Spurious Relationship of AR(P) Stable Sequences in Presence of Trends Breaks

[a]School of Science, Xi’an University of Science and Technology, Xi’an, China.

Corresponding author.

Supported by the National Natural Science Foundation of China (No.71103143), the special program of China Postdoctoral Science Foundation (No.20110491671,2012T50809) and the Scientific Research Program Funded by Shaanxi Provincial Education Department (No.12JK0858).

Received 9 March 2013; accepted 15 June 2013

Abstract

This paper analyzes spurious regression phenomenon involving AR(p) stable processes with trend breaks. It shows that when those time series are used in ordinary least squares regression, the convenient t-ratios procedures wrongly indicate that the spurious relationship is present as the pair of independent stable series contains trend changes. The spurious relationship becomes stronger as the sample size approaches to infinite. As a result, spurious effects might occur more often than we previously believed as they can arise even between AR(p) stable series in present of trend breaks.

Key words: Spurious relationship; Stable sequence; T-ratios; Trend breaks

YU Cong, WANG Xuefeng, MA Jifeng (2013). Spurious Relationship of AR(P) Stable Sequences in Presence of Trends Breaks. Advances in Natural Science, 6(2), -0. Available from: /index.php/ans/article/view/j.ans.1715787020120602.2560

DOI: /10.3968/j.ans.1715787020120602.2560

INTRODUCTION

Spurious regression is a situation in which two or more variables are statistically related, but in fact there is not any direct relation between them. It is conceived in the time series econometric literature, can be traced back to Yule (1926), who identified the phenomenon by means of a computerless Monte Carlo experience in which correlation coefficients were obtained from pairs of independent non-stationary variables. Granger and Newbold (1974) identified it again for simple least squares estimates and showed that when unrelated data series are close to the integrated processes of order one or the I(1) processes, then running a regression with this type of data will yield spurious effects. Phillips (1986) provided the theoretical framework to understand the phenomenon in the simplest case(independent driftless unit root processes (DGPs)), such as unit root with drifts, trend stationary and long range.

The above results served as a springboard to a subsequent long series of investigations of the phenomenon for different types of regression and different types of data generation process. Marmol (1998) suggested that spurious correlation generally occurs in regressions including fractionally integrated processes. Spurious regressions are also shown to occur in models with series generated by various combinations of different types of stationary processes by Granger et al. (2001). For more details about spurious regression we refer the reader to Lizeth and Daniel (2011) and Martinez-Rivera and Ventosa-Santaularia (2012), among many others.

All these studies rely on the case where variances of the sequence are finite and, therefore, demonstrate the existence of spurious regression under finite-variance. However, there is growing body of evidence showing that many economic and financial time series have volatilities that are stable sequence with infinite-variance. Many types of data from economics and finance have the same character: a heavier tail than the normal variables, and it is more suitable to model these heavy-tailed data by some processes belonging to the domain of attraction of a stable law with stable index, where the stable index can reflect the heaviness of the data. This kind of data was considered by Tasy (1999) and Phillips (1990). Later on, Rechev and Mittnik (2000) and Kokoszka and Taqqu (2001) studied linear processes with heavy-tailed distributions; Davis and Mikosch (1998) and McElroy and Politis (2002) have developed the asymptotic theory for sample autocovariances and extreme for such processes. The purpose of this paper is to investigate the asymptotic behavior of the usual diagnostic statistics when they are employed to test if there exists a relationship between two independent stable sequences with infinite-variance in presence of trend breaks. Thus, this paper is to extend the interval of tailed index from k=2(Gaussian series) to k∈(1,2](Infinite-variance sequences).

This paper is organized as follows. In Section 2, we present the data-generation processes with structural breaks in trend and the assumptions made on the various components. Section 3 deals with the asymptotic properties of the least squares estimates involving trend breaks. In Section 4, we would provide some simulation evidence, whilst conclusions are drawn in Section 5.

1. THE MODELS AND ASSUMPTIONS

Our analysis of the spurious effects are based on simple regression models where the dependent variable and the single nonconstant regressor are independent infinite-variance processes with structural breaks in mean. Before presenting these models, let us first briefly review some basic properties of the stable process.

We consider moving average of the form

(1)

where and the weights cj satisfying

(2)

This model nests causal ARMA(p,q) and .specifications. The independent identical distribution innovations εt are assumed to be mean zero and in the domain of attraction of a stable law with 1

Assumption 2.1 The innovations εt are in the domain of attraction of a stable law with tailed index k∈(1,2) and Eεt=0.

Our method also relies on the results derived by Resnick (1987).

Lemma 2.1 If Assumption 2.1 holds, then

where

and the random variable Z(r) is k― stable and W2(2) is k/2―stable Levy process in [0,1]. The notation stands for convergence in distribution.

The exact definition of the Levy process (Z(r),W2(r)) appearing in Lemma 2.1 is not needed in the following, but we recall that the quantities aT can be represented as

for some slowly varying function J.

Lemma 2.2 Suppose yt are defined by (1). If Assumption 2.1 and (2) hold, then

where Z(1),W 2(1) and aT are defined in Lemma 2.1.

Interestingly, Lemma 2.2 does not extended directly to a functional version, as it does in Lemma 2.1. This has been discovered by Avram and Taqqu (1986) and Resnick (1987). However, they still proved the results that and , which are necessary to derive the asymptotic validity of our test procedures.

