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PROFESSOR: We introduced the data last time.
These were some macroeconomic variables
that can be used for forecasting the economy in terms of growth
and factors such as inflation or unemployment.
The case note goes through analyzing just three
of these economic time series-- the unemployment rate,
the federal funds rate, and a measure of the CPI,
or Consumer Price Index.
When one fits a vector autoregression model
to this data, it turns out that the roots
of the characteristic polynomial are 1.002, then 0.9863.
And you recall when our discussion of vector
autoregressive models, there's a characteristic equation
sort of in matrix form, the determinant
is just like the univariate autoregressive case.
And in order for the process to be invertible, basically,
the roots of the characteristic polynomial
need to be less than 1 in magnitude.
In this implementation of the vector autoregression model,
the characteristic roots are the inverses
of the characteristic roots that we've been discussing.
So anyway, this particular fit of the vector autoregression
model suggests that the process is non-stationary.
And so one should be considering different series
to model this as a stationary time series.
But in terms of interpreting the regression model,
one can see-- to accommodate the non-stationarity,
we can take differences of all the series
and fit the vector autoregression
to the difference series.
So one way of eliminating any non-stationarity in time series
models, basically eliminate the random walk
aspect of the processes, is to be modeling first differences.
And so doing that with this series-- let's see.
Here is just a graph of the time series properties
of the difference series.
So with our original series, we take differences and eliminate
missing values in this R code.
And this autocorrelation function
shows us basically the correlations
and autocorrelations of individual series
and the cross-correlations across the different series.
So along the diagonals are the autocorrelation function.
And one can see that every series
is correlation one with itself.
But then at the first lag, positive for the Fed
funds and the CPI measure.
And there's also some cross-correlations
that are strong.
And whether or not a correlation is strong or not
depends upon how much uncertainty there
is in our estimate of the correlation.
And these dashed lines here correspond
to plus or minus two standard deviations of the correlation
coefficient when the correlation coefficient is equal to 0.
So any correlations that sort of go beyond those bounds
is statistically significant.
The partial autocorrelation function is graphed here.
And let's say our time series problem
set goes through some discussion of the partial autocorrelation
coefficients and the interpretation of those.
The partial autocorrelation coefficients
are the correlation between one variable
and the lag of another after explaining
for all lower degree lags.
So it's like the incremental correlation of a variable
with a lag term that exists.
And so if we are actually fitting regression models where
we include extra lags of a given variable,
that partial autocorrelation coefficient
is essentially the correlation associated with the addition
of the final lagged variable.
So here, we can see that each of these series
is quite strongly correlated with itself.
But there are also some cross-correlations
with, like, the unemployment rate and the Fed funds rate.
Basically, the Fed funds rate tends
to go down when the unemployment rate goes up.
And so this data is indicating the association
between these macroeconomic variables
and the evidence of that behavior.
In terms of modeling the actual structural relations
between these, we need several, up to about 10
or 12 variables more than these three.
And then one can have a better understanding
of the drivers of various macroeconomic features.
But this sort of illustrates the use
of these methods with this reduced variable case.
Let me also go down here and just
comment on the unemployment rate or the Fed funds rate.
When fitting these vector autoregressive models
with the packages that exist in R,
they give us output which provides the specification
of each of the autoregressive models
for the different dependent variables, the different series
of the process.
And so here is the case of the regression model for Fed funds
as a function of unemployment rate lagged,
Fed funds rate lagged, and CPI lagged.
These are all on the different scale.
When you're looking at these results, what's
important is basically how strong
the signal-to-noise ratio is for estimating
these autoregressive parameters, vector
autoregressive parameters.
And so with the Fed funds, you can look at the t values.
And t values that are larger than 2
are certainly quite significant.
And you can see that basically when the unemployment rate
coefficient is a negative 0.71, so if the unemployment
rate goes up, we expect to see the Fed rate going down
the next month.
And the Fed funds rate for the lag 1 has a t value of 7.97.
So these are now models on the differences.
So if the Fed funds rate was increased
last month or last quarter, it's likely to be increased again.
And that's partly a factor of how slow the economy is
in reacting to changes and how the Fed doesn't
want to shock the economy with large changes in their policy
rates.
Another thing to notice here is that there's actually
a negative coefficient on the lag 2
Fed funds term, a negative 0.17.
And in interpreting these kinds of models,
I think it's helpful just to think of,
if you have Fed funds sub t, that's
equal to minus 0.71 times the unemployment rate at t minus 1.
