Let {yt} be a first-order autoregressive process such that yt = φyt−1 + εt, where the white noise sequence {εt} follows a standard normal distribution. Suppose that we have observed the values {y1,...

Let
{
y
t
}
be a first-order autoregressive process such that
y
t
=
φ
y
t
−1 +
ε
t
,

where the white noise sequence
{
ε
t
}
follows a standard normal distribution.

Suppose that we have observed the values
{
y1, y2, y3, y5}
but
y4 is missing.

(a)
Calculate the joint density of
y1, y2, y3, y4, y5,
f(y1, y2, y3, y4, y5).

(b)
Find
z, the value that maximizes
f(y1, y2, y3, y4, y5) with respect to
y4.

(c)
Show that
z
corresponds to the
smoother
of
y4, that is,
z
=
E[y4|
y1, y2, y3, y5].