Let {Y_(t)} be a stationary process with model equation
Y_(t)=24+(2)/(3)Y_(t-1)+epsi lon_(t)+(1)/(2)epsi lon_(t-1),tinZ,
where {epsi lon_(t)} is a white noise process such that the epsi lon_(t) are NID(0,sigma ^(2)). Let y_(1),y_(2),dots,y_(T)
denote an observed realization of the process up to time T. Consider the forecasting
problem with T the origin of the forecasts, h the lead time and hat(y)_(T)(h) the corresponding
minimum mean square error forecast, for lead times h>=1.
(a) Show that an iteration scheme for the forecasts can be characterized by the fol-
lowing relations
hat(y)_(T)(1)=24+(2)/(3)y_(T)+(1)/(2)epsi lon_(T)
hat(y)_(T)(h)=24+(2)/(3)hat(y)_(T)(h-1),h>=2.
Without carrying out any further calculation, comment on whether a statistical
programmer would be able to implement the above algorithm in its current form. If
not, state why.
(b) i) Show that the minimum mean square error forecast at lead time 2 based
around the origin at time T is given by
hat(y)_(T)(2)=24sum_(i=0)^1 ((2)/(3))^(i)+((2)/(3))^(2)(y_(T)+0.75epsi lon_(T)).
ii) Express Y_(T+2) in terms of nothing other than a linear combination of observed
process values {y_(t)} residing within H_(T), and white noise terms {epsi lon_(t)} (which may
reside within or outside H_(T), the history of the process up to time T ).
Derive an explicit formula for the forecast error at lead time h=2, i.e. e_(T)(2)=
Y_(T+2)-hat(y)_(T)(2), formulated in terms of one or more components of the white
noise process.
iii) Specifying the value of gamma , show that the forecast error variance at lead time
h=2 is given by
V(2)=gamma sigma ^(2)
6. Let {Y} be a stationary process with model equation
Y = 24+ Y+ 1+E+
-Et-1
teZ,
where {et} is a white noise process such that the et are NID(0, o2). Let y1,Y2,...,Yt denote an observed realization of the process up to time T. Consider the forecasting problem with T the origin of the forecasts, h the lead time and yt(h) the corresponding minimum mean square error forecast, for lead times h > 1.
(a) Show that an iteration scheme for the forecasts can be characterized by the fol- lowing relations
1
24 + 3
yT(h) = 24+ yT(h -1),
h2.
Without carrying out any further calculation, comment on whether a statistical programmer would be able to implement the above algorithm in its current form. If not, state why. [4] (b) i) Show that the minimum mean square error forecast at lead time 2 based around the origin at time T is given by
yT(2)=24
(yT+0.75eT)
[3] ii) Express YT+2 in terms of nothing other than a linear combination of observed process values {yt} residing within HT, and white noise terms {et} (which may reside within or outside HT, the history of the process up to time T). Derive an explicit formula for the forecast error at lead time h = 2, i.e. er(2) = YT+2 -- yT(2), formulated in terms of one or more components of the white noise process. [6] iii) Specifying the value of y, show that the forecast error variance at lead time h = 2 is given by V(2) = yo2