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Chapter 2. Unobserved Component models

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Title: Chapter 2. Unobserved Component models

1
Chapter 2. Unobserved Component models

Esther Ruiz
2006-2007
PhD Program in Business Administration and
Quantitative Analysis
Financial Econometrics

2
2.1 Description and properties

Unobserved component models assume that the
variables of interest are made of components with
a direct interpretation that cannot be directly
observed
Applications in finance
Fads model of Potterba and Summers (1998).
There are two types of traders informed (µt) and
uninformed (et). The observed price is yt

Models for Ex ante interest differentials
proposed by Cavaglia (1992) We observe the ex
post interest differential which is equal to the
ex ante interest differential plus the
cross-country differential in inflation

Factor models simplify the computation of the
covariance matrix in mean-variance portfolio
allocation and are central in two asset pricing
theories CAPM and APT

Term structure of interest rates model proposed
by Rossi (2004) The observed yields are given by
the theoretical rates implied by a no arbitrage
condition plus a stochastic disturbance

Modelling volatility
There are two main types of models to represent
the dynamic evolution of volatilities
GARCH models that assume the volatility is a
non-linear funcion of past returns

s is the one-step ahead (conditional) variance
and, therefore, can be observed given
observations up to time t-1.
As a result, classical inference procedures can
be implemented.

Example Consider the following GARCH(1,1) model
for the IBEX35 returns
The returns corresponding to the first two days
in the sample are 0.21and -0.38

In this case, there are not unobserved components
but consider the model for fundamental prices
with GARCH errors
In this case, the variances of the noises cannot
be observed with information available at time t-1

ii) Stochastic volatility models assume that the
volatility represents the arrival of new
information into the market and, consequently, it
is unobserved
Both models are able to represent
Excess kurtosis
Autocorrelations of squares small and persistent

Although the properties of SV models are more
attractive and closer to the empirical properties
observed in real financial returns, their
estimation is more complicated because the
volatility, st, cannot be observed one-step-ahead.

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2.2 State space models

The Kalman filter allows the estimation of the
underlying unobserved components.
To implement the Kalman filter we are writting
the unobserved model of interest in a general
form known as state space model.
The state space model is given by
where the matrices Zt, Ht, Tt and Qt can evolve
over time as far as they are known at time t-1.

Consider, for example, the random walk plus noise
model proposed to represent fundamental prices in
the market.
In this case, the measurement equation is given
by
Therefore, Zt1, the state at is the underlying
level µt and Ht
The transition equation is given by
and Tt1 and Qt

Unobserved component models depend on several
disturbances. Provided de model is linear, the
components can be combined to give a model with a
single disturbance reduced form.
The reduced form is an ARIMA model with
restrictions in the parameters.

Consider the random walk plus noise model
In this case
The mean and variance of are given by

The autocorrelation function is given by
signal to noise ratio.
The reduced form is an IMA(1,1) model with
negative parameter where

When, q0, reduces to a non-invertible MA(1)
model, i.e. yt is a white noise process. On the
other hand, as q increases, the autocorrelations
of order one, and consequently, ? , decreases. In
the limit, if , is a white noise and yt
is a random walk.

Although we are focusing on univariate series,
the results are valid for multivariate systems.
The Kalman filter is made up of two sets of
equations
i) Prediction equations one-step ahead
predictions of the states and their corresponding
variances
For example, in the random walk plus noise model

ii) Updated equations Each new observation
changes (updates) our estimates of the states
obtained using past information
where

The updating equations can be derived using the
properties of the multivariate normal
distribution.
Consider the distribution of at and yt
conditional on past information up to and
including time t-1.
The conditional mean and variance are

The conditional covariance can be easily derived
by writting

Consider, once more the random walk plus noise
model. In this case,

The Kalman filter needs some initial conditions
for the state and its covariance matrix at time
t0. There are several alternatives. One of the
simplest consists on assuming that the state at
time zero es equal to its marginal mean and P0 is
the marginal variance.
However, when the state is not stationary this
solution is not factible. In this case, one can
initiallize the filter by assuming what is known
as a diffuse prior (we do not have any
information about what happens at time zero m00
and P08.

