Testing for Panel Unit Roots in the Presence of an Unknown Structural Break and Cross-Sectional Dependency

Transcription

1 Testing for Panel Unit Roots in the Presence of an Unknown Structural Break and Cross-Sectional Dependency A. Almasri, K. Månsson 2, P. Sjölander 3 and G. Shukur 4 Department of Economics and Statistics, Karlstad University, Sweden 4 Department of Economics and Statistics, Linnaeus University, Sweden 2,3&4 Department of Economics, Finance and Statistics, Jönköping International Business School, and The Swedish Retail Institute ( HUI), Sweden Abstract This paper introduces two different non-parametric tests for panel unit root based on the wavelet decomposition of time series which may be used in the presence of cross-sectional dependency and an unknown structural break in the data. These tests are compared with the parametric IPS test proposed by Im, Pesaran and Shin (997) and the ald test suggested by Taylor and Sarno (998). By means of Monte Carlo simulations, the results shown that the size and power properties of the new non-parametric tests are robust to cross sectional dependency of the error terms. Furthermore, it is shown that the tests may be used when the time series has an unknown structural break. These tests have also shown to have high power against the alternative hypothesis under the above mentioned conditioned, whiles the IPS and the ald did not have any power to reject the alternative hypothesis in the presence of structural break in the data. Key words: Panel unit root test, cross-sectional dependency, structural break, wavelet

2 . Introduction One of the most important issues in the subject of time series is testing for the non-stationarity of the series under consideration. Non-stationarity can usually be due to some kind of trend that is associated with these series, e.g., deterministic or stochastic trend. Testing for unit root is one important aspect when the trend might be of the stochastic form. The history of unit root testing dates back, at least, to the paper by Dickey and Fuller (979), who introduced their tests for unit root in an autoregressive model. These tests were later extended to other situations and have been augmented to remove autocorrelation and other sorts of misspecifications. By the middle of the eighties and later on, a confusing large number of tests were thereby available to the practitioner with the same purpose, i.e. testing whether a time series contains a unit root. The above tests, however, are only applicable in a single equation environment. Many models are expressed in terms of systems of equations, for example time series models across different units (or say countries) and, in particular, panel data models. Treating each equation of a system separately, and performing a succession of single equation unit root tests, will lead to the problem of mass significance and to a reduction in the validity of the conclusions. Moreover, due to situations for which the alternatives are near integrated series (but still stationary) and the number of observations are too small, the panel data version of unit root tests became more attractive with higher expected power over the single equation tests. In addition, a utilization of cross-sectional information makes the test more efficient especially when the time series is not long individually but very similar among the cross section of units. However, testing for unit root in panel data is not very straightforward since one should be aware about other issues, like heterogeneity in the cross sectional units or cross sectional dependency, which use to affect the properties of the tests with regard to size and power. Several panel unit root tests that allow for cross sectional dependency have been proposed and investigated in the literature; see Pesaran (2007) for an expanded overview of various panel unit root tests. Among various panel unit root tests, Im-Pesaran-Shin (IPS) (997) type of test utilizes the average of single unit test statistics among all the units and then releases the restriction of homogeneity. The original IPS test is constructed as the average of the Dickey- Fuller t test statistics for all the series under consideration. Taylor and Sarno (998) proposed 2

