MAT3379 (Winter 2015) Assignment 5 Due date (Assignment 5): April 13, 2015 Q1. Assume that Z = (Z1 , Z2 ) is a random vector with the mean vector 0 and the covariance matrix σ11 σ12 Σ= . σ21 σ22 Let A be a deterministic matrix defined by A= a11 a21 a12 a22 . Consider the random vector X = AZ. • Show that E[X] = 0. Note: You need to multiply A by Z and compute the expected value of each coordinate. • Show that the covariance matrix of the random vector X is given by AΣAT . Note: You need to multiply A by Z. The resulting vector has two components. Calculate the variance of each component and the covariances between both components. Check that what you obtain is exactly AΣAT Solution to Q1: We have T a11 Z1 + a12 Z2 a21 Z1 + a22 Z2 X = (X1 , X2 ) = . By the definition, the expected value of the vector X = (X1 , X2 )T is a vector with two components being E[X1 ] and E[X2 ]. We have E[X1 ] = E[a11 Z1 + a12 Z2 ] = 0 since E[Z1 ] = E[Z2 ] = 0. The same for E[X2 ]. Also, the covariance matrix of X is given by Var[X1 ] Cov[X1 , X2 ] Cov[X1 , X2 ] Var[X2 ] . We calculate (note that we must have σ12 = σ21 ) Var[X1 ] = Var[a11 Z1 + a12 Z2 ] = a211 Var[Z1 ] + a212 Var[Z2 ] + 2a11 a12 Cov(Z1 , Z2 ) = a211 σ11 + a211 σ22 + 2a11 a12 σ12 Var[X2 ] = Var[a21 Z1 + a22 Z2 ] = a221 Var[Z1 ] + a222 Var[Z2 ] + 2a21 a22 Cov(Z1 , Z2 ) = a221 σ11 + a221 σ22 + 2a21 a22 σ12 Cov[X1 , X2 ] = Cov[a11 Z1 + a12 Z2 , a21 Z1 + a22 Z2 ] = a11 a21 σ11 + a12 a22 σ22 + (a11 a22 + a12 a21 ) σ12 . Hence, the covariance matrix of X is given by a211 σ11 + a211 σ22 + 2a11 a12 σ12 a11 a21 σ11 + a12 a22 σ22 + (a11 a22 + a12 a21 ) σ12 a11 a21 σ11 + a12 a22 σ22 + (a11 a22 + a12 a21 ) σ12 a221 σ11 + a221 σ22 + 2a21 a22 σ12 . You obtain the same when computing AΣAT . Marking scheme for Q1: 1 point for computation of E[X], 2 points for computation of the covariance matrix of X, two points for computation of AΣAT . 1 2 2 Q2. Assume that Zt = (Zt , Zt ), where {Zt } is an i.i.d. sequence with mean vector 0 and variance σZ (yes, both components of the vector are the same). Define the bivariate linear process by Xt = Zt + ΨZt−1 , where 1 0 Ψ= . 1 θ Find Γ(0), Γ(1) and Γ(−1), where Γ(h) is the covariance matrix function of the bivariate time series Xt . Note: this is the same example as I did in class on Wednesday, April 1st. Solution to Q2: Here, there is the correct solution for Xt = Zt + ΨZt−1 , where 1 1 Ψ= We have Xt = (Xt1 , Xt2 )T = and 0 θ . Zt + Zt−1 Zt + (1 + θ)Zt−1 . 2 2 Var(Xt1 ) Cov(Xt1 , Xt2 ) 2σZ (2 + θ)σZ Γ(0) = = . 2 2 Cov(Xt2 , Xt1 ) Var(Xt2 ) (2 + θ)σZ (1 + θ)2 + 1 σZ 2 2 Cov(Xt1 , Xt+1,1 ) Cov(Xt1 , Xt+1,2 ) σZ (1 + θ)σZ Γ(1) = = . 2 2 Cov(Xt2 , Xt+1,1 ) Cov(Xt2 , Xt+1,2 ) σZ (1 + θ)σZ 2 2 Cov(Xt1 , Xt−1,1 ) Cov(Xt1 , Xt−1,2 ) σZ σZ Γ(−1) = = . 2 2 Cov(Xt2 , Xt−1,1 ) Cov(Xt2 , Xt−1,2 ) (1 + θ)σZ (1 + θ)σZ Below, there is the solution for Xt = Zt + ΨZt−1 , where Ψ= 0 0 0 θ which is the example we did in class. We have T Xt = (Xt1 , Xt2 ) = and Zt Zt + θZt−1 . 2 2 Var(Xt1 ) Cov(Xt1 , Xt2 ) σZ σZ = . 2 2 Cov(Xt2 , Xt1 ) Var(Xt2 ) σZ (1 + θ2 )σZ 2 Cov(Xt1 , Xt+1,1 ) Cov(Xt1 , Xt+1,2 ) 0 θσZ Γ(1) = = . 2 Cov(Xt2 , Xt+1,1 ) Cov(Xt2 , Xt+1,2 ) 0 θσZ 0 0 Cov(Xt1 , Xt−1,1 ) Cov(Xt1 , Xt−1,2 ) Γ(−1) = = . 2 2 Cov(Xt2 , Xt−1,1 ) Cov(Xt2 , Xt−1,2 ) θσZ θσZ Γ(0) = Marking scheme for Q2: This part will not be marked. Total number of points for Assignment 5: 5
© Copyright 2024