Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Testing for seasonal unit roots in monthly

panels of time series

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

†Institute for Advanced Studies Vienna, Stumpergasse 56, 1060Wien (Vienna), Austria (e-mail kunst@ihs.ac.at)

‡Erasmus School of Economics, Erasmus University Rotterdam,Burg. Oudlaan 50, 3062 PA Rotterdam, Rotterdam,

Netherlands (e-mail franses@ese.eur.nl)

Prepared for “Econometrics, Time Series Analysis and Systems Theory: AConference in Honor of Manfred Deistler June 18–20, 2009”

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Contents

1 Introduction

2 The testing procedures

3 An empirical example

4 Post-sample evidence on size and power

5 Discussion

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Abstract

We consider the problem of testing for seasonal unit roots inmonthly panel data. To this aim, we generalize the quarterlyCHEGY test to the monthly case. This parametric test iscontrasted with a new nonparametric test, which is the panelcounterpart to the univariate RURS test that relies on countingextrema in time series. All methods are applied to an empiricaldata set on tourism in Austrian provinces. The power properties ofthe tests are evaluated in simulation experiments that are tuned tothe tourism data.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Basics

Testing for seasonal unit roots means investigating the degreeof reversion of seasonal cycles to time-constant intra-annualpatterns (pattern reversion)

All unit-root tests on economic time series of limited lengthsuffer from low discriminatory power

Testing for unit roots in panel data may serve convenient inincreasing that power

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Literature on testing seasonal unit roots in panels

While there is a sizeable literature on panel unit roots, there areonly few contributions on seasonal unit-root tests in panels.A few exceptions:

Otero et al. (2005,2007): CHEGY

Ucar and Guler (2007): panel HEGY

Dreger and Reimers (2004): seasonal IPS

Most of this research focuses on quarterly data. Here, we considerthe monthly case.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The tests considered here

1 The CHEGY test by Otero et al.

2 A panel version of the nonparametric RURS test (Kunst,2009), which is a seasonal and modified variant of the recordunit root (RUR) test by Aparicio et al. (2006)

3 The univariate tests

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Why nonparametric tests?

Nonparametric tests are tests not derived as optimal tests fora parametric model (a possible definition). Typically, statisticsare not formed from moment estimates.

Nonparametric tests have lower power than parametric testsfor standard cases.

Nonparametric tests are robust toward some deviations fromthe assumed parametric model, such as outliers, breaks,nonlinear transformations.

Example: Testing for randomness by runs tests(nonparametric) and correlogram-based tests (parametric).

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Monte Carlo is needed for panel tests

Asymptotic results for panel tests rely on N → ∞, T → ∞,and assumptions on limits of proportions such as N/T

In time-series panels, N is often fixed and small

In micro panels, T is often fixed and small

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The post-sample simulation

Most Monte Carlo designs are based on assumed artificial andsimple structures

We fit univariate autoregressive models to the data and generatepseudo-data from the estimated structure. By construction, thisstructure belongs to the alternative hypothesis.

Shrinking a parameter τ toward zero defines designs that approachthe null. τ = 1 is the realistic design, and τ = 0 is a null model.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The testing problem

Consider a panel of N real-valued time-series variables

Xt = (X1t , . . . ,XNt)′, t = 1, . . . ,T ,

observed at monthly intervals.Suppose there exist autoregressive representations

Φj(B)Xjt = εjt ,

for j = 1, . . . ,N, with white-noise εjt .The problem is to determine whether the autoregressive operatorsΦj contain roots at the locations exp(ikπ/6) for k = 0, . . . 6.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Assumptions on homogeneity and dependence

Lag orders and shapes of the polynomials may vary across thecross-section dimension, pj ≥ 12

The covariance matrix Σ = E (εtε′

t) may be non-diagonal

Unit-root events are homogeneous across the cross-sectiondimension

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The monthly HEGY transformation

Remark. Full details on this test were developed after 1990 byFranses, Beaulieu& Miron, and others.Idea. Transform the autoregressive model

Φ(B)Xt = εt

to the equivalent representation

∆12Xt = α′ (Xt−1, . . . ,Xt−12)′ + γ′ (∆12Xt−1, . . . ,∆12Xt−p)

′ + εt .

