Trend and initial condition in stationarity tests: the asymptotic analysis

Transcription

1 Trend and initial condition in stationarity tests: the asymptotic analysis Anton Skrobotov Gaidar Institute for Economic Policy, The Russian Presidential Academy of National Economy and Public Administration May 3, 013 Abstract In this paper we investigate the behavior of stationarity tests proposed by Müller (005) and Harris et al. (007) with uncertainty over the trend and/or initial condition. As different tests are efficient for different magnitudes of local trend and initial value, we propose following Harvey et al. (01) decision rule based on the rejection of null hypothesis for multiple tests. Additionally, we propose a modification of this decision rule relying on additional information about magnitudes of local trend and/or initial value that is obtained through pre-testing. The resulting modification has good size properties under both types of uncertainty. Key words: Stationarity test, KPSS test, uncertainty over the trend, uncertainty over the initial condition, size distortion, intersection of rejection decision rule. JEL: C1, C 1 Introduction The influence of linear trend and/or the initial value can be very important in unit root testing. In recent works, Harvey et al. (009) and Harvey et al. (01) (hereafter HLT, see also Harvey et al. (008)) considered the issue of deterministic trend inclusion in the unit root test and investigated the behavior of tests with different initial values. With uncertainty over the linear trend Harvey et al. (009) showed that the best unit root test is a simple union of rejections of two tests (i.e., the null hypothesis of unit root is rejected if it rejects at least one of the tests), the first with inclusion of a linear trend, the second with only the constants. Both of these tests need to be effective for their respective types of deterministic components in the absence of a large initial value. At the same time, with knowledge of the type of deterministic components (i.e. whether the trend is present in the data or not) and uncertainty over the initial condition, the union of rejection testing strategy of two tests (one effective at small initial value, and the second is effective at a large initial We thank two anonymous referees and Marina Turuntseva for helpful comments and recommendations for improvement earlier version of this paper. antonskrobotov@gmail.com 1

2 value for a given type of deterministic component) will be the best. HLT extended the procedure, assuming uncertainty over both the trend and initial condition, and suggested a union of rejection testing strategy for all four tests. Additionally, HLT also suggested a modification of this test with pre-testing of the linear trend coefficient and the initial value. Then, if there is evidence of a large local trend and/or evidence of a large initial condition, it is favorable that the trend and/or large initial value is actually present in the data. Thus, this information can be used to construct the union of rejection testing strategy. There is a need of a similar procedure for tests for stationarity as null, as hypothesis testing opposed to unit root is important for confirmatory analysis (see, e.g., Maddala and Kim (1998, Ch. 4.6)). Harris et al. (007) (hereafter HLM) have proposed a modification of the standard Kwiatkowski et al. (199) test (hereafter KPSS) in the near integration using a (quasi) GLSdetrending 1. Asymptotic properties of the test (only in constant case) were compared with pointoptimal test proposed by Müller (005) for different initial values. The results revealed that in the case of small initial values the test proposed in Müller (005) is effective, while with a large (or even moderate) initial value this test has serious liberal size distortions, tending to unity for moderately close to a unit root process. At the same time it is strictly dominated by the HLM test with a large initial value. In this paper we consider the asymptotic properties of stationarity tests proposed by HLM and Müller (005) following the approach of HLT. In Section we introduce the HLM test and pointoptimal test proposed in Müller (005) and obtain corresponding limiting distributions assuming local behavior of the trend. Additionally, we use parametrization of the initial condition following the Müller and Elliott (003) approach. In Section 3.1 we analyze these tests in the case of asymptotically negligible initial condition, assuming the local behavior of the trend. As in HLM, in this case the asymptotic size curves show that the point-optimal tests of Müller (005) is superior to the HLM test. At the same time, the test with only constant has serious liberal size distortions by increasing the magnitude of the local trend parameter. We propose to use the intersection of rejections testing strategy of two tests with and without a trend in deterministic component, i.e., we reject the null hypothesis, if both tests simultaneously reject it. We also propose a modification of this decision rule, using pre-testing of a trend parameter and using this information to perform only the test with trend, if there is evidence of a large local trend. As simulations show, this procedure has advantages over a simple intersection rejection of two test (with and without trend). In Section 3. we analyze a similar procedure, assuming knowledge of deterministic term, but no knowledge of the magnitude of initial value. In this case, as in Section 3.1, simple intersection of rejections of corresponding tests is the best solution, as well as the modification with pre-testing of the initial value. In Section 3.3 we address the joint problem of uncertainty over both the linear trend and the initial condition. In this case, following HLT, we propose an intersection of rejection testing strategy consisting of all four test statistics, as well as the modification of pre-testing the trend magnitude and initial value. Asymptotic analysis shows superiority of the considered modifications by varying parameters in the local trend and the initial value. 1 Müller (005) showed that the use of conventional KPSS test with the bandwidth parameter in the long-run variance estimator, increasing at a slower rate than the length of the sample, leads to an asymptotic size equal to unity under the null hypothesis about near integration. Therefore, in our analysis (local to unit root) we do not consider conventional OLS-detrended KPSS test. HLT used the union of rejections term, and their test rejected the null hypothesis, if at least one of the tests rejected it. As we consider stationarity tests in our procedure, we reject the null hypothesis if all of tests reject it, and we call this strategy the intersection of rejections.

