TitleReducing the Size Distortion of the.

Transcription

1 itlereducing the Size Distortion of the Authors) Kurozumi, Eiji; anaka, Shinya Citation Issue 29-9 Date ype echnical Report ext Version publisher URL Right Hitotsubashi University Repository

2 Global COE Hi-Stat Discussion Paper Series 85 Research Unit for Statistical and Empirical Analysis in Social Sciences Hi-Stat) Reducing the Size Distortion of the KPSS est Eiji Kurozumi Shinya anaka Hi-Stat Discussion Paper September 29 Hi-Stat Institute of Economic Research Hitotsubashi University 2- Naka, Kunitatchi okyo, Japan

3 Reducing the Size Distortion of the KPSS est Eiji Kurozumi 2 Shinya anaka 3 Department of Economics Hitotsubashi University September 2, 29 Abstract his paper proposes a new stationarity test based on the KPSS test with less size distortion. We extend the boundary rule proposed by Sul, Phillips and Choi 25) to the autoregressive spectral density estimator and parametrically estimate the long-run variance. We also derive the finite sample bias of the numerator of the test statistic up to the / order and propose a correction to the bias term in the numerator. Finite sample simulations show that the correction term effectively reduces the bias in the numerator and that the finite sample size of our test is close to the nominal one as long as the long-run parameter in the model satisfies the boundary condition. JEL classification: C2; C22 Key words: Stationary test; size distortion; boundary rule; bias correction We are grateful to Professor Katsumi Shimotsu for his helpful comment. All errors are our responsibility. his research is supported by the Global COE program, the Research Unit for Statistical and Empirical Analysis in Social Sciences at Hitotsubashi University. 2 Correspondence: Eiji Kurozumi, Department of Economics, Hitotsubashi University, 2- Naka, Kunitachi, okyo, 86-86, Japan. kurozumi@stat.hit-u.ac.jp 3 stanaka@wd5.so-net.ne.jp

4 . Introduction Following longstanding and well-known studies of the unit root problem, it has been common practice to test for a unit root in time series analysis. In conjunction with testing for a unit root, the null hypothesis of stationarity has often been investigated in practical analysis, and one of the widely applied stationarity tests is the one in Kwiatkowski et al. 992; hereafter, referred to as KPSS 992)), also known as the KPSS test. Although the KPSS test is asymptotically free from nuisance parameter and hence we can asymptotically control the size of the test, it is also known that the test suffers from considerable size distortion in finite samples when the series tested is strongly serially correlated. See, for example, Caner and Killian 2) and Müller 25). In order to mitigate the size distortion problem, Rothman 997) considers the use of size-adjusted critical values, but the KPSS test with size-adjusted critical values loses its power, as pointed out by Rothman 997) and Caner and Killian 2). Since one of the reasons for size distortion is the bias in the log-run variance estimator, Sul, Phillips and Choi 25, hereafter) propose a modified KPSS test by estimating the long-run variance using the prewhitening method proposed by Andrews and Monahan 992) with the data-dependent boundary rule. Carrion-i-Silvestre and Sansó 26) investigate the finite sample properties of several stationarity tests and conclude that the test with first order autoregressive AR)) prewhitening is preferable to others in terms of size control. However, their simulations also show that the test with AR) prewhitening suffers from size distortion when the data generating process DGP) is an AR2) process. Harris, Leybourne and McCabe 27) focus on the local-to-unity model and propose the GLS-type transformation of data before constructing the test statistic. he size of their test is close to the nominal one when the pre-specified localizing parameter is close to the true one but the test is undersized when the true process is moderately serially correlated, as is shown by Aznar and Ayuda 28). Since the size distortion problem is not specific to the KPSS test but a general problem for other stationarity tests, several methods for reducing the size distortion of stationarity tests have been proposed in the literature. For example, Cheung and Chinn 997) and Kuo

