Statistical Papers 45, 249-266 (2004) Statistical Papers �9 Springer-Verlag 2004

Implementing unit roost tests in ARMA models of unknow order

Ismaei Sfinchez

Departamento de Estadfstica y Econometrfa. Avd. Universidad 30, 28911, Legands, Madrid (Spain),. e-maih ismael@est-econ.uc3m.es

Received: December 28,2001; revised version: August 5, 2002

Abstract This paper compares the performance of classical and recent unit root tests based on different estimation procedures, including fitting ARMA models of unknown orders. The article also introduces an estima- tor of the spectral density function that is based on the estimation of an ARMA model with data previously detrended by GLS. The Monte Carlo experiment shows that tests improve their performance if an ARMA model is estimated, instead of an autoregressive approximation. The best results are obtained by tests based on the estimation of the spectral density func- tion.

1 Introduct ion

The statistical analysis of models with nonstationary variables has a t t racted considerable at tention from both theoretical and applied researchers. Since the t-test (TLs) and coefficient regression test (PLs) of Dickey and Fuller (DF) (1979), many tests have been developed for testing the null hypoth- esis of a unit root against the alternative of stationarity. Except in special cases, one often assumes that the series to be tested is driven by serially correlated innovations. Tests should, therefore, take that serial correlation into account. This constitutes a severe problem, since it is the main source of power loss and size distortion. The purpose of this article is to explore the advantages of alternative ways to deal with this serial correlation. As a result, practical advises for applied researchers will be proposed.

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Typically, the more common options to deal with serial correlation are: (i) to make a test based on the pivotal t-statistic of some regression, or (ii) to use some statistic based on a consistent estimate (under the null) of the spectral density function at frequency zero. For each of this two options, the analyst needs to make a set of important decisions that can significantly affect the performance of the tests. These decisions and their consequences are briefly exposed in next subsections. It will be assumed in this paper tha t the serial correlation admits an ARMA representation.

1.1 Tests based on pivotal statistics

Regarding the first option, the construction of t4ests , the analyst should decide (i) the estimation method and (ii) whether to build an ARMA model or an autoregressive approximation. For some tests, and in order to obtain bet ter performance, the analyst would also need to make an appropriate statement about the initial conditions of the data (Elliott, 1999; S~nchez, 2002). The most extended procedure for implementing a t-test is the popular Augmented Dickey-Fuller (ADF) test (Said and Dickey, 1984), where ~-Ls is obtained from an autoregressive approximation estimated by ordinary least squares (OLS). Although the ADF with LS estimation is the most widely used procedure, it is not the most efficient. Efficient t-tests depend on the assumption made about initial conditions. If it is assumed that the initial observations are extracted from the conditional distribution (conditional case), a more efficient t-test based on LS can be obtained by applying a generalized least squares (GLS) detrending and then obtaining the DF test, as proposed by Elliott et hi. (1996) (hereafter the ~-oLs test). Conversely, if it is assumed that the first observations are extracted from the uncondi- tional distribution under the stationary alternative (unconditional case), an efficient test based on LS can be built by applying the weighted symmetric estimator to the DF t-test (Park and Fuller, 1995) (hereafter the ~w test).

The conditional case is a reasonable assumption when the t ime series has its origin at the first observed data point. Potential applications of this assumption can be found, for instance, in engineering, where the beginning of processes are known. On the other hand, the unconditional case reflects the situation of a time series with the origin in a very far past point. This assumption can be of interest, for instance, with some economic data. It is important to note that although the conditional case is the most commonly assumed condition in the unit root literature, only in the unconditional case the process is covariance-stationary under the alternative (Hwang and Schmidt, 1996).

Alternatively, efficient t-test can also be obtained by using maximum likelihood (ML) estimation (~-ML tests) (Pantula et al., 1994; Gonzalez-Farias and Dickey, 1999). This test is also sensitive to the assumption made about initial conditions. Pantula et al. (1994) show that , in general, better results are obtained if the likelihood function used in ~-ML agrees with the initial conditions assumption.

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The tests based on pivotal statistics have some important drawbacks. Although tests based on autoregressive approximations are very easy to implement, they suffer from the well known problem of size distortion. This problem can part ly be alleviated with a careful choice of the truncation lag of the autoregressive approximation. In a recent paper, Ng and Perron (2001) propose a modified information criterion (MIC) to select the lag order which significantly reduces the size inflation induced by MA components. However, in some situations this procedure can make the tests undersized and, consequently, can reduce (size unadjusted) power.

On the other hand, published works about TML in an ARMA(1,1) with known orders have reported fairly good properties of size and power (Shin and Fuller, 1998). The implementation of these tests in a more general ARMA case is, however, complex because the specific software required to deal with the correct likelihood function is not always available to the practitioner. For instance, some statistic software does not perform exact ML. Some others claim to do so, but actually perform some approximate procedure that could have unknown finite sample properties if applied to a t-test. This is a key aspect, since the stochastic properties of the t-test depend on the estimation method (Pantula et al., 1994; Agiakloglou and Newbold, 1992). Gonzalez-Farias and Dickey (1999) also extend this cau- tionary remark to %s when using a program that performs unconditional ML estimation.

1.2 Tests based on the spectral density function

Regarding the second popular option to deal with serial correlation, the most frequently used test is the Phillips-Perron test (Phillips and Perron, 1988). Paralleling the case of the ADF test, in spite of its popularity, it is not the most efficient. Efficient tests based on a spectral estimator are the PT tests of Elliott et al. (1996) and Elliott (1999). Elliott et al. (1996) built the PT test for the conditional case and Elliott (1999) extended it to the unconditional case. Also of interest are the M GLS tests of Ng and Perron (2001). These tests are extensions of the M tests of Stock (1990), which, in turn, are modifications of the popular Phillips-Perron test to GLS detrending under a local-to-unity alternative. Ng and Perron (2001) analyze the behaviour of three different M GLs tests. Since they all have comparable performance, at tention here will be restricted to the so-called MZ~ Ls test. It should be noted that Ng and Perron (2001) only analyse the M GLs tests in the conditional case.

The behaviour of these tests is highly sensitive to the estimator of the spectral density. Typically, there have been two prevalent estimators in the literature. The first one will be denoted as &2 and is the non-parametric

S C ,

sum-of-covariances estimator of Phillips (1987) and Phillips and Perron (1988). The second will be denoted as ~b2R, and is the autoregressive esti- mator of Stock (1990). It is well known that these two estimators can incur

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severe size distortions, especially in the presence of MA components with positive roots. It happens tha t this kind of MA components are very fre- quent in real data. This fact has traditionally discouraged analysts from the use of these procedures and has made the ADF test the preferred choice. Recently, Ng and Perron (2001) show tha t the application of their MIC procedure to w~R, together with a previous GLS detrending of the data, considerably improves the behaviour of the tests. However, in some situa- tions, this procedure can reduce the size; tha t will be t ranslated into a lower (size-unadjusted) power.

1.3 Scope of the paper

The purpose of this article is twofold. First, it proposes an est imator of the spectral density est imator tha t can be obtained from the efficient est imation of an ARMA model whose orders are estimated. Secondly, the article shows an empirical comparison of all the above mentioned unit root tests fitting either an AR approximation or ARMA models of est imated orders. The experiment allows comparisons in two directions. On the one hand, each test is presented under different estimation strategies. On the other hand, it shows a competi t ion between pivotal tests and tests based on the spectral density function. Overall, the experiment shows tha t the performance of the tests improve if we allow for ARMA modelling.