3 In our asymptotic analysis the size of all tests was compared with a fixed power, where critical values were obtained with the integrated process (as well as zero trend parameters and zero initial values). Thus, in Section 4 we obtain the critical values and scaling constants for some fixed amount of mean reversion under the stationary null hypothesis and discuss the behavior of tests in finite samples. Results obtained in the course of this investigation are formulated in the Conclusion section. The Model We consider the data generating process (DGP) as y t = µ + βt + u t, t = 1,..., T (1) u t = ρu t 1 + ε t, t =,..., T () where the process ε t is taken to satisfy the standard assumptions considered by Phillips and Solo (199): Assumption 1 Let ε t = γ(l)e t = γ i e t i, with γ(z) 0 for all z 1 and i=0 i γ i <, where e t is the martingale-difference sequence with conditional variance σe and sup t E(e 4 t ) <. Short-run and long-run variances ( T ) are defined as σε = E(ε t ) and ωε = lim T T 1 E t=1 ε t = σ eγ(1), respectively. In contrast to the conventional KPSS test, we use the local to unity behavior of the parameter ρ, i.e., ρ = ρ T = 1 c/t, where c 0. The reason for the consideration of the local asymptotics is that it provides a more accurate approximation for small samples than the standard asymptotics when the series contains a large autoregressive root that is frequently observed in many macroeconomic time series. We test the null of stationarity (local to unit root) H 0 : c c > 0 against alternative hypothesis H 1 : c = 0, where c is the minimal amount of mean reversion under the stationary null hypothesis. Conventional KPSS test with the bandwidth parameter in the long-run variance estimator increasing at a slower rate than the length of the sample leads to an asymptotic size equal to unity under the null hypothesis about near integration (see Müller (005)). Thus, we consider two tests having non-degenerate limiting distribution under local to unit root behavior of autoregressive parameter. The first was suggested by Müller (005). Following Müller and Elliott (003), this study proposed as asymptotically optimal test statistic Q µ ( c) for the constant case and Q τ ( c) for the trend case to discriminate between a null hypothesis ρ T = 1 c/t and ρ T = 1. This statistic is constructed as following: i=0 Q i ( c) = q1(ˆω i ε 1 T 1/ û i T ) + q(ˆω i ε 1 T 1/ û i 1) + q i 3(ˆω 1 ε T 1/ û i T )(ˆω 1 T 1/ û i 1) + q4ˆω i ε ε T T (û i t), (3) t=1 3

4 where û i t are OLS residuals from regression of y t on d t, where d t = µ in constant case and d t = µ + βt in trend case, q µ 1 = q µ = c(1 + c)/( + c), q µ 3 = c/( + c), q µ 4 = c and q1 τ = q τ = c (8 + 5 c + c )/(4 + 4 c + 8 c + c 3 ), q3 τ = c (4 + c)/(4 + 4 c + 8 c + c 3 ), q4 τ = c. Also ˆω ε is any consistent estimator of long-run variance of ε t using residuals from an AR(1) regression of û i t. 3 The second test was proposed by HLM. It uses (quasi) GLS-detrended series. Specifically, ũ i t, i = µ, τ are OLS residuals from regression of y c = y t ρ T y t 1 on Z c = z t ρ T z t 1, t =,..., T, where z t = 1 in constant case and z t = (1, t) in trend case. Then, the S i ( c) test statistic is constructed as S i ( c) = T T t= ( t j= ũi j), (4) where kernel-based long-run variance estimator ˆω ε is calculated using GLS-detrended residuals ũ i t. 4 We consider the following two assumption, specifying the behavior of the trend coefficient β and initial condition u 1. Assumption The trend coefficient β satisfies β = β T constant. ˆω ε = κω ε T 1/, where κ is some finite Assumption 3 The initial condition u 1 satisfies u 1 = = α ωε/(1 ρ T ), where ρ T = 1 c/t, c > 0. In unit root case, c = 0, the initial condition is equal to zero, i.e. all tests are invariant to α. HLM showed that in the case of small initial values the Q i ( c) test is efficient (in sense of smaller size) in comparison with the S i ( c) test. However, for large initial values Q i ( c) has serious size distortion and is strictly dominated by S i ( c) test. One could also consider the optimal Q i (g, k) test proposed by Elliott and Müller (006) and analyzed by HLM in a stationarity testing context. However, as it was shown in HLM, this test is strictly dominated by Q i ( c) test for small initial values and by S i ( c) test for large initial values. Thus, we do not use it in further consideration. Similarly, as will be shown in Section 3.1, in the absence of trend the effective tests are those that do not account for the trend in the construction, while these tests will have a serious size distortion in the presence of a trend in DGP. The following lemma summarizes limiting distributions of four tests under given Assumptions 1-3. Lemma 1 Let {y t } is generated as in (1) and (3) and Assumptions 1-3 are satisfied. Then under ρ T = 1 c/t, 0 c < ( Q µ ( c) q µ 1 K c µ (1) + κ ) ( + q µ K c µ (0) κ ) + q µ 3 ( K c µ (1) + κ ) ( K c µ (0) κ ) + q µ 4 1 Q τ ( c) q τ 1K τ c (1) + q τ K τ c (0) + q τ 3K τ c (1)K τ c (0) + q µ 4 0 ( K µ c (r) + κ(r 1 ) ) dr, (5) 1 0 K τ c (r)dr, (6) 3 Kernel-based long-run variance estimator proposed by (Müller, 005) and used by в HLM leads to very poor finite sample properties of the tests under negative moving average errors. In unreported simulations (results made available upon request) we found that autoregressive long-run variance estimator leads to satisfactory finite sample properties. 4 Note that this estimator has satisfactory finite sample properties in unreported finite sample simulations. 4