5 and Mikkola 999) use size-adjusted critical values for Leybourne and McCabe 994, 999) tests and Saikkonen and Luukkonen 993a, b) tests, respectively, but these methods, as in the case of the KPSS test, are unable to correct the loss in the power of the tests. Lanne and Saikkonen 23) and Kurozumi 29) propose to modify Leybourne and McCabe 994, 999) tests; note that though the sizes of these tests are closer to those of the original tests, they still suffer from size distortion in some cases. Aznar and Ayuda 28) develop a new test for stationarity using a local-to-unity model but this test is undersized for a process with moderate serial correlation because their test is not designed for the null of stationarity. Unfortunately, all of the above methods seem to have a problem with controlling the size of stationarity tests. In this paper, we propose a new KPSS-type test for trend) stationarity with less size distortion. We extend the boundary rule proposed by 25) to the autoregressive spectral density estimator; the long-run variance is estimated based on the AR approximation. Although it is known that the long-run variance estimator based on the least squares method results in the inconsistency of the test as pointed out by Leybourne and McCabe 994), we show that this problem can be avoided by applying the boundary rule of 25). his autoregressive spectral density estimator works relatively well but we still have another problem the numerator of the KPSS test statistic has a downward bias, and as such, the KPSS test statistic corrected by the new long-run variance estimator becomes undersized. In order to correct the size of the test, we derive the finite sample bias of the numerator of the test statistic and propose the bias-corrected version of the KPSS test. It is shown that the empirical size of our modified test can be well controlled as compared to the other tests. he paper is organized as follows. Section 2 introduces a model and briefly reviews the KPSS test and the boundary rule. We consider the application of the boundary rule to the long-run variance estimator in Section 3. We also derive the finite sample bias of the numerator of the KPSS test statistic and propose the bias corrected test statistic. Section 4 investigates the finite sample properties of our test. Section 5 gives the concluding remarks. 2

6 2. Model and Review of KPSS est Let us consider the following model: y t = d tβ + x t for t =, 2,,, ) where d t is deterministic and x t is a stochastic component. As in the literature, we consider two cases: d t = constant case) and d t =[,t] trend case). he integrated order of y t is determined by the behavior of x t. We consider the following assumption in this paper. Assumption. a) Under the null hypothesis, x t is covariance stationary with -summable autocovariances; the spectral density function of x t, given by fλ), is bounded and does not equal zero for <λ< ; the functional central limit theorem FCL) can be applied to the partial sum process of x t. b) Under the alternative hypothesis, Δx t satisfies condition a) where Δ= L with L being the lag operator. According to Assumption, y t is covariance stationary trend stationary) under the null hypothesis while it is a unit root process under the alternative. KPSS 992) propose to test for the null of trend) stationarity against the alternative of a unit root. he KPSS test statistic is defined as KPSS = t 2 t= s= ˆx s ˆω ) 2, 2) where ˆx t is the regression residual of y t on d t and ˆω is a consistent estimator of the long-run variance ω defined by ω = lim Var /2 t= x t). KPSS 992) originally proposed to estimate ω by the nonparametric method using the Bartlett kernel, but other kernels such as the quadratic spectral kennel are also applicable. By applying the FCL to the partial sum process of ˆx t, it is shown that /2 [r] t= ˆx t weakly converges to ωv r) for the constant case and to ωv 2 r) for the trend case where [a] denotes the largest integer less than a, V r) = Br) rb) and V 2 r) = Br) + 2r 3r 2 ) B) + 6r +6r 2) Bs) ds with Br) being a standard Brownian motion. 3

7 Using these results, it is shown that KPSS d {V ir)} 2 dr under the null hypothesis d with i = for the constant case and i = 2 for the trend case where signifies convergence in distribution. As is seen from the limiting null distribution, the KPSS test is free from nuisance parameter and we can asymptotically control the size of the test. However, as explained in the introduction section, it suffers from considerable size distortion in finite samples. In order to mitigate size distortion, 25) propose to estimate the long-run variance using the prewhitening method with a boundary rule. According to 25), we first estimate an ARp) modelforˆx t as ˆx t =ˆρ ˆx t + +ˆρ pˆx t p +ê t and then define the new long-run variance estimator as ˆω e ω = ρ) 2 where ρ =min ˆρ + +ˆρ p, ) and ˆω e is the long-run variance estimator based on ê t. 25) show that the KPSS test statistic 2) corrected by ω has the same limiting distribution under the null hypothesis while it diverges to infinity at rate under the alternative. In practice, it is often the case that the prewhitening method is implemented with an AR) approximation. Moreover, Carrion-i-Silvestre and Sansó 26) show that the size of the test with AR) prewhitening is close to the nominal one when the true DGP is an AR) process while it tends to be greater than the nominal size when the true DGP is an AR2) process. 3. Bias Corrected KPSS est 3.. Estimation of the long-run variance he boundary rule exploited by 25) is a clever tool to estimate the long-run variance with less bias. We apply this rule to the autoregressive spectral density estimator. As shown by Perron and Ng 996), unit root tests corrected by the autoregressive spectral density estimator perform well and we expect that this would be the case for stationarity tests. 4

