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 December 13, 2012 Abstract In this paper we investigate the behavior of stationarity tests proposed by Mller (2005) and Harris et al. (2007) with uncertainty over the trend and/or initial condition. As for different magnitudes of local trend and initial value different tests are efficient, we propose following Harvey et al. (2008) the decision rule based on the rejection of null hypothesis for multiple tests. Also we propose the modification of this decision rule using additional information obtained through pre-testing about magnitudes of local trend and/or initial value. 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: C12, C22 1 Introduction The influence of linear trend and/or the initial value can be very important in unit root testing.

In recent papers, Harvey et al. (2009) and Harvey et al. (2008) (hereafter HLT, see also Harvey et al. (2012)) considered the issue of the inclusion of a deterministic trend in the unit root test, and also investigated the behavior of tests with different initial values. With uncertainty over the linear trend Harvey et al. (2009) 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 own type of deterministic components in the absence of a large initial value. At the same time, with the knowledge of the type of deterministic components (i.e. researcher knows precisely, 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 of which is effective at low initial value, and the second is effective at a large initial value for a given type of model deterministic component) will be the best. Harvey et al. (2008) extended E-mail: antonskrobotov@gmail.com the procedure, assuming uncertainty over both the trend and initial condition, suggesting a union of rejection testing strategy for all four tests. They also suggested a modification of this test with pre-testing 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 in favor that the trend and/or large initial value actually is present in the data. Thus, you can use this information to construct the union of rejection testing strategy.

In this regard, there is a need to develop a similar procedure for tests for stationarity as null, because hypothesis testing opposed to unit root is important for confirmatory analyzes (see, e.g., Maddala and Kim (1998, Ch. 4.6)). Harris et al. (2007) (hereafter HLM) have proposed a modification of the standard Kwiatkowski et al. (1992) test (hereafter KPSS) in the nearly integration1 using a (quasi) GLS-detrending. Asymptotic properties of the test (only in constant case) were compared with point-optimal test proposed by Mller (2005) for different initial values. The results showed that in the case of small initial values the test proposed in Mller (2005) is effective, while with large (or even moderate) initial value this test has a 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 large initial value.

In this paper we consider the asymptotic properties of stationarity tests proposed by HLM and Mller (2005) following the approach of HLT. In Section 2 we introduce the HLM test and point-optimal test proposed in Mller (2005) and obtain corresponding limiting distributions assuming local behavior of the trend, as well as using the parametrization of the initial condition following the Mller and Elliott (2003). 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 the asymptotic size curves show in this case that the point-optimal tests of Mller (2005) 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 the 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 the advantage over a simple intersection rejection of two test (with and without trend). In Section 3.2 we analyze the similar procedure, assuming knowledge of deterministic term, but not knowing 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. However, the results for the constant and trend cases are somewhat different. While for the trend case we strongly recommend the use of this modification, in the constant case our modification does not lead to a gain in size, and we recommend to use only HLM test. 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 Mller (2005) showed that the using of conventional KPSS test with the bandwidth parameter in the longrun 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.

HLT used the “union of rejections” term, and their test rejected the null hypothesis, if at least one of the tests rejected it. But since 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”.

shows the superiority of the considered modifications by varying parameters in the local trend and the initial value.

Since in our asymptotic analysis the size of all tests was compared with a fixed power, i.e., critical values were obtained with the integrated process (and zero trend parameters and zero initial values) in Section 4 we obtain the critical values and scaling constants for some fixed amount of mean reversion under the stationary null hypothesis. The obtained results are formulated in the Conclusion.

2 The Model We consider the data generating process (DGP) as yt = µ + t + ut, t = 1,..., T (1) ut = ut-1 + t, t = 2,..., T (2) where the process t is taken to satisfy the standard assumptions considered by Phillips and Solo (1992):

Assumption 1 Let t = (L)et = iet-i, i= with (z) = 0 for all |z| 1 and i|i| <, where et is the martingale-difference sequence i=with conditional variance e and supt E(e4) <. Short-run and long-run variances are defined t T 2 2 -1 as = E(2) и = limT T E t = e(1)2, respectively.

t t=Also = T = 1 - c/T, where c 0. We testing the null of stationarity (local to unit root) H0 : c c > 0 against alternative hypothesis H1 : c = 0, where c is the minimal amount of mean reversion under the stationary null hypothesis.

We consider two tests. The first was suggested by Mller (2005). Following Mller and Elliott (2003), he proposed asymptotically optimal test statistic Qµ( for constant case and c) Q( for trend case to discriminate between a null hypothesis T = 1 - c/T and T = 1. This c) statistic is constructed as following:

i i Qi( = q1(u -1/2i )2 + q2(u -1/2i )c) -1T -1T T T i i + q3(u -1/2i )(u -1/2i ) + q4u -2 (i)2, (3) -1T -1T -2T T 1 t t=where i are OLS residuals from regression of yt on dt, where dt = µ in constant case and t µ µ µ µ dt = µ + t in trend case, q1 = q2 = c(1 + c)/(2 + c), q3 = -2 + c), q4 = c2 and c/( q1 = q2 = c2(8 + 5 + c2)/(24 + 24 + 82 + c3), q3 = 22(4 + c)/(24 + 24 + 82 + c3), q4 = c2.

c c c c c c Also u is any consistent estimator of long-run variance of ut using i residuals.

t The second test was proposed by Harris et al. (2007). It uses (quasi) GLS-detrending series.

More specifically let i, i = µ, are OLS residuals from regression of yc = yt - T yt-1 on t Zc = zt - T zt-1, t = 2,..., T, where zt = 1 in constant case and zt = (1, t) in trend case.

