A PANIC Attack on Unit Roots and Cointegration. July 31, Preliminary and Incomplete

A PANIC Attack on Unit Roots and Cointegration Jushan Bai Serena Ng July 3, 200 Preliminary and Incomplete Abstract his paper presents a toolkit for Panel Analysis of Non-stationarity in Idiosyncratic and Common components (PANIC). he point of departure is that the unobserved common factors can be consistently estimated, provided that the cross-section dimension is large. his allows us to decouple the issue of whether common factors exist from the issue of whether the factors are stationary. While unit root tests are imprecise when the common and idiosyncratic components have different orders of integration, direct testing on the two components are found to be more accurate in simulations. he tests are applied to a panel of real exchange rates. Keywords: Common factors, common trends, principal components, unit root, cointegration, panel data Dept. of Economics, Boston College, Chestnut Hill, MA 02467 Email Jushan.Bai@bc.edu Dept. of Economics, Johns Hopkins University, Baltimore, MD 228 Email: Serena.Ng@jhu.edu

Introduction A hallmark development in time series econometrics during the last decade is the observation that a linear combination of N individually non-stationary series can be stationary. A system of variables with these properties is said to be cointegrated. he Granger Representation heorem of Engle and Granger (987) establishes that if a N-vector time series X t has Wold representation X t = C(L)e t, then under cointegration, C() has rank k < N and X t has a vector errorcorrection representation (VECM). A large body of work has been devoted to the determination of k and efficient estimation of the parameters within the VECM framework under the assumption that N is small. An implication of cointegration is that a system with N variables only has k < N unit roots. It follows that under cointegration, non-stationarity in the data is driven by common stochastic trends. Idiosyncratic stochastic trends are possible only if there is no cointegration, in which case, C() is full rank and X t is a VAR. he fact that a cointegrated system has common trends also implies that X t can be decomposed into k non-stationary ( t ) and r = N k stationary (S t ) components. Stock and Watson (988) suggest a -S decomposition in which t is a random walk, while Gonzalo and Granger (995) obtain a decomposition in which t is serially correlated. Vahid and Engle (993) suggest a different decomposition that allows common cycles to be identified along with the common trends. Nonetheless, the common stochastic trends are generally not regarded as objects of interest. In part, this is because the non-uniqueness of the -S decomposition makes interpretation of the trends difficult. In part, this is also because under the small N assumption, consistent estimation of the cointegration parameters does not imply consistent estimation of the space spanned by the common trends. Hence even if the common trends can be given economic interpretation, the statistical properties of the estimated common trends are not well understood. he starting point of our analysis is first, that common trends and cycles can be consistently estimated, and second, that cointegration restrictions are not necessary to identify these common variations. he main appeal of stepping outside the cointegration framework is that it makes it possible to decouple the issue of whether common factors exist from the issue of whether these factors are stationary. Once the common factors are estimated, a common-idiosyncratic (I-C) decomposition of the data follows. Our main contribution is to provide testing procedures to determine whether non-stationarity in a panel of data is induced by common sources or if it is of the idiosyncratic type. Because the common variations can be stationary or non-stationary, they are referred to as common factors rather than common trends. Such an analysis is made possible by use of large dimensional panels. Because N is large by assumption, pooling across units allows

for consistent estimation of the common factors whether or not they are stationary. By virtue of the fact that N is large, more information can also be used for analysis. his not only improves statistical efficiency of the estimates, it also allows for identification of factors that are common in a more general sense. We consider a factor analytic framework X it = c i + λ if t + e it = c i + λ i F t + λ S i F s t + e it where F t is a k vector of common factors, λ i are the factor loadings, and e it is an idiosyncratic error. he common factors can be non-stationary (Ft ) or stationary (F S ). hus, common factors with different orders of integration can co-exist. A factor model with N variables can have N idiosyncratic components and an unrestricted number of common factors that can be stationary and non-stationary In a I-C decomposition, non-stationarity in X it can arise because of common stochastic trends, or because the idiosyncratic error is non-stationary. he factor model encompasses the multivariate local level model of Nyblom and Harvey (2000) as a special case. In that analysis, F s t is not permitted, N is fixed, and their object of interest is simply to determine k. Our interest is in understanding whether non-stationarity in X it is due to F t, e it, or both, based on their consistent estimates. We refer to these procedures as Panel Analysis of Non-stationarity in the Idiosyncratic and Common components (PANIC). Many macroeconomic issues of interest can be addressed using PANIC. Let X it be output across countries. With integrated world markets, real output of country i may consist of a global growth component (Ft ), a global cyclical component (Ft S ), and a idiosyncratic component which may itself be non-stationary. he framework then allows us to establish the source of non-stationarity and the relative importance of the country specific variations. In the same vein, let X it be the inflation rate of good i, and core inflation be that component of inflation that is common to all goods. he proposed framework allows us to determine if an individual inflation series is non-stationary because core inflation is non-stationary, or because the series has idiosyncratic component has a unit root. esting stationarity of the common and idiosyncratic components separately can potentially a problem that has challenged unit root and stationarity tests. he problem arise when F t is a weak random walk that is dominated by the variation of the stationary process e it. It can also arise when e it is I() but F t is stationary. In such cases, unit root tests on X it tend to be oversized while stationarity tests tend to have no power. he issue is documented by Schwert (989), and formally analyzed in Perron and Ng (996) and Ng and Perron (997). he problem arises because X it has 2

