STA 6857 VAR, VARMA, VARMAX ( 5.7)

STA 6857 VAR, VARMA, VARMAX ( 5.7)

Outline 1 Multivariate Time Series Modeling 2 VAR 3 VARIMA/VARMAX Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 2/ 16

Outline 1 Multivariate Time Series Modeling 2 VAR 3 VARIMA/VARMAX Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 3/ 16

Examples We are considering multiple observations taken simultaneously. E.g. Meteorology Economics temperature air pressure rainfall retail price index gross domestic product unemployment level Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 4/ 16

Advantages and Disadvantages of Multivariate Modeling Advantages Disadvantages more choices of models better understanding of the system harder to find the right model doesn t necessarily provide better forecasts (parameter estimation, misidentification error) Parsimonious Model vs. Complete Model Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 5/ 16

Difficulty in Economic Model Building 1 Feedback is not well-controlled like in certain physical systems such as a chemical reactor experiments cannot be conducted to test the system. 2 The economy has a complex, nonlinear structure which can change in time and suffer from limited data sets. 3 Using the wrong model is more serious than poor parameter estimation. Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 6/ 16

Open and Closed Loop Systems Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 7/ 16

Outline 1 Multivariate Time Series Modeling 2 VAR 3 VARIMA/VARMAX Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 8/ 16

Why VAR? Allows for the more dynamic closed-system model Single input/output model model is not always general enough Two or more variables may arise on an equal footing Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 9/ 16

Vector Autoregressive Time Series We start with the vector time series with the given data structure x t = (x 1t, x 2t,..., x mt ) For simplicity, we will focus on the case m = 2. The VAR(1) model (of dimension m = 2) is described the following system of linear equations { x 1t = φ 11 x 1,t 1 + φ 12 x 2,t 1 + ε 1t x 2t = φ 21 x 1,t 1 + φ 22 x 2,t 1 + ε 2t Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 10/ 16

Matrix Form of VAR The matrix form of the above VAR(1) is x t = Φ 1 x t 1 + ε t where Φ 1 is a matrix given by ( φ11 φ Φ 1 = 12 φ 21 φ 22 ) More generally, we can write down an m dimensional VAR(p) very compactly as Φ(B)x t = ε where Φ(B) = 1 Φ 1 B Φ 2 B 2 + + Φ p B p and Φ i are m m matrices of parameters. Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 11/ 16

Multivariate White Noise The multivariate white noise process ε is assumed to satisfy { Γ 0, j = 0 Cov(ε t, ε t+j ) = 0 m, j 0 where Γ 0 is any covariance matrix and 0 m denotes the m m matrix of zeros. This assumption allows for covariance between ε it and ε jt, but ε it must be uncorrelated with ε j,t+k. Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 12/ 16

Stationarity of VAR Models Similar to the AR(p) model, the VAR(p) model is stationary if the roots of det(φ(z)) all lie outlside the unit disk. Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 13/ 16

Outline 1 Multivariate Time Series Modeling 2 VAR 3 VARIMA/VARMAX Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 14/ 16

VARIMA The natural generalization to the ARIMA model is the VARIMA model compactly written as Φ(B)(1 B) d Ix t = Θ(b)ε t One should consider co-integration of the time series before applying multiple differences. Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 15/ 16

VARMAX Additionally, exogenous variables u t may be added to the ARMA model producing an ARMAX model with representation x t = Γu t + p Φ j x t j + j=1 where Γ is an m r parameter matrix. q Θ k w t k + w t k=1 Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 16/ 16