TESTING STATIONARITY AND TREND STATIONARITY AGAINST THE UNIT ROOT HYPOTHESIS

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1 TESTING STATIONARITY AND TREND STATIONARITY AGAINST THE UNIT ROOT HYPOTHESIS Herma J. BIERENS Departmet of Ecoomics, Souther Methodist Uiversity, Dallas Shegyi GUO Departmet of Ecoomics, Souther Methodist Uiversity, Dallas ad Federal Reserve Bak, Dallas * JEL Classificatio: C, C2, C5, C22 ABSTRACT I this paper we propose a family of relatively simple oparametric tests for a uit root i a uivariate time series. Almost all the tests proposed i the literature test the uit root hypothesis agaist the alterative that the time series ivolved is statioary or tred statioary. I this paper we take the (tred) statioarity hypothesis as the ull ad the uit root hypothesis as the alterative. The other differece with most of the tests proposed i the literature is that i all four cases the asymptotic ull distributio is of a well-kow type, amely stadard Cauchy. I the first istace we propose four Cauchy tests of the statioarity hypothesis agaist the uit root hypothesis. Uder H these four test statistics ivolved, divided by the sample size, coverge weakly to a o-cetral Cauchy distributio, to oe, ad to the product of two ormal variates, respectively. Hece, the absolute values of these test statistics coverge i probability to ifiity (at order ). The tests ivolved are therefore cosistet agaist the uit root hypothesis. Moreover, the small sample performace of these test are compared by Mote Carlo simulatios. Furthermore, we propose two additioal Cauchy tests of the tred statioarity hypothesis agaist the alterative of a uit root with drift. *) The views expressed i this paper are ot ecessarily those of the Federal Reserve Bak of Dallas or the Federal Reserve System.

2 2. INTRODUCTION I this paper we propose a family of relatively simple oparametric tests for a uit root i a uivariate time series. Almost all tests proposed i the literature test the uit root hypothesis agaist the alterative that the time series ivolved is statioary or tred statioary. See Fuller (976), Dickey ad Fuller (979, 98), Evas ad Savi (98, 984), Said ad Dickey (984), Phillips (987), Phillips ad Perro (988), Kah ad Ogaki (99) ad Bieres (99), amog others. For further related refereces, see Phillips (987) ad Haldrup ad Hylleberg (989, Table ). Moreover, see Schwert (989) ad DeJog et al. (99) for power problems of uit root tests. Oly Park's (99) approach allows for a reversal of the hypotheses ivolved. I this paper we first take the statioarity hypothesis as the ull ad the uit root hypothesis as the alterative, i.e., deotig the time series process by y t we test the ull hypothesis (.) H : y t ' µ % u t with µ ' E[y t ] agaist the alterative (.2) H : )y t ' u t, where u t is a statioary process. The other differece with most of the tests i the literature is that the asymptotic ull distributio is of a well-kow type, amely stadard Cauchy (c.q. the Studet distributio with oe degree of freedom). Uder H the four test statistics ivolved, divided by the sample size, coverge weakly to a o-cetral Cauchy distributio, to oe, ad to the product of two ormal variates, respectively. Hece, the absolute values of these test statistics coverge i probability to ifiity (at order ). The tests ivolved are therefore cosistet agaist the uit root hypothesis. Moreover, the small sample performace of these tests is compared by limited Mote Carlo simulatios, for sample sizes ad 3, usig replicatios. I sectio 8 of this paper we geeralize the third Cauchy test to two tests of the tred statioarity hypothesis agaist the uit root with drift hypothesis, with similar asymptotic properties.

3 3 2. LINEAR TIME TREND REGRESSION The ituitio behid our tests is that uder H the process y t has a stochastic tred ad therefore behaves (more or less) as if there is a determiistic liear tred. This suggests to regress y t o time t, i.e., estimate the auxiliary "model" y t = µ + βt + v t o the basis of the observatios t=,...,, ad use the least squares estimate of β, (2.) $ ' [ j with t' (t&t)(y t &y)] / [ j t' (t&t) 2 ], (2.2) t ' (/) j t' t ' 2 (%) ad y ' (/) j y t, t' as a basis for a test statistic. The further ituitio is that the rate of covergece i distributio of β is differet uder H o ad H, ad that this differece ca be exploited to distiguish betwee H o ad H. Followig Phillips (987, Assumptio 2.) we assume that the process (u t ) is such that: ASSUMPTION : (a) E(u t ) = for all t; (b) sup t E*u t * 6 < 4 for some 6 > 2 ; (c) F 2 ' lim 64 E[(/ ) j t' u t ] 2 exists ad F 2 > ; (d) (u t ) is "&mixig with mixig coefficiets "(s) that satisfy 4 j "(s) &2/6 < 4. s' Note that this assumptio allows for a fair amout of heterogeeity. Next, deote for λ ε [,], [8] (2.3) W (8) ' (/ ) j u j /F if 8 $, W (8) ' if 8 <, j'