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The monthly HEGY test

There will be unit roots at exp(ikπ/6) for k = 0, . . . 6 iffα = (0, . . . , 0)′

The relevant statistic is the ‘Wald’ F–statistic, with anonstandard null distribution

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Testing for individual roots: spectral transforms

1 Consider the 12–vectors ck , k = 0, . . . , 6, filled by cos(lkπ/6)for l = 1, . . . , 12.

2 Consider the 12–vectors dk , k = 1, . . . , 5, filled by sin(lkπ/6)for l = 1, . . . , 12.

3 Define the matrix M by

M = (c0, c1, d1, c2, d2, . . . , d5, c6) .

4 Transform X−

t = (Xt−1, . . . , Xt−12)′ to (Y−

t )′ = (X−

t )′M

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The autoregressive model with spectral transforms

∆12Xt = β′Y −

t + γ′ (∆12Xt−1, . . . ,∆12Xt−p)′ + εt .

permits testing for individual unit roots:

β1 = 0 ⇔ Φ(+1) = 0,β12 = 0 ⇔ Φ(−1) = 0,

β2k = β2k+1 = 0, k = 1, . . . , 5 ⇔ Φ(exp(ikπ/6)) = 0.

Corresponding t– and F–statistics (nonstandard null distributions)will be denoted by t0,F1, . . . ,F5, t6.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The monthly CHEGY test

Remark. The CHEGY or cross-sectionally augmented HEGY testis due to Otero et al. (2007) who build on Pesaran (2007).The idea. In order to make the test robust to cross-sectiondependence, cross-section averages of the Y− variables Y − and ofthe seasonal differences ∆12X serve as additional regressors to thebasic HEGY regression.Then, the HEGY regression for the monthly case reads

∆12Xjt = β′Y−

j ,t + γ′(

∆12Xj ,t−1, . . . ,∆12Xj ,t−pj

)

′

+β′Y −

t + γ′(

∆12Xt , . . . ,∆12Xt−pj

)

′

+εjt .

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The CHEGY statistics

The CHEGY statistics are defined as arithmetic averages ofindividual t– and F–statistics for the βk elements over the Nindividual values. Otero et al. (2007) provide simulatedsignificance points.

We provide additional simulations for specific N and T .Deterministic regressors contain monthly dummy constants and alinear time trend. Lag orders are determined via BIC.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Table: Empirical distribution of the CHEGY statistic for T = 406 andN = 5, 9.

mean 0.05 0.10 0.50 0.90 0.95

N = 5t0 -2.388 -3.063 -2.908 -2.394 -1.853 -1.699F1 4.117 2.219 2.569 4.022 5.780 6.327F2 4.117 2.208 2.577 4.039 5.745 6.265F3 4.103 2.199 2.549 4.040 5.725 6.274F4 4.096 2.231 2.570 4.015 5.707 6.244F5 4.085 2.190 2.551 3.995 5.760 6.276t6 -1.820 -2.567 -2.409 -1.827 -1.216 -1.039

N = 9t0 -2.386 -2.895 -2.786 -2.385 -1.998 -1.892F1 4.122 2.666 2.934 4.076 5.376 5.789F2 4.127 2.647 2.923 4.080 5.387 5.783F3 4.110 2.605 2.908 4.060 5.362 5.751F4 4.118 2.623 2.924 4.071 5.384 5.783F5 4.112 2.618 2.899 4.064 5.369 5.762t6 -1.816 -2.370 -2.254 -1.820 -1.366 -1.242

Note: Columns correspond to quantiles of the empirical distribution generated

using 10,000 replications of seasonal random walks with N(0, 1) errors.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The monthly RURS test

Idea. The RURS test by Kunst (2009) follows the ‘record unitroot’ RUR test by Aparicio et al. (2006, AES).Idea of the RUR test. The RUR test counts the new extrema ina series, starting from the beginning of the series and also from theend. Then, forward and backward counts are averaged. If theseries under investigation is a random walk, the statistic T−1/2R ,where R is the averaged number of extremum counts and T is thesample size, converges to a nonstandard distribution as T → ∞. Ifthe series is stationary, the statistic converges to 0. Too fewextrema imply rejecting the null.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Modifications of RURS relative to RUR

1 The asymptotic null distribution is known only for the case ofa pure random walk. The RURS test uses an autoregressiveadjustment to purge the variables under their null.