8 In order to see the AR expression of x t, we first express x t as an infinite order moving average MA )) process under the null hypothesis. Under Assumption a), the original process x t can be expressed, using the Wold representation and heorem of Brillinger 98), as x t = ψl)ε t = ψ j ε t j, j= where j ψ j <, 3) ψl) = j= ψ jl j is a lag polynomial with ψ = and ε t is a sequence of white noise with E[ε 2 t ]=σε. 2 Further, from heorem of Brillinger 98), the lag polynomial ψl) is invertible and hence we have φl)x t = ε t, where φl) = φ j L j j= j= and j φ j <. 4) Similarly, we can also see that x t is expressed as φ L) L)x t = ε t under the alternative because Δx t satisfies Assumption a) under the alternative. By defining φl) =φ L) L) under the alternative, we can see that the original process x t has an AR ) representation given by φl)x t = ε t under both the null and the alternative hypotheses and thus the testing problemisgivenby H : φ) > vs. H : φ) =. From 4), the long-run variance of x t is given by σ 2 ε/φ 2 ) and the natural estimator is obtained by approximating the AR ) representation by the ARp) model where p diverges to infinity at an appropriate rate as. However, as pointed out by Leybourne and McCabe 994), the long-run variance estimator based on the least squares estimation of the ARp) model results in the inconsistency of the test. In order to avoid this problem, we make use of the boundary rule by 25). We first fit the ARp) modeltoˆx t, j= ˆx t = ˆφ ˆx t + + ˆφ pˆx t p +ˆε t and then estimate the long-run variance based on the autoregressive spectral density esti- 5

9 mator as follows: ω AR = ˆσ 2 ε φ) 2 where ˆσ 2 ε = p ˆε 2 t and φ =min ˆφ j, c t= with c being some constant. We propose to test for the null of trend) stationarity using the KPSS test statistic 2) with ˆω replaced by ω AR. We call this test the modified KPSS test. Note that under the null hypothesis, φ equals p ˆφ j= j for large ; hence ω AR converges in probability to ω. For details, see Berk 974). We can see that the modified KPSS test statistic has the same limiting distribution as the original one. On the other hand, as in 25), ˆσ 2 ε still converges in probability to σ 2 ε under the alternative while φ) 2 is shown to be of order /. As a result, ω AR diverges to infinity at rate. Since the numerator of the KPSS test statistic diverges to infinity at rate 2, we can see that the KPSS test statistic corrected by ω AR diverges to infinity at rate under the alternative, and as such, the modified test is consistent. j= 3.2. Bias correction of the test statistic As expected from the case of unit root tests by Perron and Ng 996), the autoregressive spectral density estimator performs quite well in our preliminary simulation. However, once the long-run variance is well estimated, we encounter another problem the numerator of the test statistic 2) is biased downward in finite samples. As we will see in the next section, the modified KPSS test tends to under-reject the null hypothesis because of this downward bias, and hence, it loses power considerably under the alternative. In order to control the size of the test, we derive the bias in the numerator of the test statistic under the null hypothesis and consider the bias-corrected version of the modified KPSS test statistic. o calculate the bias, we first express x t in 3) by the Beveridge-Nelson decomposition as x t = ψ)ε t + v t v t, where v t = j= ψ j ε t j with ψj = ψ i. 5) i=j+ 6

10 Since ˆx t is obtained by regressing y t on d t,wecanseethat ) ˆx t = x t d t t= d t d t t= d t x t ) = ψ)ε t + v t v t d t d t d t d t ψ)ε t + v t v t ) = ψ)ˆε t Δv t t= where ˆε t and Δv t are the regression residuals of ε t and Δv t on d t, respectively note that Δv t is different from Δˆv t ). Using this expression, the numerator of the KPSS test statistic 2) is decomposed into three terms: t ) 2 2 ˆx s = ψ2 ) 2 t= s= + 2 = ψ2 ) 2 t ) 2 ˆε s t= s= t= t ) 2 Δv s 2ψ) t ) t ) 2 ˆε s Δv s. t= s= t= s= s= t ) 2 ˆε s + R R 2, say. 6) t= s= It can be shown that the first term on the right hand side of 6) is the leading term while the second and third terms are o p ). From the simulation result in KPSS 992), the finite sample distribution of the first term on the right hand side of 6), except for a scalar term ψ 2 ), is well approximated by the limiting distribution and thus we expect that the downward bias in the numerator comes from R and R 2. herefore, we define the finite sample bias in the numerator as the expectation of R R 2 up to the O ) terms. his is denoted by b : ) E[R R 2 ]=b + o. he following theorem gives the expression of the bias term b. heorem. Let γ = E[v 2 t ] and φl) be the lag polynomial given in 4). Under Assumption a), the bias term b in the numerator of the KPSS test statistic is expressed as b = b γ + σ 2 φ ) ) ε φ 3 ) 7

