Multivariable Zero-Free Transfer Functions with applications in Economic Modelling

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Multivariable Zero-Free Transfer Functionswith
applications in Economic Modelling
Systems and Control Group Seminar

• Brian D O Anderson
• Australian National University and National ICT
  Australia

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Collaborators

• Manfred Deistler, TU Wien

• Weitian Chen, ANU

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Outline

• Background and Applications
• Tall Transfer Functions and Zeros
• Left Invertibility, Impulse Responses and MFDs
• Canonical Forms
• Second Order Properties and Spectra
• Summary and further issues

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Factor Models

• Factor models were used in social science to
  capture the following idea.
• Suppose we measure for m people N attributes with
  m and N large. Can we use the data to infer that
  there exists a small number, q say, of
  meta-attributes (factors) such that
• Original application IQ, with q1.
• Here nk denotes zero mean noise with diagonal or
  almost diagonal covariance
• Key issues identifying q, identifying A,
  identifying the factors xk, using model for
  forecasting.

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Dynamic Factor Models

• Econometricians do things like
• Measure 150 quarterly variables relating to
  production, consumption, employment, interest
  rates, price inflation, in different sectors or
  states
• Build a dynamic model which has one to four white
  noise inputs, called factor variables, plus
  output noise
• When the entries of nk are independent, the model
  is a dynamic factor model. When the entries are
  dependent, it is a generalized dynamic factor
  model (GDFM)
• Models like this appear in multi-sensor signal
  processing too

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Applications areas

• Business cycle analysis
• Economy-wide and global shock identification
• Indexing and Forecasting
• Conduct of Monetary Policy
• Arbitrage Pricing
• Risk Management
• Cross-country studies (FR, IT, DE)

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Remarks

• There are two big dimensions
• The time dimension
• The cross-sectional dimension
• We are used to averaging in time
  dimensionperhaps to build a model.
• Progress in this area involves averaging also in
  the cross-sectional dimension

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Key questions

• How do we construct models from data?
• Nontrivial result 1 when the dimension of the
  output variable N goes to infinity, one can
  eliminate the effects of noise. Intuition the 1
  to 4 exciting variables affect all the outputs so
  the signal power grows with the output
  dimension, while the output noise power is
  bounded. It is as if one has measurements for the
  following model with white noise input
• Nontrivial result 2 Knowing the spectrum of yk,
  one can recover W(z) uniquely to within
  inessential right multiplication by a constant
  orthogonal matrix.
• How do we predict? Use W(z) and standard theory.
  

The key to all this is to use the fact that the
transfer function matrix W(z) is TALL.
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Outline

• Background and Applications
• Tall Transfer Functions and Zeros
• Left Invertibility, Impulse Responses and MFDs
• Canonical Forms
• Second Order Properties and Spectra
• Summary and further issues

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Tall transfer functions

• From now on we look at transfer functions
• where W(z) is p m, McMillan degree n, and
  A,B,C,D is a minimal realization. Further
• We will now forget economics and talk about such
  transfer functions.

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Tall Transfer function zeros

• Let W(z)DC(zI-A)-1B be tall with full column
  rank. We are going to talk about the zeros.
• There are finite zeros and infinite zeros
• We will note the different ways to characterize
  the finite zeros.
• Then we can look at infinite zeros.

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Tall Transfer function zeros

• Let W(z)DC(zI-A)-1B be tall with full column
  rank. Finite zeros are values of z at which
  following matrix drops normal rank
• Let W(z) U1(z)V(z)U2(z) where Ui(z) are
  polynomial with constant nonzero determinant and
  V(z)diagni(z)/di(z) 0T and ni divides ni1
  and di1 divides di. Finite zeros are zeros of
  the ni.
• Let W(z)M-1(z)N(z) with M,N coprime matrix
  polynomials. Finite zeros are where N(z) drops
  full rank

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Tall Transfer function zeros

• Let W(z)DC(zI-A)-1B be tall with full column
  rank. Finite zeros are values of z at which
  following matrix drops normal rank
• Infinite zeros arise iff D fails to have full
  column rank.
• For Smith-McMillan or MFDs, let
• where a is not a pole of W. Then W has an
  infinite zero if and only if has no zero at
  q-c.

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Tall Transfer function zeros

• Let W(z)DC(zI-A)-1B be tall with full column
  rank. Finite zeros are values of z at which
  following matrix drops normal rank
• Infinite zeros arise iff D fails to have full
  column rank.

Theorem Let W(z)DC(zI-A)-1B be tall with
generic A,B,C,D. Then W(z) has full column rank
and is zero-free.
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Outline

• Background and Applications
• Tall Transfer Functions and Zeros
• Left Invertibility, Impulse Responses and MFDs
• Canonical Forms
• Second Order Properties and Spectra
• Summary and further issues

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Left Invertibility

• A system is termed left invertible with unknown
  initial state if given a sufficiently long
  (possibly infinite) output sequence y0,y1,y2,.,
  the input sequence u0, u1,u2, and the initial
  state x0 can be computed.