2 Spectral transformations of the variables admit investigatingseasonal unit roots at all frequencies.

3 The RURS test is one-sided against stationary alternatives.Too many records would indicate instability or strongdeterministic expansion.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The panel RURS test

The RURS-p statistic is defined as the cross-section average overN RURS statistics at each frequency. If N → ∞, this statisticconverges to the first moment of the RUR distribution under thenull of a unit root.For small N, we simulate the null distribution of

J∗k = N−1N

∑

j=1

J(j)∗k ,

where J(j)∗k is the RURS statistic at frequency k for series j .

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Quantiles for the RURS and RURS-p statistics based on

T = 406 and for N = 5, 9

mean 0.05 0.1 0.5 0.9 0.95RURS0 2.369 1.613 1.757 2.330 3.047 3.263π/6 2.329 1.577 1.721 2.295 3.012 3.227π/3 2.384 1.649 1.757 2.330 3.047 3.263π/2 2.313 1.784 1.887 2.301 2.766 2.9222π/3 2.384 1.649 1.757 2.330 3.047 3.2985π/6 2.330 1.577 1.721 2.295 3.012 3.227π 2.371 1.613 1.757 2.330 3.047 3.263RURS-p N = 50 2.375 2.022 2.094 2.366 2.667 2.753π/6 2.335 1.979 2.051 2.330 2.632 2.718π/3 2.389 2.036 2.108 2.381 2.689 2.775π/2 2.314 2.058 2.115 2.311 2.513 2.5802π/3 2.390 2.029 2.108 2.381 2.682 2.7685π/6 2.335 1.972 2.051 2.323 2.632 2.725π 2.366 2.008 2.079 2.359 2.660 2.753RURS-p N = 90 2.369 2.095 2.155 2.366 2.589 2.653π/6 2.329 2.052 2.111 2.326 2.549 2.613π/3 2.384 2.111 2.167 2.382 2.601 2.665π/2 2.313 2.129 2.166 2.313 2.462 2.5082π/3 2.384 2.107 2.167 2.378 2.605 2.6695π/6 2.330 2.052 2.115 2.326 2.549 2.617π 2.371 2.103 2.159 2.370 2.585 2.653

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The data

Data are from the Austrian Wifo data base. They are monthly andcover the time range January 1970 to October 2008. Variables arethe registered overnight stays in the nine Austrian ‘lander’ (federalstates).

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The Lander of Austria

1=Burgenland 2=Carinthia 3=Lower Austria 4=Upper Austria5=Salzburg 6=Styria 7=Tyrol 8=Vorarlberg 9=Vienna

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

1975 1980 1985 1990 1995 2000 2005

02

46

8

Burgenland

Mio

.

1975 1980 1985 1990 1995 2000 2005

Carinthia

1975 1980 1985 1990 1995 2000 2005

Lower Austria

1975 1980 1985 1990 1995 2000 2005

02

46

8

Salzburg

Mio

.

1975 1980 1985 1990 1995 2000 2005

Styria

1975 1980 1985 1990 1995 2000 2005

Tyrol

1975 1980 1985 1990 1995 2000 2005

02

46

8

Upper Austria

Mio

.

1975 1980 1985 1990 1995 2000 2005

Vienna

1975 1980 1985 1990 1995 2000 2005

Vorarlberg

Figure: Overnight stays in the nine Austrian provinces, on a commonscale.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

1975 1980 1985 1990 1995 2000 2005

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Burgenland

mio

.

1975 1980 1985 1990 1995 2000 2005

01

23

45

67

Carinthia

mio

.

1975 1980 1985 1990 1995 2000 2005

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Lower Austria

mio

.

1975 1980 1985 1990 1995 2000 2005

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Salzburg

mio

.

1975 1980 1985 1990 1995 2000 2005

0.0

0.5

1.0

1.5

Styria

mio

.

1975 1980 1985 1990 1995 2000 2005

23

45

67

Tyrol

mio

.

1975 1980 1985 1990 1995 2000 2005

0.0

0.5

1.0

1.5

2.0

Upper Austria

mio

.

1975 1980 1985 1990 1995 2000 2005

0.0

0.2

0.4

0.6

0.8

1.0

Vienna

mio

.

1975 1980 1985 1990 1995 2000 2005

0.8

1.0

1.2

1.4

Vorarlberg

mio

.

Overnight stays in January (solid) and in July (dashed) in the nine Austrianprovinces, individual scales.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Visual impression

Seasonality is strong and changes persistently over time.Winter tourism gains, summer tourism loses.