And then we have plus 0.37 times the Fed funds, so t minus 1.
And this is delta.
And then minus 1.8 times the Fed funds.
So t minus 2.
In interpreting these coefficients,
notice that these two terms correspond
to 0.19 times the Fed funds change 1 lag ago plus 0.18
times the change in that rate.
So when you see multiple lags coming
into play in these models, the interpretation of them
can be made by considering different transformations
essentially of the underlying variables.
In this form, you can see that OK, the Fed funds
tends to change the way it changed the previous month.
But it also may change depending on the double change
in the previous month.
So there's a degree of acceleration in the Fed funds
that is being captured here.
So the interpretation of these models
sometimes requires some care.
This kind of analysis, I find it quite useful.
So let's push on to the next topic.
So today's topics are going to begin with a discussion
of cointegration.
Cointegration is a major topic in time series analysis, which
is dealing with the analysis of non-stationary time series.
And in the previous discussion, we
addressed non-stationarity of series
by taking first differences to eliminate
that non-stationarity.
But we may be losing some information
with that differencing.
And cointegration provides a framework
within which we characterize all available
information for statistical modeling,
in a very systematic way.
So let's introduce the context within which
cointegration is relevant.
It's relevant when we have a stochastic process,
a multivariate stochastic process, which
is integrated of some order d.
And to be integrated of order d means
that if we take the d-th difference,
then that d-th difference is stationary.
So and if you look at a time series
and you plot that over time, well, OK, a stationary time
series we know should be something that basically
has a constant mean over time.
There's some steady mean which that has.
And the variability is also constant.
With some other time series, it might increase linearly
over time.
And a series that increases linearly over time, well,
if you take first differences, that
tends to take out that linear trend.
If there are higher order differencing is required, then
that means that there's some curvature, quadratic say,
that may exist in the data that is being taken out.
So this differencing is required to result in stationarity.
If the process does have vector autoregressive representation
in spite of its non-stationarity,
then it can be represented by a polynomial lag of the x's is
equal to white noise epsilon.
And the polynomial phi of L going
to have a factor term in there of 1 minus L,
basically the first difference to the d power.
So if taking these the d-th order difference
reduces it to stationarity, then we
can express this vector autoregression in this way.
So the phi star of L basically represents
the stationary vector autoregressive process
on the d-th difference series.
Now, as it says here, each of the component series
may be non-stationary and integrated, say of order one.
But the process itself may not be jointly integrated.
In that it may be that there are linear combinations
of our multivariate series which are stationary.
And so these linear combinations basically
represent the stationary features of the process.
And those features can be apparent without looking
at differences.
So in a sense, if you just focused
on differences of these non-stationary multivariate
series, you would be losing out on information
of the stationary structure of contemporaneous components
of the multivariate series.
And so cointegration deals with this situation
where some linear combinations of the multivariate series
in fact are stationary.
So how do we represent that mathematically?
Well, we say that this multivariate time series
process is cointegrated if there exists an m-vector beta
such that, defining linear weights on the x's, and
beta prime X_t is a stationary process.
The cointegration vector of beta can be scaled arbitrarily.
So it's common practice, if one has
an interest, some primary interest, perhaps,
in the first component series of process,
to set that equal to 1.
And the expression basically says
that our time t value of the first series
is related in a stationary way to a linear combination
of the other m minus 1 series.
And this is a long-run equilibrium type relationship.
How does this arise?
Well, it arises in many, many ways in economics and finance.
The term structure of interest rates, purchase power parity.
In the terms structure of interest rates,
basically the differences between yields
on interest rates over different maturities,
those differences might be stationary.
The overall level of interest might not be stationary,
but the spreads ought to be stationary.
The purchase power parity in foreign exchange,
if you look at the value of currencies
for different countries, basically different countries
ought to be able to purchase the same goods for roughly
the same price.
And so if there are disparities in currency values,
purchase power parity suggests that things will revert back
to some norm where everybody is paying on average over time
the same amount for different goods.
Otherwise, there would be arbitrage.
Money demand, covered interest rate parity,
law of one price, spot and futures.
Let me show you another example that
will be in the case study for this chapter.
View, full screen.
Let's think about energy futures.
In fact, next Tuesday's talk from Morgan Stanley
is going to be an expert in commodity futures and options.
And that should be very interesting.
Anyway, here, I'm looking at energy futures
from the Energy Information Administration.
