Local Asymptotic Equivalence of the Bai and Ng (2004) and Moon and Perron (2004) Frameworks for Panel Unit Root Testing

Local Asymptotic Equivalence of the Bai and Ng (2004) and Moon and Perron (2004) Frameworks for Panel Unit Root Testing

Testing for unit roots is an important aspect of time series and panel data analysis.\footnote{See, for example, the textbook Choi2015 for an overview.} The presence of unit roots not only determines how to proceed for correct statistical inference but can also have serious policy implications. A well-known problem with univariate unit roots tests is their low power. In the last two decades, increased data availability led to the development of panel unit root tests that increase the statistical power by exploiting the cross-sectional data dimension.

The “first generation’’ of panel unit root tests does not allow for cross-sectional dependence, i.e., panel units are assumed to be independent of each other.\footnote{See, for example, the surveys Banerjee1999, BaltagiKao2000, Choi2006, BreitungPesaran2008, and WesterlundBreitung2013. Local asymptotic powers of first generation tests have been considered in, for example, Breitung2000, and Madsen2010. A large scale Monte Carlo study to assess finite-sample powers was conducted in HlouskovaWagner2006.} For many, if not most, empirical applications, however, the assumption of cross-sectional independence is not only empirically hard to justify but has also non-trivial implications for the properties of test statistics. In fact, as shown in OConnell1998 and Gutierrez2006, the dependence between cross-section units can compromise the validity of “first generation tests”. For this reason, a “second generation” of tests, which are also valid in case of cross-sectional dependence, gained a foothold in the literature.\footnote{Popular second generation tests are proposed in, among others, PhillipsSul2003, BaiNg2004,BaiNg2010, MoonPerron2004, BreitungDas2005 (BreitungDas2005, BreitungDas2008), Pesaran2007 and PesaranSmithYamagata2013.}

This paper reconsiders the two leading second generation classes of data generating processes, namely the PANIC framework proposed in BaiNg2004 and the framework proposed in MoonPerron2004, henceforth MP. Both setups allow for cross-sectional dependence through common, unobserved factors. MP uses an autoregressive structure with the factors appearing in the innovations (errors). For PANIC, the factors are part of the “mean specification”��. Consequently, the PANIC framework allows for non-stationarity generated by the factors and for non-stationarity generated by the idiosyncratic components. This is in contrast to the MP framework, for which the factors and the idiosyncratic components have the same order of integration. As BaiNg2010, PesaranSmithYamagata2013 and Westerlund2015, this paper will focus on testing for unit roots in the idiosyncratic components.

This paper offers four main contributions. Firstly, our results imply that for all tests (satisfying a mild regularity condition) it suffices to determine the asymptotic size and local power in one of the frameworks, since the same behaviour automatically holds for the other one.\footnote{For unit-root testing, it thus is irrelevant to make a specific choice for the data generating process. This is good news for the practitioner, who no longer must decide between two competing frameworks that are typically hard to distinguish based on finite samples.} Previous papers are based on either MP or PANIC for the construction of test statistics. However, using our first main result, the (same) local asymptotic power function is automatically obtained for the other framework as well.\footnote{In particular, we use the first main result to show that the unit root tests proposed in MoonPerron2004 are equivalent (in terms of asymptotic size and power) to the tests proposed in BaiNg2010. A first study on the comparison of the behavior of these tests is present in Bai and Ng (2010), but, to our best knowledge, the equivalence has not been observed before. } These conclusions are obtained by showing that the PANIC and MP experiments are both Locally Asymptotically Normal (LAN) with the same central sequence and Fisher information.\footnote{This means that the limit experiment (in the Le Cam sense) is a Gaussian shift experiment; see, for example, vdVaart2000. For unit root problems in (univariate) time series, limit experiment theory has been exploited by, amongst others, Jansson2008 and BDvdA. }

Secondly, exploiting the general theory for LAN experiments we easily obtain the local asymptotic power envelope, which is, in view of the first main result, the same for PANIC and MP. This result extends the work by MoonPerronPhillips2007, BDvdA, MPP2014, and JuodisWesterlund2018 on first generation frameworks, to the second generation. It turns out that the level of the local asymptotic power envelope only depends on the (local) deviation to the unit root. The level of the power envelope is thus not affected by the nuisance parameters. \footnote{Westerlund (2015) observed that the local asymptotic power of the tests considered in that paper do depend on the presence of serial and cross-sectional dependence (see Remark 2 in that paper). Consequently, these tests are not globally optimal.} We also provide a new derivation, using our LAN-result, of the local asymptotic power of the popular MoonPerron2004 and BaiNg2010 tests.\footnote{Westerlund2015 derived, via “triangular array asymptotics”, the local asymptotic power function of the tests proposed in BaiNg2010. Using our LAN-result we provide a short derivation at the expense of slightly less general conditions.} A comparison of the power functions to the power envelope shows that these original tests are only optimal in case there is no heterogeneity in the long-run variances of the idiosyncratic components.

