A Residual-Based Cointegration Test for Near Unit Root Variables

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 907 October 2007 A Residual-Based Cointegration Test for Near Unit Root Variables Erik Hjalmarsson and Pär Österholm NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/.

A Residual-Based Cointegration Test for Near Unit Root Variables (cid:3) Erik Hjalmarsson P(cid:228)r (cid:214)sterholm y z October 9, 2007 Abstract Methods of inference based on a unit root assumption in the data are typically not robust to even small deviations from this assumption. In this paper, we propose robust procedures for a residual-based test of cointegration when the data are generated by a near unit root process. A Bonferroni method is used to address the uncertainty regarding the exact degree of persistence in theprocess. Wethusprovideamethodforvalidinferenceinmultivariatenearunitrootprocesses wherestandardcointegrationtestsmaybesubjecttosubstantialsizedistortionsandstandardOLS inference may lead to spurious results. Empirical illustrations are given by: (i) a re-examination of the Fisher hypothesis, and (ii) a test of the validity of the cointegrating relationship between aggregate consumption, asset holdings, and labor income, which has attracted a great deal of attention in the recent (cid:133)nance literature. JEL classi(cid:133)cation: C12, C22. Keywords: Bonferroni test; Cointegration; Near unit roots. We have bene(cid:133)tted from comments by Meredith Beechey, David Bowman, Mike McCracken, Chris Erceg, Dale (cid:3) Henderson, Lennart Hjalmarsson, Randi Hjalmarsson, George Korniotos, Rolf Larsson, Andy Levin, Johan Lyhagen, John Rogers, Jonathan Wright, and seminar participants at the Federal Reserve Board. The views in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:135)ecting the views of the Board of Governors oftheFederalReserveSystem orofanyotherpersonassociatedwiththeFederalReserveSystem. (cid:214)sterholm gratefully acknowledges (cid:133)nancialsupport from Jan Wallander(cid:146)s and Tom Hedelius(cid:146)Foundation. Division of International Finance, Federal Reserve Board, Mail Stop 20, Washington, DC 20551, USA. email: y erik.hjalmarsson@frb.gov Phone:+1 202 452 2426 Department of Economics, Uppsala University, Box 513, 751 20 Uppsala, Sweden. email: par.osterholm@nek.uu.se z Phone:+1 202 378 4135

1 Introduction Cointegration tests have been among the most important and in(cid:135)uential tools in empirical economics since their introduction over two decades ago. In essence, cointegration tests attempt to identify commondrivingfactorsinstochasticallytrendingdata,thusidentifyinglong-runequilibriumrelationships between economic variables. The most common cointegration tests are based on the assumption that the individual variables are unit root processes. The unit root assumption, however, is often hard to fully justify for actual economic data. In (cid:133)nite samples, many economic variables appear highly, but not totally, persistent; that is, the largest autoregressive root is close to, but not necessarily equal to, unity. Unfortunately, inferencial procedures designed for unit root data tend not to be robust to even small deviations from the unit root assumption. For instance, Elliot (1998) shows that large size distortions can occur when performing inference on the cointegrating vector in a system where the individual variables follow near unit root processes rather than pure unit root processes. Unitroottestsgosomewaytowardalleviatingtheuncertaintyregardingthepersistenceinagiven time series but do not provide a de(cid:133)nitive answer. Since unit root tests have low power against local alternatives, a failure to reject the null hypothesis of a unit root does not rule out the possibility of a rootslightlydi⁄erentfromunity. Ontheotherhand,rejectingthenullofaunitrootdoesnotruleout that the process is still fairly persistent and leaves open the possibility of spurious regressions. It is thus far from obvious how to deal with a multivariate near unit root process: Standard cointegration tests will not be valid under deviations from the pure unit root assumption and the possibility of spurious regressions invalidates standard OLS inference.1 The aim of this paper is to design a test of cointegration that is robust to deviations from the pure unit root assumption. In particular, we extend the standard framework to the case where the original data possess autoregressive roots that are local-to-unity, rather than identically equal to unity. The methods developed here are useful from two di⁄erent perspectives. First, they provide a robustness checktostandardcointegrationtestsinthetypicalsituationwhereitisnotknownwithcertaintythat there is an exact unit root in the data. Second, and just as importantly, the test procedures in this paperallowforvalidinferenceinthecasewhenthedataislikelynotapureunitrootprocess,butstill highly persistent. 1In most cointegration studies,the regressors are endogenous,in which case OLS inference would be furthercomplicated and invalid even in the strictly stationary case. Stock (1997) provides a detailed discussion on many ofthe issues that arise in inference with nearunit root variables. 1

While there is a large literature on cointegrating regressions with near unit root regressors, the focus has been on inference on the slope parameter in these regressions, rather than actual tests of cointegration; see, for example, Cavanagh et al. (1995), Elliot (1998), Campbell and Yogo (2006) and JanssonandMoreira(2006). Typically,themodelsinthisliteraturehavebeenspeci(cid:133)edsuchthatunder the null hypothesis of a zero slope coe¢ cient, the dependent variable is a stationary process. Tests on the slope coe¢ cient therefore become joint tests of cointegration as well, and the issue of spurious regressions never occurs. Although this is a useful speci(cid:133)cation, for instance, in tests of stock-return predictability which motivated much of this literature, it is less convenient in most typical economic applications where both dependent and independent variables are near-integrated. The closest related literature to the current paper is the work on stationarity tests (Leybourne and McCabe, 1993, and Shin, 1994) and the work by Wright (2000). In particular, Wright (2000) develops a joint test of a speci(cid:133)chypothesisregardingthecointegratingvectorandatestofthenullhypothesisofcointegration that is robust to deviations from the pure unit root framework. We focus on a residual-based test of cointegration. Following the work of Phillips and Ouliaris (1990), we extend the asymptotic results for a residual-based test to the case of near-integrated processes. Unlikethepureunitrootcase,theasymptoticdistributionoftheteststatisticnowdepends on an unknown nuisance parameter; the local-to-unity root. Since this parameter is not consistently estimable, feasible tests cannot be directly constructed from the asymptotic distribution. Instead, we propose to replace the unknown parameter value for the local-to-unity root with a conservative estimate. Inordertounderstandtheintuitionbehindourprocedure,itisusefultoconsiderthepotentialerrors whenapplyingastandard,pureunitrootcase,cointegrationtesttoasetofnearunitrootvariables. A residual-basedcointegrationtestevaluateswhethertheresidualsfromtheempiricalregressioncontain a unit root. Now, if the original data are in fact near-integrated, with a root less than unity, the test willover-rejectsincetheresidualswillnotcontainaunitrootevenifthereisnocointegration. But,by instead using critical values based on a conservative estimate of the local-to-unity root in the original data, a valid test is obtained. Intuitively, if one views a residual-based test of cointegration as a test of whether there is less persistence in the residuals than in the original data, then this test is only valid if the persistence of the original data is not overstated.2 In a spirit similar to the Bonferroni 2Although, perhaps, less obvious, the same also holds true for non-residual-based tests, such as those of Johansen (1988,1991);see Hjalmarsson and (cid:214)sterholm (2007). 2

methods proposed by Cavanagh et al. (1995), we show how an appropriately conservative estimate of the local-to-unity root is obtained. Therestofthepaperisorganizedasfollows. Section2outlinesthemodellingassumptionsandthe theoretical results. Section 3 describes the Bonferroni methods. In Section 4, the proposed procedure is evaluated using Monte Carlo simulations. We show that once the conservative estimate for the local-to-unity parameter is chosen appropriately, the resulting test has both good size and power properties. This is in contrast to standard cointegration tests, based on the unit root assumption, which are shown to severely over-reject as the data generating process deviates from a pure unit root setup. As an illustration of the method, two empirical applications are considered in Section 5. First, we re-examine the Fisher hypothesis and show that using the robust methods proposed in this paper, one can no longer (cid:133)nd signi(cid:133)cant support for a long-run equilibrium relationship between nominal interest rates and in(cid:135)ation; using standard unit root based cointegration tests on the other hand, the null hypothesis of no cointegration is rejected. In a second illustration, we consider the robustness of the long-run relationship between aggregate consumption, asset holdings, and labor income, which was initially studied by Lettau and Ludvigson (2001) and has since received a great deal of attention in the (cid:133)nance literature. We (cid:133)nd that after controlling for the unknown persistence in the variables, there is still strong evidence of cointegration between the three variables. Section 6 concludes and the Appendix contains tables of critical values for the test statistic. 2 Theoretical framework 2.1 Model and assumptions Let f z t g 10 beanm (cid:0) vectorofnearlyintegratedprocesses,suchthatthedatageneratingprocesssatis(cid:133)es z =Az +u (1) t t 1 t (cid:0) where A=I+ C/T is an m m matrix with A=diag(a ;:::;a ) and C =diag(c ;::;c ), and T is 1 m 1 m (cid:2) the sample size. That is, each component process in z is generated as a near unit root process with t individual local-to-unity parameters c , i = 1;:::;m. The initial conditions are set at t = 0 and z is i 0 assumed randomly distributed with (cid:133)nite variance. Although none of the formal results depend upon it, we will work under the assumption that c 0 for all i, which rules out explosive processes. The i (cid:20) 3

