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Panel stationary tests against changes in persistence

  • Roy Cerqueti
  • Mauro Costantini
  • Luciano Gutierrez
  • Joakim Westerlund
Open Access
Regular Article
  • 598 Downloads

Abstract

In this paper we propose new panel tests to detect changes in persistence. The test statistics are used to test the null hypothesis of stationarity against the alternative of a change in persistence from I(0) to I(1), from I(1) to I(0), and in an unknown direction. The limiting null distributions of the tests are derived and evaluated in small samples by means of Monte Carlo simulations. An empirical illustration is also provided.

Keywords

Persistence Stationarity Panel data 

JEL Classification

C12 C22 

1 Introduction

Over the last two decades, a vast literature has investigated whether economic and financial time series may be characterized by a change in persistence between separate I(1) and I(0) regimes rather than simply I(1) or I(0) behavior. Changes of this kind in macroeconomic variables are well documented; see the literature reviews in Kim (2000) and Leybourne et al. (2003). A non-exhaustive list of the variables for which such phenomena have been observed includes inflation, real output, budgetary deficits, interest rates and exchange rates. Interestingly, while many data sets are in fact panels of multiple time series, the way that existing tests are constructed requires that the series are tested one at a time. This is wasteful in the sense that each time a test is carried out the information contained in the other series is effectively ignored. The current paper can be seen as a reaction to this. The purpose is to develop tests for changes in persistence that explores the multiplicity of series, and that can be seen as panel extensions of the time series tests of Kim (2000), Kim et al. (2002), and Busetti and Taylor (2004). The tests can be used to flexibly test the null hypothesis of stationarity against the alternative of a change in persistence not only from I(0) to I(1), and from I(1) to I(0), but also when the direction is unknown. The data generating process (DGP) considered is quite general. Some of the allowances are unit-specific constant and trend terms, cross-section heteroskedasticity, error serial correlation and cross-section dependence in the form of common factors. The asymptotic distributions of the tests are derived and evaluated in small samples using Monte Carlo simulation. An empirical illustration is also provided showing how how inflation of 20 developed countries has undergone a shift from I(0) to I(1).

The rest of the paper is organized as follows. Sections 2 and 3 present the model, the test statistics, and their asymptotic distributions, which are evaluated using simulations in Sect. 4. Section 5 reports the results from the empirical application. Section 6 concludes. Proofs of important results are provided in the Appendix.

2 Model and assumptions

Consider the panel data variable \(Y_{i,t}\), where \(i = 1,...,N\) and \(t = 1,...,T\) index the time-series and cross-sectional units, respectively. The DGP of this variable is given by
$$\begin{aligned}&\displaystyle Y_{i,t} = \theta _i'D_{t,p}+\lambda _{i}'F_t+ e_{i,t},\end{aligned}$$
(1)
$$\begin{aligned}&\displaystyle e_{i,t} = \mu _{i,t} + \varepsilon _{i,t}, \end{aligned}$$
(2)
where \(D_{t,p} = (1,t,...,t^p)'\) is a p-order trend polynomial such that \(D_{t,p} = 0\) is \(p=-1\), \(F_t\) is an \(r\times 1\) vector of common factors with \(\lambda _{i}\) being the corresponding vector of factor loadings, and \(\varepsilon _{i,t}\) is a mean zero and I(0) error term. The following three specifications of \(\mu _{i,t}\) are considered, where 1(A), \(\lfloor x\rfloor \), \(\eta _{i,t}\) and \(\tau _i^0\in [0,1]\) denote the indicator function of the event A, the integer part of x, a mean zero I(0) error term, and the break fraction, respectively:
  1. MU1.

    I(0) \(\rightarrow \) I(1): \(\mu _{i,t}=\mu _{i,t-1}+1(t>\lfloor T\tau _{i}^0\rfloor )\eta _{i,t}\).

     
  2. MU2.

    I(1) \(\rightarrow \) I(0): \(\mu _{i,t}=\mu _{i,t-1}+1(t \le \lfloor T\tau _{i}^0\rfloor )\eta _{i,t}\).

     
  3. MU3.

    Unknown direction: I(0) \(\rightarrow \) I(1) or I(1) \(\rightarrow \) I(0).

     
Under MU1 \(Y_{i,t}\) is I(0) up to and including time \(\lfloor \tau _{i}^0 T\rfloor \) but is I(1) after the break, provided that \(\sigma _{\eta , i}^2 = \mathrm {var}(\eta _{i,t}) > 0\). Under MU2 \(Y_{i,t}\) is I(1) up to and including time \(\lfloor \tau _{i}^0 T\rfloor \) but it is I(0) after the break, provided again that \(\sigma _{\eta ,i}^2 > 0\). Therefore, the hypothesis of stationarity against a shift in persistence from I(0) to I(1) or viceversa can be stated as \(H_0:\sigma _{\eta ,1}^{2}= ...= \sigma _{\eta ,N}^{2}=0\) versus \(H_1:\sigma _{\eta ,i}^{2} > 0\) for at least some i. Whenever the alternative is I(1) \(\rightarrow \) I(0) we write “\(H_1:\) I(1) \(\rightarrow \) I(0)”, whereas if the alternative is I(1) \(\rightarrow \) I(0), we write “\(H_{1}:\) I(1) \(\rightarrow \) I(0)”.

The conditions placed on the above DGP are given in Assumption 1, where \(C <\infty \), tr(A), \(||{A}|| = \sqrt{tr({A}' {A})}\), \(\rightarrow _p\) and \(\mathcal {F}_{i,t}\) denote a generic positive constant, the trace and Euclidean norm of the (generic) matrix A, convergence in probability, and the sigma-field generated by \(\{(\varepsilon _{i,n},\eta _{i,n})\}_{n=1}^t\), respectively.