3 a ald test together with a SUR estimation of the system. In this manner, the last authors estimated all the equations in the system simultaneously and conducted a multivariate ald test. The ald test is among the classical multivariate tests and has been used to test for Granger causality by for example Dolado and Lüthkepohl (996), Mantalos and Shukur (2000) and Månsson and Shukur (2009), to test for autocorrelation by Edgerton and Shukur (999). This test has also been successfully applied as a preliminary test maximum likelihood estimator for a mean vector by Kibria and Saleh (2005) and as a combination between preliminary test and ridge regression methodology by Kibria (2004). Hence, this test may be used in many different situations and in this paper we propose two non-parametric tests based on the wavelet methodology that has the same basic structure as the ald test. This test is based on the fact that in the presence of unit root the variance of the scaling coefficients dominates the variance of the wavelet coefficients. Previously, Fan and Gracay (200) utilized this idea and conducted a wavelet ratio statistic in order to test for unit root in univariate time series. In our paper we also utilize wavelet methods in the proposed nonparametric tests for panel unit root. On the other hand, when building a statistical/econometric model, the assumption of parameter constancy is widely used because of the resulting simplicity in estimation and ease of interpretation. However, in situations where a structural change may have occurred in the generation of observations, this assumption is obviously inappropriate. Particularly in the field of time series econometrics, where data are not generated under controlled conditions, the problem of ascertaining whether or not the underlying parameter structure is constant is of paramount interest. A structural break appears when we see an unexpected shift in a time series. This can lead to huge forecasting errors and unreliability of the model in general. However, when conducting panel unit root tests this issue might be of great interest since this problem might affect the size and power properties of the tests. In this paper we introduce a non-parametric multivariate test for panel unit roots based on the wavelet decomposition method. This test is able to have good size and power properties under conditions where cross-sectional dependency and un-known structural break are present. e analyse the size, power and robustness of our proposed tests and the other available tests mentioned above by Monte Carlo simulations. These tests are applied to systems of 5, 0 and 5 equations (time series or cross-sectional units) with samples ranging from 00 to 500 observations. An effective test should have correct significance level under the null hypothesis, irrespective of the values of the regression and other distributional parameters. It 3

4 should also have reasonable power against the class of alternative specifications under investigation, but low power against other alternatives. The paper is arranged as follows. In the next section we present the methodology and in Section 3 the Monte Carlo design is presented. In Section 4 we discuss the results of the study while we give brief summary and conclusions in Section Methodology This section presents the traditional test methods together with the wavelet methodology and tests based on it. 2. The Parametric Tests for Panel Unit Roots Consider the following regression model: yit = α+ ρi yit + uit, (2.) where is a first-difference operator, α and ρ are unknown parameters and the disturbances are assumed to be independently and identically distributed with a possibly nonscalar covariance matrix. For this model we want to test the following hypothesis: H : 0 i 0 ρ i =, (2.2) against the possible heterogeneous alternative: ρi < 0 for i=, K, N H : ρi = 0 for i= N+, K, N. (2.3) To test this hypothesis some different tests have been proposed. The first one denoted IPS is a panel version of the unit root test proposed by Dickey-Fuller (976) and for this test each of the ith models in equation (2.) is estimated using ordinary least squares (OLS). Then the t- tests can be calculated as t ˆ ρ se( ˆ ρ ) test statistic. =. Finally, the average value, i i i N =, is used as i i= t t N 4

5 In Taylor and Sarno (998), a ald test was proposed instead where the following test statics is used: ( (( ) ) ) U R = T tr S S N, (2.4) where S R and S U are the restricted and unrestricted residual covariance matrices respectively obtained from the residual series when estimating equation (2.) using OLS. 2.2 avelet Analysis avelet analysis, for which the theoretical development to a large extent is made by Grossman and Morlet (984), tries to reproduce a Fourier analysis method but with functions (wavelets) that are better suited to capture the local behavior of time series. The wavelet function, say ψ (.), should satisfy the following two basic properties, The integral of the real-valued function ψ (.) is zero: ψ The square of ψ (.) integrates to unity: ( u) du= 0. (2.5) ψ 2( u) du=. (2.6) Daubechies (992) derives a class of wavelets defined by two filters of positive integer width L. The high-pass filter (wavelet filter): { hl } = { h0, K, hl }. The low-pass filter (scaling filter): { gl} = { g0, K, gl } which is defined via the quadrature mirror relationship { hl = ( ) l gl l : l= 0, K, L }. Fundamental properties of the continuous wavelet functions, such as integration to zero and unit energy in (2.5) and (2.6), respectively, have discrete counterparts. A discrete wavelet filter must satisfy the following three properties: L hl = l= 0 0; L h 2 l l= 0 = ; and L hl hl+ 2n hl hl+ 2n l= 0 l= = = 0, 5

