Contents Introduction 1 The General Framework 2 3 Factorization - - PowerPoint PPT Presentation

GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY

GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY

Manfred Deistler1

1Research Group Econometrics and System Theory

TU Wien deistler@tuwien.ac.at joint work with: B.D.O. Anderson (Research School of Information Sciences and Engineering, ANU, Canberra)

• A. Filler (Vienna University of Technology and University of Vienna)
• Ch. Zinner (BAWAG P

.S.K., Market Risk, Vienna)

Linz, October 2008 Workshop on Inverse and Partial Information Problems

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY

Contents

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Introduction

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Introduction

Modeling of high dimensional time series

Multivariate time series yt ∈ RN, t = 1, ..., T Sample size T, cross-sectional dimension N “Traditional“ approach, e.g. “unstructured“ AR modeling: “Curse of dimensionality“: Dimension of parameter space N2p (p: maximal lag). Number of data points NT. Alternatives: Factor models

Comovement allows for dimension reduction in the cross-sectional dimension.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Introduction

Modeling of high dimensional time series

Time series factor models: Complexity reduction in time and cross-section. Under certain assumptions the dimension of the parameter space is linear in N.

Note, there is no symmetry in the time and the cross-sectional dimension: Stationarity in time, “similarity”

• r “comovement” of time series; permutation invariant.

Cointegration Panel-time series models Structural models; e.g. ARX models which are sparse due to “physical“ a priori knowledge. “Graphical“ time series models, where the inverse of the spectral density is sparse.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Introduction

Applications

History: Psychometrics: Intelligence factors (Burt 1909, Thurstone 1934) Great range of applications: Signal processing, marketing econometrics, finance econometrics, ... Recent applications for generalized factor models: Forecasting for macrovariables Structure and insigths for macroeconomics Cross-country studies Finance

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

GDFM’s - The Model Class

Generalized linear dynamic factor models (GDFM’s) Generalization of:

Linear dynamic factor models with (strictly) idiosyncratic noise (Geweke, in: Aigner and Goldberger (eds), Latent Variables in Socioeconomic Models 1977, Scherrer and Deistler, SIAM, J. Optim and Control 1998) Generalized static linear factor models (Chamberlain, Chamberlain and Rothschild, Econometrica 1983)

Main features:

Dynamics (here in a stationary context) Uncorrelatedness of noise components in cross-section is replaced by weak dependence (risk diversification is possible) Similarity: Information gain by adding additional time series

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

GDFM’s - The Model Class

Main references for GDFM’s Forni, Hallin, Lippi, Reichlin, RES, 2000 Forni and Lippi, Econometric Theory, 2001 Forni, Hallin, Lippi, Reichlin, JASA, 2005 Stock and Watson, JASA, 2002 Stock and Watson, Journal of Business and Economic Statistics, 2002

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

GDFM’s - The Model Class

yN

t = ˆ

yN

t + uN t

yN

t ... observations

ˆ yN

t ... latent variables, strongly dependent in the cross-sectional dimension

uN

t ... (wide sense) idiosyncratic noise, weakly dependent

Assumptions: (ˆ yN

t ), (uN t ) wide sense stationary with absolutely

summable covariances Eˆ yN

t uN′ s

= 0 Eˆ yN

t = EuN t = 0

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

GDFM’s - The Model Class

Spectral densities: fN

y (λ) = fN ˆ y (λ) + fN u (λ)

Asymptotic analysis: T → ∞, N → ∞ Sequence of GDFM’s; Nested i.e. elements of ˆ yN

t and uN t do not depend on N

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Assumptions

Additional assumptions to separate the latent variables from the noise for N → ∞: A1 fN

ˆ y is a rational spectral density with constant rank q < N,

and of McMillan degree 2n < N; q and n do not depend on N A2 Weak dependence of (uN

t ): The largest eigenvalue of fN u is

uniformly bounded for all frequencies λ and all N A3 Strong dependence of (ˆ yN

t ): The first q eigenvalues of fN ˆ y

diverge to infinity for all frequencies λ, as N → ∞

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Identifiability

GDFM’s are identifiable only for N → ∞: The elements of fN

ˆ y and ˆ

yN

t are uniquely determined from

yN

t for N → ∞

Asymptotic equivalence to dynamic PCA

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Dynamic PCA (Brillinger 1981)