11 where φ ) = dφz)/dz z= and b =5/3 for the constant case and b =9/5 for the trend case. he direction of the bias is not necessarily obvious because γ > while φ )/φ 3 ) is negative when x t is positively serially correlated. However, as is shown in the following corollary, the bias turns out to be negative when x t is an AR) process with positive serial correlation. Corollary. Assume that x t is an AR) process given by x t = φ x t + ε t. hen, when φ <, the bias term b in the numerator of the KPSS test statistic is expressed as b = b where b is the same as in heorem. σ 2 εφ φ ) 2 φ 2 ) his corollary is easily obtained by noting that γ = for the AR) case and we omit the poof. σ 2 εφ 2 φ ) 2 φ 2 ) From corollary we can see that the bias for an AR) process is always negative when φ > and that the bias takes large negative values as φ approaches. his explains why the modified KPSS test tends to be undersized when the process is strongly serially correlated. As we will see in the next section, the downward bias in the test statistic is serious when the process is strongly serially correlated and hence the power of the test can be below the significance level in some cases. We thus need to correct the bias in the numerator of the test statistic. In practice, we need to estimate the bias based on the ARp) approximation. Although we can easily estimate b for the AR) case because it is explicitly expressed as a function of the AR coefficient as given in Corollary, we have to estimate γ in general. Since γ cannot be expressed in the closed form using the AR coefficients for a general ARp) model, we need to estimate it recursively like solving the Yule-Walker equations. 8

12 We first note that the bias b includes the reciprocal of φ) and the least squares estimator of this term might take large values because of the estimation error in the AR coefficients. As a result, the estimator of the bias might take explosively large negative values through the φ )/φ 3 ) term. In order to avoid the explosive behavior of the bias term, we estimate it based on the least squares method with the same inequality constraint as the boundary rule. hat is, we estimate the ARp) model by minimizing the sum of squared residuals with the inequality constraint given by p j= φ j c/. We can see that the constrained estimator is consistent as long as the boundary rule is satisfied while it is not explosive even under the alternative because of the constraint. In order to explain how to estimate γ, let us assume that φl) be the lag polynomial of order p, so that x t = φ x t + + φ p x t p + ε t. By inserting the MA ) expression 3) into both sides, we have ψ i ε t i = φ ψ i ε t i + + φ p ψ i ε t i p + ε t. i= i= By comparing the coefficients associated with ε t i for i =,, 2,, we can observe the following relations between {ψ i } and {φ i }: ψ =,ψ = φ ψ, ψ 2 = φ ψ +φ 2 ψ,. hat is, i= ψ i = ψ i = i φ k ψ i k for i =,,p 7) k= p φ k ψ i k for i p. 8) k= Using relation 7) and the constrained estimators of φ,,φ p, we can get the estimators ˆψ,, ˆψ p. In addition, we also obtain the estimators of ψ,, ψ p using the following relation: ψ i = k=i+ ψ k = k= for i =,,p, since ψ) = /φ). ψ k ψ ψ i = φ) ψ ψ i, We next make use of the relation among the autocovariances of v t. By summing 8) over 9

13 i j + with j p, wehave ψ i = φ ψ i + φ 2 or equivalently, i=j+ i=j i=j ψ i + + φ p i=j p+ ψ j = φ ψj + φ 2 ψj φ p ψj p for j p, 9) since ψ j = i=j+ ψ i. Multiplying both sides of 9) with ψ j k and summing over j p, we get ψ j ψj k = φ ψ j ψj k + φ 2 ψ j 2 ψj k + + φ p ψ j p ψj k. ) j=p j=p j=p Noting that γ k = E[v t v t k ]=σε 2 ψ j= j+k ψj for k =,, 2, because, as given in 5), v t = ψ j= j ε t j, we can see that ) can be expressed as γ k = φ γ k + + φ k γ + φ k γ + φ k+ γ + + φ p γ p k + a k ) ψ i, j=p for k =,,,p where p k p k a k = σε 2 ψ j+k ψj φ j= p k 2 φ k+ j= j= p k ψ j+k ψj φ k ψ j ψj+ φ p ψ ψp k. j= p k ψ j+ ψj φ k j= ψ 2 j Since we have already obtained the estimators of φ,,φ p and ψ,, ψ p, we can also calculate a k for k =,,p. Since ) for k =,,p can be seen as p +simultaneous equations with respect to γ,,γ p, we can get the estimator of γ by solving a set of these equations. Once we get the estimator of γ, we can construct ˆb, the estimator of b. Finally, we construct the bias corrected version of the KPSS test statistic t KPSS BC = 2 t= s= ˆx 2 s) ˆb. ω AR Note that the bias corrected KPSS test statistic has the same limiting distribution as the original KPSS test statistic under the null hypothesis, and as such, we can use the critical values in the table given by KPSS 992).

