Point-optimal panel unit root tests with serially correlated errors
Summary
Generalizations of the point-optimal panel unit root tests of Moon, Perron and Phillips (MPP) are developed to cover cases of serially correlated errors. The resulting statistics involve two modifications relative to those of MPP: (a) the error variance is replaced by the long-run variance; (b) centring of the statistic is adjusted to correct for second-order bias effects induced by the correlation between the error and lagged dependent variable.
1. Introduction
and an (unobserved) stochastic component 
for some observable panel observations 
for individual 
in period 
satisfying
(1.1)
is an error term that has zero mean and is stationary over time, and 
for simplicity. Dynamic panel models with incidental trend components of this type arise in many applications in microeconometrics, multicountry growth studies and international finance. Empirical interest often centres on the individual dynamics and on whether there is commonality and persistence across individuals (i.e. that the autoregressive parameters 
are all unity) or whether such commonality occurs for certain subgroups of individuals.
(1.2)
is a sequence of i.i.d. random variables and κ is a parameter defining the width of the vicinity as 
. The null hypothesis of interest is then
(1.3)
(1.4)The MPP tests are point optimal in the sense of giving highest power against a specific set of 
. These tests were derived under the assumption that the error term 
is independent across individual units and over time. They represent a panel extension of the work of Elliott et al. (1996) in the time-series case where the autoregressive parameter converges to unity at a rate of 
, regardless of the deterministic component in the model.
Independence assumptions are not realistic in many empirical applications and in this work we extend the MPP tests by allowing for serially correlated errors 
. In their Section 6.4 (p. 436), MPP briefly mentioned this extension. Here, we provide explicit test statistics that have optimal asymptotic properties. The modified tests replace estimated variances of the errors in MPP with estimated long-run variances, and adjust centring terms. Our main purpose is to provide the form of the modified tests and to give their asymptotic properties so that they can be used in empirical work.
The paper is organized as follows. In Section 2., we show how to construct the tests, give results for cases with no fixed effects, fixed effects and incidental trends, and discuss implementation. In Section 3., we report some simulation findings. We conclude in Section 4, and in the Appendix we provide technical derivations and supporting lemmata.
2. Tests under Serial Correlation
Following MPP, in the following analysis, we consider three deterministic trend cases: (a) no individual effects (i.e. 
and 
); (b) fixed effects (i.e. 
); (c) heterogeneous or incidental linear trends (i.e. 
). In each case, we proceed in three steps. First, we define the likelihood ratio (
) statistic under Gaussianity, which is known to be optimal by the Neyman–Pearson lemma when the null and alternative hypotheses are simple. Then, we show that this statistic can be approximated by a simpler version with parameters that are consistently estimable. Finally, we derive the asymptotic distribution of this approximation (with appropriate recentring). In all three cases, this asymptotic distribution coincides with the one in MPP.
and U the 
observation matrices whose 
th elements are 
, 
, 
, 
and 
, respectively. Define the 
vectors 
, 
and set 
, where 
. Define 
, 
and 
, where 
. Let 
, 
, 
and 
denote the transpose of the ith row of Z, Y, 
and U, respectively. With this notation, the model has the matrix form
.Define 
, 
and 
as the variance of 
, the long-run variance of 
and the one-sided long-run variance of 
, respectively, so that 
. Let Σ, Ω and Λ be the diagonal matrices with elements 
, 
and 
, respectively. Define 
as the 
covariance matrix of 
and 
as the 
covariance matrix of 
. As in MPP, we assume that the errors 
are cross-section independent over i.
in the local alternative 1.2 is a sequence of i.i.d. random variables with bounded support.1 Let 
, 
and define the quasi-difference operator
and 
.
Throughout the paper, we assume panel linear process errors with conditions similar to those in the literature (e.g. Phillips and Moon, 1999). Let 
, let 
be the spectral density of 
, and let 
.
Assumption 2.1.(a) 
, where 
with 
and 
for some 
; (b) 
for some 
, 
, 
, 
; (c) 
and for 
and some constant M.
These conditions extend the serial dependence restrictions of Elliott et al. (1996) (e.g. their Condition A) to heterogeneous panels. Assumption 2.1(a) assumes that the error term 
follows a linear process that is heterogeneous across i. The higher moments are needed to ensure the large 
asymptotics of panel data that are heterogeneous across i and serially correlated over t. Under cross-sectional homoscedasticity, these moment conditions could be weakened. Assumptions 2.1(b) and 2.1(c) restrict the temporal dependence of the error term 
to be ‘weak’ uniformly across i. These restrictions exclude long memory type strong dependence. The conditions in Assumption 2.1 are quite weak and are satisfied by many parametric weak dependent processes, such as stationary and invertible ARMA processes.