5 Here S µ ( c) S τ ( c) H c, c,α,κ (r) dr, (7) ( H c, c,α,0 (r) 6r(1 r) 1 K c µ (r) = K c (r) K c (s)ds, 0 ( Kc τ (r) = K c µ (r) 1 r 1 ) 1 r H c, c,α,κ (r) = K c (r) + c K c (r) = 0 1 ( s 1 ) K c (s)ds, 0 H c, c,α,0 (s)ds) dr, (8) [ K c (1) + c (K c (s) + κs)ds r 0 { α(e rc 1)(c) 1/ + W c (r), c > 0 W (r), c = 0, 1 0 ] (K c (s) + κs)ds, where W c (r) = r 0 e (r s) cdw (s) is a Ornstein-Uhlenbeck process, W (r) is a standard Wiener process, and denotes weak convergence. The proof of (5) is similar to Harvey et al. (009), the proof of (7) is given in the Appendix. The proofs of (6) and (8) are standard and use FCLT and CMT. Also following HLM (see also Müller and Elliott (003) and Elliott and Müller (006)), we set c = 10 for Q µ ( c) and S µ ( c) statistics and c = 15 for Q τ ( c) and S τ ( c) statistics. Remark 1 Note that the Q τ ( c) and S τ ( c) are invariant to the trend magnitude while the limiting distributions of Q µ ( c) and S µ ( c) explicitly depend on the local drift parameter κ. Remark Under fixed nonzero trend of the form β = κω ε 0 it is easy to show that the Q µ ( c) and S µ ( c) are both O p (T ), i.e. diverge to infinity. This leads to the fact that these tests have size equal to unity for all c. 3 Asymptotic analysis of stationarity tests 3.1 Asymptotic behavior under a local trend Consider a case when initial condition is u 1 = o p (T 1/ ). Then, in limiting distributions obtained in Lemma 1 the process K c (r) is simply replaced by Ornstein-Uhlenbeck process W c (r). The figures 1(a)-(d) show asymptotic size for c [0, 0], where the critical values are obtained for comparison of the tests for c = 0 and κ = 0, that the power was 0.5 for any test as in Müller (005) and HLM 5. Comparing the size of the tests for case of κ = 0 (fig. 1(a)), i.e. under the absence of a linear trend, it is clear that the size is smaller for tests that do not take into account the presence of a trend. Additionally, the Q i dominates S i, as was evident in the results of simulations performed by Harris et al. (007) for the case of only constant in deterministic term. As a result, for stationarity testing the Q µ test will be effective of the considered tests for κ = 0. 5 Here and in the following sections results are obtained by simulations of the limiting distributions in Lemma 1, approximating the Wiener process using i.i.d.n(0, 1) random variates and with integrals approximated by normalized sums of 1,000 steps, with 50,000 replications. 5

