Use Proposition 2.18 to complete the following exercises. (a) Write a univariate AR(1) model, Y t = φY t −1+V t , in state-space form. Verify your answer is indeed an AR(1). (b) Repeat (a) for an...

Use Proposition 2.18 to complete the following exercises.

(a) Write a univariate AR(1) model, Y_t
= φY_t−1+V_t
, in state-space form. Verify your answer is indeed an AR(1).

(b) Repeat (a) for an MA(1) model, Y_t
= V_t
+θV_t−1.

(c) Write an IMA(1,1) model, Y_t
= Y_t−1 +V_t
+θV_t−1, in state-space form.

In Section 2.3, we discussed that it is possible to obtain a recursion for the gradient or score vector, −∂ lnL(θ; Y₁:n)/∂ θ. Assume the model is given by (2.1) and (2.2) and At is a known design matrix that does not depend on θ, in which case Proposition 2.3 applies. For the gradient vector, show