of the unknown parameters.

01:03:37.280 --> 01:03:44.430
And if you think about observing
a Poisson Jump Diffusion

01:03:44.430 --> 01:03:52.070
process, if you knew
where the jumps occurred,

01:03:52.070 --> 01:03:53.740
so you knew where
the jumps occurred

01:03:53.740 --> 01:03:57.080
and how many there were
per increment in your data,

01:03:57.080 --> 01:04:00.490
then the maximum
likelihood estimation

01:04:00.490 --> 01:04:03.230
would all be very, very simple.

01:04:03.230 --> 01:04:05.660
And because this sort
of is a separation

01:04:05.660 --> 01:04:07.840
of the estimation of
the Gaussian parameters

01:04:07.840 --> 01:04:10.940
from the Poisson parameters.

01:04:10.940 --> 01:04:14.100
When you haven't observed
those values then

01:04:14.100 --> 01:04:18.880
you need to deal with methods
appropriate for missing data.

01:04:18.880 --> 01:04:22.910
And the EM algorithm is a very
famous algorithm developed

01:04:22.910 --> 01:04:28.730
by the people up at Harvard,
Rubin, Laird, and Dempster,

01:04:28.730 --> 01:04:35.720
which deals with, basically if
the problem is much simpler,

01:04:35.720 --> 01:04:41.240
if you could posit there
being unobserved variables

01:04:41.240 --> 01:04:44.600
that you would observe,
then you sort of

01:04:44.600 --> 01:04:46.960
expand the problem to
include your observed

01:04:46.960 --> 01:04:50.700
data, plus the missing
data, in this case where

01:04:50.700 --> 01:04:52.310
the jumps have occurred.

01:04:52.310 --> 01:04:57.190
And you then do
conditional expectations

01:04:57.190 --> 01:04:59.850
of estimating those jumps.

01:04:59.850 --> 01:05:03.560
And then assuming that
those jumps had those--

01:05:03.560 --> 01:05:06.960
occurred with those frequencies,
estimating the underlying

01:05:06.960 --> 01:05:07.580
parameters.

01:05:07.580 --> 01:05:09.800
So the EM algorithm
is very powerful

01:05:09.800 --> 01:05:14.550
and has extensive
applications in all kinds

01:05:14.550 --> 01:05:15.840
of different models.

01:05:15.840 --> 01:05:17.740
I'll put up on the
website a paper

01:05:17.740 --> 01:05:24.240
that I wrote with David
Pickard and his student

01:05:24.240 --> 01:05:30.470
Arshad Zakaria, which goes
through the maximum likelihood

01:05:30.470 --> 01:05:31.600
methodology for this.

01:05:31.600 --> 01:05:34.980
But looking at that,
you can see how

01:05:34.980 --> 01:05:39.110
with an extended model,
how maximum likelihood gets

01:05:39.110 --> 01:05:44.390
implemented and I think
that's useful to see.

01:05:44.390 --> 01:05:44.890
All right.

01:05:44.890 --> 01:05:51.060
So let's turn next
to ARCH models.

01:05:51.060 --> 01:05:54.430
And OK.

01:05:54.430 --> 01:06:03.250
Just as a bit of motivation, the
Geometric Brownian Motion model

01:06:03.250 --> 01:06:05.640
and also the Poisson
Jump Diffusion model

01:06:05.640 --> 01:06:10.750
are models which assume
that volatility over time

01:06:10.750 --> 01:06:14.340
is essentially stationary.

01:06:14.340 --> 01:06:20.470
And with the sort of independent
increments of those processes,

01:06:20.470 --> 01:06:22.160
the volatility over
different increments

01:06:22.160 --> 01:06:23.830
is essentially the same.

01:06:23.830 --> 01:06:27.650
So the ARCH models
were introduced

01:06:27.650 --> 01:06:31.440
to accommodate the
possibility that there's

01:06:31.440 --> 01:06:33.025
time dependence in volatility.

01:06:37.790 --> 01:06:40.220
And so let's see.

01:06:47.041 --> 01:06:47.540
Let's see.

01:06:47.540 --> 01:06:50.550
Let me go.

01:06:50.550 --> 01:06:52.020
OK.

01:06:52.020 --> 01:06:55.190
At the very end, I'll go through
an example showing that time

01:06:55.190 --> 01:07:04.910
dependence with our
euro/dollar exchange rates.