4 4 where σ is defied i Assumptio (c) ad [x] meas trucatio to the earest iteger # x. The W (λ) is a stochastic elemet of the metric space D[,] of fuctios o [,] with coutably may discotiuities, edowed with the Skohorod topology [cf. Billigsley (968)]. The Skohorod orm is domiated by the sup orm (2.4) D(f,g) ' sup #8# *f(8)&g(8)*. Herrdorf (984) shows that uder Assumptio, W coverges weakly to a stadard Wieer process W (deoted by W Y W), which is a stochastic elemet of the metric space C[,] with orm (2.4) of cotiuous fuctios o [,] such that for # λ# ad # δ # -λ, (2.5) W(8) - N(,8), W(8%*)&W(8) - N(,*), W(8) ad W(8%*)&W(8) are mutually idepedet. Moreover, for ay cotiuous mappig Φ from D[,] ito C[,] we have Φ(W ) Y Φ(W). A special case of such a mappig is the itegral M(W ) ' 8 m W m (8)d8; m $. Sice, (/) j (2.6a) t' we thus have (t/) m W (t/) ' 8 m W m (8)d8 % O p (/), (/) j t' ad similarly for < µ <, (t/) m W (t/) Y 8 m W(8)d8, m [µ ] µ (2.6b) (/) j (t/) m W (t/) Y 8 m W(8)d8, m t' (2.6c) (/) j t'[µ ]% (t/) m W (t/) Y 8 m W(8)d8. m µ

5 Furthermore, it follows from lemma below that (2.7) (/ ) j t' 5 (t/) m u t /F ' W ()&m x m& W m (x)dx Y W()&m x m& W(x)dx. m LEMMA : For real umbers v,..., v, let [x] S (x)' j v t if x, [ &,], S (x) ' if x, [, & ), t' ad let F be a differetiable real fuctio o [,], with derivative f. The j F(t/)v t ' F()S () & f(x)s t' m (x)dx. PROOF: Easy ad therefore left to the reader. With these results at had we ca ow prove lemmas 2 ad 3 below: LEMMA 2: Uder Assumptio ad H, r β Y N(,σ 2 ), where r ' [(%) 3 & 3(%) 2 % 2(%)] / 2. PROOF: Observe that (2.8) $ ' j (t & ½(%))u t t' [(%) 3 & 3(%) 2 % 2(%)] /2 ' r, 2 say. It follows from (2.7) ad (2.8) that

6 6 (2.9) /( ) ' (/ ) j [(t/) & ½]u t % O p (/) t' ' ½FW () & F W m (x)dx % O p (/) Y ½FW() & F W(x)dx. m The lemma ow easily follows from the fact that ½W() & m W(x)dx - N(,/2). LEMMA 3: Uder Assumptio ad H, (r /)β Y N(,σ 2 /). PROOF: Observe that ' F 2 t t j (t&t)(y t &y) ' j t j u j & ½(%) j j u j t' t' j' t' j' {(/) j (t/)w (t/)} & F½(%) {(/) j t' t' W (t/)}. Hece by (2.6a), (2.) [r 2 /( 2 )]$ Y F (8& ½)W(8)d8, m where r is defied i Lemma 2. The limitig distributio i (2.) is ormal with zero mea. It is ot hard to show that (2.) E[F (8&½)W(8)d8] 2 ' F 2 /2. m Observig that 2r 2 / 3 6, it follows from (2.) ad (2.) that

7 7 (2.2) (r /)$ Y F 2 (8&½)W(8)d8 - N(,F 2 /). m Comparig the results of Lemmas 2 ad 3 we see that uder H the asymptotic rate of covergece i distributio of β is of order %, whereas uder H the asymptotic rate of covergece is of order %. Thus, if σ 2 were kow, the test r β /σ would be a cosistet stadard ormal test of the statioarity hypothesis agaist the uit root hypothesis, sice,r β /σ, 64 i probability uder H. However, i practice we caot use this test statistic because the variace σ 2 is ukow. (3.) > ' (/) j t' 3. CAUCHY TEST I 3. The Test We may thik of estimatig σ 2 i a similar way as i White ad Domowitz (984), Newey ad West (987), ad i particular Phillips (987) ad Phillips ad Perro (988). Actually, this is the approach take by Park (99), who uses a Newey-West (987) type variace estimator. However, due to the behavior of the Newey-West estimator uder the uit root hypothesis this approach will lead to substatial loss of power. Park's test statistic G, which is comparable with G * = [r β ] 2 /σ 2, with σ 2 replaced by a Newey-West type estimator, is such that uder the statioarity hypothesis, G coverges weakly to a chi-square distributio, whereas uder the uit root hypothesis, -+δ G 6 4 for some δ >, where δ depeds o the trucatio lag of the Newey-West estimator. Cf. Park (99, Lemma 3.3(b)). O the other had, the statistic G * is such that uder the statioarity hypothesis, G * Y χ 2 (), whereas uder the uit root hypothesis, -2 G * Y.χ 2 (). Cf. Lemma 3 for the latter result. Thus, loosely speakig, the Newey-West estimator eats up a substatial amout of asymptotic power. Therefore we propose a differet approach. The idea is to costruct a statistic with equal rates of covergece uder H ad H that also depeds o σ i a similar way as above. The we combie r β with this statistic such that σ cacels out. There are may ways to costruct such a statistic. I the first istace, the followig statistic will be used i the costructio of the tests: [½] y t & (/[½]) j y t, t' where [½] deotes the iteger part of ½. Later o, i Sectio 8, we cosider some

