an outlier robust unit root test with an application to the extended nelson-plosser data

JOURNALOF Econometrics

ELSEVIER Journal of Econometrics 66 (1995) 153- 173

An outlier robust unit root test with an application to the extended Nelson-Plosser data

An&-t: Lucas

Tinbergen Institute, Erasmus Universi@, NL-3000 DR Rotterdam. The Netherlands

(Received November 1992; final version received December 1993)

Abstract

This paper considers unit root tests based on robust estimators with a high breakdown point and high efficiency. The asymptotic distribution of these tests is derived. Critical values for the test are obtained via simulation. It is found that the size of the classical OLS based Dickey-Fuller test breaks down if the time series contains additive outliers. For innovative outliers the size of the robust test is less stable, while its size-adjusted power properties are better. An example is provided by applying the robust tests to the extended NelsonPlosser data. For four series the null hypothesis of nonstationarity is rejected.

Key words: Unit root test; Robust estimation; Additive outlier; Innovative outlier JEL classification: c12; c22

1. Introduction

The unit root issue has received considerable attention in the econometric literature over the last decade. The consequences for both statistical and economic theory have been and are being studied thoroughly. For two survey articles see Diebold and Nerlove (1990) and Campbell and Perron (1991). Several procedures to test for unit roots in a univariate time series have been proposed. References can be found in the two articles mentioned above.

Thanks go to Teun Kloek, Philip-Hans Franses. Marnix Klooster, Alex Koning. Peter Rousseeuw,

and Peter Schotman for reading the preliminary drafts. Comments of two anonymous referees and

one of the editors are also gratefully acknowledged. Herman van Dijk kindly provided the data used

in the application. All errors are of course my own.

0304-4076/95/$09.50 Q 1995 Elsevier Science S.A. All rights reserved SSD1030440769401613 5

154 A. Lucas / Journal of Economebics 66 (199.5) 153-l 73

Especially the (augmented) Dickey-Fuller (DF) r-test (Fuller, 1976, Sect. 8.5) is

well known and widely used. This test procedure makes use of ordinary least

squares (OLS) estimates of the parameters of an autoregressive (AR) model for

the time series. It is well known that the technique of least squares has some desirable optimality properties if a normal distribution is underlying the obser-

vations. On the other hand it is also known that the OLS estimates are very

sensitive to the occurrence of outliers in the data. This sensitivity of the OLS estimator is inherited by the DF t-test. Therefore the inference drawn on the

basis of this test can be incorrect in case the data set at hand contains anomalous points. It is the aim of this paper to suggest the use of an alternative unit root

test, which is less influenced by aberrant observations than the traditional DF t-test. The behavior of this alternative test will be studied using simulated data as well as the fourteen economic time series considered by Nelson and Plosser

(1982).

As the sensitivity of the DF r-test is caused by the sensitivity of the OLS estimator, it seems worthwhile to replace this estimator by one that is less

sensitive to the occurrence of outlying observations. Several of these estimators are readily available from the literature on robust statistics. One can choose

between minimax type robust estimators as proposed by, e.g., Huber (1964) bounded influence estimators as initiated by the work of Hampel (1971) and high breakdown point (HBP) estimators to be found, for example, in Rousseeuw

and Leroy (1987). In this article I suggest the use of an HBP estimator for testing the unit root hypothesis.

HBP estimators are concerned with the concept of the breakdown point of an

estimator (Donoho and Huber, 1983). Informally stated, in the iid regression case the breakdown point measures the largest proportion of outliers an estimator can tolerate before collapsing to some nonsensical value (often infinity).

As the name suggests, HBP estimators possess a high breakdown point, which means that they can still provide useful information about the characteristics of the bulk of the data in the presence of a large number of anomalous observations. It should be kept in mind that statistical breakdown is mostly concerned with extreme outlier configurations, which might be deemed unrealistic. More- over, defining a breakdown point for the case of dependent observations is not as easy as for the iid regression case, although some attempts have been made (e.g. Papantoni-Kazakos, 1984).

The choice for an HBP estimator is motivated by the dependence structure inherent to AR models. Consider a standard AR(p) process y1 = 41y,_ I + ... + Qfpy, - p + F,, with E, Gaussian white noise and p known. Furthermore assume

that one of the points ys is replaced by the outlier ys + S for some s and some large 6. Ignoring endpoint effects for the moment the outlying observation will enter the model p + 1 times: first as the left-hand-side variable and afterwards p times as a right-hand-side variable. So one outlying value of y, can cause p + 1 outlying (p + I)-tuples ( _vt,yt_ 1. . . . 9 y __p), which are used to calculate the

A. Lucas 1 Journal of Econometrics 66 (1995) 153-l 73 155

estimates of the AR parameters. Regarding the occurrence of several outlying points y, as a realistic possibility, the use of an HBP estimator does not seem overly cautious.

Several HBP estimators for the regression case are presented in the literature: repeated medians (Siegel, 1982) least median of squares (Rousseeuw, 1?84), S estimators (Rousseeuw and Yohai, 1984) MM estimators (Yohai, 1987), r estimators (Yohai and Zamar, 1988), and one-step GM estimators (Simpson et al., 1992). All of these estimators can attain a breakdown point of approximately 0.5 in large samples. Especially the earlier high breakdown estimators have a poor (relative) asymptotic efficiency if there are no outliers in the data. In this paper I use the MM estimator, which is one of the estimators that can easily achieve a high efficiency and a high breakdown point at the same time.