While Assumption 2.1 is quite general in terms of the serial correlation that is allowed, it is restrictive in that it assumes that all cross-sectional units are independent. This assumption is not reasonable for many interesting empirical data sets, such as cross-country studies where business-cycle effects are likely to induce correlation across countries. As in MPP, we conjecture that the procedures proposed below are valid after appropriate orthogonalization is applied, for example, after the removal of common factors as in Moon and Perron (2004), Bai and Ng (2004) or Phillips and Sul (2003). Moreover, the development of optimal procedures under cross-sectional dependence is beyond the scope of the current paper.
2.1. No fixed effect: 
, the model becomes
, so that the rate coefficient 
, and one-sided alternatives in which the support of 
is a bounded interval 
for some 
so that 
under this alternative. In terms of the first moment of 
, the hypotheses about 
are
(2.1)
(2.2)
are Gaussian so that 
with known 
and the initial conditions 
are all zeros.2 By the Neyman–Pearson lemma, rejecting a small value of the log-likelihood ratio test statistic
(2.3)
for 
against the simple alternative 
for 
. When the alternative is 1.4 with 2.2, this becomes a point-optimal test.In order to implement the optimal test statistic 2.3, we need an estimate of the entire 
covariance matrix 
. This is a huge high-dimensional covariance estimation problem in a non-parametric set-up. The following theorem provides an approximation of the likelihood ratio test statistic in 2.3 with a statistic where the unknown nuisance parameters are consistently estimable.
Theorem 2.1.Let Assumption 2.1 hold with 
. Assume that 
→ 0 as 
. Then, for 
, we have
(2.4)
with an adjustment of the one-sided long run variance 
. The one-sided long-run drift correction appears to be a result of the correlation between the stationary error 
and the lagged dependent variable 
. The main advantage of this formulation is that it involves quantities (Ω and Λ) that can be estimated consistently.
is the sum vector and 
.
as 
. Then, under the local alternative 
, we have
Remark 2.1.We can interpret the test statistic 
as an asymptotic version of the point-optimal test for panel unit roots with possible serial correlation of unknown form in the error term.
Remark 2.2.Compared to the corresponding statistic in MPP, which makes no allowance for serial correlation, there are two differences in 
. First, as discussed in MPP, we use the long-run covariance matrix 
instead of the variance matrix 
as the weight matrix. In addition, we recentre the statistic by subtracting the term 
, which corrects for the correlation between the stationary error 
and the lagged dependent variable 
. This term is not required for the test under temporal independence.
Remark 2.3.The limit distribution of 
is the same limit as in MPP (see their Theorem 6).
2.2. Time-invariant fixed effects: 
are fixed over time. This corresponds to the standard fixed effects model. In this case, the model has matrix form
with known 
and the initial conditions 
are all zeros. Then, rejecting a small value of the test statistic,
(2.5)
for 
and the alternative 
for 
is known as the uniformly most powerful invariant test that is invariant with respect to the transformation 
for arbitrary 
. Against the alternative in 1.4, this becomes a point-optimal invariant test (e.g. Dufour and King, 1991).As mentioned in the previous section, this statistic is difficult to implement because of the presence of 
, the full 
covariance matrix of the error. This again motivates the use of an approximation.
Theorem 2.3.Let Assumption 2.1 hold with 
and let 
as 
. Then, for 
, we have
Remark 2.4.This approximation is derived under the stronger rate condition 
→ 0 as 
in place of the condition 
as 
that is used without fixed effects.
Remark 2.5.The approximation involves the same correction for second-order bias as in the case without fixed effects.
Theorem 2.4.Let Assumption 2.1 hold and let 
as 
. Then, under the local alternative 
, we have
.
This asymptotic distribution is the same as without fixed effects and as in MPP (see their Theorem 9).
2.3. Incidental trends: 
Under heterogeneous linear trends, we follow MPP and use local neighbourhoods of unity that shrink at the slower rate of 
, so that the rate coefficient is 
. The alternative might be two-sided; that is, 
with mean 
and variance 
, with a support that is a subset of a bounded interval 
, where 
, 
. The slower rate of shrinkage in the local neighbourhoods of unity is the result of the presence of heterogeneous trend effects in the panel. The presence of these incidental trends reduces discriminatory power in testing for the presence of common stochastic trends, so wider localizing intervals are needed to attain non-trivial power functions.
with known 
and the initial conditions 
are all zeros. Then, similar to the case of time-invariant fixed effects, rejecting a small value of the test statistic,
for 
and the alternative 
for 
, is known as the uniformly most powerful invariant test (with respect to the linear transformation 
for arbitrary 
), and against the alternative in 1.4, it becomes a point-optimal invariant test. As before, we start by proving the validity of an approximation to this log-likelihood ratio.
Theorem 2.5.Let Assumption 2.1 hold with 
and let 
as 
. Then, for 
, we have
Remark 2.6.This approximation is derived under the condition 
as 
, which is a stronger rate condition than that used for the intercepts case.
Remark 2.7.As before, the correction is a result of the presence of a second-order bias term arising from the correlation between the lagged dependent variables and the error term.