The outline of the article is as follows. The notation, the model, and the proposed spectral density est imator are exposed in Section 2. Section 3 presents the Monte Carlo experiment. Section 4 summarizes the main conclusions.

2 The m o d e l and the proposed e s t i m a t o r

Let {Yt} be a discrete stochastic process. Let us assume tha t this process contains a deterministic component dt and a pure stochastic component xt; namely, Yt : dt -~-xt. I t will be assumed tha t the deterministic com- ponent consists of a mean: dt -- it; or a deterministic trend: dt = it + St. The pure stochastic par t will have the following structure: x t = px t -1 + ut,

2 The null hypothesis of a unit root cor- with E(u t ) = 0 and E ( u t ) = au. responds with the case p = 1 and the stat ionary alternative with IP[ < 1. Let us assume tha t ut is a s tat ionary and invertibte process tha t admits an ARMh(p,q) representation, r = 6(B)at , where at is a sequence of iid random variables with E(a t ) = 0 and Z(a2t) = a 2. The Data Generat- ing Process (DGP) is, therefore, Yt = # + 5t + xt; x t = px t -1 + ut; ut = r with r = r This process can also be expressed as Yt = it(1 - p ) + 5p + 5(1 - p)t + PYt-1 + r

Many unit root tests require an estimate of the spectral density at fre- quency zero w 2, where w 2 . . . . a2r 2 a2(1 Eq=l ~i)2( 1 Ej=lP r

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There are several well known estimators of w 2 in the literature. The au- toregressive est imate was first proposed by Stock (1990) and is defined as

& ~ = # 2 ~ ( 1 - Y~=l r -2, where # ~ = ( T - k ) -1 ~T=k+lg~k , with r and gtk obtained from the OLS estimation of the error correction model

k , (ECM) Ayt = dt + (~Y~-I+ ~ j = ~ e j A y t - j + ztk. There are several proce- dures in the l i terature for choosing the truncation lag k. Recently, Ng and Perron (2001) propose the estimation of w 2 using da ta previously detrended by a local-to-unity GLS estimation. This est imator of w 2 is ^ 2 (~G LS-AR

aA~ -- j , where O-AR-2 = ( T - - k) - 1 E L k + I V~' with $~ and

~)tk obtained from the OLS estimation of the autoregression Ay~ .= aye_ 1 + F k " * A c j=l (Pj Yt-j +vt~; where y~ are GLS detrended data using Pc = 1 - c/T. Ng and Perron (2001) also propose a new information criteria, MIC, to select the t runcat ion lag k. The MIC selects k between 0 and some up- per bound k ~ x a s k m i c = argminkMIC(k) where MIC(k) = In ( ~ ) + C T {~-T(k)+ k } / T - k . . . . with CT > 0 and CT/T ----+ 0 as T --* oo; 82 ( T - k ~ x ) - ' T = E ~ = ~ . ~ + ~ ~ i , and ~ ( k ) : ~ ; 2 ~ 2 ~ . Et=kmax+l (Y~--1)2 On the other hand, we could est imate the parameter w 2 through the esti- mat ion of an ARMA model. This est imator can be expressed as

1 - 4 )2

(~ARM A

-- E j = I

(i)

where 0j and r come from the estimation of the ECM

= "3 ~O~a ,_~; (2) Ayt d t + e ~ Y t - l b E r j y t - j + a t - j = l i=1

with/5, ~ being the est imated orders and ~2 is the variance of the residuals. The est imator W~MA was implicitely used in Yap and Reinsel (1995) as a par t of a likelihood ratio test statistic based on an ARMA model, and we will use it here for other unit root tests. Alternatively, an est imator based on GLS detrended data can be proposed in the same way as (1), but with parameter estimates obtained from the regression

c c = e~yt_ 1 + Ayt_ j - b r 7ir (3) j=l j=l

where y~ are GLS detrended data using Pc = 1 - c/T. Then, the proposed est imator can be wri t ten as

i - 4 )2 ^2 ^2

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with ~ being the residual variance of (3). The possibility of constructing an ARMA model to est imate w 2 can have important practical implications, since it allows to deal with MA components efficiently.

3 F i n i t e s a m p l e performance.

3.1 General considerations

In this section, the performance of the unit root tests are compared in a Monte Carlo experiment under different serial correlation structures. The underlying processes are the ARMA(1,1) process (1 - p B ) y t = (1 - OB)at, and the AR(2) process (1 - pB)(1 - CB)yt = at with at ~ N(O, 1). Sample sizes are T = 50, 100, and 150. Regarding initial conditions assumption, experiments were made for both the conditional case (Yl = al) and the unconditional case. Conclusions are similar for both cases. Therefore, for the sake of conciseness, only the results for the unconditional case will be reported here.

Regarding tests based on an est imate of w 2, the performance of the PT and the MZaa LS tests are studied. Four different estimators of w 2 were con- sidered. Namely, &2 a , •G LS-AR̂ 2 , ~ATARM A ^ 2 and the proposed estimator ~JGLS-ARMA " ^ 2

The est imator ~s2c was not included in this comparison since it is well doc- umented in the l i terature that , in this context, ~b~a is superior (Perron and Ng, 1998). In 0flGLSsAR ^ 2 and WGLS.ARMA ^ 2 data are previously detrended by GLS under the alternative Pc = 1 - c /T . Different values of c have been ana- lyzed to compute these estimators. As reported in Ng and Perron (2001), results were not very sensitive to these values. Somewhat bet ter perfor- mance (lower size distortion) was obtained in the unconditional case when the value c = 10 was used, both with dt = #, and with dt = # + 5t. In the conditional case, the value c = 10 was still preferred when dt = p, but the value c = 5 yielded bet ter performance when dt = # + St. Therefore, the reported results are based on these values.

I have analyzed t-tests based on four different estimation methods: LS, ML, GLS detrending, and the weighted symmetric estimator. Once an ECM is est imated to compute the t-test, it is straightforward to also compute the coefficient regression test based on #. The coefficient regression test has bet ter power properties than the t-test. However, it suffers from higher size distortion than the t- test (Schwert, 1989). For the sake of comparison, they have also been included in this experiment. All the tests will be implemented fitting both an autoregressive approximation and an ARMA model with est imated orders. The details of the ARMA estimation are described in the next subsection.