Then the test statistic Si( is constructed as c) T t -T ( i )t=2 j=2 j Si( =, (4) c) u where u is calculated using i residuals.

t We consider the following two assumption, specifying the behavior of the trend coefficient and initial condition u1.

-1/Assumption 2 The trend coefficient satisfies = T = T, where is some finite constant.

Assumption 3 The initial condition u1 satisfies u1 = = /(1 - 2 ), where T = 1 T c/T, c > 0. In unit root case, c = 0, the initial condition is equal to zero, i.e. all tests are invariant to.

The following lemma summarize limiting distributions of four tests under given Assumptions 1-3.

Lemma 1 Let {yt} is generated as in (1) and (3) and Assumptions 1-3 are satisfy. Then under T = 1 - c/T, 0 c < 2 µ µ µ µ Qµ( q1 Kc (1) + + q2 Kc (0) c) 2 µ µ 1 µ µ µ + q3 Kc (1) + Kc (0) - + q4 Kc (r) + (r - ) dr, (5) 2 2 µ Q( q1Kc (1)2 + q2Kc (0)2 + q3Kc (1)Kc (0) + q4 Kc (r)dr, (6) c) Sµ( Hc,c,(r)2dr, (7) c) 1 S( Hc,c,0(r) - 6r(1 - r) Hc,c,0(s)ds dr, (8) c) 0 Here µ Kc (r) = Kc(r) - Kc(s)ds, 1 µ Kc (r) = Kc (r) - 12 r - s - Kc(s)ds, 2 r Hc,c,(r) = Kc(r) + c (Kc(s) + s)ds - r Kc(1) + c (Kc(s) + s)ds, 0 (e-rc - 1)(2c)-1/2 + Wc(r), c > Kc(r) =, W (r), c = r where Wc(r) = 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. (2009), the proof of (7) is given in Appendix.

The proofs of (6) and (8) are standard and use FCLT and CMT. Also following HLM (see also Mller and Elliott (2003) and Elliott and Mller (2006)), we set c = 10 for tests Qµ( and c) Sµ( and c = 15 for tests Q( and S ( c) c) c).

3 Asymptotic analysis of stationarity tests 3.1 Asymptotic behavior under a local trend 1/Consider the case when initial condition is u1 = op(T ). Then in limiting distributions obtained in Lemma 1 the process Kc(r) is simply replaced by OrnsteinЏUhlenbeck process Wc(r). The figures 1(a)-(d) show asymptotic size for c [0, 20], where for comparison of the tests the critical values are obtained for c = 0 and = 0, that the power was 0.5 for any test as in Mller (2005) and Harris et al. (2007)3.

Comparing the size of the tests for case of = 0 (fig. 1(a)), i.e. under the absence of a linear trend, it is clear that the size is smaller for tests not taking into account the presence of a trend. Also the tests Qi dominate tests Si, as was evident in the results of simulations of Harris et al. (2007) for the case of only constant in deterministic term. As a result for stationarity testing for = 0 the test Qµ will be effective of the considered tests.

Increasing the parameter, for = 0.5 (fig. 1(b)) the results are almost similar to the case of = 0 and test Qµ is still effective except in very small interval c [0, 1], when Qµ is dominated by tests taking into account the trend (although they have a higher power the interval c [0, 1] can be considered as negligible). Note also that the size of tests with the only constant slightly increases in comparison with case of = 0. For = 1 (fig. 1(c)) the tests S and Q clearly superior to the tests Sµ и Qµ (the latter has serious size distortions, the size is never lower 0.4 for the considered interval c [0, 20]), and Q is the efficient test. The size of tests with only the constant continues to increase with increasing. For = 2 (fig. 1(d)) the size of tests Sµ and Qµ almost always equal to one.

Because each of the tests Qµ and Q is efficient (in sense of size) among the considered for some values of a local trend, then if there is uncertainty over magnitude of this local trend it is necessary to use the feasible strategy to discriminate two cases, with presence and absence of the linear trend.

Following Harvey et al. (2009) we use the following rule of intersection of rejections where we reject the null of stationarity if both tests simultaneously reject the null. More precisely, this decision rule can be written as:

Q,µ Q, IR = Reject H0 if {Qµ > mcv and Q > mcv }, (9) Q,µ Q, where cv and cv are asymptotic critical values for tests Qµ and Q for some specified value of c and significance level (for more details see Section 4), and m is some scaling constant Here and in the following sections the 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 the integrals approximated by normalized sums of 1000 steps, number of replications is 50000.

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 also possible to further improve this strategy by using information about the high value of the parameter, e.i. about clear evidence of a trend. For this end, it can use pre-tests Dan-J, t, tm2 and tRQF,4 proposed, respectively, by Bunzel and Vogelsang (2005), Harvey et al. (2007), Perron and Yabu (2009) and analyzed in HLT and Harvey et al. (2010) for various magnitudes of a local trend and initial value. More precisely, consider the following modification of the decision rule (9):

Q,µ Q, Reject H0 if {Qµ > mcv и Q > mcv }, if |s| cv IR(s) =, (10) Q, Reject H0 if {Q > cv }, if |s| > cv where s denotes some pre-test for testing = 0, and cv is corresponding critical value. The limiting distribution of these two tests follows directly from Lemma 1 and CMT, and, therefore, omitted.

The Figures 2(a)-(d) show asymptotic size of tests Qµ, Q, IR, IR(|t|), IR(|tm2|) and IR(|Dan-J|) for values {0, 1, 2, 4} and c [0, 1]. We choose the scaling constants m so that the asymptotic power of the test IR was 0.50 for = 0.

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