innovations u satisfy a general linear process. t Assumption 1 1. u t =D(L)(cid:15) t = 1j=0 D j (cid:15) t (cid:0) j ; 10 j jj D j jj < 1 ; j D(1) j = 6 0: 2. (cid:15) is iid with mean zero, variaPnce matrix (cid:6) ,Pand (cid:133)nite fourth-order moment. t (cid:15) By standard results, e.g. Phillips and Solo (1992), T 1=2 [Tr]u B(r) BM((cid:10)), where B(r) (cid:0) t=1 t ) (cid:17) isaBrownianmotionwithcovariancematrix(cid:10)=D(1)(cid:6) (cid:15) D(P1)0. Partitionz t =(y t ;x 0t )0 suchthaty t is ascalarandx t isann (cid:0) vector(n=m (cid:0) 1). LetB(r)= B 1 (r);B 2 (r)0 0,(cid:10)=[(! 11 ;! 021 );(! 21 ;(cid:10) 22 )], and C = [(c ;0);(0;C )] be conformable partitions of B(cid:0)(r), (cid:10), and C,(cid:1)respectively. We assume that 1 2 (cid:10) >0 and write (cid:10)=LL. Denote an m vector standard Brownian motion as W (r), and it follows 22 0 (cid:0) that B(r) = LW (r). Further, as T , z /pT J (r) = r e(r s)CdB(s). Partition J 0 ! 1 t ) C 0 (cid:0) C conformably with B and let JW (r)= r e(r s)CdW (s): R C 0 (cid:0) We consider residual-based tests ofRthe null of no cointegration using the regression residuals, v^, t from the following empirical regression: y =(cid:12) x +v : (2) t 0 t t 2.2 The test statistic WefocusonthetraditionalAugmentedEngle-Grangert test(EngleandGranger,1987)ofthenullof (cid:0) nocointegration, whichisprobablythemostcommonlyusedresidual-basedtestofcointegration. Our analysis could easily be extended to cover the Z and Z cointegration tests proposed by Phillips and (cid:11) t Ouliaris(1990),butforbrevitywerestrictourselvestotheAugmentedEngle-Grangertest(henceforth denoted AEG test). TheAEG testisde(cid:133)nedasthet statisticfor(cid:11)^ fromtheregression(cid:1)v^ =(cid:11) v^ + p ’ (cid:1)v^ + (cid:0) (cid:3) t (cid:3) t (cid:0) 1 i=1 i t (cid:0) i w . The below result follows from the results in Phillips and Ouliaris (1990) and the rPesults for neart integrated processes in Phillips (1987,1988). Theorem 1 Let the data generating process satisfy equation (1) for some given C =diag(c ;:::;c ), 1 m and let Assumption 1 hold. Suppose that the autoregressive order in the AEG regression satis(cid:133)es 4

p as T such that p=o T1=3 . Then, under the null of no cointegration, as T , !1 !1 !1 (cid:0) (cid:1) 1 1=2 1 JW dW AEG ) c 1 (cid:18)Z0 J 1 W (cid:1) 2;C 2 (cid:19) + R 0 1 J 1 (cid:1) W 2;C 2 1 (cid:1) 2 1=2 (3) (cid:0) (cid:1) 0 12;C (cid:18) (cid:16) (cid:1) (cid:17) (cid:19) R where 1 1 1 (cid:0) JW (r)=JW (r) JW JW JW JW JW (r) (4) 1 (cid:1) 2;C 1;c1 (cid:0) (cid:18)Z0 1;c1 2;C0 2 (cid:19)(cid:18)Z0 2;C2 2;C0 2 (cid:19) 2;C2 and 1 1 1 (cid:0) W (r)=W (r) W W W W W (r); (5) 1 (cid:1) 2 1 (cid:0) (cid:18)Z0 1 20 (cid:19)(cid:18)Z0 2 20 (cid:19) 2 are the projection residuals of JW and W on the spaces spanned by JW and W respectively. L 2 (cid:0) 1;c1 1 2;C2 2 Remark 1.1 The limiting distribution of the AEG statistic depends on the unknown parameter C, butisotherwisefreeofnuisanceparameters. ForagivenC,theasymptoticdistributioncanthuseasily be tabulated. The next section describes a feasible implementation of the test when C is unknown. Remark 1.2 E⁄ectively, the AEG test evaluates whether the persistence in the residuals is less than that predicted under the null hypothesis of no cointegration. However, since the original data is not necessarily a unit root process, the critical values re(cid:135)ect this fact. In the special case of C = 0, the limiting distribution reduces to the usual one for pure unit root variables. Remark 1.3 In empirical work, a constant or a constant and a linear trend are typically included in the empirical regression (2). As in standard cointegration analysis, this will a⁄ect the limiting distribution in a straightforward manner (e.g. Phillips and Ouliaris, 1990) and thus the critical values used, but will otherwise not alter the analysis. 3 Feasible implementation For a known C, the above test is trivial to use once critical values for the asymptotic distribution are obtained. Unfortunately, C is typically not known. We therefore consider a Bonferroni test approach, whichissimilartothatusedbyCavanaghet al. (1995)andCampbellandYogo(2006)intheirpursuit of inference in predictive regression with near-integrated variables. 5

Consider con(cid:133)dence intervals for c , i=1;:::;m, of the shape [c ;c ] m with an overall coverage i f i i gi=1 rate equal to 100 (1 (cid:11) ) percent. Let c [c ;c ] m be the set of parameter values in this (cid:2) (cid:0) 1 f i 2 i i gi=1 con(cid:133)dence region for which the critical value of the asymptotic distribution of the AEG statistic is e mostconservative,forsomegiven(cid:11) percentlevel(e.g. (cid:133)vepercent). IftheAEGstatisticisevaluated 2 using this conservative critical value, calculated at the (cid:11) percent level, the size of the resulting 2 cointegration test will be less than or equal to (cid:11)=(cid:11) +(cid:11) , by Bonferroni(cid:146)s inequality. 1 2 However, relativetotheCavanaghet al. (1995)andCampbellandYogo(2006)studies, thereisan additional complication in the current setup. In those papers, there is only one local-to-unity process, whereas here there are at least two in the simplest case with just one regressor. In the univariate case, con(cid:133)dence intervals of the local-to-unity parameter can be obtained by inverting a unit root test statistic (Stock, 1991). In the m dimensional case, a con(cid:133)dence region for C could be obtained by (cid:0) inverting individual unit root test statistics in order to obtain con(cid:133)dence intervals [c ;c ], i=1;:::;m, i i eachwithcoveragerate1 (cid:11) =m. Theoverallcon(cid:133)dencelevelof [c ;c ] m isatleast100 (1 (cid:11) ) (cid:0) 1 f i i gi=1 (cid:2) (cid:0) 1 percent,againbyBonferroni(cid:146)sinequality. Althoughtheoreticallysound,suchanapproachsu⁄ersfrom the practical disadvantage that it would be virtually impossible to tabulate the critical values for the asymptotic distribution beyond the simple two-dimensional case. We therefore propose a simpler approach that allows for tabulation of critical values and seems to give up little in robustness. Intuitively,theAEG testevaluateswhetherthepersistence,orautoregressiveroot,intheregression residuals,v ,islessthanintheoriginaldata,y . Asseeninequations(3)and(4),thecriticalvaluesof t t thetestdependonboththepersistenceinthe(cid:145)dependent(cid:146)variable,y ,andtheregressors,x ,denoted t t c and C respectively. However, it seems reasonable to conjecture that the main determinant of the 1 2 asymptotic distribution will be c , rather than C . Thus, using C^ =C^ =diag(c^ ;:::;c^ ) for some c^ , 1 2 1 1 1 1 rather than C~ = diag(c~ ;c~ ;:::;c~ ), to form critical values might not cause a large size distortion in 1 2 m the test. Although this conjecture is di¢ cult to evaluate analytically, extensive simulation evidence supports it. For instance, Figure 1 shows the critical values for the AEG test in the two-dimensional case with an intercept in the empirical regression. As is evident, the primary changes come from changingc ,whereasthecriticalvaluesarealmostconstantacrossC . Additionalevidencesupporting 1 2 this conclusion is provided by simulations in the following section. Furthermore, if C = diag(c ;:::;c ) is used to calculate the critical values for the asymptotic 1 1 1 distribution in Theorem 1, the AEGcointegration test will be more conservative as the value of c 1 6