When LS is used to est imate an ARMA model, nonlinear least squares (NLS) is performed. In this case, and following Dolado and Hidalgo-Moreno (1990) and Agiakloglou and Newbold (1992), these tests are obtained by iterating the estimation procedure until convergence. The tests based on

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Table 1 Empirical finite sample critical values for 5% level of some unit root tests. Unconditional case.

p(T 1 ) MZ~LS (i)TL(2) __(1) __(1) 7.(2) p(2S) ~(1) ~(1) IGLS IML PML PG LS

T = 50 d t : l ~ 4.82 10.1 -2.93 -2.85 -2.69 -2.57 -13.3 -12.9 -12.5

dt= ]~- ~t 3.06 -14.4 -3.50 -3.35 -3.41 -3.28 -19.7 -18.5 -19.1

T=I00 dt=l~ 4.73 - I I .4 -2.90 -2.79 -2.66 -2.55 -13.7 -13.3 -12.7

dt= ~ ~t 2.96 -16.8 -3.45 -3.26 -3.34 -3.24 20.6 ~1 9 L9

T = I 5 0 dt=Iz 4.66 -11.8 -2.90 -2.78 -2.86 -2.55 -13.9 -13.5 -13.4

dt ~ ~ ~t_ ~t 2.92 -17.7 -3.44 -3.23 -3.40 -3.23 -21.1 -19.4 -20.7

Note : (1) c r i t i ca l va lues o b t a i n e d w i t h 100.000 rep l i ca t ions . (2) pub l i shed finite s a m p l e c r i t i ca l

va lues a t T = 5 0 , 1 0 0 a n d o b t a i n e d w i th 100.000 r ep l i c a t i ons a t T = l h 0 .

GLS detrending are denoted as %Ls and PaLs. Once the data are GLS detrended, the ECM is estimated by LS. As in Pantula et al. (1994), the ~-w test is always based on an autoregressive approximation estimated by OLS. The tests based on ML are denoted as TML and PML respectively.

The choice of the truncation lag k in the ECM when an autoregressive approximation is performed was made with the AIC, BIC and also with the MIC. The performance of AIC was much worse than the other two criteria, and therefore those results are not reported here. When the BIC is used, k is restricted to be 3< k < 8, as in Elliott et al. (1996). For the MIC, the value C T ~--- 2 and the restriction 0 < k < k . . . . with kmax =int[12(T/ lO0)] 1/4

have been used, as in Ng and Perron (2001). The performance of the ~-w test was poor (extremely oversize) if the lag order is selected by minimizing the MIC along the weighted symmetric autoregressions. For this reason, when the MIC is used, "rw uses the same lag order as %Ls.

The numerator of PT and M Z ~ LS tests as well as the ECM used in % Ls require the value of c in Pc = 1 - c / T . In the conditional case, the chosen values are c = 7 if dt = #, and c = 13.5 if dt = pt q- ~t. In the unconditional case, c = 10 for both dt = # and dt = p q- St. These are the recommended values in Elliott et al. (1997) (conditional case) and Elliott (1999) (unconditional case) for the Pr and ~-aLs tests. S~nchez (2002) shows that these values of c are also valid for the M Z cLs test.

In order to ease the interpretation of the results, finite sample critical values for a size of 5% have been used for all the tests. They can be found in Table 1. Some of the finite sample critical values for T = 50, 100 were obtained from the literature but some others were obtained from 100,000 Monte Carlo replications from the model Yt = Yt 1 +at , Yl = al . At T = 150 all the critical values have been obtained from simulation. The resulting critical values were very similar to the values obtained by interpolation from the published ones. An exception is for 7ML and PML, where the simulated values are smaller than the values obtained by interpolation. This difference

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T a b l e 2 Empirical size and power for 5% level. Tests and estimators of w 2 are based on the es t imat ion of an A R M A model . The true process is an A R M A ( 1 , 1 ) or an AR(2) . Uncondit ional case. T = 1 0 0 . 5,000 replications.

Model: ARMA dr=/~ ARMA(1,1):

Test p -0.8 -0.5 0 0.5 0.8 ~ a M ^ PT 1.00 0.053 0 .057 0.049 0 .054 0.096

0.90 0.373 0.381 0.380 0.303 0.465 M Z ~ Ls 1.00 0.070 0 .074 0.062 0.062 0.099

0.90 0.221 0.431 0.433 0.319 0.470

True process 0 = AR(2): ~b=

-0.8 -0.5 0.5 0.8

G}2LS.ARMA P T 1.00 0.050 0.052 0.046 0.039 0.044 0.90 0.381 0.374 0.376 0.218 0.195

M Z ~ Ls 1.00 0.069 0.069 0.059 0.046 0.045 0.90 0.434 0.421 0.428 0.225 0 .194

t - t e s t s TLS 1.00 0.039 0.036 0.047 0.022 0.066 0.90 0.221 0.173 0.264 0.042 0.219

TCLS 1.00 0.038 0 .037 0.044 0.012 0.028 0.90 0.259 0 .204 0.315 0.035 0.157

TML 1.00 0.068 0.065 0.077 0.034 0.067 0.90 0.350 0 .294 0.406 0.046 0.189

p-tests pLS 1.00 0 .044 0.050 0.045 0.088 0.194 0.90 0.425 0.421 0.432 0.502 0.723

pQLS 1.00 0.042 0.049 0.045 0.092 0.223 0.90 0.441 0.437 0.449 0.537 0.783

PML 1.00 0.056 0.064 0.057 0.108 0.247 0.90 0.488 0.488 0.501 0.581 0.819

dt ~/~ + 6t GJARMA Pr 1.00 0.058 0.064 0.062 0.108 0.260

0.90 0.189 0.204 0.213 0.281 0.520 M Z ~ Ls 1.00 0.078 0.083 0.081 0.125 0.269

0.90 0.230 0.245 0.263 0.304 0.531

~LS-AI~MA PT 1.00 0.057 0.063 0.063 0.070 0 .104 0.90 0.205 0.214 0 .214 0.185 0.228

M Z ~ Ls 1.00 0.080 0.085 0.080 0.080 0.106 0.90 0.256 0.262 0.270 0.206 0.232

t-tests TLS 1.00 0.025 0.018 0.031 0.018 0.108 0.90 0.060 0.043 0.093 0.040 0.225

"/-Gas 1.00 0.023 0.016 0.034 0.008 0.055 0.90 0.096 0.059 0.139 0.026 0.135

"TML 1.00 0.029 0.022 0.043 0.017 0.084 0.90 0 .074 0.054 0.126 0.032 0.174

p-tests pLs 1.00 0 .047 0.056 0.043 0.125 0 .384 0.90 0.183 0.200 0.199 0.371 0.712

pGLS 1.00 0.048 0.057 0.046 0.125 0.360 0.90 0.208 0.225 0.225 0.391 0.707

pML 1.00 0.052 0.064 0.056 0.152 0.458 0.90 0.209 0.235 0.240 0.431 0.800

0.015 0.045 0 .057 0 .094 0.140 0.315 0 .342 0.339 0.016 0.060 0.079 0.128 0.155 0.318 0.401 0.383

0.012 0.033 0.056 0.102 0.111 0.259 0.350 0 .343 0.014 0.041 0.073 0 .140 0.117 0.279 0.399 0.389 0.047 0.036 0.041 0.052 0.253 0.153 0.161 0 .162

0 .044 0.029 0 .037 0 .039 0.294 0.186 0.196 0 .187

0.081 0.063 0.074 0 .087 0.380 0.247 0.266 0 .252 0.063 0.048 0.050 0.059 0.490 0.402 0.339 0.305

0.061 0.045 0.052 0 .064 0.493 0.410 0.346 0.319

0.079 0.063 0.063 0 .072 0.557 0.468 0 .394 0 .357

0.024 0.071 0.076 0 .120 0.085 0.213 0.215 0.201 0.029 0.085 0.100 0.181 0.103 0.246 0.258 0 .264