8 8 alteratives to (3.). The followig two lemmas describe the limitig joit distributio of β ad ξ uder H ad H. LEMMA 4: Uder Assumptio ad H, (r β, ξ %) T Y N [,σ 2 Ω], where (3.2) S ' ½ 3 ½ 3. PROOF: From (2.9) ad (3.) ad it easily follows (3.3) r $ > ' F 2[½W () & W m (x)dx] % O p (/) W () & 2((½)/[½])W (½) Y F 2[½W() & W(x)dx] m W() & 2((½)/[½])W(½) ' F x x 2, say. Now (x,x 2 ) T is ormally distributed with zero mea vector. We have already see i Lemma 2 that E[x 2 ] =. It is easy to verify that E[x 2 2] = ad E [x x 2 ] = ½%3. LEMMA 5: Uder Assumptio ad H, (r β /, ξ /%) T Y N 2 (,Γ), where (3.4) ' ' / 5/(32 3) 5/(32 3) /2. PROOF: Observe that (3.5) > ' (/) j t' j t j' u j & (/[½]) j [½] t' j t u j j' ' (F ){(/) j t' [½] W (t/) & 2(/) j W (t/)} % o p ( ), t'

9 9 hece by (2.6b-c), ½ (3.6) > / Y F{ W(8)d8 & 2 W(8)d8}. m m Together with (2.2) this result establishes that ((r /)β, ξ /%) σ(z,z 2 ), where ½ (3.7) z ' 2 (8&½)W(8)d8; z m 2 ' W(8)d8 & 2 W(8)d8. m m We have already see i (2.) that E[z 2 ] = /. It is easy to verify that E[z 2 2] = /2 ad, E[z z 2 ] = 5/(32%3). Q.E.D. It follows ow straightforwardly from Lemmas 4 ad 5: LEMMA 6: Let Assumptio hold ad deote γ = 2[r β - ½(%3)(%)ξ ]; γ 2 = (%)ξ. Uder H, (γ,γ 2 ) T Y N 2 [,σ 2 I 2 ], whereas uder H, (γ,γ 2 ) T / Y N 2 [,σ 2 ], where (3.8) ) ' /4 3/48 3/48 / 2. Now if we use γ /γ 2 as a test statistic the asymptotic ull distributio is stadard Cauchy, ad therefore does ot deped o σ 2. O the other had, the rate of covergece uder H ad H is the same so that the test will be icosistet. However, the followig lemma provides a solutio to this cosistecy problem. LEMMA 7: Let Assumptio hold, let α^ be the OLS estimator of α i the regressio model y t = F + αy t- + u t ad let (3.9)., ' [&exp(&( (&$")) 4 )]( 2 % exp(&( (&$")) 4 )( 2 /. Uder H, (γ,ζ, ) T Y N 2 [,σ 2 I 2 ], whereas uder H, (γ /,ζ, ) T Y N 2 [,σ 2 ], where is defied i (3.8). PROOF: It follows from Theorem of Phillips ad Perro (988) that uder H,

10 (3.) ($"&) Y ½[W() 2 &F 2 u /F2 ] & W() W(8)d8 m W(8) 2 d8 & [ W(8)d8] 2 m m whereas it is easy to verify that uder H, (3.) plim 64 $" <. Hece, (3.2) plim 64, (&$"), ' 4 if H is true, ' if H is true. It follows ow from (3.)-(3.2) that uder H, (3.3) plim 64 [&exp(&( (&$")) 4 )] ' ; plim 64 exp(&( (&$")) 4 ) ', whereas uder H, (3.4) [&exp(&( (&$")) 4 )] ' O p ( &2 ); plim 64 exp(&( (&$")) 4 ) '. Combiig (3.3) - (3.4) with Lemmas 6, the lemma uder review easily follows. Q.E.D. REMARK: The power 4 i expressio (3.9) is somewhat arbitrary. It follows from the proof of Lemma 7 that ay power larger tha 2 will do. Deotig (3.5) S, ' ( /.,, the theorem below follows straightforwardly from Lemma 7:

11 THEOREM : Let Assumptio hold. Uder H, S, Cauchy (,), whereas uder H, S, /.25%3 + (.5%5)g, where g is distributed as Cauchy(,). Note that uder H the limitig distributio of S, / is cotiuous, hece plim 64 *S, * = 4. Thus the two-sided Cauchy test ivolved is cosistet agaist the uit root hypothesis. Moreover, i comparig Park's (99) test with our test we should compare Park's test statistic G with S 2,, i.e., uder the uit root hypothesis, -+δ G 6 4 for some δ >, whereas -2 S 2, Y [.25%3+ (.5%5)g] 2. This meas that, asymptotically, our test is more powerful tha Park's test, due to the fact that we have avoided the use of the Newey-West (988) estimator of σ 2. The same applies to the other three Cauchy tests itroduced below. Table : Mote Carlo Simulatio of the Test i Theorem J 2 D = l 5% % 5 ad % rejectio frequecies D =.95 5% % D =.9 5% % D =.8 5% % D = 5% %