Assumption 1

  1. (i)

    \(\varepsilon _{i,t} = \gamma _i(L) v_{i,t}\), where \(v_{i,t}\) is independent and identically distribution (iid) with \(E(v_{i,t}) = 0\), \(E(v_{i,t}^2) = 1\), \(E(v_{i,t}^8) \le C\), \(\gamma _i(L) = \sum _{j=0}^\infty \gamma _{ji}L^j\), \(\sum _{j=0}^\infty j||\gamma _{ji}|| \le C\) and \(\gamma _i(1)^2 > 0\);

     
  2. (ii)

    \(\eta _{i,t} = \phi _i(L) w_{i,t}\), where \(w_{i,t}\) is iid with \(E(w_{i,t}) = 0\), \(E(w_{i,t}^2) = 1\), \(E(w_{i,t}^8) \le C\), \(\phi _i(L) = \sum _{j=0}^\infty \phi _{ji}L^j\), \(\sum _{j=0}^\infty j||\phi _{ji}|| \le C\) and \(\phi _i(1)^2 > 0\);

     
  3. (iii)

    \(F_t\) is I(0) such that \(E(||F_t||^4) \le C\) and \(T^{-1} \sum _{t=1}^{T} F_t F'_t \rightarrow _p \Sigma _F > 0\);

     
  4. (iv)

    \(\varepsilon _{i,t}\), \(\eta _{i,t}\) and \(F_t\) are mutually independent;

     
  5. (v)

    \(\mu _{1,0} = ... = \mu _{N,0} = 0\);

     
  6. (vi)

    \(\lambda _i\) is deterministic such that \(||\lambda _i||^4 \le C\), \(N^{-1}\sum _{i=1}^N\lambda _i\lambda _{i}' \rightarrow \Sigma _\lambda > 0\) as \(N \rightarrow \infty \).

     

Remark 1

Assumption 1 puts restrictions on the time series and cross-sectional properties of \(\varepsilon _{i,t}\) and \(\eta _{i,t}\). The restrictions are very similar to the ones of Bai and Ng (2004), and we therefore refer to this other paper for a detailed discussion. The main difference when compared to Bai and Ng (2004) is that here \(F_t\) cannot be I(1). Thus, while \(Y_{i,t}\) may be cross-correlated, it cannot be affected by common stochastic trends. However, we would like to point out that this assumption is mainly for ease of interpretation of the test outcome, for if \(F_t\) is allowed to be I(1) the persistence of \(Y_{i,t}\) cannot be inferred from \(e_{i,t}\) alone, and in the present paper we focus on the testing of \(e_{i,t}\). Hence, analogous to the PANIC approach of Bai and Ng (2004), if \(F_t\) is permitted to be I(1), then we also need to test this variable.

3 The test statistics

The general testing idea is to first purge the effect of \(F_t\), and then to submit the resulting residuals to a test for a change in persistence. The implementation of the first step depends on whether \(F_t\) is known or not.

3.1 \(F_t\) known

Consider the generic variable \(X_{i,t}\). The detrended version of this variables is henceforth denoted \(X_{i,t}^p = X_{i,t} - \sum _{n=1}^T X_{i,n}a_{n,t,p}\), where \(a_{n,k,p}=D_{n,p}' (\sum _{t=1}^T D_{t,p}D_{t,p}' )^{-1}D_{k,p}\) and \(p\ge 0\). If \(p= -1\), then we define \(X_{i,t}^p = X_{i,t}\). In this notation, the detrended and defactored version of \(Y_{i,t}\) is given by \(\hat{e}_{i,t} = Y_{i,t}^p - \hat{\lambda }_i'F_t^p\), where \(\hat{\lambda }_i\) is the least squares (LS) slope estimator in a regression of \(Y_{i,t}^p\) onto \(F_t^p\). Thus, while in this section \(F_t\) is assumed to be known, \(\lambda _i\) is still treated as unknown. Consider the following test statistic, which is suitable for testing if cross-section unit i is I(0) versus I(1) \(\rightarrow \) I(0) (see, for example, Kim 2000; Kim et al. 2002; Busetti and Taylor 2004):
$$\begin{aligned} K_{i,T}(\tau )=\frac{ (\lfloor T\tau \rfloor )^{2}}{(T-\lfloor T\tau \rfloor )^{2}} \frac{\sum _{t=\lfloor T\tau \rfloor +1}^T S_{i,t}^{ 1 }(\tau )^2}{\sum _{t=1}^{\lfloor T\tau \rfloor } S_{i,t}^{ 0 }(\tau )^2}, \end{aligned}$$
where \(\tau \in [0,1]\), \(S_{i,t}^{0}(\tau )=\sum _{n=1}^{t} \hat{e}_{i,n}\) and \(S_{i,t}^{1}(\tau )=\sum _{n= \lfloor T\tau \rfloor +1}^t \hat{e}_{i,n}\). The error sequences \(\{\hat{e}_{i,n}\}_{n=1}^{\lfloor T\tau \rfloor }\) and \(\{\hat{e}_{i,n}\}_{n=\lfloor T\tau \rfloor + 1}^T\) come from two separate regressions; while the former uses only the first \(\lfloor T\tau \rfloor \) observations, the latter uses only the last \(\lfloor T(1-\tau )\rfloor \) observations.

Remark 2

The \(K_{i,T}(\tau )\) test considered here is in the spirit of Kwiatkowski et al. (1992) in which the constant I(0) null is tested versus the constant I(1) alternative. An alternative approach is to follow Banerjee et al. (1992) and Leybourne et al. (2003) who use the Dickey–Fuller statistic, in which the null and the alternative hypotheses are reversed. Panel variants of these can be constructed in the same way as the one suggested below for \(K_{i,T}(\tau )\) (see Demetrescu and Hanck 2013, for such a proposal).

