,ELSEVIER Journal of Econometrics 81 (1997) 7-27

JOURNAL OF Econometrics

Rank tests for unit roots

J6rg Brcitung"*, Christian Gouri6roux b

a Institute oJ" Statistics and Econometrics. tlumboldt Unirersity Berlin, Spandauer Strasse !. D-!0178 Berlin, Germany

b CREST-CEPREMAP, Paris, France

Abstract

In order to obtain exact, distributional results without imposing restrictive paramet- ric assumptions, several rank counterparts o f the Dickey-Fu l le r statistic are considered. In particular, a rank counterpart o f the score statistic is suggested which appears to have attractive theoretical properties. Assuming i.i.d, errors, an exact test is obtained for a ran- dom walk model with drift and under assumptions similar to Phillips and Pert'on (1988) the test is asymptotical ly valid. In a Monte Carlo study the rank tests are compared with their parametric counterparts. © 1997 Elsevier Science S.A.

Key word.v: Unit roots; Rank tests; Oufliers; Nonl inear models J E L classification: C22

!. Introduction

Unit-root tests play an important role in the analysis o f economic time series. The performance o f such tests depends on a number o f assumptions, which are often questionable in empirical applications. As argued by Granger and Hallman (1991) , the assumption that the generating process is linear seems too restrictive in many circumstances. In fact, the time series to be tested arc often transformed to logarithms before the unit-root test is applied. As another example, Franses and McAleer (1995) consider the B o x - C o x class o f transformations for unit- root testing. Nevertheless, such transforms are often chosen for convenience and

The research for this paper was carried out while the second author was a guest of the "Sonderforschungsbcreich 373', Humboldt University Berlin. We wish to thank the "Deutsche ForschungsgemeinschaiV (DFG) for financial support. Furthermore, the comments and suggcstion~ of the Editor and an anonymous referee are gratefully acknowledged.

0304-4076/97/$17.00 © 1997 Elsevier Science S.A. All dghls reserved PII S 0 3 0 4 - 4 0 7 6 ( 9 7 ) 0 0 0 3 I - 6

J. Breitung. C. GouridrouxlJournal o f Econometr&'s 81 (1997) 7-27

altemative transforms may be considered plausible as well. Therefore, a test which is unaffected by the choice o f the initial transformation is highly desirable.

Most unit-root tests are based on the assumption of normally distributed errors. Although the asymptotic theory applies also for a wide class o f alternate assump- tions provided that the second moment exists, the critical values for small samples are computed using normally distributed data. However, it is well known that in particular the distributions o f financial data exhibit much fatter tails than is ex- pected by assuming normality and in some cases not even the : : cond moments seem to exist.

Another reason to question the use o f parametric unit-root tests is that in many cases the economic development reveals evidence for a structural change. In a random walk model the shocks have a persistent effect so that structural breaks can be modeled allowing for outlying observations (Perron, 1989). Since rank tests reduce the influence of outlying observations, they are expected to perform better than parametric tests in such situations.

To overcome these difficulties it is interesting to consider robust versions o f standard unit-root tests, such as tests based on the ranks o f the observations. Using ranks instead o f the original observations has two major advantages over parametric procedures. First, ranks are maximal invariants under monotonic trans- formations of the data and, thus, their distribution does not change if a monotonic transformation is applied to the original data. Second, ranks are invariant with respect to the distribution if they are applied to a sequence of exchangeable obser- vations. As a conseq,tence, rank tests are robust against a wide class o f "outliers'. In this paper both advantages are exploited by two different rank procedures.

The first test is the natural rank counterpart o f the conventional Dickey-Ful ler test (1979). The observations o f the series arc replaced by their ranks and the first differences of the ranks are regressed on the sequence o f ranks and a con- stant. This test was suggested by Granger and Hallman (1991) and its asymptotic properties are considered in Section 2. Clearly, the test is invariant with respect to a monotonic transformation of the data. However, since under the null the ob- servations are not exchangeable, the null distribution depends on the distribution of the errors. Moreover, it is not clear how to generalise such a test to a random walk model with dritt a~.,~ correlated errors.

Campbell and Dufour (1995) suggest a different approach. Let {Yt}t~{t ..... r} denote the observed time series o f sample size T. Instead of the original series the products T t - - O ' t - y t - I ) Y t - t are ranked and it is shown that for series generated by the simple random walk model yt = y t - t + et, where e.t is a strong white noise series symmetrically distributed around zero, the Wilcoxon test for a zero median can be applied to xt. In contrast to the ranked Dickey-Ful ler test this test is invariant w:th respect to the error distribution whereas a monotonic transformation of the time series will affect the null distribution, in general. Again, such a test has no straightforward generalisation to random walk models with drift and correlated e r r o r s .

J. Breitung, C. GouriOrouxlJournal of Econornetric~ 81 (1997) 7-27 9

In Section 3 a version o f a rank test for a unit-root is suggested which can also be applied for a random walk with drift and correlated errors. In the former case the exact finite sample distribution can be computed while in the latter case an asymptot ic correction similar to the one suggested by Phillips and Perron (1988) can be used. This test is invariant with respect to the error distribution but is affected by a monotonic t ransformation o f the data, in general. Moreover , the test can be seen to be robust against outliers and a structural break in the intercept o f the trend function.

To assess the small sample properties o f the test, several Monte Carlo experi- ments are performed. The results, which are presented in Section 4, suggest that the ranked score test performs well in some situations but lacks power in others. Section 5 concludes.