Theorem 2.6.Let Assumption 2.1 hold and let 
as 
. Then, under the local alternative 
, we have
As before, 
reduces to the statistic from MPP when there is no serial correlation, and it has the same asymptotic distribution as in Theorem 13 of MPP.
2.4. Implementation of the tests
The test statistics 
, 
and 
depend on unknown parameters 
, 
and 
. Let 
, 
and 
be consistent estimators of 
, 
and 
, respectively. Similarly define the diagonal matrices of these elements as 
, 
and 
. To implement these tests, we can replace Σ, Ω and Λ in 
, 
and 
with 
and 
and we denote the test statistics as 
, 
and 
. We assume the following regarding these estimators.
Assumption 2.2.Under the local alternative, 
, 
and 
.
Remark 2.8.An example of 
that satisfies Assumption 2.2 is the time-series sample variance of 
Remark 2.9.When kernel spectral density estimation is used for 
and 
with bandwidth h, Assumption 2.2 is satisfied if: (a) the kernel function 
is continuous at zero and all but a finite number of other points, satisfying 
and 
for some 
where parameter m is defined in Assumption 2.1(b); (b) the bandwidth h satisfies
(2.8)
and 
; see, e.g., Moon and Perron (2004). If 
for some 
and 
, then the bandwidth condition 2.8 is satisfied if
with 
we have 
, 
and 
under the local alternative.
. MPP have shown that the optimal choice of 
is 
. With this choice, the likelihood ratio statistics above attain the power envelope. However, this choice is infeasible because 
are the parameters under test. MPP proposed the assumption of a common 
for all cross-sectional units, and they called this test a common point-optimal (CPO) test. With this choice, 
, and we can deduce from Theorems 2.2, 2.4 and 2.6 that under the null hypothesis,
The surprising result here is that neither distribution depends on the choice of c used to construct the test. This feature implies that the power is the same for all choices of c asymptotically, although that choice matters in finite samples. Based on the simulation evidence provided in MPP, we set 
in the simulation below. Of course, this choice of 
is not optimal unless the alternative hypothesis is homogeneous (
for all i) and results in a power loss relative to the power envelope.
3. Monte Carlo simulations
), positive AR (
and 
), negative AR (
and 
), positive MA (
and 
) and negative MA (
and 
).In all cases, we allow for heterogeneity and draw the idiosyncratic variance 
from a uniform distribution, 
. This variance is scaled such that the scale of 
is the same for all cases. In both the incidental intercepts case 
and the incidental trends case 
, the parameters are drawn from 
.
We focus the study on the size and size-adjusted power of the CPO test with 
for all i, because MPP advocated that choice. For size calculations, we set 
for all i, which corresponds to 
for all i in our local-to-unity framework. For power, we draw 
from a uniform distribution between 0 and 8, as in one of the experiments of MPP. This specification ensures that power should be roughly constant as N and T increase.
We draw comparisons with two existing tests, those of Levin et al. (2002), hereafter LLC, and Im et al. (2003), hereafter IPS. We take three values for n (10, 25 and 100) and two values of T (100 and 250). All tests are conducted at the 5% significance level, and the number of replications is set at 2,000.
Estimation of the long-run variance and one-sided long-run variance is critical to the performance of the CPO test. We estimate these quantities in two ways based on demeaned first differences, as in Remark 2.8. The first method is a non-parametric estimator with quadratic spectral kernel and bandwidth selected in a data-based manner using the Andrews (1991) rule with no pre-whitening (PW). The second method uses pre-whitening where the appropriate model is chosen using BIC among the AR(1), MA(1), ARMA(1,1) and constant-only models. In the case of the LLC test, we follow the recommendation of Levin et al. (2002) and use a Bartlett kernel with a bandwidth equal to 
. Westerlund (2009) has shown that this choice gives the LLC test higher power than selecting the bandwidth in a data-dependent way. Similarly, for the IPS and LLC tests, the choice of lag augmentation is critical for performance. We choose this in a data-dependent way by BIC with a maximum of six lags. For both of these tests, we use the finite-sample adjustments provided in the original papers.
The size results are reported in Tables 1 and 3 for the incidental intercepts and trends cases respectively, while size-adjusted power is in Tables 2 and 4. For each of the five serial correlation specifications, each row corresponds to a different test.