There are several examples of the use of low breakdown point estimators in time series problems. Martin (1979, 1981) suggests the use of GM and conditional mean M estimators, Bustos and Yohai (1986) use outlier robust estimates of residual autocovariances to estimate the parameters of ARMA models and Allende and Heiler (1992) employ recursive GM estimators for ARMA models. There is also some work on the performance of HBP estimators in a time series context. Some references are Rousseeuw and Leroy (1987, Ch. 7) and Lucas (1992, Ch. 4) which discuss the least median of squares estimator for AR and ARMA models, respectively. Most of the literature on robust statistics in time series is however concerned with stationary time series. The notable exception is Knight (1989, 1991) who treats the case of M estimators for integrated infinite variance processes.

In a recent article Andrews (1993) discusses exactly median-unbiased estimation in the AR(l) model. The term median might suggest a link with the present approach using robust estimators. The two approaches are however quite distinct. Andrews proposes methods to correct for the downward small sample bias of the (nonrobust) OLS estimator for the AR( 1) model. His methods even seem to work in specific outlier situations (innovative outliers, see Section 2) as is demonstrated by his Table 4. However, in other situations Andrews’ approach is quite sensitive to the occurrence of discordant observations (especially additive outliers). In principle, Andrews’ methods can also be used to correct for the finite sample bias of the robust estimators used in the present setup. Therefore, his paper and the present article can be viewed as complements rather than substitutes.

In the next section I introduce several basic concepts. Furthermore 1 give a definition of the MM estimator and present the algorithm used for computing the estimates. In Section 3 some asymptotic results are obtained for the DF t-test statistic for M estimators under the null hypothesis of a unit root. Section 4 contains the results of a few simulation experiments illustrating the size and power of the test for samples with and without outliers. In Section 5 the presented techniques are applied to the extended NelsonPlosser data

156 A. Lucas 1 Journal of Econometrics 66 (1995) 153-I 73

(Schotman and van Dijk, 1991). Because there exists a link between the occurrence of outliers and structural breaks, a brief discussion of the relation of the present article with the work of Perron (1989) is given in that section. Some concluding remarks are contained in Section 6. The proofs of the theorems in Section 3 are contained in the Appendix.

2. Concepts and definitions

Modeling outliers in a time series context is more complicated than in the ordinary regression case. This is due to the fact that for time series not only the magnitude of the outlier matters, but also its place in time. There are two outlier generating models for time series that are often employed in the literature (compare Martin and Yohai, 1986). The first one is the innovation outlier (IO) model, which for an autoregressive moving average (ARMA) process reads

W)Y, = ~W)~,~ u, = E, + w,. (1)

Here L is the lag operator, Ly, = y,_ 1, 4(L) and 8(L) are polynomials in L, E, is white noise, and w, is a contaminating random variable. The second model is the additive outlier (AO) model, which reads

yt = Z, + w,, 4(L)& = W)G. (21

If W, = 0 V’t, (1) and (2) are identical and are referred to as the central model. This corresponds to a situation without outliers. Typically w, equals zero for most values oft. The remaining w, are obtained by drawing from a (possibly degener- ate) contaminating distribution.

I illustrate the effect of one extreme outlier on the OLS estimator and the DF t-test statistic for the simple AR(l) model without intercept and trend. Let Z, = 4z, _ 1 + E, for t = 1, . , T, where E, is white noise. I only consider outliers that lie approximately halfway the sample, say at t = k, thus avoiding the intricacies of endpoint effects. First construct the new time series yt = zr + I$‘, with W, = (5 if t = k and zero otherwise. The series y, now contains an A0 [see (211. The OLS estimate 4 of d, in the regression model yt = +yt_ 1 + u, equals

$p&l JL, +ys+l)ii+O(l) Cr:-l P+o(s)

(3)

So 6 + 0 for 6 + cc The DF r-test statistic in this special case equals (6 - 1). (&+_ ,/s^“,“2, where s*’ is the estimated variance of the residuals. Using some straightforward algebra it can shown that

*2 s = T-‘C(y, - q?yt_I)2 = h2/T + O(6). (4)

A. Lucas / Journal of Econometrics 66 (1995) 153-173 157

Combining (3) and (4) gives the result that the DF t-test statistic approaches - T1j2 for 6 --t co. If on the other hand we construct a new series y, = z, + w,

with w, = 0 for t < k and w, = @-‘, otherwise, the time series contains an innovative outlier. Using the same strategy as before one can show that for 6 + co the OLS estimate 6 approaches the true parameter #, while the DF t-test statistic tends to - (T(l - +)( 1 - 4 2(T-k))/(l + 4))“‘. So for moderate sample sizes one extreme A0 results in rejection of the unit root hypothesis even in case 4 = 1, while an IO can result in either rejection or nonrejection, depending on the true value 4. Franses and Haldrup (1993), using a somewhat more general and less extreme form of contamination, generalize the above findings for AO’s and state some asymptotic results.

The sensitivity of the traditional OLS estimator initiated the development of formal robust estimation procedures. Huber (1964, 1973) introduced the class of M estimators. An M estimator for the AR(l) parameter is the minimand of

where p( .) is a function from [w to !R and (r2 is the variance of the innovations E,. For p(u) = w2 one obtains the OLS estimator. Differentiating (5) with respect to 4 one obtains the first-order condition

CIc/((Yl - $Y*- 1)/4Y,-1 = 0, (6)

where $(v) = dp(u)/du. Two often used specifications for $ are the Huber II/ function, $(u) = min(c,max( - c, u)), and the bisquare $ function, $(u) = u(c’ - u2)2 l[-&c,, where c is a tuning constant and lA is the indicator function of the set A. If g is unknown, one can add an additional equation to (6) in order to estimate (p and c simultaneously. Alternatively, one can use a preliminary robust and consistent estimate of 4 in order to construct residuals. These residuals can be used to estimate 0 using a robust scale estimator, e.g., the median absolute deviation.