Actually, for this course, trying
to get data that's freely available to students
is one of the things we do.
So this data is actually available from the Energy
Information Administration of the government, which
is now open, so I guess that'll be updated over time.
But basically these energy futures
are traded on the Chicago Mercantile Exchange.
And basically CL is crude, West Texas intermediate crude,
light crude, which we have here, a time series from 2006
to basically yesterday.
And you can see how at the start of the period around $60
and then went up to close to $140,
and then it dropped down to around $40.
And it's been hovering around $100 lately.
The second series here is gasoline, RBOB gasoline.
Always have to look this up.
This is that reformulated blend stock for oxygenated blending
gasoline.
Anyway, futures on this product are traded at the CME as well.
And then heating oil.
And what's happening with these data
is that we have basically a refinery which processes
crude oil as an input.
And it basically refines it, distills it,
and generates outputs, which include heating oil, gasoline,
and various other things like jet fuel and others.
So if we're looking at the prices,
the futures prices of, say, gasoline and heating oil,
relating those to crude oil, well, certainly,
the cost of producing these products should depend
on the cost of the input .
So I've got in the next plot, a translation of these futures
contracts into their price per barrel.
Turns out crude is quoted in dollars per barrel.
And the gasoline heating oil are in cents per gallon.
So one multiplies.
There are 42 gallons in a barrel.
So you multiply those previous years by 42.
And this shows the plot of the prices of the futures
where we're looking at essentially the same units
of output relative to input.
And what's evident here is that while the futures for gasoline,
the blue, is consistently above the green, the input, and same
for heating oil.
And those vary depending on which is greater.
So if we look at the difference between, say,
the price of the heating oil future and the crude oil
future, what does that represent?
That's the spread in value of the output minus the input.
Ray?
AUDIENCE: [INAUDIBLE] cost of running the refinery?
PROFESSOR: So cost of refining.
So let's look at, say, heating oil minus CL and, say,
this RBOB minus CL.
So it's cost of refining.
What else could be a factor here?
AUDIENCE: Supply and demand characteristics [INAUDIBLE].
PROFESSOR: Definitely.
Supply and demand.
If one product is demanded a lot more than another.
Supply and demand.
Anything else?
AUDIENCE: Maybe for the outputs, if you
were to find the difference between the outputs,
it would be something cyclical.
For example, in the winter, heating oil
is going to get far more valuable as gasoline,
because people drive less and people demand more
for heating homes.
PROFESSOR: Absolutely.
That's a very significant factor with these.
There are seasonal effects that drive supply and demand.
And so we can put seasonal effects in there
as affecting supply and demand.
But certainly, you might expect to see seasonal structure here.
Anything else?
Put on your traders hat.
Profit, yes.
The refinery needs to make some profit.
So there has to be some level of profit that's
acceptable and appropriate.
So we have all these things driving basically
these differences.
Let's just take a look at those differences.
These are actually called the crack spreads.
Cracking in the business of refining
is basically the breaking down of oil
into components, products.
And on the top is the gasoline crack spread.
And the bottom is the heating oil crack spread.
And one can see that as time series,
these actually look stationary.
There certainly doesn't appear to be a linear trend up.
But there are, of course, many factors that could affect this.
So with that as motivation, how would we model such a series?
So let's go back to our lecture here.
All right, View, full size.
This is going to be a very technical discussion,
but it's, at the end of the day, I think fairly straightforward.
And the objective actually of this lecture
is to provide an introduction to the notation here, which
should make it seem like it's a very straightforward derivation
process of these models.
So let's begin with just a recap of the vector autoregressive
model of order p.
This is the extension of the univariate case where
we have a vector C of constants, m constants,
and matrices phi_1 to phi_p corresponding
to basically how the autoregression of one series
depends on all the other series.
And then there's multivariate white noise eta_t,
which has mean 0 and some covariance structure in it.
And the stationarity-- if this series were stationary,
then the determinant of this matrix polynomial
would have roots outside the unit circle for complex z.
And if it's not stationary, then some of those roots
will be on the unit circle or beyond.
So let's actually go to that non-stationary case
and suppose that the process is integrated of order one.
So if we were to take first differences,
we would have stationarity.
Well, the derivation of the model
proceeds by converting the original vector autoregressive
equation into an equation that's mostly
relating to differences but with also some extra terms.
So let's begin the process by just subtracting
the lagged value of the multivariate vector
from the original series.
