The power of tests IR(|s|) for = 0 does not deviate by more than 0.02 from 0.50, therefore we compare only the size of tests varying the parameter. The figures show similar behavior of tests, more precisely, the modified test IR(s) clearly dominate IR. For small both types of tests behave almost the same way (with very little dominance of IR(s)), and for < 1 are the same due to the principle of construction of the stationary test Qµ and Q. For = 0 the size of tests IR as expected is between Qµ and Q. The size of tests IR(s) almost identical with IR. With increasing the size of tests IR(s) approaches to effective test in this case Q.

3.2 Asymptotic behavior under a various initial values Similar to the previous section we consider the tests Qi and Si, i = µ,, varying parameters of initial value from 0 to 65 and parameter c {2.5, 5, 10}. Also we suppose a knowledge about type of the deterministic component. The Figures 3(a),(c),(e) show asymptotic size of the tests Qµ, Sµ, IR and IR(s). The last two tests will be discussed later. The results show that for small initial values the test Qµ dominate Sµ, while with increasing the asymptotic size of test Qµ goes to one (for moderate values of c). At the same time, only for small values of c the size of Sµ increases, e.g., for c = 10 (Fig. 3(e)) it remains constant for all, even though it is dominated by Qµ for small initial values ( < 2.6). Thus for small the test Qµ is efficient, while for large values of it is strongly oversized. In this case having information about large initial value it is necessary to use the test Sµ.

The results for the trend case are summarized in Figures 3(b),(d),(f) and completely similar, although the size distortions for Q and S are not as strong for large and small c, as in Qµ and Sµ.

As in Harvey et al. (2009) the following strategy of intersection of rejection can be applied:

The tests t and tRQF are asymptotic equivalent therefore we consider only the first in a study of the asymptotic behavior of the tests.

As these tests are symmetric around no need to consider a negative initial values.

Q,i S,i IR = Reject H0 if {Qi > mi cv and Si > mi cv }, (11) Q,i S,i where cv and cv, i = µ, are the asymptotic critical values of tests Qi and Si for some specified value c and significance level, and mi is the some scaling constant that the asymptotic size was at the level for a given c.

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 Si in this case:

Q,i S,i Reject H0 if {Qi > mi cv and Si > mi cv }, if s cv IR(s) =, (12) S,i REject H0 if {Si > cv }, if s > cv where s denotes some test statistic for testing = 0, and cv 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 it can be use the test statistic, proposed by HLT:

QD, cv s = DF -QD - DF -OLS, (13) OLS, cv where DF -QD and DF -OLS are ADF tests for GLS and OLS detrending, respectively, QD, OLS, and cv and cv are corresponding critical values. Large values of the upper tail of distribution for this test statistic indicate a large initial value of ||. HLT obtained the critical values for c = 30, i.e. only in this case test statistics s has correct size. For smaller values c it has liberal size distortions that increase with decreasing c.

Figures 3(a)-(f) also show asymptotic size of tests IR and IR(s) (the critical values for s were calculated at c = 20). As in the previous section the critical values of Qi and Si (i = µ, ) and scaling factors m were calculated in such way that the tests have power equal to 0.50. We also analyzed the behavior of the tests, if the critical values for s were obtained at different c (the results available upon request). In constant case (Fig. 3(a), 3(c) and 3(e)) for large c the test IR(s) has asymptotic size that close to Sµ, although the size of Sµ somewhat lower at 2.5. With decreasing c for small the size of IR(s) is much smaller than Sµ and close to IR. However only for large the size of IR(s) is much smaller than IR, although not too close to Sµ, i.e. effective in this case. In general, the difference between IR(s) and Sµ is negligible, in the sense that the possible gain in size of IR(s) in comparison to Sµ for small compensated by a loss for large. Therefore we recommend to use only test Sµ in constant case.