Thirdly, we propose a new test that is asymptotically uniformly most powerful irrespective of possible heterogeneity in the long-run variance of the idiosyncratic components. Our test is motivated by our expansion of the likelihoods, underlying the LAN results, and its optimality is easily proved by exploiting LAN-theory. Compared to the tests proposed in MoonPerron2004 and BaiNg2010, the size of the power gains depends on the amount of heterogeneity in the long-run variances of the idiosyncratic components. We report numerical asymptotic powers for commonly encountered amounts of heterogeneity and use Monte-Carlo experiments to show that the new test compares favorably also in finite samples.

Finally, to obtain the LAN result for the PANIC case, we first show that the model in which we observe both the panel units and the common factors is equivalent to that where the factors are unobserved. This in contrast to other data generating processes, used in the literature on panel unit roots, where observing factors or correlated covariates does yield additional power; see, for example, PesaranSmithYamagata2013, Becheri:2015gi, and JuodisWesterlund2018. Moreover, for both the MP and PANIC framework, our results imply that the local asymptotic power envelope for the setting in which all nuisance parameters (this includes incidental intercepts, factor loadings, and coefficients of the linear filters generating serial dependence) are known, can be attained. In other words, we demonstrate that we are in an adaptive setting.

To obtain the local and asymptotic equivalence of the MP and PANIC frameworks, we need to impose some restrictions. First, we assume that the driving innovations are Gaussian. Second, we impose the deviations to the unit root, under the alternative hypothesis, to be the same for all panel units. And third, we do not allow for (incidental) trends. The Gaussianity facilitates a relatively easy proof of the LAN-result and it seems to be rather difficult to generalize this assumption; even for first-generation frameworks no results are available yet. Nevertheless, for the proposed asymptotically uniformly most powerful test, we stress that Gaussianity is not required to obtain asymptotic size, implying validity of the test in a non-Gaussian environment. In view of BDvdA we do not expect that a generalization to random deviations from the unit root, under the alternative hypothesis, would affect our main results. The Monte Carlo results seem to confirm this conjecture for finite-samples. Allowing for incidental trends leads to a different convergence rate (see MoonPerronPhillips2007) and thus requires a new asymptotic expansion of the log-likelihood ratios, which is beyond the scope of this paper.

The paper is organized as follows. Section (ref) presents the model and assumptions. Section (ref) derives the common approximation to the local likelihood ratios in the two experiments and derives its limiting distribution. (ref) introduces our new UMP test based on the limit experiment. (ref) computes the local asymptotic power functions of the tests proposed in MoonPerron2004 and BaiNg2010 and (ref) compares their asymptotic and finite-sample power to those of the new UMP test. (ref) concludes. All proofs are organized in several appendices.

Setup and Assumptions

Data-generating processes

We consider observations $Z_{it}$, $i=1,\dots,n$ and $t=1,\dots,T$, generated by the components specification

with $\lambda_{ki}$ the loading of (unobserved) factor $\{F_{kt}\}$ on panel unit $i$, and $K\in\mathds{N}$ being the fixed number of factors. The $m_i$ are fixed effects and we assume zero starting values: $E_{i0}=0$ and $F_{k0}=0$.\footnote{BaiNg2004 assume that the initial values are bounded while MoonPerron2004 assume zero starting values for the $\varepsilon_{it}$ (see (ref)). We refer to Section 6.2 in MoonPerronPhillips2007 for a discussion on why relaxing initial conditions can be problematic in a panel context and do not pursue this issue further, except by noting that our tests are invariant with respect to the $m_i$.} The assumptions on the innovations $\eta_{it}, f_{kt}$ and factor loadings $\lambda_{ki}$ are discussed in (ref) below. This setup covers the most widely used setups for second-generation panel unit root tests: for $\rho_k=1, k=1,\dots,K,$ we obtain the PANIC framework of BaiNg2004 (`PANIC') and with $\rho_k=\rho, k=1,\dots,K,$ we obtain the framework of MoonPerron2004 (`MP'), in which we can also rewrite the DGP as

In both frameworks, the hypotheses will be phrased in terms of $\rho$.

