Chapter 2. Unobserved Component models

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Chapter 2. Unobserved Component models

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

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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

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• 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

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• 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

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• 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

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• 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

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• 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.

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• 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

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• 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

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• 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

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• 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.

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• 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

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• 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.

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

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

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• 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.

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• 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

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

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• 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

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• The conditional covariance can be easily derived
  by writting

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

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• 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.

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• 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.

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• 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

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• 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).

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• 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

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• 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).

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• 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).

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

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

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• 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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• QML estimates of the Random walk plus noise
  parameters
• Diagnostics of innovations

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• Diagnostics of auxiliary residuals

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Hewlett Packard
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• QML estimates of the Random walk plus noise
  parameters
• Diagnostics of innovations

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• Diagnostics of auxiliary residuals

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