6 for all nonzero integers n, and where we define h l = 0 for l< 0 and l L so that { } h is an infinite sequence with at most L nonzero values. This means that a wavelet filter must sum to zero; must have unit energy; and must be orthogonal to its even shifts. l Furthermore, there are two main approaches of conducting the wavelet filtering. The first method is called discrete wavelet transformation () and the second one is called maximal overlap discrete wavelet transformation (MO). Both of these are considered by Fan and Gencay (200) and both will be employed in this paper The Discrete avelet Transform The key idea of the is to decompose a data set orthogonally into different new data sets. Let {X t, t = 0,, N-} be a data vector of length N, where we assume that N is an integer divisible by 2 J, where J is a positive integer. The wavelet and scaling filters are used in parallel to define the, i.e., we have two types of coefficients in the based on these two types of filter: The scaling coefficients which represent the smoothed version of the original data. These coefficients can help us in detecting the number of clusters. The wavelet coefficients. The is calculated using Mallat s algorithm, introduced by Mallat (989), which uses linear filtering operations. The transform coefficients, Vj,k and j,k at different scales, are calculated using the following convolution-like expressions: There are J subsequent stages of the pyramid algorithm. The scaling coefficients for level j (j,,j) are given by L j, k l j, 2k+ l mod N j j l= 0 V = g V for k = 0, K, N, (2.7) and the wavelet coefficients for level j are given by where 0 L j, k l j, 2k+ l mod N j j l= 0 = h V for k = 0, K, N, (2.8) V X and N N 2 j. The modulus operator in (2.7) and (2.8) is required in order j to deal with the boundary of a finite length vector of observations. This operator circularly filters the data, by using a fast filtering algorithm of order O(N). e see from (2.7) and (2.8) that at each step we filter the previous level scaling coefficients using either the scaling or 6

7 wavelet filter, and then subsample the resulting sequence. The can be defined also by matrix calculation. [,, K, ], j=, 2, K, J and V [,0,,,,, ] T J VJ VJ K VJ N. Let T j j,0 j, j, N j The elements of the sub-vectors j correspond to that in (2.8) and the sub-vector Vj correspond to that in (2.7). e then have the analysis equation = X, where contains the coefficients, i.e., and is an orthonormal N 2 M = j, M J VJ J (2.9) N real-valued matrix whose rows depend on the wavelet filter h l, i.e., = T, so T = T = I N (see, Percival and alden, 2000, Ch. 4). A partial will be obtained by stopping the algorithm after j 0 < J repetitions. The partial s are more commonly used in practice than the full, due to the flexibility they offer in specifying a scale beyond which a wavelet analysis into individual large scales is no longer of real interest The Maximal Overlap Discrete avelet Transform An alternative wavelet transform which has been used in this paper is the Maximal Overlap Discrete avelet Transform (MO). The MO coefficients can be obtained via a pyramid algorithm, as in the case of the, except that no down-sampling is involved. The MO gives up orthogonality in order to gain features which the does not possess. The MO of level J for a time series X is a highly redundant non-orthogonal transform yielding the column vectors w ~, w~,, w~ 2 L J and v ~ J, each of dimension N. The vector w ~ j contains the MO coefficients associated with changes in X between scale j- and j, while v ~ J contains the MO scaling coefficients associated with the smooth of X at scale J, or equivalently the variations of X at scale J+ and higher. The MO transform uses 7

8 the pyramid algorithm and instead of using the wavelet and scaling filters, the MO utilizes the rescaled filters, the rescaled wavelet and scaling filters (j =,,J) j h ~ = j h j / 2 and g j j g j / 2 ~ =. 2.3 avelet Ratio Tests hen a unit root is present in the time series the variance of the scaling coefficients dominates the variance of the wavelet coefficients. Therefore, Fan and Gracay (200) chosen to use the following test statistic in order to test for unit root in a univariate time series: 2 2 V s T =. (2.0) V + 2 and the following test statistic based on MO: ~ ~ 2 V = ~. (2.) 2 s T ~ 2 V + Many different wavelet families may be applied to obtain the scaling and wavelet coefficients used in equations (2.0) and (2.). In this paper we choose to use the most simple wavelet filter, namely the Haar filter. This filter is used, firstly, since it is the simplest filter where weighted differences and averages are calculated over contiguous pairs of observations for and as a moving average and difference process for MO. Secondly, we do not have the problem of boundary coefficients which is a problem when analyzing non-stationary time series, especially in the presence of structural breaks since it will mistakenly classify some of these coefficients as outliers. As a multivariate generalization of these tests we, in this paper, propose the following ald type of test statistic based on the wavelet and scaling coefficients calculated using : (( ' ) ' ) = tr H H E E N, (2.2) 8