14 4. Simulation Study In this section, we investigate the finite sample properties of the bias corrected version of the modified KPSS test statistic through Monte Carlo simulations. he DGP we considered is given as follows: y t = d tβ + x t, x t = φ x t + φ 2 x t 2 + ε t where ε t i.i.d.n, ) and β = throughout the simulations because the test statistic is invariant to the true values of β. For the AR) case, we set φ to be from to. in increments of. and take φ 2 =. On the other hand, for the AR2) case, we take φ 2 =.3 or.3 andsetφ such that φ + φ 2 ranges from to.. he significance level is.5 and the number of replications is 5,. o obtain the estimate of the long-run variance, ω AR, we need to determine the lag length in practice. For both of the AR) and AR2) cases we choose the lag length using the Bayesian information criterion ) 4. In addition, in order to apply the boundary rule, we have to preset the value of c; however, the localizing parameter c is not necessarily interesting in practical analysis. he boundary value, c/, is of greater importance in finite samples because we truncate the long-run parameter φ + + φ p at the boundary value. For example in the AR) case, if we set the boundary value as, we expect that the size of the test would be close to the significance level when φ is less than. On the other hand, when φ is greater than, the null hypothesis would tend to be rejected. In our simulations, we choose c such that c/ equals 5, and 5 for = 5, and 3; further, the boundary value of 8 is considered for =, 3 and 5. he above boundary value is also used to obtain the bias term ˆb. We used the GAUSS- CML routine to estimate the model by the least squares method with the inequality constraint given by φ + + φ p c/. Figure provides the rejection frequencies of the tests for the constant AR) case where the horizontal axis corresponds to φ. In each figure, denotes the rejection frequencies 4 We also used the Akaike information criterion to choose the lag length. he results are similar to those of the.

15 i-a) = 5, boundary= 5 ii-a) = 5, boundary= i-a) =, boundary= 5 ii-a) =, boundary= i-a) = 3, boundary= 5 ii-a) = 3, boundary= Figure : he finite sample performance; constant case AR) model) 2

16 i-a) = 5, boundary= 5 ii-a) =, boundary= i-a) =, boundary= 5 ii-a) = 3, boundary= i-a) = 3, boundary= 5 ii-a) = 5, boundary= 8 Figure : continued) 3

17 i-a) = 5, boundary= 5 ii-a) = 5, boundary= i-a) =, boundary= 5 ii-a) =, boundary= i-a) = 3, boundary= 5 ii-a) = 3, boundary= Figure 2: he finite sample performance; constant case AR2) with φ =.3 model) 4

18 i-a) = 5, boundary= 5 ii-a) =, boundary= i-a) =, boundary= 5 ii-a) = 3, boundary= i-a) = 3, boundary= 5 ii-a) = 5, boundary= 8 Figure 2: continued) 5

19 i-a) = 5, boundary= 5 ii-a) = 5, boundary= i-a) =, boundary= 5 ii-a) =, boundary= i-a) = 3, boundary= 5 ii-a) = 3, boundary= Figure 3: he finite sample performance; constant case AR2) with φ =.3 model) 6

20 i-a) = 5, boundary= 5 ii-a) =, boundary= i-a) =, boundary= 5 ii-a) = 3, boundary= i-a) = 3, boundary= 5 ii-a) = 5, boundary= 8 Figure 3: continued) 7

21 i-a) = 5, boundary= 5 ii-a) = 5, boundary= i-a) =, boundary= 5 ii-a) =, boundary= i-a) = 3, boundary= 5 ii-a) = 3, boundary= Figure 4: he finite sample performance; trend case AR) model) 8

22 i-a) = 5, boundary= 5 ii-a) =, boundary= i-a) =, boundary= 5 ii-a) = 3, boundary= i-a) = 3, boundary= 5 ii-a) = 5, boundary= 8 Figure 4: continued) 9

23 i-a) = 5, boundary= 5 ii-a) = 5, boundary= i-a) =, boundary= 5 ii-a) =, boundary= i-a) = 3, boundary= 5 ii-a) = 3, boundary= Figure 5: he finite sample performance; trend case AR2) with φ =.3 model) 2

24 i-a) = 5, boundary= 5 ii-a) =, boundary= i-a) =, boundary= 5 ii-a) = 3, boundary= i-a) = 3, boundary= 5 ii-a) = 5, boundary= 8 Figure 5: continued) 2

