“Spurious rejection using recursive, rolling and sequential tests in the presence of a break under the null” Badillo, R. a Belaire, J. b Reverte, C. a a Universidad Politécnica de Cartagena b Universitat de València This work is an outcome of the research project (05838/PHCS/07) financed by ‘Programa de Generación de Conocimiento Científico de Excelencia de la Fundación Séneca, Agencia de Ciencia y Tecnología de la Región de Murcia’.

“Spurious rejection using recursive, rolling and sequential tests in the presence of a break under the null” 1.Introduction 2.The models and statistics 3.Critical values for the recursive, rolling and sequential tests 4.Unit root with a break in level 5.Unit root with a break in drift 6.Conclusions

Motivation Examining the order of integration of economic time series is a familiar feature of applied econometric research in the last four decades. There are substantial implications for empirical modelling when one or more of time series being used are found to contain a unit root. One of the unit root tests more widely used in applied econometric research is the Dickey-Fuller (1979) test (DF). However, the properties of this test are altered when structural breaks occur in economic data sets. 1. INTRODUCTION Perron (1989, 1990) shows that a rejection of the unit root hypothesis is possible for many macroeconomic time series once allowance is made for a one-time shift in the trend function.

Perron phenomenon (Perron,1989) “The usual Dickey-Fuller tests of the unit root null hypothesis can have little power when the true generating process is stationary about a broken linear trend.” Converse Perron phenomenon (Leybourne, Mills and Newbold, 1998) “An I(1) process with break can ‘appear to be’ I(0) when Dickey-Fuller tests are routinely applied” The null hypothesis of an I(1) generating process can be rejected very frequently, leading to the spurious conclusion of trend stationarity, specially if the break occurs very early in the series. 1. INTRODUCTION

Objective To analyse whether the ‘converse Perron phenomenon’ holds when a break in the time series is occurred and when the structural break is identified endogenously. 1. INTRODUCTION We analyse whether routine application of standard DF tests can lead to a serious problem of spurious rejection of the unit root null hypothesis, when a break occurs in the series and, contrary to Leubourne et al. (1998), when the break date is data dependent.

Christiano (1992), Banerjee, Lumsdaine and Stock (1992), Zivot and Andrews (1992), Perron and Vogelsand (1992) ad Chu and White (1992) consider that if there is a break, its date is not known a priori rather is gleaned from the data. 1. INTRODUCTION Motivation about the endogenous identification of the break data

2. THE MODELS AND STATISTICS As Banerjee et al. (1992), we consider three classes of standard DF statistics that control endogenously for structural breaks. Recursive DF testRolling DF testSequential DF test In the recursive and rolling DF tests, we estimate a traditional DF regression, like this: (1) RECURSIVE TEST ( ): We take subsamples t =1,…, , where  =  0,  0+1,…,T and using as criteria the minimum values of the t-ratio evaluating.  0 is the starting value of the recursive estimation and T is the size of the full sample. ROLLING TEST ( ): The rolling test is based on subsamples of fixed size T s, rolling through the sample. We choice the minimum value of the statistic between all subsamples.

2. THE MODELS AND STATISTICS SEQUENTIAL TEST ( ) : This test allows for a possible single shift or break at every point in the sample for the mean in the following equation, which is estimated using the whole sample: (2) Following Perron (1989, 1990), we consider a shift in mean, which is also referred by this author as the ‘crash’ model, where: (3) and the break fraction is denoted as  =  /T. The t-stastistic testing d=0 provides information about whether there has been a break or jump in the mean. The t-stastistic evaluating is used to test for the order of integration of the series, assuming that the date of the hypothetical break increases progressively. We choose the break date where the t-statistic takes the minimum value.

3. CRITICAL VALUES FOR THE RECURSIVE, ROLLING AND SEQUENTIAL TESTS We report finite critical values of recursive, rolling and sequential : Motivation Christiano (1992), Banerjee, Lumsdaine and Stock (1992), Zivot and Andrews (1992), Perron and Vogelsand (1992) ad Chu and White (1992) consider that if the identification of the break date may not be unrelated to the data and the critical values of the unit root tests assume the opposite, there may be substantial size distortions. Newbold and Kuan (1997) suggest there may be some size distortions where critical values are used in the presence of structural breaks under the null.

3. CRITICAL VALUES FOR THE RECURSIVE, ROLLING AND SEQUENTIAL TESTS All the calculations have been programmed in Ox 4.1 ( The critical values are computed using data generated for the null model and are based on 10,000 Monte Carlo replications. The critical values are well below the full-sample standard DF critical values (Fuller, 1976)

4. UNIT ROOT WITH A BREAK IN LEVEL We analyse the possibility of spurious rejection of the unit root null hypothesis when recursive, rolling and sequential tests are applied, and when there is a break in a I(1) generating process. In line with Perron (1989), we permit just a single break and we shall concentrate on additive outlier models. Particularly, we discuss the simplest possible case, where monotonic trend or drift is assumed to be absent. In that case, the H 1 would be stationarity about a fixed mean, and the null would be I(1) with zero mean change. The data generating process (DGP) is the same as Leybourne et al. (1998): (5) where:  is the break size and (6)

