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A look at the quality of the approximation of the functional
central limit theorem
a ,
*
bPierre 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 ARMA(1,1) 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 followingt convergence result as the sample size, T, increases:
[Tr ] 21 21 / 2
X (r);s T
O
x ⇒W(r) (1)T t
t51
for r[[0,1], where W(r) 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: [email protected] (P. Perron)
21 T 2
lim T E(o x ) which in the case of a stationary process is equivalent to (2p) times the spectral
T→` t51 t
density function at frequency zero of the series x . There are a wide variety of sufficient conditionst 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 W(r) approximates the finite sample distribution of X (r) in finite samples, especially in the context whereT serial correlation is present. In this note, we try to partially fill this gap using simple ARMA(1,1) processes for the data x . The framework is kept very simple to better highlight the importance of thet 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 some required results concerning the distributions of X (r) and W(r) in relation to the data-generatingT processes used. Section 3 discusses the special case with i.i.d. data. Section 4 presents the results in the general case. Section 5 concludes.
2. The experimental design
What is of interest here is to highlight the nature of the serial correlation which can cause the approximation to be adequate or not in finite samples. To that extent, a useful starting point is to consider a simple ARMA(1,1) process for the x s, i.e.t
x 5ax 1e 1ue , (2)
t t21 t t21
where the sequence of innovations e is assumed to be i.i.d. N(0,1), x 50 and e 50. To ensure that
t 0 0
the process satisfies the conditions such that (1) holds, we impose uau,1 and uuu,1. In this case,
X (r) is normally distributed with mean 0 as is W(r). Hence, the only difference between theT
distribution of X (r) and W(r) relates to the variance of each variable. As is well known, Var(W(r))5r.
T
Note that we start the process at x 50. This minimizes the discrepancies due to the initial condition
0
since W(0)50. Using the fact that
As special cases, we have that with an MA(1) process:
‘approximation error’ we use the percentage relative deviation in variance:
(Var(X (r))2Var(W(r))) (Var(X (r))2r)
3. The case with an i.i.d. process
When the process is i.i.d. N(0,1), we have
[Tr] [Tr]
S
]]D
S
]]D
e(T,r,0,0)5100 2r /r5100 21 . (4)
T Tr
It is to be noted that the error is non-positive for all T and r, so that the variance of the Wiener process
W(r) over-estimates the variance of X (r). An interesting feature of the error function (4) is that it isT
not, in general, a monotonically decreasing function of the sample size. Take for example the case
r50.5, the error is 0 for an even-valued sample size but is 100*(0.5 /T ) for any odd-valued sample
size. There is, hence, a seesaw pattern in the mean-squared error which increases in importance as r decreases. The error caused by this feature can be quite large. Take for example the case r50.1. The
errors are as follows: 0% at T520, 24.76% at T521 up to 231.03% at T529, back to 0 at
T530, 23.23% at T531 and up to 223.08% at T539.
In Fig. 1, we present the percentage error function for values of T ranging from 10 to 100 (in steps of 1) and for r between 0.05 and 1.0 (in steps of .05). The seesaw pattern is seen to be quite important when T and r are small. When r increases, it rapidly becomes negligible. However, when r is small (e.g. .1), the non-monotonic pattern remains important even when T is as large as 100. This feature due to the truncation argument holds as well in the general ARMA case. We shall, however, not dwell
upon it and choose values of T and r such that [Tr]5Tr to concentrate on the effect of serial
correlation.
4. Numerical results for ARMA(1,1) processes
We present results of the error function (3) for a wide range of values fora andu and values of T
Fig. 1. Relative error in in the i.i.d. Case (negative of the value).
hr,Tj selected avoid the non-monotonic behavior of the error caused by the truncation [Tr] relative to
Tr.