12 Mote Carlo Results I order to see how the test works i practice we have coducted the test o replicatios of data geeratig processes of the type: (3.6) y t ' Dy t& %u t ; (&JL)u t '(&2L)e t ; e t - NID(,), where ρ, τ ε {,.8,.9,.95} ad θ ε {,½}. The sample sizes we use correspod to 25 years of quarterly ad mothly data: ε {, 3}. The results are give i Table. It is clear from Table that the test is hardly able to distiguish a ear-uit root from a geuie oe whe the error process u t has a log memory amely i the cases ρ, τ =.95,.9 ad.8. O the other had, these cases are close to I(2) processes, so that we may expect a substatial size distortio. The cases ρ=, τ =.95,.9 ad.8 are more modest ear uit root cases, ad the actual size of the test i these cases is quite close to the theoretical size, apart from the case ρ=, τ=.95, θ=, where the size is much too high i compariso with the case θ=.5. A reaso for the latter may be that the ARMA process y t =.95y t- +e t -.5e t- is less a ear uit root process tha the AR process y t =.95y t- +e t. Table 2: Mea of "^ ad w^ for D=, J = ( replicatios) 2 = 2 = ½ "^ w^ "^ w^ A more puzzlig result is that the small sample power of the test i the case τ = differs dramatically betwee θ= ad θ=.5. This is possibly due to `misbehavior' of α^, by which the weight $w ' & exp(&( (&$")) 4 ) remais too large i small samples. This ca be observed from Table 2, where we have calculated the mea of α^ ad ^ for the case ρ=, τ = o the basis of w replicatios. The reaso for this pheomeo is probably the size of σ 2 u /σ 2 i the limitig distributio (3.) of (α^-), where σ 2 u = E[u 2 t]. See also Perro (99) ad Nabeya ad Perro (99). It is ot hard to verify that the umerator of the

13 3 asymptotic distributio (3.) has mathematical expectatio -½σ 2 u /σ 2, ad that for u t = τu t- + e t - θe t-, (3.7) F 2 u / F2 ' &J %J % 22(&J) (&2) 2. The values of σ 2 u /σ 2 for the cases uder review are give i Table 3. We see that i the case τ=, θ=½ the egative bias -½σ 2 u /σ 2 = -2.5 is relatively large. This may explai the low value of α^ uder H i the case τ=, θ=½. Table 3: F 2 u/f 2 J 2 F 2 u/f 2 J 2 F 2 u/f ½.9.9 ½ ½ ½ CAUCHY TEST II The argumet i Sectio 3. also suggests the followig alterative Cauchy test. Let (4.) S 2, ' ( /. 2,. [Cf. (3.5)], where ow (4.2). 2, ' [&exp(&( (&$")) 4 )]( 2 % [exp(&( (&$")) 4 )]( /. [Cf.(3.9)]. The it is easy to prove alog the lies of Sectios 2 ad 3 that the followig theorem holds. THEOREM 2: Let Assumptio hold. Uder H, S 2, Y Cauchy(,), whereas uder H, plim 64 S 2, / =.

14 4 The result uder H implies that the test may be coducted oe-sided. However, for reasos of compariso we have i the umerical applicatios below coducted the test two-sided. We have coducted a similar Mote Carlo aalysis for this test as for the oe i Theorem. The results are give i Table 4. Comparig Tables ad 4 we see that the small sample properties of the two tests are about the same. Table 4: Mote Carlo Simulatio of the Test i Theorem 2 J 2 D = 5% % D =.95 5% % 5 ad % rejectio frequecies D =.9 5% % D =.8 5% % D = 5%% CAUCHY TEST III A coceptual disadvatage of the tests i Theorems ad 2 is that the power of the test is maily determied by the behavior of the Dickey-Fuller-type test statistic α^. I particular, the test statistic (3.5) ca be writte as (5.) S, ' $w( /( 2 ' $wd, say, where

15 5 (5.2) $w ' / [ % ( & &)exp(&( (&$")) 4 )]. It follows from the proof of Theorem that (5.3) D Y g if H is true; D Y.25 3 % (.5 5)g if H is true, where g is Cauchy (,) distributed, whereas (5.4) plim 64 $w ' if H is true; plim 64 $w/ ' if H is true. Thus, the actual power of the test is determied by w^ via α^, ad therefore the test does ot differ much from the Phillips-Perro (988) test of H agaist H. The mai differece is that we do ot eed to estimate σ 2 /σ 2 u, whereas i the latter case oe eeds to modify (α^-) i order to accout for the ukow ratio σ 2 /σ 2 u. We ow propose a third Cauchy test that does ot employ the estimator α^. Let (5.5) ' ( 2 /, (5.6) $F 2 u ' (/(&)) j (y t &y t& ) 2 t'2 ad (5.7). 3, ' % 2 / $F2 u. The reaso for usig σ^ 2 u i (5.7) is to make ζ 3, ivariat for liear trasformatios of y t. Sice uder H, % Y N(,σ 2 ) [Cf. Lemma 4] ad σ^ 2 u Y E(u 2 -u ) 2, it follows that ζ 3, - % Y. Moreover, it follows from the proof of Lemma 7 that uder H, (5.8). 3, ' % 2 /$F2 u ' / / % ( 2 /)/$F2 u Y F2 u Fv 2 ad (5.9) (( /,. 3, ) Y (Fv, F 2 u /(Fv 2 )), where (v,v 2 ) T is distributed as N 2 [, ], with defied by (3.8). These results suggest