two components with different degrees of integration. However, if the non-stationary component is common to a large number of series, PANIC can be used to disentangle F t from the idiosyncratic component. We can test whether these components are individually I() or I(0). Non-stationarity in any one component will then imply X it is non-stationary. PANIC makes use of information in a panel and yet it is not designed specifically for panel unit root tests, though they can be performed. he testing is applied to the estimated common and idiosyncratic components, one at a time. hus, it can determine the proportion of the series in the panel that has unit roots. While pooling of the the tests based on observed data is invalid when the cross-sections are correlated, the I-C decomposition also allows us to pool on the basis of tests on the idiosyncratic components. hus, panel unit root tests can also be obtained to test the joint stationarity of these idiosyncratic components. he rest of the paper is organized as follows. he PANIC procedures are presented in Section 2. Asymptotic properties of the proposed tests are given in Section 3. Simulations are presented in Section 4, along with an application to real exchange rate data. Section 5 concludes. Proofs are given in the Appendix. 2 he PANIC Procedures We consider the model X it = c i + λ if t + e it () F mt = α m F mt + u mt m =,... k (2) e it = ρ i e it + ɛ it i =,... N. (3) Equation () represents a series X it as the sum of a common component λ i F t and an idiosyncratic component. he common factors F t are k dimensional. Factor m is non-stationary if α m =. Our analysis permits some, none, or all of the factors to be non-stationary. he idiosyncratic component is stationary if ρ i < and has a unit root if ρ i =. Furthermore, ɛ it is allowed to be weakly correlated in the cross section dimension, so that () is an approximate factor model in the sense of Chamberlain and Rothschild (983). We observe X it, i =,... N, t =,.... Non-stationarity in X it can be due to the presence of common stochastic trends, a non-stationary idiosyncratic component, or both. hese components are not, in general, identifiable from the observed data. Accordingly, the key to PANIC is an I-C decomposition of the data into common and idiosyncratic components. We use the method of principal components (PC) to estimate F t. he PC has been suggested by Harris (997) as an estimator of cointegrating vectors, but estimation of F t is not considered. In fact, consistent 3

estimation of F t is not possible when N is small. However, this is no longer the case when N is large. his is because the idiosyncratic variations must be dominated by those of the common factors when the data are averaged across N. he principal components estimator effectively provides the weights for averaging. Asymptotic properties of the estimator when N and are both large are analyzed in Forni, Hallin, Lippi and Reichlin (2000), Stock and Watson (998), and Bai (200a,b). PANIC has two modules. he PANIC-UR allows us to test the null hypothesis that e it has a unit root, while the PANIC-S takes stationarity as the null hypothesis. Specific implementation of the principal component estimator depends on the null hypothesis to be tested. 2. he PANIC-S he PANIC-S procedure is set up to test H 0 : ρ i < 0, or e it is I(0) for all i. H : ρ i =, or e it is I() for some i. Since e it is stationary for every i, estimation of F t is based on the data in level form. S: Estimate F t by the method of principal components. Denote this by F t. S2: est the null hypothesis that demeaned F mt, m =,... k is stationary. Denote the statistic by S F (m). S3: Given F t, obtain ê it as the residual from the regression X it = c i + λ F i t + e it. S4: est the null hypothesis that ê it is stationary, for each i =,... N. Denote the test by S e (i). We use the KPSS statistic of Kwiatkowski, Phillips, Schmidt and Shin (992) to test stationarity. 2.2 he PANIC-UR he PANIC-UR procedure is set up to test H 0 : ρ i = or e it is I() for all i. H : ρ i < or e it is I(0) for some i. Since e it is non-stationary for every i under the null hypothesis, we apply the method of principal components to the first differenced data, X it = λ i F t + e it. (4) 4