The M estimators as described above are not necessarily robust against outliers in the space of the explanatory variables. This causes no problems if only IO’s occur, which only result in large residuals. In case of an A0 however, a large residual caused by an extreme value of y1 is followed by a bad leverage point. This is a result of the fact that the yt enters both as a left- and right-hand- side variable in the model, as was mentioned in the introduction. Martin (1979, 1981) therefore proposed to use generalized M estimators (GM) and conditional mean M estimators. A GM estimator is a solution of Cq((yl - ~JJ,__ 1)i ~.y~_~)y~_~ = 0, where q(‘, .) is a function satisfying the conditions stated in. e.g., Hampel et al. (1986. p. 315). The function 9 does not only downweigh large residuals, but it also serves to reduce the influence of large values of y,_ ,, i.e., leverage points.


For more general AR models, the percentage of outliers these GM estimators can cope with decreases with the order p of the AR polynomial, For this reason HBP estimators were introduced, the breakdown point of which remains approximately fixed in large samples when the number of regressors (or AR lags) increases. There have been several interesting developments in the field of robust estimation with a high breakdown point over the last decade. After the repeated median of Siegel (1982), Rousseeuw (1984) introduced the least median of squares (LMS) estimator for the regression context. The estimator searches to minimize the median instead of the sum (or mean) of the squared residuals. This estimator has poor convergence properties as it is T 1’3-consistent instead of the usual T l”. Several alternatives with T “’ -consistency were proposed in the same paper, but they did not receive much attention in the literature.

Rousseeuw and Yohai (1984) introduced the class of S estimators. It has the same nice properties as the LMS estimator and moreover enjoys T “‘-consistency. They also showed that S estimators satisfy the same first order condition as M estimators. In order to calculate an S estimator one proceeds as follows. Assume for the rest of this section that the model is y, = $yt_ 1 + ct, although the S and MM estimator are also defined for the more general regression context. One starts by choosing a bounded function p1 [see (5)]. For a given C#J one defines the scale estimate as(#) > 0 as the solution of

n-‘C&((JJ* - 4y,-1)las) = 0.5suP{Pi(x)ls E RJ.

The S estimate of C$ is the value that minimizes ~~(4). Because pl is bounded there can be many local optima. A global search algorithm has to be employed to find the global minimum (see Rousseeuw and Yohai, 1984). Because of this the computation of the estimator is rather time consuming.

Apart from this computational issue, a second disadvantage of the S estimator is that its high breakdown point is counterbalanced by a low efficiency at the central model. Therefore Yohai (1987) presented the MM estimator. It uses one of the high breakdown point estimators described above as starting values. In this paper I used the S estimator for this purpose. As a first step in the calculation of the MM estimator a scale estimate cHM is calculated at the high breakdown estimate of C#J using (7). This scale estimate is kept fixed during the remaining calculations. Note that for the S estimator used in this paper the scale estimate equals the objective function of the estimator, GMM = rrs. The second step consists of minimizing CpZ((y, - 4yt- 1)/0 MM) with respect to 4, resulting in the MM estimate &,,M. The functions pl and p2 have to satisfy certain conditions (Yohai, 1987). I use the following specification:

(3C4X2 - 3$x4 + x6)/6 for 1x1 6 CL,

Picx) = i = 1,2, (8) for 1x1 > Ciq

A. Lucas / Journal of Econometrics 66 (1995) 153-l 73 159

with cl = 1.547 and c2 = 4.685 (Rousseeuw and Yohai, 1984; Yohai, 1987). Note that the derivative Of pi(x) gives the bisquare $ function mentioned below (6). By using the constant ci one obtains an S estimator with a breakdown point of approximately 0.5. The value of c2 serves to improve the efficiency of the S estimator at a central Gaussian model. Using c2 = 4.685 the MM estimator has an asymptotic efficiency of 95% with respect to the maximum likelihood estimator in that specific situation. The remaining details concerning the MM estimator can be found in the article of Yohai.

An important concept in robustness is the influence function (Hampel et al., 1986, Ch. 2). It measures the (appropriately standardized) effect of an infini- tesimally small number of outliers on the estimator. A bounded influence function is desirable because it implies that a small number of outliers only has a limited influence on the final estimates. Although the MM estimator has not got a bounded influence function, the estimates can only have a finite standardized bias if there is a small but positive number of outliers (Yohai, 1987, p. 650).

3. The Dickey-Fuller test for M estimators

The asymptotics of the OLS estimator for the autoregressive parameter of an AR(l) under the assumption of a unit root have been extensively studied in the literature (see the references in Diebold and Nerlove, 1990). In this section the asymptotic distribution of the DF t-test statistic for M estimators is discussed. At the central model, that is in a situation without outliers, the robust and classical DF t-test must be comparable in terms of size and power. In general the power of the robust test will be somewhat below that of the classical test at the central model. This is a result of the tradeoff between efficiency and robustness for robust estimators (Hampel et al., 1986, Sect. 2.1~). The lack of power is compensated for by a gain in insensitivity of the size and power to departures from the central model. Whereas the size and power of the DF f-test using OLS estimates can be spoilt completely by one extreme additive outlier, its robust counterpart is less sensitive to anomalous observations.

In this section the asymptotic behavior of the robust DF f-test will be studied at the central model. The effects of contamination by outliers will be studied in the next section by means of several simulation experiments. First consider the simple AR(l) case, y, = $yl_ 1 + E,. Assume the variance of the innovations 0’ is known. If 0’ is unknown, the techniques mentioned in the previous section can be used to obtain a preliminary estimate of it. As long as one uses a consistent estimate of 02, the results of this section do not change (compare Phillips, 1987; Knight, 1989, Sect. 4.3).