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| 100 | 250 | 500 | 100 | 250 | 500 | 100 | 250 | 500 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| White noise | This paper, no PW | 1.6 | 2.2 | 2.6 | 2.8 | 3.2 | 3.6 | 4.2 | 3.7 | 5.2 |
| This paper, PW | 2.4 | 2.4 | 2.5 | 3.0 | 3.8 | 4.5 | 0.6 | 2.5 | 4.4 | |
 |
MPP (2007) | 2.9 | 2.4 | 2.5 | 4.8 | 3.7 | 4.4 | 4.3 | 3.6 | 4.7 |
 |
IPS | 6.4 | 3.7 | 4.9 | 7.0 | 5.2 | 5.2 | 8.4 | 6.9 | 5.1 |
| LLC | 5.8 | 4.4 | 4.1 | 5.8 | 5.3 | 3.4 | 6.0 | 5.0 | 3.5 | |
| Positive AR | This paper, no PW | 1.7 | 1.8 | 2.9 | 2.4 | 2.8 | 3.3 | 4.5 | 4.5 | 4.9 |
| This paper, PW | 2.3 | 2.1 | 3.0 | 1.7 | 2.3 | 3.0 | 0.7 | 0.4 | 2.4 | |
 |
MPP (2007) | 0.3 | 0.5 | 0.8 | 0.1 | 0.1 | 0.1 | 0.0 | 0.0 | 0.0 |
 |
IPS | 5.0 | 4.4 | 3.6 | 4.7 | 3.9 | 3.5 | 3.1 | 3.1 | 3.5 |
| LLC | 5.0 | 4.2 | 4.0 | 4.3 | 4.3 | 3.7 | 2.6 | 2.8 | 3.9 | |
| Negative AR | This paper, no PW | 1.7 | 2.1 | 2.2 | 2.5 | 3.2 | 4.0 | 5.4 | 4.2 | 5.2 |
| This paper, PW | 2.0 | 2.5 | 2.1 | 1.2 | 2.6 | 3.3 | 0.3 | 1.5 | 2.2 | |
 |
MPP (2007) | 12.7 | 13.2 | 13.6 | 30.7 | 30.3 | 32.4 | 82.2 | 81.6 | 83.9 |
 |
IPS | 12.8 | 7.3 | 5.4 | 20.4 | 9.4 | 7.2 | 42.6 | 15.3 | 9.3 |
| LLC | 9.0 | 5.8 | 4.2 | 12.0 | 7.0 | 4.0 | 20.8 | 8.5 | 4.3 | |
| Positive MA | This paper, no PW | 1.4 | 1.7 | 2.6 | 2.1 | 3.0 | 3.7 | 4.3 | 4.5 | 4.8 |
| This paper, PW | 2.5 | 2.6 | 3.0 | 2.7 | 3.2 | 4.1 | 1.6 | 3.5 | 4.5 | |
 |
MPP (2007) | 0.7 | 0.7 | 0.6 | 0.3 | 0.2 | 0.6 | 0.0 | 0.0 | 0.0 |
 |
IPS | 6.6 | 6.1 | 5.7 | 6.8 | 6.4 | 6.0 | 7.1 | 7.4 | 8.4 |
| LLC | 6.1 | 5.9 | 4.7 | 5.4 | 6.0 | 4.6 | 5.2 | 7.2 | 5.6 | |
| Negative MA | This paper, no PW | 1.3 | 2.0 | 2.6 | 2.3 | 3.5 | 3.6 | 4.6 | 4.5 | 4.3 |
| This paper, PW | 1.5 | 2.4 | 3.0 | 0.9 | 3.1 | 4.0 | 0.0 | 1.6 | 3.5 | |
 |
MPP (2007) | 17.0 | 16.7 | 17.6 | 40.7 | 44.1 | 40.7 | 93.0 | 91.8 | 93.1 |
 |
IPS | 25.0 | 13.8 | 10.2 | 40.4 | 22.6 | 14.3 | 85.0 | 50.0 | 32.9 |
| LLC | 15.6 | 9.4 | 6.9 | 22.2 | 12.3 | 6.5 | 56.6 | 25.9 | 14.4 | |
Note
- The table reports the rejection frequency (in %) of a 5% test for a panel unit root. “This paper, no PW” refers to the common point optimal (CPO) tests proposed in this paper with no pre-whitening used when estimating long-run variances and c = 1. “This paper, PW” refers to the CPO tests in this paper with pre-whitening when estimating long-run variances. “MPP (2007)” refers to the CPO tests with c = 1 in MPP that do not allow for serial correlation. “IPS” is the t-bar test of Im et al. (2003) and “LLC” is the test of Levin et al. (2002).