Matrix notation

To write the model in matrix form we need some additional notation. We write $I_n$ and $I_T$ for identity matrices of dimension $n$ and $T$, respectively, while $\iota$ denotes a $T$-vector of ones. Introduce the $n$-vectors $\lambda_{k}=(\lambda_{k1},\dots,\lambda_{kn})^\prime$, $k=1,\dots,K$ and the $n\times K$ matrix $\Lambda=(\lambda_{1},\dots,\lambda_{K})$. Collect the observations as $Y= (Y_{11},Y_{12}, \dots,Y_{1T},\dots,Y_{n1},\dots,Y_{nT})^\prime$. We also write $Y_{-1} =(Y_{10},Y_{11}, \dots,Y_{1,T-1},\dots,Y_{n0},\dots,Y_{n,T-1})^\prime$, $\Delta Y = Y-Y_{-1}$, and define $\varepsilon$, $\eta$, $E$, $E_{-1}$, $\Delta E$, $Z$, $Z_{-1}$, and $\Delta Z$ analogously. Write $m=(m_1,\dots,m_n)^\prime$, $\eta_i = (\eta_{i1},\dots,\eta_{iT})^\prime$, $i=1,\dots,n$, $f_k =(f_{k1},\dots,f_{kT})^\prime$, $k=1,\dots,K$, and denote their corresponding covariance matrices by $ \Sigma_{f ,k}=\operatorname{var}_{}f_k\in\mathds{R}^{T\times T}$ and

The long-run variances of $\{f_{kt}\}$ and $\{\eta_{it}\}$ are denoted by $\omega_{f,k}^2$ and $\omega_{\eta,i}^2$, respectively. In addition, we define the approximate long-run variances $\omega_{f,k,T}^2= \iota'\Sigma_{f k} \iota/T$ and $\omega_{\eta,i,T}^2= \iota'\Sigma_{\eta ,i} \iota/T$. For a given $T$, these ignore the contribution of any autocovariances further than $T$ apart. We will use the approximate long-run variances to simplify notation and the structure of our proofs. We add the subscript T to the approximate versions to emphasize the difference and define

In addition to this `vectorized' notation, it will also be useful to consider the observations as $T\times n$ matrices. Thus, let $\tilde \eta=(\eta_1,\dots,\eta_n)$, and define $\tilde \varepsilon$, $\tilde Y$, $\tilde Z$, $\tilde E$, $\tilde f=(f_1,\dots,f_K)$, and $\tilde F$ analogously. With this notation, (ref) can be rewritten as

while for the vectorized versions we have

Finally, we introduce the $T \times T$ matrix $A$ by $A_{st}:=1$ if $s>t$ and $0$ otherwise and we put $\mathcal{A}:= I_n\otimes A\in\mathds{R}^{nT\times nT}$, i.e.

The matrix $A$ can be considered a cumulative sum operator and premultiplying the vectorized panel with $\mathcal{A}$ takes the cumulative sum in the time direction for each panel unit, i.e., we have $\tilde Y_{-1}=A \Delta \tilde Y$ and $Y_{-1}=\mathcal{A} \Delta Y$. It is also related to `approximate one-sided long-run variances', which we can define by $\delta_{\eta,i,T}=\operatorname{tr}[ A\Sigma_{\eta,i}/T]$ and $\delta_{f,k,T} =\operatorname{tr}[A\Sigma_{f,k}/T]$. Note $A+A'= \iota\iota'-I_T$, so that, analogous to the long-run variances, we have $2\delta_{\eta,i,T} =\omega_{\eta,i,T}^2 -\gamma_{\eta,i}(0)$.

Assumptions

Now we can formally state the full specifications of our DGPs (ref). The distributional assumptions on the time series of the factors $\{f_{kt}\}$ and idiosyncratic shocks $\{\eta_{it}\}$ are given in (ref) and we formulate the assumptions on the (deterministic) factor loadings $\lambda_{ki}$ in (ref). (ref) specifies the joint asymptotics we consider in this paper. Finally, (ref) differentiates between the two setups discussed in (ref).

As already announced, we also need to impose some stability on the factor loadings $\lambda_{ki}$, which we assume to be fixed. (ref) is standard in the literature, c.f.\ Assumption A in BaiNg2004 or Assumption 6 in MoonPerron2004. It is commonly referred to as the factors being `strong'.