The trend case is completely different. Figures 3(b), 3(d) and 3(f) demonstrates that the size of test IR(s) approximately is equal to the size of IR for small and approximately is equal to the size of S for large (the critical values for s were calculated at c = 20). In some intermediate cases the size may be less than two tests. Thus IR(s) the test in the trend case 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 discussed tests, the first of which is a stationarity test with uncertainty over linear trend with the known initial value equal to zero, 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, researchers can not be known a priori nor the magnitude of the initial value or the value of a trend parameter. In this case following the HLT (see also Harvey et al. (2012)), we can apply the strategy of intersection of rejections, which reject stationarity, if each of the four tests, Qi and Si (i = µ, ), rejects the null hypothesis of stationarity. This liberal decision rule can be written as follows:

Q,µ Q, IR4 = Reject H0 if {Qµ > mcv and Q > mcv S,µ S, and Sµ > mcv and S > mcv }, (14) where m is the scaling constant. It can also improve the test pre-identifying possible large initial value or significant linear trend as in Sections 3.1 and 3.2. However, as shown in HLT, the tests Dan-J, t and tm2 very sensitive to the magnitude of initial value. Harvey et al.

(2012) used a modified test t :

t = (1 - )t0 + t1, (15) where in contrast to Harvey et al. (2007), t1 is constructed using autocorrelation-corrected t-ratio for testing T = 0 in the following regression yt - T yt-1 = µ(1 - T ) + T (t - T (t - 1)) + t, t = 2,..., T, (16) where T = 1 - c/T, and corresponding long-run variance is calculated by using residuals t.

Harvey et al. (2012) 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. Harvey et al. (2012) set s = |t | with c = 30 and used standard normal critical value. Then the test s will be oversized as c decreases towards zero, but at c = 30 will be corectly sized.

Thus as in HLT the modified liberal decision rule can be written as follows:

Definition 1 The modified intersection of rejections strategy IR4 is defined as follows:

• If s cv и s cv, then use the liberal decision rule IR(Qµ, Q, Sµ, S), defined in (14);

• If s cv и s > cv, then use the liberal decision rule IR(Sµ, S), the corresponding scaling constant is defined as m;

• If s > cv и s cv, then use the liberal decision rule IR(Q, S), the corresponding scaling constant is defined as m;

S, • If s > cv и s > cv, then use the decision rule reject H0, if S < cv ;

Figures 4-6 demonstrates for = 0, 0.5, 1, respectively, asymptotic size of tests S, IR and IR4 for c {0, 1,..., 20}, fixing power at 0.50. All figures (a)-(i) show the results for = {-4, -2, -1, -0.5, 0, 0.5, 1, 2, 4}, respectively. Only S is considered in figures among all four original stationarity tests because it is robust to both forms of uncertainty (trend and initial condition). Note that it is necessary to correct the testing strategy IR4 that it controls Q,µ Q, S,µ size asymptotically as in Harvey et al. (2012). Therefore, we replace cv, cv, cv and S, Q,µ Q, S,µ S, cv by cv, cv, cv and cv, respectively, where is the scaling constant that the power has never been lower than 0.50.

When = 0 the test IR4 everywhere superior to S. Only in case of = 4 for small c the size of IR4 slightly higher. In comparison with IR4 the size of the latter is almost always lower S and IR4, although in case of large || this test have serious size distortions for small c. When = 0.5 for negative the size of IR4 slightly higher than S (except for c > 5 in case of = -4), but increasing, even at = 0.5 their size curves intersect, and for = 2 the size of IR4 lower than S. For || < 1 the size of IR4 behaves almost as well as S, but for large || its size curves become strongly nonmonotonic.

For higher the size curve of IR4 lies between S and IR4 curves, and size distortions of IR4 are greatly enhanced, especially for large ||, while the size of IR4 approaches to size of effective test S even for moderate c.

Because it is unclear which of the test strategy dominates another among different and, for comparison, consider the asymptotic integrated size of all three tests, similar to Harvey et al. (2009), which examined the asymptotic integrated power. Integrated asymptotic size is the area below each curve in Figures 9-11 for c {0, 1,..., 20}. Table 1 show 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 over all and the maximum value is 0.69 (in the case of = 0 and = 0), for IR4 – 0. (in the case of = 1 and = -2), for IR4 – 0.40 (in the case of = 1 and = -0.5), which proves superiority of the latter test, as the maximum losses for it is minimal.