0.015 0.053 0.073 0 .120 0.058 0.174 0.211 0 .202 0.019 0.065 0.099 0.193 0.069 0.199 0 .267 0 .272 0.032 0.020 0.025 0.038 0.095 0.048 0.052 0 .063

0.032 0.017 0.026 0.032 0.129 0.069 0.081 0.090

0.043 0.028 0 .037 0.049 0.123 0.065 0.070 0.082 0.076 0.059 0.066 0.079 0.288 0.213 0.193 0.189

0.066 0.051 0 .067 0.088 0.272 0.211 0.196 0.201

0.087 0.068 0.077 0.094 0.329 0.253 0.218 0 .214

GLS ^2 Note : T h e s t a t i s t i c s PT a n d MZ a use the e s t i m a t o r of 0) 2 shown a t the i r left side. tOARMA ^2 a n d 0JGLS.AaMA are b a s e d on an A R M A m o d e l in E C M form us ing AIC a n d exac t ML. In

&2 GLB-ARMA, d a t a a re p rev ious ly local ly d e t r e n d e d by GLS us ing c = 5 . T h e A R M A orde r s in

"/--tests a n d p - t e s t s are b a s e d on BIC. TLS , TCLS, pLS, a n d pc, Lsa re based on E C M e s t i m a t e d

by n o n l i n e a r LS. TMLand pML are based on E C M es t ima ted by exac t ML. The s t a t i s t i c s P T , GLS M Z ~ , TGLS, a n d pGLS use d a t a local ly d e t r e n d e d by GLS wi th c = 1 0 .

is related with the above mentioned influence of the estimation algorithm in the stochastic properties of the t- tests based on ML. It will then be interesting to compare the relative performance of ~-ML and P~L when T = 50, 100, where published critical values are used, with T = 150, where the

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critical values are obtained with exactly the same algorithm used in the computation of tests.

3.2 Estimation of the ARMA model

The estimation of an AR approximation is a convenient procedure, both to obtain t-tests and estimate w 2. The AR approximation does not requiere the identification of the ARMA structure and it only assumes a very general linear process. The estimation of an AR approximation using some informa- tion criteria also constitutes a robust procedure in cases where the dynamic structure of the model is unclear to the analyst. There are many situations, though, where the practitioner can identify an ARMA model. In this cases, the posibility of using an ARMA model for unit root testing is appealing, since the presence of MA components is the main source of inefficiency of tests based on an AR approximation.

The estimators WARMA ^ 2 and WQLS.AaM A ^ 2 have the advantage of being based on an ARMA(~, ~) model, where ~ =/~ + 1. Similarly, the t-tests and the regression coefficient tests can also be obtained from an ARMA model. The orders and the parameters of this ARMA model need to be estimated. There are numerous techniques for ARMA model identification and practitioners frequently take their decisions after the analysis of more than one proce- dure, usually combined with some degree of judgment. Therefore, in our Monte Carlo experiment we can only approximate the decision process of an analyst. In this experiment, the selected orders ~, ~ will be the ones that minimize an information criterion among the set of values (p*, q*) such that 0 ~ p* _< Pmax and 0 ~ q* < q . . . . When the goal of the ARMA modelling is the estimation of w 2, better results were obtained using AIC. Conversely, when the goal of the ARMA model is the computation of the t-tests and regression coefficient tests, bet ter results were obtained using BIC. The reported results for each case are, then, based on their most favourable cri- terion. Since in real applications of moderate sample size it is infrequent to identify a non-seasonal ARMA(r, q) model where r, q > 3 (with r ---- p + 1), the restriction Pm,x ---- 2, qm~x -= 3 has been used. A limited experiment with Pm~x, qm~x -- 6 yielded similar results.

When the process is an ARMA(p + 1, q) with q > 0, there is a high risk that an information criteria behaves differently than an analyst. For instance, in the ARMA(1,1) case with p ~ 0, an information criteria could select an AR(1) with parameter estimate close to zero. This parsimonious AR(1) would hardly be selected in a real situation, where a practitioner takes the final decision, and would negatively affect the performance of the unit root tests. In these situations, and since the analyst is testing for a unit root, it is more likely that the possibly non-stationary ARMA(1,1) is selected instead of an AR(1) that is far from the nonstationarity. In order to avoid this situations, and following the same fashion as in Galbraith and Zinde-Walsh (1999), the identification procedure is made with the series

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in first differences. Once an ARIMA(~, 1, ~) is identified according to the information criterion, the final estimation of the unrestricted ARMA(~ + 1, ~) is performed, and the tests or the spectral estimators are computed. An unreported experiment has shown tha t the performance of the tests using this procedure was bet ter than using the original series in the identification stage. For instance, the empirical size of Pr test using WGLS AaMA ^ 2 in the ARMA(1,1) case with T = 100, dt = #, and 0 = 0.8, with orders selected using the original series was 0.122; whereas doing the identification with the differenced series, the size is 0.044.

The est imation of a given ARMA(p*, q*) is made in three steps. The first two steps are aimed to obtain consistent initial estimates. The third step consists on the application of an efficient estimation algorithm to those ini- tial estimates. In the first step, and following Hannan and Rissanen (1982), a long autoregression, est imated by OLS, is used to obtain an initial es- t imate of the innovations. The order of this high-order autoregression has been selected with the BIC. In the second stage, an initial ARMA(p*, q*) is est imated using the GLS procedure of Koreisha and Pukkila (1990). This procedure is applied to the est imated innovations obtained in the first stage. The feasible GLS estimation used in the Koreisha and Pukkila (1990) pro- cedure has been i terated until convergence. In the third step, nonlinear least squares or exact maximum likelihood is performed with the NAG routine G13bef, using the previous initial values.

3.3 Results of the experiment

Empirical size and power, based on 5,000 replications, are summarized in Tables 2 to 6. Tables 2 to 4 show the results for T = 100, Table 5 shows the results for T = 50, and Table 6 shows the results for T = 150. To save space, the results for T = 50 and T = 150 do not include the BIC, since it is seen at T = 100 tha t it is surpassed by MIC. Similarly, the results for T ---- 50 and T ----- 150 only include the case dt = #, since the extension of the conclusions to dt = p + 5t are similar as in T = 100. Table 2 and Panel I in Tables 5 and 6 show the results of tests based on fitting an ARMA model of est imated orders. In these tables, the estimators of w 2 are based on ARMA models with orders selected with AIC, and using exact ML in the third step of the estimation procedure. The regression tests are based on ARMA models with orders est imated with BIC. Table 3 and Panel II in tables 5 and 6 show the results based on an autoregressive approximation with order selection made with MIC. Finally, Table 4 shows the results for autoregressive approximations made with BIC. The same seeds are used for all the tests. The results can be summarized as follows:

Conclusions regarding autoregressive approximations: Tests based on an AR approximation using BIC can exhibit excessively

inflated sizes, especially in the ARMA(1,1) case with 0 > 0. This consti- tutes a serious shortcoming, since many real t ime series do exhibit a positive

259

moving-average root. The t-tests based on BIC, however, have good perfor- mance in the AR(2) case, especially ~-w.