So we subtract X_(t-1) from both sides,
and we get delta X_t is equal to C plus phi_1 minus I_m X_(t-1)
plus the rest.
So that's a very simple step.
We're just subtracting the lagged multivariate series
from both sides.
Now, what we want to do is convert
the second term in the middle line into a difference term.
So what do we do?
Well, we can subtract and add phi_1 minus I_m times X_(t-2).
If we do that, subtract and add that,
we then get the delta X_t is C plus a multiple of delta
X_(t-1) plus this multiple of X_(t-2).
So we basically reduced the equations
to differences in the first two terms
or in the current series and the lagged.
But then we have the original series for lags t minus 2.
We can continue this process with the third.
And then at the end of the day, we
end up getting this equation for the difference of the series
is equal to a constant plus a matrix multiple
of the first difference multivariate series,
plus another matrix times the second difference,
all the way down to the p-th difference,
or the p minus first difference.
But at the end, we're left with terms
at p lags that have no differences in them.
So we've been able to represent this series
as an autoregressive function of differences.
But there's also a term on the undifferenced series
at the end that's left over.
And or this argument can actually
proceed by eliminating differences in the reverse way,
starting with the p-th lag and going up.
And one then can represent this as delta X_t
is C plus some matrix times just the
lagged series plus various matrices times the differences
going back p minus 1 lags.
And so at the end of the day, this model
basically for delta X_t is a constant
plus a matrix times the previous lagged series
or the first lag of the multivariate time series,
plus various autoregressive lags of the differenced series.
So these notes give you the formulas for those,
and they're very easy to verify if you go through them
one by one.
And when we look at this expression for the model,
this expresses the stochastic process model
for the difference series.
This difference series is stationary.
We've eliminated the non-stationarity
in the process.
So that means the right-hand side
has to be stationary as well.
And so while the terms which are matrix multiples of lags
of the differenced series, those are
going to be stationary because we're just
taking lags of the stationary multivariate time
series, the difference series.
But this pi X_t term has to be stationary as well.
So this pi X_t contains the cointegrating terms.
And fitting a sort of cointegrated vector
autoregression model involves identifying this term, pi X_t.
And given that the original series had unit roots,
it has to be the case that pi, the matrix, is singular.
So it's basically a transformation
of the data that eliminates that unit
root in the overall series.
So the matrix pi is of reduced rank,
and it's either rank zero, in which case
there's no cointegrating relationships,
or its rank is less than m.
And the matrix pi does define the cointegrating
relationships.
Now, these cointegrating relationships
are the relationships in the process that are stationary.
And so basically there's a lot of information
in that multivariate series with contemporaneous values
of the series.
There is stationary structure at every single time
point, which can be the target of the modeling.
So this matrix pi is of rank r less than m.
And so it can be expressed as basically alpha beta
prime, where these matrices are of rank r, alpha and beta.
And the columns of beta define linearly independent vectors
which cointegrate x.
And the decomposition of pi isn't unique.
You can basically, for any invertible r by r matrix g,
define another set of cointegrating relationships.
So in the linear algebra structure of these problems,
there's basically an r-dimensional space
where the process is stationary, and how
you define the coordinate system in that space is up to you
or subject to some choice.
So how do we estimate these models?
Well, rather nice result of Sims, Stock, and Watson.
Actually, Sims, Christopher Sims,
he got the Nobel Prize a few years ago for his work
in econometrics.
And so this is a rather significant work that he did.
Anyway, he, together with Stock and Watson,
prove that if you're estimating a vector autoregression model,
then the least squares estimator of the original model
is basically sufficient to do an analysis
of this cointegrated vector autoregression process.
The parameter estimates from just fitting
the vector autoregression are consistent for the underlying
parameters.
And they have asymptotic distributions
that are identical to those of maximum likelihood estimators.
And so what ends up happening is the least squares estimates
of the vector autoregression parameters lead
to an estimation of the pi matrix.
And the constraints on the pi matrix which are basically pi
is of reduced rank, those will hold asymptotically.
So let's just go back to the equation before,
to see if that looks familiar here.
So what that work says is that if we basically
fit the linear regression model regressing the difference
series on the lag of the series plus lags of differences,
the least squares estimates of these underlying parameters
will give us asymptotically efficient estimates
of this overall process.
So we don't need to use any new tools to specify these models.
There's an advanced literature on estimation methods
for these models.
Johansen does describe maximum likelihood estimation
when the innovation terms are normally distributed.