We want to approximate fy(λ) by a spectral density fˆ

y(λ) of

rank q for all λ s.t. Eu′

tut is minimal

fy(λ) = O1(e−iλ)Ω1(λ)O1(e−iλ)∗ + O2(e−iλ)Ω2(λ)O2(e−iλ)∗ Model: yt = O1(z)O∗

1(z)yt

• ˆ

yt

+ O2(z)O∗

2(z)yt

• ut

Note Here dimension reduction is in cross-section only; even for rational fy, fˆ

y may be non-rational

O1(z)O∗

1(z) may be non-causal

Estimation commences from a nonparametric estimator of fy.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Characterization of GDFM’s

yN

t follows a generalized dynamic factor model if and only if

the first q eigenvalues of fN

y diverge to infinity for all

frequencies λ, as N → ∞ the q + 1 − th eigenvalue of fN

y is uniformly bounded for all

frequencies λ and all N

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Aims of our Analysis

Structural insight Obtain a state space or an ARMA model and in particular an AR model for (ˆ yt) from the second moments of (yt)

estimation of the integer valued parameters such as q (dimension of dynamic factors), r (dimension of static factors) and n (state dimension) estimation of the real valued parameters such as (F , G, H)

forecasting of yt based on forecasts of ˆ yt and eventually of ut

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The General Framework

Major Steps

Factorization of fˆ

y

Realization of a “tall“ spectral factor by a state space or an ARMA model Emphasise the zeroless case Averaging out of (weakly) idiosyncratic noise

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra

fˆ

y(λ) = w(e−iλ) N×q

w(e−iλ)∗ (1) Theorem (i) Every rational spectral density fˆ

y of constant rank q can be

factorized as in (1) where w(z) =

∞

• j=0

wjzj; wj ∈ RN×q is rational, analytic in |z| ≤ 1 and has rank q for all |z| ≤ 1. (ii) For given fˆ

y, w is unique up to postmultiplication by constant

• rthogonal matrices.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra

Wold decomposition

(Smith Mc Millan form) w = uℓv u, v ... unimodular, polynomial ℓ ... diagonal, diagonal elements display poles and zeros of w w corresponds to the Wold decomposition: There exists (εt), white noise, with Eεtε′

t = 2πI, s.t.:

ˆ yt = w(z)εt =

∞

• j=0

wjεt−j εt = w−(z)ˆ yt causal left inverse w− = v−1(ℓ′ℓ)−1ℓ′u−1

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

Realization

Realization: Find a system for a transfer function

State space system - (F , G, H) ARMA system a(z)ˆ yt = b(z)εt w(z) = a(z)−1b(z) Right MFD (Lippi) w(z) = c(z)d(z)−1

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

ARMA Realizations

a(z)

• N×N

ˆ yt = b(z)

• N×q

εt (a, b) left coprime Stability: det a(z) = 0, |z| ≤ 1 Miniphase condition: b(z) has full rank, |z| ≤ 1

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

State space Realizations

xt+1 = Fxt + Gεt+1 ˆ yt = Hxt F ∈ Rn×n, G ∈ Rn×q, H ∈ RN×n xt ... state w(z) = H(I − Fz)−1G (F , G, H) is minimal (i.e. controllable and observable) Stability: |λmax(F)| < 1 Miniphase condition: M(z) = I − Fz −G H

• has rank n + q, |z| ≤ 1

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

(F , G, H) State space systems

The state is unique up to basis changes: ¯ F = TFT −1, ¯ G = TG, ¯ H = HT −1, det T = 0 Note that here xt is a static factor but not necessarely a minimal

• ne. xt is a minimal static factor if and only if rk(H) = n holds.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

Realization: Kalman-Akaike procedure

     ˆ yt ˆ yt+1|t ˆ yt+2|t . . .     

• ˆ

Yt

=       HG HFG HF 2G . . . HFG HF 2G HF 3G . . . . . . . . .      