01:07:04.910 --> 01:07:09.230
So the set up for this
model is basically

01:07:09.230 --> 01:07:13.370
we look at the log of
the price relatives y_t

01:07:13.370 --> 01:07:19.610
and we model the
residuals to not

01:07:19.610 --> 01:07:27.400
be of constant volatility,
but to be multiples of sort

01:07:27.400 --> 01:07:32.900
of white noise with mean 0
and variance 1, where sigma_t

01:07:32.900 --> 01:07:38.920
is given by this essentially
ARCH function, which

01:07:38.920 --> 01:07:42.200
says that the volatility
at a given period t

01:07:42.200 --> 01:07:48.020
is a weighted average
of the squared residuals

01:07:48.020 --> 01:07:50.870
over the last p lags.

01:07:50.870 --> 01:07:56.400
And so if there's a
large residual then

01:07:56.400 --> 01:08:02.810
that could persist and
make the next observation

01:08:02.810 --> 01:08:04.660
have a large variance.

01:08:04.660 --> 01:08:10.310
And so this accommodates
some time dependence.

01:08:10.310 --> 01:08:18.229
Now this model actually
has parameter constraints,

01:08:18.229 --> 01:08:21.790
which are never a
nice thing to have

01:08:21.790 --> 01:08:24.410
when you're fitting models.

01:08:24.410 --> 01:08:28.290
In this case the parameters
alpha_one through alpha_p

01:08:28.290 --> 01:08:30.279
all have to be positive.

01:08:30.279 --> 01:08:33.400
And why do they
have to be positive?

01:08:36.166 --> 01:08:37.262
AUDIENCE: [INAUDIBLE].

01:08:37.262 --> 01:08:37.970
PROFESSOR: Right.

01:08:37.970 --> 01:08:39.460
Variance is positive.

01:08:39.460 --> 01:08:42.089
So if any of these
alphas were negative,

01:08:42.089 --> 01:08:44.130
then there would be a
possibility that under this

01:08:44.130 --> 01:08:46.250
model that you could
have negative volatility,

01:08:46.250 --> 01:08:47.020
which you can't.

01:08:47.020 --> 01:08:55.109
So if we estimate this
model to estimate them

01:08:55.109 --> 01:08:57.729
with the constraint that
all these parameter values

01:08:57.729 --> 01:09:00.979
are non-negative.

01:09:00.979 --> 01:09:06.529
So that does complicate
the estimation a bit.

01:09:06.529 --> 01:09:12.580
In terms of understanding
how this process works

01:09:12.580 --> 01:09:17.880
one can actually see how
the ARCH model implies

01:09:17.880 --> 01:09:23.492
an autoregressive model for
the squared residuals, which

01:09:23.492 --> 01:09:24.450
turns out to be useful.

01:09:24.450 --> 01:09:28.380
So the top line there
is the ARCH model

01:09:28.380 --> 01:09:31.149
saying that the variance
of the t period return

01:09:31.149 --> 01:09:35.000
is this weighted average
of the past residuals.

01:09:35.000 --> 01:09:41.229
And then if we simply add
a new variable u_t, which

01:09:41.229 --> 01:09:51.399
is our squared residual minus
its variance, to both sides

01:09:51.399 --> 01:09:59.020
we get the next line, which says
that epsilon_t squared follows

01:09:59.020 --> 01:10:07.230
an autoregression on itself,
with the u_t value being

01:10:07.230 --> 01:10:10.800
the disturbance in
that autoregression.

01:10:10.800 --> 01:10:17.230
Now u_t, which is epsilon_t
squared minus sigma squared t,

01:10:17.230 --> 01:10:21.150
what is the mean of that?

01:10:21.150 --> 01:10:23.220
The mean is 0.

01:10:23.220 --> 01:10:25.030
So it's almost white noise.

01:10:25.030 --> 01:10:28.920
But its variance is maybe
going to change over time.

01:10:28.920 --> 01:10:31.130
So it's not sort of
standard white noise,

01:10:31.130 --> 01:10:34.950
but it basically
has expectation 0.

01:10:34.950 --> 01:10:38.040
It's also conditional
independent,

01:10:38.040 --> 01:10:40.850
but there's some possible
variability there.

01:10:40.850 --> 01:10:42.930
But what this implies
is that there basically

01:10:42.930 --> 01:10:45.770
is an autoregressive
model where we just

01:10:45.770 --> 01:10:50.190
have time-varying variances
in the underlying process.