25 i-a) = 5, boundary= 5 ii-a) = 5, boundary= i-a) =, boundary= 5 ii-a) =, boundary= i-a) = 3, boundary= 5 ii-a) = 3, boundary= Figure 6: he finite sample performance; trend case AR2) with φ =.3 model) 22

26 i-a) = 5, boundary= 5 ii-a) =, boundary= i-a) =, boundary= 5 ii-a) = 3, boundary= i-a) = 3, boundary= 5 ii-a) = 5, boundary= 8 Figure 6: continued) 23

27 of the bias corrected version of the modified KPSS test while is the test with AR) prewhitening, to which the above boundary rule is applied. o see the effect of the bias correction, we also show the result of the modified KPSS test in Subsection 3. with the true lag length, which is denoted as. From Figure, we can see that the bias correction term b effectively reduces the downward bias of the modified KPSS test statistic when the boundary value is 5, and 5. For the boundary value of 8, our method overly corrects the modified KPSS test statistic when = ; however, as the sample size increases, the size of our test gets closer to the nominal size when the AR) parameter is less than the boundary value. Figures 2 and 3 give the rejection frequencies of the tests for the constant AR2) case. In this case, the horizontal axis corresponds to φ + φ 2. Figure 2 shows that the test with the AR) prewhitening suffers from size distortion when φ 2 =.3 even if the process is not strongly serially correlated, while the modified KPSS test tends to under-reject the null hypothesis. Again, the size of the bias corrected version of the modified KPSS test is much closer to the nominal one than the other tests except when is small and the boundary values are large. On the other hand, the test tends to under-reject the null hypothesis when φ 2 =.3, and as such, it loses power considerably. Figures 4 to 6 give the results for the trend case. he relative performance of the tests is preserved compared to the constant case but it becomes more difficult to control the size of the tests for the trend case. When the sample size is small and the boundary is close to one, our method tends to correct the downward bias too much, so that our test also suffers from size distortion in some cases. However, this over-rejection is mitigated as the sample size increases. 5. Concluding Remarks In this paper, we proposed a new KPSS-type test for stationarity with less size distortion. he distinctive features of our test are summarized in the following: First, we parametrically estimate the long-run variance imposing the boundary condition. Second, we correct the 24

28 downward bias in the numerator of the KPSS test statistic. he simulation study showed that our method can mitigate the size distortion problem effectively. Our method could be extended to the panel stationarity test proposed by Hadri 2) and it is our future research. It is worth noting that we can control the empirical size only up to the boundary value and that the rejection frequencies of the test tend to be greater than the significance level when the long-run parameter is above the boundary value. his may be a natural result in view of heorem 2 of Müller 28), who pointed out Discussion 4.3, Müller 28)) that Consistent stationarity tests should be thought of as testing jointly the I) property... and additional restrictions on the behavior of the process. he boundary rule in this paper is one such restriction. 25

29 Appendix Proof of heorem : In the following, we will show that E [R ] = ) b γ + o E [R 2 ] = 2) b σεφ 2 ) φ 3, 3) ) where b =5/3 for the constant case and it is 9/5 for the trend case. Let us first consider the trend case. Since Δv t is the regression residual of Δv t on d t,we can expand R as R = t 2 Δv s = t= s= t s= d s t ) 2 Δv s 2 t= s= 2 ) d t d t d t Δv t t= t= s= ) Δv t d t d t d t t= t= = R R 2 + R 3, say. t Δv s t t= s= t= t s= d s t d s d s s= 2 ) ) d t d t d t Δv t t= t= ) ) d t d t d t Δv t t= t= Since t s= Δv s = v t v, the expectation of R becomes E[R ] = 2 t= = 2 γ 2 2 E [ vt 2 2v v t + v 2 ] t= γ t = 2 ) γ + O 2 4) because t= γ t t= γ t <. In order to evaluate R 2, we express E[R 2 ]as E[R 2 ]= 2 [ 2 tr ) d t d t) E d t Δv t t= t= t= s= t Δv s t s= d s )]. 5) 26