• H Hankelmatrix of the transfer function

     εt εt−1 εt−2 . . .     

• E−

t

ˆ yt+r|t ... best linear least squares predictor of ˆ yt+r from the infinite past ˆ yt, ˆ yt−1, ...

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Realization

Xt = S

• n × ∞

rk(S)=n

     ˆ yt ˆ yt+1|t ˆ yt+2|t . . .      = S H E−

t

= S    HFG HF 2G . . . HF 2G HF 3G . . . . . .   

• F S H

E−

t−1

• F Xt−1

+ S    HG HFG . . .   

• G

εt with H S H = HG HFG . . .

S ... Selector matrix Special choice for S: Select the first basis rows of H: Echelon form, selection described by Kronecker indices

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Latent Variables and Minimal Static Factors

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Latent Variables and Minimal Static Factors

Latent Variables and Minimal Static Factors

ˆ yt = HTT −1xt = (H1, 0)T −1xt = H1zt zt ... r dimensional minimal static factor, rk H1 = r In general n ≥ r ≥ q; q is the dimension of minmal dynamic factors Note: Minimal static factors are unique up to premultiplication by constant nonsingular matrices

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Latent Variables and Minimal Static Factors

Static factors are obtained from Σˆ

y = Eˆ

ytˆ y′

t :

Σˆ

y = MM′, M = H1R, M ∈ RN×r, rk M = r

as: zt = (M′M)−1M′ˆ yt. Static factors zt and latent variables ˆ yt are related by a linear static relation and thus have the same dynamics zt = (M′M)−1M′w(z)εt = k(z)εt zt has smaller dimension, thus we prefer to model (zt) Echelon case: S = S1 S2

• , S1 ∈ Rr×∞, S2 ∈ R(n−r)×∞

xt = S ˆ Yt, zt = S1 ˆ Yt

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Latent Variables and Minimal Static Factors

Clearly (zt) has a rational spectral density State space model for (zt): (F , G, C) where C = (M′M)−1M′H Identification of an ARMA model for (zt), (Zinner, PhD-thesis TU Wien)

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Zeroless Transfer Functions and (Singular) AR Systems

A transfer function w(z) is called zeroless if all numerator polynomials in the diagonal of the matrix ℓ in the Smith-Mc-Millan form are equal to 1 Note: k(z) is zeroless if and only if w(z) is zeroless Theorem (Anderson and Deistler, SICE J. Control 2008) Consider a rational transfer function w(z) with minimal state space realization (F , G, H) with state dimension n. If N > q holds, then the transfer functions are zeroless for generic values of (F , G, H).

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Theorem (Anderson and Deistler, CDC, 2008) The following statements are equivalent: (i) The stable miniphase spectral factors k of the spectral density fz of (zt) are zeroless (ii) There exists a polynomial left inverse k− for k (iii) (zt) is a stable AR-process, i.e. zt = e1zt−1 + · · · + epzt−p + νt where det (I − e1z − · · · − epzp)

• e(z)

= 0, |z| ≤ 1 and rk Σν = q, Σν = Eνtν′

t.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Let Γm =       γ0 · · · · · · γ′

m−1

. . . γ0 . . . . . . ... . . . γm−1 · · · · · · γ0       where γj = Ezt+jz′

t

If Σν is nonsingular, then Γm is nonsingular for all m. If Σν is singular, then Γp+1 will be singular and Γp may be singular. Yule Walker Equations: (e1, . . . , ep)Γp = (γ1, . . . , γp) Σν = γ0 − (e1, . . . , ep)(γ′

1, . . . , γ′ p)′

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Zeroless Transfer Functions and (Singular) AR Systems

Solution of the Yule Walker equations may not be unique: Description of the class of all obervationally equivalent AR systems for given p Theorem (Anderson and Deistler, CDC, 2008) (i) Every singular AR system with rk Σν = q can be written as e(z)zt = fεt, f ∈ Rr×q where (εt) is white noise with Eεtε′

t = Iq and where e(z)

and f are relatively left prime.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Theorem (Anderson and Deistler, CDC, 2008) (ii) Let (e(z), f) be relatively left prime, then the class of all