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| 100 | 250 | 500 | 100 | 250 | 500 | 100 | 250 | 500 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| White noise | This paper, no PW | 41.6 | 49.3 | 52.9 | 53.0 | 62.6 | 62.8 | 66.4 | 75.2 | 71.7 |
| This paper, PW | 43.2 | 54.8 | 52.5 | 45.1 | 62.2 | 62.6 | 40.5 | 73.0 | 73.4 | |
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MPP (2007) | 50.5 | 55.1 | 54.1 | 59.4 | 66.4 | 62.5 | 76.7 | 78.7 | 74.6 |
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IPS | 13.2 | 18.1 | 12.9 | 16.8 | 17.5 | 16.9 | 21.6 | 18.6 | 21.3 |
| LLC | 3.7 | 1.9 | 1.1 | 5.3 | 3.3 | 2.8 | 5.7 | 3.8 | 3.1 | |
| Positive AR | This paper, no PW | 42.9 | 51.8 | 45.3 | 54.8 | 62.8 | 63.6 | 63.7 | 69.5 | 73.4 |
| This paper, PW | 48.8 | 53.1 | 48.0 | 59.5 | 64.1 | 62.2 | 63.5 | 73.5 | 74.8 | |
 |
MPP (2007) | 52.0 | 54.5 | 45.4 | 62.4 | 60.7 | 63.9 | 72.2 | 73.1 | 73.7 |
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IPS | 12.0 | 13.0 | 16.1 | 12.4 | 17.2 | 14.8 | 21.3 | 19.5 | 20.7 |
| LLC | 3.1 | 2.3 | 1.8 | 3.5 | 2.7 | 1.6 | 6.0 | 5.2 | 3.5 | |
| Negative AR | This paper, no PW | 42.8 | 48.3 | 51.3 | 53.2 | 60.3 | 61.1 | 63.3 | 72.0 | 72.4 |
| This paper, PW | 45.0 | 52.5 | 51.4 | 47.2 | 59.8 | 61.0 | 40.0 | 69.6 | 76.3 | |
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MPP (2007) | 48.4 | 47.8 | 50.5 | 62.9 | 59.6 | 60.6 | 70.1 | 71.5 | 74.7 |
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IPS | 13.0 | 14.6 | 14.4 | 19.7 | 16.1 | 15.8 | 22.0 | 20.0 | 19.0 |
| LLC | 5.2 | 2.6 | 1.3 | 6.1 | 2.3 | 2.4 | 7.9 | 4.6 | 3.2 | |
| Positive MA | This paper, no PW | 44.2 | 48.0 | 51.3 | 52.8 | 63.2 | 59.8 | 63.6 | 72.8 | 73.0 |
| This paper, PW | 47.6 | 51.8 | 53.5 | 59.0 | 64.5 | 61.2 | 58.4 | 74.7 | 74.8 | |
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MPP (2007) | 52.2 | 54.0 | 50.9 | 61.3 | 66.4 | 64.2 | 72.3 | 75.8 | 75.2 |
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IPS | 10.4 | 12.7 | 12.1 | 14.3 | 16.4 | 17.4 | 17.9 | 20.8 | 18.7 |
| LLC | 2.6 | 1.9 | 1.1 | 4.6 | 2.7 | 1.9 | 5.5 | 4.0 | 3.4 | |
| Negative MA | This paper, no PW | 41.4 | 49.4 | 47.5 | 54.8 | 55.5 | 59.9 | 65.9 | 72.7 | 77.9 |
| This paper, PW | 46.7 | 51.8 | 48.7 | 43.0 | 57.6 | 62.2 | 32.6 | 68.1 | 77.8 | |
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MPP (2007) | 48.6 | 48.7 | 46.3 | 57.9 | 57.2 | 58.4 | 68.8 | 70.0 | 72.1 |
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IPS | 15.0 | 16.7 | 14.2 | 21.2 | 15.3 | 18.8 | 23.9 | 26.7 | 22.5 |
| LLC | 8.1 | 2.6 | 1.6 | 8.5 | 4.3 | 3.7 | 11.8 | 6.7 | 3.6 | |
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| 100 | 250 | 500 | 100 | 250 | 500 | 100 | 250 | 500 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| White noise | This paper, no PW | 0.0 | 0.1 | 0.5 | 0.0 | 0.6 | 1.5 | 0.0 | 0.5 | 0.9 |
| This paper, PW | 1.0 | 1.0 | 1.4 | 1.8 | 2.3 | 2.7 | 1.4 | 4.1 | 4.5 | |
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MPP (2007) | 1.7 | 1.3 | 1.4 | 4.6 | 2.5 | 2.9 | 7.2 | 5.1 | 4.4 |
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IPS | 8.2 | 5.5 | 5.2 | 10.6 | 5.6 | 4.4 | 14.9 | 7.5 | 6.4 |
| LLC | 6.4 | 4.5 | 2.1 | 5.8 | 4.3 | 1.1 | 4.8 | 4.4 | 0.5 | |
| Positive AR | This paper, no PW | 0.0 | 0.0 | 0.5 | 0.0 | 0.4 | 1.5 | 0.0 | 0.2 | 1.2 |
| This paper, PW | 0.8 | 0.4 | 1.0 | 1.6 | 2.2 | 2.2 | 2.1 | 2.1 | 2.0 | |
 |
MPP (2007) | 0.0 | 0.0 | 0.1 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
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IPS | 5.3 | 3.5 | 4.4 | 4.2 | 3.0 | 4.0 | 3.8 | 2.8 | 3.3 |
| LLC | 3.7 | 2.7 | 1.7 | 2.0 | 2.6 | 0.9 | 0.7 | 1.2 | 0.2 | |
| Negative AR | This paper, no PW | 0.0 | 0.1 | 0.5 | 0.0 | 0.2 | 1.0 | 0.0 | 0.3 | 1.3 |
| This paper, PW | 0.4 | 0.6 | 0.7 | 0.5 | 1.2 | 1.9 | 0.2 | 1.7 | 2.1 | |
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MPP (2007) | 22.2 | 19.8 | 20.3 | 61.7 | 58.4 | 56.8 | 99.3 | 99.3 | 99.1 |