decreases; that is, as c becomes more negative, so do the corresponding critical values, as shown in 1 Table A3. Only the lower bound on c , say c , is therefore of interest in constructing a conservative 1 1 test; for a given con(cid:133)dence level, such a lower bound can be obtained from a one-sided con(cid:133)dence interval for c , [c ;+ ). 1 1 1 By restricting the attention to the parameter c , and calculating critical values based on C = 1 1 diag(c ;:::;c ), it now becomes easy to implement the Bonferroni method. The lower con(cid:133)dence 1 1 bound for c , c , is obtained by inverting a unit root test statistic for the variable y . Based on this 1 1 t lowerboundofc , thetestisevaluatedusingthecorrespondingcriticalvalueforC =diag(c ;:::;c ). 1 1 1 1 If the lower bound c has con(cid:133)dence level 1 (cid:11) and the AEG test is evaluated at the (cid:11) level, the 1 (cid:0) 1 2 resulting test will have a size no larger than (cid:11)=(cid:11) +(cid:11) .3 1 2 In general, Bonferroni(cid:146)s inequality is strict, and the size of the test will be less than (cid:11). To obtain a correctly sized test of size (cid:11)~, which is distinct from (cid:11) = (cid:11) +(cid:11) , we (cid:133)rst (cid:133)x (cid:11) at some level and 1 2 2 then (cid:133)nd (cid:11) such that the resulting test has size (cid:11)~. Finding (cid:11) is e⁄ectively a trial and error exercise. 1 1 In the simulations below, we let (cid:11)~ = (cid:11) = 0:05 and show that setting (cid:11) equal to 50 percent will 2 1 approximately result in an overall (cid:133)ve percent test. Thus, by e⁄ectively using a median unbiased estimate of c , an approximately correctly-sized test is obtained. These results are discussed more 1 extensively in conjunction with the Monte Carlo simulations in the next section. In terms of practical implementation, we follow Campbell and Yogo (2006) and invert Elliot et al.(cid:146)s (1996) DF-GLS unit root test statistic to obtain a lower bound for c . Table A1 provides the 1 lower 95th, 75th, 50th, 25th, and 5th percent con(cid:133)dence bounds of c , given a value of the DF-GLS 1 test statistic.4 For instance, the lower con(cid:133)dence bound that corresponds to (cid:11) =0:05 is given in the 1 100 (1 (cid:11) )%=95%column. TableA2providesthecorrespondingboundswhenatrendisallowed 1 (cid:2) (cid:0) forintheDF-GLSregression. TableA3tabulatesthe(cid:133)vepercentcriticalvaluesfortheAEG statistic, for c =0 to c = 60, assuming that c =c =:::=c ; values for one to (cid:133)ve regressors are provided 1 1 1 2 m (cid:0) for the cases of no intercept, intercept, and intercept and a linear trend in the empirical regression. Henceforth,wewillrefertothecointegrationtestconstructedinthemanneraboveastheBonferroni AEG test,withtheadditionalspeci(cid:133)cationofthevalueof(cid:11) whennecessary. Unlessotherwisenoted, 1 we let (cid:11) =0:05. 2 3Since C2 is assumed not to play an important role in the distribution of the test-statistic, the only uncertainty regarding the persistence of the data comes from uncertainty regarding c1. The con(cid:133)dence levelof the lower bound C 1 is therefore 1 (cid:11)1 ratherthan 1 m (cid:11)1,as discussed above. 4Note that (cid:0) , for instance, the tw (cid:0) o lo (cid:2) wer con(cid:133)dence bounds at the 5 percent and 95 percent level provide a two-sided con(cid:133)dence intervalwith con(cid:133)dence level90 percent. 7

4 Finite-sample properties 4.1 Size properties Weanalyzethe(cid:133)nite-samplepropertiesoftheproposedtestprocedurethroughaseriesofMonteCarlo simulations. Starting with the size properties, it is assumed that the data generating process (DGP) is given by equation (1), with the innovations u drawn from a multivariate normal distribution such t that E[u ] = 0 and E[u u ] = I. The sample size is set to either T = 100 or 500 and the number of t t 0t regressors, n=m 1, is equal to either one or three. The regression (cid:0) y =(cid:11)+(cid:12) x +(cid:29) (6) t 0 t t is estimated, which is a spurious regression given the above DGP, and the cointegration tests are applied to the (cid:133)tted residuals, v^. Each simulated m dimensional time-series z is thus partitioned t t (cid:0) as z t = (y t ;x 0t )0, as described previously. When all components in z t are ex-ante identical, i.e. have the same persistence c, the (cid:133)rst component series is set to y and the remainder to x . When c varies t t i between each series, we describe explicitly which series are set as y and x . All tests are performed at t t the (cid:133)ve percent signi(cid:133)cance level and are evaluated using the critical values given in Table A3. The results are based on 10;000 repetitions. Inthe(cid:133)rstroundofsimulations,weletthelocal-to-unitymatrixforz begivenbyC =diag(c;:::;c), t so that all the series have identical persistence. The local-to-unity parameter c varies from 0 to 30. (cid:0) Figure 2 shows the size properties for the traditional AEG cointegration test, which by de(cid:133)nition is evaluated at c = 0, as a function of the local-to-unity parameter c. The nominal size of the test is (cid:133)ve percent, and for c close to zero, the actual rejection rate is also close to (cid:133)ve percent. However, as c decreases in value, the test starts over-rejecting and the rejection rates already approach ten percent for c = 5. The rejection rates become even larger and approach one as c becomes even smaller. It (cid:0) should be stressed that this is not a small-sample bias, but a re(cid:135)ection of the inconsistency of the test when c<0. Since the autoregressive root of the residual in equation (6) is less than one for c<0, the AEG test, evaluated under the assumption of c=0, will reject the null of a unit root in the residuals morefrequentlythanitsnominalsize. Fortimeseriesthatdonotnecessarilyhaveaunitroot,standard cointegrationtestscanthusbehighlymisleading. Thisraisesquestionsregardingpreviousstudiesthat have relied on cointegrating methods, despite having found evidence of stationarity of the included 8

variables; see, for example, Crowder and Ho⁄man (1996). We next consider the size properties of the Bonferroni AEG test using a conservative estimate of C. As discussed in the previous section, we use C = diag(c ;:::;c ) where c is the lower bound on 1 1 1 1 thepersistenceiny . AdirectapplicationoftheBonferronimethodsuggestschoosingc suchthatthe t 1 one-sidedcon(cid:133)denceinterval[c ;+ )hascon(cid:133)dencelevel100 (1 (cid:11) )percent,andthenevaluating 1 1 (cid:2) (cid:0) 1 the AEG test-statistic at the (cid:11) percent level for a total size of (cid:11) = (cid:11) +(cid:11) percent. In practice, 2 1 2 however, such an approach will deliver extremely conservative tests. For instance, if (cid:11) = (cid:11) = 0:05, 1 2 the rejection rate for the resulting test is virtually identical to zero in the simulations considered here. Instead, we follow the approach outlined above and (cid:133)x (cid:11) = 0:05 and choose (cid:11) such that the size 2 1 of the overall test is close to (cid:133)ve percent. In particular, we consider setting (cid:11) = 0:25;0:50 and 0:75. 1 Thatis,c ischosenasthelowerboundinone-sidedcon(cid:133)denceintervalswithcon(cid:133)dencelevelsequalto 1 75, 50, and 25 percent, respectively. To obtain these values for c , the DF-GLS unit root test-statistic 1 is inverted, using the values in Table A1.5 Figure 3 shows the results for the Bonferroni AEG test using these di⁄erent estimates of C . It 1 is immediately apparent that for small values of c, the test tends to over-reject when (cid:11) = 0:75, and 1 under-reject when (cid:11) = 0:25. For (cid:11) = 0:50, the test still tends to under-reject somewhat, except 1 1 for small values of c in the case of T = 500 and n = 1, where there is instead a slight over-rejection. Overall, however, for (cid:11) = 0:50, the rejection rate is typically between two and (cid:133)ve percent. One 1 could achieve rejection rates that are somewhat closer to the nominal size by letting (cid:11) vary with c 1 1 in some manner, but at the cost of a substantially more cumbersome procedure. Using a (cid:133)xed value of (cid:11) =0:50, for all values of c , yields a very simple test to implement. The procedure would simply 1 1 be given as: (i) Obtain the value of the AEG test statistic from a standard implementation of the Engle and Granger test. (ii) Calculate the DF-GLS unit root statistic for the y variable and obtain the corresponding value t of c from Table A1 or A2. 1 (iii) Compare the AEG test statistic to the critical value corresponding to c in Table A3. 1 5The number of lags included in the DF-GLS test is chosen using the Schwarz (1978) information critierion, with a maximum number of two allowed in order to keep the simulation times managable. The same number of lags is also included in the AEGregression;that is,in (cid:1)v^t=(cid:11) (cid:3) v^t (cid:0) 1+ p i=1 ’ i (cid:1)v^t (cid:0) i+wt. P 9