On the other hand, tests tha t use MIC tend to be undersized. This low size is more acute in the AR(2) case with r < 0, with PT and MZ~ Ls having extremely low size and size-unadjusted power in this setting. I t should, however, be said tha t t ime series of this type are less frequent than t ime series with a positive moving average root. The t-tests also show low size when 0 < 0, especially when T = 50. This lower size also decreases the empirical (size-unadjusted) power of the tests. The p-tests are still highly oversized when 0 > 0.

The best testing procedure based on autoregressions depends on the un- certainty about the underline DGP. If the analyst is completely sure tha t the underline model is a pure autoregression, the best testing procedures would be the Tw obtained from BIC and PML with MIC. However, if there is evidence of the presence of an MA component, the analyst is bet ter advice

^ 2 ^ 2 with MIC. For small samples to chose the PT test based on WAa or WaLS.AR (T = 50) the &~L~ A~ is preferred, whereas for moderate and large samples (T = 100, 150) 05~a yields bet ter performance.

Conclusions regarding fitting an ARMA model:

Tests based on an identified ARMA model have, in general, bet ter prop- erties than the autoregressive approximations. The p-tests have an excellent performance in the pure AR case. However, they still have inflated sizes with 0 > 0, even at T = 150. The problem of size inflation with 0 > 0 can par t ly be alleviated using a t-test. These tests, however, can incur just the oposite problem: low size an low size unadjusted power, especially with 0 < 0. I t should be noted tha t this effect is milder than using MIC. The ~-ML still tends to have some inflated size when dt = # at T = 50, 100, which corre- spond to the situations where published critical values are used. This effect is not observed at T = 150, where simulated critical values were use in- stead. Therefore, the inflated size at T = 50, 100 is provoked by the ML estimation algorithm. This result is in agreement with the cautionary note made in the introduction regarding the influence between regression tests and the available software. In a similar fashion, it can be observed that the PML also tends to have bigger size than the others p-tests. It can also be seen tha t these conclusions can be extended to the estimation of autoregressive approximations. For T = 50, 100 the remaining t-tests, %s and %Ls, have, in general, low size and low size-unadjusted power. When T = 150, these t-tests still show low size and power with 0 > 0, especially %Ls.

^ 2 and ^ 2 have In the ARMA(1,1) case, the tests based on WGLS.AaMA WAaMA very good relative size and power properties, especially when applied to the PT test. The est imator WGLS.Aa~A ^ 2 shows the best relative performance at T = 50, 100; whereas WAR~A ^ 2 is the preferred procedure at T = 150. In the

^ 2 and ^ 2 have size distortions AR(2) case, the tests based on WaLS.AaMA WAaMA when Ir = 0.8, with inflated size at r = 0.8 and low size when r = -0 .8 .

260

Their relative performance is, however, very good when r has a moderate value. Again, the best results are obtained with PT.

4 C o n c l u d i n g r e m a r k s

This paper compares the performance of unit root tests based on differ- ent approaches to deal with serial correlation. It can be concluded that the identification of an ARMA model helps to improve the performance of unit root tests. Although some regression tests based on autoregressive approx- imations can have good performance in a purely autoregressive case, their performance deteriorates in presence of moving average components. On the contrary, tests based on the spectral density function using an iden- tiffed ARMA model, have very good relative performance in the ARMA case, especially the PT test, and still have competitive advantage in the pure autoregressive case. The estimator WA~A ^ 2 yields tests with good size and power properties in large samples; whereas ^ 2 shows the best (A}G L S . A R M A

relative performance with moderate and small samples.

Acknowledgments

I would like to thank two anonymous referees for their useful comments and suggestions. This research has been sponsored by DGES (Spain) under project BEC2000-0167.

261

T a b l e 3 Empir ical size and power for 5% level. Tests and es t imators of w 2 are based on an A R using MIC. The t rue process is an ARMA(1,1) or an AR(2). Uncondi t ional case. T=100 . 5,000 replications.

Model: A R ( M I C ) True process at=# ARMA(1,1) : 0 = AR(2): 4 )=

Test p -0.8 -0.5 0 0.5 0.8 -0.8 -0,5 0.5 0.8 ~ a PT 1.00 0.056 0.042 0.029 0.045 0.071 0.005 0.021 0.042 0.061

0.90 0.255 0.286 0.302 0.288 0.353 0.060 0.243 0.316 0.266 M Z ~ as 1.00 0.072 0.049 0.035 0.053 0.077 0.005 0 .027 0 .054 0 .074

0.90 0.283 0.326 0.354 0.316 0.363 0.069 0.278 0.356 0.296

CO~LS_AR PT 1.00 0 .042 0 .034 0 .027 0 .027 0.021 0.004 0.018 0.039 0.060 0.90 0.181 0.235 0.271 0.163 0.084 0.036 0.185 0.287 0 .242

M Z ~ Ls 1.00 0.049 0.043 0.033 0.032 0.021 0.003 0.021 0.050 0.079 0.90 0.196 0.260 0.319 0.179 0.083 0.035 0.210 0.322 0 .282

t - t e s t s % s 1.00 0.008 0.013 0.019 0.030 0.074 0.016 0.018 0.016 0.015 0.90 0.055 0.096 0.178 0.190 0.340 0.138 0.153 0.116 0.098

TGLS 1.00 0.011 0.019 0.027 0.029 0.045 0.90 0.052 0.101 0.191 0.163 0.183

TML 1.00 0.021 0.033 0.042 0.050 0.096 0.90 0.101 0.166 0.281 0.260 0.380

TW 1.00 0.019 0.025 0.029 0.042 0.092 0.90 0.135 0 .207 0.308 0.287 0.429

/>tests pLs 1.00 0.055 0.042 0.032 0.078 0.231 0.90 0.277 0.288 0.315 0.470 0.755

pGLS 1.00 0.071 0.046 0.030 0.083 0.223 0.90 0.309 0.304 0.327 0.475 0.702

pML 1.00 0.068 0.057 0.041 0.095 0.254 0.90 0.320 0 .334 0.374 0.516 0 .784

0.019 0.022 0.024 0.022 0.126 0.152 0.139 0.109

0.035 0.038 0.039 0.041 0.216 0.241 0.196 0.162

0 .024 0.028 0.029 0.030 0.260 0.285 0.245 0.188 0.036 0.039 0.037 0.043 0.327 0.345 0.279 0.221

0.036 0.039 0.042 0 .045 0.330 0 .357 0.296 0 .237

0.046 0.052 0.050 0 .054 0.383 0.405 0.329 0.263

d t = ~ t + 6 t

~A~ P T 1.00 0.055 0.035 0.022 0.050 0.145 0.90 0.121 0.118 0.106 0.181 0.370

MT.G Ls ~ a 1.00 0.088 0.059 0.038 0.073 0.165 0.90 0.175 0 .174 0.171 0.231 0.398

GJ~ LS.AR PT 1.00 0.90

MZC~ Ls 1.00 0.90

t - t e s t s TLS 1.00 0.90

TGL s 1.00 0.90

TML 1.00 0.90

TW 1.00 0.90

p - t e s t s pLs 1 2 0 0.90

pGLS 1.00 0.90

pML 1.00 0.90

0.001 0.016 0.047 0.082 0.006 0.077 0.153 0.167 0.002 0.027 0.076 0.126 0.010 0 .117 0.222 0 .230