2. The r anked D i c k e y - F u l l e r tes t

First we consider the ranked Dickey -Fu l l e r ( rDF) test as suggested by Granger and Hal lman (1991) . We assume the existence o f a monotonic function, such that zt = h ( y t ) is generated by a first-order autoregressive model

z t = o ~ z t - z -t-~t, t = l , 2 . . . . T,

where ~:t is an i.i.d, white noise sequence (i.e. "strong white no i se ' ) with p . d . f . f . It is important to consider the identification o f the two functional parameters (h , f ) and the scalar parameter ~. Assume that there exist two representat ions h ( y t ) -- ~ h ( y t - t )+~.t and h * ( y t ) : ~ * h * ( y t - t ) + ~ , where h, h* are two monotonic functions and ~t, ~-t are strong white noise sequences. I f h and h* are differentiable we deduce that

f [ h ( y , ) - ~ h ( y , - t )] Oy

I h*ty,)l vy,, y,_, -- f * [ h * ( y t ) - ~ * h * ( y t - i ) ] . ~y ,

by considering the form o f the conditional p.d.f, o f Yt given its past. It follows that h and h* are in a one to one affine relationship and ~*----~. 1

Under the null hypothesis it is assumed that the t ransformed time series zt is generated by a random walk process with ~ - - 1 , i.e.

H0 - - { 3h monotonic, ~t i.i.d.: h ( y t ) = h ( y t - i ) ÷ ~t},

t A possible way to show this result is to make use of the fact that for a function f ( y -- 0L, c) the partial derivatives with respec:, to x and y are proportional. To retain proportional derivatives of an alternative functional representation g [ h ( y ) - ~*h(r)], it follows that the derivativu of h must he constant and ,, = ,,*.

1 0 J. Breitun,q. C. G o u r i k r o u x l J o u r n a i ofl Econometr ics 81 (1997) 7 -27

while the alternative is

Hi = { 3h monotonic, I~1 < 1,e, i.i.d.: h ( y t ) = o t h ( y , _ l ) + et}.

From the previous considerat ions it is seen that ~t is an identified parameter and, consequent ly, Ho is an identifiable hypothesis on ~.

A testing procedure invariant with respect to monotonic t ransformations is nec- essari ly based on the ranks o f h(y t ) or, equivalently, on the ranks o f Yt:

r,. r -- Rank[o f y, among Yo . . . . . Yr] T + 2

,

An ordinary Dickey-Fu l l e r t-test can be applied to the sequence o f ranks with

t~=o~- I ( c , - c o ) / V 7 o ,

where @ )--~tr_o I rt+j.rrt.r and A2 I - - a , - - ( T - - 1 ) - - ~- '~ tT_! [gt, T - - ( C l / C O ) ~ - - I . T ] 2. This version is sl ightly different from the test used by Granger and Halhnan (1991) , who include a constant in the regression in order to correct for the mean o f the ranks. The difference is, however , negligible even in fairly small samples.

Since the ranks o f ( y , ) are the same as the ranks o f (z,) the distributional properties o f the test statistic may be deduced from the ones o f the strong random walk. The asymptot ic theory for the distribution o f the parametr ic Dickey-Fu l l e r ( D F ) test is based on the wel l -known fact that

T-1/2z[.r I ==¢. trW(a) , (z)

where a 2 = lim T - I ~_,E(e~), O~<a~< 1, [-] represents the integer part, '=~-" means weak convergence o f the associated probabil i ty measure and W(a) stands for the Brownian motion defined on [0, 1].

Unfor tunate ly , this asymptot ic result does not carry over to the sequence o f ranks. From (1) we deduce that

T-lr{~rl . r = T - l ~ ~(z, <Z[ari) t

( - b ' ) = r - ' E z, <

' ) [ ; ' '1

I' I' =~, R(a ) = l t [ a W ( u ) < ~ W ( a ) ] d u -- 11[ W(u) < W(a ) l du.

Therefore, the limit ranks define a stochastic process indexed by a E [0, 1], such that R(a ) is the occupat ion t ime .e,f the set ] - e~, W(a) ] by the Brownian motion.

J. Breitung. C. Gouri~roux lJournal of Econometrics 81 (1997) 7-27 11

0.0 0.1 02

I i I t [ I | • ! *

0.3 0.4 0.5 0.6 0.7

RIo)

O.8 03 1.0

Fig. !. p.d.f, o f the arcsin~ distribution.

We may now easily deduce some distributional properties o f this process. The first property is well known and may be found for instance in Revuz and Yor (1991).

Property 1. The distribution o f R(O) = f2 O[W(u) < 0 ] du is the arcsine distribu- tion with c.d.f. 2/~z a r c s i n [ ~ ] , and p.d.f. [ l t ~ x / ! - R ( O ) ] - t

This distribution equals the beta distribution with parameters (0.5, 0.5) and its p.d.f, is depicted in Fig. 1. Note that as R(0) tends to zero or one the density goes to infinity.

With this result we can deduce the marginal distribution o f R(a), which is stated in the following property.

Propeay 2. The distribution o f R(a) is the same as the distribution o f aRl(O)+ (! --a)Re(O), where Ri(O) and R2(O) are two independent random variables with an are'sine distribution.

Proof. We have

R(a) = Z i U [W(u) < W(a)] du

f0 ° Z' = U[W(u)<W(a) ]du + n [ W ( u ) < W ( a ) ] d u .