It may seem surprising that using, for instance, a lower bound with only a 25 percent con(cid:133)dence level, does not result in a larger size distortion. Figure 4 helps shed some light on this puzzle. The results in the (cid:133)gure are based on 10;000 simulations of a univariate local-to-unity process, with localto-unity parameter c, iid normal innovations and sample size T = 500. It shows estimates of the lower bounds of c, with con(cid:133)dence levels of 25, 50, and 75 percent, using the inversion of the DF-GLS statistic in Table A1. The panels in Figure 4 show the densities for the lower-bounds estimates for c= 5; 10; 20; and 30. As expected, the bounds estimates at the 75 percent con(cid:133)dence level are (cid:0) (cid:0) (cid:0) (cid:0) furthest to the left. However, the densities are far from symmetric, especially for c close to zero; the density for the 25 percent con(cid:133)dence bound is also less symmetric than the density for the 75 percent bound. Thus, although the density is shifted further to the right as the con(cid:133)dence level decreases, which leads to estimates of c closer to zero, the shift is not symmetric and the risk of vastly overestimating c is not increased dramatically. This explains, to some extent, why the rejection rates in the cointegration test only increase slowly as the con(cid:133)dence level of the lower bound is decreased. In the last set of size simulations, shown in Figure 5, we analyze the properties of the Bonferroni AEG test when the local-to-unity parameters c , i=1;:::;m are not identical; i.e. when the processes i in z do not have the same persistence. Two di⁄erent cases are considered. In the (cid:133)rst case, there t are two regressors with persistence parameters equal to 10 and 20. In the second setup, there are (cid:0) (cid:0) three regressors with persistence parameters 0; 10, and 20. In both cases, it is assumed that the (cid:0) (cid:0) persistence in y , c , varies between 0 and 30. Thus, in the (cid:133)rst case, C = diag(c ; 10; 20), and t 1 1 (cid:0) (cid:0) (cid:0) in the second case C = diag(c ;0; 10; 20). The same methods as in the case with identical c s 1 i (cid:0) (cid:0) are used and the results for (cid:11) = 0:25;0:50; and 0:75, are shown. Overall, the results in Figure 5 1 are very similar to those in Figure 3. Using (cid:11) = 0:50 and a nominal size of (cid:133)ve percent results in 1 actual rejection rates around three percent. Given the results shown previously in Figure 1, it is not surprising that the test also performs well when the c s are not identical. i Insummary, theproposedprocedurefortestsofcointegrationindatawithanunknownC appears to work well in (cid:133)nite samples, once the con(cid:133)dence level of the lower bound is chosen appropriately. Additional (cid:133)ne tuning of this con(cid:133)dence level could be done to bring the actual size even closer to the nominal size, but at the cost of adding some complexity. 10

4.2 Power properties We next perform a second Monte Carlo simulation to evaluate the (cid:133)nite-sample rejection rates under the alternative of cointegration. The (cid:145)independent(cid:146)variable x is still generated according to equation t (1) using iid standard normal innovations. However, the (cid:145)dependent(cid:146)variable y , is now generated as t y =(cid:12) x +v ; (7) t 0 t t where v is an AR(1) process with an auto-regressive root (cid:26); the innovations to this AR process are t iid standard normal. (cid:12) is set to an n-vector of ones. The same empirical regression, including the constant, as in the size simulations is estimated, and the Bonferroni AEG test with (cid:11) = 0:50 is 1 applied to the estimated residuals v^. The critical values that are used are thus for the case with a t constant in the regression. Two di⁄erent sample sizes, T = 100 and 500, and n = 1 and 3 regressors, are considered. In the case of one regressor, the persistence in x is set equal to either C = 2; 10; t 2 (cid:0) (cid:0) or 20. In the case of three regressors, it is assumed that C =diag(0; 10; 20). 2 (cid:0) (cid:0) (cid:0) Figure 6 shows the results in four sub-plots corresponding to the di⁄erent combinations of sample size and number of regressors. The vertical axes of the graphs show the power of the Bonferroni AEG testplottedagainstthepersistence(cid:26)intheerrortermv . InthecaseofT =100,resultsfor(cid:26) [0:5;1] t 2 are shown and for the T = 500, results for (cid:26) [0:8;1] are shown. As is to be expected, power is a 2 monotone and declining function of the persistence, (cid:26). It should be noted that for very large values of (cid:26), we expect the test to have low power; for example, in the bivariate case, a residual that is less persistent than y cannot be generated by regressing y on x when (cid:26)>1+C =T. For most values of t t t 2 (cid:26), however, the test appears to exhibit good power properties and appears su¢ ciently powerful that it would be a useful tool in many empirical applications, including those with relatively small sample sizes. 5 Empirical illustrations To illustrate the empirical use of the Bonferroni AEG test, we next consider two applications where the variables in question are all fairly persistent, but not necessarily pure unit root processes. As a comparison to the robust methodology proposed in this paper, we will also conduct the traditional AEG test. 11

5.1 The Fisher hypothesis It is well known that both nominal interest rates and in(cid:135)ation are fairly persistent in most countries. Accordingly, cointegration techniques have been a popular approach to test the Fisher hypothesis in more recent years; see, for example, Mishkin (1992), Wallace and Warner (1993), Evans and Lewis (1995), and Crowder and Ho⁄man (1996). However, the assumption made in most of these studies of exactunitrootsinbothnominalinterestratesandin(cid:135)ationcanbequestionedonboththeoreticaland empirical grounds.6 It is therefore worth re-interesting this issue using the Bonferroni AEG test. A common formulation of the Fisher hypothesis is that the m-period nominal interest rate (im) is t related to the real interest rate (rm) and in(cid:135)ation ((cid:25)m) according to t t im =E (rm)+E ((cid:25)m): (8) t t t t t Relying on the commonly made assumption of a constant or mean-reverting real interest rate, an empirical version of the Fisher hypothesis can be written as im =(cid:11)+(cid:12)(cid:25)m+v ; (9) t t t where the constant (cid:11) has the interpretation of the (constant) equilibrium real interest rate, the error term v is assumed to be a stationary ARMA process and (cid:12), in the most traditional interpretation, t should be equal to unity.7 Monthly data on the short nominal interest rate (cid:150)given by the three month treasury bill (cid:150)and CPI in(cid:135)ation from January 1955 to October 2006 in the United States were provided by the Board of Governors of the Federal Reserve System. Table 1 shows the results from the DF-GLS unit root test and the KPSS stationarity test, as well as the median unbiased estimate of c, denoted c^, and a 90 percent con(cid:133)dence interval for c; the estimates and con(cid:133)dence intervals of c are derived using the values in Table A1 and linear interpolation.8 As can be seen, the evidence for a unit root in the interest rate appears reasonably strong; the DF-GLS test fails to reject the null of a unit root whereas the KPSS test rejects the null of stationarity. For in(cid:135)ation, on the other hand, the evidence is more 6See,forexample,Wu and Zhang (1996),Culverand Papell(1997),and Wu and Chen (2001). 7Note that in the estimations below, time t in(cid:135)ation is given as future in(cid:135)ation between t and t+m. This can be motivated by assuming rationalexpectations;see,forexample,Mishkin (1992). 8Regarding the speci(cid:133)cation of deterministic terms in the unit root tests, it should be noted that we test for mean reversion around a constant level. 12