UR: Estimate F t and λ i by the method of principal components. UR2: Given F t, define for each m =,... k, F mt = F ms. s=2 UR3: est for the null hypothesis that demeaned F mt has a unit root for each m =,... k. Denote this test by UR F (m). UR4: For each i =,... N, denote by UR e (i) the test for the null hypothesis that there is a unit root in ê it, where ê it is defined as follows: If F mt is I(0) for every m =,... k, then for t = 2,..., let ẽ it = X it λ F i t and define ê it = ẽ it ẽ i2. If we cannot reject F mt is I() for some m, then obtain ê it as the residual from the regression X it = c i + λ i F t + e it. We consider the ADF test of Said and Dickey (984) based on ê it. 3 Properties of PANIC he validity of PANIC-S and PANIC-UR hinges on the fact that consistent estimates of F t can be obtained by the method of principal components when N is large. he factors and their loadings are not, in general, separately identifiable. Assuming that F F/ = I k, we construct the principal component estimator for F t as times the eigenvector corresponding to the largest eigenvalue of the matrix XX. Denote this by F t. hen N k matrix λ is estimated as Λ = F X/. Consistency results are obtained in Bai (200a) for stationary panels and in Bai (200b) when the factors are non-stationary. he convergence rate of a rescaled version of F t is min[ n, ] for the stationary case and min[ n, 3/2 ] in the non-stationary case. Since the space spanned by F t is preserved after scaling, these convergence results imply that PANIC can treat F t as though it was known. Once consistent estimation of the factors is granted, the limiting distributions of the statistics are then obtained by application of the functional central limit theorem. hat is, for a time series z t satisfying mixing conditions stated in Phillips and Perron (988) z s σ z W z (r), s= 5

where σ z = lim E( z j ) 2 and W z (r) is a standard Brownian motion. Following convention, limits for the demeaned and detrended processes are denoted σ z W c z (r) and σ z W τ z (r) respectively. Furthermore, let z = t= z t. hen is a Brownian bridge. ) (z s z) σ z (W z (r) rw z () σ z V z (r) s= heorem PANIC-S: Suppose the KPSS statistic developed in Kwiatkowski et al. (992) is used to test stationarity. Let V um (m =,... k) be Brownian bridges independent of V ei (i =,... N), which are N mutually independent Brownian bridges.. If F mt is stationary, then S F (m) 0 V um (r) 2 dr. 2. If F mt is I(0) for every m, then for each i =,... N, S e (i) 0 V ei (r) 2 dr. 3. Suppose k of the factors are I(). hen S e (i) has the same limiting distribution as the statistic developed in Shin (994) for testing the null hypothesis of cointegration in a equation with k integrated regressors. Since F t can be consistently estimated, the test can treat the estimated factors as though they were known. As stated in part () of the heorem, S F (m) has the same distribution as derived in Kwiatkowski et al. (992). At the 5% level, this is 0.463. he limiting distribution for the S e (i) test depends on whether F t is I() or I(0). Part (2) of the theorem states that if F mt is I(0) for every m =,... k, S e (i) has the same limit as the KPSS stationary test. At the 5% level, this is also 0.463. If k factors are I(), stationarity of e it implies cointegration between X i and a subset of F of dimension k. hen S e (i) has the same limiting distribution as the Shin test for the null hypothesis of cointegration, as indicated in part (3) of the heorem. he critical values are those in Shin (994) for k regressors and a mean included in the cointegrating regression. At the 5% level, this is 0.34 for k = and 0.22 when k = 2. In each case, the null hypothesis is rejected when the test statistic exceeds the critical value. heorem 2 PANIC-UR: Suppose the ADF statistic of Said and Dickey (984) is used. Let W um (m =,... k) and W ei (i =,... N) be standard Brownian motions. 6