Let \(\mathcal {C} = [\tau _{min},\tau _{max}]\subseteq (0,1)\). In this paper, we consider three transformations to eliminate the dependence on \(\tau \) in \(K_{i,T}(\tau )\) (see, for example, Kim, 2000);
  1. T1.
    The maximum-Chow transformation:
    $$\begin{aligned} K_{i,T}^1 = \max _{s = \lfloor T\tau _{min}\rfloor ,..., \lfloor T\tau _{max}\rfloor } K_{i}(s/T). \end{aligned}$$
     
  2. T2.
    The mean-exponential transformation:
    $$\begin{aligned} K_{i,T}^2 = \ln \left( (\lfloor T(\tau _{max} - \tau _{min})\rfloor +1)^{-1}\sum _{s = \lfloor T\tau _{min}\rfloor }^{\lfloor T\tau _{max}\rfloor } \exp [K_{i}(s/T)] \right) . \end{aligned}$$
     
  3. T3.
    The mean score transformation:
    $$\begin{aligned} K_{i,T}^3 = (\lfloor T(\tau _{max} - \tau _{min})\rfloor +1)^{-1}\sum _{s = \lfloor T\tau _{min}\rfloor }^{\lfloor T\tau _{max}\rfloor } K_{i}(s/T). \end{aligned}$$
     
Table 1

Simulated mean and standard deviation normalization factors

T

\(K_{NT}^1\)

\(K_{NT}^2\)

\(K_{NT}^3\)

\(R_{NT}^1\)

\(R_{NT}^2\)

\(R_{NT}^3\)

\(M_{NT}^1\)

\(M_{NT}^2\)

\(M_{NT}^3\)

Mean, \(p=0\) (constant)

   50

1.839

1.626

6.218

1.825

   1.612

6.190

2.792

2.633

9.218

   100

1.795

1.563

6.387

1.811

1.566

6.401

2.742

2.536

9.419

   150

1.801

1.560

6.525

1.793

1.543

6.487

2.735

2.516

9.568

   500

1.795

1.546

6.801

1.802

1.560

6.856

2.738

2.521

9.996

Standard deviation, \(p=0\) (constant)

   50

1.607

2.355

5.960

1.575

2.262

5.799

1.757

2.883

6.821

   100

1.528

2.135

5.755

1.528

2.082

5.661

1.663

2.594

6.478

   150

1.530

2.129

5.842

1.521

2.088

5.750

1.664

2.599

6.585

   500

1.541

2.098

5.966

1.540

2.121

6.027

1.683

2.599

6.797

Mean, \(p=1\) (constant and trend)

   50

2.498

2.586

9.317

1.058

0.757

3.618

2.719

2.816

10.066

   100

2.448

2.574

9.897

1.081

0.786

3.935

2.699

2.841

10.791

   150

2.475

2.684

10.569

1.062

0.764

3.970

2.711

2.928

11.386

   500

2.367

2.527

9.9038

1.069

0.828

3.903

2.916

1.196

10.339

Standard deviation, \(p = 1\) (constant and trend)

   50

1.447

2.528

6.288

0.707

0.879

2.973

1.327

2.477

6.022

   100

1.416

2.578

6.622

0.761

0.967

3.332

1.294

2.531

6.322

   150

1.421

2.625

6.853

0.723

0.906

3.291

1.291

2.563

6.519

   500

1.480

2.645

6.447

0.799

0.910

3.247

1.442

2.724

6.571

Let \(Q_{NT}^j = \sigma _{Q,j}^{-1}N^{-1/2} \sum _{i=1}^N (Q_{i,T}^j-\mu _{Q,j} )\) be one of the nine test statistics considered, where \(j\in \{1,2,3\}\) and \(Q\in \{K,R,M\}\). The values reported in the table refer to the appropriate mean and standard deviation correction factors, \(\mu _{Q,j}\) and \(\sigma _{Q,j}\), respectively, needed to construct \(Q_{NT}^j\)

In Appendix (Proof of Theorem 1), we show that \(K_{i,T}(\tau ) \rightarrow _w K_i(\tau )\) as \(T\rightarrow \infty \), where \(\rightarrow _w\) signifies weak convergence and \(K_i(\tau )\) is a certain ratio of stochastic integrals. Since \(K_1(\tau ),...,K_N(\tau )\) are iid, we may define \(\mu _{K,j}= E(K_{i}^j)\) and \(\sigma _{K,j}^2 = \mathrm {var}(K_{i}^j)\) for \(j\in \{ 1,2,3 \}\). Numerical values of \(\mu _{K,j}\) and \(\sigma _{K,j}\) are reported in Table 1. The proposed panel test statistic for testing \(H_0\) versus \(H_1:\) I(0) \(\rightarrow \) I(1) is given by
$$\begin{aligned} K_{NT}^j = \frac{1}{\sigma _{K,j} \sqrt{N}} \sum _{i=1}^N( K_{i,T}^j-\mu _{K,j}). \end{aligned}$$
For testing if cross-section unit i is I(0) versus I(1) \(\rightarrow \) I(0), the following “reverse” test statistic can be used (see Kim 2000; Kim et al. 2002; Busetti and Taylor 2004):
$$\begin{aligned} R_{i}(\tau )=(K_{i}(\tau ))^{-1}, \end{aligned}$$
which can be transformed using T1–T3 to eliminate the dependence on \(\tau \). The resulting transformed statistic is written in an obvious notation as \(R_{i}^j\). Based on this test statistic, we may define \(R_{NT}^j = \sigma _{R,j}^{-1} N^{-1/2} \sum _{i=1}^N (R_{i,T}^j-\mu _{R,j} )\) with obvious definitions of \(\sigma _{R,j}^{2}\) and \(\mu _{R,j}\). When the direction of the persistency is unknown, the following maximum statistic may be used:
$$\begin{aligned} M_{i,T}^j = \max \{K_{i,T}^j,R_{i,T}^j\}, \end{aligned}$$
which can again be normalized to obtain \(M_{NT}^j = \sigma _{M,j}^{-1}N^{-1/2} \sum _{i=1}^N (M_{i,T}^j-\mu _{M,j} )\).

Theorem 1

Under \(H_0\) and Assumption 1, as \(N,\,T \rightarrow \infty \) with \(N/T\rightarrow 0\),
$$\begin{aligned} K_{NT}^j,\,R_{NT}^j,\,M_{NT}^j \rightarrow _d N(0,1), \end{aligned}$$
where \(\rightarrow _d\) signifies convergence in distribution.