6 If parameter κ is increased, e.g., for κ = 0.5 (fig. 1(b)) the results are largely similar to the case of κ = 0 and Q µ is still effective except in a very small interval c [0, 1], when Q µ is dominated by tests that take into account the trend (though they have a higher power in the interval c [0, 1] and this interval can be considered as negligible). Notably, that the size of tests with the only constant slightly increases in comparison with case of κ = 0. For κ = 1 (fig. 1(c)) the S τ and Q τ is clearly superior to the S µ and Q µ (the latter has serious size distortions, the size is never lower than for the considered interval c [0, 0]), and Q τ is the efficient test. The size of tests with only the constant continues to increase with increasing κ. For κ = (fig. 1(d)) the size of S µ and Q µ almost always equals to one, which confirms the behavior Q µ and S µ under a fixed nonzero trend (see Remark ). As each of the considered Q µ and Q τ tests is efficient (in terms of size) for some values of a local trend, there is uncertainty over magnitude of this local trend and it is necessary to use a feasible strategy to discriminate two cases, with presence and absence of the linear trend. Following Harvey et al. (009) we use the following rule of intersection of rejections where we reject the null of stationarity if both tests reject the null simultaneously. Specifically, this decision rule can be written as: IR = Reject H 0 if {Q µ > m cv Q,µ and Q τ > m cv Q,τ }, (9) where cv Q,µ and cv Q,τ are asymptotic critical values for Q µ and Q τ for some specified value of c and significance level (for more details see Section 4), and m is some scaling constant ensuring that asymptotic size equals for a given value c (in case of absence of scaling the size and power decreases, so we call the decision rule with scaling liberal). It is possible to further improve this strategy by using information about the high value of parameter κ, specifically, about the clear evidence of a trend. As noted by J. Breitung in rejoinder of Harvey et al. (009) there is no need to use the same scaling constant m in all cases. If there is strong evidence for a trend, then the probability of rejection of Q µ test tends to unity. Consequently, it is possible to improve the strategy IR by using pre-tests Dan-J, t λ, t m λ and t RQF β, 6 proposed, respectively, by Bunzel and Vogelsang (005), Harvey et al. (007), Perron and Yabu (009) and analyzed in HLT and Harvey et al. (010) for various magnitudes of a local trend κ and initial value α. I.e. under strong evidence for the trend the critical value of the Q τ can be used. More precisely, consider the following modification of the decision rule (9): { Reject H 0 if {Q µ > m cv Q,µ and Q τ > m cv Q,τ }, if s β cv β IR(s β ) = Reject H 0 if {Q τ > cv Q,τ, (10) }, if s β > cv β where s β denotes some pre-test for testing β = 0, and cv β is the corresponding critical value. The limiting distribution of these two tests follows directly from Lemma 1 and CMT, and is, therefore, omitted. Notably, under a fixed nonzero trend each of s β statistics diverges to infinity so that asymptotically the Q τ test will be selected. On the other hand, in the absence of a trend the Q µ with scaling critical value will actually be selected. The Figures (a)-(d) show asymptotic size of tests Q µ, Q τ, IR, IR( t λ ), IR( t m λ ) and IR( Dan-J ) for values κ {0, 1,, 4} and c [0, 0]. We chose the scaling constants m so that the asymptotic power of the test IR was 0.50 for κ = 0. 6 The t λ and t RQF β statistics are asymptotic equivalents, therefore we consider only the first in a study of the asymptotic behavior of the tests. 6

7 The power of IR( s β ) tests for κ = 0 does not deviate by more than from 0.50, therefore we compare only the size of tests varying the parameter κ. Figures show similar behavior of tests, specifically, the modified test IR(s β ) clearly dominates IR. For small κ both types of tests behave almost the same way (with very little dominance of IR(s β )), and for κ < 1 are the same due to the principle of construction of the stationary test Q µ and Q τ. For κ = 0 the size of IR tests is between Q µ and Q τ as expected. The size of tests IR(s β ) is almost identical with IR. With increasing κ the size of tests IR(s β ) approaches to effective test in this case Q τ. 3. Asymptotic behavior under a various initial values We consider tests Q i and S i, i = µ, τ, in a fashion similar to the previous section, varying parameters of initial value α from 0 to 6 7 and parameter c {.5, 5, 10}. Additionally, we assume knowledge of the type of deterministic component. Figures 3(a) and (c) show asymptotic size of the tests Q µ, S µ, IR and IR(s α ). The last two tests will be discussed below. Our results show that for small initial values the Q µ test dominates S µ, while with increasing α the asymptotic size of Q µ test goes to one (for moderate values of c). At the same time, the size of S µ increases with increasing α only for small values of c, but for c = 10 (Fig. 3(c)) it remains constant for all α, even though it is dominated by Q µ for small initial values (α <.6). Thus, for small α the Q µ test is efficient, while for large values of α it is strongly oversized. Therefore, if information is available about the large initial value, it becomes necessary to use the S µ test. Our results for the trend case are summarized in Figures 3(b) and (d) for c = 10 and c = 15, respectively, and appear to be identical, although the size distortions for Q τ and S τ are not as strong for large α and small c, as in Q µ and S µ (results available on request). As in Harvey et al. (009) the following strategy of intersection of rejection can be applied: IR = Reject H 0 if {Q i > m i cv Q,i and S i > m i cv S,i }, (11) where cv Q,i and cv S,i, i = µ, τ are the asymptotic critical values of tests Q i and S i for some specified value c and significance level, and m i is the some scaling constant that the asymptotic size was at the level for a given c. In an approach that is similar to the previous section, this strategy can be modified by using additional information about the large initial value, in order to use only the test S i in this case: { Reject H 0 if {Q i > m i IR(s α ) = cvq,i and S i > m i cvs,i }, if s α cv α Reject H 0 if {S i > cv S,i, (1) }, if s α > cv α where s α denotes some test statistic for testing α = 0, and cv α is the corresponding critical value. As in the previous section the limiting distribution of these two tests directly follows from Lemma 1 and CMT and omitted for brevity. As s α it can be used as s the test statistic, proposed by HLT: s α = DF -QD i cvqd,i cv OLS,i DF -OLS i, (13) where i = µ, τ depending on the type of the deterministic term, DF -QD τ and DF -OLS τ are ADF tests for GLS and OLS detrending, respectively, and cv QD,τ and cv OLS,τ are corresponding 7 As these tests are symmetric around α there is no need to consider negative initial values. 7

