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

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY Manfred Deistler 1 1 Research 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 Linz, October 2008 .S.K., Market Risk, Vienna) Workshop on Inverse and Partial Information Problems 1 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY Contents Introduction 1 The General Framework 2 3 Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors 4 Zeroless Transfer Functions and (Singular) AR Systems The Yule Walker Equations 5 Estimation of Integers 6 Removing the (weakly) idiosyncratic noise from the 7 observations Conclusions 8 2 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY Introduction Outline Introduction 1 The General Framework 2 3 Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors 4 Zeroless Transfer Functions and (Singular) AR Systems The Yule Walker Equations 5 Estimation of Integers 6 Removing the (weakly) idiosyncratic noise from the 7 observations Conclusions 8 3 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY Introduction Modeling of high dimensional time series Multivariate time series y t ∈ R N , t = 1 , ..., T Sample size T , cross-sectional dimension N “Traditional“ approach, e.g. “unstructured“ AR modeling: “Curse of dimensionality“: Dimension of parameter space N 2 p ( p : maximal lag). Number of data points NT . Alternatives: Factor models Comovement allows for dimension reduction in the cross-sectional dimension. 4 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY 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” or “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. 5 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY 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 6 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework Outline Introduction 1 The General Framework 2 3 Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors 4 Zeroless Transfer Functions and (Singular) AR Systems The Yule Walker Equations 5 Estimation of Integers 6 Removing the (weakly) idiosyncratic noise from the 7 observations Conclusions 8 7 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY 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 8 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY 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 9 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework GDFM’s - The Model Class y N y N t + u N t = ˆ t y N t ... observations y N t ... latent variables, strongly dependent in the cross-sectional dimension ˆ u N t ... (wide sense) idiosyncratic noise, weakly dependent Assumptions: y N t ) , ( u N (ˆ t ) wide sense stationary with absolutely summable covariances t u N ′ y N E ˆ = 0 s y N t = E u N E ˆ t = 0 10 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework GDFM’s - The Model Class Spectral densities: f N y ( λ ) = f N y ( λ ) + f N u ( λ ) ˆ Asymptotic analysis: T → ∞ , N → ∞ Sequence of GDFM’s; y N t and u N Nested i.e. elements of ˆ t do not depend on N 11 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework Assumptions Additional assumptions to separate the latent variables from the noise for N → ∞ : A1 f N y is a rational spectral density with constant rank q < N , ˆ and of McMillan degree 2 n < N ; q and n do not depend on N A2 Weak dependence of ( u N t ) : The largest eigenvalue of f N u is uniformly bounded for all frequencies λ and all N y N t ) : The first q eigenvalues of f N A3 Strong dependence of (ˆ y ˆ diverge to infinity for all frequencies λ , as N → ∞ 12 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework Identifiability GDFM’s are identifiable only for N → ∞ : The elements of f N y N y and ˆ t are uniquely determined from ˆ y N t for N → ∞ Asymptotic equivalence to dynamic PCA 13 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework Dynamic PCA (Brillinger 1981) We want to approximate f y ( λ ) by a spectral density f ˆ y ( λ ) of rank q for all λ s.t. E u ′ t u t is minimal f y ( λ ) = O 1 ( e − iλ )Ω 1 ( λ ) O 1 ( e − iλ ) ∗ + O 2 ( e − iλ )Ω 2 ( λ ) O 2 ( e − iλ ) ∗ Model: y t = O 1 ( z ) O ∗ + O 2 ( z ) O ∗ 1 ( z ) y t 2 ( z ) y t � �� � � �� � u t y t ˆ Note Here dimension reduction is in cross-section only; even for rational f y , f ˆ y may be non-rational O 1 ( z ) O ∗ 1 ( z ) may be non-causal Estimation commences from a nonparametric estimator of f y . 14 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework Characterization of GDFM’s y N t follows a generalized dynamic factor model if and only if the first q eigenvalues of f N y diverge to infinity for all frequencies λ , as N → ∞ the q + 1 − th eigenvalue of f N y is uniformly bounded for all frequencies λ and all N 15 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY 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 y t ) from the second moments of ( y t ) 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 y t based on forecasts of ˆ y t and eventually of u t 16 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY The General Framework Major Steps Factorization of f ˆ 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 17 / 50

G ENERALIZED L INEAR D YNAMIC F ACTOR M ODELS - A S TRUCTURE T HEORY Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Outline Introduction 1 The General Framework 2 3 Factorization of Rational Singular Spectra and Realization of Tall Spectral Factors Factorization of Rational Singular Spectra Realization Latent Variables and Minimal Static Factors 4 Zeroless Transfer Functions and (Singular) AR Systems The Yule Walker Equations 5 Estimation of Integers 6 Removing the (weakly) idiosyncratic noise from the 7 observations Conclusions 8 18 / 50