. If F mt is I(), then UR F (m) 0 W um(r)dw c um (r) r 0 W um(r) c 2. dr 2. if F mt is I(0) for every m, then for every i =,... N, UR e (i) 0 W ei(r)dw ei (r) 0 W. ei(r) 2 dr 3. If F mt is I() for some k factors, UR e (i) has the same limiting distribution as the residuals based cointegration test of Phillips and Ouliaris (990) with k integrated regressors. PANIC-UR is based on the the first differenced model which satisfies the assumptions of Bai (200a). Since F t can be consistently estimated, UR F (m) has the same limiting distribution as derived in Fuller (976) for the constant only case. he 5% asymptotic critical value is -2.86. he critical values of UR e (i) depend on whether F t has a unit root. If F mt is I(0) for every m, the critical values are those of a unit root test and are tabulated in Fuller (976) for the case of no constant. At the 5% significance level, this is -.95. Essentially, when F t is I(0), ĉ i is consistent and its effects on ê it are asymptotically negligible. However, when k of F t series are I(), testing the null hypothesis that e it is I() is the same as testing the null hypothesis that X it does not cointegrate with k integrated factors. UR e (i) then becomes the residuals based test for no cointegration with k regressors. he critical values are tabulated in Phillips and Ouliaris (990). When a mean is in the regression, the critical value at the 5% level is -3.37 for k = and -3.77 for k = 2. As defined, ê i = ê i2 = 0 for every i, which provides an initial condition assumption. simpler alternative computationally is to take ê it is the cumulative sum of e it. Another method is to obtain ê it as the residuals from a regression of X it on a constant and F t, which effectively sets the mean of the process to zero. he latter two methods appear to have similar properties in finite samples, though our proof is based explicitly on the assumption that ê i2 = 0. he number of factors k is unknown. N, when F mt is stationary for every m if k is the minimizer of One Bai and Ng (2000) showed that P ( k = k) as PC(m) = log N N σ i 2 (m) + kg(n, ) (5) i= where σ i 2(m) = t= ê 2 it (m) is the sample variance from estimation of a model with m factors, and g(n, ) 0 as N, and min[n, ]g(n, ). Bai (200b) showed if we use the first differenced model and let σ 2 i = e 2 it, the PC criterion remains valid. his is also true even 7

if F t or e it is over-differenced. Furthermore, the PC consistently estimates the total number of factors whether or not the factors are non-stationary. Bai and Ng (2000) showed that g(n, ) = N + [ ] N N log N + satisfies the conditions required for consistent estimation of k. 3. Panel ests hus far, we have introduced PANIC as univariate tests on the common and idiosyncratic components. A common criticism of univariate unit root tests is low power, especially when is small. his has generated substantial interest to improve power. A popular method is to pool information across units, leading to panel unit root tests. A survey of panel unit root tests is offered by Maddala and Wu (999). he early test developed in Levin and Lin (993) imposes substantial homogeneity but independence across units. Subsequent developments have led to tests that allow for heterogeneous intercepts and slopes, while maintaining the assumption of independence across units. his assumption is restrictive, and if violated, can lead to over-rejections of the null hypothesis. O Connell (998) provides a solution to this problem assuming that all cross-section correlation is of the stationary type. If cross-section correlation can be adequately represented by common factors, the idioyscractic components will more likely satisfy the assumption of independence across units. hus, while panel unit root tests applied to X it may be inapproriate, panel unit root tests can be applied to e it. PANIC allows consistent estimates of e it to be obtained. o allow as much heterogeneity across units as possible, we consider panel unit root tests based on meta analysis. heorem 3 Let p Se (i) and p URe (i) be the p-value associated with the stationarity and unit root tests on ê it, respectively. Consider pooled tests defined by If F mt is I(0) for every m =,... k, P Se = n 2 log p Se (i) i= P URe = n 2 log p URe (i). i= P Se 2N 4N d N(0, ) and P URe 2N 4N d N(0, ). heorems and 2 show tests involving ê it are functionals of Brownian motions that are independent across i. he p-values are thus independent U[0,] random variables. aking logarithms, summing 8