{ W ( u ) < W(a), u < a } and { W ( u ) < W(a), u > a } are independent events since the Brownian motion is with independent increments. Then we have with obvious notation R ( a ) = R l ( a ) + R2(a), where Rt(a) and R2(a) are independent random

1 2 J. Breitun.q. C. Gouridroux l Journal o f Econometrics 81 (1997) 7-27

variables. Now

Rl(a) :=- fO a

~J~o a

"f° "f°

~[W(u)<W(a)]du

I ] [ W ( a ) - W ( u ) > O ] d u

~ [ W ( a - u ) > O ] du

[ W(u) > O] du

d fo~ = a U [ W ( u ) > O ] d u

d aR(O) ,

where d stands for the equality in distribution. []

In Appendix A some properties o f the distribution of R ( a ) are considered and Fig. 2 presents the p.d.f, for selected values of a. In order to assess the impact o f these properties on the null distribution of the ranked Dickey-Ful ler test, a Monte Carlo simulation was conducted. For the sample sizes T - - 5 0 , T = 100 and T = 500 we generate 10,000 artificial random walk paths with different error distributions and zt = yr. To represent skewed and heavy-tailed distributions we generate errors having a (centered) Z 2 distribution with one degree o f freedom and a t distribution with two degrees o f freedom. The critical values o f a Dickey- Fuller test with a constant term and a significance level o f 0.05 as tabulated in Fuller (1976) were used for both the parametric and the ranked test,

From the results presented in Table 1 it emerges that the actual size o f the ranked Dickey-Ful ler test depends on the error distribution in finite samples, while the asymptotic distribution of the ranks is independent o f the distribution of the errors, it is seen that the large sample distribution appears to be (slightly) different from the one derived by Dickey and Fuller (1979). Even for T - - 5 0 0 , the application of Fuiler 's (1976) critical values yields a significant over-rejection o f the null hypotb.esis, when the errors are normally distributed. This was also observed by Granger and Hallman (1991).

It is important to note that in our simulations we have assumed h to be known. As was shown by Granger and Hallman (1991), the parametric Dickey-Fuller test applied to the raw series may perfonn poorly if the series is generated by a nonlinear unit root process. In particular, a alisspecification of h affects the rate

13

a:0.!

Z

0 . q , i . t . i . i • | . m . t . , .

o00 0.1 0.2 0.5 0.4 0.5 0.6 03 08 0.9 i.O

o:0.25

0

0 . i ~ , . , . , . ~ . t I I I I I I I

00.0 O.I 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9

o :0 .4

0

0

t t

o. °O.O 0.1

• I. Breitung. C G o u r i 6 r o a x l J o u r n a l oJ Econometr ics 81 ( 1 9 9 7 ) 7 -27

, t i f i I i i * I t I J

02 n3 ,~4 o.~ 06 07 o~ ~.9 io

0:0.5

ol , °

Oi . • • i . , . i t . • i 2 . r , .w . ,, .

60. 0 O" ! 0"2 O~ 014 O~ 0--6 0" 7 018 0"9 ? *0

Fig. 2. p.d.f, o f R(a) for selected values o f a.

Table i Empirical sizes for the DF and the rDF test

T = 50 T = 100 T = 500

f DF rDF DF rDF DF rDF

. I "(0, I ) 0.055 0.054 0.053 0.059 0.048 0.075 X 2 ( ! ) 0.042 0.033 0.045 0.044 0.046 0.065 t (2) 0.061 0.036 0.055 0.042 0.052 0.052

Note: Rejection frequencies computed from 10,000 replications o f the random walk model without drift. DF indicates the parametric Dickey-Ful ler h, test including a constant term and rDF signifies the ranked Dickey-Fuller test for the mean adjusted ranks. ~2( 1 ) and t(2) denote Z 2 and t distributed errors with one and two degrees o f freedom, respectively.

14 J. Breitunt.l, C. Gouri6rouxlJournal of Econometrics 81 (1997) 7-27

o f divergence o f the parametric D ickey-Fu l l e r test statistic, so that the test may fail to be consistent.

To conclude, the analysis shows that the sequence o f ranks on a random walk behave asymptot ica l ly as a weighted average o f two independent random vari- ables with arcsine distribution. Since this distribution does not depend on nui- sance parameters, an asymptot ic test can be constructed using the rank version o f the D ickey -Fu l l e r test. However , such a test possesses some undesirable prop- erties. First, the results o f the Monte Carlo simulations suggest that the limiting behaviour o f the test may provide a poor approximat ion to the small sample properties. Second, there is no simple way to extend the test procedure to random walk models with drift and correlated errors. In the next section we therefore consider an alternative test procedure.

3. The r a n k e d s c o r e tes t

In this section we consider a rank counterpart o f the score statistic suggested by Sehmidt and Phillips (1992) . This test can be used to test the "difference stat ionary model" given by

Yt = b + ~ Y t - I + £,t with ~ = 1 , (2)

against the "trend stationary model '

y t = c + b t + ~ Y t - I +~;t with [ ~ l < l . (3)

In what follows, we assume that the errors are independent and identically dis- tributed with E ( e , t ) = 0 and c . d . f . F . As will be discussed below, it is possible to relax this assumption in order to al low for heteroskedastic or serially correlated errors. In these cases, however, it is no longer possible to obtain the exact null distribution.

As shown by Schmidt and Phillips (1992) the score (or Lagrange Mult ipl ier) principle gives rise to the fol lowing statistic 2

T ~ T = ~ t ~ 2 x t S t - 1 (4)

2There are two possible tbrms of the score statistic. Here we use the form (15B) of Schmidt and Phillips (1992). The form (15A) which they prefer is not appropriate to construct a rank statistic. The properties of the test according to (4) is also considered in Schmidt and Lee (1991).

J. Breitunff. C. Gouri~rouxlJournal o f Econometrics 81 (1997) 7-27 15

where

xt = A y , -- b,

T b= r-t ay , = ( y r - y0)/r ,

t = |

t

s , = xi. i = !

Under the null hypothesis o f a random walk with drift, - - ( 2 T ~ r ) - I is asymp-

totically distributed as f0 t W(a) 2 da, where W ( a ) = W ( a ) - a W ( ! ) represents the standard Brownian bridge (cf. Schmidt and Lee, 1991 ).