• bservationally equivalent (¯

e(z), ¯ f) satisfying the degree restrictions δ(¯ e(z)) ≤ p, δ( ¯ f) = 0 is given by

• ¯

e(z), ¯ f

• = u(z)(e(z), f)

where the polynomial matrix u(z) satisfies det u(z) = 0, |z| ≤ 1 u(0) = I δ(u(z)e(z)) ≤ p δ(u(z)f) = 0

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Zeroless Transfer Functions and (Singular) AR Systems

Theorem (Anderson and Deistler, CDC, 2008) (iii) e(z) is unique if and only if rk(ep, f) = r holds.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The Yule Walker Equations

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The Yule Walker Equations

Let ˆ γT

j = 1/T T t=1 ˆ

yt+j ˆ y′

t and let

(ˆ e1, . . . , ˆ ep)ˆ Γp = (ˆ γ1, . . . , ˆ γp), ˆ Γp =       ˆ γ0 · · · · · · ˆ γ′

p−1

. . . ˆ γ0 . . . . . . ... . . . ˆ γp−1 · · · · · · ˆ γ0       be the corresponding Yule Walker equations. Typically ˆ Γp will be nonsingular, even if Γp is singular, however, truncation is appropriate in such a case (ˆ e1, ...ˆ ep) = (ˆ γT

1 , ..., ˆ

γT

p )OT Λ−1 T O′ T

where ˆ ΓT

p = OT ΛT O′ T , OT ∈ Rpr×s, ΛT ∈ Rs×s, s = rk Γp

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY The Yule Walker Equations

The Yule Walker Equations

Theorem (Anderson, Deistler, Filler and Zinner, ECC, 2009) (i) If rk Γp = pr holds, then the YW estimators correspond to a stable autoregression (ii) For rk Γp = s < pr, the truncation procedure above yields a stable autoregression

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Estimation of Integers

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Estimation of Integers

Estimation of Integers

Work in progress: Estimation of r, q, p and s for the AR case, or r, n, q and the Kronecker indices for the state space case.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Removing the (weakly) idiosyncratic noise from the observations

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Removing the (weakly) idiosyncratic noise from the observations

Removing the (weakly) idiosyncratic noise from the

• bservations

Note: ˆ yt is not directly observed, how can we get rid of ut? N → ∞ There are several approaches. Here we only describe the (static) PCA based procedure (i) Commence from ˆ ΓN

y = T −1 yN t yN′ t

Now, under suitable assumptions, for T, N → ∞ the first r eigenvalues of ˆ ΓN

y tend to infinity, and the other eigenvalues of

ˆ ΓN

y converge to finite values.

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Removing the (weakly) idiosyncratic noise from the observations

Removing the (weakly) idiosyncratic noise from the

• bservations

This is used to estimate r and zt :

ˆ ΓN

y = OT ΛT

O′

T + remainder

first r eigenvalues ˆ zt = O′

T yt

ˆ r and ˆ zt are consistent for T, N → ∞ (ii) Use ˆ zt to estimate the AR order p and the autoregressive coefficients

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Conclusions

Outline

1

Introduction

2

The General Framework

3

Factorization of Rational Singular Spectra and Realization

• f Tall Spectral Factors

Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors

4

Zeroless Transfer Functions and (Singular) AR Systems

5

The Yule Walker Equations

6

Estimation of Integers

7

Removing the (weakly) idiosyncratic noise from the

• bservations

8

Conclusions

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Conclusions

Conclusions

Aims:

• btain „structural“ insights

direct estimation procedure based on estimation of the second moments of the observations (computational simplicity) treatment of case more general case (compared to existing literature) forecasting Zeroless transfer functions and spectra make things easier. Further open questions: Properties of the estimators for the autoregression and of the estimators for (F , G, H) beyond consistency Properties of the Estimation of the integer-valued parameters such as r, q, p, s

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GENERALIZED LINEAR DYNAMIC FACTOR MODELS - A STRUCTURE THEORY Conclusions

Thank You

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