And that methodology applies reduced rank regression
methodology and yields tests for what
the rank is of the cointegrating relationship.
And these methods are implemented in our packages.
Let's see.
Let me just go back now to the-- so let's see.
The case study on the crack spread data
actually goes through sort of testing for non-stationarity
in these underlying series.
And actually, why don't I just show you that?
Let's go back here.
If you can see this, for the crack spread data,
looking at the crude oil futures,
basically the crude oil future can be evaluated
to see if it's non-stationary.
And there's this augmented Dickey-Fuller test
for non-stationarity.
And it basically has a null hypothesis that the model
or the series is non-stationary, or it has a unit root,
versus the alternative that it doesn't.
And so testing that null hypothesis
that it's non-stationary yields a p-value of 0.164
for CLC1, the first nearest contract,
near month contract of the futures for crude.
And so the data suggests that crude
has a distribution that's non-stationary, integrated
order 1.
And the HOC1 also basically has a test for-- p-value
for non-stationarity of 0.3265.
So we can't reject non-stationarity or unit root
in those series with these test statistics.
In analyzing the data, this suggests that we basically
need to accommodate that non-stationarity when
we specify the models.
Let me just see if there's some results here.
For this series, actually the case notes
will go through actually conducting this Johansen
procedure for testing for the rank
of the cointegrated process.
And that test basically has different test statistic
for testing whether the rank is 0, 1, less than or equal to 1,
or less than or equal to 2.
And one can see that there's marginal-- the test
statistic is almost significant at the 10% level
for the overall series.
It's not significant for the rank
being less than or equal to 1.
And so these results, it doesn't suggest there's
strong non-stationarity.
But certainly with that non-stationarity
is no more than rank one for the series.
And the eigenvector corresponding
to the stationary relationship is
given by these coefficients of 1 on the crude oil future,
1.3 on the RBOB and minus 1.7 on the heating oil.
So what this suggests is that there's
considerable variability in these energy futures contracts.
What appears to be stationary is some linear combination
of crude plus gasoline minus heating oil.
And in terms of why does it combine that way,
well, there are all kinds of factors
that we went through-- cost of refining, supply and demand,
seasonality, which affect things.
And so when analyzed, sort of ignoring seasonality,
these would be the linear combinations that appear
to be stationary over time.
Yeah?
AUDIENCE: Why did you choose to use the futures
prices as opposed to the spot?
And how did you combine the data with actual [INAUDIBLE]?
PROFESSOR: I chose this because if refiners are wanting
to hedge their risks, then they will go to the futures market
to hedge those.
And so working with these data, one
can then consider problems of hedging refinery production
risks.
And so that's why.
AUDIENCE: [INAUDIBLE]
PROFESSOR: OK, well, the Energy Information Administration
provides historical data which gives the first month,
the second month, the third month available for each
of these contracts.
And so I chose the first month contract
for each of these features.
Those 10 are the most liquid.
Depending on what one is hedging,
one would use perhaps longer periods for those.
There's some very nice finance problems
dealing with hedging, hedging these kinds of risks,
and as well as trading these kinds of risk.
Traders can try to exploit short term movements in these.
Anyway, I'll let you look through these,
the case note later.
And it does provide some detail on the coefficient estimates.
And one can basically get a handle
on how these things are being specified.
So let's go back.
The next topic I want to cover is linear state-space models.
It turns out that many of these time series
models appropriate in economics and finance
can be expressed as a linear state-space model.
I'm going to introduce the general notation first and then
provide illustrations of this general notation
with a number of different examples.
So the formulation is we have basically an observation vector
at time t, y_t.
This is our multivariate time series that we're modeling.
Now, I've chosen it to be k-dimensional
for the observations.
There's an underlying state vector
that's of m dimensions, which basically characterizes
the state of the process at time t.
There's an observation error vector at time t, epsilon_t.
So it's k by 1 as well, corresponding to y.
And there's a state transition innovation error vector,
which is n by 1, which actually can
be different from m, the dimension of the state vector.
So we have-- in the state space specification,
we're going to specify two equations, one
for how the states evolve over time and another for how
the observations or measurements evolve,
depending on the underlying states.
So let's first focus on a state equation
which describes how the state progresses
from the state at time t to the state at time t plus 1.
Because this is a linear state-space model,
basically the state at t plus 1 is
going to be some linear function of the states at time
t plus some noise.
And that noise is given by eta_t,
being independent identically distributed white noise,
or normally distributed with some covariance matrix
Q_t, positive definite.