The M estimator & is a solution of

C$((.vt - &Yr- 1)lNYt- 1 = 0, (9)

160 A. Lucas / Journal of Econometrics 66 (1995) 153-l 73

where $ is a function satisfying the conditions below. A unit root corresponds to the case 4 = 1. In order to calculate the DF t-test statistic one needs an estimate of the variance of &. Following Hampel et al. (1986, Sect. 2.3) I use

sf = ~*(~Jr%Yt - &Yr- twJ)Y:- M&wYl - &Y1-1)/4YLK (10)

where $‘(x) = d$(x)/dx. We are thus looking at the (asymptotic) behavior under the null hypothesis of a unit root of the statistic

4$ = (&I - 1)/s,. (11)

Note that if $(x) = x one just obtains the traditional DF t-test.’ I make use of the following assumptions.

Assumption 1. The errors E, are iid with mean zero and finite positive variance a*; dy, is stationary.

Assumption 2. $( .) is bounded; $‘( .) is bounded and Lipschitz continuous;

E(i&,/4) = 0; c: = E($*(s,/a)) and 0 < ci < cc; c2 = E($‘(r,/a)) and o<c*< 00.

All expectations are taken with respect to the central model. Assumption 1 can be relaxed by allowing more general innovation sequences like mixing processes (e.g., Phillips, 1987) at the cost of additional complexity. Probably also the finite variance assumption can be dispensed with by using the results of Knight (1991). It is worth mentioning that the results obtained here partly overlap those of Knight. This is especially true for the first part of Theorem 1. However, because Theorem 1 is used to introduce Theorem 2 and because the assumptions of Knight and the ones used in this article are not identical, I choose to present the theorem and its proof in full.

In order to study the asymptotic behavior of t4 I define the following partial sum processes:

tsT1 IsTl

ET(S) = T-l’* x E,, ST(s) = T - “* c $(&t/d,

1=1 i=l

for all s E [0, 11, where [x] is the largest integer smaller than or equal to x. Let XT * X denote that the associated probability measure(s) of XT converge weakly to that (those) of X. Now the following lemma follows directly from Corollary 2.2 of Phillips and Durlauf (1986)

Lemma 1. Under Assumption I,

(BT(-hST(‘)) - (aB(‘hclS(-))>

with B and S denoting two correlated standard Brownian motions.

’ As the covariance estimator in (10) is of the form suggested by White (1980), choosing $4~) = x

actually gives a heteroskedasticity robust variant of the traditional DF r-test.

A. Lucas /Journal of Econometrics 66 (1995) 153-l 73 161

The next theorem states the main asymptotic result. It shows how the statistic td is asymptotically distributed.

Theorem 1. lfthe Assumptions 1 and 2 are satisfied and if(& - 1) = Op(T-‘/2), then under the null hypothesis H,: cf, = 1,

(12)

T -2C11/2(Ay,/a)y,2_1 * c:o’J B’dt, (13)

T - 3’2 C $‘(A yJa) y:_ 1 =z- c202 j B2 dt ,

t0 * SBdS/{JB2dt}“2.

(14)

(15)

The condition (4, - 1) = O,(T - ‘I’) may seem strange at first sight. It ensures that the ‘right’ solution of (9) is chosen. This is important if there are multiple solutions to (9), as is the case for S and MM estimators. Because the p function defining these estimators has to be bounded, e.g., (8), its derivative $(x) vanishes for large values of x. Therefore (9) can be satisfied for values &, very different from the true value 1. This is prevented by the additional condition of the theorem.

The final result in (15) is very similar to the expression as found in Phillips (1987). Only the numerators of the two expressions differ. This can be seen by choosing $(x) = x. In that case c1 = 1 and S = B in Lemma 1, yielding j B dB/[ S B2 dt] ‘I2 for (15). The last expression equals part (e) from Theorem 3.1 of Phillips (1987) for the iid case. One can make the relation between the DF t-test statistic for the M estimator (TM) and for the OLS estimator (toLS) even more explicit. Defining Y as the correlation between E and 51/(,+), r = E(EI,&/cJ))/(c, CJ), it is proved in Lucas (1993) that tM - rtoLs *

dmN(O; l), where N(0; 1) . IS a standard normal random variable. Estima- tors, like the MM estimator, which have a high efficiency at the Gaussian model also have values of r close to 1. Therefore the differences between the critical values of tM and toLs are expected to be small in these cases.

Theorem 1 is of little practical importance because it only states a result for the AR(l) case. Therefore a theorem is necessary to generalize the result to the AR(p) case with general p. First some additional definitions are required. Let y1 be generated by the AR(p) model

Y, = byt-I + +,AY,-I + ... + WY,-,+I + 6, (16)

(compare Fuller, 1976, p. 374). The null hypothesis of a unit root corresponds to H0:4,=1.Let~=($, ,..., ~p)‘andx,=(y,-,,Ay,_l, . . . . Ayt_p+l)‘,thenan M estimate $ of 4 is the solution to C +((y, - x~$)/,)x~ = 0. The covariance

162 A. Lucas / Journal of Econometrics 66 (1995) 153-I 73

matrix of (4 - 4) can be estimated by oZA- ‘CA- ‘, with

A = C $‘((Y, - 4~)l~)x,x;, c = c$2((Yt - Jml~)x,x; (see Hampel et al., 1986, Sect. 6.3). Now the following theorem can be proved.

Theorem 2. Given the conditions of Theorem 1 and given model (16), then under the null hypothesis HO: $1 = 1,

t+r =E= JBdS/{JB2dt}“2:

where tbl = (4, - l)/sg and sb is the square root of the (1, I) element of a2A- ‘CA- ‘.

