Economics Letters 68 2000 225–234
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A look at the quality of the approximation of the functional central limit theorem
a , b
Pierre Perron , Sylvie Mallet
a
Department of Economics , Boston University, 270 Bay State Road, Boston, MA 02215, USA
b
Transport Canada , Economic Analysis and Research-ACAD, Place de Ville, Tower C, Ottawa, Ontario, K1N-0N5
Canada Received 24 September 1999; accepted 7 January 2000
Abstract
This note investigates the adequacy of the finite sample approximation provided the Functional Central Limit Theorem when the errors are allowed to be dependent. We compare the distribution of the scaled partial sums of
some data with the distribution of the Wiener process to which it converges. Our setup is, on purpose, very simple in that it considers data generated from an ARMA1,1 process. Yet, this is sufficient to bring out
interesting conclusions about the particular elements which cause the approximations to be inadequate in even quite large sample sizes.
2000 Elsevier Science S.A. All rights reserved.
Keywords : Central Limit Theorem; Wiener process; Weak convergence in distribution; Time series
JEL classification : C1
1. Introduction
The Functional Central Limit Theorem FCLT is by now a standard tool in econometrics. It states that, under suitable conditions on the data generating process of some series x , we have the following
t
convergence result as the sample size, T, increases:
[Tr ] 2
1 2
1 2
X r ; s T
O
x ⇒
Wr 1
T t
t 51
for r [ [0,1], where Wr is the standard Wiener process, ⇒
denotes weak convergence in distribution
2
usually from the space D[0,1] to C[0,1] under the sup or the Skorohod metrics and s 5
Corresponding author. Tel.: 11-617-353-4389; fax: 11-617-353-4449. E-mail address
: perronbu.edu P. Perron 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 2 5 3 - 6
226 P
. Perron, S. Mallet Economics Letters 68 2000 225 –234
2 1
T 2
lim T
E o
x which in the case of a stationary process is equivalent to 2p times the spectral
T →
` t 51
t
density function at frequency zero of the series x . There are a wide variety of sufficient conditions
t
available in the literature which ensure that the result 1 holds. For example, a class of linear processes with martingale difference innovations Phillips and Solo, 1992, mixing conditions as used
in Phillips and Perron 1988, Herrndorf 1984, and so on. For a comprehensive review, see Davidson 1994.
This asymptotic result has been widely used in time series analysis, in particular in relation to the asymptotic distribution of estimators and test statistics related to integrated processes. Yet, despite its
popularity, little has been documented on how well the distribution of the Wiener process Wr approximates the finite sample distribution of X r in finite samples, especially in the context where
T
serial correlation is present. In this note, we try to partially fill this gap using simple ARMA1,1 processes for the data x . The framework is kept very simple to better highlight the importance of the
t
nature of the serial correlation for the adequacy or lack of it of the asymptotic approximation. The rest of this note is structured as follows. Section 2 discusses the experimental design and states