To examine the spurious effects, we first define two independent series ut and vt, which satisfy Assumption 2.1 with tailed indices ku and kv, respectively. Now, we consider two stable processes xt (explanation variable) and yt (depedent variable) generated from the following DGP:

(3)

(4)

where lag polynomials, A(L)and B(L), have their roots lying outside the unit circle; xt is the explanation variable and yt is the depedent variable; μx and μy are the intercept of xt and yt; θx and γx are, respectively, the permanent trend and the transitory trend, resulting from a break, of the process xt; θy and γy are, respectively, the permanent trend and the transitory trend, resulting from a break, of the process yt. Both and equal 1 when t >[Tτx] and t >[Tτy], or equal 0. The below Theorem 3.1 indicates that (3) and (4) exist the

spurious regression, but in fact there is not any direct relation between them.

Let us define the following inverse lag operators:

and

, with and . Because the roots of A(L) and B(L) are outside the unit circle. Therefore, it follows from the BN (Beveridge and nelson) decomposition can be used as following, yields

Where and . BN decomposition yields directly the martingale approximation to the partial sum process of a stationary time series, see Hall and Heyde(1980).

We assume, without loss of generality, that the initial values of the stable processes ,, and are all zero. Hence, and can be rewritten as

(5)

(6)

In the view of Lemma 2.1 and 2.2, if we define and ,

then we have

and

.

2. MAIN RESULTS

Given the preceding discussion, we consider the

(7)

Let, and denote the ordinary least square estimates from a regression of yt on a constant, the trend t and xt respectively. Their respective ‘variance’ are estimated by, and from which we have the diagnostic statistics , and .

In order to determine the limit behavior of the -ratios, the following Lemma is needed.

Lemma3.1 Suppose that (xt , yt) is generated by (5) and (6). The sequence, ut and vt, are independent and satisfy Assumption 2.1. Then, as ,

By applying Lemma 3 .1, one can immediately derive the limits of individual terms in the above equations. Theorem 3.1 has been proved.

3. SIMULATION

In this section we use Monte Carlo simulation methods to examine the sample performance of our theoretical results in Section 2 and 3. We compute rejection frequency of thet-ratios for testing the null hypotheses H0:β=0, in equations (7). All results are obtained by 3000 replications using a 1.96 critical value (5% level) for a standard normal distribution.

We consider the properties of the t-ratios when the data-generating processes exhibit structural breaks in trend. For the simulation, we let

Where and γy=0.2 The values of autoregressive parameters are still chosen to be{0.0,0.2,0.5,0.8,1.0}. The innovation processes ut and vt. The spurious regression of generated by the program of STABLE are independent of each other and satisfy Assumption 2.1 with tailed indexes kuand kv varying among {1.2,1.3,1.8}. Moreover, the program STABLE is available from J. P. Nolans website: academic2.american.edu/.jpnolan. We just report the results for f=fx=fy, and the other cases have similar results.

Table1

Table 1 report the simulated empirical power for the case of structural breaks in trend. There are some conclusions should be mentioned. Firstly, we will find the phenomenon of spurious regression driven by trend breaks is serious, since the magnitude of the probability limit of increases even further. Hence, the rejection rate of 100% is wel predicted. Second, as T increases, or either or increass, the rejection powe increases. The rejection rate for the T=1000, are 58.82% for k1=1.3,k2=1.2, and 91.27% for k1=1.3,k2=1.8, confirming the consistency results of Theroem 3.1. Finally, it is clear that the less tailed indexes provide a lower empirical power. It is mainly because both DGP xt and yt have more ‘outliers’ when the tailed indexes decrease. This conclusion that the test statistics are sensitive to the tailed index, and the similar results can been seen in the paper of Jin et al. (2009).

In order to give further intuitive idea for the influence of autoregressive parameter, break fraction and tailed index, we provide the rejection frequency of two pairs of tailed indexes with

and sample size T=1000in Figure 1-2.

Figure 1

The First Panel Represents Rejection Frequency For Autoregressive Parameters and the Last Three for , and (, ) Respectively

Figure 2

The First Panel Represents Rejection Frequency for Autoregressive Parameters and the Last Three for , and (, ) Respectively

Figure 1-2 report that the null hypothesis of will almost be rejected with certainty even if in the case of trend breaks. In a word, our simulation experiments confirm our motivation a spurious relationship is present in these regressions in the case of structural breaks in trend.

CONCLUSION

The research of Tsay (1999) examined the possibility of spurious relationship between two independent integrated errors processes belonging to the domain of attraction of a stable law with tailed index, and showed that the t-ratios diverge at the rate, which is identical to what Phillips (1986) has obtained for the Gaussian case where. This paper has been extended to consider the properties of t-ratios allowing for structural breaks in the regressive relationship when applied to independent infinite-variance series subject to breaks in trend. We find that using this fairly standard AR(p) framework allows us to successfully address the questions whether spurious regctural breaks in trend is found to be (). A fairly extensive Monte Carlo study has also been conducted to verify the performance of our test procedures, especially those of convergence rate established in the paper. Hence, it is likely to find a spuriously statistical significant relationship between two independent stable AR(p) processes subject to trend breaks when the regression model includes a linear trend in its deterministic specification.

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