mixed since the DF-GLS test rejects a unit root but the KPSS test rejects stationarity.9 Table 1: Unit root tests. i (cid:25) t t DF-GLS -1.40 -2.54 (cid:3) KPSS 0.53 0.52 (cid:3) (cid:3) c^ -3.40 -12.91 90% CI for c [-9.06, 2.00] [-21.37, -3.46] Notes: * indicates signi(cid:133)cance at the (cid:133)ve percent level. The cointegration tests are conducted using a signi(cid:133)cance level of (cid:133)ve percent. For the Bonferroni AEG test, based on the simulation results in the previous section, we set (cid:11) = 0:5; thus c^= 3:40, 1 (cid:0) themedianunbiasedestimateforthenominalinterestrate,isusedtoestablishthecriticalvalueinthe Bonferroni AEG test. The results from the cointegration tests based on the speci(cid:133)cation in equation (9) are given in Table 2.10 Asymptotic critical values are used for both the standard Engle-Granger test (denoted AEG) and the Bonferroni AEG test (denoted AEGC) and are provided in Table 2; the AEGC critical value is obtained from Table A3 and linear interpolation. Table 2: Cointegration tests. Test statistic -3.43 Critical value AEGC -3.47 Critical value AEG -3.34 Notes: Nominalsize is 0.05. As can be seen, the null hypothesis of no cointegration is rejected if the standard method is used, as the test statistic is smaller than the critical value for the traditional AEG test. However, when the Bonferroni AEG test is used, the null hypothesis is not rejected. Thus, performing inference using robust methods, there is no strong evidence of cointegration, or co-movement, between the nominal interest rate and in(cid:135)ation in U.S. data. This raises doubts about the validity of the Fisher hypothesis, and also illustrates the importance of controlling for the unknown degree of persistence in the data; assuming unit roots in the data, the cointegration test would have resulted in evidence favorable of 9Lag length in the DF-GLS testwasdetermined using the Schwarz (1978)information criterion. Forthe KPSS test, a Newey-West estimatorwas employed to correct forserialcorrelation. 10As in the DF-GLS test,lag length in the test equation is determined using the Schwarz (1978) criterion. 13

the Fisher hypothesis. Having looked at a traditional application from the macroeconomic literature, we next turn to a recent issue from (cid:133)nancial economics. 5.2 Consumption, aggregate wealth and stock returns Many studies argue that (cid:133)nancial valuation ratios such as the dividend- and earnings-price ratios may have predictive power for excess stock returns over the risk-free rate. In a novel attempt to tie macroeconomic variables more closely to (cid:133)nancial markets, Lettau and Ludvigson (2001) argue that consumption is a function of aggregate wealth. Based on this claim, they suggest that aggregate consumption (k ), asset holdings (a ) and labour income (y ) are cointegrated and that the deviation t t t from equilibrium is useful in terms of predicting both excess stock returns and real stock returns. The empirical speci(cid:133)cation used by Lettau and Ludvigson accordingly takes its starting point in a cointegrating relationship of the type k =(cid:22)+(cid:18)a +(cid:21)y +(cid:31) ; (10) t t t t where the error term (cid:31) is assumed to be a stationary ARMA process which has predictive power for t future returns. However, there is no strong a priori reason to assume that the above variables contain pure unit roots.11 WethereforeinvestigatethesensitivityofLettauandLudvigson(cid:146)sresultswhentheuncertainty regarding the persistence in the data is taken into account. Quarterly data on US consumption, asset holdingsandlabourincomerangingfromthe(cid:133)rstquarter1952tothefourthquarter2006wereobtained from Professor Ludvigson(cid:146)s web page;12 all variables are given by the natural logarithm of real, per capita data. Table 3 shows the results from unit root tests and stationarity tests for all variables and also provides the median unbiased estimates of c, c^, as well as 90 percent con(cid:133)dence intervals.13 The 11As was shown above, the persistence of the dependent variable is of special importance when using the AEG test. Theassumptionofaunitrootinconsumptionisthusofparticularinterest. Althoughthisconjecture(cid:133)ndssomesupport (cid:150)see, for example, Hall (1978) and Gali (1993) (cid:150)the opinion in the litterature is far from unanimous. For instance, the vast literatue that uses linear trends to detrend consumption (cid:150)see, for example, Cooper and Ejarque (2000) and Casares (2007) (cid:150)implicitly or explicitly assumes that consumption is trend stationary rather than generated by a unit root process. Furthermore, it has been argued that consumption and output should be integrated of the same order. Thus, if output is trend stationary (e.g. Flavin, 1981 and Diebold and Senhadji, 1996) then consumption should be as well. 12http://www.econ.nyu.edu/user/ludvigsons/ 13Note that in this application, the unit root tests have both constant and trend included in the speci(cid:133)cation. Thus, the estimates and con(cid:133)dence intervals ofcare derived using the values in Table A2;again,linearinterpolation is used. 14

evidence for unit roots in consumption and labour income seems strong, whereas it is mixed for asset holdings. Table 3: Unit root tests. k a y t t t DF-GLS -1.95 -2.54 -0.78 KPSS 0.36 0.20 0.38 (cid:3) (cid:3) (cid:3) c^ -4.06 -9.98 2.32 90% CI for c [-12.28, 3.35] [-19.63, 2.32] [-2.18, 4.44] Notes: * indicates signi(cid:133)cance at the (cid:133)ve percent level. As in the previous application, we choose a signi(cid:133)cance level of (cid:133)ve percent for the cointegration tests and set (cid:11) =0:5. The results from the AEG and Bonferroni AEG cointegration tests are shown 1 in Table 4. The null hypothesis of no cointegration is rejected regardless of which test is used. The robust cointegration methods developed here thus support the conclusion of Lettau and Ludvigson that US consumption, asset holdings and labour income are cointegrated. Table 4: Cointegration tests. Test statistic -4.03 Critical value AEGC -3.86 Critical value AEG -3.77 Notes: Nominalsize is 0.05. 6 Conclusion For many economic time series, it is di¢ cult to justify theoretically that they are generated by unit root processes. This is problematic from an empirical point of view since cointegration tests may be misleading when the data follow near-integrated, rather than pure unit root, processes. The size distortions of cointegration tests relying on the unit root assumption (cid:150)combined with the fact that standardOLSinferencecouldleadtospuriousresults(cid:150)makesitunclearhowtoanalyzeamultivariate time series of near-integrated variables. 15

Inthispaper,wehaveextendedastandardresidual-basedcointegrationtesttoallowforanunknown local deviation from the unit root assumption. This more robust test is easy to implement and Monte Carlo simulations show that it works well in (cid:133)nite samples. Unlike standard cointegration tests, the methods developed in this paper thus provide a means of performing valid inference on a multivariate near unit root process. The framework suggested in this paper therefore takes another step towards addressing the problems associated with inference when variables are near-integrated. The methods presented here take their starting point in the work of Engle and Granger (1987). In future research it would also be of interest to see Johansen(cid:146)s (1988,1991) VAR-based framework extended to a setting with near-integrated variables. 16