Remark 3

While the test statistics considered here are independent of \(\tau _1^0,...,\tau _N^0\), in applications it is sometimes useful to be able to estimate these parameters. This can be accomplished using the proposal of Kim (2000, Sect. 3.2), which basically amounts to setting \(\hat{\tau }_i^0\) equal to the suitably maximizing or minimizing value of \(K_{i,T}(\tau )\), depending on whether it is I(0) \(\rightarrow \) I(1) or I(1) \(\rightarrow \) I(0) that is being tested. Alternatively, we may follow Busetti and Taylor (2004, Sect. 6.2), who suggest setting \(\hat{\tau }_i^0\) equal to the value of \(\tau _i^0\) that minimizes the sum of squares of \(\hat{e}_{i,t}\).

Remark 4

The requirement that \(N/T \rightarrow 0\) is sufficient but not necessary and is needed to make sure that certain remainder terms are negligible. However, the order of these terms is not the sharpest possible. A more elaborate asymptotic analysis would be required to obtain the exact order. In Sect. 4, we use Monte Carlo simulation to evaluate the effect of N / T in small samples.

3.2 \(F_t\) unknown

The estimation of \(F_t\) can be performed in two ways; (i) unrestrictedly, or (ii) restricted under \(H_0\). In both cases, we follow the bulk of the previous literature and use the principal components method (see, for example, Bai and Ng 2004). The restricted estimator of \(F = ( F_{1},..., F_{T})'\), denoted \(\hat{F}^0 = (\hat{F}_{1}^0,...,\hat{F}_{T}^0)'\), is \(\sqrt{T}\) times the eigenvectors corresponding to the first r largest eigenvalues of the \(T \times T\) matrix \(Y^p(Y^p)'\), where \(Y^p = ( Y_{1}^p,..., Y_{N}^p)\) and \(Y_i^p = ( Y_{i,1}^p,..., Y_{i,T}^p)'\) are \(T\times N\) and \(T\times 1\), respectively. Under the normalization \(T^{-1}\hat{F}^0(\hat{F}^0)'= I_r \), the estimated loading matrix is \((\hat{\lambda }^0)' = (\hat{\lambda }_1^0,...,\hat{\lambda }_N^0)=T^{-1}(\hat{F}^0)' Y^p\). The restricted estimator of \(e_{i,t}\) that we will be considering can now be constructed as
$$\begin{aligned} \hat{e}_{i,t}^0 = Y_{i,t}^p - (\hat{\lambda }_i^0)'\hat{F}_t^0. \end{aligned}$$
(3)
Let \(X_{i,t}^{p-1}\) be \(X_{i,t}\) when detrended using a trend polynomial of order \(p-1\). Hence, \(X_{i,t}^{p-1} = X_{i,t}\) if \(p = 0\). Let \(f_t = \Delta F_t\) and \(y_{i,t} = \Delta Y_{i,t}\) (for \(t=2,...,T\)). The unrestricted estimators \(\hat{f}_t^1\) and \(\hat{\lambda }_i^1\) of (the space spanned by) \(f_t^{p-1}\) and \(\lambda _i\) are \(\hat{F}_t^0\) and \(\hat{\lambda }_i^0\), respectively, but with \(Y_{i,t}^p\) replaced by \(y_{i,t}^{p-1}\). Let
$$\begin{aligned} \tilde{e}_{i,t}^1 = \sum _{n=2}^t \left[ y_{i,n}^{p-1} - (\hat{\lambda }_i^1)'\hat{f}_n^1\right] , \end{aligned}$$
(4)
where \(\tilde{e}_{i,1}^1 = 0\). The unrestricted estimator \(\hat{e}_{i,t}^1\) of \(e_{i,t}\) is given by \(\hat{e}_{i,t}^1 = (\tilde{e}_{i,t}^1)^p\). The appropriate test statistics to consider when \(F_t\) is unknown, henceforth denoted \(K_{hNT}^j\), \(R_{hNT}^j\) and \(M_{hNT}^j\) for \(h \in \{ 0,1 \}\), are given by \(K_{NT}^j\), \(R_{NT}^j\) and \(M_{NT}^j\), respectively, with \(\hat{e}_{i,t}\) replaced by \(\hat{e}_{i,t}^h\).
Table 2

5% size and power when testing I(0) \(\mathrm \rightarrow \) I(1) and \(\rho = 0.3\)

T

N

\(\sigma _\eta \)

\(K_{NT}^1\)

\(K_{NT}^2\)

\(K_{NT}^3\)

\(R_{NT}^1\)

\(R_{NT}^2\)

\(R_{NT}^3\)

\(M_{NT}^1\)

\(M_{NT}^2\)

\(M_{NT}^3\)

\(\sigma _{\varepsilon , i} = 1\)

   50

5

0.000

0.055

0.040

0.061

0.073

0.058

0.077

0.046

0.042

0.040

   50

5

0.250

0.103

0.099

0.132

0.090

0.046

0.085

0.082

0.073

0.094

   50

5

0.500

0.210

0.205

0.270

0.162

0.027

0.148

0.133

0.132

0.165

   50

10

0.000

0.070

0.050

0.073

0.075

0.050

0.064

0.057

0.047

0.036

   50

10

0.250

0.141

0.126

0.171

0.142

0.035

0.092

0.112

0.099

0.122

   50

10

0.500

0.338

0.343

0.467

0.306

0.112

0.285

0.225

0.227

0.284

   50

20

0.000

0.055

0.038

0.067

0.103

0.053

0.075

0.044

0.041

0.059

   50

20

0.250

0.204

0.177

0.252

0.295

0.098

0.146

0.124

0.117

0.149

   50

20

0.500

0.539

0.553

0.724

0.596

0.439

0.645

0.295

0.319

0.419

   100

5

0.000

0.077

0.057

0.079

0.088

0.070

0.074

0.070

0.055

0.057

   100

5

0.250

0.278

0.288

0.374

0.186

0.066

0.146

0.225

0.237

0.283

   100

5

0.500

0.432

0.450

0.540

0.326

0.059

0.292

0.342

0.363

0.431

   100

10

0.000

0.099

0.071

0.080

0.112

0.065

0.087

0.086

0.067

0.051

   100

10

0.250

0.359

0.361

0.462

0.342

0.162

0.241

0.313

0.329

0.385

   100

10

0.500

0.633

0.680

0.815

0.547

0.348

0.583

0.482

0.532

0.646

   100

20

0.000

0.100

0.067

0.077

0.115

0.078

0.087

0.090

0.067

0.049

   100

20

0.250

0.506

0.551

0.634

0.510

0.297

0.380

0.387

0.430

0.484

   100

20

0.500

0.884

0.924

0.968

0.748

0.639

0.854

0.706

0.762

0.847

\(\sigma _{\varepsilon , i} \sim U(1,2)\)