8 critical values. Large values of the upper tail of distribution for this test statistic indicate a large initial value of α. HLT obtained the critical values for c = 30, as only in this case the test statistic s α has the correct size. For smaller values of c it has liberal size distortions that increase with decreasing c. Figures 3(a)-(d) also show asymptotic size of tests IR and IR(s α ) (the critical values for s α were calculated at c = 0). As in the previous section the critical values of Q i and S i (i = µ, τ) and scaling factors m were calculated in such way that the tests have power equal to We also analyzed the behavior of the tests, if the critical values for s α were obtained at different c (the results available upon request). We use the critical values for s α at c = 10 for the mean case and at c = 0 for the trend case. Furthermore, we use an additional correction of critical values, so that the IR(s α ) strategy has power equal to As expected, the size curve of IR lies between the size curves of S i and Q i, although the size distortion is quite substantial for large α. However, the modification IR(s α ) has a size that is close enough to effective S i test for large α and significant gain in size for small α in comparison to S i. Thus, IR(s α ) effectively discriminates the cases of small and large initial values. 3.3 Asymptotic behavior under uncertainty over both the trend and initial condition Previous sections of this work discussed two testing strategy, the first of which is a stationarity test with uncertainty over linear trend with the known asymptotically negligible initial value, and the second is also stationarity test with the knowing the exact specifications of the deterministic component, but with uncertainty over the magnitude of the initial condition. However, the magnitude of the initial value or the value of a trend parameter are not known in advance. In this case following the HLT method (see also Harvey et al. (008)), we can apply the strategy of intersection of rejections, which reject stationarity, if each of the four tests, Q i and S i (i = µ, τ), rejects the null hypothesis of stationarity. This liberal decision rule can be written as follows: = Reject H 0 if {Q µ > m cv Q,µ and Q τ > m cv Q,τ and S µ > m cv S,µ and S τ > m cv S,τ }, (14) where m is the scaling constant. It can also improve the test pre-identifying the possible large initial value or significant linear trend as in Sections 3.1 and 3.. However, as shown in Harvey et al. (008), the tests Dan-J, t λ and tλ m are very sensitive to the magnitude of initial value. HLT used a modified test t λ : t λ = (1 λ )t 0 + λ t 1, (15) where in contrast to Harvey et al. (007), t 1 is constructed using autocorrelation-corrected t-ratio for testing β T = 0 in the following regression y t ρ T y t 1 = ˆµ(1 ρ T ) + ˆβ T (t ρ T (t 1)) + û t, t =,..., T, (16) where ρ T = 1 c/t, and corresponding long-run variance is calculated by using residuals û t. HLT obtained the limiting distribution of modified test statistic and shown that for c = c this statistic is asymptotically invariant to the initial condition at point c = c and asymptotically standard normal. 8