over i, and applying central limit theorem gives the result stated. he pooled test 2 N i= ln p Sx (i) was first proposed in Maddala and Wu (999) for fixed N. Choi (200) extended the analysis to allow N. But pooling is invalid when the units are not independent. heorem 3 shows if the factors are stationary, pooling over tests based on ê it is yields a test statistic that is standard normal. However, when a subset of F t is I(), the limiting distributions of the pooled tests will be mixture of χ 2 random variables 4 Monte Carlo Simulations We simulate data using Equations ()-(3). We let λ be a N r matrix, ɛ a N matrix of N(0, ) variables, and u t N(0, )σ F. he following variations are considered: σ F =20, 0, 5, and. α=0,.5,.8,.9,.95, ; ρ=0,.5,.8,.9,.95, ; k = 0,, 2; N=0, 20, 30, 00; =00, 200. Simulations are performed assuming k (the total number of common factors) and k (the number of I() common factors) are known. Naturally, k = k if α m = for every m =, ldotsk. We report results for k =, =200 and N=20. able reports rejection rates for unit-root tests applied to {X t }, { F t }, and {ê t }. he number of lags for the ADF test is set to 4(/00) /4 throughout. UR x should have a rejection rate of.05 when α = or ρ =. his is true only when α = but is not the case when ρ =, α <. he problem is once again that the random walk component is small relative to the stationary variations of the idiosyncratic component, so that X it has a moving average root that is close to unity. he consequence of the near common factor is severe over-rejections of the unit root hypothesis. Such size distortions evidently arise because X it has two components with different degrees of integration. hus, if one of these components is common, PANIC will be able to consistently estimate it and separate unit root tests then become possible. he last six rows of able most clearly illustrate what PANIC can deliver. he column UR F is the rejection rate of the ADF test applied to F. It is around.05 when α is and exhibit little size distortion. At other values of α, it reflects power. he column UR e is the rejection rate when the test is applied to ê. It too should be.05 when ρ =, and the results show that the finite sample rejection rates are indeed close to the nominal size. In contrast, the UR x column shows that the 9

ADF test applied to the observed data does not give the correct inference. able 2 reports results with different values for the variance σ F. he finding is similar to that of able. More tables will be added. 5 Application to PPP he PANIC can be used to shed new light on a much research issue:- the validity of the purchasing power parity (PPP) hypothesis. Under PPP, real exchange rates should be mean reverting and thus stationarity. he low power of unit root tests over short span has often been used to rationalize nonrejections of the unit root null. his has prompted the development of panel unit root tests which exploit information in the cross section dimension to increase power. But a major shortcoming of panel unit root tests is the assumption of independence across units. Indeed, because real exchange rates are often defined using the same base country, cross-section correlation arises almost by construction. Such strong cross-section correlation amounts to a common factor that cannot be aggregated away. Panel unit root tests based on pooling in the cross-section dimension will depend on this nuisance cross-section correlation parameter. As O Connell (998) found, the tests are biased towards the alternative hypothesis. hat is, if the null hypothesis is I(), we tend to accept stationarity. If the null is stationary, we tend to accept nonstationarity. O Connell suggest removing the cross-section correlation by a GLS transformation of the data. his presupposes that the common variation is stationary, which need not be the case. A PANIC allows us to test if real exchange rates are non-stationary because of a common non-stationary component, or if country-specific variations are the source of non-stationarity. 0

Appendix Lemma Consistency of F t. Suppose F t is I(0) and Assumptions A to G of Bai (200a) hold.. hen F t is N consistent if N/ 0. If n/ τ > 0, F t is consistent. Suppose F t is I() and Assumptions A-F of Bai (200b) hold. hen F t is N consistent if N/ 3 0. If N/ 3 τ > 0, F t is consistent at rate 3/2. Stationarity and unit root applied to F t have the same distributions as though F t is known. heorems. and 2. are stated in terms of the KPSS and the ADF test, but other tests are equally valid. he remaining proofs assume that F t is known. Proof of the heorem, part 2 For a given i, consider the regression X it = c i + γ if t + e it. Because F t is stationary and an intercept is allowed in the regression, without loss of generality, we assume E(F t ) = 0. Let ê it be the estimated residuals. By direct calculations, ê it = e it (ĉ i c i ) ( λ λ) F t. Note that ĉ i and λ i are consistent. ê ij = = = e ij t (ĉ i c i ) ( λ i λ i ) F j e ij t (ĉi c i ) ( λ i λ i ) e ij t (ĉi c i ) O p ( ) F j But ĉ i = X i λ i F. We have (ĉ i c i ) = e it + o p (). hus, ê ij = e ij t e ij + o p (). Let t = [ r] and t/ r. By the functional central limit theorem, t e ij W ei (r)σ ei. hus, ê ij [ ] W ei (r) rw ei () σ ei.

