Ch. 22 Unit root in Vector Time series 1 Multivariate Wiener Processes and multivari- ate FCLT Section 2.1 of Chapter 21 described univariate standard Brownian motion W(r) as a scalar continuous-time process(W: rE0, 1-R). The variable W(r) has a N(O, r)distribution across realization, and for any given realization, w(r) is continuous function of the date r with independent increments. If a set of k such independent processes, denoted Wi(r), W2(r),.,Wk(r), are collected in a (kx1) vector w(r), the results is k-dimentional standard Brownian motion Definition 1 A k-dimensional standard Brownian motion w( is a continuous-time process as- sociating each dater E0, 1] with the(k x 1)vector w(r)satisfying the following (b). For any dates0≤n1<r2<…<Tk≤1, the changes w(r2)-w(r1)],w(r3)- w(r2),,w(rk)-w(rk-1)) are independent multivariate Gaussian with w(s) w()~N(0,(s-t)Ik) (c). For any given realization, w(r) is continuous in r with probability 1 Analogous to the univariate case. we can define a multivariate random walk as follows Definition Let the k x 1 random vector yt follow yt= yt-1+Et, t= 1, 2, .. where yo =0 and Et is a sequence of ii d random vector such that E(Et)=0 and E(EtE=n,a finite positive definite matrix. Then yt is a multivariate(k-dimensional)random form the rescaled partial sums as g-12T- t=1
Ch. 22 Unit Root in Vector Time Series 1 Multivariate Wiener Processes and Multivariate FCLT Section 2.1 of Chapter 21 described univariate standard Brownian motion W(r) as a scalar continuous-time process (W : r ∈ [0, 1] → R 1 ). The variable W(r) has a N(0, r) distribution across realization, and for any given realization, W(r) is continuous function of the date r with independent increments. If a set of k such independent processes, denoted W1(r), W2(r), ..., Wk(r), are collected in a ( k×1) vector w(r), the results is k−dimentional standard Brownian motion. Definition 1: A k-dimensional standard Brownian motion w(·) is a continuous-time process associating each date r ∈ [0, 1] with the (k×1) vector w(r) satisfying the following: (a). w(0) = 0; (b). For any dates 0 ≤ r1 < r2 < ... < rk ≤ 1, the changes [w(r2)−w(r1)], [w(r3)− w(r2)], ..., [w(rk) − w(rk−1)] are independent multivariate Gaussian with [w(s) − w(v)] ∼ N(0,(s − v)Ik); (c). For any given realization, w(r) is continuous in r with probability 1. Analogous to the univariate case, we can define a multivariate random walk as follows. Definition: Let the k×1 random vector yt follow yt = yt−1+εt , t = 1, 2, ..., where y0 = 0 and εt is a sequence of i.i.d. random vector such that E(εt) = 0 and E(εtε 0 t ) = Ω, a finite positive definite matrix. Then yt is a multivariate (k-dimensional) random walk. We form the rescaled partial sums as wT (r) ≡ Ω −1/2 T −1/2 [Tr] X∗ t=1 εt . 1
The components of wr(r) are the individual partial sums Wr()=T2∑ t=1 where Eti is the jth element of Q2-2E The Function Central Limit Theorem(FCLT) provides conditions under which wr(r) converges to the multivariate standard Wiener process w(r). The simplest multivariate Fclt is the multivariate Donsker's theorem Theorem 1(Multivariate Donsker) Let et be a sequence of ii d random vector such that e(Et)=0 and E(E,E)=S a finite positive definite matrix. Then wr()=w( Quite general multivariate FCLTs are available. For example, we may applied FCLT to serially dependent vector processes using a generalization of(70) and Theorem 12 of Chapter 21 Theorem 2(FCLT when ut is a vector M A(oo) then Wr()=w() where wr()=业(1)-1g-1/r-12∑mε1, Et is a k dimensional i id,rand vector with variance covariance nL, and if v,(s denote the row i, column j element of亚 ∑s1。1<∞ for eac roo Using multivariate Beveridge-Nelson decomposition and from that to derive the long run variance matrix of ut to be tE(ut)]=4-(1)2