9 HLT set s β = t λ with c = 30 and used standard normal critical value. Then the test s β will be oversized as c decreases towards zero, but at c = 30 will be correctly sized. Thus, as in HLT the modified liberal decision rule can be written as follows, where each IR(, ) strategy reject the null hypothesis if all tests in the brackets reject it: Definition 1 The modified intersection of rejections strategy IR 4 is defined as follows: 1) If s β cv β and s α cv α, then use the liberal decision rule IR(Q µ, Q τ, S µ, S τ ), defined in (14); ) If s β cv β and s α > cv α, then use the liberal decision rule IR(S µ, S τ ), the corresponding scaling constant is defined as m ; 3) If s β > cv β and s α cv α, then use the liberal decision rule IR(Q τ, S τ ), the corresponding scaling constant is defined as m τ ; 4) If s β > cv β and s α > cv α, then use the decision rule reject H 0, if S τ > cv S,τ ; The basic concepts of this strategy are as follows. Under 1) there is no reason to assume that the values of the local trend and the initial value are large, and there is no reason to argue that they are necessarily small. Thus, the strategy should be applied. Under ) there may be some evidence in favor of a large initial value, so that Q µ and Q τ tests will be ineffective (then can have size approaching to unity) and can be excluded from the intersection of rejections. But since we can not be sure of the magnitude of the local trend, it is necessary to use both tests S µ и S τ. Under 3) there is evidence of a large magnitude of a local trend and in this case, Q µ and S µ are ineffective and should be excluded from the strategy. However, we should consider both Q τ and S τ tests, as there is no reason to believe that the initial value is large. Finally, under 4) there is evidence of both a large local trend and large initial value so that only S τ will be effective test in this case. Thus, the null hypothesis will be rejected if only this test will be reject it. Figures 4-6 respectively demonstrate for κ = 0, 0.5, 1, the asymptotic size of tests S τ, and for c {0, 1,..., 0}, fixing power at All figures (a)-(i) show the results for α = { 4,, 1, 0.5, 0, 0.5, 1,, 4}, respectively. Only S τ is considered in the figures among the four original stationarity tests because its size is never below a certain level across considered α and κ. Thus, this test can be considered robust to both forms of uncertainty (trend and initial condition). Notably, it is necessary to correct the testing strategy that it controls size asymptotically as in HLT. Therefore, we replace cv Q,µ, cv Q,τ, cv S,µ and cv S,τ by τ cv Q,µ, τ cv Q,τ, τ cv S,µ and τ cv S,τ, respectively, where τ is the scaling constant that the power has never been lower than When κ = 0 the test is superior to S τ. Only in case of α = 4 for small c the size of is slightly higher. In comparison with the size of the latter is almost always lower than S τ and, although in case of large α this test has serious size distortions for small c. When κ = 0.5 for negative α the size of is slightly higher than S τ (except for c > 5 in case of α = 4), but increasing α, even at α = 0.5 causes their size curves to intersect, and for α = the size of is lower than S τ. For α < 1 the size of behaves almost as well as S τ, but for large α its size curves become strongly nonmonotonic. 8 There is no need to consider results for larger κ because s β will always reject the null hypothesis in these cases, so that the size of IR 4 will no longer vary with κ. 9

10 For higher κ the size curve of IR 4 lies between S τ and curves, and size distortions of are greatly enhanced, especially for large α, while the size of IR 4 approaches to size of effective test S τ even for moderate c. Because it is unclear which of the test strategy dominates another among different α and κ, consider the asymptotic integrated size of all three tests for comparison, similar to Harvey et al. (009), which examined the asymptotic integrated power. Integrated asymptotic size is the area below each curve in Figures 9-11 for c {0, 1,..., 0}. Table 1 shows a loss in the asymptotic integrated size relative to the effective test (with a minimum size) in a particular case. Bold type indicates the maximum size losses for each specific strategy. For the test S τ over all κ and α the maximum value is 9 (in the case of κ = 0 and α = 0), for (in the case of κ = 1 and α = ), for IR 4 0 (in the case of κ = 1 and α = 0.5), which proves superiority of the latter test, as the maximum losses for it are minimal. At the same time, if we consider integrated size relative to the maximum value (Table ), it is evident that for the S τ test and the test there exists κ, where they are always dominated by IR 4 (for the first in the case of κ = 0, for the second κ = 1). On the other hand, the maximum gain does not differ much for all tests, except for the case κ = 0 and α = 0, when the size gain for the test equals to 0.5, as expected by theoretical results. Thus, based on the asymptotic results, we strongly recommend the use of the modified decision rule IR 4 if there is uncertainty over both the trend and the initial condition. 4 Critical values In this section we discuss obtaining critical values for practical application 9, because previously we compared the tests, fixing the power at The critical values for tests Q µ ( c) and S µ ( c) are given in Table 3. We obtain them at c = 10 for tests Q µ ( c) and S µ ( c), as in Müller and Elliott (003) and Elliott and Müller (006), and for test Q τ ( c) and S τ ( c) at c = 15. We note (see HLM), that in this case c = c, and tests S i ( c), (i = µ, τ) have standard KPSS limiting distribution, so the critical values are the same as for conventional KPSS tests. However, the finite sample critical values slowly converge to the asymptotic, so we provide additional critical values of all tests for T = 150, 300, 600. Asymptotic scaling constants, however, are appropriate for finite samples. Also, Q i ( c) is not invariant to the initial value α, when c > 0, so the critical values obtained for α = 0. Critical values for the initial condition test s α were obtained at c = 0 and are listed in Table 4. It is necessary to obtain scaling constants for all intersection of rejections testing strategy, which were considered in Sections 3. However, there is some difficulty with obtaining the scaling constants, as the critical values for the tests with trend are constructed with c = 15, while critical values for the tests, only with constant, are constructed with c = 10. Therefore, if the intersection of rejections testing strategy includes tests with different types of deterministic components, we obtain the scaling constants with c = 1.5 (i.e. at c = 1.5 tests has correct size). Otherwise, scaling is performed using the specific c for each type of deterministic component. All scaling constants are given in Table 5 (the simulation code is available upon request). We also conduct finite sample simulations for proposed strategies (fixing the power at 0.50 to allow comparison), by using QS kernel and automatic bandwidth selection of Newey and West 9 In this section, we obtain the results using the normalized sum of 5000 steps and 100,000 replications. 10