There may be seasonal unit roots in some lander.

The lander are quite heterogeneous.

No single individual dominates.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Results of the parametric tests

Table: HEGY statistics for individual series in the panel and CHEGYstatistics for the panel.

Region p t0 F1 F2 F3 F4 F5 t6Burgenland 2.00 -1.64 0.54 6.37 15.50* 23.08* 31.62* -6.85*Carinthia 2.00 -1.97 1.15 5.08 7.00* 13.26* 37.78* -7.01*L. Austria 3.00 -1.51 0.16 7.55* 10.91* 29.74* 41.24* -5.33*Salzburg 10.00 -3.04 3.27 3.90 5.52 12.16* 33.15* -5.63*Styria 2.00 -1.13 1.13 2.86 7.69* 11.74* 31.24* -5.96*Tyrol 1.00 -2.53 2.37 6.18 7.02* 18.42* 38.55* -6.66*U. Austria 2.00 -1.52 0.90 1.52 12.05* 13.10* 28.90* -6.88*Vienna 2.00 -2.31 0.76 3.29 6.93* 7.12* 17.63* -5.14*Vorarlberg 1.00 -2.21 2.06 6.82 6.86* 21.37* 34.59* -6.63*CHEGY -2.40 5.78 5.95* 7.87* 9.93* 8.21* -2.09

Note: Asterisks denote significance at 5%.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Summary of the parametric tests

Univariate HEGY tests support unit roots at frequency 0 andat longer seasonal frequencies. They reject athigher-frequency seasonals.

The panel CHEGY tests support unit roots at frequency 0, atthe annual cycle, and at the Nyqvist frequency.

The discrepancy is rooted in Styria and Carinthia, where theCHEGY conditioning eliminates the evidence against a unitroot at the Nyqvist frequency.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Results of the nonparametric tests

Table: RURS statistics for all countries and RURS-p statistics.

0 π/6 π/3 π/2 2π/3 5π/6 πBurgenland 3.585 2.976 2.868 1.474 3.621 2.976 1.291Carinthia 3.119 2.151 2.868 1.887 3.083 2.868 2.079L. Austria 3.729 3.012 2.581 1.577 3.513 2.581 1.183Salzburg 3.513 3.119 3.191 1.474 3.693 3.191 1.291Styria 3.191 2.940 2.223 1.603 3.298 2.223 1.434Tyrol 3.693 3.513 3.693 1.525 3.693 3.693 1.542U. Austria 3.047 2.402 2.581 1.784 3.083 2.581 1.398Vienna 2.653 2.008 2.474 1.810 2.474 2.330 1.613Vorarlberg 2.761 2.617 2.617 1.577 2.617 2.617 1.398RURS-p 3.255 2.749 2.788 1.635 3.231 2.784 1.470

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Summary of the nonparametric tests

Seasonal unit roots are rejected at π/2 and at π but they aresupported at the other frequencies.

Frequent occasions of high values for the statistics mayindicate instabilities that are outside the autoregressive frame.

The panel version coincides qualitatively with the univariatetests.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

The simulation design

1 Autoregressive models are fitted to the data. This determinesΦj and Σ.

2 10,000 replications are drawn for Φj and Σ. This is the‘alternative hypothesis’ design (τ = 1).

3 Imposing β = 0 defines the ‘null hypothesis’ design (τ = 0).

4 Other values for τ define alternative-hypothesis design on anarc between these two models.

5 All statistics are evaluated for all replications.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Graphical representation of the simulation design

θr

θ

HA

H0

τ = 1

τ = 0

Power is studied along the line segment [θr , θ] or τ ∈ [0, 1]. Notethat θr is not the restricted ML estimate.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Rejection frequency of CHEGY test around the null

τ 0.00 0.20 0.40 0.60 0.80 1.00unstable unrestricted estimatet0 0.166 0.063 0.023 0.069 0.221 0.486F1 0.164 0.020 0.005 0.001 0.000 0.000F2 0.153 0.050 0.196 0.450 0.716 0.908F3 0.178 0.315 0.774 0.987 1.000 1.000F4 0.169 0.645 0.999 1.000 1.000 1.000F5 0.164 0.994 1.000 1.000 1.000 1.000t6 0.069 0.979 1.000 1.000 1.000 1.000stabilized designt0 0.166 0.062 0.025 0.074 0.231 0.500F1 0.164 0.036 0.048 0.074 0.108 0.156F2 0.153 0.050 0.201 0.466 0.735 0.917F3 0.178 0.316 0.776 0.986 1.000 1.000F4 0.169 0.646 0.999 1.000 1.000 1.000F5 0.164 0.994 1.000 1.000 1.000 1.000t6 0.069 0.979 1.000 1.000 1.000 1.000

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Average rejection frequency of HEGY test around the null.