16 6 the followig alterative (two-sided) Cauchy test: (5.) S 3, Y ( /. 3,. THEOREM 3: Let Assumptio hold. Uder H, S 3, Y Cauchy(,), whereas uder H, S 3, / Y (σ 2 /σ 2 u)v v 2, where (v, v 2 ) T is distributed as N 2 (, ) with defied by (3.8). Table 5: Mote Carlo Simulatio of the Test i Theorem 3 J 2 D = 5% % D=.95 5% % 5 ad % rejectio frequecies D =.9 5% % D =.8 5% % D = 5% % The results of a similar Mote Carlo simulatio as i Tables ad 4 are preseted i Table 5. We see that i geeral the size distortio i the ear uit root cases is less (although still substatial) tha for the tests i Theorems ad 2, but that also the power agaist the uit root case ρ= is lower, i particular i the case

17 7 τ=, θ=. 6. CAUCHY TEST IV The result i Theorem 3 for the uit root hypothesis idicates that the power of the test i Theorem 3 depeds o the ratio σ 2 /σ 2 u. We ca make the power idepedet of this ratio by replacig σ^ 2 u i (5.7) ad (5.8) by a Newey-West (987) type estimator of σ 2. Thus, replace (5.6) by (6.) $F 2 ' (/(&)) j t'2 % 2(/(&)) j m (y t &$"y t& ) 2 j' w j,j (y t &$"y t& )(y t&j &$"y t&&j ). t'j%2 where w j, = - j/(m +), ad m 6 4 at rate o( ¼ ). I the umerical applicatios below we have chose m = +[5 /5 ]. Deotig (6.2) S 4, ' ( /. 4,, where ow (6.3). 4, ', % 2 /$F2 it follows from Theorem 3 ad the cosistecy of σ^ 2 that: THEOREM 4: Let Assumptio hold. Uder H, S 4, Y Cauchy(,), whereas uder H, S 4, / Y v v 2, where (v, v 2 ) T is distributed as N 2 (, ) with defied by (3.8).

18 8 Table 6: Mote Carlo Simulatio of the Test i Theorem 4 5 ad % rejectio frequecies J 2 D = 5% % D =.95 5% % D =.9 5% % D =.8 5% % D = 5% % I Table 6 we preset the Mote Carlo results ivolved. Comparig Tables 5 ad 6 we see that ideed the power i the case τ =, θ = ½, has bee improved (although less tha expected o the basis of Table 3), but at the expese of lower power i the cases with τ >. Thus, i geeral the test i Theorem 4 does ot perform better tha the oe i Theorem 3. Summarizig, the tests i Theorems ad 2 perform the best whe the process u t has a short memory, ad the test i Theorem 3 performs the best i the ear uit root cases. 7. THE NEAR-STATIONARITY CASE Sice the ull hypothesis of our tests is statioarity, it makes sese to compare their performace for the followig ear-statioarity cases:

19 9 (7.) y t ' y t& % e t & 2e t&, e t - NID(,), 2 '.8,.9,.95,.99. Note that for θ = this process is statioary, as the the uit root cacels out. Thus for θ close to oe we may cosider (7.) as a ear-statioary process. Table 7: The ear-statioarity case 5 ad % rejectio frequecies; D=, J= 2 =.8 5% % 2 =.9 5% % 2 =.95 5% % 2 =.99 5% % Test Th Th Th Th.4 The Mote Carlo results for our four tests o the basis of replicatios are preseted i Table 7. We see from Table 7 that our tests have oly trivial power agaist these ear-statioarity cases. 8. TREND STATIONARITY VERSUS THE UNIT ROOT HYPOTHESIS The statioarity hypothesis is ofte too restrictive for macro ecoomic time series. Time series like real GNP usually seem to have a determiistic liear tred rather tha beig statioary. Such a liear determiistic tred may occur because the series is tred statioary, or because it is uit root process with drift. I this sectio we shall therefore exted the test i Theorem 3 to testig the ull hypothesis of tred statioarity, (8.) H + : y t ' µ % $t % u t,

20 2 agaist the uit root hypothesis with (or without) drift, (8.2) H + : y t & y t& ' µ % u t, where i both cases the process (u t ) satisfies Assumptio. Agai we base our test o the OLS estimates of the parameters i a auxiliary tred regessio, but because our ull is ow tred statioarity we ca o loger use the OLS estimate of the parameter of time t. Therefore we propose to ru the followig auxiliary regressio: (8.3) y t = F + βt + δt ν + e t, where ν>, ν. A atural choice of ν would be ν = 2, but ay positive ν uequal to will work too. Now let δ (ν) be the OLS estimator of δ i the auxiliary regressio (8.3). I the Appedix we shall prove: LEMMA 8: Let Assumptio hold. Uder H +, ν+½ δ (ν) Y N(,σ 2 s 2 (ν)), where (8.4) s 2 2 (<%)(<%2) (<) ' (2<%). <(<&) Uder H +, ν-½ δ (ν) Y N(,σ 2 s 2 2(ν)), where s 2 2(ν) ca be calculated as idicated i the Appedix. I order to costruct a Cauchy test, we eed a secod statistic havig the same rate of covergece uder H ad H, with a limitig ormal distributio with zero mea ad variace proportioal to σ 2 uder H. Note that the statistic (3.) is o loger suitable, due to the fact that the ull hypothesis is ow tred statioarity. A possible solutio to this problem is to replace the y t i (3.) by the OLS residuals ^g t of the auxiliary regressio (8.5) y t = F + βt + g t.