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IPS | 25.3 | 10.7 | 7.1 | 40.6 | 15.6 | 8.9 | 84.1 | 31.7 | 13.7 |
| LLC | 16.9 | 7.6 | 2.8 | 24.5 | 9.4 | 1.6 | 53.8 | 14.5 | 0.5 | |
| Positive MA | This paper, no PW | 0.0 | 0.2 | 0.5 | 0.0 | 0.6 | 1.2 | 0.0 | 0.2 | 1.8 |
| This paper, PW | 1.0 | 1.3 | 1.2 | 3.7 | 3.4 | 2.6 | 6.5 | 6.5 | 5.3 | |
 |
MPP (2007) | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.0 | 0.0 | 0.0 | 0.0 |
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IPS | 8.1 | 6.6 | 6.0 | 10.7 | 7.2 | 6.7 | 12.4 | 11.4 | 8.4 |
| LLC | 5.5 | 5.5 | 1.7 | 5.6 | 5.9 | 1.8 | 2.6 | 7.0 | 0.6 | |
| Negative MA | This paper, no PW | 0.0 | 0.1 | 0.4 | 0.0 | 0.3 | 1.1 | 0.0 | 0.4 | 0.9 |
| This paper, PW | 0.1 | 0.7 | 0.9 | 0.4 | 2.4 | 2.9 | 0.2 | 4.0 | 3.9 | |
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MPP (2007) | 31.7 | 29.1 | 29.7 | 77.1 | 74.8 | 73.3 | 100.0 | 100.0 | 100.0 |
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IPS | 48.2 | 23.5 | 15.3 | 77.6 | 42.2 | 26.6 | 99.9 | 86.0 | 63.0 |
| LLC | 38.1 | 17.7 | 6.1 | 61.1 | 30.1 | 7.0 | 96.6 | 67.8 | 12.2 | |
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| 100 | 250 | 500 | 100 | 250 | 500 | 100 | 250 | 500 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| White noise | This paper, no PW | 10.8 | 17.0 | 15.8 | 12.9 | 17.3 | 17.2 | 17.5 | 23.2 | 26.1 |
| This paper, PW | 15.6 | 17.6 | 16.2 | 14.2 | 20.5 | 19.3 | 19.9 | 27.9 | 28.2 | |
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MPP (2007) | 18.6 | 19.1 | 15.6 | 17.7 | 20.9 | 19.4 | 28.1 | 27.5 | 28.9 |
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IPS | 10.5 | 10.1 | 10.1 | 9.8 | 12.8 | 12.7 | 13.9 | 14.0 | 11.9 |
| LLC | 8.5 | 6.8 | 4.9 | 9.2 | 7.9 | 5.1 | 11.6 | 8.4 | 5.8 | |
| Positive AR | This paper, no PW | 11.0 | 15.6 | 16.4 | 10.9 | 18.1 | 19.5 | 18.1 | 26.0 | 27.5 |
| This paper, PW | 17.1 | 18.7 | 16.5 | 18.8 | 22.3 | 19.8 | 26.2 | 29.9 | 29.5 | |
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MPP (2007) | 17.7 | 16.8 | 15.4 | 18.7 | 20.3 | 19.3 | 27.6 | 27.7 | 28.6 |
 |
IPS | 8.5 | 9.9 | 8.7 | 11.1 | 13.5 | 11.3 | 12.1 | 12.6 | 13.3 |
| LLC | 6.2 | 6.6 | 3.9 | 8.5 | 7.0 | 6.0 | 8.6 | 5.6 | 5.0 | |
| Negative AR | This paper, no PW | 10.7 | 15.1 | 15.2 | 14.1 | 16.8 | 18.0 | 15.8 | 23.3 | 23.8 |
| This paper, PW | 13.9 | 16.6 | 16.9 | 16.7 | 20.1 | 19.0 | 20.1 | 28.6 | 25.9 | |
 |
MPP (2007) | 16.5 | 16.6 | 16.3 | 21.8 | 17.5 | 19.2 | 26.5 | 24.3 | 27.0 |
 |
IPS | 12.0 | 9.6 | 10.4 | 11.2 | 9.9 | 10.9 | 11.9 | 10.9 | 13.8 |
| LLC | 10.7 | 7.3 | 4.6 | 10.7 | 6.1 | 5.5 | 11.0 | 7.5 | 6.8 | |
| Positive MA | This paper, no PW | 12.0 | 14.6 | 18.4 | 11.9 | 18.0 | 22.3 | 19.3 | 25.8 | 30.0 |
| This paper, PW | 20.2 | 17.9 | 18.3 | 18.6 | 21.2 | 23.8 | 25.8 | 30.4 | 31.0 | |
 |
MPP (2007) | 19.0 | 16.2 | 19.3 | 19.4 | 20.7 | 24.4 | 32.1 | 28.1 | 30.3 |
 |
IPS | 9.1 | 11.2 | 10.3 | 9.9 | 11.2 | 11.0 | 14.0 | 12.6 | 16.0 |
| LLC | 8.3 | 7.7 | 6.0 | 7.8 | 8.0 | 5.1 | 11.5 | 7.0 | 6.9 | |
| Negative MA | This paper, no PW | 10.7 | 16.3 | 18.0 | 13.0 | 19.3 | 22.0 | 15.0 | 23.2 | 29.6 |
| This paper, PW | 14.3 | 16.7 | 17.5 | 11.8 | 18.7 | 20.1 | 12.2 | 23.7 | 29.4 | |
 |
MPP (2007) | 14.0 | 14.6 | 16.3 | 18.1 | 18.3 | 21.2 | 24.1 | 23.1 | 27.7 |
 |
IPS | 9.9 | 9.4 | 10.2 | 11.2 | 12.2 | 11.7 | 15.5 | 12.6 | 14.2 |
| LLC | 11.1 | 6.7 | 5.2 | 11.5 | 8.3 | 5.5 | 12.7 | 7.6 | 6.9 | |
In general, we see that our CPO test is conservative. This is especially true in the incidental trends case. The test is better behaved without pre-whitening in estimating the long-run variance in the incidental intercepts case, but the reverse is true in the trends case. It is evident that the original MPP test is not robust to the presence of serial correlation with size that can be as low as 0 or as high as 1, and that the extensions proposed here are therefore needed.