9 where the tth row of H and E equals [ v K v ] and [ w + v K w + v ],, t This test will also be applied using the MO coefficients: MO (( ' ) ' ) Nt,, t t Nt Nt, respectively. = tr H% H% E% E% N, (2.3) where the tth row of H % and E % equals [ v% K v% ] and [ w% + v% K w% + v% ],, t nt,, t t nt nt, respectively. hen the time series has a unit root the H (or H % ) and E (or E % ) will be approximately equal. However, under the alternative E (or E % ) will dominate H (or H % ) and the null hypothesis will be rejected. Due to the non-diagonal structure of the test statistics this test is robust to crosssectional dependency. Furthermore, since we base our tests on averages and differences, the dominance structure between H and E will be the same under the null and alternative hypothesis regardless whether we have a structural break or not. 3. The Monte Carlo study The following DGP (which also used in Breitung and Candelon (2003)) is adopted here to investigate the properties of the different tests for panel unit roots: y = ρ y + θd + u, (3.) it it t it where d t 0 for t< B =. (3.2) for t B For the calculation of the size of the tests we put ρ to be equal to one, while for the power calculations,ρ will take on the values 0.95, 0.9 and 0.8. As in Breitung and Candelon (2003), θ is set to be equal to 3. The sample size, T, is equal to 00, 200 and 500 observations. Furthermore, the break point ( B ) is set to happen in the following intervals; T 4, T 2 and 3T 4. e also vary the number of observations and the number of cross sectional units. e choose to use 5, 0 and 5 different time series in the system of equations. Finally, we generate the time series both with and without cross sectional dependency. hen we do not 9

10 have any cross sectional dependency, the error term will be generated from a N ( 0,) distribution. In the case of cross sectional dependency, the error term will be generated through the relation ut = Lη where u ( u, u, u ) =, iidn( 0, I) t t 2t Nt toeplitz matrix. The different Σ matrices used can be found below: N=5: toeplitz(,0.7, 0.5, 0.3, 0.) N=0: toeplitz(,0.7,0.6,0.5,0.4,0.3,0.2,0.,0.05,0.0 ) η, LL ' =Σ and Σ is a N=5:,0.7,0.65,0.6,0.55,0.5,0.45,0.4 toeplitz,0.35,0.3,0.25,0.2,0.5,0.,0.05 For each model we performed Monte Carlo repetitions and studied the tests properties at the nominal size of 5%. The calculations have been done using the R program Package. 4. Result Discussion In this section we present and discuss the results from the simulation study regarding the size and power properties of the different panel unit root tests. To save space we follow Breitung and Candelon (2003) and we only present the result when the break point corresponds to T 2. This is due to the fact that the same pattern can be seen regardless when the structural break occurs. However, full results are available from the authors upon request. 4. Analysis of the Size of the Panel Unit Root Tests In Tables and 2, results for the estimated size of the different tests are presented. New critical values are simulated under the condition of no structural break and no cross correlation between the error terms of the different time series (or cross-sectional units). Hence, in this situation the tests are expected to be always unbiased. hen adding crosscorrelation, the IPS test severely over-rejects the true null-hypothesis yielding rejection rates as high as 20.%. The ald test under-rejects the true null-hypothesis and this under-rejection increases with increasing the number of cross-section units and number of observations. On 0

