Table of contents Inference of high-dimensional VAR models Linear - PDF document

Linear time series Structural analysis with VAR models Estimation of VAR models Numerical examples Table of contents Inference of high-dimensional VAR models Linear time series 1 Basics Univariate time series Seminar: Multivariate time series ”Modeling, Simulation and Inference of Complex Biological Structural analysis with VAR models 2 Systems” Granger-Causality Impulse response analysis Estimation of VAR models 3 Katharina Schneider Overview Maximum likelihood estimation Bayes estimation 07.07.2006 Numerical examples 4 Simulations VAR of U.S. Economy Concluding remarks Katharina Schneider Inference of high-dimensional VAR models Linear time series Linear time series Basics Basics Structural analysis with VAR models Structural analysis with VAR models Univariate time series Univariate time series Estimation of VAR models Estimation of VAR models Multivariate time series Multivariate time series Numerical examples Numerical examples Stochastic process Linear time series and stationarity Sequence of random variables Y = { Y t , t ∈ T } One finite realization of a stochastic process y = { y t , t ∈ T } Classical decomposition model: y t = µ t + s t + u t Trend: µ t = E ( Y t ) Variance: σ t = E [( Y t − µ t ) 2 ] (= trend + seasonal component + stationary random noise) Autocovariance: γ t , s = E { [ Y t − µ t ][ Y s − µ s ] } Stationarity Y t stationary ⇔ y t stationary Stationarity Y t strongly stationary : ⇔ descriptive analysis of stationarity with graphs and correlograms: ∀ n , t 1 , . . . , t n , h : F x t 1 ,..., x tn ( x 1 , . . . , x n ) = F x t 1+ h ,..., x tn + h ( x 1 , . . . , x n ) no trend Y t weakly stationary : ⇔ no systematic change of variance µ t = µ = const no strictly periodic fluctuations t = σ 2 = const σ 2 Tests on stationarity: γ t , s = γ t − s = γ k with k = t − s (Lag) Unit Root Tests (Dickey-Fuller-Test, Augmented DF-Test) Autocovariance function: γ k = E { [ Y t − µ ][ Y t − k − µ ] } Autocorrelation function (ACF): ̺ k = γ k γ 0 = γ k approaches to obtain stationarity: differentiation, integration, σ 2 (by standardization with σ 2 = γ 0 ) filtering Katharina Schneider Inference of high-dimensional VAR models Katharina Schneider Inference of high-dimensional VAR models Linear time series Linear time series Basics Basics Structural analysis with VAR models Structural analysis with VAR models Univariate time series Univariate time series Estimation of VAR models Estimation of VAR models Multivariate time series Multivariate time series Numerical examples Numerical examples Basics Linear time series models Lag operator L (Backshift operator) univariate multivariate L 0 y t = y t L 1 y t = y t − 1 L 2 y t AR VAR = y t − 2 stationary MA VMA L k y t . . . . . . = y t − k ARMA VARMA non-stationary ARIMA VARIMA White noise ǫ t a series of iid random variables (”innovations”, ”shocks”) Modeling a time series E ( ǫ t ) = µ t = 0 1 diagnosis (stationarity, autocorrelation, etc.) σ 2 ǫ ( Σ ǫ ) 2 model identification γ t , s = 0 for t � = s d : order of integration � = number of differentiations for Properties ACF stationarity ̺ ( k ) = ̺ ( − k ) p , q : with Box-Jenkins(ACF, PACF, etc.), AIC, Bayes-Schwarz, − 1 ≤ ̺ ( k ) ≤ 1 etc. Y ( t ) and Y ( t − k ) independent ⇒ ̺ ( k ) = 0 3 estimation of the parameters (LS, ML, etc.) Correlogram: graph of ̺ 4 model selection Katharina Schneider Inference of high-dimensional VAR models Katharina Schneider Inference of high-dimensional VAR models Linear time series Linear time series Basics Basics Structural analysis with VAR models Structural analysis with VAR models Univariate time series Univariate time series Estimation of VAR models Estimation of VAR models Multivariate time series Multivariate time series Numerical examples Numerical examples Autoregressive process of order p Moving average process of order q AR( p ) MA( q ) p � q � y t = φ i y t − i + ǫ t ⇔ Φ ( L ) y t = ǫ t y t = ǫ t + ⇔ y t = Θ ( L ) ǫ t θ j ǫ t − j i =1 j =1 Properties Properties E ( y t ) = 0 V ar ( y t ) = const . E ( y t ) = 0 � = � p V ar ( y t ) = σ 2 � q i =0 θ 2 γ k l =1 φ l γ k − l : k = 1 , 2 , . . . = � p � i Yule − Walker σ 2 � q − k : k = 1 , 2 , . . . ̺ k l =1 φ l ̺ k − l : k = 0 , 1 , . . . , q i =0 θ i + k θ i γ k = Stationarity 0 : k > q Characteristic equation: Φ ( u ) = 0 with u ∈ C Stationarity AR( p ) stationary: | u | > 1 ↔ if all (complex) solutions of the E ( y t ) , V ar ( y t ) , γ k independent of t ⇒ MA( q ) weakly characteristic equation lie outside the unit circle stationary AR( p ) nonstationary: | u | = 1 (unit root) Katharina Schneider Inference of high-dimensional VAR models Katharina Schneider Inference of high-dimensional VAR models