0.040 0.033 0.019 0.030 0.049 0.001 0.011 0.040 0.073 0.092 0.098 0.092 0.102 0.128 0.004 0.059 0.137 0.143 0.064 0.048 0.034 0.041 0.054 0.002 0.018 0.066 0.112 0.126 0.146 0 .147 0.134 0.138 0.005 0 .084 0.197 0.207 0.005 0.009 0.021 0.041 0.150 0.023 0.029 0.102 0.142 0.362

0.002 0.010 0.021 0.027 0.055 0.017 0.038 0.104 0.093 0.123

0.008 0.013 0.028 0.048 0.153 0.031 0.042 0.129 0.155 0.367

0.007 0.014 0.026 0.038 0.095 0.026 0.053 0.121 0.130 0.247 0.061 0.047 0.033 0.128 0.399 0 .164 0.145 0.153 0.383 0.748

0.068 0.048 0.028 0.095 0.283 0.172 0.159 0.146 0.325 0.564

0 .069 0.053 0.040 0.141 0.419 0 .182 0.162 0.175 0.405 0.760

0.019 0.020 0.015 0.021 0.080 0.088 0.056 0.062

0.012 0.016 0.017 0.020 0.055 0.075 0.068 0.066

0.025 0 .027 0.023 0.032 0.100 0.108 0.070 0.079

0.016 0.020 0.023 0 .027 0.088 0.105 0.082 0.076 0.049 0.051 0.050 0.064 0.195 0.208 0.150 0.146

0.036 0.040 0.050 0 .057 0.159 0.186 0.158 0.145

0.055 0.058 0.058 0.064 0.215 0.228 0.165 0.156

Note : T h e s t a t i s t i c s PT a n d M Z ~ Ls use the e s t i m a t o r of 0J "~ shown a t the i r left side. ~ a

a n d GJ 2 GLS.AR are b a s e d on an A R in E C M form us ing MIC a n d OLS. In ~ 2 GLS-AR~ d a t a a re

p rev ious ly local ly d e t r e n d e d by GLS us ing c=5 . The A R o rde r s in T - t e s t s and p - t e s t s are

b a s e d on MIC. TLS, TGLS, pLS, a n d POLS are b a s e d on E C M e s t i m a t e d by OLS. TML a n d

pML are based on E C M e s t i m a t e d by exac t ML. Twis based on the w e i g t h e d s y m m e t r i c LS GLS

e s t i m a t o r . T h e s t a t i s t i c s P w , M Z a , TGLs. a n d pGLsuse d a t a local ly d e t r e n d e d b y G L S w i t h

c = 1 0 .

262

T a b l e 4 E m p i r i c a l s i z e a n d p o w e r f o r 5 % l eve l . T e s t s a n d e s t i m a t o r s o f w 2 a r e

b a s e d o n a n A R u s i n g B I C . T h e t r u e p r o c e s s is a n A R M A ( 1 , 1 ) o r a n A R ( 2 ) .

U n c o n d i t i o n a l c a s e . T = 1 0 0 . 5 , 0 0 0 r e p l i c .

M o d e l : A R ( B I C ) T r u e p r o c e s s d t = t t A R M A ( 1 , 1 ) : 0 = A R ( 2 ) : r

T e s t p -0.8 -0.5 0 0.5 0.8 -0.8 -0.5 0.5 0.8

& ~ P T 1.00 0.90

M Z ~ Ls 1.00 0.90

G)g 5S.Aa PT 1.00 0.90

MZ~ Ls 1.00 0.90

t - t e s t s ~-Ls 1.00 0.90

T a a s 1.00 0.90

TML 1.00 0.90

0.148 0.112 0.090 0.092 0.419 0.619 0.595 0.546 0.565 0.938 0.170 0.132 0.108 0.106 0.450 0.653 0.633 0.589 0.602 0.950

0.037 0.073 0.100 0 .122 0.284 0.486 0.531 0 .437 0.044 0.085 0.119 0 .142 0.310 0.527 0.569 0.473

0.112 0.091 0.074 0.070 0.270 0.023 0.054 0 .084 0 .112 0.522 0.512 0.465 0.434 0.675 0.186 0.396 0.469 0.396 0.128 0.110 0.087 0.076 0.287 0.024 0.059 0.098 0 .137 0.559 0.562 0.515 0.470 0.715 0.197 0.433 0.520 0.440 0.075 0.058 0.053 0.071 0.444 0.329 0.283 0.261 0.405 0.972

0.061 0.050 0.043 0.067 0.432 0.319 0.301 0.278 0.420 0.949

0.117 0.097 0.087 0.110 0.519 0.420 0.385 0.361 0.519 0.986

0.050 0.051 0 .047 0.046 0.270 0.270 0.233 0.169

0.040 0.041 0.039 0.039 0.282 0.285 0.247 0.180

0.082 0.083 0.086 0.095 0.380 0.377 0.323 0.254

TW 1.00 0.073 0.064 0.052 0.082 0.496 0.90 0.445 0.433 0.414 0.588 0.962

p - t e s t s pLs 1.00 0.150 0.118 0.101 0.142 0.637 0.90 0.608 0.573 0.545 0.716 0 .997

p a l s 1.00 0.140 0.116 0.097 0.136 0.617 0.90 0.595 0.576 0.547 0.712 0.991

pML 1.00 0.163 0.133 0.117 0.164 0.662 0.90 0.645 0.619 0.595 0.762 0.997

0.052 0.050 0.053 0.056 0.445 0.443 0.376 0.281

d t ~ / ~ + 6 t

QAa PT

0.101 0.101 0.099 0 .104 0.573 0.568 0.498 0 .402

0.097 0.099 0.094 0.105 0.571 0.571 0 .504 0 .407

0.117 0.118 0.114 0 .116 0.623 0.613 0.542 0.443

1.00 0.282 0.192 0.145 0.130 0.513 0.047 0 .114 0.175 0 .224 0.90 0.550 0.475 0.407 0.387 0.838 0.155 0.339 0.422 0.378

M Z ~ as 1.00 0.343 0.249 0.194 0.171 0.580 0.065 0.150 0.229 0.295 0.90 0.619 0.555 0.486 0.463 0.881 0.195 0.410 0.502 0 .450

&2 Pr 1.00 0.224 0.158 0.121 0.092 0.347 a L S - A R 0.90 0.469 0 .417 0.347 0.303 0.627

M Z ~ Ls 1.00 0.278 0.208 0.155 0.124 0.407 0.90 0.539 0.495 0.423 0.368 0.690

t - t e s t s TLS 1.00 0.084 0.060 0.045 0.078 0.572 0.90 0.229 0.180 0.154 0.268 0.923

T a a s 1.00 0 .067 0.055 0.038 0.072 0.532 0.90 0.215 0.185 0.162 0 .264 0.873

TML 1.00 0.105 0.078 0.063 0.096 0.602 0.90 0.258 0.208 0.181 0.298 0.932

TW 1.00 0.086 0.068 0.053 0.089 0.620 0.90 0.271 0.230 0.201 0.333 0.938

0.279 0.208 0.169 0.246 0.840 0.553 0.488 0.444 0.624 0.991

0.228 0.175 0.142 0.206 0.772 0.510 0.459 0.418 0.581 0.973

0.277 0.212 0.178 0.259 0.850 0.564 0.501 0.462 0.643 0.991

p - t e s t s pLs 1.00 0.90

p a L s 1.00 0.90

pML 1.00 0.90

0.031 0.087 0.150 0.197 0.104 0.277 0.371 0.342 0.040 0.116 0.198 0.269 0.130 0.341 0.451 0.418 0.048 0.049 0.049 0.048 0.162 0.160 0.141 0.112