Table 2 reports results for size-adjusted power in the intercept case. Size adjustment is needed, given some of the large size distortions reported in Table 1. We see that the size-adjusted power of the CPO tests robust to serial correlation is typically lower than that of uncorrected tests, but the difference becomes smaller as N and T increase, as predicted by theory because the tests have the same asymptotic distribution. Also, we see that these tests have much higher size-adjusted power than either the LLC or IPS tests. The LLC test has poor power because it corrects for bias by adjusting the numerator of the pooled OLS estimator, as pointed out by Moon and Perron (2008) and Breitung and Westerlund (2013).3
Table 4 presents size-adjusted power for the incidental trends case. Note that the alternative considered in this scenario is further from the unit root null than in Table 2 because of the different definition of local neighbourhoods. While the CPO tests have lower power in this case, the same conclusions remain as in the intercept case.
4. Conclusion
In this paper we develop generalizations of the point-optimal panel unit root tests of MPP to cover the case where the error term is serially correlated. The resulting statistics have two simple modifications relative to those in MPP. First, the variance of the errors is replaced by the long-run variance. Second, the centring of the statistic is adjusted to accommodate the second-order bias induced by the correlation between the error and lagged values of the dependent variable. Simulations show that these two adjustments lead to appropriately sized tests in most cases.
Acknowledgements
We thank Vanessa Smith for raising with us questions about the performance of the original point-optimal statistics in MPP when there are serial correlated errors and about the need for possibly different correction factors in that case. B. Perron acknowledges financial support from the SSHRC and FQRSC. P. C. B. Phillips acknowledges partial support from the National Science Foundation under Grant Nos. SES-0956687 and SES-1258258.
is made for convenience, and could be relaxed at the cost of stronger moment conditions. It is also convenient to assume that 
are identically distributed, and this assumption could be relaxed as long as cross-sectional averages of the moments 
have well-defined limits.
statistic as equivalent to the LLC statistic, while they should have labelled their 
statistic (which has lower power) as equivalent to the LLC statistic.APPENDIX A:: PROOFS OF THE APPROXIMATIONS IN THEOREMS 2.1, 2.3 AND 2.5
We provide three appendices. Here, in Appendix A, we provide proofs of Theorems 2.1, 2.3 and 2.5 that approximate the Gaussian log-likelihood ratio statistic. In Appendix B, we provide sketches of the proofs of the limit distribution results in Theorems 2.2, 2.4 and 2.6. In Appendix C, we provide a heuristic proof of Theorem 2.7. We only provide sketches of the proofs in Appendices B and C because the details are similar to those of the corresponding theorems in MPP and can be established with only minor modifications. Throughout the appendices, M denotes a generic (finite) constant.
Proof of Theorem 2.1.Here, 
. Let Assumption 2.1 hold and 
. By definition, we can write
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
, we deduce that
Proof of Theorem 2.3.Here, 
. Let Assumption 2.1 hold and 
. By definition, we have
(A.6)
such that 
for some constant M. The proof of (A.6) is available in the following subsection. 
Proof of Theorem 2.5.Here, 
. Let Assumption 2.1 hold and 
. By definition, we have
(A.7)
such that 
for some constant M. Then, we have
A.1 Supplementary results
A.1.1. A useful lemma
(where 
denotes the maximum eigenvalue), 
and 
(where 
is the 
th element of A). It is well known that
is 
. Let 
be the 
matrix whose 
element 
is 
if 
and zero, if 
. Let
(A.8)Lemma A.1.Let Assumption 2.1 hold. Then, 
for some constant M.