01:10:50.190 --> 01:10:53.820
Now because of that
one can sort of quickly

01:10:53.820 --> 01:10:57.120
evaluate whether there's
ARCH structure in data

01:10:57.120 --> 01:10:58.960
by simply fitting an
autoregressive model

01:10:58.960 --> 01:11:00.970
to the squared residuals.

01:11:00.970 --> 01:11:02.750
And testing whether
that regression

01:11:02.750 --> 01:11:04.660
is significant or not.

01:11:04.660 --> 01:11:08.940
And that formally is a
Lagrange multiplier test.

01:11:08.940 --> 01:11:13.490
Some of the original papers by
Engle go through that analysis.

01:11:13.490 --> 01:11:17.880
And the test statistic
turns out to just

01:11:17.880 --> 01:11:24.620
be the multiple of the r
squared for that regression fit.

01:11:24.620 --> 01:11:33.690
And basically under,
say, a null hypothesis

01:11:33.690 --> 01:11:35.820
that there isn't
any ARCH structure,

01:11:35.820 --> 01:11:40.290
then this regression model
should have no predictability.

01:11:40.290 --> 01:11:41.820
This ARCH model
in the residuals,

01:11:41.820 --> 01:11:45.920
basically if there's no time
dependence in those residuals,

01:11:45.920 --> 01:11:52.620
that's evidence of there being
an absence of ARCH structure.

01:11:52.620 --> 01:11:56.640
And so under the null
hypothesis of no ARCH structure

01:11:56.640 --> 01:11:59.580
that r squared statistic
should be small.

01:11:59.580 --> 01:12:04.260
It turns out that sort of n
times the r squared statistic

01:12:04.260 --> 01:12:08.230
with p variables
is asymptotically

01:12:08.230 --> 01:12:11.370
a chi-square distribution
with p degrees of freedom.

01:12:11.370 --> 01:12:17.760
So that's where that test
statistic comes into play.

01:12:17.760 --> 01:12:23.180
And in implementing this, the
fact that we were applying

01:12:23.180 --> 01:12:25.630
essentially least squares
with the autoregression

01:12:25.630 --> 01:12:30.900
to implement this Lagrange
multiplier test, but we were

01:12:30.900 --> 01:12:33.530
assuming, well,
we're not assuming,

01:12:33.530 --> 01:12:35.980
we're implicitly assuming the
assumptions of Gauss-Markov

01:12:35.980 --> 01:12:37.730
in fitting that.

01:12:37.730 --> 01:12:42.920
This corresponds to the notion
of quasi-maximum likelihood

01:12:42.920 --> 01:12:47.040
estimates for
unknown parameters.

01:12:47.040 --> 01:12:51.870
And quasi-maximum
likelihood estimates

01:12:51.870 --> 01:12:56.570
are used extensively in some
stochastic volatility models.

01:12:56.570 --> 01:12:59.140
And so essentially situations
where you sort of use

01:12:59.140 --> 01:13:04.180
the normal approximation, or
the second order approximation,

01:13:04.180 --> 01:13:05.870
to get your estimates,
and they turn out

01:13:05.870 --> 01:13:11.930
to be consistent and decent.

01:13:16.570 --> 01:13:18.590
All right.

01:13:18.590 --> 01:13:20.700
Let's go to Maximum
Likelihood Estimation.

01:13:20.700 --> 01:13:23.800
OK Maximum Likelihood
Estimation basically

01:13:23.800 --> 01:13:28.350
involves-- the hard part
is defining the likelihood

01:13:28.350 --> 01:13:33.700
function, which is the
density of the data given

01:13:33.700 --> 01:13:35.290
the unknown parameters.

01:13:35.290 --> 01:13:40.743
In this case, the data are
conditionally independent.

01:13:50.630 --> 01:13:55.080
The joint density is the product
of the density of y_t given

01:13:55.080 --> 01:13:56.570
the information at t minus 1.

01:13:56.570 --> 01:14:00.520
So basically the joint
probability density

01:14:00.520 --> 01:14:03.479
is the density at each time
point conditional on the past,

01:14:03.479 --> 01:14:06.020
and then the density times the
density of the next time point

01:14:06.020 --> 01:14:07.130
conditional on the past.

01:14:07.130 --> 01:14:09.730
And those are all
normal random variables.

01:14:09.730 --> 01:14:13.760
So these are the normal
PDFs coming into play here.