0.039 0.041 0.045 0.043 0.153 0.160 0.142 0.113

0.062 0.063 0.065 0.070 0.191 0.190 0.165 0 .134

0.053 0.058 0.055 0.055 0.213 0.211 0.178 0.135 0.170 0.172 0.173 0.195 0.464 0.460 0.416 0.355

0.136 0.143 0.154 0.175 0.410 0.429 0.392 0 .334

0.180 0.182 0.175 0.182 0.486 0.480 0.425 0 .354

Note : T h e s t a t i s t i c s PT a n d MZaa Ls use the e s t i m a t o r of 02 z shown a t the i r left side. ~ ) ~

a n d ~]2 aLS-AR are b a s e d on an A R in E C M form us ing BIC a n d OLS. In ~ 2 G L S - A R ~ d a t a a r e

p rev ious ly local ly d e t r e n d e d by GLS us ing c=5 . The A R o rde r s in T - t e s t s and p - t e s t s are

b a s e d on BIC. TLS, TGLS, p n s , a n d P G L S are based on E C M es t ima ted by OLS. TML a n d

p M L a r e based on E C M e s t i m a t e d by exac t ML. Twis based on the we ig thed s y m m e t r i c LS

e s t i m a t o r . T h e s t a t i s t i c s PT , M z G L S , TGLS, a n d P G L S u s e d a t a local ly d e t r e n d e d by GLS

w i t h c = 1 0 .

263

Table 5 Empirical size and power for 5% level. Tests and &2 based on the esti- mation of an ARMA model. Unconditional case. T=50. 5,000 replic.

] '=50 'l~'ue process Panel I ARMA(1,1): 0 = AR(2): r Model: ARMA

Test p -0.8 -0.6 0 0.5 0.8 -0.8 -0.5 0.5 0.8 ^~ PT 1.00 Lt}ARM A

0.90 M Z ~ Ls 1.oo

0.90

~ ; P T 1.00 GLS-ARMA 0.90

MZGa Ls 1.00 0.90

t - t e s t s TLS 1.00 0.90

TGL s 1.00 0.90

TML 1.00 0.90

p-tests pLS 1.00 0.90

PGLS 1.00 0.90

pML 1.00 0.90

0.064 0.074 0.052 0.098 0.293 0.037 0.063 0.091 0 .166 0.178 0.183 0.141 0.225 0.498 0.092 0.145 0.194 0.228 0 . I01 0.106 0.078 0.115 0.302 0.041 0.077 0.126 0.214 0.250 0.246 0.197 0.257 0.512 0.100 0 .172 0.252 0.293

0.062 0.069 0.052 0.080 0.208 0.032 0 .057 0.099 0.226 0.176 0.180 0.136 0.189 0.385 0.072 0.132 0.200 0.278 0.098 0.101 0.076 0.094 0.215 0.035 0.135 0.067 0.290 0.251 0.249 0.190 0 .212 0 .394 0.080 0.263 0.152 0.343 0.032 0.029 0.040 0.062 0.168 0.064 0.050 0.090 0.088 0.288

0.026 0.022 0.036 0.037 0.116 0.057 0 .044 0.090 0.070 0.221

0.056 0.052 0.067 0.074 0.155 0.104 0.090 0.149 0.101 0.256 0 .047 0 .057 0.047 0.131 0.394 0.149 0.159 0.150 0.330 0.699

0.048 0.056 0.049 0.150 0.466 0.150 0 .164 0.159 0.375 0.791

0 .054 0.064 0.058 0.160 0.469 0 .172 0.189 0.189 0.401 0.790

Panel II Model: AR(MIC)

C)2R PT 1.00 0.050 0.029 0.028 0.051 0.156 0.90 0.084 0.068 0.085 0.156 0.340

M Z G a as 1.00 0.073 0 .047 0.038 0.069 0.175 0.90 0.121 0.089 0.128 0.195 0.369

~GLS-AR^2 PT 1.00 0.047 0 .027 0.021 0.035 0.077 0.90 0.072 0.050 0.075 0.107 0.165

M Z Q a as 1.00 0.065 0.043 0.031 0.047 0.084 0.90 0.099 0.077 0.111 0.134 0.176

0.003 0.004 0.015 0.033 0.142 0.008 0.012 0.049 0.102 0.311

0.001 0.003 0.009 0.016 0.065 0.003 0.006 0.035 0.060 0.148

0.009 0.011 0.027 0.047 0.159 0.021 0.030 0.086 0.132 0.338

0.008 0.009 0.025 0.042 0.111 0.025 0.031 0 . I00 0.136 0.235 0.042 0 .027 0.028 0.096 0.313 0.096 0.062 0.098 0.276 0.590

0.050 0.032 0.026 0.086 0.267 0.114 0.072 0.095 0.251 0.504

0.053 0.036 0.036 0.115 0.330 0.118 0.083 0.130 0.308 0.615

t - t e s t s TLS 1.00 0.90

TGL s 1.00 0.90

TML 1.00 0.90

Tw 1.00 0.90

p-tests PLS 1.00 0.90

PaLS 1.00 0.90

pML 1.00 0.90

0.042 0.033 0.036 0.048 0.082 0.052 0.047 0.066

0.033 0.019 0.024 0.030 0.075 0.043 0.045 0.062

0.067 0.050 0.058 0.081 0.137 0.082 0.087 0 .117 0.079 0.066 0.058 0.076 0 .227 0.182 0.140 0.162

0 .077 0.068 0.061 0.088 0.231 0.193 0.150 0.170

0.095 0.082 0.067 0.090 0.265 0.224 0,166 0.186

0.005 0.022 0.046 0.101 0.010 0.065 0.080 0.153 0.009 0.033 0.068 0.135 0.016 0.092 0 .117 0.200

0.004 0.018 0.043 0.103 0.007 0.050 0 .077 0.147 0.006 0.026 0.066 0.142 0.010 0.069 0.111 0 .200 0.014 0.014 0.007 0.013 0.042 0.043 0.014 0.027

0.007 0.008 0.006 0.009 0.021 0.027 0.011 0.023

0.025 0.024 0.018 0.028 0.065 0.074 0.031 0.058

0.021 0.023 0 .019 0.025 0.070 0.086 0 .044 0.068 0.044 0.048 0.035 0 .060 0.143 0.145 0.070 0.121

0.042 0.043 0.038 0.063 0.125 0.131 0.076 0.120

0.056 0.060 0.045 0.066 0.169 0.178 0.085 0.138

Note : T h e s t a t i s t i c s PT a n d M Z ~ Ls use the e s t i m a t o r of 022 shown a t the i r left side. In

P a n e l I, the e s t i m a t o r s of ~d 2 are b a s e d on a n A R M A m o d e l us ing AIC a n d ML, w h e r e a s in ^2 ^2

P a n e l II t hey a re based on an A R us ing MIC a n d OLS. In 02GLS.AaMA a n d r , d a t a a re

p rev ious ly local ly d e t r e n d e d by GLS using c = 5 . T h e T- tes t s a n d p - t e s t s in P a n e l I a re b a s e d

on an A R M A m o d e l us ing BIC, and in P a n e l II t hey are based on an A R us ing MIC . TLS,

TaLS, pLS, a n d P a L s a re b a s e d on E C M e s t i m a t e d by OLS. TML a n d PMLare b a s e d on E C M

e s t i m a t e d by exac t ML. TwlS b a s e d on the w e i g t h e d s y m m e t r i c LS e s t i m a t o r . T h e s t a t i s t i c s

PT , M Z ~ Ls, a n d TCLS use d a t a local ly d e t r e n d e d by GLS wi th c = 1 0 .