21 2 Thus, let for τ ε (,), (8.6) > (J) ' [J] j [J] t' Note that uder H +, g t = u t, whereas uder H, $g t. g t ' j t j' u j. The joit asymptotic distributio of δ (ν) ad ξ (τ) is give i Lemma 9: LEMMA 9: Let Assumptio hold. Uder H +, ( ν+½ δ(ν), ½ ξ (τ)) T Y N 2 (,σ 2 Q (ν,τ)), where Q (<,J) ' s 2 (<) s 3 (<,J), s 3 (<,J) s 2 3 (J) with s 2 3 (J) ' J& & 3(J&) 2 &, s 3 (<,J) ' s 2 2(<&)&3J< (<) (<%)(<%2) % J< <%. Uder H +, ( ν-½ δ (ν), -½ ζ (τ)) T Y N 2 (,σ 2 Q 2 (ν,τ)), where Q 2 ca be calculated as idicated i the Appedix. Now let the matrix L(ν,τ) be such that (8.7) L(ν,τ)L(ν,τ) T = Q (ν,τ), ad defie (8.8) (γ (ν,τ), γ 2 (ν,τ)) T = L(ν,τ) - ( ν+½ δ (ν), ½ ζ (τ)) T. Note that s 3 (ν,τ) = for ν = 2, τ = ½, so that the

22 22 (8.9) L(2, ½) ' s 2 (2) s 2 3 (½) ' 6 ½ ½. Similarly to Lemma 6, it follows from Lemma 9: LEMMA : Let Assumptio hold. Uder H +, (γ (ν,τ),γ 2 (ν,τ)) T Y N 2 (,σ 2 I 2 ), whereas uder H +, (γ (ν,τ),γ 2 (ν,τ)) T / Y σ(v (ν,τ),v 2 (ν,τ)) T, where (v (ν,τ), v 2 (ν,τ)) T is bivariate ormally distributed. With this result at had we ca ow easily modify the tests i Theorems -4 to tests of the tred statioarity hypothesis agaist the uit root hypothesis. Here we shall oly focus o the modificatio of the test i Theorem 3. Thus defie the test statistic S 5, (ν,τ) similarly to S 3, [cf. (5.)], where ^σ 2 u i (5.7) is replaced by (8.) $F 2 )$g ' & j t'2 ($g t &$g t& ) 2, with the ^g t 's the OLS residuals of the auxiliary regressio (8.5). The THEOREM 5: Let Assumptio hold. Uder H +, S 5, (ν,τ) Y Cauchy(,), whereas uder H +, S 5, (ν,τ)/ Y (σ 2 /σ 2 u)v (ν,τ)v 2 (ν,τ). A alterative to the above approach is to base the test o the OLS estimators of two superfluous regressios t ν ad t η, say, ad orthogoalise these OLS estimators i order to obtai the statistics γ ad γ 2. If we choose for ν ad η atural umbers, say 2 ad 3, respectively, the orthogoalizatio ad estimatio ca be doe joitly by usig orthogoal polyomials. Thus let for t=,..,, P, (t) ' ; P, (t) ' t & % 2, P 2, (t) ' ( t &a 2, )P, (t) & b 2, P, (t), P 3, (t) ' ( t &a 3, )P 2, (t) & b 3, P, (t),

23 23 with a i, ' j (t/)p 2 i&, (t) t', b ' i, j P 2 i&, (t) t' j (t/)[p i&2, (t)p i&, (t)] t' (i ' 2,3). j P 2 i&2, (t) t' These polyomials are exactly orthogoal. It ca be show that lim 64 a 2, = lim 64 a 3, = ½, lim 64 b 2, = /2, lim 64 b 3, = /5. For x ε [,] we the have: lim 64 P, ([x]) ' q (x) ' ; lim 64 P, ([x]) ' q (x) ' x& 2 ; lim 64 P 2, ([x]) ' q 2 (x) ' (x& 2 )2 & 2 ; lim 64 P 3, ([x]) ' q 3 (x) ' (x& 2 )3 & 3 2 (x& 2 ). Next, defie the correspodig system of orthoormal polyomials by: P ( i, (t) ' P i, (t) j P 2 i, (j) j'. Note that lim 64 P ( i, ([x]) ' m q i (x) q 2 i (z)dz ' q ( i (x), i ',,2,3. Now ru the orthoormal regressio (8.) y t ' j 3 i' b i P ( i, (t) % e t, ad let ^b i be the OLS estimator of b i. Stackig the ^b i ad b i i vectors ^b ad b, respectively, we have:

24 24 LEMMA : Let Assumptio hold. Uder H +, ^b - b Y N 4 (,σ 2 I 4 ), whereas uder H +, ( $b&b) Y F W(r)dr, q ( m m (r)w(r)dr, q ( 2 m (r)w(r)dr, q ( 3 T. m (r)w(r)dr Sice uder the ull hypothesis H +, b i = for i = 2, 3, we may ow choose (8.2) ( ' $b 2, ( 2 ' $b 3, ad agai we ca costruct a Cauchy test S 6, similarly to (5.), where ow σ 2 u i (5.7) is replaced by (8.3) $F 2 )$e ' & j t'2 ($e t & $e t& ) 2, with the ^e t 's the OLS residuals of (8.). THEOREM 6: Let Assumptio hold. Uder H +, S 6, Y Cauchy(,), whereas uder H +, S 6, / Y (F 2 /F 2 u ) q ( 2 m (x)w(x)dx q ( 3 m (x)w(x)dx. We have coducted a similar Mote Carlo aalysis as for the test i Theorem 3. The small sample properties of the tests i Theorems 5 ad 6 appear to be very similar to those of the test i Theorem 3. The results are ot reported here, but are available from the secod author o request. 9. SUMMARY AND DISCUSSION I this paper we have proposed four Cauchy tests of the statioarity hypothesis H : y t = µ + u t, agaist the uit root hypothesis H : y t = y t- + u t, where u t is a zero mea α-mixig process. These tests have better asymptotic power properties tha Park's (99) statioarity test. All four tests are based o the OLS estimator β of the

25 ( )/(%8 2 ) Y N(,F2 ) uder H, Y /(8z) uder H 25 parameter β i the auxiliary tred model y t = µ + βt + u t. I particular, we exploit the fact that the speed of covergece i distributio of β uder H ad H differs by a factor equal to the sample size, i.e., r β N(,σ 2 ) uder H, r β / N(,σ 2 /) uder H, where r is give i Lemma ad σ 2 is the log ru variace of u t. The mai ovelty of the approach i this paper is the way we have avoided the use of the Newey-West (987) estimator of σ 2 i costructig the tests, amely by costructig a secod statistic ζ which has the same rate of covergece i distributio uder H ad H ad is, uder H, asymptotically N(,σ 2 ) distributed. The statistics r β ad ζ ca the be combied ito a test statistic S which uder H takes, asymptotically, the form of a ratio of two idepedet N(,σ 2 ) distributed variates. Hece, σ the cacels out like i the case of t- ad F- statistics, ad the asymptotic distributio of S is thus stadard Cauchy. Moreover, if H is true the S / coverges weakly to a (possibly degeerated) distributio. The tests i Theorems ad 2 use the OLS estimator α of the coefficiet α i the auxiliary model y t = µ + αy t- + u t to equalize the speed of covergece of the statistic ζ uder H ad H. Although the fiite sample performace of these tests i compariso with the Mote Carlo results i Phillips-Perro (988) ad Schwert (989) is ot bad at all, the fact that they use the OLS estimate α is a coceptual disadvatage because the performace of the tests ivolved is almost completely determied by the behavior of α uder H ad H. The tests i Theorems 3 ad 4 equalize the speed of covergece of the statistic ζ uder H ad H i a completely differet way. The trick is actually very simple: if % Y N(,σ 2 ) uder H ad /% Y z uder H the for ay λ >, Cf. (5.7) ad (5.8). Thus, i this way we ca costruct a statistic ζ that has the same speed of covergece uder H ad H ad is asymptotically N(,σ 2 ) distributed uder H. The mai differece betwee the tests i Theorems 3 ad 4 is that uder H, S / Y (σ 2 /σ 2 u)v v 2 for the test i Theorem 3 ad S / Y v v 2 for the test i Theorem 4, where v ad v 2 are ormal variates ad σ 2 u is the variace of u t. Sice the test i Theorem 3 is the simplest oe of the two ad σ 2 /σ 2 u will likely be larger tha oe for ecoomic time series, the test i Theorem 3 is our favorite. Moreover, the Mote

26 26 Carlo results ivolved show that the tests i Theorems 3 ad 4 are less sesitive to size distortio caused by a ear uit root tha the tests i Theorems ad 2, ad that i geeral the power of the test i Theorem 3 is better tha the power of the test i Theorem 4. I Sectio 8 we geeralize the test i Theorem 3 to testig the tred statioarity hypothesis agaist the uit root hypothesis with drift, i two directios. The first extesio employs istead of β the OLS estimator δ of the parameter δ i the auxiliary regressio y t = F + βt + δt ν + e t, where ν, together with a statistic similar to ζ. The secod extesio is based o the OLS estimates of the regressio of y t o secod ad third order orthoormal polyomials i t. Fially, oe may woder what the practical sigificace is of testig (tred) statioarity agaist the uit root hypothesis, rather tha testig the uit root hypothesis agaist (tred) statioarity. First, there are situatios where the statioarity hypothesis is a more atural ull hypothesis that the uit root hypothesis, for example i testig for coitegratio. The same applies to tred statioarity. But eve if the uit root hypothesis is the atural ull, it makes sese to coduct (tred) statioarity tests as well. As is show by Phillips ad Perro (988) ad Schwert (989), the power of uit root tests is ofte quite low ad the size distortio ca be substatial. Therefore, coductig (tred) statioarity tests after havig coducted uit root tests provides a double check of the decisio to reject or accept the uit root hypothesis. Of course, the same applies the other way aroud. If the atural ull hypothesis is (tred) statioarity, oe should also coduct uit root tests as a double check, because the Mote Carlo results i this paper show that also our Cauchy tests may suffer from size distortio ad low power. ACKNOWLEDGMENT The very helpful commets of Loures Broersma, Esfadiar Maasoumi ad two referees are gratefully ackowledged. A previous versio of this paper, etitled "Testig Statioarity Agaist the Uit Root Hypothesis", was preseted by the first author at the 6 th World Cogress of the Ecoometric Society, Barceloa, 99. A substatial part of this research was doe while the first author was affiliated with the Free Uiversity, Amsterdam.