   50

5

0.000

0.071

0.059

0.097

0.090

0.060

0.100

0.062

0.058

0.070

   50

5

0.250

0.073

0.069

0.100

0.080

0.049

0.089

0.071

0.057

0.071

   50

5

0.500

0.169

0.154

0.206

0.110

0.040

0.119

0.094

0.085

0.132

   50

10

0.000

0.074

0.051

0.065

0.088

0.054

0.088

0.059

0.046

0.044

   50

10

0.250

0.093

0.078

0.107

0.095

0.051

0.071

0.085

0.064

0.083

   50

10

0.500

0.208

0.204

0.319

0.233

0.093

0.202

0.146

0.150

0.199

   50

20

0.000

0.047

0.030

0.068

0.109

0.062

0.084

0.044

0.035

0.046

   50

20

0.250

0.118

0.085

0.142

0.169

0.069

0.102

0.093

0.071

0.088

   50

20

0.500

0.328

0.317

0.477

0.405

0.251

0.405

0.171

0.176

0.246

   100

5

0.000

0.093

0.069

0.092

0.099

0.060

0.089

0.068

0.064

0.056

   100

5

0.250

0.176

0.162

0.218

0.142

0.048

0.104

0.146

0.139

0.166

   100

5

0.500

0.371

0.388

0.485

0.298

0.056

0.247

0.275

0.293

0.353

   100

10

0.000

0.084

0.069

0.069

0.104

0.055

0.070

0.070

0.056

0.043

   100

10

0.250

0.232

0.245

0.307

0.223

0.083

0.118

0.199

0.204

0.230

   100

10

0.500

0.543

0.577

0.727

0.479

0.274

0.497

0.421

0.462

0.558

   100

20

0.000

0.076

0.059

0.075

0.105

0.062

0.073

0.089

0.073

0.053

   100

20

0.250

0.329

0.354

0.427

0.384

0.181

0.245

0.246

0.273

0.315

   100

20

0.500

0.754

0.805

0.916

0.669

0.552

0.757

0.560

0.614

0.706

\(\sigma _\eta \) and \(\sigma _{\varepsilon , i}\) refer to the standard deviation of \(\eta _{i,t}\) and \(\varepsilon _{i,t}\), respectively, while \(\rho \) refers to the autoregressive coefficient of \(F_t\). The results are based on setting \(p = 0\) (constant) and using the restricted factor estimation method, which assumes that the null hypothesis is true

Table 3

5% size and power when testing I(0) \(\mathrm \rightarrow \) I(1) and \(\rho = 0.6\)

T

N

\(\sigma _\eta \)

\(K_{NT}^1\)

\(K_{NT}^2\)

\(K_{NT}^3\)

\(R_{NT}^1\)

\(R_{NT}^2\)

\(R_{NT}^3\)

\(M_{NT}^1\)

\(M_{NT}^2\)

\(M_{NT}^3\)

\(\sigma _{\varepsilon , i} = 1\)

   50

5

0.000

0.049

0.042

0.062

0.080

0.062

0.071

0.058

0.047

0.041

   50

5

0.250

0.106

0.098

0.134

0.084

0.045

0.082

0.084

0.077

0.097

   50

5

0.500

0.221

0.203

0.256

0.151

0.031

0.144

0.146

0.146

0.164

   50

10

0.000

0.069

0.045

0.073

0.079

0.046

0.065

0.053

0.038

0.039

   50

10

0.250

0.153

0.135

0.181

0.135

0.039

0.096

0.106

0.082

0.121

   50

10

0.500

0.342

0.348

0.452

0.313

0.138

0.304

0.246

0.247

0.293

   50

20

0.000

0.064

0.040

0.071

0.111

0.053

0.090

0.044

0.036

0.054

   50

20

0.250

0.203

0.179

0.255

0.314

0.121

0.178

0.132

0.110

0.154

   50

20

0.500

0.543

0.543

0.706

0.591

0.446

0.641

0.307

0.314

0.408

   100

5

0.000

0.080

0.060

0.077

0.088

0.068

0.068

0.064

0.057

0.057

   100

5

0.250

0.265

0.285

0.353

0.202

0.067

0.151

0.231

0.236

0.264

   100

5

0.500

0.405

0.432

0.531

0.338

0.061

0.281

0.333

0.350

0.403

   100

10

0.000

0.114

0.079

0.084

0.110

0.062

0.082

0.092

0.068

0.056

   100

10

0.250

0.343

0.347

0.439

0.339

0.153

0.229

0.285

0.292

0.348

   100

10

0.500

0.582

0.640

0.785

0.545

0.354

0.559

0.481

0.522

0.626

   100

20

0.000

0.094

0.062

0.066

0.113

0.073

0.083

0.088

0.072

0.055

   100

20

0.250

0.502

0.532

0.618

0.535

0.332

0.415

0.374

0.416

0.454

   100

20

0.500

0.875

0.909

0.960

0.765

0.638

0.854

0.708

0.752

0.827

\(\sigma _{\varepsilon , i} \sim U(1,2)\)