(ref) below specifies the asymptotic framework we consider throughout this paper. We follow MoonPerron2004, BaiNg2010, and Westerlund2015 in considering large `macro panels', where both $n$ and $T$ go to infinity, but $T$ will be the larger dimension. We derive all our results using joint asymptotics, which yields more robust results than taking sequential limits where first $T\to\infty$ and subsequently $n\to\infty$.

Finally, (ref) below specifies that we either operate in the PANIC (case (ref)) or in the MP (case (ref)) framework. In the PANIC framework, we allow the long-run variance of the factor innovations to be zero, so that we consider both integrated and and stationary factors. This is ruled out in the MP case to enforce that the factors have the same order of integration as the idiosyncratic parts.

Limit Experiment and Power Envelope

In this section we show that the likelihood ratios related to the unit root hypothesis, for the MP and for the PANIC framework, exhibit the same local asymptotic expansion. Both experiments are proved to be Locally Asymptotically Normal (LAN) with the same central sequence and (asymptotic) Fisher information. This result allows us to treat the two setups jointly and to obtain three main results. Firstly, we derive the asymptotic power envelopes. Secondly, in (ref) we obtain asymptotically optimal, feasible tests. Thirdly, the LAN result allows us to show that any test, satisfying a mild regularity condition, has the same, perhaps nonoptimal, local asymptotic power function under both data generating processes.

We phrase our hypotheses about $\rho$ in (ref) using the local parameterization

As shown below, these rates lead to contiguous alternatives, which allow us to obtain the (local) power of our tests. The unit root hypothesis can be reformulated in terms of the “local parameter” $h$:

In both setups, we start by considering the likelihood ratio for observing $Z_{it}$ in case $\rho$ is the only unknown parameter. Hence, the number of factors $K$, the factor loadings $\lambda_{ki}$, the autocovariance functions, and the fixed effects $m_i$ are considered as known in this section. We will first show, for each model separately, that its likelihood ratio satisfies an expansion, under the null hypothesis, of the form $\log \mathrm{d} \mathrm{P}_{h,n,T}/\mathrm{d} \mathrm{P}_{0,n,T}= h\Delta_{n,T} - h^2 J/2 + o_P(1) $ with Fisher-information $J=1/2$. In (ref), we consider the limiting distribution of their common central sequence $\Delta_{n,T}$ and will conclude that both experiments enjoy the LAN-property. In (ref) we demonstrate that the conclusions of this section also hold for the model of interest, where all the parameters are unknown, i.e., that the nuisance parameters can be adaptively estimated.

Expanding the likelihood in the PANIC setup

For the PANIC case, for now, consider the factors as known. Just as for the other parameters, we show in (ref) that the resulting likelihood ratio can still be approximated by an observable version (up to a negligible term).\footnote{ The result implies that observing the factors in the PANIC framework will not lead to an increase in power. This contrasts with the situation in the MP setting, for which Becheri:2015gi report higher powers with observed factors and JuodisWesterlund2018 show power gains when observing covariates correlated to the innovations. } Denote the joint law of $F$ and $Z$ under (ref) by $\mathrm{P}_{h,n,T}^{\text{PANIC}}$. Using $\eta\sim N(0,\Sigma_{\eta })$ and $\eta=\Delta E- h E_{-1}/(\sqrt{n}T)$, we obtain the log-likelihood ratio

Note, from (ref), $\Delta\tilde E =\Delta\tilde Y -\Delta\tilde F \Lambda'$ , implying $\Delta E$ is indeed observable in this PANIC framework (with observed factors as considered here). Moreover, under $\mathrm{P}_{0,n,T}^{\text{PANIC}}$, $\Delta E=\eta$. We now show that we can replace variances by long-run variances, to obtain simpler versions of the central sequence and empirical Fisher information.

In the following subsections we show that $\Delta_{n,T}$ also approximates the central sequence in the MoonPerron2004 setup.