And R_t is some linear transformation
of those, which characterize the uncertainty
in the particular states.
So there's a great deal of flexibility
here in how things depend on each other.
And right now, it will appear just like a lot of notation.
But as we see it in different cases,
you'll see how these terms come into play.
And they're very straightforward.
So we're considering simple linear transformations
of the states plus noise.
And then the observation equation or measurement
equation is a linear transformation
of the underlying states plus noise.
So the matrix Z_t is the observation coefficients
matrix.
And the noise or innovations epsilon_t are, we'll assume,
independent identically distributed
normal, multivariate normal random variables
with some covariance matrix H_t.
To be fully general, the subscript t
means the covariance can depend on time t.
It doesn't have to, but it can.
These two equations can be written together
in a joint equation where we see that the underlying
state at time t, s, gets transformed with T sub t
to the state at t plus 1 plus residual innovation term.
And the observation equation y_t is Z_t s_t plus that.
So we're representing how the states evolve over time
and how the observations depend on the underlying
states in this joint equation.
And the structure of basically this sort
of linear function of states plus error, the error term u_t
here is normally distributed with covariance matrix omega,
which has this structure.
It's a block diagonal.
We have the covariance of the epsilons as the H.
And the covariance of R_t eta_t is R_t Q_t R_t transpose.
So you may recall when we take a covariance matrix
of linear function of random variables given by a matrix,
then it's that linear function R times the covariance matrix
times the transpose.
So that term comes into play.
So let's see how a capital asset pricing
model with time-varying betas can be represented
as a linear state-space model.
You'll recall, we discussed this model a few lectures ago,
where we have the excess return of a given stock, r_t,
is a linear function of the excess return of the market
portfolio, r_(m,t), plus error.
What we're going to do now is extend that previous model
by adding time dependence, t, to the regression parameters.
The alpha is not a constant.
It is going to vary by time.
And the beta is also going to very by time.
And how will they vary by time?
Well, we're going to assume that the alpha_t is
a Gaussian random walk.
And the beta is also a Gaussian random walk.
And with that set up, we have the following expression
for the state equation.
OK, the state equation, which is just the unknown parameters--
it's the alpha and the beta at given time t.
The state at time t gets adjusted
to the state at time t plus 1 by just adding these random walk
terms to it.
So it's a very simple process.
We have the identity times the previous state
plus the identity times this vector of these innovations.
So s_(t+1) is equal to T_t s_t plus R_t eta_t,
where this matrix, T sub t and R sub t are trivial;
they're just the identity.
And eta_t has a covariance matrix
which is just given by Q_t, sigma squared nu,
sigma squared epsilon.
This is a complex way, perhaps, of representing this model.
But it puts this simple model into that linear state-space
framework.
Now, the observation equation is given by this expression
defining the Z_t matrix as the unit element and r_(m,t) So
it's basically a row vector, or a row matrix, one-row matrix.
And epsilon_t is the white noise process.
Now, putting these equations together,
we basically have the equation for the state transition
and the observation equation together.
We have this form for that.
So now, let's consider a second case
of linear regression models where
we have a time varying beta.
In a way, this case we just looked at
is a simple case of that.
But let's look at a more general case
where we have p independent variables, which
could be time-varying.
So we have a regression model almost
as we've considered it previously.
y_t is equal to x_t transpose beta_t plus epsilon_t.
The difference now is our regression coefficients
beta are allowed to change over time.
How do they change over time?
Well, we're going to assume that those also
follow independent random walks with variances
of the random walks that may depend on the component.
So the joint state-space equation
here is given by the identity times s_t plus eta_t.
That's basically the random walk process for the underlying
regression parameters.
And y_t is equal to x_t transpose
times the same regression parameters plus the observation
error.
I guess needless to say, if we consider the special case where
the random walk process is degenerate
and they're basically steps of size zero,
then we get the normal linear regression model coming out
of this.
If we were to be specifying the linear state-space
implementation of this model and consider successive estimates
of the model parameters over time,
then these equations would give us recursive estimates
for updating regressions as we add
additional values to the data, additional observations
to the data.
Let's look at autoregressive models of order p.
The autoregressive model of order p for a univariate time
series has the setup given here.
It's a polynomial lag of the response
variable y_t is equal to the innovation epsilon_t.
And we can define the state vector
to be equal to the vector of p values, p successive values
of the process.
And so we basically get a combination
here of the observation equation and state equation joining
where basically one of the states
is actually equal to the observation.
