A Meta Analytic Approach to Testing for Panel Cointegration

Abstract

We propose new tests for panel cointegration by extending the panel unit root tests of Choi (2001) and Maddala and Wu (1999) to the panel cointegration case. The tests are flexible, intuitively appealing, and relatively easy to compute. We investigate the finite sample behavior in a simulation study. Several variants of the tests compare favorably in terms of both size and power with other widely used panel cointegration tests.

Keywords:

• Meta analysis
• Monte Carlo study
• Panel cointegration tests

Mathematics Subject Classification:

• 62P20
• 62M10
• 62F03

Acknowledgments

The article was partially supported by DFG under Sonderforschungsbereich 475 and Ruhr Graduate School in Economics is gratefully acknowledged. I am indebted to an anonymous referee whose suggestions greatly helped improve the article.

Notes

¹Both Maddala and Wu (1999) and Choi (2001) suggested extending their panel unit root tests to the cointegration case. However, to the best of our knowledge, they do not provide an actual implementation nor do they investigate the performance of the tests. Furthermore, our approach is more general and likely to be more accurate in some respects to be discussed below.

²For an overview of panel data models relying on N → ∞ asymptotics see Hsiao (2003).

³MacKinnon improves upon his prior work by using a heteroscedasticity and serial correlation robust technique to approximate between the estimated quantiles of the response surface regressions. Our application is based on a translation of James MacKinnon's Fortran code into a GAUSS procedure which is available upon request. The procedure implements all panel data tests developed in this section.

⁴Uniform random numbers are generated using the KM algorithm from which Normal variates are created with the fast acceptance-rejection algorithm, both implemented in GAUSS. Part of the calculations are performed with COINT 2.0 by Peter Phillips and Sam Ouliaris.

Note: ρ = 1, ψ = 0, σ = 1, and a ₁ = 0. M = 5,000 replications. 5% nominal level. ADF and λ_trace are the underlying time series tests.

⁵The full set of results of the finite sample study are available upon request. Broadly speaking, a lower σ seems to have little, if any, systematic effect. Correlation in the error processes (ψ ≠ 0) has a slightly negative effect on power.

⁶We also investigate whether using MacKinnon's (MacKinnon 1996) p-values improves the behavior of the tests relative to obtaining quantiles by generating only one set of replicates. For smaller panels, the latter approach (with 50,000 replications) exhibits non negligible upward size distortions even when using quantiles specifically generated for the sample sizes considered. Interestingly, however, there does not seem to be a trend towards higher distortions with increasing N. For medium- and large-dimensional panels neither approach has a clear advantage over the other.

⁷Gutierrez (2003) provided a power study of these tests. He does, however, not analyze the finite sample size, whence our study should be viewed as complementary to his.

⁸These size distortions are well in line with results found by Kao (1999) and Larsson et al. (2001) (cf. their Tables 4 and 2, respectively). Pedroni (2004) conducts experiments for T ≥ 40 only. Any remaining differences are due to differences in the underlying DGP. As these results suggest that the asymptotic approximations should be used with care for very small T, future research might attempt to provide correction factors similar to those of Cheung and Lai (1993).

Note: ρ = 1, ψ = 0, σ = 1, and a ₁ = 0. M = 5,000 replications. 5% nominal level.

⁹See below for their performance under other DGPs.

Note: ρ = 0.9, ψ = 0, σ = 1, δ = 0.5, and a ₁ = 0. M = 5,000 replications. 5% nominal level. ADF and λ_trace are the underlying time series tests.

¹⁰It is, however, not clear whether this attractive performance would be available in practice, as these numbers are based on size-adjusted critical values and as the test appears to be rather undersized at least for small T (cf. Table 2).

Note: ρ = 0.9, ψ = 0, σ = 1, and a ₁ = 0. M = 5,000 replications. 5% nominal level.

¹¹I am grateful to an anonymous referee for suggesting this extension.

¹²We also parameterized DGP B with other combinations of distributions for the and ϕ _{i,p i} . The additional results, which were qualitatively very similar, are available upon request.

¹³Indeed, Tables 3–8 report that the λ_trace-based P tests and the ϒ _LR test are generally less powerful than the residual-based tests under DGPs A and B.

Note: ρ _i  = 1, ψ = 0, σ = 1, and a ₁ = 0. M = 5,000 replications. 5% nominal level. ζ _i ∼𝒰[1,2], i = 1,…,δ N and ϕ _i,p ∼𝒰[0.1,0.35].

Note: ρ _i  = 1, ψ = 0, σ = 1, and a ₁ = 0. M = 5,000 replications. 5% nominal level. ζ _i ∼𝒰[1,2], i = 1,…,δ N, and ϕ _i,p ∼𝒰[0.1,0.35].

Note: Half of the series has ρ _i ∼𝒰[0.9,1], ρ _i  = 1 else. ψ = 0, σ = 1, and a ₁ = 0. M = 5,000. 5% nominal level. ζ _i ∼𝒰[1,2], i = 1,…,δ N and ϕ _i,p ∼𝒰[0.1,0.35].

Note: Half of the series has ρ _i ∼𝒰[0.9,1], ρ _i  = 1 else. ψ = 0, σ = 1, and a ₁ = 0. M = 5,000. 5% nominal level. ζ _i ∼𝒰[1,2], i = 1,…,δ N, and ϕ _i,p ∼𝒰[0.1,0.35].

¹⁴Results for the other tests considered above are available upon request. Broadly speaking, the other P tests behave very much like the P _Φ⁻¹ test when using the same time series test, with the P _χ² again being slightly less powerful. Furthermore, the other tests by Kao and Pedroni are somewhat less powerful than the ones reported here.

Note: M = 5,000 replications. 5% nominal level. Power results ( > 0) are size adjusted. ADF is the underlying time series test for P _Φ⁻¹,DF , and λ_trace for P _Φ⁻¹,λ.