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Testing for Unit Roots in Dynamic Panels with Smooth Breaks and Cross-Sectionally Dependent Errors

Abstract

We develop the extended unit root testing procedure for dynamic panels characterised by slowly moving trends (SMT) and cross-section dependence (CSD). We allow SMT to follow the smooth logistic transition function and the components error terms to contain the unobserved common factors. We propose the two panel unit root test statistics, one derived by the extended common correlated effects (CCE) estimator and the other based on the Sieve bootstrap. We have conducted extensive simulation exercises and document that the failure to take into account SMT and CSD may lead to misleading inference. On the other hand, we find that both bootstrap and CCE-based tests maintain good power properties in small samples in the presence SMT and CSD. We apply our proposed tests to real interest rates for 17 OECD countries and find overwhelming evidence in favour of the Fisher hypothesis.

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Notes

  1. 1.

    Banerjee (1999), Baltagi and Kao (2000) and Breitung and Pesaran (2005) provide a comprehensive review of the literature on unit roots and cointegration in panels.

  2. 2.

    See also the earlier contributions by Bacon and Watts (1971) and Maddala (1977), which pioneered the smooth transition-type models.

  3. 3.

    The simulation studies by Kapetanios et al. (2011) confirm that CCE estimators have better small sample properties than the PC-based ones. See also Westerlund and Urbain (2011) and Pesaran and Tosetti (2011).

  4. 4.

    Gradually changing mean in Model A could be proxied by a linear trend especially if \(\gamma _{\mathrm{i}}\) is relatively small. For Model A, therefore, we can make a “fair” comparison. However, Models B and C include highly nonlinear deterministic components that cannot be captured by conventional unit root tests. We have also conducted simulation exercises for Models B and C, and find that the small sample performance of our proposed test statistics is fairly satisfactory, suggesting that these results are qualitatively similar to those reported in the paper (see “Appendix 2”).

  5. 5.

    Heteroskedasticity may affect the power of standard panel unit root tests, though asymptotic size of tests should be intact. However, this issue is expected to affect our tests and existing tests in a similar manner. Therefore, we will leave this issue to future studies.

  6. 6.

    It is well-established that failure to allow for positive (negative) residual serial correlation results in severe under-rejection (over-rejection) of the null hypothesis, e.g. IPS.

  7. 7.

    The transition function \(S_{i,t} \left( {\gamma _{i},\tau _{i} }\right) \) is close to a straight line for these ranges of \(\gamma _{i}\) in which case \(y_{i,t}\) becomes close to a stationary process around a linear trend. Moreover, we have to estimate two more parameters for evaluating \(\bar{{t}}_{\alpha } \) Hence, it is not surprising to find that the power of \(\bar{{t}}_{ IPS} \) is slightly better.

  8. 8.

    Further simulation results and the number of technical discussions on the power of the IPS test in the presence of SMT are available in the supplementary online Technical Annex.

  9. 9.

    For example, assuming that the true number of factors is 2, Pesaran et al. (2013) find that real equity price is stationary when oil price and the long-term interest rates are used as additional regressors, but it is not stationary when inflation rate, deflated exchange rate and nominal GDP are employed.

  10. 10.

    Smith and Fuertes (2010) argue that “apparent structural changes may result from having left out an unobserved global variable”. If all of the cross-section entities share the same deterministic components, then such components can be regarded as common factors. In the special case where both \(\gamma \) and \(\tau \) parameters are homogenous, then the cross-section averages of the series of interest can be served as good proxy for such common factors. As the degree of heterogeneity in the deterministic components increases, however, cross-section averages will deviate from individual deterministic components. We have also conducted the graphical analysis exhibiting the relationship between cross-section averages and nonlinear trend functions under the various degrees of heterogeneity of deterministic trend functions, which is available in the supplementary online Technical Annex.

  11. 11.

    We have also conducted the simulation studies for models B and C with more general and complex trend functions. The results reported in “Appendix 2”. Further simulation results clearly show that that the CCE-based tests advanced by Pesaran (2007) and Pesaran et al. (2013) are not suitable to accommodate the presence of nonlinear and slow moving trends.

  12. 12.

    We have also considered the bootstrap version of the CCE-based statistics, denoted as \({\overline{BCt}}_{\alpha }\). But, such combination does not improve the power over either \({\overline{Ct}}_{\alpha }\) or \({\overline{Bt}}_{\alpha }\) tests. In particular, these simulation results (available upon request) show that \({\overline{Bt}}_{\alpha }\) test dominates \({\overline{BCt}}_{\alpha }\) tests in terms of power.

  13. 13.

    See Omay et al. (2016) for more details about the empirical application and possible extension.

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Acknowledgements

We would like to thank the editor and two anonymous referees for most constructive and helpful comments and suggestions, which help us to greatly improve the quality of the paper. The usual disclaimer applies.

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Correspondence to Tolga Omay.

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Appendices

Appendix 1: Sieve Bootstrap Method

We describe the Sieve bootstrap method in multi-steps as below:

  1. 1.

    Consider the individual ADF regression with heterogeneous lag order, \(p_{i}\):

    $$\begin{aligned} \Delta \hat{{x}}_{i,t} =\rho _{i} \hat{{x}}_{i,t-1} +\sum \limits _{j=1}^{p_{i} } {\delta _{i,j} \Delta \hat{{x}}_{i,t-j} } +e_{i,t} \end{aligned}$$
    (21)

    where \(\hat{{x}}_{i,t}\) are detrended series obtained from by (5a)–(5c).