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dnert emit a tuohtiw ,citsitats SLG-FD eht fo eulav nevig a roF .citsitats SLG-FD eht no desab c rof sdnuob ecned(cid:133)noc rewoL :1A elbaT 5 dna ;52;05;57;59 fo slevel ecned(cid:133)noc htiw c retemarap ytinu-ot-lacol eht fo sdnuob ecned(cid:133)noc rewol evig snmuloc gniwollof eht ,dedulcni .ylevitcepser ,tnecrep %5 %52 %05 %57 %59 SLG-FD %5 %52 %05 %57 %59 SLG-FD 13.0- 15.4- 44.7- 14.01- 09.41- 0.2- 32.4 93.2 74.1 27.0 92.0- 0.1 88.0- 12.5- 52.8- 73.11- 59.51- 1.2- 91.4 43.2 14.1 56.0 04.0- 9.0 63.1- 49.5- 90.9- 53.21- 41.71- 2.2- 51.4 92.2 53.1 75.0 05.0- 8.0 89.1- 47.6- 79.9- 83.31- 43.81- 3.2- 21.4 42.2 92.1 94.0 36.0- 7.0 55.2- 84.7- 29.01- 24.41- 75.91- 4.2- 80.4 91.2 32.1 04.0 67.0- 6.0 82.3- 23.8- 98.11- 84.51- 48.02- 5.2- 40.4 31.2 51.1 03.0 19.0- 5.0 59.3- 91.9- 19.21- 16.61- 51.22- 6.2- 00.4 70.2 70.1 02.0 70.1- 4.0 96.4- 60.01- 59.31- 87.71- 35.32- 7.2- 59.3 20.2 99.0 90.0 52.1- 3.0 25.5- 60.11- 30.51- 89.81- 39.42- 8.2- 09.3 49.1 09.0 30.0- 64.1- 2.0 82.6- 60.21- 31.61- 02.02- 43.62- 9.2- 58.3 78.1 08.0 71.0- 66.1- 1.0 41.7- 80.31- 92.71- 94.12- 17.72- 0.3- 08.3 97.1 07.0 13.0- 98.1- 0.0 79.7- 21.41- 54.81- 18.22- 72.92- 1.3- 57.3 17.1 95.0 64.0- 41.2- 1.0- 88.8- 12.51- 26.91- 71.42- 68.03- 2.3- 96.3 26.1 84.0 36.0- 14.2- 2.0- 48.9- 33.61- 78.02- 35.52- 44.23- 3.3- 36.3 35.1 43.0 28.0- 27.2- 3.0- 38.01- 25.71- 51.22- 49.62- 60.43- 4.3- 75.3 24.1 81.0 30.1- 50.3- 4.0- 08.11- 07.81- 94.32- 93.82- 87.53- 5.3- 15.3 03.1 20.0- 92.1- 54.3- 5.0- 78.21- 78.91- 58.42- 78.92- 34.73- 6.3- 04.3 51.1 32.0- 06.1- 48.3- 6.0- 69.31- 61.12- 42.62- 44.13- 90.93- 7.3- 82.3 89.0 74.0- 49.1- 13.4- 7.0- 90.51- 84.22- 56.72- 89.23- 58.04- 8.3- 71.3 87.0 57.0- 23.2- 78.4- 8.0- 52.61- 28.32- 11.92- 55.43- 96.24- 9.3- 60.3 45.0 01.1- 87.2- 44.5- 9.0- 25.71- 81.52- 26.03- 22.63- 25.44- 0.4- 19.2 82.0 74.1- 72.3- 40.6- 0.1- 17.81- 55.62- 71.23- 78.73- 53.64- 1.4- 17.2 60.0- 09.1- 97.3- 37.6- 1.1- 78.91- 39.72- 07.33- 05.93- 42.84- 2.4- 94.2 93.0- 53.2- 73.4- 54.7- 2.1- 22.12- 44.92- 13.53- 72.14- 41.05- 3.4- 92.2 67.0- 88.2- 79.4- 91.8- 3.1- 75.22- 49.03- 49.63- 70.34- 41.25- 4.4- 10.2 81.1- 04.3- 66.5- 40.9- 4.1- 98.32- 54.23- 85.83- 68.44- 69.35- 5.4- 47.1 56.1- 79.3- 33.6- 09.9- 5.1- 12.52- 00.43- 32.04- 86.64- 80.65- 6.4- 83.1 51.2- 06.4- 50.7- 28.01- 6.1- 95.62- 76.53- 59.14- 45.84- 02.85- 7.4- 30.1 27.2- 32.5- 58.7- 57.11- 7.1- 50.82- 92.73- 07.34- 93.05- 72.06- 8.4- 95.0 72.3- 49.5- 56.8- 87.21- 8.1- 35.92- 09.83- 05.54- 13.25- 83.26- 9.4- 22.0 68.3- 96.6- 15.9- 48.31- 9.1- 20

SLG-FD eht fo eulav nevig a roF .dedulcni dnert emit raenil a htiw citsitats SLG-FD eht no desab c rof sdnuob ecned(cid:133)noc rewoL :2A elbaT ecned(cid:133)nochtiwcretemarapytinu-ot-lacolehtfosdnuobecned(cid:133)nocrewolevigsnmulocgniwollofeht,dnertemitraenilarofgniwolla,citsitats .ylevitcepser ,tnecrep 5 dna ;52;05;57;59 fo slevel %5 %52 %05 %57 %59 SLG-FD %5 %52 %05 %57 %59 SLG-FD 72.3 96.1 65.4- 51.8- 09.21- 0.2- 42.5 27.3 70.3 36.2 02.2 0.1 21.3 82.1 45.5- 51.9- 40.41- 1.2- 02.5 96.3 40.3 06.2 61.2 9.0 59.2 01.2- 84.6- 71.01- 62.51- 2.2- 61.5 56.3 10.3 75.2 21.2 8.0 67.2 22.3- 74.7- 82.11- 25.61- 3.2- 31.5 26.3 79.2 35.2 90.2 7.0 85.2 62.4- 94.8- 04.21- 58.71- 4.2- 90.5 85.3 39.2 05.2 50.2 6.0 93.2 03.5- 95.9- 55.31- 41.91- 5.2- 50.5 55.3 09.2 64.2 20.2 5.0 91.2 33.6- 76.01- 77.41- 94.02- 6.2- 10.5 15.3 68.2 24.2 79.1 4.0 69.1 14.7- 08.11- 40.61- 79.12- 7.2- 79.4 74.3 28.2 83.2 39.1 3.0 16.1 74.8- 89.21- 53.71- 44.32- 8.2- 39.4 24.3 87.2 43.2 88.1 2.0 55.1- 26.9- 02.41- 76.81- 79.42- 9.2- 88.4 83.3 47.2 03.2 38.1 1.0 01.3- 57.01- 74.51- 20.02- 55.62- 0.3- 48.4 33.3 07.2 62.2 87.1 0.0 72.4- 19.11- 87.61- 84.12- 41.82- 1.3- 97.4 92.3 56.2 22.2 27.1 1.0- 55.5- 91.31- 01.81- 79.22- 68.92- 2.3- 57.4 42.3 16.2 71.2 46.1 2.0- 86.6- 84.41- 15.91- 94.42- 46.13- 3.3- 07.4 02.3 65.2 21.2 65.1 3.0- 19.7- 08.51- 69.02- 50.62- 24.33- 4.3- 46.4 51.3 25.2 70.2 74.1 4.0- 21.9- 51.71- 54.22- 76.72- 12.53- 5.3- 95.4 01.3 74.2 20.2 23.1 5.0- 03.01- 35.81- 59.32- 73.92- 90.73- 6.3- 45.4 50.3 24.2 59.1 18.0- 6.0- 26.11- 39.91- 65.52- 90.13- 99.83- 7.3- 94.4 10.3 63.2 98.1 85.1- 7.0- 69.21- 24.12- 91.72- 58.23- 79.04- 8.3- 34.4 59.2 13.2 28.1 92.2- 8.0- 43.41- 79.22- 58.82- 46.43- 60.34- 9.3- 73.4 98.2 62.2 57.1 59.2- 9.0- 97.51- 75.42- 65.03- 05.63- 81.54- 0.4- 13.4 28.2 81.2 16.1 07.3- 0.1- 13.71- 81.62- 43.23- 54.83- 81.74- 1.4- 52.4 67.2 01.2 54.1 34.4- 1.1- 77.81- 98.72- 31.43- 53.04- 63.94- 2.4- 71.4 96.2 30.2 27.0- 51.5- 2.1- 91.02- 65.92- 10.63- 73.24- 66.15- 3.4- 90.4 06.2 29.1 58.1- 10.6- 3.1- 38.12- 13.13- 09.73- 64.44- 19.35- 4.4- 10.4 25.2 08.1 57.2- 38.6- 4.1- 44.32- 51.33- 38.93- 06.64- 72.65- 5.4- 19.3 24.2 36.1 26.3- 47.7- 5.1- 00.52- 40.53- 98.14- 47.84- 47.85- 6.4- 18.3 13.2 63.1 64.4- 96.8- 6.1- 66.62- 99.63- 49.34- 89.05- 02.16- 7.4- 96.3 91.2 65.1- 33.5- 76.9- 7.1- 25.82- 69.83- 70.64- 23.35- 87.36- 8.4- 65.3 60.2 96.2- 22.6- 56.01- 8.1- 82.03- 59.04- 92.84- 46.55- 52.66- 9.4- 24.3 98.1 46.3- 71.7- 67.11- 9.1- 21