• Theorem Consider a square or tall W(z) with
  minimal realization DC(zI-A)-1B, and with no
  zeros in the finite complex plane. Then it is
  left invertible. In fact
• There exists L n, such that for arbitrary k,
  xk and uk
• are computable from yk,yk1,,ykL-1.

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Left Invertibility (ctd)

• Theorem Consider a square or tall W(z) with
  minimal realization DC(zI-A)-1B and with no
  zeros in the finite complex plane. Then it is
  left invertible. In fact
• There exists L n, such that for arbitrary k,
  xk and uk
• are computable from yk,yk1,,ykL-1.

• If also there is no zero at infinity, one can
  additionally recover xk1,xk2,xkL and the
  input segment uk1, uk2,...ukL-1.
• Further result in case there is no zero at
  infinity but there is a zero at zero.
• Main result builds on a Theorem of Moylan of
  1977.

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Left Invertibility (ctd)
Theorem (a) No finite zeros means xk and uk are
computable from yk,yk1,,ykL-1. (b) No infinite
or finite zeros means the u interval and the x
interval plus a prediction are computable
xk
xk1
xk2
xkL-1
xkL
uk
uk1
ukL-1
uk2
ukL
yk1
yk
ykL-1
yk2
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Left Invertibility (ctd)
Theorem (a) No finite zeros means xk and uk are
computable from yk,yk1,,ykL-1. (b) No infinite
or finite zeros means the u interval and the x
interval plus a prediction are computable
No zeros finite or infinite except at origin mean
that lose left hand end of the computable
intervals
xk
xk1
xk2
xkL-1
xkL
uk
uk1
ukL-1
uk2
ukL
yk1
yk
ykL-1
yk2
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Left Invertibility (ctd)
Theorem (a) No finite zeros means xk and uk are
computable from yk,yk1,,ykL-1. (b) No infinite
or finite zeros means the u interval and the x
interval plus a prediction are computable
No zeros finite or infinite except at origin mean
that lose left hand end of the computable
intervals
xk
xk1
xk2
xkL-1
xkL
uk
uk1
ukL-1
uk2
ukL
yk1
yk
ykL-1
yk2
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Indication of a derivation

• Suppose W(z) M-1(z)N(z) is a polynomial coprime
  factorization. Zero-free tall W(z) implies N(z)
  has full rank for all z and so there exists N2(z)
  so that
• is (square and) unimodular. Let Q(z) be an
  inverse. It is also polynomial.
• From ykW(z)uk, we obtain
• or

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Impulse response description
NL

• If W(z) is tall and zero-free, then there exists
  L n with L at least equal to the
  observability index of (C,A) such that the
  matrix NL has full column rank.
• Variations possible for when W(z) has a zero at
  infinity, or a zero at z0.
• Now use the fact that xk is producible from prior
  uj.

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Impulse response description ctd

• W(z) is zero free iff given yk,yk1,ykL-1, this
  equation can be solved to give unique uk,,ukL-1
• Write the matrix above as W1 W2. Let S be a
  nonsingular matrix premultiplying W1 to generate
  row echelon form
• Zero-free is equivalent to has full
  column rank.

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Outline

• Background and Applications
• Tall Transfer Functions and Zeros
• Left Invertibility, Impulse Responses and MFDs
• Canonical Forms
• Second Order Properties and Spectra
• Summary and further issues

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Matrix Fraction Descriptions

• In multivariate time-series analysis, one works
  often with Autoregressive (AR) and Autoregressive
  Moving Average (ARMA) models
• The associated transfer function matrix is
  A-1(z-1)B(z-1) but it is easier to define qz-1
  to recover true matrix polynomials. Thus the
  system has transfer function W(q)A-1(q)B(q).
• The zerofree property for B(q) for all finite q
  means uk can be recovered from a finite interval
  of outputs that maybe starts before time k.

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Matrix Fraction Descriptions ctd

• Assume tall W(q)A-1(q)B(q) and the zero free
  property for B(q) for all finite q.
• Because B(q) is zero free there exists B2(q) so
  that B(q) B2(q) is unimodular, with inverse
  Q(q).
• Then W(q) can always be assumed to have a
  nonstandard AR description

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Matrix Fraction Descriptions ctd

• There exist unimodular V(q) with V(q) I 0T
  I 0T . For such a V(q), W(q)V(q)D(q)-1I
  0T is another matrix fraction description.
• Hence it makes sense to look for a canonical
  representation. For identification, you usually
  want the smallest number of parameters and
  unique parameterisation, i.e. canonical.
• V(q) must have the following form
• It transforms D(q) according to

Key Question What can be done with unimodular
V22 and arbitrary V12 to bring D to a canonical
form?
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Matrix Transfer Descriptions ctd

• V22(q) reduces D2 to row (polynomial) echelon
  form,
• Then V12 is used to make entries of D1V12D2
  above pivot index entries of V22D2 to have
  degrees less than degree of the pivot entries
• Gives a unique new D
• One wants to control the degree of all the
  polynomial terms.