  2. 2.

    We generate the residuals under the null hypothesis of unit root (Basawa et al. 1991) by

    $$\begin{aligned} e_{i,t} =\Delta \hat{{x}}_{i,t} -\sum \limits _{j=1}^{p_{i} } {\delta _{i,j} \Delta \hat{{x}}_{i,t-j} } \end{aligned}$$
    (22)
  3. 3.

    We apply the centering to the residuals as follows (e.g. Stine 1987):

    $$\begin{aligned} \tilde{{e}}_{t} =\hat{{e}}_{t} -\left( {T-p-2} \right) ^{-1}\sum \limits _{t=p+2}^T {\hat{{e}}_{t} } \end{aligned}$$
    (23)

    where \(\hat{{e}}_{t} =\left( {\hat{{e}}_{1,t},\hat{{e}}_{2,t} ,\ldots ,\hat{{e}}_{N,t} } \right) ^{\prime }\) and \(p={\mathrm {max}}(p_{i})\). We then generate the \(N\times T\) matrix of residuals, denoted \(\left[ {\tilde{{\varepsilon }}_{\hat{\iota },t} } \right] \). To preserve the CSD structure of errors, we randomly draw the column vector of residual with replacement, and denote them by \(\tilde{{e}}_{i,t}^{*}\) for \(t=1,2,\ldots ,T^{*}\). We generate the bootstrap samples, \(\Delta x_{i,t}^{*}\) recursively from

    $$\begin{aligned} \Delta \hat{{x}}_{i,t}^{*} =\sum \limits _{j=1}^{p_{i} } {\hat{{\delta }}_{i,j} \Delta x_{i,t-j}^{*} } +\tilde{{e}}_{i,t}^{*} \end{aligned}$$
    (24)

    where the initial values of \(\Delta x_{i,t-j}^{*}\) set to zero. We then generate \(x_{i,t}^{*}\) as the partial sum process:

    $$\begin{aligned} x_{i,t}^{*} =\sum \limits _{k=1}^t {\Delta x_{i,k}^{*}} \end{aligned}$$
    (25)
  4. 4.

    We discard 50 initial values (burn-in) and run the following regression:

    $$\begin{aligned} \Delta x_{i,t}^{*} =\rho _{i} x_{i,t-1}^{*} +\sum \limits _{j=1}^{p_{i} } {\theta _{i,j} \Delta x_{i,t-j}^{*} } +\nu _{i,t} \end{aligned}$$
    (26)

    and compute the bootstrap statistics per each replication.

In this paper, we use 2000 replications, derive the empirical distribution of the bootstrap test statistics and compute the rejection probabilities. The same procedure is used in Ucar and Omay (2009) and Omay et al. (2014).

Appendix 2: Power Analysis for Models B and C

To evaluate powers of tests under models B and C, we consider following DGPs:

$$\begin{aligned} y_{i,t}= & {} 1.0+10.0S_{i,t} \left( {\gamma _{i},\tau _{i} } \right) +2t+x_{i,t}\nonumber \\ x_{i,t}= & {} 0.8x_{i,t-1} +\varepsilon _{i,t},x_{i,0} =0 \end{aligned}$$
(27)

which allows a linear trend under Model B, and

$$\begin{aligned} y_{i,t}= & {} 1.0+10.0S_{i,t} \left( {\gamma _{i},\tau _{i} } \right) +2t+1.5tS_{i,t} \left( {\gamma _{i},\tau _{i} } \right) +x_{i,t}\nonumber \\ x_{i,t}= & {} 0.8x_{i,t-1} +\varepsilon _{i,t},x_{i,0} =0 \end{aligned}$$
(28)

which also allows a smooth break in the trend function under Model C. We focus on the strongly heterogeneous specification with \(\tau _{i} \sim { iid.U}\left[ {0.2,0.8} \right] \) and \(\gamma _{i} \sim { iid.U}\left[ {0.1,5.5} \right] \). Empirical powers of tests are now presented in Tables 9 and 10 below.

Table 9 Powers of the \({\overline{Ct}}_{\alpha \left( \beta \right) } \) and \({\overline{{ CADF}}} \) test statistics

First, consider the Model B. Overall, the power of the \({\overline{{ CADF}}}\) test is mostly negligible, though surprisingly, it tends to display substantial power under high CSD especially only with large T. On the other hand, the power of the \({\overline{Ct}}_{\alpha \left( \beta \right) } \) test is much more satisfactory, and rises monotonically with the sample sizes, albiet much faster with T.

Table 10 Powers of the \({\overline{Ct}}_{\alpha \beta } \) and \({\overline{{ CADF}}} \) test statistics

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Omay, T., Hasanov, M. & Shin, Y. Testing for Unit Roots in Dynamic Panels with Smooth Breaks and Cross-Sectionally Dependent Errors. Comput Econ 52, 167–193 (2018). https://doi.org/10.1007/s10614-017-9667-7

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Keywords

  • Slow moving trends
  • Cross-section dependence
  • Common correlated estimator
  • Bootstrap
  • Panel unit root tests

JEL Classification

  • C12
  • C22
  • O47