01:14:13.760 --> 01:14:16.170
And so what we want
to do is basically

01:14:16.170 --> 01:14:18.150
maximize this
likelihood function

01:14:18.150 --> 01:14:19.440
subject to these constraints.

01:14:22.040 --> 01:14:23.820
And we already went
through the fact

01:14:23.820 --> 01:14:25.890
that the alpha_i's have
to be greater than zero.

01:14:25.890 --> 01:14:29.080
And it turns out you
also have to have

01:14:29.080 --> 01:14:33.124
that the sum of the
alphas is less than one.

01:14:33.124 --> 01:14:35.040
Now what would happen
if the sum of the alphas

01:14:35.040 --> 01:14:36.578
was not less than one?

01:14:40.036 --> 01:14:41.852
AUDIENCE: [INAUDIBLE].

01:14:41.852 --> 01:14:42.560
PROFESSOR: Right.

01:14:42.560 --> 01:14:45.680
And you basically could have
the process start diverging.

01:14:45.680 --> 01:14:50.570
Basically these
autoregressions can explode.

01:14:50.570 --> 01:14:52.590
So let's go through and see.

01:14:57.580 --> 01:14:58.080
Let's see.

01:15:00.828 --> 01:15:06.570
Actually, we're going to
go to GARCH models next.

01:15:06.570 --> 01:15:08.800
OK.

01:15:08.800 --> 01:15:09.300
Let's see.

01:15:17.464 --> 01:15:18.880
Let me just go
back here a second.

01:15:18.880 --> 01:15:19.379
OK.

01:15:19.379 --> 01:15:20.200
Very good.

01:15:20.200 --> 01:15:21.160
OK.

01:15:21.160 --> 01:15:23.780
In the remaining few minutes
let me just introduce you

01:15:23.780 --> 01:15:28.320
to the GARCH models.

01:15:36.560 --> 01:15:43.960
The GARCH model is
basically a series

01:15:43.960 --> 01:15:46.610
of past values of the
squared volatilities,

01:15:46.610 --> 01:15:55.860
basically the q sum of
past squared volatilities

01:15:55.860 --> 01:16:00.620
for the equation for the
volatility sigma t squared.

01:16:00.620 --> 01:16:06.300
And so it may be
that very high order

01:16:06.300 --> 01:16:09.760
ARCH models are
actually important.

01:16:09.760 --> 01:16:17.410
Or very high order ARCH terms
are found to be significant

01:16:17.410 --> 01:16:20.680
when you fit ARCH models.

01:16:20.680 --> 01:16:25.610
It could be that
much of that need

01:16:25.610 --> 01:16:29.540
is explained by adding
these GARCH terms.

01:16:29.540 --> 01:16:33.660
And so let's just consider
a simple GARCH model where

01:16:33.660 --> 01:16:40.020
we have only a first order ARCH
term and a first order GARCH

01:16:40.020 --> 01:16:40.830
term.

01:16:40.830 --> 01:16:43.230
So we're basically
saying that this

01:16:43.230 --> 01:16:48.470
is a weighted average of
the previous volatility,

01:16:48.470 --> 01:16:51.210
the new squared residual.

01:16:51.210 --> 01:16:58.060
And this is a very
parsimonious representation

01:16:58.060 --> 01:17:01.630
that actually ends up fitting
data quite, quite well.

01:17:01.630 --> 01:17:07.710
And there are various
properties of this GARCH model

01:17:07.710 --> 01:17:12.270
which we'll go
through next time,

01:17:12.270 --> 01:17:14.260
but I want to just
close this lecture

01:17:14.260 --> 01:17:20.750
by showing you fits of the ARCH
models and of this GARCH model

01:17:20.750 --> 01:17:25.500
to the euro/dollar
exchange rate process.

01:17:25.500 --> 01:17:31.080
So let's just look at that here.

01:17:34.521 --> 01:17:35.021
OK.

01:17:40.700 --> 01:17:41.200
OK.

01:17:41.200 --> 01:17:44.160
With the euro/dollar
exchange rate,

01:17:44.160 --> 01:17:47.110
actually there's
the graph here which

01:17:47.110 --> 01:17:51.380
shows the
auto-correlation function

01:17:51.380 --> 01:17:54.070
and the partial
auto-correlation function

01:17:54.070 --> 01:17:57.290
of the squared returns.

01:17:57.290 --> 01:18:02.600
So is there dependence in
these daily volatilities?

