Lecture 6: Vector Autoregression∗

Lecture 6: Vector Autoregression∗ In this section, we will extend our discussion to vector valued time series. We will be mostly interested in vector autoregression (VAR), which is much easier to be estimated in applications. We will fist introduce the properties and basic tools in analyzing stationary VAR process, and then we’ll move on to estimation and inference of the VAR model. 1 Covariance-stationary VAR(p) process 1.1 Introduction to stationary vector ARMA processes 1.1.1 VAR processes A VAR model applies when each variable in the system does not only depend on its own lags, but also the lags of other variables. A simple VAR example is: x1t = φ11x1,t−1 + φ12x2,t−1 + 1t x2t = φ21x2,t−1 + φ22x2,t−2 + 2t where E(1t2s) = σ12 for t = s and zero for t 6= s. We could rewrite it as x φ φ x 0 0 x 1t = 11 12 1,t−1 + 1,t−2 + 1t , x2t 0 φ21 x2,t−1 0 φ22 x2,t−2 2t or just xt = Φ1xt−1 + Φ2xt−2 + t (1) and E(t) = 0,E(ts) = 0 for s 6= t and 2 0 σ1 σ12 E(tt) = 2 . σ21 σ2 As you can see, in this example, the vector-valued random variable xt follows a VAR(2) process. A general VAR(p) process with white noise can be written as xt = Φ1xt−1 + Φ2xt−2 + ... + t p X = Φjxt−j + t j=1 or, if we make use of the lag operator, Φ(L)xt = t, ∗Copyright 2002-2006 by Ling Hu. 1 where p Φ(L) = Ik − Φ1L − ... − ΦpL . The error terms follow a vector white noise, i.e., E(t) = 0, Ω for t = s E( 0 ) = t s 0 otherwise with Ω a (k × k) symmetric positive definite matrix. Recall that in studying the scalar AR(p) process, φ(L)xt = t, we have the results that the process {xt} is covariance-stationary as long as all the roots in (2) 2 p 1 − φ1z − φ2z − ... − φpz = 0 (2) lies out side of the unit circle. Similarly, for the VAR(p) process to be stationary, we must have that the roots in the equation p |Ik − Φ1z − ... − Φpz | = 0 all lies outside the unit circle. 1.1.2 Vector moving average processes Recall that we could invert a scalar stationary AR(p) process, φ(L)xt = t to a MA(∞) process, −1 xt = θ(L)t, where θ(L) = φ(L) . The same is true for a covariance-stationary VAR(p) process, Φ(L)xt = t. We could invert it to xt = Ψ(L)t where Ψ(L) = Φ(L)−1 The coefficients of Ψ can be solved in the same way as in the scalar case, i.e., if Φ−1(L) = Ψ(L), then Φ(L)Ψ(L) = Ik: 2 p 2 (Ik − Φ1L − Φ2L − ... − ΦpL )(Ik + Ψ1L + Ψ2L + ...) = Ik. j Equating the coefficients of L , we have Ψ0 = Ik,Ψ1 = Φ1,Ψ2 = Φ1Ψ1 + Φ2, and in general, we have Ψs = Φ1Ψs−1 + Φ2Ψs−2 + ... + ΦpΨs−p. 1.2 Transforming to a state space representation Sometime, it is more convenient to write a scalar valued time series, say an AR(p) process, in vector form. For example, p X xt = θjxt−j + t. j=1 2 where ∼ N(0, σ2). We could equivalently write it as xt φ1 φ2 . φp−1 φp xt−1 t xt−1 1 0 ... 0 0 xt−2 0 = + . . . . . . . . xt−p+1 0 ...... 1 0 xt−p 0 0 If we let ξt = (xt, xt−1, . , xt−p+1) , ξt−1 = (xt−1, xt−2, . , xt−p), t = (t, 0,..., 0), and let F denote the parameter matrix, then we can write the process as: ξt = F ξt−1 + t 2 where ∼ N(0, σ Ip). So we have rewrite an AR(p) scalar process as an vector autoregression of order one, denoted by VAR(1). Similarly, we could also transform a VAR(p) process to a VAR(1) process. For the process xt = Φ1xt−1 + Φ2xt−2 + ... + Φpxt−p + t, let xt xt−1 ξ = , t . . xt−p+1 Φ1 Φ2 ... Φp−1 Φp Ik 0 ... 0 0 0 Ik ... 0 0 F = , . . 0 0 ...Ik 0 t 0 v = , t . . 0 Then we could rewrite the VAR(p) process in state space notations, ξt = F ξt−1 + vt. (3) 0 where E(vtvs) equals Q for t = s and equals zero otherwise, and Ω 0 ... 0 0 0 ... 0 Q = . . . ... 0 0 ... 0 3 1.3 The autocovariance matrix 1.3.1 VAR process For a covariance stationary k dimensional vector process {xt}, let E(xt) = µ, then the autocovariance is defined to be the following k by k matrix 0 Γ(h) = E[(xt − µ)(xt−h − µ) ]. 