EVALUATING THE STATISTICAL PROPERTIES OF TIME SERIES NON PARAMETRIC ESTIMATORS BY MEANS OF SMOOTHING

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EVALUATING THE STATISTICAL PROPERTIES OF TIME
SERIES NON PARAMETRIC ESTIMATORS BY MEANS OF
SMOOTHING MATRICES

• Estela Bee Dagum, Alessandra Luati
• Department of Statistics
• University of Bologna, Italy
• e-mail beedagum_at_stat.unibo.it,
  luati_at_stat.unibo.it

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OUTLINE OF THE STUDY

• Contents theoretical and empirical study on the
  algebraic and statistical properties of smoothing
  matrices in time series.
• Motivation the short-term trend-cycle estimation
  of seasonally adjusted series.
• Purpose to generalize the theory of symmetric
  projection matrices to smoothing matrices (not
  symmetric) and to define matrix-based measures of
  goodness of fit and smoothness.

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THE PROBLEM

• Linear filters in time series smoothing methods
  have always been studied by means of classical
  spectral analysis techniques, i.e. gain and
  phaseshift functions (Priestley, 1981).
• Recently, weight-based measures of bias, variance
  and mean square error have been proposed (Dagum
  and Luati, 2002 SMA, 2004 SNDE).

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THE WEIGHT-BASED MEASURES

• are easy to apply
• are independent on the time series they are
  applied on
• provide information on the bias and variance
  produced on the estimates by the smoothing
  procedure
• agree with the classical spectral ones

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MATRIX REPRESENTATION

• Linear estimators can be represented in matrix
  form through the corresponding representative
  matrix, called smoothing matrix.
• Matrix-based measures of bias, variance and mean
  square error can be defined
• The theory of symmetric smoother matrices in
  multivariate statistics (Hastie and Tibshirani,
  1990) can be generalized to non symmetric
  smoothing matrices in time series.

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SMOOTHING MATRICES IN TIME SERIES

• A linear operator acting on a time series
• to produce smooth estimates of the underlying
  trend can be represented in matrix form as
• where S is the SxS smoothing matrix of the form
  (Dagum and Luati, 2000)

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SMOOTHING MATRICES IN TIME SERIES

• where
• Ws band matrix of 2m 1 symmetric weights to be
  applied to central observations
• Wa, Wa matrices whose row elements are the
  asymmetric weights for first and last
  observations, respectively.

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• With respect to symmetric filters,
• ASYMMETRIC FILTERS
• concern the most recent data
• produce phaseshifts in the output
• are not time invariant
• are often obtained based on different assumptions.

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• It is important to distinguish between symmetric
  and asymmetric filters when studying the smoother
  properties.

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CENTROSYMMETRIC STRUCTURE OF SMOOTHING MATRICES

• The matrix S is centrosymmetric, i.e. wij
  wN1-j,N1-j
• Centrosymmetric matrices are symmetric with
  respect to their geometric center.
• The matrix Ws is rectangular centrosymmetric
• The matrices Wa and Ws are the t-transform of
  each other, i.e.
• Wa t (Wa)

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THE t-TRANSFORMATION

• t is the linear transformation (Dagum and Luati,
  2004 LAA)
• t Rmxn? Rmxn , A ? t (A), aij ? am1-,n1-j
• or equivalently
• t(A) EmAEn
• where En?Rn?n is the permutation matrix with
  ones on the cross diagonal (bottom left to top
  right) and zeros elsewhere.

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EXAMPLE smoothing matrix associated to a 13-term
Gaussian kernel estimator
0.347 0.302 0.199 0.100 0.038 0.011 0.002 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.232
0.267 0.232 0.153 0.077 0.029 0.008 0.002 0.000
0.000 0.000 0.000 0.000 0.000 0.000 0.133 0.201
0.231 0.201 0.133 0.066 0.025 0.007 0.002 0.000
0.000 0.000 0.000 0.000 0.000 0.062 0.125 0.189
0.217 0.189 0.125 0.062 0.024 0.007 0.001 0.000
0.000 0.000 0.000 0.000 0.023 0.061 0.122 0.184
0.212 0.184 0.122 0.061 0.023 0.007 0.001 0.000
0.000 0.000 0.000 0.007 0.023 0.060 0.121 0.183
0.210 0.183 0.121 0.060 0.023 0.007 0.001 0.000
0.000 0.000 0.001 0.007 0.023 0.060 0.121 0.183
0.210 0.183 0.121 0.060 0.023 0.007 0.001 0.000
0.000 0.000 0.001 0.007 0.023 0.060 0.121 0.183
0.210 0.183 0.121 0.060 0.023 0.007 0.001
0.000 0.000 0.000 0.001 0.007 0.023 0.060 0.121
0.183 0.210 0.183 0.121 0.060 0.023 0.007
0.001 0.000 0.000 0.000 0.001 0.007 0.023 0.060
0.121 0.183 0.210 0.183 0.121 0.060 0.023
0.007 0.000 0.000 0.000 0.000 0.001 0.007 0.023
0.061 0.122 0.184 0.212 0.184 0.122 0.061
0.023 0.000 0.000 0.000 0.000 0.000 0.001 0.007
0.024 0.062 0.125 0.189 0.217 0.189 0.125
0.062 0.000 0.000 0.000 0.000 0.000 0.000 0.002
0.007 0.025 0.066 0.133 0.201 0.231 0.201
0.133 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.002 0.008 0.029 0.077 0.153 0.232 0.267
0.232 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.002 0.011 0.038 0.100 0.199 0.302 0.347
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ALGEBRAIC PROPERTIES OFCENTROSYMMETRIC MATRICES