The components of wT (r) are the individual partial sums WTj(r) = T −1/2 [Tr] X∗ t=1 ε˜tj, j = 1, 2, ..., k, where ε˜tj is the jth element of Ω −1/2 εt . The Function Central Limit Theorem (FCLT) provides conditions under which wT (r) converges to the multivariate standard Wiener process w(r). The simplest multivariate FCLT is the multivariate Donsker’s theorem. Theorem 1(Multivariate Donsker): Let εt be a sequence of i.i.d. random vector such that E(εt) = 0 and E(εtε 0 t ) = Ω, a finite positive definite matrix. Then wT (·) =⇒ w(·). Quite general multivariate FCLTs are available. For example, we may applied FCLT to serially dependent vector processes using a generalization of (70) and Theorem 12 of Chapter 21. Theorem 2 (FCLT when ut is a vector MA(∞) process): Let ut = X∞ s=0 Ψsεt−s, then wT (·) =⇒ w(·), where wT (r) ≡ Ψ(1)−1Ω −1/2T −1/2 P[Tr] ∗ t=1 εt , εt is a k dimensional i.i.d. random vector with variance covariance Ω, and if ψ (s) ij denote the row i, column j element of Ψs, X∞ s=0 s · |ψ (s) ij | < ∞ for each i, j = 1, 2, ..., k. Proof: Using multivariate Beveridge-Nelson decomposition and from that to derive the long run variance matrix of ut to be 1 T E[ P(ut) 2 ] = Ψ2 (1)Ω. 2
2 Vector Autoregression Containing Unit Roots Let yt be an(k x 1)vector autoregressive process(VAR(p),i.e k-重1L-重22-…-重,Ly= The scalar algebra in(33)of Chapter 21 works perfectly well for matrices, es- tablishing that for any value重1,重2,…,更p, the following polynomials are equiv I-重1L-重2L2-…-重門] (Ik-pL-(5,L+52L n1Dm-1)1-L) 更1+更 重+1+重+2+…+重]fors=1,2, It follows that any VAR(p) process(1) can always be written in the form (Lk-pLy-(1L+E2L2+…+5-1L-1)(1-L yt=1△y-1+2△yt-2+…+5p-yt-p+1+c+pyt-1+Et There are tow meanings of a V AR process contains unit roots First, if the first difference of yt follows a V AR(p-1)process 1△yt-1+52△ Sp-1yt-p+1+C+ requiring from(4)that
2 Vector Autoregression Containing Unit Roots Let yt be an (k × 1) vector autoregressive process (V AR(p)), i.e. [Ik − Φ1L − Φ2L 2 − ... − ΦpL p ]yt = c + εt . (1) The scalar algebra in (33) of Chapter 21 works perfectly well for matrices, establishing that for any value Φ1, Φ2,..., Φp, the following polynomials are equivalent: [Ik − Φ1L − Φ2L 2 − ... − ΦpL p ] = (Ik − ρL) − (ξ1L + ξ2L 2 + ... + ξp−1L p−1 )(1 − L), where ρ ≡ Φ1 + Φ2 + ... + Φp (2) ξs ≡ −[Φs+1 + Φs+2 + ... + Φp] for s = 1, 2, ..., p − 1. (3) It follows that any V AR(p) process (1) can always be written in the form (Ik − ρL)yt − (ξ1L + ξ2L 2 + ... + ξp−1L p−1 )(1 − L)yt = c + εt or yt = ξ14yt−1 + ξ24yt−2 + ... + ξp−1yt−p+1 + c + ρyt−1 + εt . (4) There are tow meanings of a V AR process contains unit roots. First, if the first difference of yt follows a V AR(p − 1) process: 4yt = ξ14yt−1 + ξ24yt−2 + ... + ξp−1yt−p+1 + c + εt , requiring from (4) that ρ = Ik 3
or from(2)that Ik Second, recalling from(8)of Chapter 18 that a V AR() such as in(1) will be said to contain at least one unit root(2=1) if the following determinant is zero 更p Note that() implies(6) but(6)does not imply(5). Vector autoregression for which(6)holds but(5) does not will be considered in Chapter 23