Letting

T + l ~ , r = Rank[of A y t among A y t . . . . . A y T ] 2 "

t S R

s-~l

a rank counterpart o f the score statistic is

~ = y~~L2Pt 'TS f - I 'T (5) T R 2

Note that the ranks o f the observations are not affected by subtracting the mean of the series so that the mean of the differences b can be neglected.

The numerator of (5) does not depend on the data but is merely a function of T. Indeed

T T i - - I

t = 2 i = 2 j = l

so that the expression is the sum o f the products given by all non-redundant combinations o f the ranks. This quantity does not change by permutation o f the ranks ~Lr,--- ,~r,r and, thus,

T T i - -1

rt .rSt n - , , r "- ~ ~ i j , t = 2 i - - - 2 j = l

which is a function of T only. As a consequence, it is sufficient to consider the statistic

T ._ t s R X2, I T ~ , ,.T ,

t=l

1 6 J. Breitun.q. C. Gour i~roux lJourna l o f Econometrics 81 (1997) 7 -27

where we add R 2 (St, r ) in order to symmetr i se the expression. As in the previous subsection, the ranked score statistic is defined by analogy with some standard statistic based on the y t ' s themselves. The test statistic only depends on the ranks o f the observed differences and, thus, it is invariant with respect to monotonic t ransformations o f Ayt. Similarly, the null hypothesis can be shown to have such an invariance property. Let us consider the strong random walk hypothesis cor- responding to (2):

H~) - -{3 strong white noise ~ such that: Ayt =e t } ,

where et is not necessari ly centered. It is immedia te ly seen that H~} satisfies the invariance property and, as a consequence, the ranked score statistic has a null distribution independent o f F , even in finite samples.

It should be noted that the hypothesis H~} is strictly included in the hypothesis H0 considered in the previous section. Therefore, the two rank tests are associated with different implicit null hypotheses. In the fol lowing theorem, the l imiting distribution o f the test statistic is presented and several remarks are made which may be useful for practical applications.

Theorem. Le t {~:1 . . . . . or} be a sequence o f independently anti identically dis- tr ibuted random variables. Then, under H~ atut T--> c~:

i ~ ! ~ z -~-"~'AT =~ ~ W(a ) 2 da

where W ( a ) denotes the s tandard B r o w n k m bridge on [0, 1].

P r o o f Let the normal ized ranks be defined as gt.r = T-IF,,r. We have

___ (cR x2 = T 2 t = l t-----I s = l t = ! s----I

Furthermore, let ~ . r be a sequence o f independent random variables with

P { ~ t . T = T - I [ j - - ( T + I ) / 2 ] } = T - I for j = 1 . . . . . T and t - - 1 . . . . ,T.

Then, the distribution o f the normalized ranks (Qt.r . . . . . gr . r ) is asymptot ical ly

equivalent to the distribution o f (~t.r . . . . . ~r.r) conditional on ~,r=l ¢t.r = 0, since the probabil i ty o f equali ty o f two variables ~t.r and es.r (t ~ s) becomes asymp- totically negligible. Asymptot ica l ly , ?~LT . . . . . c,r.r tend to independent ly uniformly U [ - 0 . 5 , 0 . 5 ] distributed random variables and, thus, T -I/2 x-,l,rl z.-,=l ~,.r :=> ( 12)-! /2

W(a) . Since the distribution o f T -1/2 ~;-.(,,rl z_.,-_, ~:.r is asymptot ica l ly the same as the

distribution o f T - !/2 ~-,[,r] z_-,=1 ¢t.r condit ioned by

T T - I /2 ~ et.r = 0,

t = l

J. Breitunff, C. GouridrouxlJournai o f Econometrics 81 (1997) 7-27 17

we have

taT] i/ '2[ T-I~2 ~ , O,.r ==> ( 1 2 ) - W ( a ) - a W ( 1 ) ] = ( 1 2 ) - t / 2 - W ( a ) .

t----!

Finally, it follows that

T-2~-'~ Os, r =~ (12) - I W ( a ) 2 da . [] t = l

R e m a r k A. This result suggests to use the test statistic

T t 2 r ( u n i ) = T -2 ~ ~ ( v / ~ 0 s . r ) 2, (6 )

t = l s = l

where Ot, r = T - I~ t , r is the normalized rank. Under the null hypothesis 2 r con- verges to f0 t W ( a ) 2 da . Ill Appendix B critical values for 2 r are given.

R e m a r k B. From the literature on nonparametr ic tests it is known that the test might be improved by using nonlinear transformations o f the ranks such as the "inverse normal scores" ( INS) transformation:

~,* = q~-I(Qt, r + 0.5), (7 )

where @(-) is the c.d.f, o f the standard normal distribution (cf. van der Waerden, 1952). Let us assume that we know a priori that (2) is satisfied with a Gaussian white noise, i.e. that we know that F ( z ) is the standard normal c.d.f.. Then, the idea behind the INS transformation is that

O,.r = FT(/lY, ) r + 1 ~ FT( A)'t ) -- 0.5, 2 T

where ~!__z~ denotes the empirical distribution function o f Ayt given by F r ( 6 ) = T -~ O(Ayt <~6). Since Fr converges pointwise to the distribution o f the errors, F ( z ) , the transformation F - ~ [Ot, r + 0.5] renders a series with similar dis- tributional properties as Ayt. Therefore, for normally distributed errors the INS transformation renders a test which behaves similar to the parametric test. Criti- cal values for the modified statistic can be found in Appendix B. O f course, this procedure may be extended by introducing any other transformation in order to approximate the inverse o f the error distribution as closely as possible. However, since in most applications the error distribution is unknown such a transformation is difficult to choose.