It is well known (Fuller, 1976, p. 378) that including a constant or trend into the regression model (16) changes the asymptotic distribution of the DF statistic. This also holds in the present context of M estimators. The above two theorems can be adjusted in order to give similar results in those situations.

A last point to be mentioned here is that Theorems 1 and 2 are also valid for both S and MM estimators. This may seem strange as the breakdown behavior of ordinary M estimators is completely different from that of the other two estimators (Hampel et al., 1986, Sect. 6.4) which suggests that the derivations given above cannot cover all three estimators at once. However, both S and MM estimators satisfy the same first-order condition as ordinary M estimators (Rousseeuw and Yohai, 1984; Yohai, 1987). Because only the first-order condition is used in the theorems, the results hold for all three classes of estimators.

4. Size and power robustness using the MM estimator

In this section the model

z,=&z,-,+~+yt+E* (17)

is considered. A constant and trend are included in (17), because they are needed in the empirical illustration contained in the next section. It was shown in the previous section for #J = 1 that the DF t-statistic converges in distribution to the random variable given in (15). I place however more emphasis on the finite sample distributions of the (non)robust DF t-test statistic. The critical values for this test statistic will be obtained by means of a simulation experiment in the present section. The behavior of the test is also studied in case AO’s or IO’s are present, both for C$ = 1 and 4 # 1.

The simulation experiment is set up as follows. First the following parameters are fixed: y = p = z. = 0. Next T normal iid variables E, are generated2 with

21 used the random number generator drand48() present in the standard library of the C pro-

gramming language on a SUN (System IV) workstation. This generator has a large period of approximately 231 - 1.

A. Lucas J Journal of Econometrics 66 (1995) 153-173 163

mean zero and unit variance. For the IO situations a new series & is constructed using random numbers u, uniformly distributed over the interval [0, 11. The variable Et equals E, except when ut < v] for some prespecified q, in which case CE, = E, + w,. Here w, is a random variable with distribution function G. Using this new innovations sequence E, a time series y, is constructed using yt = @yt_ 1 + Et and y, = 0. In A0 situations the original sequence E, is used to construct an uncontaminated time series z, using (17). Next a new time series y, is constructed using uniform random numbers u, with y, = z, + w, if ut < q and y, = z, otherwise.

I have used several choices for the contaminating distribution G of w,. The first one is G = 1 iX > o ), so no outliers are generated. The second one generates outliers of equal size 5, G = l,,, 5). The third and fourth one are a Gaussian with mean zero and variance 9 and a standard Cauchy distribution, respectively. For each of these choices of G I used q = 0.05. Note that the last contaminating distribution violates Assumption 1. It is only included to illustrate the effect of extreme outliers.

For the estimation of the critical value of the robust and nonrobust unit root test at the 5% significance level the possibly contaminated series y, was used along with a constant and a trend in order to compute both the OLS and the MM estimator. The values of the associated DF t-test statistics were stored over 10,000 replications and the final estimate of the critical value was the 0.05th quantile of these 10,000 values, i.e., the 500th observation.3 In order to study the effect of the sample size, the simulations were carried out for T = 50, 100, 200. The results are contained in Table 1.

Before discussing the details of Table 1, something must be said about the illustrative purpose of its entries. When testing the unit root hypothesis for some empirical time series, a researcher in general does not know which observations are outliers with respect to the chosen model, nor does he know the kind of the contamination he is dealing with (IO or AO). Therefore he will use the critical value simulated for an uncontaminated error process. For example for a time series of length 100 he can use - 3.46 for the nonrobust test and - 3.67 for the robust test, suggesting a size of 5%. Now, if the considered time series contains several outliers, the actual size of the test can be far different from 5%. This is of course undesirable. One would like to have a test with an approximately constant size under a variety of contaminations. It is the aim of the table to show that the MM estimator is more suited to achieve this objective than the OLS estimator.

3 Estimated standard errors of the entries in Table 1 are in general below 0.05. The exceptions are the entry for the MM estimator for T = 50 and 6s A0 contamination, and the entries for the OLS estimator for T = 100,200 with C(0; 1) A0 contamination. The estimated standard error for these three numbers is approximately 0.10.

164 A. Lucas / Journal of Econometrics 66 (1995) 153-l 73

Table 1

Upper 5% points for the robust and nonrobust DF r-test for contaminated and uncontaminated processes

Contaminating distribution

T = 50 T=lOO T = 200

OLS MM OLS MM OLS MM

No outliers

- 3.50 - 4.03

Additive outliers

- 3.46 - 3.67 - 3.42 - 3.52

65

N(0; 9) C(0; 1)

- 5.44 - 6.10 - 4.61 - 4.43 - 6.66 - 4.25

Innovative outhers

- 5.72 - 3.61 - 5.85 - 3.36 - 4.63 - 3.79 - 4.62 - 3.10 - 9.04 - 3.70 - 12.77 - 3.63

65 - 3.46 - 3.11 - 3.51 - 3.47 - 3.44 - 3.08 N(O;9) - 3.45 - 3.46 - 3.54 - 3.86 - 3.44 - 3.33 C(0; 1) - 3.54 - 3.91 - 3.49 - 3.45 - 3.45 - 3.16

6, is a point mass at x; N(0;9) is a normal distribution with mean 0 and variance 9; C(0; 1) is a standard Cauchy distribution.

The entries under the heading OLS in the first row of Table 1, i.e., for uncontaminated processes, are similar to the entries in the table of Fuller (1976, p. 373). The critical values for the robust test are somewhat larger in this case than those for the nonrobust test. The difference however decreases for increasing values of the sample size T. This is to be expected in view of the high efficiency of the MM estimator at the central (Gaussian) model (Lucas, 1993). For the uncontaminated process I also performed simulations with T = 400, resulting in two almost identical critical values of - 3.41 and - 3.42 for the OLS and MM estimator, respectively.