or from (2) that Φ1 + Φ2 + ... + Φp = Ik. (5) Second, recalling from (8) of Chapter 18 that a V AR(p) such as in (1) will be said to contain at least one unit root (z = 1) if the following determinant is zero: |Ik − Φ1 − Φ2 − ... − Φp| = 0. (6) Note that (5) implies (6) but (6) does not imply (5). Vector autoregression for which (6) holds but (5) does not will be considered in Chapter 23. 4
3 Spurious Regression 3.1 Asymptotics for Spurious Regression Consider a regression of the form yt=x6+ut, for which elements of yt and xt might be nonstationary. If there does not exist some population value for B for which the disturbance ut=yt-x'tB is I(0), then OLS is quite likely to produce spurious results. In a extreme condition that Yt and xt are independent random walks, as we shall see, the OLS estimator of B, B is not consistent for B=0 but instead converge to a particular random variable. Because there is truly no relation between Yt and xt, and because Br is incapable of revealing this, we call this a case of"spurious regression". Thi phenomenon was first considered by Yule(1926), and the dangers of spurious re- gression were forcefully brought to the economists by the Monte Carlo studies of Granger and Newbold(1974)and latter explained theoretically by Phillips(1986) Theorem 3(Spurious Regression, two independent random walks) Let Xt and Yt be independent random walks, Xt=Xt-I+nt and Yt=Yt-1+St and nt is independent of zeta. Consider the regression equation for Yt in terms of X,, formally as Yt= XtB+ut, where B=0 and ut= Yt, reflecting the lack of any relations between Yt and Xt. Then the OLS estimator of 6, Br (oa/o1 wi()dr Jowi(r)Wa(r)dr, where o?=E(ne)and o2=E(S) To proceed, we write Wr(-1)=m∑m/01=T12x-1/ Wxr(r1-1)=T-1∑s/02=m-1Y-1/2 T-1P2Xt-1=01W1r(Tt-1
3 Spurious Regression 3.1 Asymptotics for Spurious Regression Consider a regression of the form yt = x 0 tβ + ut , (7) for which elements of yt and xt might be nonstationary. If there does not exist some population value for β for which the disturbance ut = yt −x 0 tβ is I(0), then OLS is quite likely to produce spurious results. In a extreme condition that Yt and xt are independent random walks, as we shall see, the OLS estimator of β, βˆ T is not consistent for β = 0 but instead converge to a particular random variable. Because there is truly no relation between Yt and xt , and because βˆ T is incapable of revealing this, we call this a case of ”spurious regression”. This phenomenon was first considered by Yule (1926), and the dangers of spurious regression were forcefully brought to the economists by the Monte Carlo studies of Granger and Newbold (1974) and latter explained theoretically by Phillips (1986). Theorem 3 (Spurious Regression, two independent random walks): Let Xt and Yt be independent random walks, Xt = Xt−1 + ηt and Yt = Yt−1 + ζt , and ηt is independent of zetat . Consider the regression equation for Yt in terms of Xt , formally as Yt = Xtβ + ut , where β = 0 and ut = Yt , reflecting the lack of any relations between Yt and Xt . Then the OLS estimator of β, βˆ T L−→ (σ2/σ1) hR 1 0 W1(r) 2dri−1 R 1 0 W1(r)W2(r)dr, where σ 2 1 = E(η 2 t ) and σ 2 2 = E(ζ 2 t ). Proof: To proceed, we write W1T (rt−1) = T −1/2X t−1 s=1 ηs/σ1 = T −1/2Xt−1/σ1, W2T (rt−1) = T −1/2X t−1 s=1 ζs/σ2 = T −1/2Yt−1/σ2 or T −1/2Xt−1 = σ1W1T (rt−1) (8) 5