R e m a r k C. As usual in the literature on rank tests, the i.i.d, assumption on the errors may be relaxed by al lowing for 'exchangeable" errors, i.e., F(e l . . . . . ~ r ) - -

18 J. Breituml, C. GouriOrouxlJournal o f Econometrics 81 (1997) 7-27

F(ea, . . . . . ear) for all pe rmuta t ions d l , . . . , d r , where F ( - ) is the jo in t dis t r ibut ion funct ion o f el . . . . . e r (e.g. McCabe , 1989). Th i s a s sumpt ion wou ld a l low for the inc lus ion o f a s tochast ic drif t term. However , s ince the dis t r ibut ion is not affected b y the w a y the drif t t e rm is specified, such a genera l i sa t ion is o f no pract ical use here.

R e m a r k D. It is easy to see that the test is robus t aga ins t a s t ructural break in the intercept o f the t rend function. For the pa ramet r i c vers ion o f the test this p rope r ty was noted by A m s l e r and Lee (1995) . A s s u m e that the value o f the pa rame te r c in mode l (3 ) changes to c + t~ at t ime 1 < To < T. Then, under the null hypo thes i s we have Ayt -- b + et for t ~ To and Ayro -" b + 6 + er,,. I f t$ < c~ it is not difficult to ver i fy that the l imit ing dis t r ibut ion g iven in the theorem is the same for all - -oo < 6 < + o~. Hence the test m a y a l low for a s tructural break in c.

S imi la r ly it can be seen that the l imi t ing dis t r ibut ion o f the test does not change in the p resence o f the "additive" and the "innovative" out l iers (see, e.g., Tsay , 1988, for a d i scuss ion o f this out l ier c lass i f icat ion) .

R e m a r k E. It is poss ib le to relax the a s sumpt ion o f i.i.d, e r rors in o rder to a l low for a wide class o f he te roskedas t ic and ser ia l ly corre la ted errors. However , in such cases on ly a sympto t i c results are avai lable . Since the sequence Pt.r + 0.5 conve rges to F ( A y t ) as T--~ cx~, the no rma l i zed ranks can be seen as a mono ton ic m a p p i n g o f the differences o f the original series on to [0, I ]. Unde r the

a s sumpt ions g iven in Phi l l ips ( 1 9 8 7 ) it can be shown that 2 r =~ tr2 e f 2 W ( a ) 2 d a , where o~ = l i m r _ . ~ E [ T - t ¢S n r , r )2] . A val id es t imate for a 2 can be obta ined by us ing sample au tocovar i ances and app ly ing a N e w e y - W e s t we igh t ing scheme, for example .

4. Small sample properties

In this sect ion the resul ts o f several Mon te Car lo s imula t ions are p resen ted in o rder to compare the small s ample proper t ies o f the rank stat is t ics with their para- metr ic counterpar ts . The first test o f the compar i son is the (pa rame t r i c ) D i c k e y - Ful ler t -s tat is t ic ( D F ) compu ted f rom a first o rder au toregress ion wi th a l inear t ime t rend as g iven in (3) . Second, the score- type statistic sugges ted by Schmid t and Phil l ips ( S P ) is c o m p u t e d as g iven in (4) . For the DF statistic, the cri t ical va lues o f Ful ler ( 1 9 7 6 ) are used and for the SP test, we app ty the cri t ical va lues for the INS t rans format ion given in Tab le 6 o f the append ix to - - ( 2 T ~ r ) - ! .

T w o vers ions o f the score- type rank statistic are considered. First, we compu te ) . r (un i ) as in (6) . Second, the ranks are t r ans fo rmed using the INS t rans format ion and are used instead o f the normal i zed ranks. The resul t ing test statist ic is deno ted

3'. Breituno, C. Gouri~rouxlJournal o f Econometrics 81 (1997) 7-27 19

by ).T(~rNS). For these statistics, the critical values presented in Appendix B are used.

For all experiments the rejection frequencies are computed from 5000 Monte Carlo replications and the nominal significance level for the tests is 0.05.

4.1. S m a l l s a m p l e p r o p e r t i e s f o r d i f ferent error d i s t r ibu t ions

In the first Monte Carlo experiment we compute the rejection frequencies for the case that zt -- .vt is a random walk with drift letting • = 1 (the null hypothesis) , as well as for the trend stationary model with ~ - - 0 . 9 and ~ = 0.8 ( the alternative). Three different error distributions are considered. First the normal distribution for which the parametric tests are asymptot ical ly optimal. As an example for a skewed distribution we compute errors generated by a (centered) X 2 distribution wi~h two degrees o f freedom, and t distributed errors with two degrees o f freedom are used to represent a heavy tailed distribution.

To investigate the relative performance in the presence o f outliers we con- taminate the data with normally distributed errors by an additive and an inno- vative outlier at To = T/2 of size 5¢. The exper iment with an additive outlier is indicated by A D D ( T / 2 , 5 ¢ ) and an innovative outlier in the data is indicated by INNO( T/2, 5o).

Tables 2 and 3 present the results for T = 50 and T = 100, respectively. It turns out that the score type test o f Schmidt and Phillips (1992) is more powerful than the Dickey-Ful le r test in small samples. Under normali ty, the power o f the rank tests is similar to their parametric counterpart. For alternate distributions a different picture emerges. While the Dickey-Fu l le r test turns out to be quite robust against violations o f the normali ty assumption, the parametric Schmidt-Phi l l ips statistic tends to be conservative.