30 We evaluate each element of the expectation on the right hand side of 5). he,) element becomes [ ] E[the,) element] = E v v ) v t v )t = tγ t γ t= t= t= t tγ t + γ t= t= = 2 2 γ + o 2 ) 6) because γ t is absolutely summable and γ = o). In exactly the same way, we can see that [ ] t E[the,2) element] = E v v ) v t v ) s = γ t t= s= t= t s γ [{ E[the 2,) element] = E +)v t= s= s= t s γ t t= s= t t s + γ t s t= s= = 3 6 γ + o 3 ), 7) = +) + } ] v t v v t v )t t= tγ t +)γ t= γ t t= t= t tγ t + γ t= t= t= t= t E [ v t t= t= tv t ] t = o 3 ) 8) because E[ t= v t t= tv t] E[ t v t) 2 ]E[ t tv t) 2 ] O )O 3 ) by the Cauchy- Schwarz inequality, and [{ } ] t E[the 2,2) element] = E +)v v t v v t v ) s = +) + γ t t= s= t= γ t t= s= t s t= t s +)γ γ t t= s= 27 t s + γ t= s= t t= s= s= [ t s E v t v t t= t= s= ] t s s = o 4 ). 9)

31 Since direct calculation yields 22 +) d t d t) = ) 6 t= ) 6 ) 2 2 ), 2) we have, using 5) 9), E[R 2 ]= 2 γ + o ). 2) he expectation of R 3 is obtained in a similar manner. We first express E[R 3 ]as E[R 3 ]= ) 2 tr ) t t [ d t d t d s d s d t d t) ] E d t Δv t Δv t d t. t= t= t= s= Since it is shown that ) d t d t = [ E d t Δv t t= t t= s= s= t d s d s s= ) +) 5 ) ) t= ] Δv t d t = t= ) d t d t t= ) t= t= ) 6 2 +), 22) 5 2 ) [ 2γ + o) γ + o ) γ + o ) 2 γ + O ) we have E[R 3 ]= 9 ) 5 γ + o. 23) Hence, we obtain 2) from 4), 2) and 23). he evaluation of the expectation of R 2 proceeds in the same way but is more compli- ], 28

32 cated. We first expand R 2, except for the scalar, 2ψ)/ 2,as t ) t ) ˆε s Δv s t= s= s= t t ) = ε s d s d t d t d t ε t t= s= s= t= t= t t ) Δv s d s d t d t d t Δv t = s= t t= s= + t= s= ε s s= t s= t Δv s t ε s d s s= t= t= s= t Δv s t= t s= d s ) d t d t d t ε t t= ) d t d t d t Δv t t= ) ε t d t d t d t t= t= t t= s= = R 2 R 22 R 23 + R 24, say. t= t d s d s s= t= ) d t d t d t Δv t We evaluate each term. he expectation of R 2 becomes [ t ] t E[R 2 ]= E ε s v t v ) = σε 2 ψ t s = σε 2 t= s= t= s= t= t= t= t ) ψ t. 24) In order to evaluate the expectations of R 22, R 23 and R 24, we use the following lemma. Lemma. Let f t and g t be deterministic sequences for t =,,. hen, [ ) )] t ) E f t ε t g t v t = σε 2 f s g s+t ψ t, 25) t= t= f t t= s= t ε s = t= s= ) f s ε t. 26) We omit the proof of Lemma because it is directly obtained by noting that E[ε t s v t ]= σ 2 ε ψ s for s ande[ε t+s v t ]=fors>. t= For E[R 22 ], note that [ E[R 22 ]=tr d t d t) E d t ε t t= 29 t= s=t t= s= t Δv s t s= d s ]. 27)

33 Each element of the expectation on the right hand side of 27) is evaluated as E[the,) element] = E [ ε t t= t= [ E[the 2,) element] = E tε t E[the,2) element] = E E[the 2,2) element] = E t= t= t ε t v t t= t= s= t tε t v t t= t= s= [ [ ] tv t = σε 2 2 t 2 ψ t + O ), 2 t= ] tv t = σε t 2 ) 2 + t3 ψ t + O 2 ), 6 t= ] s = σε 2 3 t 3 ψ t + O 2 ), 6 t= ] s = σε t4 24 t 3 ) ψ t + O 3 ), 6 where we used 25) with f t = and g t = t for the,) element, f t = t and g t = t for the 2,) element, f t = and g t = tt +)/2 for the,2) element, and f t = t and g t = tt +)/2 for the 2,2) element. hen, using 2) it is shown that E[R 22 ]=σ 2 ε t= t= 4t t ) + t4 ψ 4 t + O). 28) For R 23, we note that t ε s t= s= s= t d s = = [ t t t ε s, s t= s= t= s= ] t ε s s= [ 2 t 2 ) + O ) ε t, 2 t= 3 t 3 t= 6 ) ] + O 2 ) ε t where the last expression is obtained by using 26). Hence, using 25), it is shown that [ ) E d t Δv t t= t t= s= [ = σεe 2 t= t= t ε s d s s= 2 t) 2 2 )] ψ t + O ) t 2 t3 6 ) ψt + O 2 ) t= t= 3 t) t4 + t 3 2 t2 2 4 ψ t + O ) 2 ) ψt + O 3 ) ] Using this result and 2), we have E[R 23 ]=σ 2 ε t= 4t t ) + t4 ψ 4 t + O). 29) 3

