TEST OF PERSISTENCE CHANGE UNDER THE INFLUENCE OF VARIANCE SHIFT

Jiawei Sun 1 and Hao Jin 2 . 1. College of Science, Xi’an University of Science and Technology, Xi’an, China. 2. Energy and Economic Research Center, Xi’an University of Science and Technology, Xi’an, China. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 11 September 2017 Final Accepted: 13 October 2017 Published: November 2017

This paper studies the test of persistence change under the circumstance that the innovations have time-varying variances. We consider processes both shifting from a stationary to a unit root and the contrary change direction, the innovations of which have time-varying variances. The statistics applied in this paper is derived by S. Leybourne, R. Taylor and T. H. Kim[2007]. The limiting distributions are given under the null hypothesizes that have only variance change but persistence change and numerical simulation shows that there are severe size distortion and the power loss cannot be ignored either.

Introduction:-
Since the paper of Page[1955], a vast amount of relevant articles dealing with parameter breaks have appeared in the literature and the problem of testing for and estimating of change points have attracted more and more attention among many researchers.
In recent years, a change in the persistence of a time series, which means a change in the order of integration, has come more and more into the focus of empirical and theoretical researchers. Examples of time series having change in persistence are found in De Long and Summers(1988), real output series of the U.S. and the European countries. They conjectured that these series shifted from stationarity to a unit root after WWⅡ. They performed informal tests to find the evidence in favor of their conjecture. Beginning with Banerjee et al. (1992), several authors proposed tests for a change in persistence in the classical framework. A popular stationarity test against a break in persistence was introduced by Kim(2000). He has developed the residual-based ratio test against changes in persistence in a time series, focusing on the case of a shift from stochastic stationarity to difference stationarity, at some point in the sample. But Kim's test has the disadvantage to reject the null if the data generating process is constantly during the whole sample what is theoretically correct but not desirable. Leybourne et al.(2007) suggest a CUSUM-squares based test to solve this problem. Sibbertsen and Kruse(2009) generalized this test to the long memory framework by allowing for fractional degrees of integration.
Variance change, as what we understand intuitively, means the variance of a time series remains unchanged at a level until the change point, at which the variance changes to another level. In linear processes, a variance change in Naturally, we may think that since the CUSUM based statistics can be applied on both persistence change test and variance change test, is there any influence if the variance shift occurs in the innovations of a series which has a persistence change. In other words, we want to make it clear that whether the variance shift brings some disturbance when we test the persistence change and how the relationship is.
Our paper is organized as follows. In section 2, we give the models and assumptions that used in later theorems. Section 3 first provides a brief review of persistence change test by Kim and Leybourne et al and then gives the theoretical results under both the null hypothesizes and alternative hypothesizes. In section 4, Monte Carlo simulation methods are used to show the size distortion and power loss. Section 5 concludes. All proofs are given in the Appendix.

Models and assumptions:-
Like many other papers, ,and ,is the integer part of . More specifically, in order to taking variance change into consideration, the innovation series * + is defined as: , where ( ) and the component, , is defined in the following two ways according to the different variance change types: In both the two types, * + is the indicator function and ,is the variance break point. and are set to be constants.
Assumption 1:-The variance term * + satisfies the relation: (, -) ( ) where ( ) is a non-stochastic function with a finite number of points of discontinuity; more over, ( ) and satisfied a (uniform) first-order Lipschiz condition except at the points of discontinuity.
Assumption 2:-The process * + satisfies the following conditions:

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The null hypothesizes in our paper is as follows: is a unit root process throughout the sample period, denoted as . The two corresponding alternative hypothesizes are: maintains stationarity of constant persistence until some period, after which it becomes a unit root process, denoted by . is a unit root process until , -, after which it becomes a constant persistence series, denoted by . In all the cases we explain above, the two types of variance change have always exists. Now we write these hypothesizes below: Theoretical results:-Before giving the limiting distributions under the null hypothesizes, we first briefly introduce the statistics that derived by Leybourne et al [2006] used to test change in persistence. In their paper, The denominator of the above two statistics are estimators of the long-run variance and defined as followed: Due to the true break point is usually unknown, the consider the two-tailed test which rejects for large or small values of the statistic formed from the minimized CUSUMs of squared sub-sample OLS residuals obtained from the forward and backward realizations of the process; i.e.

( ) ( )
For the test statistic above, is a compact subset of (0, 1). Where 799 ( ), and , equals to before ,and after , -. The ratio of variance change takes the values from 2, 3, 1/2 and 1/3. The size simulation results of an abrupt variance and time-varying variance change are provided in Table 1 and 2 separately. For a convenient discussion, we introduce the following definition: we say the variance change in a positive direction, which means ⁄ and the negative direction is ⁄ .   Table 3, 4, 5 and 6 give the power of and with abrupt and time-varying variance changes, separately. In our simulation, is set to take value from 0.7, 0.8, 0.9 and 0.95. is the persistence change point position and equals to 0.5. From Table 3, first we can easily get that the test power decreases as tends to 1, no matter the variance change's direction is positive or negative. This phenomenon isn't difficult to understand: a larger means our series is more similar to a unit root process and it is certainly hard for the test statistics to distinguish * + from a stationary series. Second, for all the change directions, test power will decrease as . Besides, when the change direction is negative, the test power has positive correlation with the change extent of variance. While for the positive change direction, things seem to be complex. When , power has the same change trend with negative cases; but when , power has negative correlation with change extent of variance. From Table 4, the power values under a that is not so close to 1 are larger than those in the same position in Table 3 generally, which isn't hard to understand: the influence brought by a time-varying variance 800 change is more gentle than by an abrupt variance change. A special attention should be paid that power values in positive change direction and , still stay different with other cases.
Next we turn to Table 5 and 6. First we can conclude in Table 5 the test power will increase as the growth of sample size, especially for the positive change direction. The influence brought by the variance change extent looks less significant compared with Table 3 and 4. For the positive change direction, power values have a significant growth as , while it is negligible for the negative change cases. But we should pay special attention to the fact that the power values in Table 6 when change direction is negative have an opposite change direction. That is to say, power value has negative correlation with when there is an abrupt variance change under .   And the probability limit is finite and positive.
Combining the above results, we therefore have for ( ) Now consider the case where . We decompose the numerator of ( ) into its constituent ( ) and ( ) part. Basing on the discussion above, we know that the ( ) part is asymptotically negligible. Next turn to the behavior of ̂ ( ), observe that the s estimated autocovariance, ̂ , can be expressed as a weighted average of ( ) and ( ) components as follows: Where ̂ is the sth estimated autocovariance when is ( ); i.e. To summarize, we obtain the limit function of ( ) is given by * + * +, which is therefore minimized at . Then we have