An Intersection Test for Panel Unit Roots

Abstract

This article proposes a new panel unit root test based on Simes’ (1986) classical intersection test. The test is robust to general patterns of cross-sectional dependence and yet is straightforward to implement, only requiring p-values of time series unit root tests of the series in the panel, and no resampling. Monte Carlo experiments show good size and power properties relative to existing panel unit root tests. Unlike previously suggested tests, the new test allows to identify the units in the panel for which the alternative of stationarity can be said to hold. We provide an empirical application to real exchange rate data.

Keywords:

• Cross-sectional dependence
• Multiple testing
• Panel unit root test

JEL Classification:

• C12
• C23

ACKNOWLEDGMENTS

I am grateful to two anonymous referees, conference and seminar participants in Cologne, Graz, and Maastricht as well as to Matei Demetrescu and Franz Palm for constructive and helpful comments.

Notes

Recently, procedures taking multiplicity into account have begun to find their way into the econometrics literature. Romano et al. (2008) provide a survey of available methods and Hanck (2009) an application.

Most tests are also silent about the number of stationary units n ₁. Ng (2008) is a recent exception, discussed in Section 3.2.

We assumed the number of common factors to be known. In practice, one could and should employ information criteria (IC) such as those of Bai and Ng (2002). Bai and Ng (2004, p. 1145) show that this does not materially affect the accuracy of their procedure. In view of the relatively large T and the order of magnitude of calculations 𝒪(T ³) involved in defactoring the y _{i, t} a full search over a number of possible factors using IC would have been rather demanding in terms of computation time.

If F _t  ∼ I(1), the tests can be seen as panel non-cointegration tests, as λ _j y _{i, t}  − λ _i y _{j, t}  ∼ I(0) if e _{i, t}  ∼ I(0) (Bai and Ng, 2004).

To investigate nonzero intercepts, we simulate μ _i ∼ 𝒰[0, 0.2] and calculate τ-statistics from regressions of y _{i, t} on 1, y _{i, t−1}. The results, which we do not report for brevity but which are available upon request, are similar to those to be reported. As expected, fitting a constant results in loss of power throughout. Furthermore, we analyzed φ _n  = 0.95·ı _n for a homogenous and for a fully stationary alternative.

Nominal level α = 0.05, 10,000 draws. S is by Simes (1986), P _χ² by Maddala and Wu (1999) and by Demetrescu et al. (2006).

This result is robust to other methods to choose the lag lengths. Unreported results show that if these are chosen so high that (seemingly slightly) undersized time series unit root tests result, P _χ² will accumulate these distortions to become increasingly conservative with n, thus wasting power. (Also see subpanel (b) for DGP B, which shows that P _χ² is undersized by 2.7% for n = 8 and T = 30 under DGP B, but already by 3.9% for n = 48.)

See notes to Table 2. No serial correlation.

Here, we focus on the case of no serial correlation. Unreported additional simulations, which are available upon request, draw a qualitatively similar picture under serial correlation and size-adjusted critical values.

10000 replications. is the mean of Hommel's estimator and its MSE. and are Ng's.

Since the singular value decompositions involved in defactoring the y _{i, t} for DGP B are of order 𝒪(T ³), they become computationally very burdensome for T ≥ 1000. We therefore waive to analyze DGP B for very large T.

Further research could investigate whether recent powerful resampling-based multiple tests by, e.g., Romano and Wolf (2005) can mitigate the bias of .

See, e.g., Breitung and Pesaran (2008), Gengenbach et al. (2010) or the working paper version (available from the author's website) for surveys and comparisons of the tests used here.

Nominal level α = 0.05, 10,000 draws. In C, δ = 0.98. In D, ψ = 0.75. S is by Simes (1986), P _χ² by Maddala and Wu (1999), C* by Pesaran (2007), t _rob by Breitung and Das (2005), by Demetrescu et al. (2006) and and by Moon and Perron (2004).

See notes to Table 5.

See notes to Table 5.

Results for the remaining power experiments mentioned in footnote 5, which were similar, are available upon request. Overall, the C* performs marginally best there, and the Moon and Perron (2004) tests are substantially better. For near-integrated homogenous alternatives, the power of S test is relatively lower. This is unsurprising because we then have p ₍₁₎ ≈ p _(n), and it is unlikely that p _(n) ≤ α for φ _n close to ı _n .

See notes to Table 5.

See notes to Table 5.

See notes to Table 5. Two cross-unit cointegrating relationships, .

“ p-Values” are for the DF-GLS^μ time series unit root tests (Elliott et al., 1996). “Simes crit.” is the value that needs to exceed the adjacent p-value in at least once for a rejection of H ₀, cf. (3).

See notes to Table 5. Nominal level is α = 0.05. The column “critical points” gives the α level critical value. r _{i, t} gives the test statistics applied to the real exchange rates and those applied to extracted idiosyncratic components. CIPS is by Pesaran (2007).