5 X; as in previous problem. Show that n Ε(Σ(Χ - μ;) Σ(Χ - μ;)) = Σκ IE i=1 where K; is the covariance matrix of X;.

5. Show that the residual vector, e, can be written as e = (I, - H)y, where H is a hat matrix. Furthermore, show both H and (I, – H) are idempotent matrices. 6. Find E(e) and Cov(e). What is the Variance of e,? What is the Variance of Cov(e, e)?

b. Suppose that e, is zero mean white noise with var(et) = o. Consider the process: i. ii. iii. iv. Y₁ = 1+0.4Y-1 + et - 0.3e-1 0.15€t-2 Write the model using lag operator notation. Assess if the process is covariance stationary. Identify this model as an ARIMA (p, d, q) process; that is, specify p, d, and q. Find μ = E(Y).

Express the following as linear combinations of = (2, 1,4), v = (1, – 1, 3), and w= 8. = (3, 2, 5). %3D (a) (-9, –7, –15) (b) (6, 11, 6) (c) (0, 0, 0) (d) (7, 8, 9)