In case o f additive outliers the parametric tests possess a substantial size bias while the rank tests turn out to be much more robust. By contrast, in case o f innovative outliers all tests have empirical sizes close to the nominal ones. The reason for this finding is that a random walk with an additive outlier can be seen as a random walk with two consecutive innovative outliers o f the same size but different sign. Accordingly, the errors at these t ime periods are negat ively corre- lated and imply a negative bias in the est imation o f the autoregressive coefficient.

With respect to the performance under the alternative, the rank statistic reveals a substantial loss o f power, in particular under heavy tailed distributions. The At ( INS) statistic performs better than ~.r(uni) but a considerable loss o f power remains.

4.2. S m a l l s a m p l e p r o p e r t i e s f o r s o m e non l inear t r a n s f o r m a t i o n s

The data are generated such that z t - - - -h(y t ) is a random walk without a drift. We apply the (monotonic) transformations zt = ( y t ) 3, zt = ( y t ) !/3, zt = ln(y t ) , and z t - - t an(y t ) . The increments o f zt are normal ly distributed.

20 J. Breitzozq. C. GouridrouxlJournal o/ 'Econometrics 81 (1997) 7--27

Table 2 Rejection frequencies ( F = 50)

~c DF SP 2 r ( u n i ) ,;.z'(INS)

N(0 , 1 ) ! .00 0.056 0.050 0.048 0.049 0.90 0.08 ! 0.097 0.097 0.092 0.80 0.190 0.230 0.200 0.220

Z2(2) ! .00 0.053 0.042 0.054 0.053 0.90 0.077 0.090 0.072 0.074 0.90 O. 189 0.252 O. 120 O. 127

z(2) i .00 0.053 0.045 0.049 0.047 0.90 0.079 0.088 0.062 0.074 0.80 0.174 0.250 0.101 0.137

A D D ( 7 7 2 . 5 a ) 1.00 0.315 0.227 0.065 0.085 0.90 0.445 0.360 0.122 0. ! 51 0.80 0.684 0.603 0.247 0.3 ! 0

INNO( T/2, 5a ) ! .00 0.049 0.05 i 0.051 0.045 0.90 0.082 0.105 0.084 0.087 0.80 0.214 0.257 0.158 0.181

Note: Rejection frequencies for the random walk with drift under different error distributions. Z2(2) denotes the central Z 2 distribution with two degrees o f freedom and t (2) indicates the t distribution w,th two degrees o f freedom. "DF" denotes the Dickey-Ful ler t-statistic and "SP" is the Schmidt - Phillips statistic given in (4) .

From the results in Table 4 it appears that the parametric tests suffer substan- tially from the misspecif ication o f the process as a linear random walk, while the rank tests perform much better. As an extreme case, the parametric tests reject the unit-root hypothesis for the logarithmic transformation in more than 70% o f the cases whi le the actual s izes o f the rank tests are close to the nominal ones.

On a less rigorous level the performance o f the tests can be understood using a linear approximation for Yt = h - t (zt) around z t - t :

Yt ~ Y t - t + [ d h ( z z - t ) / d z t - I ] - l e , t

and, thus, the approximation suggests that the first differences o f the observed series behave approximately like conditionally heteroskedastic white noise. For the rank tests to be valid it is required that the differences behave like a strong white noise process. Although, in general this assumption is not satisfied here, it

J. Breitung. C. Gouridroux I Journal o f Econmnetrics 81 (1997) 7-27 21

Table 3 Rejection frequencies (T = 100)

DF SP ,;,r(uni) )-TONS)

N(0, ~ ) 1.00 0.047 0.044 0.048 0.048 0.90 0.186 0.227 0.198 0.216 0.80 0.641 0.623 0.5 i 0 0.596

z2(2) 1.00 0.050 0.044 0,048 0.050 0.90 0.192 0.236 0,099 0.092 0.80 0.660 0.663 0,208 0.228

t(2) 1.00 0.055 0.04l 0.052 0.053 0.90 0.177 0.256 0.072 0.110 0.80 0.665 0.725 0. i 29 0.241

ADD( 7"/2, 5a ) 1.00 0.174 0.145 0.057 0.066 0.90 0.509 0.543 0.263 0.338 0.80 0.927 0.890 0.595 0.725

INNO( ?'/2, 5a ) 1.00 0.049 0.047 0.047 0.041 0.90 0. ! 90 0.236 0. ! 68 0. ! 94 0.80 0.68 ! 0.678 0.413 0.521

Note: See Table 2.

appears that for a wide range o f possible transformations the sizes o f the rank tests are affected only marginally.

4.3. The case o f correlated errors

It is wel l known that parametric unit root tests run into serious problems, i f the errors are generated by a M A process with a root close to one (Schwert 1989, Agiakloglou and Newbold 1992). In particular the Phil l ips-Perron test has been shown to suffer from a tremendous size bias in this case. Thus, it is interesting to consider the performance o f the rank test in a model with M A ( 1 ) errors.

In order to correct for nuisance parameters both the parametric and the rank test employ the N e w e y - W e s t weighting scheme to the covariances as suggested by Phillips (1987) and Phillips and Perron (1988) . In analogy to the parametric tests the "long-run variance' %2 is estimated using the residuals from a regression o f the ranks ~t.r on S~_ i, r- Table 5 presents the rejection frequencies o f the ranked score statistic for 0c = ! (i.e. the actual s ize) and ~ = 0.8 (i.e. the empirical power)

22 J. Brei tung, C. G o u r i ~ r o u x l J o u r n a l o f Econometric.~ 81 ( 1 9 9 7 ) 7 - 2 7

Table 4 Empirical sizes for some nonlinear transformations

h(y t ) DF SP ,;.r(uni) 2T(INS)

zt = O't )3 0.402 0.21 ! 0.049 0.062 zt = (Y t ) i /3 O. i 59 0. i 96 0.058 0.081 :t = ln(yt ) 0.848 0.754 0.05 ! 0.082 =t ---- tan(yt ) 0.421 0.172 0.047 0.067

Note: The series are generated by zt = h - t ( y , ) and yt = y t - I + ct, where ~:t ~-- N(0, I ). The nominal size is 0.05 and the sample size is T = I00. The rejection frequencies are based on 5000 replications each. The test statistics are labeled as in Table !.