0 For simplicity, assume that µ = 0. Then we have Γ(h) = E(xtxt−h). Because of the lead-lag effect, we may not have Γ(h) = Γ(−h), but we have Γ(h)0 = Γ(−h). To show this, 0 0 Γ(h) = E(xt+hxt+h−h) = E(xt+hxt), taking transpose 0 0 Γ(h) = E(xtxt+h) = Γ(−h). Similar as in the scalar case, we define the autocovariance generating function of the process x as ∞ X h Gx(z) = Γ(h)z h=−∞ where z is again a complex scalar. Let ξt as defined in (3). Assume that ξ and x are stationary, and let Σ denote the variance of ξ, 0 Σ = E(ξtξt) xt xt−1 = E x0 x0 ... x0 . t t−1 t−p+1 . xt−p+1 Γ(0) Γ(1) ... Γ(p − 1) Γ(1)0 Γ(0) ... Γ(p − 2) = . . . ... Γ(p − 1)0 Γ(p − 2)0 ... Γ(0) Postmultiplying (3) by its transpose and taking expectations gives 0 0 0 0 0 E(ξtξt) = E[(F ξt−1 + vt)(F ξt−1 + vt) ] = FE(ξt−1ξt−1)F + E(vtvt), or Σ = F ΣF 0 + Q. (4) To solve for Σ, we need to use the Kronecker product, and the following result: let A, B, C be matrices whose dimensions are such that the product ABC exists. Then vec(ABC) = (C0 ⊗ A) · vec(B). 4 where vec is the operator to stack each column of a matrix (k × k) into a k2-dimensional vector, for example, a11 a11 a12 a21 A = vec(A) = . a21 a22 a12 a22 Apply vec operator on both sides of (4), we get vec(Σ) = (F ⊗ F ) · vec(Σ) + vec(Q), which gives −1 vec(Σ) = (Im − F ⊗ F ) vec(Q), where m = k2p2. We can use this equation to solve for the first p order of autocovariance of x, Γ(0),..., Γ(p). To derive the hth autocovariance of ξ, denoted by Σ(h), we can postmultiplying 0 (3) by ξt−h and take expectations, 0 0 0 E(ξtξt−h) = FE(ξt−1ξt−h) + E(vtξt−h), then Σ(h) = F Σ(h − 1), or Σ(h) = F hΣ. Therefore we have the following relationship for Γ(h) Γ(h) = Φ1Γ(h − 1) + Φ2Γ(h − 2) + ... + ΦpΓ(h − p). 1.3.2 Vector MA processes We first consider the MA(q) process. xt = t + Ψ1t−1 + Ψ2t−2 + ... + Ψqt−q. Then the variance of xt is 0 Γ(0) = E(xtxt) 0 0 0 0 0 = E(tt) + Ψ1E(t−1t−1)Ψ1 + ... + ΨqE(t−qt−q)Ψq 0 0 0 = Ω + Ψ1ΩΨ1 + Ψ2ΩΨ2 + ... + ΨqΩΨq. and the autocovariances 0 0 ΨhΩ + Ψh+1ΩΨ1 + Ψh+2ΩΨ2 + ... + ΨqΩΨq−j for h = 1, . , q. 0 0 0 0 Γ(h) = ΩΨ−h + Ψ1ΩΨ−h+1 + Ψ2ΩΨ−h+2 + ... + Ψq+hΩΨq for h = −1,..., −q. 0 for |h| > q As in the scalar case, any vector MA(q) process is stationary. Next consider the MA(∞) process xt = t + Ψ1t−1 + Ψ2t−2 + ... = Ψ(L)t. 5 ∞ A sequence of matrices {Ψs}−∞ is absolutely summable if each of its element forms an absolutely summable scalar sequence, i.e. ∞ X (s) |ψij | < ∞ for i, j = 1, 2, . n, s=0 (s) where ψij is the row i column j element (will use ijth for short) of Ψs. Some important results about MA(∞) process is summarized as follows: Proposition 1 Let xt be a k × 1 vector satisfying ∞ X xt = Ψjt−j, j=0 where t is vector white noise and Ψj is absolutely summable. Then (a) The autocovaiance between the ith variable at time t and the jth variable s periods earlier, E(xitxj,t−s) exists and is given by the ijth element of ∞ X 0 Γ(s) = Ψs+vΩΨv for s = 0, 1, 2,... ; v=0 ∞ (b) {Γ(h)}h=0 is absolutely summable. ∞ If {t}t=−∞ is i.i.d. with E|i1,ti2,ti3,ti4,t| < ∞ for i1, i2, i3, i4 = 1, 2, . , k then we also have (c) E|xi1,t1xi2,t2xi3,t3xi4,t4| < ∞ for i1, i2, i3, i4 = 1, 2, . , k and all t1, t2, t3, t4. −1 Pn (d) n t=1 xitxj,t−s →p E(xitxj,t−s) for i, j = 1, 2, . , k and for all s. All of these results can be viewed as extensions from the scalar case to vector case, and its proof can be found on page 286-288 in Hamilton’s book. 1.4 The Sample Mean of a Vector Process Let xt be a stationary process with E(xt) = 0 and E(xtxt−h) = Γ(h), where Γ(h) is absolutely summable. Then we consider the properties of the sample mean n 1 X x¯ = x .

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