Monte Carlo Analysis All simulations are based on 5,000 replications using sample sizes of 100 and 50 observations. An additional initial 100 observations were discarded to remove the influence of the initial condition, y 0 =0. In order to compare our results with those of Leybourne et al. (1998), the values   {2.5, 5, 10} were chosen for the break size. The break in level was therefore imposed after observation  T= . For each replication, the tests are estimated using regression (1): under the assumption  =0, and the statistic is estimated using regression (2):, under the same assumption and for D t (  ) defined in (3): The (false) rejections of the unit root hypothesis are noted at the 5% level of significance using the critical values calculated in Section 3 (see Table 1, break in level columns). 4. UNIT ROOT WITH A BREAK IN LEVEL

Where:  : break size  : break fractions Spurious rejections of the H 0 are below the nominal size

4. UNIT ROOT WITH A BREAK IN LEVEL Where:  : break size  : break fractions Ignoring the possibility of a break in level produces many rejections of the H 0, specially when the break point is closer to the middle of the sample and when  increases.

4. UNIT ROOT WITH A BREAK IN LEVEL Where:  : break size  : break fractions Spurious rejections of the H 0 are very close to zero

4. UNIT ROOT WITH A BREAK IN LEVEL Where:  : break size  : break fractions Spurious rejections of the H 0 are below the nominal size

4. UNIT ROOT WITH A BREAK IN LEVEL Where:  : break size  : break fractions Ignoring the possibility of a break in level produces many rejections of the H 0, specially when the break point is closer to the middle of the sample, when  increases and when T decreases.

4. UNIT ROOT WITH A BREAK IN LEVEL Where:  : break size  : break fractions Spurious rejections of the H 0 are virtually zero

4. UNIT ROOT WITH A BREAK IN LEVEL Conclusions Using and tests, the spurious rejections of the null hypothesis are below the nominal size, and, even, the latter test has a size very close to zero. In the case of the test, ignoring the possibility of a break produces many rejections of the null, especially when the break point is closer to the middle of the sample, when  increases and when T decreases.

We examine the behaviour of the recursive, rolling and sequential unit root tests assuming a different case, i.e., a trend is permitted under the alternative hypothesis and a drift is allowed under the null. Specially, we generate data from and I(1) process where the mean experiences a single abrupt shift, corresponding under the H 1 to the two segments of the trend function joined at the break point. As Leybourne et al. (1998), we consider the following DGP:,(7) where:  is the break size and 5. UNIT ROOT WITH A BREAK IN DRIFT

Monte Carlo Analysis All simulations are based on 5,000 replications using sample sizes of 100 and 50 observations. An additional initial 100 observations were discarded to remove the influence of the initial condition. As Leybourne et al. (1998), the sizes of the drift break considered are   {0.5, 1, 2}. For each replication, the and tests are estimated under the assumption  0 in models (1): and (2):, respectively. The (false) rejections of the unit root hypothesis are noted at the 5% level of significance using the critical values calculated in Section 3 (see Table 1, break in drift columns). 5. UNIT ROOT WITH A BREAK IN DRIFT

Where:  : break size  : break fractions Ignoring the possibility of a break in drift produces a severe phenomenon of spurious rejection of the H 0, specially when the break point is relatively early in the time series and when  increases.

5. UNIT ROOT WITH A BREAK IN DRIFT Where:  : break size  : break fractions Ignoring the possibility of a break in drift produces many rejections of the H 0, specially when the break point is closer to the middle of the sample and when  increases.

5. UNIT ROOT WITH A BREAK IN DRIFT Where:  : break size  : break fractions There is not spurious rejections of the H 0.

5. UNIT ROOT WITH A BREAK IN DRIFT Where:  : break size  : break fractions Ignoring the possibility of a break in drift produces many rejections of the H 0, specially when the break point is relatively early in the time series and when  and T increase.

5. UNIT ROOT WITH A BREAK IN DRIFT Where:  : break size  : break point Spurious rejections of the H 0 are bellow the nominal size. However, when the break point is closer to the middle of the sample and when  increases spurious rejections frequencies are close to the 5% level of significance.

5. UNIT ROOT WITH A BREAK IN DRIFT Where:  : break size  : break point There is not spurious rejections of the H 0.

5. UNIT ROOT WITH A BREAK IN DRIFT Conclusions Using the test, there is not spurious rejections of the null hypothesis when a break in drift is occured. For and tests, a severe phenomenon of spurious rejection of the null emerges when  increases and, contrary to the above section, when T increases. The spurious rejection problem is even more severe for the test than test. For the former, the size distortion is higher for a break relatively early in the time series. For the latter, this distortion is greater for a break occurred relatively in the middle of the sample.

6. CONCLUSIONS Applying recursive, rolling and sequential DF type tests that control endogenously for structural breaks: Unlike Leybourne et al. (1998), we find no evidence of the ‘converse Perron phenomenon’ when using the sequential procedure. We find some distortion in the test size (evidence of the ‘converse Perron phenomenon’) when using both recursive and rolling DF-type tests. The spurious rejection of the null depends on the break type (in level or in drift), the break size, the location of the break point in the sample and the sample size.