Table 5 Rejection frequencies for MA( 1 ) errors

fl DF-PP SP ,;.T(uni ) ~;.T(INS)

= 1.0 (size) -- 0.50 0.0 ! 3 0.007 0.005 0.004

0.00 0.063 0.037 0.035 0.036 0.20 0.218 0.104 0.107 0.111 0.40 0.594 0.289 0.288 0.314 0.60 0.951 0.644 0.595 0.663 0.80 1.000 0.943 0.846 0.925

~. = 0.8 (power) --0.50 0.241 0.235 O. 179 0.219

O.0O 0.691 0.575 0.46 i 0.564 0.20 0.975 0.779 0.650 0.769 0.40 1.000 0.9 i 5 0.789 0.899 0.60 i .000 0.970 0.866 0.949 0.80 ! .000 0.984 0.900 0.968

Note: Rejection frequencies from 5000 replications of model ( ! ) with MA( 1 ) errors generated by rt = ut - - f lu t - I , where ut is generated by a Gaussian white noise process. The sample size is F- - IO0 and the nominal significance level is 0.05.

a n d u s i n g n o r m a l l y d i s t r i b u t e d e r ro r s . T h e t r u n c a t i o n l ag f o r t h e P h i l l i p s - P e r r o n

i ~ p c o f c o r r e c t i o n is e i g h t . I," t u r n s o u t t h a t t h e P h i l l i p s - P e r r o n t e s t ( D F - P P ) is

s e r i o u s l y b i a s e d e v e n fo r m o d e r a t e v a l u e s o f t h e m o v i n g a v e r a g e p a r a m e t e r /L

T e S c h m i d t - P h i l l i p s t e s t p e r f o r m s s l i g h t l y b e t t e r a l t h o u g h it s t i l l h a s a s e v e r e

s i z e d i s t o r t i o n f o r f l>~0.4 . W i t h r e s p e c t to t h e s i ze , t h e r a n k v e r s i o n s o f t h e

S c h m i d t - P h i l l i p s t e s t p e r f o r m s i m i l a r to t h e p a r a m e t r i c c o u n t e r p a r t .

F o r t r e n d s t a t i o n a r y a l t e r n a t i v e s w i t h at = 0.8 , t h e p o w e r o f ',he r a n k s t a t i s t i c

, ;~r(INS) is s o m e w h a t s m a l l e r t h a n t h e p a r a m e t r i c c o u n t e r p a r t , b u l t h e d i f f e r e n c e s

a r e n o t v e r y l a r g e f o r m o d e r a t e v a l u e s o f ft. In s u m m a r y , e v e n in t h e c a s e o f

J. BreRun~l. C. GouridrouxlJournal o f Econometrics 81 (1997) 7-27 23

a known transform h, the rank tests appear to perform roughly similar to the parametric tests so that these tests should be used with care whenever the errors have a substantial negative correlation.

5. Concluding remarks

In this paper several aspects o f rank tests are addressed. First, we consider a rank version o f the Dickey-Ful le r test suggested by Granger and Hal lman (1991) to test the hypothesis that the series is a monotonic transformation o f a random walk. It is shown, that the sequence o f ranks built f rom the levels o f the t ime series does not converge to a Brownian motion. Consequently, the asymp- totic properties o f the ranked Dickey-Ful le r test are different from its parametric counterpart.

Second, for testing the null hypothesis o f a strong random walk with drift a new rank test is suggested which can be seen as a ranked counterpart o f the score test suggested by Schmidt and Phillips (1992). An asymptotic correction is suggested in order to account for serially correlated errors. We also consider a modification o f the rank statistic by t ransforming the ranks using the inverse o f the normal c.d.f. This approach, known as "inverse normal scores ' t ransformation, seems to improve the power o f the test in many cases.

Concluding the results o f the Monte Carlo experiments it turns out that i f the transform h is known and the errors are i.i.d, there may be a substantial loss in power when using ranks instead o f the original observations. While the size o f the rank test is robust against departures f rom the normali ty assumption, the p o w e r o f the test may deteriorate considerably when the error distribution is skewed or heavy tailed. In order to improve the robustness with respect to the power, alternative transformations o f the ranks may be useful. For example, applying the inverse o f a heavy tailed distribution, such as the t-distribution with a small number o f degrees o f freedom, may help to improve the power o f the rank test.

On the other hand, when the monotonic t ransform h is unknown or the errors are correlated, the parametric tests may perform poorly and the ranked score tests are attractive competi tors to standard testing procedures. Accordingly, rank tests may serve as a useful tool for specifying nonlinear t ime-series processes. Once the unit-root hypothesis is accepted for the transformed process z t - h(yt), it is interesting to estimate the function h from the data. For a Markov process o f order one we have E[h(y t ) - h(y t_l ) ly t_l ,y t_2, . . . ]=O and, thus, the function h is an eigenfunction o f the transition operator h(-) --~ E[h(yt ) [y t_ !, Yt-2,. . .] associated with a unit eigenvalue (see Conley et ai., 1995, for a related problem arising in the estimation o f stochastic differential equations), Such problems, however, are beyond the scope o f this paper and arc left for research.