Expanding the likelihood in the MoonPerron2004 setup

Let us denote the law of $Z$ under (ref) by $\mathrm{P}_{h,n,T}^{\text{MP}}$. Then the log-likelihood ratio of $\mathrm{P}_{h,n,T}^{\text{MP}}$ with respect to $\mathrm{P}_{0,n,T}^{\text{MP}}$ is given by, using $\varepsilon\sim N(0,\Sigma_\varepsilon)$ and $\varepsilon=\Delta Y- h Y_{-1}/(\sqrt{n}T)$,

In this more complicated model, we simplify the central sequence and also the Fisher information in two steps. The first is analogous to the approximation in the PANIC setup, i.e., we replace variances by long-run variances. Note that thanks to our independence assumptions, the $nT\times nT$ covariance matrix of the $\varepsilon$ can be written as

Replacing $\Sigma_{f ,k}$ by $\omega_{f,k,T}^2 I_{T}$ and $\Sigma_{\eta ,i}$ by $\omega_{\eta,i,T}^2I_{T}$ in (ref) we obtain the simplified versions of central sequence

where the $nT\times nT$ matrix $\Psi_\varepsilon$ is defined by

with $\Omega_\eta=\operatorname{diag}( \omega_{\eta,1,T}^2,\dots,\omega_{\eta,n,T}^2)$ and $\Omega_{F}=\operatorname{diag}(\omega_{f,1,T}^2,\dots,\omega_{f,K,T}^2)$. The following lemma demonstrates that applying these replacements to the central sequence and Fisher information do not affect their asymptotic behavior.

Exploiting the Sherman-Morrison-Woodbury formula we obtain

Note that removing $\Omega_{F}^{-1}$ from (ref) yields a projection matrix corresponding to `projecting out the factors'. Thus, basing a central sequence on such a projection matrix would simplify approximating it based on observables by removing the need to estimate $\Omega_{F}^{-1}$ and, more importantly, by ensuring that the factors are projected out. The next lemma shows that using such a projection version ${{\psi^*_{\varepsilon }}}^{-1}$ of $\psi_{\varepsilon }^{-1}$ in the central sequence does not change its asymptotic behaviour.

Asymptotic normality

Having simplified each framework's central sequence and Fisher information separately, we are now ready to show that they are asymptotically equivalent and the central sequences converge to a normal distribution. We begin this section by showing that the central sequence in the MP framework is asymptotically equivalent to the one in the PANIC framework.

Finally, we consider the weak limit of the central sequence $\Delta_{n,T}$ (and therefore also of $\Delta^*_{n,T}$), showing that both experiments are locally asymptotically normal.

(ref) is an important result as it establishes that the unit root testing problem in both models is locally asymptotically normal, i.e., it is asymptotically equivalent to testing $h=0$ against $h<0$ based on one observation $X\sim N(Jh,J)$. This equivalence prescribes how to perform asymptotically optimal inference and yields the asymptotic local power envelope and the power functions of various test statistics: The asymptotic representation theorem vdVaart2000 implies that in our framework no unit root test can have higher power than the optimal test in the limit experiment. This best test is clearly rejecting for small values of $X$, leading to a power (for a level-$\alpha$ test) of $ \Phi(\Phi^{-1}(\alpha)-J^{1/2}h).$ Thus, with $J=1/2$, this constitutes the power envelope for our unit root testing problems:\footnote{As this section assumes the nuisance parameters to be known, for now we can only present an upper bound on the attainable power. In (ref) we show that the power envelope of (ref) can be attained.}

The above power envelope would be reached by any of our previously introduced central sequences.\footnote{This always holds in LAN experiments and follows from Le Cam's Third Lemma vdVaart2000.} In the next section we show that we can approximate these central sequences based on observables, yielding a feasible test that attains the asymptotic power envelope.

An Asymptotically UMP Test

In the previous section we derived a testing procedure that reaches the power envelope for the unit root testing problem. This test, however, is not feasible when the nuisance parameters are unknown. In this section, we demonstrate how to estimate the nuisance parameters to obtain a feasible version that also attains the power envelope. We provide a feasible version of $\Delta^*_{n,T}$, which is motivated by the likelihood ratio in the MP experiment. As (ref) projects out the factors, basing our feasible version on $\Delta^*_{n,T}$ instead of $\Delta_{n,T}$ spares us the approximation of the idiosyncratic parts.

Recalling our LAN results in (ref) and that the central sequences are asymptotically equivalent across the two setups (see (ref)) it is clear that a feasible version of $\Delta^*_{n,T}$ would be optimal. Therefore, we show that replacing all nuisance parameters with estimates does not change the limiting behavior of $\Delta^*_{n,T}$. Specifically, we need estimates $\hat\Lambda$ of the factor loadings, as well as estimates $\hat{\delta}_{\eta,i}$ and $\hat{\omega}_{\eta,i}^2$ of the (one-sided) long-run variances of each idiosyncratic part. The feasible test statistic is then

Under suitable restrictions on the bandwidth and the kernel, (ref) hold for kernel spectral density estimates; see Remark 2.9 in MPP2014. (ref) is stronger that the results in MoonPerron2004, so we show in (ref) that it indeed holds under our assumptions.