01:18:02.600 --> 01:18:07.470
And basically these blue
lines are plus or minus

01:18:07.470 --> 01:18:12.580
two standard deviations of
the correlation coefficient.

01:18:12.580 --> 01:18:16.950
Basically we have highly
significant auto-correlations

01:18:16.950 --> 01:18:20.900
and very highly significant
partial auto-correlations,

01:18:20.900 --> 01:18:24.290
which suggests if you're
familiar with ARMA process

01:18:24.290 --> 01:18:27.820
that you would need a very
high order ARMA process

01:18:27.820 --> 01:18:34.360
to fit the squared residuals.

01:18:34.360 --> 01:18:38.730
But this highlights how
with the statistical tools

01:18:38.730 --> 01:18:45.610
you can actually identify this
time dependence quite quickly.

01:18:45.610 --> 01:18:52.530
And here's a plot of the ARCH
order one model and the ARCH

01:18:52.530 --> 01:18:54.200
order two model.

01:18:54.200 --> 01:18:56.570
And on each of
these I've actually

01:18:56.570 --> 01:19:00.940
drawn a solid line where the
sort of constant variance model

01:19:00.940 --> 01:19:01.440
would be.

01:19:01.440 --> 01:19:05.300
So ARCH is saying that we
have a lot of variability

01:19:05.300 --> 01:19:09.270
about that constant mean.

01:19:09.270 --> 01:19:14.070
And a property, I guess,
of these ARCH models

01:19:14.070 --> 01:19:18.340
is that they all have
sort of a minimum value

01:19:18.340 --> 01:19:21.330
for the volatility that
they're estimating.

01:19:21.330 --> 01:19:23.020
If you look at
the ARCH function,

01:19:23.020 --> 01:19:26.770
that alpha_0 now is--
the constant term

01:19:26.770 --> 01:19:31.460
is basically the minimum
value, which that can be.

01:19:31.460 --> 01:19:35.790
So there's a constraint
sort of on the lower value.

01:19:35.790 --> 01:19:48.080
Then here's an
ARCH(10) fit which,

01:19:48.080 --> 01:19:51.190
it doesn't look like it sort of
has quite as much of a uniform

01:19:51.190 --> 01:19:54.940
lower bound in it, but one could
go on and on with higher order

01:19:54.940 --> 01:19:58.970
ARCH terms, but rather than
doing that one can also fit

01:19:58.970 --> 01:20:01.030
just a GARCH(1,1) model.

01:20:01.030 --> 01:20:04.250
And this is what it looks like.

01:20:04.250 --> 01:20:10.170
So basically the time varying
volatility in this process

01:20:10.170 --> 01:20:12.310
is captured really,
really well with just

01:20:12.310 --> 01:20:16.040
this two-parameter GARCH model
as compared with a high order

01:20:16.040 --> 01:20:17.990
autoregressive model.

01:20:17.990 --> 01:20:20.760
And it sort of
highlights the issues

01:20:20.760 --> 01:20:25.090
with the Wold decomposition
where a potentially infinite

01:20:25.090 --> 01:20:29.010
order autoregressive
model will effectively

01:20:29.010 --> 01:20:30.770
fit most time series.

01:20:30.770 --> 01:20:32.510
Well, that's nice
to know, but it's

01:20:32.510 --> 01:20:34.360
nice to have a parsimonious
way of defining

01:20:34.360 --> 01:20:36.390
that infinite
collection of parameters

01:20:36.390 --> 01:20:39.130
and with the GARCH model
a couple of parameters

01:20:39.130 --> 01:20:40.580
do a good job.

01:20:40.580 --> 01:20:42.820
And then finally here's
just a simultaneous plot

01:20:42.820 --> 01:20:47.270
of all those volatility
estimates on the same graph.

01:20:47.270 --> 01:20:53.010
And so one can see the
increased flexibility basically

01:20:53.010 --> 01:20:56.490
of the GARCH models compared to
the ARCH models for capturing

01:20:56.490 --> 01:20:59.280
time-varying volatility.

01:20:59.280 --> 01:21:00.840
So all right.

01:21:00.840 --> 01:21:02.970
I'll stop there for today.

01:21:02.970 --> 01:21:05.080
And let's see.

01:21:05.080 --> 01:21:10.290
Next Tuesday is a presentation
from Morgan Stanley so.

01:21:10.290 --> 01:21:15.570
And today's the last day to
sign up for a field trip.