27 27 TECHNICAL APPENDIX Deote D ', x t ' (,t), x 2t '(t < ), X T i ' [x T i,...,x T i ], i',2, u ' (u,...,u ) T, g ' (g,...,g ) T, $g ' ($g,...,$g ) T, e ' (e,...,e ) T, u ( ' (u,u %u 2,..., j u t ) T, t' ad let s be the dimesioal vector with 's as the first s etries ad zeros as the rest. Usig simple algebra ad Lemma, it ca be easily show that D & (X T X )D& 6 /2 /2 /3 ' M, say, ( <%½ D ) & X T X 2 6 (<%)& ' N, say, & (<%2) &(<%/2) X T 2 u Y F r < dw(r) ' FS m, say, &(<%3/2) (X T 2 u( ) Y F r < W(r)dr ' FS ( m, say, D & X T u Y F dw(r), rdw(r) m m T ' FS 2, say, (D ) & X T u( Y F W(r)dr, rw(r)dr m m T ' FS ( 2, say, T [J] X ( ½ D ) & 6 J J 2 /2 ' P(J), say.

28 28 Before presetig the proofs, we first give the followig lemma which is useful for calculatig the covariace matrix of fuctios of a Browia motio. LEMMA A: For ay determiistic cotiuous fuctios f ad g o [,], ad ay τ ε [,], J J (a) E f(r)dw(r) g(r)dw(r) ' f(r)g(r)dr, m m m J J r J (b) E f(r)w(r)dr g(r)w(r)dr ' f(r) g(s)sds % r g(s)ds dr m m m m m r J % f(r) sg(s)ds dr. m m J Proof: Sice the Browia motio W has idepedet icremets, it follows that E[dW(r)dW(s)] ' if r s dr if r ' s ; E[W(r)W(s)] ' mi(r,s). Usig this result, Lemma A easily follows. (A) E[S S 2 ] ' N, E[S 2 ] ' (2<%)&, E[S 2 S T 2 ] ' M. Q.E.D. As a applicatio of Lemma A, we ca calculate the followig covariaces: Proof of Lemma 8: Observe that <%/2 * (<) ' d, (<) d 2, (<), say, with d, (<) ' &(<%/2) X T 2 e & (<%/2 D ) & (X T 2 X )[D& (X T X )D& ]& (D & X e),

29 29 d 2, (<) ' &(2<%) j t 2< t' & (X T 2 X &(<%/2) D & )[D& (X T X )D& ]& ( &(<%/2) D & X T X 2 ). Moreover it easily follows that lim 64 d 2, (<) ' (2<%) & & N T M & N. Uder H, e = u, so that d, (<) Y F(S &N T M & S 2 ) hece (<%/2) * (<) Y F[(2<%) & & N T M & N] & [S &N T M & S 2 ] - N(,F 2 s 2 ), where, usig (A), s 2 ' [(2<%)& & N T M & N] & ' (2<%) (<%)(<%2) <(<&) 2. Uder H, e = u *, so that d, (<)/ Y F(S ( & N T M & S ( 2 ), hece <&½ * (<) Y F[(2<%) & &N T M & N] & [S ( &N T M & S ( 2 ] ' F(2<%) (<%)(<%2) <(<&) 2 & r < W(r)dr% 2(<&) W(r)dr m (<%)(<%2) m 6< rw(r)dr - N(,F 2 s 2 2 (<%)(<%2) m ), where s 2 2 ca be calculated usig Lemma A. Q.E.D.

30 3 Proof of Lemma 9: Observe that T s $ g ' T s g & (T s X D& )[D&(X T X )D& ]& (D & X T g) ad ½ > (J) ' [J] &½ T [J] $ g, Uder H, g = u, so that &½ T [J] [J] g ' &½ j u t Y FW(J), t' hece &½ T [J] $ g Y F(W(J) & P(J) T M & S 2 ), Cosequetly, ½ > (J) Y FJ & [W(J) & P(J) T M & S 2 ] J ' FJ & [ dw(r)&(4j&3j 2 ) dw(r)%(6j&6j 2 ) rdw(r)] m m m - N(,F 2 s 2 3 ), where s 2 3 ' J&2 [J&P(J) T M & P(J)] ' J & &3(J&) 2 &. It the trivially follows that the joit asymptotic distributio is bivariate ormal, with covariace matrix, Q ' s 2 s 3 s 3 s 2 3

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