11 the other hand, both of the wavelet ratio tests are virtually unaffected by introducing crosssectional dependency and the estimated sizes of both tests are close to the nominal size. In Table 2, in the presence of a structural break, we can see that the IPS test never rejects the true null-hypothesis while the ald test almost always rejects it. For the wavelet ratio tests we may see that the version of the test is more robust than the MO version. But both tests slightly underestimate the nominal size in the presence of a structural break. However, this under-rejection in magnitude is much smaller than the amount of the overrejection in the cases of the parametric IPS and ald tests. Hence, when looking at the size of the tests the new wavelet ratio tests should be preferred due to the robustness to crosssectional dependency and structural break. 4.2 Analysis of the Power of the Panel Unit Root Tests Even if the estimated size of the panel unit root test correctly corresponds to the nominal size it will be of little use if the test does not have sufficient power to reject a false null hypothesis. Thus a correctly given size is not sufficient to ensure the good performance of the test even though it is a prerequisite. hen looking at the power of the tests for panel unit roots in Tables 3-6 we can clearly see that the IPS test should be the preferred option when we have the empirically unlikely situation that the error terms are not cross-correlated and there are no structural breaks. Then, in the presence of cross-correlated error terms and no structural break the ald test should be the preferred option since it has the highest power among the tests for panel unit root that does not over-reject the true null hypothesis. However, in the presence of structural break we can see that both the parametric IPS and ald test never rejects the false null-hypothesis. Hence, these tests are not able to separate between a non-stationary and a stationary time series. Therefore, in this situation they are not useful. Instead, we should use the statistical tests that are based on the wavelet decomposition method. These two tests have the power to detect when a time series is stationary. Among these two options the MO version should be preferred since it has a higher power. This is in line with what we expected since it has been shown by Percival and Mjofeld (997) that the wavelet variance is more efficient for MO than.

12 Table : Estimated size with no structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO Table 2: Estimated size with structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO

13 Table 3: Estimated power when ρ=0.95 with no structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO Table 4: Estimated power when ρ=0.95 with structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO

14 Table 5: Estimated power when ρ=0.9 with no structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO Table 6: Estimated power when ρ=0.9 with structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO

15 Table 7: Estimated power when ρ=0.8 with no structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO Table 8: Estimated power when ρ=0.8 with structural change Uncorrelated error terms Cross correlated error terms N=5 T IPS IPS MO MO N=0 T IPS IPS MO MO N=5 T IPS IPS MO MO

16 5. Summary and Conclusions In this paper, two different non-parametric tests for panel unit root based on the wavelet decomposition of time series are proposed. These tests are compared by means of Monte Carlo simulations with the parametric IPS test proposed by Im, Pesaran and Shin (997) and the ald test suggested by Taylor and Sarno (998). The time series are generated both with and without a structural break. Furthermore, we simulate error terms of each series to be both with and without cross-correlation. In the design of the experiment we also choose to vary the number of time periods and cross sectional units. Based on the results from the Monte Carlo study we may conclude that the IPS and ald test are highly non-robust to cross-sectional dependency and structural breaks. Furthermore, they do not have any power to separate between a stationary and non-stationary time series in the presence of an unknown structural change Instead, one may use the new wavelet-based tests where the data is filtered using the Haar wavelet.. The result from the simulation study shows that the wavelet test based on the MO decomposition has the highest power. Hence, this test may be recommended to practitioners. References Breitung, J. and Candelon, B. (2005). Purchasing power parity during currency crisis A panel unit root test under structural breaks, Review of orld Economics, 4, pp Daubechies, I. (992). Ten Lectures on avelets, Vol. 6 of CBMS-NFS Regional Conference Series in Applied Mathematics. Philadelphia: Society for Industrial and Applied Mathematics. Dickey, D.A. and Fuller,.A. (979). Distributions of the Estimators for Autoregressive Time Series with a Unit root, Journal of American Statistical Association, 74, pp Dolado, J. and Lüthkepohl, H (996). Making ald tests work for cointegrated VAR systems, Econometric Reviews, 5, Edgerton, D. and Shukur, G. (999). Testing autocorrelation in a system perspective testing autocorrelation, Econometric Reviews, 8, pp

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