Let s 2 ei be a consistent estimate of σ2 ei, the long run variance of e i. hen ( t ê ij ) 2 S e (i) = s 2 ei Proof of heorem 2, part 2 0 (W ei (r) rw ei ()) 2 dr 0 V ei (r) 2 dr. Again, we assume F t is known. he model is X it = c i +λ i F t +e it = ĉ i + λ i F t +ê it. Set ê i = ê i2 = 0. hen X i2 = ĉ i + λ F 2 which implies ĉ i = X i2 λ i F 2. It follows that ê it = X it ĉ i λ i F t = ẽ it ẽ i2, where ẽ it = X it λ i F t by definition. We have ê it = e it ( λ i λ i ) F t (ĉ i c i ) = [ ( λ i λ i ) (ĉ i c i )] e it F t b ζ t. ê it = e it ( λ i λ i ) F t [ = [ ( λ i λ i ) ] e it F t ] ĥη t. Note that b b (, b 2, b 3 ) and ĥ h (, h 2). We want the limiting distribution of ρ i, where ( ρ i ) is obtained a first order autoregression in ê it. he numerator is ( ρ i ) = t=2 ê it ê it 2 t=2 ê 2. it ê it e it = b ζ t η tĥ t=2 t=2 = b t=2 e it e it 3/2 t=2 F t e it t=2 e it σ 2 b ɛi 0 W ɛi(r)dw ɛi (r) 0 0 0 0 0 Analogous calculations give Combining the results, he t statistic follows. 2 t=2 ê 2 it = b [ 2 t=2 ζ t ζ t] b σ 2 ɛi 3/2 t=2 e it F t 5/2 t=2 F t F t 3/2 t=2 F t ĥ h = σ 2 ɛi W ɛi (r)dw ɛi (r). 0 ( ρ i ) W ɛi(r)dw ɛi (r) 0 W. ɛi(r) 2 dr 0 0 W ɛi (r) 2 dr. 2

able : Rejection rates for H 0 : e it is I() σ F = 20 σ F = 5 N ρ α UR x UR F UR e URe UR x UR F UR e URe 200 20 0.00 0.00.00.00 0.7 0.70.00.00 0.7 0.72 200 20 0.00 0.50.00.00 0.70 0.69.00.00 0.68 0.68 200 20 0.00 0.80 0.96 0.96 0.67 0.67 0.98 0.97 0.70 0.7 200 20 0.00 0.90 0.68 0.67 0.68 0.69 0.7 0.68 0.72 0.72 200 20 0.00 0.95 0.26 0.25 0.69 0.68 0.32 0.27 0.72 0.70 200 20 0.00.00 0.07 0.07 0.74 0.74 0.08 0.07 0.75 0.74 200 20 0.50 0.00.00.00 0.80 0.79.00.00 0.8 0.8 200 20 0.50 0.50.00.00 0.8 0.8.00.00 0.8 0.80 200 20 0.50 0.80 0.98 0.98 0.8 0.8 0.98 0.97 0.8 0.79 200 20 0.50 0.90 0.65 0.64 0.8 0.80 0.73 0.67 0.80 0.79 200 20 0.50 0.95 0.30 0.27 0.82 0.82 0.32 0.24 0.8 0.8 200 20 0.50.00 0.07 0.06 0.90 0.89 0.08 0.05 0.88 0.89 200 20 0.80 0.00.00.00 0.8 0.8.00.00 0.8 0.82 200 20 0.80 0.50.00.00 0.8 0.8 0.99.00 0.8 0.82 200 20 0.80 0.80 0.97 0.97 0.82 0.84 0.97 0.97 0.82 0.82 200 20 0.80 0.90 0.68 0.66 0.83 0.83 0.72 0.67 0.83 0.83 200 20 0.80 0.95 0.28 0.26 0.8 0.82 0.33 0.25 0.8 0.8 200 20 0.80.00 0.07 0.06 0.76 0.76 0. 0.07 0.78 0.77 200 20 0.90 0.00 0.97.00 0.78 0.76 0.9.00 0.77 0.77 200 20 0.90 0.50 0.98.00 0.74 0.75 0.93.00 0.75 0.74 200 20 0.90 0.80 0.95 0.97 0.76 0.76 0.9 0.97 0.76 0.76 200 20 0.90 0.90 0.67 0.68 0.76 0.77 0.69 0.69 0.76 0.75 200 20 0.90 0.95 0.29 0.28 0.75 0.75 0.34 0.27 0.74 0.74 200 20 0.90.00 0.06 0.05 0.3 0.33 0. 0.06 0.33 0.32 200 20 0.95 0.00 0.9.00 0.53 0.54 0.65.00 0.52 0.52 200 20 0.95 0.50 0.94.00 0.52 0.53 0.75.00 0.52 0.5 200 20 0.95 0.80 0.9 0.97 0.52 0.53 0.74 0.97 0.5 0.50 200 20 0.95 0.90 0.6 0.65 0.53 0.52 0.54 0.66 0.49 0.5 200 20 0.95 0.95 0.26 0.24 0.54 0.53 0.26 0.26 0.53 0.53 200 20 0.95.00 0.06 0.06 0.0 0.0 0.07 0.06 0.09 0.09 200 20.00 0.00 0.73.00 0.06 0.06 0.27 0.98 0.06 0.06 200 20.00 0.50 0.79.00 0.05 0.06 0.4.00 0.04 0.05 200 20.00 0.80 0.80 0.96 0.05 0.05 0.47 0.95 0.05 0.05 200 20.00 0.90 0.56 0.68 0.06 0.06 0.32 0.62 0.06 0.06 200 20.00 0.95 0.23 0.27 0.05 0.06 0.7 0.28 0.04 0.04 200 20.00.00 0.05 0.05 0.02 0.02 0.07 0.07 0.02 0.0 3