tnecrep ev(cid:133) eht ta citsitats GEA eht rof seulav lacitirc eht sevig elbat sihT .citsitats GEA eht rof seulav lacitirc tnecrep eviF :3A elbaT eht edivorp seulav fo tes tsr(cid:133) ehT .srosserger ev(cid:133) ot eno rof dna ,c= c=:::= c taht noitpmussa eht rednu c fo seulav tnere⁄id rof ,level m 1 on tub ,tpecretni na nehw seulav eht sedivorp tes dnoces ehT .noisserger gnitargetnioc eht ni dedulcni si tpecretni on nehw seulav lacitirc no desab era seulav ehT .dnert emit raenil a dna tpecretni na htob htiw esac eht tneserper seulav fo tes driht eht dna dedulcni si dnert emit .000;1= T htiw snoititeper 000;001 dnert dna tnatsnoC tnatsnoC tnatsnoc oN 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 c 00.5- 37.4- 44.4- 41.4- 97.3- 27.4- 24.4- 01.4- 77.3- 43.3- 14.4- 90.4- 37.3- 03.3- 77.2- 0 10.5- 27.4- 64.4- 41.4- 97.3- 37.4- 34.4- 21.4- 67.3- 73.3- 24.4- 90.4- 47.3- 23.3- 08.2- 1- 10.5- 47.4- 54.4- 61.4- 28.3- 37.4- 44.4- 21.4- 87.3- 04.3- 24.4- 01.4- 57.3- 53.3- 88.2- 2- 30.5- 67.4- 84.4- 81.4- 68.3- 57.4- 64.4- 51.4- 28.3- 54.3- 44.4- 21.4- 87.3- 93.3- 69.2- 3- 30.5- 87.4- 05.4- 12.4- 98.3- 67.4- 74.4- 71.4- 68.3- 05.3- 54.4- 51.4- 28.3- 54.3- 50.3- 4- 50.5- 97.4- 35.4- 42.4- 49.3- 77.4- 94.4- 12.4- 98.3- 65.3- 64.4- 71.4- 68.3- 15.3- 51.3- 5- 70.5- 18.4- 65.4- 72.4- 89.3- 08.4- 35.4- 42.4- 49.3- 26.3- 15.4- 12.4- 19.3- 85.3- 32.3- 6- 90.5- 58.4- 85.4- 23.4- 30.4- 18.4- 65.4- 82.4- 00.4- 86.3- 35.4- 62.4- 69.3- 46.3- 23.3- 7- 11.5- 98.4- 36.4- 63.4- 80.4- 68.4- 06.4- 33.4- 50.4- 57.3- 75.4- 03.4- 10.4- 27.3- 14.3- 8- 61.5- 19.4- 66.4- 14.4- 41.4- 98.4- 46.4- 73.4- 11.4- 28.3- 16.4- 43.4- 70.4- 97.3- 05.3- 9- 81.5- 39.4- 17.4- 44.4- 91.4- 29.4- 86.4- 34.4- 61.4- 98.3- 56.4- 04.4- 31.4- 68.3- 85.3- 01- 12.5- 89.4- 47.4- 15.4- 62.4- 59.4- 27.4- 74.4- 22.4- 79.3- 96.4- 54.4- 91.4- 39.3- 86.3- 11- 32.5- 20.5- 97.4- 65.4- 23.4- 99.4- 67.4- 25.4- 92.4- 30.4- 37.4- 05.4- 62.4- 10.4- 57.3- 21- 82.5- 60.5- 48.4- 06.4- 73.4- 30.5- 18.4- 85.4- 43.4- 01.4- 87.4- 65.4- 33.4- 80.4- 48.3- 31- 03.5- 90.5- 88.4- 66.4- 44.4- 70.5- 58.4- 46.4- 14.4- 81.4- 38.4- 06.4- 83.4- 61.4- 29.3- 41- 43.5- 41.5- 39.4- 17.4- 05.4- 11.5- 09.4- 96.4- 74.4- 52.4- 88.4- 66.4- 44.4- 32.4- 10.4- 51- 93.5- 91.5- 79.4- 87.4- 55.4- 61.5- 69.4- 47.4- 45.4- 03.4- 49.4- 27.4- 05.4- 03.4- 60.4- 61- 24.5- 32.5- 20.5- 28.4- 16.4- 02.5- 00.5- 08.4- 06.4- 83.4- 89.4- 87.4- 75.4- 73.4- 51.4- 71- 64.5- 72.5- 70.5- 88.4- 76.4- 52.5- 50.5- 58.4- 66.4- 44.4- 30.5- 38.4- 36.4- 34.4- 22.4- 81- 05.5- 23.5- 21.5- 49.4- 37.4- 92.5- 11.5- 09.4- 27.4- 05.4- 70.5- 98.4- 96.4- 05.4- 92.4- 91- 55.5- 73.5- 81.5- 99.4- 97.4- 43.5- 61.5- 79.4- 77.4- 85.4- 31.5- 49.4- 67.4- 75.4- 73.4- 02- 95.5- 14.5- 12.5- 40.5- 58.4- 93.5- 12.5- 10.5- 38.4- 56.4- 91.5- 00.5- 18.4- 26.4- 44.4- 12- 26.5- 54.5- 72.5- 80.5- 19.4- 24.5- 52.5- 70.5- 98.4- 07.4- 32.5- 40.5- 78.4- 96.4- 15.4- 22- 76.5- 94.5- 33.5- 51.5- 79.4- 74.5- 03.5- 31.5- 69.4- 77.4- 82.5- 11.5- 39.4- 67.4- 75.4- 32- 27.5- 45.5- 83.5- 91.5- 30.5- 25.5- 53.5- 91.5- 00.5- 38.4- 43.5- 61.5- 00.5- 18.4- 56.4- 42- 67.5- 95.5- 24.5- 52.5- 80.5- 75.5- 14.5- 32.5- 60.5- 98.4- 93.5- 22.5- 40.5- 88.4- 17.4- 52- 18.5- 46.5- 74.5- 92.5- 41.5- 26.5- 54.5- 92.5- 11.5- 59.4- 44.5- 72.5- 11.5- 39.4- 77.4- 62- 58.5- 96.5- 35.5- 63.5- 91.5- 76.5- 05.5- 43.5- 81.5- 10.5- 94.5- 23.5- 61.5- 00.5- 48.4- 72- 88.5- 27.5- 65.5- 14.5- 52.5- 07.5- 55.5- 93.5- 32.5- 80.5- 35.5- 83.5- 12.5- 60.5- 09.4- 82- 39.5- 77.5- 36.5- 74.5- 03.5- 67.5- 06.5- 54.5- 92.5- 31.5- 95.5- 34.5- 82.5- 21.5- 69.4- 92- 89.5- 28.5- 66.5- 25.5- 63.5- 18.5- 46.5- 94.5- 43.5- 91.5- 46.5- 74.5- 23.5- 81.5- 20.5- 03- 22