The remaining entries in Table 1 demonstrate the size robustness or non- robustness of the DF t-test calculated with the MM estimates and the OLS estimates. Except for small samples, T = 50, the critical values for the robust unit root test do not fluctuate much under different forms of additive outlier contaminations. This stands in sharp contrast to the results obtained with the traditional OLS estimator. Consider the case T = 100 with critical value - 3.46. One believes one is working with a significance level of 5%. If however the time series is contaminated with 5% outliers, the actual size of the test is 27% if N(0; 9) is used as a contaminating distribution, or even 57% if b5 is used. It is noteworthy that even in the extreme case of a contaminating Cauchy distribution the critical values for the robust DF t-test at a 5% significance level are very

A. Lucas / Journal of Econometrics 66 (1995) 153-I 73 165

similar to the ones for the uncontaminated process, while those for the nonrobust estimator are very much affected. Note also that the critical values computed with the OLS estimator decrease for increasing sample size T for a5 and C(0; l), whereas they increase for the MM estimator.

The results are reversed if one looks at IO situations. In these cases the size of the traditional OLS-based DF t-test seems more stable than that of its robust counterpart. An important difference however is that the size of the robust test at the fixed critical value given in the top row of Table 1 is often decreased by the occurrence of the innovative outliers, while that of the nonrobust test was increased by the additive outliers.

A final thing to note about the entries of Table 1 concerns the results for the additive outliers. When looking at the A0 generating model one can see that in the present case the true model is given by y, = y,_ 1 + E, + w, - w,_ Ir which is an ARIMA(0, 1,l) model. Therefore it would be more appropriate to use the Phillips-Perron (1988) Z(t,) test, which corrects for the MA behavior of the disturbance term. Using the same simulation framework it turned out however that the results obtained with this latter test statistic were comparable to those obtained with the DF t-test.

Because of the large fluctuations in the significance levels of the tests at fixed critical values under different outlier configurations, it seems useless to look at the raw power properties of the two procedures. In order to obtain some insight, a small simulation study is set up to calculate the size-adjusted power of the two procedures for the cases given in Table 1. Again cc, y, zO, and y,, are equal to zero. For several values for 4 between 0.5 and 1.10 a (possibly contaminated) series y, of length 100 is constructed using the method described above. Next the OLS and MM estimates are calculated using the first lag of y,, a constant, and a trend as regressors. This procedure is repeated 100 times. Using the critical values from Table 1 a record is kept of the number of rejections of the null hypothesis of a unit root for both the robust and nonrobust DF t-test. The rejection percentages are plotted in Fig. 1 for the uncontaminated process and a process with innovative outliers using the contaminating distribution N(0; 9).

The first thing to be noted in Fig. 1 is the similarity in power behavior of the robust and nonrobust test for uncontaminated processes. This is due to the high efficiency of the MM estimator at the central model. Furthermore the figure clearly demonstrates that the size-adjusted power of the robust test is superior to that of the traditional test in the IO case. This is of little practical importance however, because in reality one will not be working with a size-adjusted critical value, but just with the top entries of Table 1. This entails a poorer power behavior of the robust test for at least some values of (p, because Table 1 indicates that the distribution of the robust test statistic has shifted leftward in the IO situation. The results for IO’s with contaminating distribution a5 are similar to the ones described above. For the A0 situations the size-adjusted

166 A. Lucas / Journal of Econometrics 66 (1995) 153-I 73

P 0 w c

1

0.8 - MM(&) - _ OLS(N(0; 9)) MM(N(O;9))

0.6 -

0.4 -

0.2 -

0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

4

Fig. 1. Simulated size-adjusted power functions for processes containing innovative or no outliers.

power of the robust test is somewhat inferior to that of the nonrobust test. Again the argument of practical importance given above can be raised, now favoring the use of the robust estimator over the nonrobust one. Finally, it is worth mentioning that for 4 = 0.5 the size-adjusted power of the traditional test in the extreme case of C(0; 1) additive outliers is only 0.15, while that of the robust test is as high as 0.95.

5. Application to the Nelson-Plosser data

Nelson and Plosser (1982) looked at the behavior of 14 economic time series. Using the testing methodology of Dickey and Fuller they showed that 13 of these series were better described by an AR process with a unit root than by a stationary AR process around a deterministic trend. By now most authors agree that their findings were rather premature. Several researchers were able to obtain different conclusions in a Bayesian (see for example the October- December 1991 issue of the Journal of&plied Econometrics) or nonparametric framework (e.g., Bierens 1992). In the classical parametric context Perron (1989) argued that most of the series considered by Nelson and Plosser could be regarded as stationary fluctuations around a linear trend if allowance was made for several structural breaks in this trend function. A number of authors have criticized Perron’s findings. They indicated that if one does not fix the date of the structural break a priori, the evidence against the unit root hypothesis is much weaker (see the July 1992 issue of the Journal of Business and Economic Statistics).

A. Lucas / Journal of Econometrics 66 (I 995) 153-I 73 167

It is interesting to note that there exists a link between outliers and structural breaks. This is clearly exhibited in Section 2 of Perron and Vogelsang (1992). Both additive and innovative outliers can cause structural breaks under the unit root hypothesis. Perron (1989) suggests to remedy this problem by introducing the appropriate dummy variables. As will be clear from the previous sections, one could also try to use robust estimators to eliminate the effect of outlying observations (or innovations). In this way one can circumvent the problem of choosing the specific points in time for the dummy to equal one. This should overcome the size problem of the test of Perron signaled by Christian0 (1992). A noticeable difference with the approach of Perron however is that the structural breaks can be present only under the null hypothesis of a unit root. Under the alternative hypothesis the effect of outliers is only temporary due to the stationary character of the series in that case.