Fig. 2 presents the error function e(T,r,0,u) corresponding to the case of an MA(1) process. The
graphs are constructed using values ofu between 20.95 and 0.95 in steps of 0.05. For any value of
the MA parameter, the error function is positive and decreasing as r increases. A notable feature is the fact that when the MA component is close to 21, the error is very large reaching as high a 14 000%
for T530 when r50.1 andu5 20.95. Even, when r51, it is of the order of 2000%. As expected,
the error function decreases as T increases but very slowly when the MA parameter is large and
negative. Even at T55000, the error with u5 20.95 is between 10% and 80% depending on the
value of r.
Fig. 3 presents a similar graph for the case of an AR(1) process again for values of a between
20.95 and 0.95 in steps of 0.05. Here the results are quite different. First, the errors are nowhere near
as large as in the MA case. The largest error is of the order of 160% when T530. An interesting
feature is that the error is negative when a is positive and vice versa. The error declines in absolute
value as r increases. Also, the errors are larger for negative values ofa. The largest errors occur when a is either close to 21 or 11 in which case the rate of decrease as T increases is rather slow. The
rate of decrease is slower whena is positive, so that when T is either 500 or 5000, the error is now
larger (in absolute value) for positive than for negative values of a.
We now consider the behavior of the error as a function of the MA and AR parameters jointly. We consider the case r51 for illustration since the qualitative results are unchanged using other values of
Fig. 2. The error function in the MA case.
where both the MA and the AR coefficients are large and negative. In the extreme, with a5u5 2
0.95, the error is of the order of 50 000% when T530 and is still as high as 300% when T55000.
The error in this region is so large that we fail to see variations in the error function for other parameter configurations. To that effect, we present in Fig. 5, the error function in the region where the AR coefficient a is positive. One can see that there are still important variations when the MA
coefficient is negative. For example, with T530, a50 and u5 20.95, the error is sill around
1400% and decreases to 0% whena50.95 andu5 20.95 (in which case the process is i.i.d. since
the roots cancel). The error decreases quite rapidly but when T55000 it is still around 8% when
a50 andu5 20.95. Hence, we see that the error function is highest, by a wide margin when the
MA coefficient is close to 21. The effect induced by the presence of an autoregressive component is
to reduce the error function when a is positive and to increase it when a is negative reaching
extremely high values as both the MA and AR coefficients get close to 21.
Fig. 3. The error function in the AR case.
is first of interest is the fact that variations in the MA coefficient from 0 to 0.95 basically induce no variation in the error function (except somewhat when the AR coefficient takes a high negative value in which case an increase in the MA coefficient decreases somewhat the error). Hence, the behavior of the error function as a varies is basically the same as that presented in Fig. 2 for r51.
In summary, the results show that when the data exhibit little serial correlation, the Wiener process is a good approximation to the finite sample distribution of the scaled partial sums X (r), except forT some non-monotonicities caused by truncation which, however, dissipate rather quickly as the sample size increases. However, the presence of a negative moving average component causes serious distortions to the approximation which become very important when the moving average coefficient approaches the boundary 21. The distortions are mitigated somewhat when a positive autoregressive
component is present but exacerbated to a great extent with a negative autoregressive component. When only an autoregressive component is present, the errors can be large for sample sizes like
T5100 but become small as T increases further. We have performed similar experiments with the
Fig. 4. The error function in the ARMA(1,1) case.
qualitative results are very similar when r50.5. The difference is that the magnitude of the errors
decreases as r approaches either 0 or 1, but the general shape remains the same. The interested reader may consult the full set of results in Mallet (1995).
5. Conclusion
Fig. 5. The error function in the ARMA(1,1) case (positive AR coefficients).
Fig. 6. The error function in the ARMA(1,1) case (positive MA coefficients).
Acknowledgements
´ ´
This paper is drawn from Sylvie Mallet MSc. essay completed at the Universite de Montreal, Mallet (1995). Perron acknowledges financial support from the Social Sciences and Humanities Research Council of Canada (SSHRC), the Natural Sciences and Engineering Council of Canada (NSERC), and
` ´
the Fonds pour la Formation de Chercheurs et l’Aide a la Recherche du Quebec (FCAR).
References
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´ ´ ´ ` ´ ´
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´ ´
Unpublished MSc. Essay, Universite de Montreal.
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