11 (1994) for long-run variance estimator of S i ( c) statistics and autoregressive long-run variance estimator of Q i ( c) statistics. Various DGP were examined (i.i.d., AR(1) and MA(1) processes for ε t ). The results show that the asymptotic analysis provides a good approximation of the behavior of tests in finite samples (results available on request). Notably, only S i ( c) and Q i ( c) requires finite sample critical values while the asymptotic scaling constants are similar to the asymptotic. 5 Conclusion In this paper we consider the problem of stationarity testing with uncertainty over the trend and/or initial condition. We proposed the intersection of rejections testing strategy of several tests (similar to using a union of rejection testing strategy proposed by HLT), if there is uncertainty over the trend and/or initial condition. Simulation evidence shows that the testing strategy based on the rejection of all tests (each of which is effective for small/large initial value and/or for small/large parameter of the local trend) suggests size performance in the presence of uncertainty over both the trend and the initial condition. Additionally, we showed that pre-testing of the trend parameter and of the initial value could improve the procedure if any of these parameters will be significantly nonzero. We conclude that our procedure is useful in empirical applications in conjunction with the HLT test, as stationarity testing is necessary for confirmatory analysis. A Appendix Proof of Lemma 1: Due to the invariance, we set µ = 0 without loss of generality, thus y t = βt+u t. We consider the model without the trend when it is actually present and using GLS-detrending we obtain that Let z t = u t u 1. Then r t = y t ρy t 1 µ, µ = 1 T (y t ρy t 1 ) T 1 r t = u t ρu t 1 + βt ρβ(t 1) 1 T 1 = z t + u 1 ρz t 1 ρu 1 1 T 1 t= t= T t= (u t ρu t 1 ) β T 1 T (z t + u 1 ρz t 1 ρu 1 ) + βt ρβ(t 1) β T (t ρt + ρ) T 1 t= [ ] [ = z t ρz t 1 1 T (z t ρz t 1 ) + βt ρβt β T 1 T 1 t= The second term can be written as β ( t T + 11 ) ρβ ( t T + T (t ρt + ρ) t= ] T (t ρt) t= ) = T 1 cβ ( ) t T +.

12 Then r t = T 1/ [rt ] i= T 1/ [rt ] ( z t ρz t 1 1 T 1 i= T )) (z t ρz t 1 t= + T 1/ [rt ] T 1 cβ i= ( i T + ) The first term of obtaining expression has a limiting distribution obtained in HLM and corresponds to the case of κ = 0 (more precisely, converges to ω ε H c, c,α,0 (r)). Consider the second term, responsible for the behavior of test statistics under local trend: T 1/ [rt ] T 1 cβ i= ( i T + ) ([rt ] 1)([rT ] + ) ([rt ] 1)(T + 1) = cω ε κt cω ε κt This expression converges to cω ε κ ( ) r r, which proves the lemma using CMT and simple integral transformations, because the long-run variance estimators (kernel-based in our case) ω ε are still consistent under the local trend misspecification. 1