Nelson and Plosser used yearly data up to 1970 for 14 economic time series. In this paper I use the extended Nelson-Plosser data of Schotman and van Dijk (1991) which contain observations up to 1988. The first difficulty arises in fixing the order of the AR polynomial for each series. I use the model selection strategy as suggested by Perron (1989) and Perron and Vogelsang (1992). First an AR(8) is estimated, with a trend and a constant included in the regression. If the coefficient of the longest lag differs significantly from zero at the 5% significance level, the present model is used for testing the unit root hypothesis. Otherwise the order of the AR polynomial is reduced by 1 and the selection strategy is applied to this lower-order AR model. It is clear that the modeling strategy described above may give rise to different orders of the AR polynomials, depending on whether the OLS or the MM estimator is used. Therefore I present the results for both models. Let model I be the model chosen with the OLS estimator and model II the one chosen with the MM estimator. The results are shown in Tabies 2 and 3. The length of the time series is given by T and the chosen order of the AR polynomial by p. Furthermore the number of points with scaled residuals greater than 3 in absolute value is counted for both the OLS and the MM estimator, where the scaling is done using an estimate of the standard error of the regression. These numbers are given under the label outliers.

A nice way to visualize the outliers is to use added variable plots (Cook and Weisberg, 1991). In order to illustrate that robust methods are also very useful for model selection, I discuss the stock prices series. Let y, denote stock prices at time t and let P be the projection matrix, projecting on the space spanned by (1, t, y,_ J. The added variable plot in Fig. 2 plots (I - P)y, versus (I - P)y,_ 1. Three outliers can be observed in this figure. Running an OLS regression on all the observations in Fig. 2 yields the solid line, which has a significant positive slope. However, a regression without the three outlying observations results in the dotted line, which has an insignificant slope coefficient. This explains the difference between the AR orders (2 versus 1) chosen with the OLS and the MM

168 A. Lucas / Journal of Econometrics 66 (1995) 153-173

Table 2

Tests for autoregressive unit roots in model I

T P &LS t+0,, Outliers 4, tPM Outliers

Real GNP Nominal GNP Real per capita GNP Industrial production Employment Unemployment rate GNP deflator Consumer prices Wages Real wages Money stock Velocity Interest rate Common stock prices

80 2 0.82 - 3.45 1 80 2 0.94 - 2.02 1 80 2 0.82 - 3.52” 1

129 6 0.85 - 2.49 1 99 2 0.85 - 3.41 1 99 4 0.72 - 3.92” 1

100 2 0.97 - 1.59 2 129 2 0.99 - 1.01 3 89 7 0.91 - 2.41 0 89 2 0.93 - 1.68 0

100 2 0.94 - 2.86 1 120 1 0.96 - 1.60 0 89 6 0.96 - 1.11 2

118 2 0.92 - 2.41 1

0.81 - 4.70” 4 0.96 - 1.38 5 0.80 - 4.77” 4 0.89 - 1.71 9 0.84 - 3.27 8 0.75 - 3.80” 8 0.99 - 0.88 5 0.99 - 1.33 10 0.95 - 1.00 12 0.94 - 1.56 1 0.95 - 2.13 3 0.94 - 2.39 6 0.90 - 1.97 11 0.92 -2.44 2

“Significant at the 5% level using the first row of Table 1.

Table 3

Tests for autoregressive unit roots in model II

Series T P 60,s tmom Outliers $, t&l Outliers

Real GNP Nominal GNP Real capita GNP per Industrial production Employment Unemployment rate GNP deflator Consumer prices Wages Real wages Money stock Velocity Interest rate Common stock prices

80 8 0.75 - 3.22 1 0.68 - 4.83” 6 80 6 0.95 - 1.47 1 0.97 - 1.37 9 80 8 0.73 - 3.39 1 0.65 - 4.38” 7

129 4 0.81 - 3.38 3 0.83 - 3.17 9 99 4 0.84 - 3.43 0 0.85 -4.13” 12 99 4 0.72 - 3.92” 1 0.75 - 3.80” 8

100 6 0.96 - 1.51 2 0.98 - 1.45 5 129 4 0.99 - 1.20 3 0.99 - 1.50 9 89 2 0.94 - 2.36 0 0.94 - 2.84 12 89 1 0.96 - 0.97 0 0.97 - 0.90 1

100 2 0.94 - 2.86 1 0.95 - 2.13 3 120 1 0.96 - 1.60 0 0.94 - 2.39 6 89 8 0.96 - 0.61 2 1.04 1.53 13

118 1 0.94 - 1.82 1 0.93 - 2.44 2

“Significant at the 5% level using the first row of Table 1.

estimator. The robust MM estimator automatically recognizes the three points as being outliers and gives them less weight accordingly. Therefore the MM estimator is more suited for model selection in an outlier context.

The results of the DF t-tests are quite remarkable. While there is some doubt about the stationarity of real GNP and GNP per capita if one uses the OLS

A. Lucas J Journal of Econometrics 66 (I 995) 153-l 73 169

0.4 , I , I I I I

0.2

(I - P)Y, O -0.2

-0.4

-0.6

with outliers - without outliers

0

-0.8 I I I I I I -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

(I- WY,-,

Fig. 2. Added variable plot for laged differenced common stock prices.

estimator, the DF r-test obtained with the MM estimator strongly rejects the unit root hypothesis at the 5% significance level, both in model I and II. Similar findings hold for the employment series, although rejection of the unit root hypothesis is now only obtained in model II. Another result is the larger value of the test statistic for industrial production in model II. Although the unit root cannot be rejected at the 5% significance level, this series is more a borderline case than suggested by the results obtained with OLS. For the remaining series there is no difference in the conclusions that are obtained with either the OLS or the MM estimator.