.)deunitnoc( citsitats GEA eht rof seulav lacitirC :3A elbaT dnert dna tnatsnoC tnatsnoC tnatsnoc oN 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 c 20.6- 68.5- 27.5- 75.5- 14.5- 58.5- 07.5- 55.5- 93.5- 52.5- 96.5- 35.5- 93.5- 32.5- 90.5- 13- 70.6- 29.5- 67.5- 26.5- 74.5- 09.5- 57.5- 16.5- 64.5- 03.5- 47.5- 95.5- 44.5- 03.5- 41.5- 23- 11.6- 59.5- 28.5- 76.5- 25.5- 49.5- 08.5- 66.5- 15.5- 63.5- 97.5- 36.5- 94.5- 53.5- 02.5- 33- 51.6- 10.6- 68.5- 27.5- 75.5- 99.5- 58.5- 07.5- 65.5- 24.5- 38.5- 96.5- 45.5- 14.5- 62.5- 43- 02.6- 50.6- 19.5- 67.5- 26.5- 40.6- 09.5- 67.5- 16.5- 74.5- 98.5- 57.5- 06.5- 64.5- 23.5- 53- 42.6- 01.6- 69.5- 28.5- 76.5- 90.6- 59.5- 18.5- 66.5- 35.5- 49.5- 08.5- 56.5- 15.5- 83.5- 63- 82.6- 41.6- 10.6- 78.5- 27.5- 31.6- 99.5- 68.5- 17.5- 85.5- 89.5- 48.5- 17.5- 65.5- 44.5- 73- 33.6- 91.6- 60.6- 29.5- 97.5- 81.6- 40.6- 19.5- 77.5- 46.5- 30.6- 09.5- 67.5- 36.5- 05.5- 83- 63.6- 42.6- 01.6- 79.5- 48.5- 22.6- 90.6- 59.5- 28.5- 96.5- 70.6- 49.5- 18.5- 86.5- 55.5- 93- 14.6- 82.6- 51.6- 10.6- 88.5- 72.6- 31.6- 00.6- 78.5- 47.5- 21.6- 99.5- 68.5- 37.5- 06.5- 04- 64.6- 23.6- 02.6- 60.6- 39.5- 13.6- 81.6- 50.6- 29.5- 97.5- 71.6- 40.6- 19.5- 87.5- 56.5- 14- 94.6- 73.6- 42.6- 11.6- 99.5- 53.6- 22.6- 01.6- 79.5- 48.5- 22.6- 90.6- 69.5- 48.5- 17.5- 24- 45.6- 14.6- 82.6- 61.6- 40.6- 04.6- 72.6- 51.6- 20.6- 09.5- 62.6- 31.6- 10.6- 98.5- 77.5- 34- 85.6- 54.6- 33.6- 02.6- 80.6- 44.6- 23.6- 91.6- 60.6- 49.5- 03.6- 81.6- 60.6- 39.5- 18.5- 44- 36.6- 94.6- 83.6- 52.6- 31.6- 94.6- 63.6- 42.6- 11.6- 00.6- 63.6- 32.6- 11.6- 89.5- 78.5- 54- 86.6- 55.6- 24.6- 03.6- 81.6- 45.6- 14.6- 92.6- 71.6- 40.6- 14.6- 82.6- 61.6- 40.6- 29.5- 64- 17.6- 95.6- 64.6- 53.6- 22.6- 85.6- 54.6- 33.6- 22.6- 90.6- 44.6- 33.6- 02.6- 90.6- 69.5- 74- 57.6- 36.6- 15.6- 04.6- 72.6- 16.6- 94.6- 83.6- 62.6- 41.6- 94.6- 73.6- 62.6- 41.6- 10.6- 84- 08.6- 86.6- 65.6- 44.6- 23.6- 76.6- 55.6- 34.6- 23.6- 91.6- 45.6- 34.6- 03.6- 91.6- 70.6- 94- 48.6- 27.6- 06.6- 84.6- 73.6- 27.6- 95.6- 84.6- 63.6- 52.6- 95.6- 74.6- 63.6- 32.6- 31.6- 05- 98.6- 67.6- 56.6- 45.6- 24.6- 67.6- 36.6- 35.6- 14.6- 92.6- 36.6- 15.6- 04.6- 92.6- 71.6- 15- 29.6- 18.6- 96.6- 85.6- 74.6- 08.6- 86.6- 65.6- 54.6- 53.6- 76.6- 65.6- 44.6- 33.6- 22.6- 25- 59.6- 58.6- 47.6- 36.6- 15.6- 38.6- 27.6- 26.6- 05.6- 93.6- 17.6- 06.6- 94.6- 93.6- 72.6- 35- 00.7- 98.6- 87.6- 76.6- 55.6- 88.6- 67.6- 66.6- 55.6- 34.6- 67.6- 56.6- 45.6- 34.6- 13.6- 45- 40.7- 39.6- 28.6- 17.6- 06.6- 29.6- 18.6- 07.6- 95.6- 94.6- 08.6- 96.6- 85.6- 74.6- 83.6- 55- 90.7- 89.6- 68.6- 67.6- 46.6- 79.6- 68.6- 47.6- 46.6- 25.6- 58.6- 57.6- 26.6- 25.6- 14.6- 65- 31.7- 10.7- 19.6- 08.6- 96.6- 10.7- 98.6- 08.6- 86.6- 85.6- 98.6- 87.6- 86.6- 75.6- 64.6- 75- 61.7- 60.7- 59.6- 58.6- 47.6- 50.7- 49.6- 38.6- 37.6- 26.6- 39.6- 38.6- 27.6- 26.6- 15.6- 85- 12.7- 01.7- 00.7- 88.6- 87.6- 90.7- 89.6- 98.6- 77.6- 76.6- 89.6- 78.6- 77.6- 66.6- 65.6- 95- 52.7- 51.7- 40.7- 39.6- 28.6- 31.7- 30.7- 39.6- 28.6- 17.6- 20.7- 29.6- 28.6- 17.6- 06.6- 06- 23

−3 −3.5 −4 −4.5 −5 −5.5 0 −5 0 −10 −15 −20 −20 −15 −10 −5 −25 −25 −30 −30 c 1 c 2 eulaV lacitirC c 2 −30 −25 −20 −15 −3.4 −3.6 −3.8 −4 −4.2 −4.4 −4.6 −4.8 −5 −10 −5 0 −5 −10 −15 −20 −25 −30 c 1 Figure 1: Critical values at the five percent level for the AEG test as a function of c and c . The 1 2 top panel shows the surface describing the five percent critical values of the AEG test, in the case of an intercept and one regressor, when c and c are non-identical. The bottom panel shows the 1 2 corresponding contour plot. The values are based on 10,000 repetitions with T =1,000.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 −5 −10 −15 −20 −25 −30 eziS c T=100, n=1 T=100, n=3 T=500, n=1 T=500, n=3 Figure 2: Size properties of the Engle and Granger (1987) test of cointegration, as a function of the local-to-unity parameter c. The graph shows the average rejection rates under the null hypothesis of nocointegrationfortheEngleandGrangertestofcointegration,i.e. thestandardAEGtestevaluated under the assumption that c = 0, for different true values of c. The sample size is equal to either T = 100 or 500, and the number of regressors equal to either n=1 or 3. The true persistence in the dataisequaltoC =diag(c,...,c), wherecvariesbetween0and 30. Theresultsarebasedon10,000 − repetitions.

T=100, n=1 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 eziS T=100, n=3 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 c eziS c T=500, n=1 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 eziS T=500, n=3 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 c eziS c α = 0.75 α = 0.50 α = 0.25 1 1 1 Figure 3: Size properties of the Bonferroni AEG test when the variables all have equal persistence. The graphs show the average rejection rates for the Bonferroni AEG test, under the null hypothesis of no cointegration, for α = 0.75,0.50, and 0.25. The sample size is equal to either T = 100 or 500, 1 and the number of regressors is equal to either n = 1 or 3. The true persistence in the data is equal to C =diag(c,...,c), where c varies between 0 and 30. The results are based on 10,000 repetitions. −

c=−5 c=−10 0.12 0.15 0.1 0.08 0.1 0.06 0.05 0.04 0.02 0 0 −20 −15 −10 −5 0 5 10 −25 −20 −15 −10 −5 0 5 c c c=−20 c=−30 0.08 0.06 0.05 0.06 0.04 0.04 0.03 0.02 0.02 0.01 0 0 −50 −40 −30 −20 −10 0 −60 −50 −40 −30 −20 −10 0 c c 75 Percent 50 Percent 25 Percent Figure 4: Estimates of the lower bounds of c. The graphs show the density of the estimates of the lower bounds of c, with confidence levels of 75, 50, and 25 percent, based on inversion of the DF-GLS statistic. The results are obtained from10,000 simulations of a univariate local-to-unityprocess, with local-to-unity parameter c, iid normal innovations and sample size T =500.

T=100, n=2 0.12 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 eziS T=100, n=2 0.12 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 c eziS c T=500, n=3 0.12 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 eziS T=500, n=3 0.12 0.1 0.08 0.06 0.04 0.02 0 0 −5 −10 −15 −20 −25 −30 c eziS c α = 0.75 α = 0.50 α = 0.25 1 1 1 Figure5: SizepropertiesoftheBonferroniAEGtestwhenc isnotidenticalforalli. Thegraphsshow i theaveragerejectionratesfortheBonferroniAEGtest,underthenullhypothesisofnocointegration, for α = 0.75,0.50, and 0.25. The sample size is equal to either T = 100 or 500, and the number 1 of regressors is equal to either n = 2 or 3. For n = 2, the true persistence in the data is equal to C = diag(c , 10, 20), and for n = 3, C = diag(c ,0, 10, 20), where c varies between 0 and 1 1 1 − − − − 30. The results are based on 10,000 repetitions. −

Equal Persistence: T=100, n=1 1 0.75 0.5 0.25 0 0.5 0.6 0.7 0.8 0.9 1 rewoP ρ Equal Persistence: T=500, n=1 1 0.75 0.5 0.25 0 0.8 0.9 1 rewoP Different Persistence: T=100, n=3 1 0.75 0.5 0.25 0 0.5 0.6 0.7 0.8 0.9 1 c=−2 c=−10 c=−20 ρ rewoP ρ Different Persistence: T=500, n=3 1 0.75 0.5 0.25 0 0.8 0.9 1 rewoP ρ Figure 6: Power properties of the Bonferroni AEG test. The graphs show the average rejection rates oftheBonferroniAEGtest,forα =0.50,underthealternativeofcointegration. Thepowerisplotted 1 as a function of ρ, the AR(1) persistence parameter in the cointegrating residuals. The sample size is set equal to either T =100 or 500. The left column gives results for the case of one regressor with persistenceC = 2, 10,or 20. Therightcolumngivestheresultsforthecasewiththreeregressors 2 − − − and C =diag(0, 10, 20). The results are based on 10,000 repetitions. 2 − −