Although (ref) only concerns adaptivity under the null hypothesis $\mathrm{H}_0$, we can use Le Cam's First Lemma to obtain that, thanks to contiguity, also under $\mathrm{P}_{h,n,T}^{\text{MP}}$ or $\mathrm{P}_{h,n,T}^{\text{PANIC}}$, $\hat{\Delta}_{n,T}$ has the same limiting distribution as $\Delta^*_{n,T}$, so that tests based on $\hat{\Delta}_{n,T}$ will be uniformly most powerful. Formally, the size and power properties of our optimal test follow from the following theorem.

Here is one way to obtain the UMP test in practice:

Comparing Powers Across Tests and Frameworks

This section derives the asymptotic powers of commonly used tests in both the MoonPerron2004 and the BaiNg2004 frameworks. We start by formalizing our observation that local powers are equal across the two frameworks.

Once again, our result on the asymptotic equivalence of the two experiments allows us to obtain results for both frameworks at the same time. By demonstrating the joint normality under the null as in (ref) we obtain simple proofs of the powers of commonly used tests in these frameworks, without ever relying on triangular array calculations.

To show the elegance of this approach, we include here the full proof of the first part of this lemma. The second part follows immediately from a more specific version of Le Cam's third lemma, which directly prescribes the desired normal distribution under alternatives. We can use this simple way to obtain powers under local alternatives thanks to our LAN results of (ref).

Before we apply (ref) to derive asymptotic powers, we first describe the relevant test statistics in some detail. We focus on the tests proposed in BaiNg2010 (`BN tests') and MoonPerron2004 (`MP tests'). Following these papers, we denote

all assumed to be positive, and their estimated counterparts

Finally, we define $\omega^4 = (\omega^2)^2$ and $\hat\omega^4 = (\hat\omega^2)^2$.

Both the MP and BN tests rely on a two stage procedure. In the first stage, the unobserved idiosyncratic innovations $E$ are estimated. Subsequently, a pooled regression procedure is used to estimate the (pooled) autoregression parameter. This pooled estimator is then used to construct a $t$-test. The main difference between the MP and the BN procedures lies in the way the idiosyncratic innovations are estimated.

BaiNg2010 propose to estimate the idiosyncratic errors $E$ by the PANIC approach introduced in BaiNg2004, which in turn relies on principal component analysis applied to the differences $\Delta Y_{it}$. Denoting this estimator of $E_{i}$ by $\hat E_{i}$, the BN tests are

is a bias-corrected pooled estimator for the autoregressive coefficients.

The MP tests are based on a different estimator of $\rho$. The idiosyncratic components $E_{i}$ are estimated by projecting the data on the space orthogonal to the common factors. Let $\hat\Lambda$ be a consistent estimators for $\Lambda$ as defined in MoonPerron2004, and $Y_{\cdot,t} = (Y_{1t}, \dots, Y_{nt})^\prime$. Then the MP test statistics are given by

We are now ready to compute the asymptotic behaviour of the MP and BN tests under local alternatives by an application of (ref). The power of the MP tests in the MP framework has been derived in MoonPerron2004 and that of the BN tests in the PANIC framework has been derived in Westerlund2015. Given our LAN result, we can provide simple independent proofs of these results. These rely on the second part of (ref); we demonstrate the required joint asymptotic normality in a supplementary appendix. More importantly, our approach also leads to new results, namely the asymptotic powers of the MP test in the PANIC framework and the asymptotic powers of the BN tests in the MP framework. In fact, those results can be considered an immediate consequence of the first part of (ref) and the existing power results in the literature.

The Cauchy-Schwarz inequality implies $\frac{\omega^4}{\phi^4} \leq 1$, thus Proposition (ref) shows that, in general, the local asymptotic power of the MP and BN tests lies below the power envelope. In fact, they are all asymptotically UMP only when $\frac{\omega^4}{\phi^4} = 1$. This condition is satisfied when the long-run variances of the idiosyncratic shocks $\eta_{it}$ are homogeneous across $i$. The proposed test $t_{\text{UMP}}$ is asymptotically UMP irrespective of possible heterogeneity. In (ref) we assess whether the asymptotic power gains, compared to the MP and BN tests, are also reflected in finite samples for realistic parametric settings.