Proof.For the desired result, we show
under Assumption 2.1 and by the triangle inequality.First, we show that 
. Define
, we have
, the last inequality uses 
and 
and the equality uses the Taylor representation of the exponential function. Then,
and 
and the last inequality holds by the mean-value theorem and 
. Then,
. Finally, we have
, we have the required result
follows in a similar fashion and is omitted. 
Proof of (A.5).We prove the required result when 
and 
. Because
(A.9)
(A.10)For A.9, we follow similar arguments used in proving 
on p. 831 (in the proof of Lemma A2) of Elliott et al. (1996), and have for some constant M
and by Lemma A.1, and the last inequality holds because 
For A.10, we also follow similar arguments to those used in proving 
on p. 831 (in the proof of Lemma A2) of Elliott et al. (1996), and have for some constant M
Proof of (A.4).We prove the required result when 
and 
. By replacing 
in 
with 
, we can decompose 
as
, we can show that
, we show
(A.11)
, the desired result follows. Because 
, we have
, we have
, we can bound
, as required. 
A.1.2. More preliminary results
In this section, 
and let Assumption 2.1 hold. Define 
to be the 
matrix whose 
th element is 
, where 
is defined in Assumption 2.1.
. Direct calculations show that
(A.12)
is the tth element of the vector x and 
Lemma A.2.(a) 
for all k; (b) 
for some finite constant M.
Proof.Part (a). By definition, for 
, we have
Part (b). By definition,
Lemma A.3.Suppose that 
and 
are 
vectors such that 
is bounded, where 
is the (t)th element of 
. Then, (a) 
; (b) 
, where 
is defined in A.8.
Proof.Part (a). The proof is similar to that of Lemma A1 of Elliott et al. (1996) and is omitted.
Part (b). We replace 
in the proof of Lemma A.1 with 
. Then, the required result follows if we show
For Part (b1), by definition, we have
For Part (b2), we have
, we define
to be the 
th element of 
and 
to be the kth element of 
, where 
. Similarly, we define 
and 
.
Lemma A.4.Under Assumptions 2.1, the following hold: (a) 
; (b) 
, 
; (c) 
, 
; (d) 
, 
; (e) 
; (f) 
; (g) 
.
Proof.Part (a). A direct calculation shows that 
. We bound 
by
Part (b) is an immediate corollary of Part (a).
Part (c). First, note that under Assumption 2.1 we have
. It follows immediately that
Part (d). The desired result 
follows from (A.12) and by direct calculation. For the second desired result, note that 
. First, 
since 
by 
and by Part (g), which we prove later. Next, by definition,
, where 
is defined above Lemma A.1, we have
Part (e). Note that
Part (f) follows by Lemma A.3(b2).
Part (g). By definition, we have
where 
is defined above Lemma A.1, and 
is defined in A.8 :
and 
we have the desired result for Part (g). 
Proof of (A.6).The required result follows if we show
. The second term is bounded by
, and the last equality holds because 
. Combining these two, we have the required result
Proof of (A.7).The required result follows, if we show
- Step 1. We show
- Step 2. By (A.6), we have
- Step 3. We show
Proof of Step 1.Note that because 
is a diagonal matrix,
Proof of Step 3.We show
follows in similar fashion and we omit the derivation. Note that
. For the second term, note that
. Then, we have all of the desired results for Part (c). 
APPENDIX B:: PROOFS OF THE LIMIT DISTRIBUTION RESULTS, THEOREMS 2.2, 2.4 AND 2.6
In this section, we provide proofs of Theorems 2.2, 2.4 and 2.6. These proofs are very similar to the proofs of the corresponding results in MPP and therefore we provide just an outline of the proofs here.
Proof of Theorem 2.2.Because 
, we can write
Proof of Theorem 2.4.For the required result of the theorem, it is enough to show that
. Then, 
, and we can rewrite 
as
with 
, which proves the desired result. 
Proof of Theorem 2.6.The required result is a consequence of Lemmata A.5 and A.6. 
Lemma A.5.Let Assumption 2.1 hold. Then, as 
with 
, we have
Proof.The proof is similar to the proof of Lemma 11 of MPP and is omitted. 
with 
the following hold:
Proof.The proofs of Parts (b) and (c) are similar to those of Lemma 12(b) and (c) of MPP, and are omitted.
Part (a). First, note from
we have
APPENDIX C:: PROOF OF THEOREM 2.7
Proof of Theorem 2.7.We provide a sketch of the proof. Note that under Assumption 2.2, the following hold:
and 
Then,
under Assumption 2.1, we have
satisfies the properties in Lemmata 8, 10 and 14 of MPP, while 
and 
satisfy the properties in Lemmata 8(a), 8(b), 10(a) and 14(a)–(d) of MPP. The desired results follow by similar arguments to those used in Theorems 8, 10 and 15 of MPP. 