Now we are in the position to run the Kalman
filter. Consider, for example, the random walk
plus noise model and that we want to obtain
estimates of the underlying level of the series.
In this case, the equations are given by

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Consider, for example, that we have observations
of a time series generated by a random walk with
and
1.14, 0.59, 1.58,.

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The Kalman filter gives
One-step ahead and updated estimates of the
unobserved states and their associated mean
squared errors at/t-1, Pt/t-1, at and Pt
One-step ahead estimates of yt
One-step ahead errors (innovations) and their
variances, ?t and Ft

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Smoothing algorithms

There are also other algorithms known as
smoothing algorithms that generate estimates of
the unobserved states based on the whole sample
The smoothers are very useful because they
generate estimates of the disturbances associated
with each of the components of the model
auxiliary residuals.

For example, in the random walk plus noise model

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The auxiliary residuals are useful to
identify outliers in different components Harvey
and Koopman (1992)
test whether the components are heteroscedastic
Broto and Ruiz (2005a,b)
This test is based on looking at the differences
between the autocorrelations of squares and the
squared autocorrelations of each of the auxiliary
residuals Maravall (1983).

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Prediction

One of the main objectives when dealing with time
series analysis is the prediction of future
values of the series of interest. This can easily
be done in the context of state space models by
running the prediction equations without the
updating equations
In the context of the random walk plus noise
model

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Estimation of the parameters

Up to now, we have assumed that the parameters of
the state space model are known. However, in
practice, we need to estimate them. In Gaussian
state space models, the estimation can be done by
Maximum Likelihood. In this case, we can write
The expression of the log-likelihood is then
given by

The asymptotic properties of the ML estimator
are the usual ones as far as the parameters lie
on the interior of the parameter space. However,
in many models of interest, the parameters are
variances, and it is of interest to know whether
they are zero (we have deterministic components).
In some cases, the asymptotic distribution could
still be related with the Normal but is modified
as to take into account of the boundary see
Harvey (1989).

If the model is not conditionally Gaussian, then
maximizing the Gaussian log-likelihood, we obtain
what is known as the Quasi-Maximum Likelihood
(QML) estimator. In this case, the estimator
looses its eficiency. Furthermore, droping the
Normality assumption tends to affect the
asymptotic distribution of all the parameters. In
this case, the asymptotic distribution is given
by
where

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Unobserved component models for financial time
series

We are considering two particular applications of
unobserved component models with financial data
Dealing with stochastic volatility
Heteroscedastic components

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Stochastic volatility models

Understanding and modelling stock volatility is
necessary to traders for hedging against risk, in
options pricing theory or as a simple risk
measure in many asset pricing models.
The simplest stochastic volatility model is given
by

Taking logarithms of squared returns, we obtain a
linear although non-Gaussian state space model
Because, log(yt)2 is not truly Gaussian, the
Kalman filter yields minimium mean square linear
estimators (MMSLE) of ht and future observations
rather than minimum mean square estimators
(MMSE).

If y1, y2,,yT is the returns series, we
transform the data by
The Kalman filter is given by

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SP500
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Canadian/Dollar
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Unobserved heteroscedastic components

Unobserved component models with heteroscedastic
disturbances have been extensively used in the
analysis of financial time series for example,
multivariate models with common factors where
both the idiosyncratic and common factors may be
heteroscedastic as in Harvey, Ruiz and Sentana
(1992) or univariate models where the components
may be heteroscedastic as in Broto and Ruiz
(2005b).

To simplify the exposition, we are focusing on
the random walk plus noise model with ARCH(1)
disturbances given by
where

The model can be written in state space form as
follows
where

The model is not conditionally Gaussian, since
knowledge of past observations does not, in
general, imply knowledge of past disturbances.
Nevertheless we can proceed on the basis that the
model can be treated as though it were
conditionally Gaussian, and we will refer to the
Kalman filter as being quasi-optimal.

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Nikkei
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