able 2: Rejection rates for H 0 : e it is I() σ F = 0 σ F = N ρ α UR x UR F UR e UR x UR F UR e 200 20 0.00 0.00.00.00 0.7.00.00 0.70 200 20 0.00 0.50.00.00 0.70.00.00 0.68 200 20 0.00 0.80 0.97 0.96 0.67 0.99 0.97 0.70 200 20 0.00 0.90 0.68 0.67 0.68 0.82 0.68 0.72 200 20 0.00 0.95 0.27 0.25 0.69 0.47 0.27 0.7 200 20 0.00.00 0.08 0.07 0.74 0.9 0.07 0.7 200 20 0.50 0.00.00.00 0.80.00.00 0.8 200 20 0.50 0.50.00.00 0.82.00.00 0.8 200 20 0.50 0.80 0.98 0.98 0.8 0.99 0.97 0.80 200 20 0.50 0.90 0.67 0.64 0.8 0.86 0.68 0.79 200 20 0.50 0.95 0.32 0.28 0.82 0.55 0.25 0.8 200 20 0.50.00 0.09 0.06 0.90 0.22 0.05 0.88 200 20 0.80 0.00.00.00 0.8 0.98.00 0.8 200 20 0.80 0.50 0.99.00 0.8 0.98.00 0.8 200 20 0.80 0.80 0.97 0.97 0.82 0.97 0.97 0.83 200 20 0.80 0.90 0.70 0.66 0.83 0.86 0.68 0.83 200 20 0.80 0.95 0.3 0.26 0.8 0.62 0.28 0.8 200 20 0.80.00 0.08 0.06 0.76 0.3 0.07 0.77 200 20 0.90 0.00 0.95.00 0.78 0.74 0.99 0.77 200 20 0.90 0.50 0.96.00 0.74 0.77.00 0.75 200 20 0.90 0.80 0.93 0.97 0.76 0.78 0.96 0.75 200 20 0.90 0.90 0.67 0.68 0.76 0.68 0.68 0.76 200 20 0.90 0.95 0.3 0.28 0.75 0.46 0.27 0.74 200 20 0.90.00 0.08 0.05 0.3 0.24 0.06 0.33 200 20 0.95 0.00 0.82.00 0.53 0.32 0.87 0.52 200 20 0.95 0.50 0.88.00 0.52 0.35 0.96 0.52 200 20 0.95 0.80 0.86 0.97 0.52 0.42 0.92 0.5 200 20 0.95 0.90 0.57 0.65 0.53 0.36 0.62 0.50 200 20 0.95 0.95 0.26 0.24 0.54 0.26 0.27 0.53 200 20 0.95.00 0.07 0.06 0.0 0.2 0.06 0.09 200 20.00 0.00 0.52.00 0.06 0.06 0.28 0.06 200 20.00 0.50 0.64.00 0.05 0.08 0.50 0.04 200 20.00 0.80 0.67 0.96 0.05 0. 0.59 0.05 200 20.00 0.90 0.46 0.69 0.06 0.09 0.42 0.06 200 20.00 0.95 0.2 0.26 0.05 0.06 0.2 0.04 200 20.00.00 0.06 0.05 0.02 0.07 0.06 0.02 4

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