"Frontmatter". In: Analysis of Financial Time Series

VECTOR ARMA MODELS 327interest-rate series appear to be unit-root nonstationary. Using the back-shift operatorB, the model can be rewritten approximately as(1 − B)r 3t = 0.025 + (1 + 0.47B)a 3t(1 − B)(1 − 0.6B)r 1t = 0.02 − 0.55B(1 − B)r 3,t + (1 − 0.6B)a 1t + 1.17Ba 3,t .8.4.1 Marginal Models of ComponentsGiven a vector model for r t , the implied univariate models for the components r itare the marginal models. For a k-dimensional ARMA(p, q) model, the marginalmodels are ARMA[kp,(k − 1)p + q]. This result can be obtained in two steps. First,the marginal models of a VMA(q) model is univariate MA(q). Assume that r t isaVMA(q) process. Because the cross-correlation matrix of r t vanishes after lag q(i.e., ρ l = 0 for l>q), the ACF of r it is zero beyond lag q. Therefore, r it is an MAprocess and its univariate model is in the form r it = θ i,0 + ∑ qj=1 θ i, jb i,t− j ,where{b it } is a sequence of uncorrelated random variables with mean zero and varianceσ 2ib . The parameters θ i, j and σ ib are functions of the parameters of the VMA modelfor r t .The second step to obtain the result is to diagonalize the AR matrix polynomialof a VARMA(p, q) model. For illustration, consider the bivariate AR(1) model[ 1 − 11 B − 12 B− 21 B 1 − 22 B][r1tPremultiplying the model by the matrix polynomialwe obtain]=r 2t[ ]1 − 22 B 12 B, 21 B 1 − 11 B[a1t].a 2t[ ] [ ][ ][(1 − 11 B)(1 − 22 B) − 12 22 B 2 r1t 1 − 22 B −] =12 B a1t.r 2t − 21 B 1 − 11 B a 2tThe left-hand side of the prior equation shows that the univariate AR polynomialsfor r it are of order 2. In contrast, the right-hand side of the equation is in aVMA(1) form. Using the result of VMA models in step 1, we show that the univariatemodel for r it is ARMA(2, 1). The technique generalizes easily to the k-dimensionalVAR(1) model, and the marginal models are ARMA(k, k − 1). More generally, for ak-dimensional VAR(p) model, the marginal models are ARMA[kp,(k − 1)p]. Theresult for VARMA models follows directly from those of VMA and VAR models.The order [kp,(k − 1)p + q] is the maximum order (i.e., the upper bound) for themarginal models. The actual marginal order of r it can be much lower.