24 J. Breituny. C. Gouru;rouxlJournal of Econometrics 81 (1997) 7-27

A p p e n d i x A. T h e distr ibut ion o f R ( a )

! A. 1. Symmetry with respect to

By the symmet ry o f the arcsine distribution we have

R(a) d aRl(O) + (1 -- a )R2(0)

a_. a l l -- R l (0 ) ] + (1 -- a ) [ l -- R2(0)]

=d ! -- [ aR l (0 ) + (1 -- a)R2(0)]

d= I -- R l ( a ) ,

which is the required result.

A.2. Form o f the p.d.f, o f R(a )

We get

f ( y ) ~ _ ~----fiP{R(a)E [y, y + d y ] }

= ~P{aRl (O) - t - (1 - -a )R2(O)E[y ,y + dy]}

= l_.]_P(dy R I ( O , E [y -- (1-- "Y- - ( l - -a 'R2(O'+d~Ya]}a

min[ I,y/( 1 --a)] I 1 1 1 dz

dmax[O,(y--a)/ll-a)] 7t2 x/~ lx/T -Z-~-z V ~'v-(l-a):a ~ l - ,v--tl--a).-.aa

The plots o f f ( y ) presented in Fig. 2 are computed by numerical integration. Let us assume a < ½. The p.d.f, takes different forms depending on the position

o f the a rgument y. Three eases have to be considered: y < a , a < y < ! - - a , and y > ! - - a , where the third one is the symmetr ic counterpart o f the first one f rom Section A. 1.

Case: y < a. We get

y/(t-a) 1 1 1 ! f ( y ) = dz .

.,o a, 2 v ' i V/Y--(l--a)..- ~/1- v--(l--a)z U U

lfo' f ( Y ) = 1 I 1

v / - f fv" i - u V/£~a_ _ u ~ + u du.

Let us introduce the change o f variable: u = ( 1 - - a ) z / y . W e obtain

J. Breitunff. C. Gour i~roux l Journa i o f Econometr ics 81 (1997) 7 - 2 7 25

Case: a < y < 1 -- a. We get

[ y/(n -a ) I 1 f ( y ) =

.J(v-a),'tl-a) a~z2 v / z ~

1

¢ ¢ y--( I --a ~z 1 y - ( I --a )z a a

dz.

Let us introduce the change of variable: z - - ~i-a + uTs-J-~". Then,

I__ f ' l l l f ( y ) - -

a ~ J0 v ~ v q - ,, d , , + ( y - a)la j ( l - y ) l a - u

du.

A.3. Behav iour o f the p . d . f in a n e i o h b o u r h o o d o f y = a

It is directly seen that if y is close to a we get

1 f t 1 1 ! f ( Y ) Jo du, arc2 u ~ x/(! - a ) / a - u

which is equal to +c~ .

,4.4. Behav iour o f the p . d . f f o r y = 0

For this ease we get

1 ~o z 1 1 f ( Y ) : - ~ x/-ff ~ x/rd x/q - - a

1 1

m du

A p p e n d i x B: C o m p u t a t i o n o f c r i t i c a l v a l u e s

The critical values for the statistic using uniform ranks, 2 r (un i ) and the "inverse normal scores transformation" 2 r ( I N S ) are computed from 10,000 Monte Carlo replications of the random walk model (2) with et--, IN(0, 1 ) for a grid o f 116 sample sizes letting T ---- 15, ! 6, ! 7 . . . . . 100, 105, 110 . . . . . 250.

For the estimated critical values the following regression model was fitted by OLS:

1 1 c~ = bo + bl V I~ + b2"~ + er ,

where c~ denotes the critical value for the significance values 0c = 0.01, 0.05, 0.10 observed at sample size T. Table 6 presents the fitted values ~ , - bo +bu ~ r + / ~ n for selected sample sizes.

26 J. Breitunff, C. Gouridroux/J~ournal o f Econometrics 81 (1997) 7-27

Table 6 Critical values

T 0.01 0.05 0.10 T 0.0 ! 0.05 0. I 0

2(uni)

20 0.0296 0.0404 0.0491 70 0.026 ! 0.0376 0.0469 25 0.0286 0.0397 0.0486 80 0.0260 0.0375 0.0468 30 0.0279 0.0392 0.0482 90 0.0258 0.0374 0.0467 35 0.0274 0.0388 0.0479 I O0 0.0257 0.0373 0.0466 40 0.0271 0.0385 0.0477 150 0.0254 0.0370 0.0463 45 0.0268 0.0383 0.0475 200 0.0253 0.0368 0.0462 50 0,0266 0.0381 0.0473 250 0.0252 0.0367 0.0461 60 0.0263 0.0378 0.0471 oo 0.0250 0.0362 0.0453

:.(INS) 20 0.03 ! ! 0.0424 0.0516 70 0.0264 0.038 ! 0.0475 25 0.0298 0.0412 0.0504 80 0.0262 0.0379 0.0473 30 0.0289 0.0404 0.0496 90 0.0260 0.0377 0.047 I .~5 0.0283 0.0400 0.0490 lO0 0.0259 0.0376 0.0470 ~;0 0.0278 0.0393 0.0486 150 0.0255 0.0373 0.0467 ~ ; 0.0274 0.0390 0.0483 200 0.0253 0.037 ! 0.0466 50 0.02 71 0.03 88 0.04 8 ! 250 0.0252 0.0370 0.0465 60 0.0267 0.0384 0.0477 oo 0.0250 0.0368 0.0464

Note: Entries report es t imated critical values for the significance levels 0.01, 0.05, 0.10 resulting from the response surface analysis. ,;.(uni) and 2( INS) denote the rank statistic based on the original ranks and the "invcrse normal scores" t ransformation given in (7). The est imated standard deviat ions are roughly 0.0005.

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