1. MOTIVATION

In the past decade or so, there has been much interest in testing for the presence of a unit root in panel data. Many researchers have proposed statistics to test the hypothesis of a common unit autoregressive root. Recent surveys by Baltagi and Kao (2000), Choi (2004), Hurlin and Mignon (2004), and Breitung and Pesaran (2005) provide an overview of these developments.

In this context, Breitung (2000) proposed a t-ratio-type test statistic for testing a panel unit root. Through numerical analysis, he claimed that his test has “nice” power properties within a certain local neighborhood of unity. The present paper investigates analytically the asymptotic power properties of Breitung's test and clarifies some of the analytic results in Breitung (2000). Specifically, we show that the limiting distribution of the Breitung test is the same under the null and the O(n^−1/2T⁻¹) local alternatives considered by Breitung, so that the test has trivial power in such narrow neighborhoods. We provide expressions for the local asymptotic power of this test for wider n^−1/4T⁻¹ local departures from the null, discuss comparative results for other procedures such as point optimal tests, and study the accuracy of the asymptotic approximations in finite samples.

2. BREITUNG'S TEST STATISTIC AND CLAIMED POWER

Suppose that panel data y_it is generated by the following simple components model:

where the unobserved error term x_it follows

Our main interest is in testing the presence of a unit root in all cross-sectional units, namely,

For this, we assume that ε_it ∼ iid(0,σ²) with E(ε_it⁴) < ∞, the initial observations x_i0 are independent and identically distributed (i.i.d.) across i with E(x_i0⁴) < ∞ and independent of ε_it for all t ≥ 1 and i.¹

Notice that this testing problem is invariant to the following linear transformation: y_it* = y_it + μ_i* + β_i*t. To construct a test that is invariant to the transformation, Breitung (2000) suggested the use of the following transformed data:

for t = 1,…,T − 1, where s_t² = (T − t)/(T − t + 1), and

for t = 2,…,T.²

The panel unit root test for the null hypothesis (1) proposed by Breitung (2000) is to reject the null for the small values of the following statistic:

where

is a consistent estimator of σ². This is a t-ratio statistic from a pooled regression of the transformed data (Δy_it)* on y_it−1*. Define

. Under the null hypothesis, a direct calculation (given in the Appendix) shows that for 2 ≤ t ≤ T − 1

Hence, by central limit theory (see, e.g., Phillips and Moon, 1999), it is possible to show that

as n,T → ∞. On the other hand,

so that

as n,T → ∞. Therefore, under the null hypothesis, it follows that as n,T → ∞,

Breitung's test for H₀ with size α rejects H₀ if

, where z_α is the (1 − α) quantile of the N(0,1) distribution.

To analyze the local power of the test based on

, Breitung (2000) considered the following local parameterization:

and he claimed (see Breitung, 2000, Thm. 5) that the test B_nT has asymptotically significant local power against the local alternative

To verify the claim, because

under the local alternative (2), we can express the transformed variables as follows:

Note that A_it and C_it do not depend on c. Then, the preceding formulas lead to the following decomposition of

:

LEMMA 1. The following conditions hold under the local alternative ρ_i = 1 − (c/n^1/2T) as n,T → ∞.

It follows from Lemma 1 that

. Together with the limit of

, under the local alternative in (2), we therefore obtain

and have the following result.

THEOREM 2. Under the local alternative ρ_i = 1 − (c/n^1/2T), assumed in Breitung (2000), the Breitung test statistic

has asymptotic power equal to the size of the test.

The preceding result does not imply a mistake in Breitung (2000). Specifically, Breitung stated that the power of his test depends on a quantity that in our notation is

. However, the preceding results demonstrate that this expectation has zero limit. Instead of an analytical evaluation of the limit, Breitung relied on a numerical approximation to the expectation that suggested nonnegligible power against alternatives of the form (2). Our results indicate that the numerical approximation is poor and that consequently the implications regarding power in neighborhoods of the form (2) are misleading.

3. ASYMPTOTIC LOCAL POWER OF BREITUNG'S TEST

The main problem with Theorem 2 is that the local neighborhood in (2) shrinks too fast to unity and so the asymptotic power is trivial. To correct the result, this section considers local neighborhoods that shrink to unity more slowly and derives an asymptotic local power function for the Breitung test.

First, consider the local-to-unity region defined by

where the c_i may be defined as the realizations of a sequence of i.i.d. random variables whose support lies in an interval of the form [−M_lc,M_uc] for some 0 < M_lc < ∞ and 0 ≤ M_uc < ∞. Let c_i be independent of ε_js for all i,j,s. Under these conditions, the numerator of Breitung's test statistic can be decomposed as

where the components A_it, B_it, C_it−1, and D_it−1 are defined in the previous section.

LEMMA 3. Under the local alternative (4), the following conditions hold as n,T → ∞.

Using Lemma 3, the limit distribution under the local alternative hypothesis (4) is

from which we deduce the following theorem.

THEOREM 4. Against the local alternative in (4), the asymptotic local power of Breitung's test is

, where Φ( ) is the standard normal cumulative distribution function.

Remarks.

A small-scale simulation was conducted to assess the accuracy of these asymptotic results in finite samples. We use the following data generating process:

The heterogeneous trend coefficients are taken to be iid N(0,1). We assume that the error term is independent in both time and cross-section dimensions with a Gaussian distribution and identical variances. We consider four values for n (10, 25, 100, and 250) and three values for T (50, 100, and 250). All tests are carried out at the 5% significance level, and the number of replications is set at 10,000.

The autoregressive parameters are generated according to (4). We consider the following nine distributions for the local-to-unity parameters:

These distributions enable us to examine performance of the tests as the mass of the distribution of the localizing parameters moves away from the null hypothesis. We can also look at the effect of homogeneous versus heterogeneous alternatives (cases (1) and (4) versus (7) and cases (2) and (5) versus (8)) together with the role of the higher order moments of the distribution. For instance, case (7) has the same mean as cases (1) and (4) but smaller higher order moments than the other two cases. The same situation arises for cases (2), (5), and (8) and for cases (3) and (6). Note that the alternatives with χ² distributions do not fit our asymptotic framework because they have unbounded support.

Table 1 presents the results. The second column provides the size and power predicted by our asymptotic theory using the moments of c_i. The other columns in the table report the size and size-adjusted power of the tests for the various combinations of n and T. If asymptotic theory were a reliable guide to finite-sample behavior, all columns in the table would be very close.

Overall, we see that the test performs much more closely to the asymptotic theory as both n and T increase. The test underrejects for large n relative to T. Power is usually below what is predicted by asymptotic theory, especially for the more distant alternatives, but the discrepancy diminishes with increases in either n or T or both. Finally, experiments where the local-to-unity parameters have a fatter tailed distribution tend to have higher power as predicted. Thus, for a given mean autoregressive parameter, more heterogeneous alternatives are easier to detect (despite the pooling approach used in the test). There also seems no sign that the unboundedness of the χ² distributions affects the validity of the asymptotic theory.