Simulation results

This section reports the results of a Monte-Carlo study with three main goals: firstly, to assess the finite sample performance of our proposed test $t_{\text{UMP}}$, secondly, to see how the asymptotic equivalence between the MoonPerron2004 and PANIC setups is reflected in finite samples, and, finally, to check the robustness of our results to deviations from our assumptions.

The DGPs

We generate the data from (ref) with $m_i=0$.\footnote{Recall that our tests are invariant with respect to $m_i$.} Using sample sizes $n=25,50,100$ and $T=n,2n,4n$, we simulate both the MP and the PANIC setups. Recall that, for a local alternative $h$, we take $\rho=1+\frac{h}{\sqrt{n}T}$ in both setups. In the MP case we also set $\rho_k=\rho$, whereas in the PANIC case we set $\rho_k=1$ under the null and all alternatives. The factor loadings $\Lambda$ are drawn from a normal distribution with mean $K^{-1/2}$ and covariance matrix $K^{-1}I_K$.\footnote{As done in MoonPerron2004, we scale by $\sqrt{K}$ to ensure the contribution of the factors is comparable across specifications.} Most of the simulations are run with $K=1$ but we also explore what happens with more factors. Throughout this section we assume the number of factors to be known.\footnote{This number can be estimated consistently, so this makes no difference for the asymptotic analysis. See, for example, Section 2.3 in MoonPerron2004 and Section 5 in BaiNg2010 for a discussion of this issue.} For the innovation processes $f_{kt}$ and $\eta_{it}$ we examine Gaussian i.i.d., MA(1), and AR(1) processes. We fix the MA or AR parameter at 0.4 and set the variance such that the long-run variances of the $f_{kt}$ equal one, and the long-run variance of the $\eta_{it}$ is $\omega_i^2$. The $\omega_i^2$ are drawn i.i.d.\ from a lognormal distribution whose parameters are chosen to match different values of $\omega^4/\phi^4$ and a mean of one.\footnote{Recall from (ref) that the asymptotic relative efficiency of the existing tests compared to our UMP test depends on the heterogeneity of the long-run variances and more specifically on the ratio $\omega^4/\phi^4$. Therefore, the sample size at which it becomes worthwhile to estimate the heterogeneous long-run variances (i.e., use the asymptotically UMP tests suggested here) mainly depends on this ratio. We present simulation results for $\sqrt{\omega^4/\phi^4}$ between 0.6 and 1, where lower values indicate more heterogeneity. A cursory look at a few typical applications reveals that these ratios are mostly between 0.6 and 0.8 and match the skewed nature of the lognormal distribution.}

The test statistics

In addition to the tests proposed in (ref), $t_{\text{UMP}}$ and $t_{\text{UMP}}^{\text{emp}}$, we consider the MP tests of MoonPerron2004 and the BN tests of BaiNg2010. However, the powers and sizes of the (MP) $t_b$ and (BN) $P_b$ tests were very similar also in finite samples, so we only report results for $P_b$. We omit the comparison with $P_a$ and $t_a$ since they tend to show large biases in terms of size (see, for example, the Monte Carlo studies in GengenbachPalmUrbain2010 and BaiNg2010).

The sizes of all considered tests are highly sensitive to estimation of the (one-sided) long-run variances. We have considered a variety of methods, for example, using a Bartlett or quadratic spectral kernel and selection of the bandwidth according to the NeweyWest1994 or the Andrews1991 rule with/without various forms of prewhitening. Whereas the differences from using different kernels are small, the selection of both the bandwidth and the prewhitening are essential. Our preferred method employs a Bartlett kernel with prewhitening.\footnote{As in MPP2014, the prewhitening model is selected based on the BIC between four simple ARMA models. } There is a size-power tradeoff between using the Andrews1991 and the NeweyWest1994 bandwidth selection: The Andrews1991 bandwidth leads to higher powers for the smallest sample sizes, but an oversized test when the innovations have a strong MA component. The decision which bandwidth to use thus depends on the preferences of the researcher. In this section, all results are based on the Andrews1991 bandwidth. However, the sizes and powers based on the NeweyWest1994 bandwidth can be found in a supplementary appendix.