13 References Bunzel, H. and Vogelsang, T.J. (005). Powerful Trend Function Tests That Are Robust to Strong Serial Correlation with an Application to the Prebisch-Singer Hypothesis. Journal of Business and Economic Statistics, 3, Elliott, G. and Müller, U. (006). Minimizing the Impact of the Initial Condition on Testing for Unit Roots. Journal of Econometrics, 135, Harris, D., Leybourne, S., and McCabe, B. (007). Modified KPSS Tests for Near Integration. Econometric Theory, 3, Harvey, D.I., Leybourne, S.J., and Taylor, A.M.R. (007). A Simple, Robust and Powerful Test of the Trend Hypothesis. Journal of Econometrics, 141, Harvey, D.I., Leybourne, S.J., and Taylor, A.M.R. (008). Testing for Unit Roots in the Presence of Uncertainty over Both the Trend and Initial Condition. Granger Centre Discussion Paper No. 08/03, University of Nottingham. Harvey, D.I., Leybourne, S.J., and Taylor, A.M.R. (009). Unit Root Testing in Practice: Dealing with Uncertainty over the Trend and Initial Condition (with Commentaries and Rejoinder). Econometric Theory, 5, Harvey, D.I., Leybourne, S.J., and Taylor, A.M.R. (010). The Impact of the Initial Condition on Robust Tests for a Linear Trend. Journal of Time Series Analysis, 31, Harvey, D.I., Leybourne, S.J., and Taylor, A.M.R. (01). Testing for Unit Roots in the Presence of Uncertainty over Both the Trend and Initial Condition. Journal of Econometrics, 169, Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., and Shin, Y. (199). Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root: How Sure Are We That Economic Time Series Have a Unit Root? Journal of Econometrics, 54, Maddala, G.S. and Kim, I-M (1998). Unit Roots, Cointegration, and Structural Change. Cambridge University Press, Cambridge. Müller, U. (005). Size and Power of Tests for Stationarity in Highly Autocorrelated Time Series. Journal of Econometrics, 18, Müller, U. and Elliott, G. (003). Tests for Unit Roots and the Initial Condition. Econometrica, 71, Newey, W.K. and West, K.D. (1994). Automatic Lag Selection in Covariance Matrix Estimation. Review of Economic Studies, 61, Perron, P. and Yabu, T. (009). Estimating Deterministic Trends with an Integrated or Stationary Noise Component. Journal of Econometrics, 151, Phillips, P.C.B. and Solo, V. (199). Asymptotics for Linear Processes. Annals of Statistics, 0,

14 Table 1: Maximum relative asymptotic size losses over c {0, 1,..., 0} κ = 0 κ = 0.5 κ = 1 α S τ S τ S τ Table : Maximum relative asymptotic size gains over c {0, 1,..., 0} κ = 0 κ = 0.5 κ = 1 α S τ S τ S τ

15 Table 3: Asymptotic and finite sample critical values for Q i and S i, i = µ, τ at the significance level T = 0.10 = 5 = 1 Q µ Q τ S µ S τ Table 4: Asymptotic critical values for s α at the significance level = 0.10 = 5 = Table 5: Asymptotic scaling constants for intersection of rejections testing strategies at the significance level = 0.10 = 5 = 1 IR(Q µ, Q τ ) m IR(Q µ, S µ ) m µ IR(Q τ, S τ ) m τ IR(Q µ, Q τ, S µ, S τ ) m IR(S µ, S τ ) m τ

16 Q Μ Q Τ S Μ Q Μ Q Τ S Μ (a) κ = 0 (b) κ = 0.5 Q Μ Q Τ S Μ Q Μ Q Τ S Μ (c) κ = 1 (d) κ = Figure 1: Asymptotic size and power for different values of κ, β T = κω ε T 1/ 16

17 Q Τ IR IRtΛ IRt m Λ Q Τ IR IRtΛ IRt m Λ (a) κ = 0 Q Μ Q Τ IR IRtΛ IRt m Λ IRDanJ c (c) κ = (b) κ = 1 Q Μ Q Τ IR IRtΛ IRt m Λ IRDanJ c (d) κ = 4 Figure : Asymptotic size and power for different values of κ, β T = κω ε T 1/ 17

18 Q Μ S Μ IR IRs Α Q Τ IR IRs Α (a) c = 5, mean case (b) c = 10, trend case Q Μ S Μ IR IRs Α 8 Q Τ IR IRs Α (c) c = 10, mean case (d) c = 15, trend case Figure 3: Asymptotic size for different initial values α 18

19 (a) α = 4 (b) α = (c) α = 1 (d) α = 0.5 (e) α = 0 (f) α = 0.5 (g) α = 1 c (h) α = c (i) α = 4 Figure 4: Asymptotic size and power for stationarity tests, κ = 0 19

20 (a) α = 4 (b) α = (c) α = 1 (d) α = 0.5 (e) α = 0 (f) α = 0.5 (g) α = 1 c (h) α = c (i) α = 4 Figure 5: Asymptotic size and power for stationarity tests, κ = 0.5 0

21 (a) α = 4 (b) α = (c) α = 1 (d) α = 0.5 (e) α = 0 (f) α = 0.5 (g) α = 1 c (h) α = c (i) α = 4 Figure 6: Asymptotic size and power for stationarity tests, κ = 1 1