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Row polynomial echelon form
All degrees lt ki
Row degrees ki are monotone decreasing. Row
properness holdsSki is minimal.
Monic, degree ki
Row i, row degree ki
Degree 0 ??nonzero constant Degree lt 0 ??zero
entry
Pivot index pi
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The full D matrix
Pivot index pi
D12
Monic, degree ki
All degrees lt ki
D22
Row i, row degree ki
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Controlling degrees

• Theorem The row degrees of D2 sum to n nz
  where n, nz are the McMillan degree and number of
  zeros of W(q).
• Proof Trick The description WD-1I 0T
  ensures that D2W0 and when D2 is row proper, its
  rows form a minimal basis for left kernel of W.
  There is no nontrivial right kernel for W. Then
  appeal to literature on sum of row degrees of
  left kernel and column degrees of right kernel.
• This controls the degree of everything in D2 and
  entries in those columns of D1 above pivot index
  in D2.
• What about the entries of D1 which are not in
  pivot index columns (there are m such
  non-pivot-index columns), so m2 entries in all?
  W(q) is p m

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Controlling degrees

• What about the entries of D1 which are not in
  pivot index columns (there are m such
  non-pivot-index columns), so m2 entries in all?
• Separate argument gives a bound on their degree
  of (n-1)p2n.
• D is a p p matrix. We can bound the degrees.
  What is a bound on the total number of
  parameters?

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Counting parameters

• D is a p p matrix. We can bound the
  degrees. What is bound on the total number of
  parameters?
• The sum of the row degrees of D2 are bounded by
  n-nz and so no matter how big the number of rows
  is, the number with nonzero degree is bounded.
• The polynomial echelon form then forces lots of
  zero elements
• The total parameter count in D is linear in p,n
  but quadratic in m.

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Nesting Property

• Suppose we have a W(q) which is p m. Then
  another output becomes available. The new W(q) is
  (p1) m. What is the relation between the two
  matrix fraction descriptions?
• Assume the McMillan degree does not get any
  bigger. So we have an output from the same state
  variables.
• The nesting property of the two transfer
  functions matrices propagates.

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Nesting Property
The nesting property of the two transfer
functions matrices propagates.
x(q) obeys the degree constraints, and Dp1(q)
canonical means that the top left submatrix D(q)
is also canonical.
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Outline

• Background and Applications
• Tall Transfer Functions and Zeros
• Left Invertibility, Impulse Responses and MFDs
• Canonical Forms
• Second Order Properties and Spectra
• Summary and further issues

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Recall

• Econometricians do things like
• .
• Build a dynamic model with 150 or so outputs
  which has one to four inputs, called factor
  variables, plus output noise
• uk is a vector of independent white noise
  processes.
• When the entries of nk are independent, the model
  is a dynamic factor model. When dependent, it is
  a generalized dynamic factor model (GDFM).

• The starting point is the spectrum of the yk
  process.
• We assume that the noise part of the spectrum is
  removed (this is subject of theoretical and
  computational consideration, and ultimately
  possible when yk dimension is very large).

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Second order properties

• Let S(z) be a power spectrum associated with a
  tall zeroless rational W(z). Thus
• The zeroless property on W(z) is reflected in a
  zeroless property of S(z).
• Key result Starting with a minimal
  state-variable description of S(z), a finite
  number of rational calculations will determine up
  to right multiplication by a real orthogonal
  matrix the unique W(z) of minimum McMillan degree
  satisfying the above equation.

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Insights

• Kalman filtering and spectral factorization are
  two sides of the one coin.
• Consider a linear system with white noise input
  uk and measurement process yk
• The yk process has a spectrum of the form
• Using the spectrum of yk we can build a Kalman
  filter for the state.
• For a zerofree spectrum, the filter will recover
  the state and indeed the input exactly after a
  finite measurement interval. The Riccati equation
  converges in finite time to steady state. The
  Riccati equation gives the spectral factor.

The spectral factor is found with a finite number
of rational calculations (finite number of
iterations of Riccati equation), under the
zerofree assumption.
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Testing for zeroless property

• The zeroless property can be tested using a state
  variable description of the spectrum matrix, such
  as
• It can also be tested by looking at the ranks of
  successively larger Toeplitz matrices, once the
  McMillan degree of the spectrum (or the spectral
  factor) is known.

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Testing for zeroless property
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Outline

• Background and Applications
• Tall Transfer Functions and Zeros
• Left Invertibility, Impulse Responses and MFDs
• Canonical Forms
• Second Order Properties and Spectra
• Summary and further issues

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Summary and Further issues

• For transfer function matrices arising in
  econometric problems
• Very tall means we can get rid of noise
• Tall means generically spectrum and spectral
  factor are zerofree. There are various
  characterizations of this.
• Transfer function matrix can be recovered from
  spectrum with finite number of rational
  calculations
• Nonstandard autoregressive canonical forms can be
  found
• Input and state can be recovered from output

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Summary and Further issues

• Further issues to be addressed
• Greater precision over canonical forms
• Yule-Walker connections
• Connecting matrix fraction canonical forms to
  state variable A,B,C,D canonical forms
• Connecting to static factors
• Tall ARMA processes
• Effect of errors in spectrum estimation on the
  model
• Mixing quarterly and monthly time series

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Questions?