   50

5

0.000

0.075

0.058

0.092

0.092

0.058

0.099

0.063

0.058

0.068

   50

5

0.250

0.087

0.073

0.107

0.084

0.042

0.089

0.073

0.059

0.073

   50

5

0.500

0.168

0.157

0.200

0.118

0.042

0.120

0.112

0.105

0.136

   50

10

0.000

0.074

0.047

0.071

0.097

0.058

0.092

0.049

0.047

0.046

   50

10

0.250

0.109

0.084

0.115

0.103

0.053

0.086

0.070

0.063

0.091

   50

10

0.500

0.209

0.200

0.315

0.216

0.098

0.194

0.145

0.149

0.205

   50

20

0.000

0.054

0.029

0.074

0.111

0.064

0.094

0.049

0.036

0.041

   50

20

0.250

0.133

0.105

0.171

0.182

0.069

0.117

0.104

0.072

0.097

   50

20

0.500

0.320

0.309

0.453

0.382

0.247

0.392

0.205

0.197

0.279

   100

5

0.000

0.093

0.076

0.096

0.106

0.061

0.095

0.083

0.069

0.067

   100

5

0.250

0.186

0.173

0.214

0.142

0.056

0.114

0.160

0.146

0.153

   100

5

0.500

0.322

0.352

0.448

0.283

0.065

0.238

0.272

0.280

0.331

   100

10

0.000

0.088

0.064

0.069

0.098

0.050

0.070

0.070

0.059

0.043

   100

10

0.250

0.226

0.230

0.275

0.225

0.085

0.124

0.203

0.199

0.218

   100

10

0.500

0.504

0.542

0.677

0.458

0.261

0.449

0.398

0.422

0.519

   100

20

0.000

0.089

0.069

0.069

0.106

0.065

0.075

0.080

0.063

0.051

   100

20

0.250

0.328

0.347

0.401

0.388

0.181

0.233

0.249

0.260

0.294

   100

20

0.500

0.678

0.734

0.872

0.623

0.498

0.701

0.529

0.578

0.668

See Table 2 for an explanation

Table 4

5% size and power when testing I(1) \(\mathrm \rightarrow \) I(0) and \(\rho = 0.3\)

T

N

\(\sigma _\eta \)

\(K_{NT}^1\)

\(K_{NT}^2\)

\(K_{NT}^3\)

\(R_{NT}^1\)

\(R_{NT}^2\)

\(R_{NT}^3\)

\(M_{NT}^1\)

\(M_{NT}^2\)

\(M_{NT}^3\)

\(\sigma _{\varepsilon , i} = 1\)

   50

5

0.000

0.055

0.040

0.061

0.073

0.058

0.077

0.046

0.042

0.040

   50

5

0.250

0.068

0.063

0.094

0.150

0.118

0.105

0.111

0.107

0.076

   50

5

0.500

0.079

0.042

0.111

0.209

0.182

0.184

0.142

0.135

0.109

   50

10

0.000

0.070

0.050

0.073

0.075

0.050

0.064

0.057

0.047

0.036

   50

10

0.250

0.109

0.077

0.110

0.149

0.117

0.105

0.124

0.118

0.084

   50

10

0.500

0.176

0.046

0.158

0.426

0.413

0.313

0.303

0.302

0.171

   50

20

0.000

0.055

0.038

0.067

0.103

0.053

0.075

0.044

0.041

0.059

   50

20

0.250

0.182

0.105

0.129

0.137

0.127

0.097

0.139

0.126

0.081

   50

20

0.500

0.514

0.290

0.463

0.670

0.674

0.630

0.487

0.478

0.360

   100

5

0.000

0.077

0.057

0.079

0.088

0.070

0.074

0.070

0.055

0.057

   100

5

0.250

0.126

0.080

0.103

0.291

0.283

0.224

0.254

0.249

0.160

   100

5

0.500

0.227

0.059

0.220

0.492

0.474

0.454

0.403

0.400

0.351

   100

10

0.000

0.099

0.071

0.080

0.112

0.065

0.087

0.086

0.067

0.051

   100

10

0.250

0.258

0.121

0.188

0.403

0.418

0.316

0.360

0.370

0.221

   100

10

0.500

0.452

0.164

0.423

0.798

0.797

0.752

0.694

0.699

0.592

   100

20

0.000

0.100

0.067

0.077

0.115

0.078

0.087

0.090

0.067

0.049

   100

20

0.250

0.371

0.258

0.228

0.547

0.601

0.388

0.541

0.588

0.346

   100

20

0.500

0.646

0.418

0.589

0.922

0.919

0.913

0.843

0.834

0.762

\(\sigma _{\varepsilon , i} \sim U(1,2)\)

   50

5

0.000

0.071

0.059

0.097

0.090

0.060

0.100

0.062

0.058

0.070

   50

5

0.250

0.071

0.061

0.099

0.122

0.081

0.105

0.076

0.070

0.066

   50

5

0.500

0.077

0.050

0.092

0.167

0.136

0.157

0.122

0.111

0.086

   50

10

0.000

0.074

0.051

0.065

0.088

0.054

0.088

0.059

0.046

0.044

   50

10

0.250

0.079

0.052

0.087

0.103

0.071

0.072

0.086

0.066

0.064

   50

10

0.500

0.148

0.059

0.125

0.221

0.202

0.163

0.147

0.142

0.093

   50

20

0.000

0.047

0.030

0.068

0.109

0.062

0.084

0.044

0.035

0.046

   50

20

0.250

0.113

0.073

0.100

0.135

0.107

0.099

0.116

0.090

0.084

   50

20

0.500

0.297

0.141

0.259

0.397

0.395

0.325

0.266

0.266

0.144

   100

5

0.000

0.093

0.069

0.092

0.099

0.060

0.089

0.068

0.064

0.056

   100

5

0.250

0.117

0.090

0.094

0.213

0.201

0.139

0.201

0.183

0.106

   100

5

0.500

0.162

0.051

0.149

0.420

0.415

0.359

0.323

0.319

0.252

   100

10

0.000

0.084

0.069

0.069

0.104

0.055

0.070

0.070

0.056

0.043

   100

10

0.250

0.180

0.092

0.136

0.283

0.262

0.183

0.245

0.258

0.160

   100

10

0.500

0.352

0.107

0.284

0.724

0.734

0.659

0.643

0.649

0.506

   100

20

0.000

0.076

0.059

0.075

0.105

0.062

0.073

0.089

0.073

0.053

   100

20

0.250

0.275

0.154

0.158

0.319

0.324

0.200

0.291

0.300

0.158

   100

20

0.500

0.655

0.376

0.556

0.928

0.932

0.903

0.858

0.860

0.721

See Table 2 for an explanation

Table 5

5% size and power when testing I(1) \(\mathrm \rightarrow \) I(0) and \(\rho = 0.6\)