Concerning the nature of the outliers in the series, most of them appear to be of the innovative type. As explained in the introduction, an isolated additive outlier is directly followed by a number of additional outliers. These ‘patches’ of outliers rarely occur in the 14 Nelson-Plosser series. Therefore one can suspect that the outliers are more often of the innovative than the additive type. One can try to visualize the outliers using the added variable plots discussed earlier. For the real GNP series such a plot reveals that the difference between the DF t-statistics from Tables 2 and 3 is due to six outliers, which are most likely of the innovative type.

A final thing to note is that the MM estimator indicates a structural break for the interest rate series after 1980. This conclusion is based on the observation that 7 of the 13 outliers occur during the last 9 years of the sample. All the standardized residuals of these outliers exceed 4.5 in absolute value. The OLS estimator on the other hand only indicates the year 1980 as being an outlier with an absolute standardized residual of about 3. It is indeed true that the interest rate series shows an increased error variance after 1980. This suggests that the MM estimator is also more suited for detecting specific model inadequacies than the OLS estimator.

170 A. Lucas J Journal of Econometrics 66 (1995) 153-173

6. Conclusions

In this paper an outlier robust alternative was considered for the well-known Dickey-Fuller r-test for testing the unit root hypothesis. This alternative test was obtained by replacing the nonrobust ordinary least squares estimator in the Dickey-Fuller procedure by the high breakdown point MM estimator. A simulation study shows that the size of this robust test is much less influenced by the occurrence of additive outliers than its nonrobust counterpart. The fact that the use of robust estimators may lead to different outcomes, both for model selection procedures and for the unit root test, is demonstrated using the macro-economic time series considered in Nelson and Plosser (1982) and extended by Schotman and van Dijk (1991). For one of these series, namely the interest rate, it appeared that the robust estimator is also more suited for detecting model inadequacy.

The results are promising, although several problems remain to be tackled. First of all the robust estimation of ARMA models could be considered as an alternative to the AR models discussed in this paper. Another option is to consider alternative robust estimators or to extend the present testing methodology to the multivariate context of cointegration. Second the interpretation of the outliers is important. Are there reasons to expect certain years not to fit the pattern set out by the bulk of the data? If there are, one can feel fairly comfortable when discarding these observations. Otherwise it might prove difficult to accept the fact that these data points are not fully taken into account when estimating the model. If there are several outliers for which one cannot find an explanation, respecifying the model might be deemed a more plausible solution than assigning less weight to several observations.

Appendix

Proof of Theorem I

Without loss of generality, assume that 0’ = 1. Note that (12) is just the definition of a stochastic integral. To prove (13) we have

T-2C$2(A~r)~f-r = T-2C($2(A~t) - c:)y:-r + T-2C~:y:_1. (A.l)

The first term on the right-hand side of (A.l) is O,( T - ‘12). The result (13) now follows from Phillips (1987, Theorem 3.1). The result in (14) is proved similarly.

To prove (15), take a first order Taylor expansion of (9) around & = 1. One obtains

0 = CVWY,)Y,- I - (6~ - l)&Wy,)y:- 1 + R&). 64.2)

A. Lucas / Journal of Econometrics 66 (1995) 153-l 73 171

Taking a first-order expansion of the numerator and denominator of (1 l), we get

CvQ’(yt - &yt- I)Y:- I = C &(AY~)Y~“- I + &k&d (A.3)

and

1 J/‘(Y~ - &yt- I)Y:- I = 1 P@Y,)Y~- 1 + W&I).

Because of the continuity of $ and t,V we have

min(IRJ&)I,IR3(&)I) < R(& - l)Cb-~l~,

and due to the Lipschitz continuity

IW,h)l G W&I - 1)2Cl~z-~13>

for some constant K. Now

(A.4)

(A.3

64.6)

T-‘CWYJY~-~ + RI&~/T ” = CT-2C$2@~,)~f-1 + R2(&Y~21”2

(1 + U#hL (A.7)

with R4(@) = R3(~)/[C$‘(Ay,)y?- I]. Using the assumption (6, - 1) = Op(T-1/2) and the continuous mapping theorem (Billingsley, 1968, Sect. 5) (A.2) to (A.7) and Phillips (1987) give the desired result.

Proof of Theorem 2

The proof of this theorem follows the same lines as the proof of Theorem 1. Again we assume without loss of generality that o2 = 1. We have

= C KY, - x3)x, - 46 - 4) + rt&,

where A is obviously defined. Note that all entries in the rows 2 to p of A are O,( T ). Defining

c = c *2(YI - d#h4 2

one also has that the (1,l) element of C is 0,(T2), while the remaining entries are O,(T ). I now proceed in a heuristic fashion and omit the proof that the remainder terms converge to zero (which in fact happens under the current assumptions) in probability. Let a be a column vector containing the cofactors of the elements in the first row of A and let el be the p-vector (LO, . . . , 0)'. One obtains


Multiplying the numerator and denominator by T -p one obtains

U1Ct4Yt - X#)Yt- 1 + o,(l) JBdS

“I = [a:T-‘C$‘(yt - x;@Y,~-~ + 0,(1)]“~ * [JB2dr]1’2’ (A-8)

by the results of Theorem 1. Here a, is the first element of a divided by TP- ‘. By including the remainder terms it can be shown that (A.8) holds with equality rather than with approximate equality.

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an outlier robust unit root test with an application to the extended nelson-plosser data

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