Mathematical and Statistical Methods for Actuarial Sciences and Finance

236 Marcella Niglio and Cira Perna

90-day US T-bill secondary market rates. Some concluding remarks are given at the
end.

2 Nonparametric kernel predictors

LetXT={X 1 ,X 2 ,...,XT}be a real-valued time series. We assume that:
A1: XTis a realisation of a strictly stationary stochastic process that belongs to
the class:
Xt=f(Xt− 1 ,Xt− 2 ,...,Xt−p)+t, 1 ≤t≤T, (1)

where the innovations{t}are i.i.d. random variables, independent from the past of
Xt, withE(t)=0,E(^2 t)=σ^2 <+∞,andpis a nonnegative integer.
In class (1),f(Xt− 1 ,Xt− 2 ,...,Xt−p)is the conditional expectation ofXt,given
Xt− 1 ,...,Xt−p, that can be differently estimated. When the Nadaraya-Watson (N-
W)-type estimator (among others see [2]) is used:

fˆ(x 1 ,x 2 ,...,xp)=

∑T

t=p+ 1

∏p

i= 1

K

(

xi−Xt−i

hi

)

Xt

∑T

t=p+ 1

∏p

i= 1

K

(

xi−Xt−i

hi

) , (2)

whereK(·)is a kernel function andhiis the bandwidth, fori= 1 , 2 ,...,p.
Under mixing conditions, the asymptotic properties of the estimator (2) have been
widely investigated in [18], and the main properties, when it is used in predictive
context, have been discussed in [9] and [2].
When the estimator (2) is used, the selection of the “optimal” bandwidth, the
choice of the kernel function and the determination of the autoregressive orderpare
needed. To solve the latter problem, many authors refer to automatic methods, such
as AIC and BIC, or to their nonparametric analogue suggested by [19].
Here we propose a procedure based on one-step-ahead kernel predictors.
The estimator for the conditional mean (2) has a large application in prediction
contexts. In fact, when a quadratic loss function is selected to find a predictor for
XT+, with lead time>0, it is well known that the best predictor is given by
XˆT+=E[XT+|XT], obtained from the minimisation of

arg min

XˆT+∈R

E[(XˆT+−XT+)^2 |XT], with > 0.

It implies that when N-W estimators are used to forecastXT+, the least-squares
predictorXˆT+becomes:

XˆT+=

T∑−

t=p+ 1

∏p

i= 1

K

(

xi−Xt−i

hi

)

Xt+

T∑−

t=p+ 1

∏p

i= 1

K

(

xi−Xt−i

hi

). (3)