Suppose yt follows a stationary AR(1) process with AR parameter and innovation variance v with ( v) uncertain. At any time t write Dt for the past data and information, including all past observations.

If no additional information arises over the time interval (t 1 t] then Dt sequentially updates as the new observation is made via simply Dt = Dt 1 yt Now suppose you are standing at the end of time interval t 1 so that you have current information set Dt 1 The current posterior for ( v)

based on this information has a conjugate normal-inverse gamma form written as

( vDt 1)

(v 1 Dt 1)

N(mt 1Ct 1(v st 1))

G(nt 1 2nt 1st 1 2)

with known de ning parameters. This would be the case, for example, of a reference posterior based on the rst t 1 observations. Here mt 1 and st 1 > 0 are natural point estimates of and v respectively, while Ct 1 >0 and nt 1 >0 relate to uncertainty.

(a) What is the current marginal posterior for namely p( Dt 1)?

(b) Show that, conditional on v and marginalizing over the implied 1 step ahead forecast distribution for yt given v is

(ytvDt 1) N(ft qtv st 1)

with ft = mt 1yt 1 and qt = st 1 +Ct 1y2 t 1

(c) Now marginalize also over v to nd the implied 1 step ahead forecast distribution for yt, namely p(yt Dt 1), i.e., the distribution you will use in practice to predict yt 1 step ahead. What is this distribution?

(d) Now move to time t and observe the outcome yt Show that the time t posterior p( v Dt) is also normal-inverse gamma, having the same form as in at time t 1 above but now with t 1 updated to t and updated de ning parameters mtCt nt st that can be written in the following forms:
mt =mt 1+Atet Ct =rt(Ct 1 A2 tqt)
nt = nt 1+1 st = rtst 1 with rt = (nt 1 +e2 t qt) nt where et = yt ft is the realized 1 step ahead (point) forecast error, and At =Ct 1yt 1 qt is the adaptive coe cient.

(e) Comment on these expressions, giving particular attention to the fol lowing:
i. How (mtCt) depend on the new data yt relative to the prior values (mt 1Ct 1).
ii. The role of the adaptive coe cient in the update of (mt 1Ct 1)
to (mtCt)
iii. The updates for the degrees of freedom nt and point estimate st and how they depend on yt

(f) Consider an example in which the forecast error is very large relative to expectation, resulting in a value of e2 t qt much greater than 1.
Comment on how the posterior for ( v) responds.