T

N

\(\sigma _\eta \)

\(K_{NT}^1\)

\(K_{NT}^2\)

\(K_{NT}^3\)

\(R_{NT}^1\)

\(R_{NT}^2\)

\(R_{NT}^3\)

\(M_{NT}^1\)

\(M_{NT}^2\)

\(M_{NT}^3\)

\(\sigma _{\varepsilon , i} = 1\)

   50

5

0.000

0.049

0.042

0.062

0.080

0.062

0.071

0.058

0.047

0.041

   50

5

0.250

0.070

0.059

0.089

0.145

0.119

0.106

0.098

0.102

0.078

   50

5

0.500

0.099

0.047

0.115

0.190

0.171

0.177

0.141

0.134

0.109

   50

10

0.000

0.069

0.045

0.073

0.079

0.046

0.065

0.053

0.038

0.039

   50

10

0.250

0.098

0.074

0.106

0.167

0.115

0.114

0.129

0.099

0.077

   50

10

0.500

0.177

0.045

0.172

0.411

0.393

0.315

0.304

0.299

0.173

   50

20

0.000

0.064

0.040

0.071

0.111

0.053

0.090

0.044

0.036

0.054

   50

20

0.250

0.182

0.097

0.130

0.157

0.124

0.097

0.141

0.129

0.086

   50

20

0.500

0.483

0.278

0.450

0.613

0.611

0.584

0.434

0.428

0.336

   100

5

0.000

0.080

0.060

0.077

0.088

0.068

0.068

0.064

0.057

0.057

   100

5

0.250

0.133

0.084

0.108

0.269

0.262

0.217

0.271

0.261

0.160

   100

5

0.500

0.209

0.060

0.212

0.483

0.467

0.447

0.392

0.386

0.325

   100

10

0.000

0.114

0.079

0.084

0.110

0.062

0.082

0.092

0.068

0.056

   100

10

0.250

0.264

0.138

0.182

0.377

0.383

0.285

0.355

0.370

0.212

   100

10

0.500

0.437

0.150

0.387

0.750

0.743

0.710

0.628

0.643

0.539

   100

20

0.000

0.094

0.062

0.066

0.113

0.073

0.083

0.088

0.072

0.055

   100

20

0.250

0.364

0.283

0.252

0.484

0.528

0.350

0.518

0.575

0.320

   100

20

0.500

0.639

0.410

0.590

0.909

0.901

0.887

0.800

0.797

0.719

\(\sigma _{\varepsilon , i} \sim U(1,2)\)

   50

5

0.000

0.075

0.058

0.092

0.092

0.058

0.099

0.063

0.058

0.068

   50

5

0.250

0.084

0.067

0.100

0.118

0.080

0.106

0.078

0.070

0.076

   50

5

0.500

0.082

0.046

0.107

0.155

0.132

0.153

0.120

0.109

0.101

   50

10

0.000

0.074

0.047

0.071

0.097

0.058

0.092

0.049

0.047

0.046

   50

10

0.250

0.074

0.045

0.086

0.116

0.080

0.084

0.083

0.066

0.072

   50

10

0.500

0.149

0.063

0.125

0.236

0.213

0.167

0.165

0.154

0.106

   50

20

0.000

0.054

0.029

0.074

0.111

0.064

0.094

0.049

0.036

0.041

   50

20

0.250

0.124

0.076

0.111

0.162

0.102

0.104

0.109

0.093

0.090

   50

20

0.500

0.297

0.152

0.246

0.371

0.356

0.294

0.257

0.254

0.161

   100

5

0.000

0.093

0.076

0.096

0.106

0.061

0.095

0.083

0.069

0.067

   100

5

0.250

0.110

0.080

0.081

0.204

0.179

0.138

0.182

0.179

0.113

   100

5

0.500

0.160

0.056

0.154

0.402

0.390

0.341

0.325

0.330

0.237

   100

10

0.000

0.088

0.064

0.069

0.098

0.050

0.070

0.070

0.059

0.043

   100

10

0.250

0.177

0.079

0.122

0.287

0.268

0.192

0.247

0.231

0.127

   100

10

0.500

0.343

0.108

0.261

0.659

0.664

0.598

0.576

0.581

0.458

   100

20

0.000

0.089

0.069

0.069

0.106

0.065

0.075

0.080

0.063

0.051

   100

20

0.250

0.279

0.177

0.175

0.299

0.303

0.167

0.279

0.297

0.154

   100

20

0.500

0.588

0.339

0.494

0.875

0.896

0.831

0.781

0.793

0.654

See Table 2 for an explanation

Table 6

Empirical test results

Statistic

Unrestricted

Restricted

\(K_{NT}^1\)

\(-0.921\)

\(-0.881\)

\(K_{NT}^2\)

\(-0.658\)

\(-0.645\)

\(K_{NT}^3\)

\(-0.985\)

\(-0.947\)

\(R_{NT}^1\)

4.002***

2.439**

\(R_{NT}^2\)

4.565***

2.636***

\(R_{NT}^3\)

4.844***

3.156***

\(M_{NT}^1\)

3.027***

1.662*

\(M_{NT}^2\)

3.293***

1.744*

\(M_{NT}^3\)

3.862***

2.293**

***, ** and * denote significance at the 1, 5 and 10% levels, respectively. While the restricted factor estimation method assumes that the null hypothesis is true, the unrestricted method does not

Theorem 2

Under \(H_0\) and Assumptions 1, as \(N,\,T \rightarrow \infty \) with \(N/T\rightarrow 0\),
$$\begin{aligned} K_{hNT}^j,\,R_{hNT}^j,\,M_{hNT}^j \rightarrow _d N(0,1). \end{aligned}$$

As Theorem 2 makes clear, the factors can be unknown and still the asymptotic distributions of the test statistics are N(0, 1). This is in agreement with the results reported by Bai and Ng (2004) for their pooled panel unit root tests.

