Functional Coef±cient Regression Models
for Non-linear Time Series: A Polynomial
Spline Approach
JIANHUA Z. HUANG
University of Pennsylvania
HAIPENG SHEN
University of North Carolina at Chapel Hill
ABSTRACT. We propose a global smoothing method based on polynomial splines for the esti-
mation of functional coefFcient regression models for non-linear time series. Consistency and rate
of convergence results are given to support the proposed estimation method. Methods for automatic
selection of the threshold variable and signiFcant variables (or lags) are discussed. The estimated
model is used to produce multi-step-ahead forecasts, including interval forecasts and density
forecasts. The methodology is illustrated by simulations and two real data examples.
Key words:
forecasting, functional autoregressive model, non-parametric regression, threshold
autoregressive model, varying coeﬃcient model
1. Introduction
For many real time series data, non-linear models are more appropriate than linear models for
accurately describing the dynamic of the series and making multi-step-ahead forecasts (see e.g.
Tong, 1990; Franses & van Dijk, 2000). Recently, non-parametric regression techniques have
found important applications in non-linear time series analysis (Tjøstheim & Auestad, 1994;
Tong, 1995; Ha
¨ rdle
et al.
, 1997). Although the non-parametric approach is appealing, its
application usually requires an unrealistically large sample size when more than two lagged
variables (or exogenous variables) are involved in the model (the so-called ‘curse of dimen-
sionality’). To overcome the curse of dimensionality, it is necessary to impose some structure
to the non-parametric models.
A useful structured non-parametric model that still allows appreciable ﬂexibility is the
functional coef±cient regression model described as follows. Let
f
Y
t
;
X
t
;
U
t
g
1
ÿ1
be jointly
strictly stationary processes with
X
t
¼
(
X
t
1
,
…
,
X
td
) taking values in
R
d
and
U
t
in
R
. Let
E
ð
Y
2
t
Þ
<
1
. The multivariate regression function is de±ned as
f
ð
x
;
u
Þ¼
E
ð
Y
t
j
X
t
¼
x
;
U
t
¼
u
Þ
:
In a pure time series context, both
X
t
and
U
t
consist of some lagged values of
Y
t
. The
functional coef±cient regression model requires that the regression function has the form
f
ð
x
;
u
X
d
j
¼
1
a
j
ð
u
Þ
x
j
;
ð
1
Þ
where
a
j
(
Æ
)s are measurable functions from
R
to
R
and
x
¼
(
x
1
,
…
,
x
d
)
T
.As
U
t
2
R
, only one-
dimensional smoothing is needed in estimating the model (1).
The functional coef±cient regression model extends several familiar non-linear time series
models such as the exponential autoregressive (EXPAR) model of Haggan & Ozaki (1981) and
Ozaki (1982), threshold autoregressive (TAR) model of Tong (1990), and functional
ȑ
Board of the Foundation of the Scandinavian Journal of Statistics 2004. Published by Blackwell Publishing Ltd, 9600 Garsington
Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA Vol 31: 515–534, 2004