At the same time, if we consider integrated size relative to the maximum value (Table 2), it is evident that for the test S and the test IR4 there exists, where they always dominated by IR4 (for the first in the case of = 0, for the second – = 1). At the same time, the maximum gain does not differ much differ for all tests, except for the case = 0 and = 0, when the size gain for the test IR4 equal to 0.52, as expected by theoretical results.

Thus based on the asymptotic results, we strongly recommend the use of the modified decision rule IR4, if there is uncertainty over both the trend and the initial condition.

4 Critical values In this section we discuss the obtaining critical values for practical application6, because previously we compared the tests, fixing the power at 0.50.

The critical values for tests Qµ( and Sµ( are given in Table 3. We obtain them at c = c) c) for tests Qµ( and Sµ( as in Mller and Elliott (2003) and Elliott and Mller (2006), and c) c), for test Q ( and S( at c = 15. We note (see HLM), in this case c = c, and tests Si( c) c) c), (i = µ, ) have standard KPSS limiting distribution, so the critical values are the same as for In this section, we obtain the results using the normalized sum of 5000 steps and 100,000 replications.

conventional KPSS tests. Also, Qi( is not invariant to the initial value when c > 0, so the c) critical values obtained for = 0.

The critical values for the initial condition test s were obtained at c = 20 and are listed in Table 4.

It is necessary to obtain a scaling constants for all intersection of rejections testing strategy, which were considered in Sections 3. However, there is some difficulty, because 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 = 12.5 (i.e. at c = 12.5 tests has correct size). Otherwise, the scaling is performed using the specific c for the own type of deterministic component. All scaling constants are given in Table 5 (the simulation code is available upon request).

5 Conclusion In this paper we consider the problem of stationarity testing with uncertainty over the trend and/or initial condition. We have 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. Simulations evidence show that 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) suggest size performance in the presence of uncertainty over both the trend and the initial condition. In addition, we showed that pretesting of the trend parameter and the initial value could improve the procedure if any of these parameters will be significantly nonzero. Thus our procedure is useful in empirical applications in conjunction with the HLT test, because the stationarity testing is necessary for confirmatory analysis.

A Appendix The proof of Lemma 1: Due to the invariance, we set µ = 0 without loss of generality, thus yt = t + ut.

We consider the model without the trend when it is actually present and we obtain using GLS-detrending that rt = yt - yt-1 - µ, T µ = (yt - yt-1) T - t=Let zt = ut - u1. Then T T rt = ut - ut-1 + t - (t - 1) - (ut - ut-1) - (t - t + ) T - 1 T - t=2 t=T = zt + u1 - zt-1 - u1 - (zt + u1 - zt-1 - u1) T - t=T + t - (t - 1) - (t - t + ) T - t=T T = zt - zt-1 - (zt - zt-1) + t - t - (t - t) T - 1 T - t=2 t=T +2 T +2 -1 T +The second term can be written as t - - t - = T c t -.

2 2 Then [rT ] [rT ] T -1/T -1/2 zt - zt-1 - 1 (zt - zt-1) T rt = T - i=2 i=2 t= [rT ] T -1/2 T -1c i - T + + i=The first term of obtaining expression have a limiting distribution obtaining in HLM and corresponds the case of = 0 (more precisely, converges to Hc,c,0(r)). Consider the second term, responsible for the behavior of test statistics under local trend:

[rT ] T + 2 ([rT ] - 1)([rT ] + 2) ([rT ] - 1)(T + 1) -1/2 -1 -2 -T T c i - = cT - cT 2 2 i=r2 r This expression converges to c -, which proves the lemma using CMT and simple 2 integral transformations, because the long-run variance estimator (kernel-based in our case) are still consistent under the local trend misspecification.

References Bunzel, H. and Vogelsang, T.J. (2005). 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, 23, 381–394.

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