4 Monte Carlo simulations

A small-scale Monte Carlo study was conducted to investigate the properties of the new tests in small samples. The DGP is given by a restricted version of (1)–(2) that sets \(\varepsilon _{i,t} \sim N(0,\sigma _{\varepsilon , i}^2)\), \(\eta _{i,t} \sim N(0,\sigma _{\eta }^2)\), \(\sigma _{\eta }\in \{0,0.25,0.5\}\), \(\tau _i^0 \sim U(0.3,0.7)\), \(r = 1\), and \(F_t=\rho F_{t-1} + v_t\), where \(v_t\sim N(0,1)\) and \(\rho \in \{0.3,0.6\}\) (see, for example, Gengenbach et al. 2010, for a similar parametrization). For \(\sigma _{\varepsilon , i}\), we consider two cases. In the first, \(\sigma _{\varepsilon , i} = 1\) for all i, while in the second, \(\sigma _{\varepsilon , i} \sim U (1,2)\). Since a more volatile idiosyncratic error will make \(F_t\) more difficult to discern, we expect that the results for the second case will deteriorate when compared to the first. All results are based on 1,000 replications of samples of size \(N\in \{ 5, 10, 20 \}\) and \(T\in \{50,100\}\). Also, following Kim (2000), \(\mathcal {C}=[0.20,0.80]\). Results were obtained for \(p \in \{ 0,1\}\), although in this paper we focus on the results for the empirically most common specification with \(p = 0\) (a constant but no trend). The results for \(p=1\) (constant and trend) can be obtained upon request. Both the restricted and unrestricted factor estimation methods were simulated. Interestingly, the restricted method led to better results in terms of both size accuracy and power. In this paper, we therefore only report the results for the restricted method, where the number of common factors is determined using the \(IC_2\) criterion of Bai and Ng (2002) with a maximum of three factors.1

The 5% size and power results are reported in Tables 2, 3, 4 and 5. While Tables 2 (\(\rho = 0.3\)) and 3 (\(\rho = 0.6\)) contain the results for the tests of I(0) \(\rightarrow \) I(1), Tables 4 (\(\rho = 0.3\)) and 5 (\(\rho = 0.6\)) contain the corresponding results for I(1) \(\rightarrow \) I(0). The information content of these tables may be summarized as follows.
  • All tests have good size accuracy when \(\sigma _{\varepsilon , i} = 1\) and \(\rho = 0.3\). This is true for all constellations of T and N considered, although the distortions do have a tendency to increase slightly in N, which is consistent with the previous panel unit root literature (see Westerlund and Breitung 2013, for a discussion). While there are no big differences, the best size accuracy is generally obtained by using \(K_{NT}^2\), \(R_{NT}^2\) and \(M_{NT}^2\), whereas \(K_{NT}^3\), \(R_{NT}^1\) and \(R_{NT}^3\) generally leads to the worst accuracy.

  • As expected, increases in \(\rho \) and/or \(\sigma _{\varepsilon , i}\) generally lead to reduced size accuracy, although the distortions are never very large. This is true regardless of the direction of the change in persistence. In fact, the results are remarkably stable, given that the test statistics do not require any corrections to account for nuisance parameters.

  • All tests perform quite well in terms of power, and there are clear improvements as N and/or T increases. The fact that power is not only increasing in T, but also in N illustrates the advantage of accounting for the cross-sectional variation of the data. Power is also increasing in the distance to the null, as measured by \(\sigma _\eta \), which is again just as expected.

5 Empirical illustration

The question of whether inflation should be considered as I(0) or I(1) has been subject to a long debate. According to recent studies (see, for example, Kim 2000; Busetti and Taylor 2004), however, inflation may be better characterized by a change in persistence between separate I(1) and I(0) regimes rather than simply I(1) or I(0) behavior. The purpose of this illustration is to test this hypothesis using a large panel of quarterly CPI inflation data covering 20 countries (Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Italy, Japan, Korea, the Netherlands, New Zealand, Norway, Spain, Sweden, Switzerland, the UK and the US) between 1970:1 and 2013:4. All data are taken from OECD Main Economic Indicators.

The number of common factors is determined in the same way as in the simulations. As is customary when dealing with inflation (see, for example, Leybourne et al. 2003), the tests are fitted with a constant but no trend. The results are reported in Table 6. The first thing to note is that while in case of \(K_{NT}^1\), \(K_{NT}^2\) and \(K_{NT}^3\) there is no evidence against the I(0) null, \(R_{NT}^1\), \(R_{NT}^2\) and \(R_{NT}^3\) all lead to a clear rejection. This is true even at the most conservative 1% level. We therefore conclude that inflation has been subject to a change in persistence from I(1) to I(0), which is in agreement with the recent empirical literature based on US data (see, for example, Busetti and Taylor 2004; Harvey et al. 2006). A common explanation for the observed change in persistence of inflation in the US is that it is due to the stock market collapse of the late 1980s and the recession that followed it. One interpretation of the results reported in the current paper is therefore that they reflect the worldwide recession of the early 1990s, which was to a large extent triggered by the recession in the US. Another possibility is that the results reflect in part monetary policy shifts (see, for example, Davig and Doh 2014, and the references provided therein).

6 Conclusion

This paper develops panel tests that are suitable for testing the null hypothesis of stationarity against the alternative of a change in persistence from I(0) to I(1), from I(1) to I(0), or when the direction is unknown. The DGP used for this purpose is quite general and allows unit-specific constant and trend terms, cross-section heteroskedasticity, error serial correlation and cross-section dependence in the form of common factors.

Footnotes

  1. 1.

    See Westerlund and Mishra (2016) for a more elaborate selection approach that uses a data-driven penalty.

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Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Roy Cerqueti
    • 1
  • Mauro Costantini
    • 2
  • Luciano Gutierrez
    • 3
  • Joakim Westerlund
    • 4
    • 5
  1. 1.University of MacerataMacerataItaly
  2. 2.Brunel UniversityLondonUK
  3. 3.University of SassariSassariItaly
  4. 4.Lund UniversityLundSweden
  5. 5.Centre for Financial EconometricsDeakin UniversityBurwoodAustralia

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