Sizes

(ref) reports the sizes of our tests for the baseline DGP based on the Andrews bandwidth. Many other specifications can be found in the supplemental appendix. Recall that the sizes depend considerably on how the long-run variances are estimated. Using the method described above, the sizes of $t_{\text{UMP}}^{\text{emp}}$ reasonable across most DGPs and generally comparable to those of $P_b$. $t_{\text{UMP}}$, on the other hand, is undersized in many specifications, so that we focus on its empirical version $t_{\text{UMP}}^{\text{emp}}$ in the remainder. Only in the MA(1) example, both $t_{\text{UMP}}^{\text{emp}}$ and $P_b$ are oversized ($t_{\text{UMP}}^{\text{emp}}$ is more oversized for the smallest sample sizes and marginally less oversized in the larger ones). Thus, when a strong MA component is suspected, we recommend to use tests based on the NeweyWest1994 bandwidth. Generally, the NeweyWest1994 bandwidth provides better sizes, especially in the MA case. However, small sample powers are slightly lower. Both sizes and powers based on the NeweyWest1994 bandwidth can be found in a supplementary appendix.

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Powers

We start this subsection by investigating the finite-sample differences between the MP and the PANIC setups. Recall that we have shown that the asymptotic, local power functions are the same and that (under some regularity conditions) all tests have the same asymptotic power in the MP framework as they do in the PANIC framework. (ref) compares the powers of $t_{\text{UMP}}^{\text{emp}}$ and $P_b$ across the two frameworks. Indeed, also in small samples the powers are very similar. Moreover, both a larger $n$ and a larger $T$ contribute to reduce the difference. When the factor is stationary under the hypothesis, the difference is considerably smaller still. Noting the small scale on the $y$ axis in these plots, in the remainder we will only present results for the PANIC framework, as the lines would otherwise be mostly indistinguishable.

We now turn to comparing the performance of the UMP tests to existing ones. As discussed in (ref), we need to estimate the individual long-run variance of each idiosyncratic part in order to attain the power envelope. Of course, this becomes easier with a larger time series dimension and is more beneficial when the long-run variances differ substantially between series.

(ref) presents the baseline power results for a medium amount of heterogeneity ($\sqrt{\omega^4/\phi^4}=0.8$). It is evident that even for relatively small samples using the optimal test pays off: except for $n=T=25$, the power of $t_{\text{UMP}}^{\text{emp}}$ is uniformly higher than that of $P_b$.

Next, (ref) presents the power difference between the optimal test and $P_b$ for varying degrees of heterogeneity. As expected, the higher the amount of heterogeneity, the more beneficial it is to use the optimal test, also in finite samples. In the case of perfect homogeneity, the losses from estimating individual long-run variances are minor, except for the $n=T=25$ case.

In the supplemental appendix we investigate the effects of serial correlation and multiple factors. Qualitatively, the power results are not affected by these variations in the DGP. We also consider the robustness of our results to deviations of our assumptions: we consider the power against heterogeneous alternatives and investigate the effects of non-Gaussian innovations.

Conclusion and Discussion

This paper shows that the MP and PANIC frameworks are equivalent, for unit root testing, from a local and asymptotic point of view. Using the underlying LAN-result, the local asymptotic power envelope for the MP and PANIC frameworks readily follows. We show that the tests proposed in MoonPerron2004 and BaiNg2010 only attain this bound in case the long-run variances of the idiosyncratic component are sufficiently homogeneous. We develop an asymptotically uniformly most powerful test; a Monte Carlo study demonstrates that this test also improves on existing tests for finite-samples.

To obtain the local and asymptotic equivalence of the MP and PANIC frameworks, we need to impose some restrictions. First, we assume that the driving innovations are Gaussian. Second, we impose the deviations to the unit root, under the alternative hypothesis, to be the same for all panel units. And third, we do not allow for (incidental) trends. The Gaussianity facilitates a relatively easy proof of the LAN-result and it seems to be rather difficult to generalize this assumption; even for first-generation frameworks no results are available yet. For the proposed asymptotically uniformly most powerful test, we stress that Gaussianity is not required for its validity. In view of BDvdA we do not expect that imposing constant deviations to the unit root, under the alternative hypothesis, affects our main results. The Monte Carlo results seem to confirm this conjecture for finite-samples. To allow for incidental trends the proper strategy seems to be to first determine the maximal invariant (i.e. determine which part of the observations is invariant with respect to the incidental trends), and to analyze if the resulting maximal invariant satisfies a LAN-result (yielding the power envelope). On basis of MoonPerronPhillips2007 we expect that the reduction of the data to the maximal invariant will result in a different localizing rate compared to the situation in which there are no incidental trends. This indicates that the generalization to incidental trends really requires a separate analysis.

References