Nonparametric prediction in time series analysis 237The properties of (3) in the presence of strictly stationary and Markovian processes
of orderpare discussed in [9] and [2].
Under well defined assumptions on the generating process, [9] shows that when
=1 the predictor (3) is a strong consistent estimator forE[XT+ 1 |XT] and this
result has been subsequently generalised in [2] to the case with≥1.
In presence of real data, [5], [16], [10] and recently [23] evaluate the forecast
accuracy of (3) and give empirical criteria to define confidence intervals forXˆT+.
In the following, taking advantage of (3), we propose the use of the estimated risk
of prediction (ERP), discussed in [13], to select the orderpof the the autoregres-
sion (1).
In particular we further assume that:
A2: XTis a realisation of a strong mixing (orα-mixing) process. Under conditions
A1 and A2, the ERP can be estimated through resampling techniques and in particular
using the subsampling approach as proposed by [13].
The subsampling has a number of interesting advantages with respect to other
resampling techniques: in particular it is robust against misspecified models and gives
consistent results under weak assumptions.
This last remark makes the use of subsampling particularly useful in a nonpara-
metric framework and can be properly applied in the context of model selection.
LetXˆT+ 1 be the N-W predictor (3); its mean square error is defined as
T=E[(XT+ 1 −XˆT+ 1 )^2 ].The algorithm we propose to selectpis established on the estimation ofTthat,
as described in Procedure 1, is based on the overlapping subsampling. Note that in
this procedure Step 2 implies the choice of the subsample lengthb. A large number
of criteria have been proposed in the statistical literature to selectb(inter alia [17]).
Here we refer to the proposal in [14], which describes an empirical rule for estimating
the optimal window size in the presence of dependent data of smaller length (m)than
the original (T). The details are given in Procedure 2.
Procedure 1: Selection of the autoregressive orderp
- Choose a grid forp∈( 1 ,...,P).
- Select the subsample lengthb(Procedure 2).
- For eachp, compute the estimated risk of prediction (ERP):
ˆT,b=(T−b+ 1 )−^1T∑−bi= 0(
Xˆ(i+i)b−Xi+b) 2
,
whereXˆ(i+i)bis the one-step-ahead predictor (3) ofXi+b, based on the subsample
(Xi+ 1 ,Xi+ 2 ,...,Xi+b− 1 )of lengthb.- Selectpˆwhich minimisesˆT,b.