264

T a b l e 6 E m p i r i c a l s i z e a n d p o w e r f o r 5 % l eve l . T e s t s a n d &2 b a s e d o n t h e e s t i -

m a t i o n o f a n A R M A m o d e l . U n c o n d i t i o n a l c a s e . T = 1 5 0 . 5 , 0 0 0 r e p l i c .

' l ~ u e p r o c e s s T = 150 P a n e l I A R M A ( 1 , 1 ) : 0 = A R ( 2 ) : r M o d e l : A R M A

T e s t p -0.8 -0.5 0 0.5 0.8 -0.8 -0.5 0.5 0.8 2 PT 1.00 O-JAR M A

0.90 M Z % Ls 1.oo

0.90

~ L S - A R M A PT 1.00 0.90

M Z ~ ~s 1.oo 0.90

t - t e s t s TLS 1.00 0.90

"/-G L S 1 .00 0.90

TML 1.00 0.90

p - t e s t s pLS 1.00 0.90

pOLS 1.00 0.90

pML 1.00 0.90

P a n e l I I M o d e l : A R ( M I C )

cS~a P T 1.00 0.90

MZG~ Ls 1.oo 0.90

~ 2 PT 1.00 GLS-AR 0.90

M Z a ~ Ls 1.oo 0.90

TLS 1.00 0.90

TGLS 1.00 0.90

TML 1.00 0.90

Tw 1.00 0.90

p - t e s t s PLs 1.00 0.90

pGLS 1.00 0.90

pML 1.00 0.90

t - t e s t s

0.060 0.061 0.053 0.048 0.040 0.680 0.632 0.788 0.479 0.454 0.070 0.073 0.062 0.053 0.040 0.702 0.653 0.810 0.487 0.453

0.058 0.056 0.052 0.036 0 .017 0.662 0.613 0.785 0.348 0.144 0.067 0.067 0.061 0.037 0.016 0.683 0.633 0.804 0.343 0.139 0.042 0.037 0.044 0.015 0.035 0.477 0.392 0.605 0.036 0.161

0.043 0.039 0.046 0.011 0.008 0.558 0.479 0.682 0.039 0.145

0.046 0.041 0.051 0.016 0.031 0.479 0.408 0.634 0.033 0.125 0 .047 0.050 0.044 0.076 0.134 0 .737 0.713 0.771 0.726 0.792

0.050 0.052 0.045 0.075 0.125 0.756 0.729 0.786 0.732 0.746

0.053 0.056 0.052 0.087 0.164 0.769 0.751 0.801 0.771 0.866

0.061 0.041 0.033 0.043 0.042 0.496 0.544 0.616 0.494 0.415 0.071 0.048 0.036 0.048 0.043 0.511 0.562 0.638 0.502 0.417

0.020 0.043 0.067 0 .084 0.414 0.609 0.641 0.575 0.020 0.051 0.074 0 .102 0.420 0.626 0.663 0 .592 0.017 0.039 0.063 0 .082 0.344 0.552 0.635 0 .567 0.016 0.044 0.071 0.100 0.335 0.567 0.658 0.592 0 .044 0.039 0 .044 0.045 0.497 0.381 0.347 0.301

0.046 0.039 0.043 0 .044 0.571 0.443 0.413 0.365

0.050 0.043 0.048 0.051 0.520 0.396 0.365 0 .324 0.060 0.052 0.053 0.057 0.781 0.706 0.600 0.515

0.060 0.053 0.051 0.061 0.787 0.716 0.616 0.528

0.071 0.060 0.058 0 .064 0.813 0.744 0.636 0.545

0.010 0.029 0 .047 0.052 0.241 0.526 0 .577 0 .454 0.010 0.033 0.052 0.058 0.242 0.538 0.606 0.476

0.048 0.035 0.033 0.032 0.016 0.009 0.026 0 .044 0.056 0.392 0.466 0.570 0.343 0.118 0.180 0.451 0.530 0 .434 0.054 0.038 0.035 0.034 0.014 0.007 0.028 0.047 0.067 0.395 0.487 0.589 0.350 0.110 0.163 0.463 0.553 0.455 0.014 0.017 0.023 0.029 0.055 0.150 0.250 0 .387 0.337 0.460

0.022 0.025 0.032 0.041 0.062 0.186 0.291 0.433 0 .364 0.419

0.018 0.022 0.029 0.031 0.054 0.152 0.260 0.409 0.337 0.443

0.026 0.028 0.036 0.046 0.094 0.342 0.457 0.603 0.531 0.652 0.061 0.040 0.032 0.074 0.195 0.511 0.542 0.607 0.685 0.885

0.073 0.046 0.035 0.078 0.205 0.547 0.566 0.628 0.704 0.856

0.068 0.046 0.039 0.082 0.206 0.539 0.572 0.638 0.707 0.893

0.020 0.022 0.022 0.018 0.320 0.348 0.295 0.205

0.027 0.032 0.033 0.035 0.349 0.391 0.343 0 .249

0.023 0.025 0.026 0.022 0.332 0.308 0.308 0.217

0.032 0.033 0.036 0.038 0.534 0.567 0.506 0.381 0.039 0.041 0.041 0 .042 0.594 0.617 0.545 0.403

0.044 0.047 0.042 0.050 0.615 0.637 0 .564 0 .422

0.048 0.048 0.046 0.047 0.626 0.647 0.576 0.430

Note : T h e s t a t i s t i c s PT a n d M Z ~ Ls use the e s t i m a t o r of W :z shown a t the i r left s ide. In

P a n e l I, the e s t i m a t o r s of O) 2 a re b a s e d on a n A R M A m o d e l us ing AIC a n d ML, w h e r e a s in

P a n e l II t hey a re b a s e d on an A R us ing MIC a n d OLS. In ~ 2 A2 GLS-ARMA a n d 0JGLS_AR , d a t a a r e

prev ious ly local ly d e t r e n d e d by GLS us ing c = 5 . T h e T- tes t s a n d p - t e s t s in P a n e l I a re b a s e d

on an A R M A m o d e l us ing BIC, a n d in P a n e l II t hey are b a s e d on an A R us ing MIC . TLS,

TCLS, PLS, a n d PaLS are b a s e d on E C M e s t i m a t e d by OLS. TML a n d p M g a r e b a s e d on E C M

e s t i m a t e d by e x a c t ML. Twis b a s e d on the w e i g t h e d s y m m e t r i c LS e s t ima to r . T h e s t a t i s t i c s GLS P T , M Z a , a n d TGLS use d a t a local ly d e t r e n d e d by GLS w i t h c = 1 0 .

265

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