And basically, with this definition
for a state of the vector at the next time point t,
that is equal to this linear transformation
of the lagged state vector plus that innovation term.
I dropped the mic.
So the notation here shows the structure
for how this linear state-space model is evolving.
Basically, the observation equation
is the linear combination of the five
multiples of lags of the values plus the residual.
And the previous lags of the states
are just simply the identities times those values, shifted.
So it's a very simple structure for the autoregressive process
as a linear state-space model.
We have, as I was just saying, for the transition matrix T sub
t, this matrix and the observation equation
is essentially picking out the first element of the state
vector, which has no measurement error.
So that simplifies that.
The moving average model of order q
could also be expressed as a linear state-space model.
Remember, the moving average model
is one where our response variable, y, is simply
some linear combination of innovations,
q past innovations.
And this state vector, if we consider
the state vector just being basically q
lags of the innovations, then the transition
of those underlying states is given by this expression here.
And we have a state equation, an observation equation,
which has these forms for these various transition matrices
and for how the innovation terms are related.
Let me just finish up with example
showing with the autoregressive moving average model.
And many years ago, it was actually
very difficult to specify the estimation
methods for autoregressive moving average models.
But the implementation of these models
as linear state-space models facilitated that greatly.
And with the ARMA model, the setup basically
is a combination of the autoregressive
moving average processes.
We have an autoregression of the y's
is equal to a moving average of the residuals
or the innovations.
And it's convenient in the setup for linear state-space models
to define the dimension m, which is the maximum of p and q
plus 1, and think of having basically a possibly m order
polynomial lag for each of those two series.
And we can basically constrain those values
to be 0 if m is greater than p or m is greater than q.
And Harvey, in a very important work in '93,
actually defined a particular state-space representation
for this process.
And I guess it's important to know
that with these linear state-space models,
we're dealing with characterizing structure
in m-dimensional space.
There's often some choice in how you represent your underlying
states.
You can basically re-parametrize the models
by considering invertible linear transformations
of the underlying states.
So let me go back here.
In expressing the state equation generally
is T sub t s_t plus R_t eta_t.
This matrix T sub t and st-- basically s_t
can be replaced by a linear transformation of s_t,
so long as we multiply the T sub t by the inverse
of that transformation.
So there's flexibility in the choice
of our linear state-space specification.
And so there really are many different equivalent linear
state-space models for a given process depending
on exactly how you define the states
and the underlying transformation matrix T.
And the beauty of Harvey's work was coming up
with a nice representation for the states,
where we had very simple forms for the various matrices.
And the lecture notes here go through the derivation
of that for the ARMA process.
And this derivation is-- I just want
to go through the first case just
to highlight how the argument goes.
We basically have this equation, which is the original equation
for an ARMA(p,q) process.
And Harvey says, well, define the first--
or the state at time t to be equal to the observation
at time t.
If we do that, then how does this equation relate
to the basically-- this is the state at the next time point, t
plus 1, is equal to phi_1 times the state at time t,
plus a second state at time t and a residual innovation
eta_t.
So by choosing the first state to be the observation value
at that time, we can then solve for the second state,
which is given by this expression,
just by rewriting our model equation in terms of s_(1,t),
s_(2,t) and eta_t.
So this s_(2,t) is this function of the observations and eta_t.
So it's a very simple specification
of the second state.
Just what is that second state element
given this definition of the first one?
And one can do this process iteratively
getting rid of the observations and replacing them
by underlying states.
And at the end of the day, you end up
with this very simple form for the transition matrix T.
Basically, the T has the autoregressive components
as the first column of the T matrix.
And this R matrix has this vector of the moving
average components.
So it's a very nice way to represent the model.
Coming up with it was something very clever that he did.
But what one can see is that this basic model where
you have the states transitioning according
to a linear transformation of the previous state plus error,
and the observation being some function of the current states,
plus error or not, depending on the formulation,
is the representation.
Now, with all of these models, a reason
why linear state-space modeling is in fact effective
is that their specification is fully specified with the Kalman
filter.
So with this formulation of linear state-space models,
the Kalman filter as a methodology is
the recursive computation of the probability density
functions for the underlying states at basically
t plus 1 given information up to time t,
as well as the joint density of the future state
and the future observation at t plus 1, given information up
to time t.
And also just the marginal distribution
of the next observation given the information up to time t.
So what I want to do is just go through with you
how the Kalman filter is implemented and defined.