τ 0.00 0.20 0.40 0.60 0.80 1.00unstable unrestricted estimatet0 0.049 0.236 0.562 0.549 0.362 0.228F1 0.056 0.040 0.043 0.057 0.155 0.194F2 0.060 0.069 0.120 0.181 0.268 0.377F3 0.061 0.131 0.282 0.477 0.675 0.838F4 0.065 0.273 0.633 0.858 0.947 0.980F5 0.058 0.558 0.964 0.998 1.000 1.000t6 0.043 0.403 0.883 0.984 0.999 1.000stabilized designt0 0.049 0.236 0.562 0.549 0.363 0.229F1 0.056 0.042 0.044 0.054 0.070 0.088F2 0.060 0.069 0.121 0.181 0.269 0.378F3 0.061 0.131 0.282 0.477 0.675 0.838F4 0.065 0.273 0.633 0.858 0.947 0.980F5 0.058 0.558 0.964 0.998 1.000 1.000t6 0.043 0.403 0.883 0.984 0.999 1.000

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Average rejection frequency of RURS test around the null

τ 0.00 0.20 0.40 0.60 0.80 1.00unstable unrestricted estimate0 0.031 0.000 0.000 0.006 0.018 0.066π/6 0.198 0.005 0.006 0.024 0.050 0.141π/3 0.143 0.010 0.024 0.017 0.064 0.209π/2 0.074 0.082 0.137 0.205 0.309 0.4162π/3 0.215 0.006 0.001 0.021 0.042 0.1285π/6 0.145 0.000 0.000 0.005 0.018 0.077π 0.040 0.202 0.439 0.632 0.754 0.836stabilized design0 0.031 0.000 0.000 0.006 0.018 0.044π/6 0.198 0.005 0.006 0.024 0.050 0.122π/3 0.143 0.010 0.024 0.017 0.064 0.177π/2 0.074 0.081 0.128 0.200 0.295 0.4052π/3 0.215 0.006 0.001 0.021 0.041 0.1065π/6 0.145 0.000 0.000 0.005 0.018 0.060π 0.040 0.204 0.445 0.635 0.762 0.843

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Rejection frequency of RURS-p test around the null

τ 0.00 0.20 0.40 0.60 0.80 1.00unstable unrestricted estimate0 0.049 0.000 0.000 0.000 0.000 0.001π/6 0.478 0.000 0.000 0.000 0.000 0.034π/3 0.296 0.000 0.000 0.000 0.000 0.096π/2 0.222 0.282 0.486 0.705 0.890 0.9742π/3 0.479 0.000 0.000 0.000 0.000 0.0045π/6 0.347 0.000 0.000 0.000 0.000 0.002π 0.134 0.893 0.997 1.000 1.000 1.000stabilized design0 0.049 0.000 0.000 0.000 0.000 0.000π/6 0.478 0.000 0.000 0.000 0.000 0.026π/3 0.296 0.000 0.000 0.000 0.000 0.059π/2 0.222 0.278 0.471 0.697 0.882 0.9692π/3 0.479 0.000 0.000 0.000 0.000 0.0035π/6 0.347 0.000 0.000 0.000 0.000 0.001π 0.134 0.894 0.997 1.000 1.000 1.000

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Some summary remarks

We confirm that the power of the CHEGY test is acceptablebut it is not immune to size bias effects.

The panel CHEGY test has higher power than univariateHEGY tests.

The discriminatory power of the nonparametric tests isdisappointing.

The nonparametric tests are sensitive to slight super-lineartrends, which imply failure to reject unit roots.

Similarly, near-unit roots at seasonal frequencies together withdeterministic dummies can generate near-unstable expansionsof seasonal cycles.

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series

Introduction The test procedures An empirical example Post-sample evidence on size and power Discussion

Thank you for your attention

ROBERT M. KUNST† and PHILIP HANS FRANSES‡

Testing for seasonal unit roots in monthly panels of time series