Economics Letters 69 2000 267–276
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Small sample properties of the conditional least squares estimator in SETAR models
George Kapetanios
National Institute of Economic and Social Research , Smith Square, 2 Dean Trench Str., London SW1P 3HE, UK
Received 30 April 1999; received in revised form 2 February 2000; accepted 25 May 2000
Abstract
This note considers the small sample performance of the conditional least squares estimator of the threshold parameters in nonlinear threshold and particularly self exciting threshold autoregressive SETAR models. It is
shown that despite the superconsistency of the threshold parameter estimates the estimator performs poorly in samples of sizes usually encountered in macroeconomics.
2000 Elsevier Science S.A. All rights reserved.
Keywords : Threshold models; SETAR models; Monte Carlo; Superconsistency
JEL classification : C13; C15; C32
1. Introduction
The investigation of nonlinearity in macroeconomic series has attracted considerable attention recently. The class of threshold autoregressive models introduced by Tong 1978 and Tong and Lim
1980 provides a widely used framework for such investigation. The theoretical appeal and relative computational tractability are the major reasons for its use.
This note considers the small sample performance of the conditional least squares estimator in threshold and particularly SETAR models with particular focus on the threshold parameter estimator.
It is well known that under certain regularity conditions, this estimator is superconsistent see Chan 1993. However, we find that despite this theoretical property, the estimator performs poorly in
samples of sizes usually encountered in macroeconomics.
Tel.: 144-207-2227-665; fax: 144-207-6541-900. E-mail address
: gk205niesr.ac.uk G. Kapetanios. 0165-1765 00 – see front matter
2000 Elsevier Science S.A. All rights reserved.
P I I : S 0 1 6 5 - 1 7 6 5 0 0 0 0 3 1 4 - 1
268 G
. Kapetanios Economics Letters 69 2000 267 –276
2. Theory
We consider the following self exciting threshold autoregressive SETAR model y 5 f
1 f y
1 ? ? ? 1 f y
1 s e ,
j 5 1, . . . ,m, t 5 p, . . . ,T,
s . 0 1
t j,0
j,1 t 21
j, p t 2p
j t j
The model has m regimes. The process is in regime j if r y
, r where d is an integer valued
j 21 t 2d
j
delay parameter. r 5 2 ` and r 5 `. hr . . . r j is a strictly increasing sequence of parameters to
m 1
m 21
be estimated. e is an i.i.d. zero mean process with unit variance. For simplicity, throughout this paper,
t
we will assume that e has a standard normal distribution. This model will be denoted as SETARm, p,
t 1
d . The number of regimes, m, and the delay parameter, d, are assumed known in our setup . We note that p is the true lag order for all the m regimes. Estimation is carried out by constructing a grid of
possible values for r , j 5 1, . . . ,m 2 1 and running the regressions
j
y 5 X f 1 e ,
j 5 1, . . . ,m 2
j j
j j
for each point in the threshold parameter grid, where y and X are a vector and matrix, respectively,
j j
containing the observations for regime j. f and e are the coefficient and error vectors for regime j. In
j j
matrix notation, y 5 y , y , . . . , y 9, X 5 x , . . . ,x 9, x 5 y , y
, . . . , y 9,
f 5
j j
j j
j j
j j
j 21 j 22
j 2p j
1 2
T 1
T i
i i
i
j j
f , . . . ,f 9 e 5 e , . . . ,e 9 and h j , j , . . . , j j are the time indices of the observations
j,1 j, p
j j
j 1
2 T
1 T
j
j
belonging to regime j, j 5 1, . . . ,m. The grid point which minimises the sum of squared residuals from the m regressions is adopted as the estimate for the threshold parameters. Chan 1993 proves
that under geometric ergodicity and some other regularity conditions the threshold parameters are consistent, tend to their true value at rate T, and suitably normalised follow asymptotically a
21 2
compound Poisson process. The other parameters of the model are T consistent and are
asymptotically normally distributed. To the best of our knowledge investigation of the small samples properties of the parameter estimates and particularly the threshold parameter estimates has not been
undertaken. A Monte Carlo investigation is carried out in the next section.