And the implementation of the Kalman filter
requires us to have some notation that's a bit involved,
but we'll hopefully explain it so it's very straightforward.
There are basically conditional means of the states.
s sub t given t is the mean value
of the state at time t given the information up to time t.
If we condition on t minus 1, then
it's the expectation of the state
at time t given the information up to t minus 1.
And then y t t minus 1 is the expectation
of the observation given information up to t minus 1.
There's also conditional covariances
and mean squared errors.
All these covariances are determined by omegas.
The subscript corresponds to states s, or observation y.
And basically, the conditioning set
is either information up to time t, t minus 1 or t minus 1
in the second case.
And we want to compute basically the covariance matrix
of the states given whatever the information is, information
up to time t, t minus 1.
So these covariance matrices are the expectation
of the state minus their expectation
under the conditioning times the state minus the expectation
transpose.
That's the definition of that covariance matrix.
So the different definitions here
correspond to just whether we're conditioning
on different information.
And then the observation innovations or residuals
are the difference between an observation y_t
and its estimate given information up to t minus 1.
So the residuals in this process are the innovation residuals,
one period ahead.
And the Kalman filter consists of four steps.
We basically want to, first, predict the state vector
one step ahead.
So given our estimate of the state vector at time t minus 1,
we want to predict this state vector at time t.
And we also want to predict the observation
at time t given our estimate at state vector time t minus 1.
And so at time t minus 1, we can estimate these quantities.
[INAUDIBLE]
At t minus 1, we can basically predict
what the state is going to and predict what
the observation is going to be.
And we can estimate how much error there's
going to be in those estimates, by these covariance matrices.
The second step is updating these predictions
to get our estimate of the state given the observation at time t
and to update our uncertainty about that state given
this new observation.
So basically, our estimate of the state at time t
is an adjustment to our estimate given information up
to t minus 1, plus a function of the difference between what we
observed and what we predicted.
And this T_t function matrix is called the filter gain matrix.
And basically, it characterizes how
do we adjust our prediction of the underlying state
depending on what happened.
So that's the filter gain matrix.
So we actually do gain information
with each observation about what the new value of the process
is.
And that information is characterized
by filter gain matrix.
You'll notice that the uncertainty
in the state at time t, this omega_s of t given t, that's
equal to the covariance matrix given t minus 1.
So it's our beginning level of uncertainty adjusted
by a term that tells us how much information did we
get from that new information.
So notice that there's a minus sign there.
We're basically reducing our uncertainty
about the state given the information in the innovation
that we now have observed.
Then, there's a forecasting step which
is used to forecast the state one period forward,
is simply given by this linear transformation
of the previous state.
And we can also update our covariance matrix
for future states given the previous state
by applying this formula which is a recursive formula
for estimating covariances.
So we have forecasting algorithms
that are simple linear functions of these estimates.
And then finally, there's a smoothing step
which is characterizing the conditional expectation
of underlying states, given information in the whole time
series.
And so ordinarily with Kalman filters, Kalman filters
are applied sequentially over time
where one basically is predicting ahead
one step, updating that prediction,
predicting ahead another step, updating the information
on the states.
And that overall process is the process
of actually computing the likelihood
function for these linear state-space models.
And so the Kalman filter is basically ultimately applied
for successive forecasting of the process
but also for helping us identify what the underlying model
parameters are using maximum likelihood methods.
And so the likelihood function for the linear state-space
model is basically the-- or the log-likelihood
is the log-likelihood of the entire data series,
give the unknown parameters.
But that can be expressed as the product
of the conditional distributions of each successive observation,
given the history.
And so basically, the likelihood of theta
is the likelihood of the first observation
times the density of the second observation given
the first times and so forth for the whole series.
And so the likelihood function is basically
a function of all these terms that we were computing
with the Kalman filter.
And with the Kalman filter, it basically
provides all the terms necessary for this estimation.
If the error terms are normally distributed,
then the means and variances of these estimates
are in fact characterizing the exact distributions
of the process.
Basically, we're taking-- if the innovation series are
all normal random variables, then
the linear state-space model, all
it's doing is taking linear combinations of normals
for the underlying states and for the actual observations.
And normal distributions are fully
characterized by their mean vectors
and covariance matrices.
And the Kalman filter provides a way
to update these distributions for all these features
of a model, the underlying states
as well as the distributions of the observations.
So that's a brief introduction the Kalman filter.
Let's finish there.
Thank you.