• Centrosymmetric matrices inherit desirable
  properties from the properties of the permutation
  matrices En which are
• symmetric En ETn
• orthogonal E-1n ETn
• reflections E2n In
• Any set of square centrosymmetric matrices, Cn,
  is an algebra (Dagum, Guidotti, Luati, 2004).

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THE ALGEBRA OF SQUARECENTROSYMMETRIC MATRICES

• Being Cn an algebra is crucial on an (1)
  algebraic and a (2) statistical point of view. In
  fact
• (1) centrosymmetric matrices are the sole
  structured matrices, respect to some symmetry, to
  constitute an algebra
• (2) the convolution of smoothing matrices gives
  a smoothing matrix as well.

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LIMITATIONS OF CENTROSYMMETRICMATRICES (respect
to symmetric, algebraically)

• The degrees of freedom of a smoother (df, Hastie
  and Tibshirani, HT, 1990)
• trS
• do not make sense for non symmetric matrices for
• complex eigenvalues and
• asymmetric filters.
• Using trSTS trSST overcomes (a) but not (b).

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THE DECOMPOSITION trSST trWsWsT2trWaWaT

• trWsWsT df of the symmetric filter
• trWaWaT df of the asymmetric filters

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THE STANDARDIZED DECOMPOSITION

• (N-2m2m) trSST
• (N-2m) trWsWsT/(N-2m)(2m)2trWaWaT/(2m)
• (N-2m) (trSST/N- trWsWsT/(N-2m))
• (2m) (trSST/N-2trWaWaT/2m) 0
• trWsWsT/(N-2m) df of the symmetric filter
• trWaWaT/2m df of the asymmetric filters

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LOCAL BIAS OF A SYMMETRIC SMOOTHER

• HT a smoother introduces zero bias if
• We define the local bias of a symmetric smoother
  as
• D? R (N 2m)N 2k 1-band matrix such that
  dij1 if j - i m k,m k, dij0 otherwise

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• The local bias measures provides a measure of
  bias for symmetric smoothers that cannot be
  inferred by classical spectral analysis.

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LOCAL VARIANCE OF A SYMMETRIC SMOOTHER

• We define the local variance of a symmetric
  smoother as
• Global variance
• PN NxN symmetric (centrosymmetric) Toeplitz
  matrix of the process.

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LOCAL VARIANCE OF A SYMMETRIC SMOOTHER

• To evaluate the goodness of the local
  approximation
• Best approximation for I1?1 and I0 ?0

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• Analogous measures of local bias and local
  variance for asymmetric smoothers have been
  defined, as well as a local mean square error
  given by square local bias plus local variance.
  We restrict here to measures for symmetric
  filters.

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EMPRICAL ANALYSIS

• Local bias we compare standardized Hastie and
  Tibshirani degrees of freedom with our local bias
  measures
• Local variance we apply I0 and I1 to different
  smoothers (L1, L2, CSS, GK, H13) and different
  ARMA(p,q) models (p,q1,0 2,0 0,1 0,21,1).

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LOCAL BIAS
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LOCAL VARIANCE

• Best approximation for
• Smoothers ? concentrated (unbiased)
• Models ? white noise, AR
• In particular,
• AR(1) with ??0
• MA (1) with ??0
• ARMA(1,1) with ?gt0
• MA(2) with ?2gt?1
• AR(2) with ?1gt?2

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CONCLUSIONS

• A theory for centrosymmetric smoothing matrix is
  required due to asymmetry.
• A decomposition of the smoothing matrix allows to
  separately consider the effects of symmetric and
  asym-metric weights.
• Matrix-based measures of bias and variance can be
  defined such that
• local bias can be defined also for symmetric
  filters
• local variance is a good approximation of the
  global variance left by the smoothing procedure

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FURTHER RESEARCH

• To relate the spectral properties of the
  smoothing matrix with the fitting and smoothing
  properties of the associated filters.