The experimental design Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol68.Issue3.Sep2000:

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 some required results concerning the distributions of X r and Wr in relation to the data-generating T 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 ARMA1,1 process for the x s, i.e. t x 5 ax 1 e 1 ue , 2 t t 21 t t 21 where the sequence of innovations e is assumed to be i.i.d. N0,1, x 5 0 and e 5 0. To ensure that t the process satisfies the conditions such that 1 holds, we impose uau , 1 and uu u , 1. In this case, X r is normally distributed with mean 0 as is Wr. Hence, the only difference between the T distribution of X r and Wr relates to the variance of each variable. As is well known, VarWr 5 r. T Note that we start the process at x 5 0. This minimizes the discrepancies due to the initial condition since W0 5 0. Using the fact that 2 T 2 1 1 u 2 2 1 ]]] s 5 lim T E O x 5 , F G t 2 T → ` 1 2 a t 51 tedious but straightforward algebra shows that 2 [Tr ]21 1 2 a [Tr] u 1 a [Tr] 2 1 a1 2 a ]]] ]] ]]] ]]] ]]]]] VarX r 5 1 2 2 F G H T 2 T T 1 2 a T1 2 a 1 1 u 2 [Tr ]21 2 2 [Tr ]21 u 1 a [Tr] 2 1 a1 2 a a 1 2 a ]]] ]]] ]]]]] ]]]]]] 1 2 2 1 . F GJ 2 2 T T1 2 a 1 2 a T1 2 a P . Perron, S. Mallet Economics Letters 68 2000 225 –234 227 As special cases, we have that with an MA1 process: [Tr] 2 1 1 ]]] ]]] VarX r 5 1 , T 2 T T1 1 u and with an AR1 process: [Tr ] 2 2[Tr ] [Tr] 2a1 2 a a 1 2 a ]] ]]]]] ]]]]] VarX r 5 2 1 . T 2 T T1 2 a T1 2 a It is easily seen, from each expression, that as T → ` , VarX r → r 5VarWr. To assess, the T ‘approximation error’ we use the percentage relative deviation in variance: VarX r 2VarWr VarX r 2 r T T ]]]]]]]] ]]]]] eT,r,a,u 5 100 5 100 . 3 r VarWr

3. The case with an i.i.d. process