34 Exactly in the same way, for R 24,wehave, [ ) )] [ E d t Δv t ε t d t = σ ψ ε 2 t= t t= t) ψ t t= t ψ t + O) ψ t + O ) t= t= t= 2 t 2 2 ]. hen, using 22), it is shown that ) 9 E[R 24 ]=σε 2 3 9t2 ψ 5 2 t + O). 3) t= From 24), 28), 29) and 3), we obtain E[R 2 ]= 2σ2 εψ) t= 9 3 3t ) + 7t2 5 2 t4 ψ 4 t + o ). Furthermore, since j= ψ j <, the above summation converges to j= 9/3) ψ t and we get Noting that ψ) = /φ) and E[R 2 ]= 9σ2 εψ) 5 ) ψ j + o. j= j= ψ j = ψ ) = ) = φ ) φ) φ) 2, we finally obtain 3). We obtain a similar expression for the constant case in exactly the same manner and we omit the proof. 3

35 References [] Andrews, D., and J. C. Monahan 992). An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator, Econometrica 6, [2] Aznar, A. and M. Ayuda 28). A Point Optimal est for the Null of Near Integration, presented at EEA-ESEM meeting 28, University of Bocconi. [3] Berk, K. N. 974). Consistent Autoregressive Spectral Estimates, Annals of Statistics 2, [4] Brillinger, D. R. 98). ime Series Data Analysis and heory. Holden-Day, San Francisco. [5] Caner, M., and L. Kilian 2). Size Distortions of ests of the Null Hypothesis of Stationarity: Evidence and Implications for the PPP Debate, Journal of International Money and Finance 2, [6] Carrion-i-Silvestre, J. L., and A. Sansó 26). A Guide to the Computation of Stationarity ests, Empirical Economics 3, [7] Cheung, Y. W. and M. D. Chinn 997). Further Investigation of the Uncertain Unit Root in GNP, Journal of Business and Economic Statistics 5, [8] Hadri, K. 2). esting for Stationarity in Heterogeneous Panel Data. Econometrics Journal 3, [9] Harris, D., S. Leybourne and B. McCabe 27). Modified KPSS ests for Near Integration, Econometric heory 23, [] Kuo, B. S. and A. Mikkola 999). Re-Examining Long-Run Purchasing Power Parity, Journal of International Money and Finance 8, [] Kurozumi, E. 29). Construction of Stationarity ests with Less Size Distortions, Hitotsubashi Journal of Economics 5,

36 [2] Kwiatkowski, D., P. C. B. Phillips, P. Schmidt and Y. Shin 992). esting the Null Hypothesis of Stationarity against the Alternative of a Unit Root, JournalofEconometrics 54, [3] Lanne, M. and P. Saikkonen 23). Reducing Size Distortions of Parametric Stationarity ests, Journal of ime Series Analysis 24, [4] Leybourne, S. J. and B. P. M. McCabe 994). A Consistent est for a Unit Root, Journal of Business and Economic Statistics 2, [5] Leybourne, S. J. and B. P. M. McCabe 999). Modified Stationarity ests with Data- Dependent Model-Selection Rules, Journal of Business and Economic Statistics 7, [6] Müller, U. K. 25). Size and Power of ests of Stationarity in Highly Autocorrelated ime Series, Journal of Econometrics 28, [7] Müller, U. K. 28). he Impossibility of Consistent Discrimination between I) and I) Processes, Econometric heory 24, [8] Perron, P. and S. Ng 996). Useful Modifications to Some Unit Root ests with Dependent Errors and heir Local Asymptotic Properties, Review of Economic Studies 63, [9] Rothman, P. 997). More Uncertainty About the Unit Root in U.S. Real GNP, Journal of Macroeconomics 9, [2] Saikkonen, P. and R. Luukkonen 993a). esting for a Moving Average Unit Root in Autoregressive Integrated Moving Average Models, Journal of the American Statistical Association 88, [2] Saikkonen, P. and R. Luukkonen 993b). Point Optimal ests for esting the Order of Differencing in ARIMA Models, Econometric heory 9,

37 [22] Sul, D., P. C. B. Phillips and C. Y. Choi 25). Prewhitening Bias in HAC Estimation, Oxford Bulletin of Economics and Statistics 67,