Testing for fiscal sustainability: New evidence from the G-7 and some European countries

Economic Modelling 37 (2014) 1–15

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Testing for fiscal sustainability: New evidence from the G-7 and some European countries Shyh-Wei Chen Department of International Business, Center for Applied Economic Modeling, Chung Yuan Christian University, No. 200, Chung Pei Rd, Chung Li 32023, Taiwan

a r t i c l e

i n f o

Article history: Accepted 15 October 2013 JEL classification: E62 H62 C32 Keywords: Debt Sustainability Threshold Unit root

a b s t r a c t Whether or not a government deficit is sustainable has important implications for policy. If the debt of a nation is sustainable, then it implies that the government should have no incentive to default on its internal debt. In this article we examine whether or not the debt-GDP ratios of the G-7 and some European countries can be characterized by a unit root process with the non-linear trend and asymmetric adjustment. The econometric methodology allows us to determine whether the stationarity holds for the government's debt–GDP ratio after considering the non-linear trend. Among the main results, it is found that it is very likely that the debt–GDP ratios of Canada, Germany, the US and Italy are stationarity after taking account of the non-linear trend in the long run. Nevertheless, it is model-dependent for the debt–GDP ratios of these countries to be asymmetrically adjusted after taking the non-linear trend into consideration. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The concept of fiscal deficit sustainability has long been a focus of research and policy debate in economics and public finance. In principle, a government will be able to sustain fiscal deficits as long as it can raise the necessary funds by borrowing. Although such a policy may be feasible in the short run, the ability of the government to service its debt by resorting to further borrowing is likely to be questioned once the deficits become persistent. Therefore, the government is required to balance its budget intertemporally by setting the current value of debt equal to the discounted sum of expected future surpluses. Violation of the intertemporal budget constraint (IBC) would indicate that the fiscal policy cannot be sustained forever because the value of debt will explode at a rate faster than the growth rate of the economy. A myriad of studies have devoted many efforts to this issue. According to Hamilton and Falvin (1986) and Wilcox (1989),1 early studies that examined the government's deficit sustainability used conventional unit root tests to investigate the mean-reverting behavior of the public deficit and debt (Feve and Henin, 2000; Getzner et al., 2001; Makrydakis et al., 1999; Uctum and Wickens, 2000). However, soon the cointegration approach either with or without a structural break become predominant (e.g., Afonso, 2005; Baharumshah and Lau, 2007; Bajo-Rubio et al., 2004; Gabriel and Sangduan, 2011; Goyal et al., 2004;

E-mail addresses: [email protected], [email protected]. Trehan and Walsh (1988, 1991) show that the stationarity of the government deficit is a sufficient condition for the intertemporal budget constraint to hold. 1

0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.10.024

Kalyoncu, 2005; Lusinyan and Thornton, 2009; Martin, 2000; Quintos, 1995).2 Motivated by the statistical power of the advances in panel unit root and panel cointegration tests (Maddala and Wu, 1999; Westerlund, 2006), an increasing number of authors have applied these new tools to test whether or not the government deficit is sustainable in the long run, for example, Westerlund and Prohl (2008), Afonso and Rault (2010) and Mahdavi and Westerlund (2011), to name a few. An important feature of previous studies is that distinct results based on previous research are due to differences in methodology, approaches and samples and are subject to diverse interpretations, thus making it difficult to reach a corroborative position on the stationarity property of the government deficit. Another potential problem with previous studies is that if the government deficit is adjusted in an asymmetric or non-linear way, then conventional unit-root and cointegration tests suffer from a loss of power that may lead to the acceptance of nonstationarity when the government deficit is actually sustainable. Therefore, a growing body of research (see, for example, Arestis et al., 2004; Arghyrou and Luintel, 2007; Bajo-Rubio et al., 2006, 2010; Bohn, 1998; Cipollini, 2001; Davig, 2005; Payne et al., 2008; Sarno, 2001) has turned its attention to the adoption of more sophisticated non-linear models to test the government's ability to sustain its deficit. The empirical

2 The univariate approach focuses on the stochastic properties of the stock of debt. The multivariate approach focuses on the long-run properties of the flow of expenditures and revenues, i.e., on the stochastic properties of the deficit. Therefore, sustainability through the debt implies sustainability through the deficit but the reverse is not true. Recently, Berenguer-Rico and Carrion-i-Silvestre (2011) have unified these approaches by testing for multicointegration in the flow-stock system. Escario et al. (2012) examine Spanish long-run fiscal sustainability by using the multicointegration methodology.

2

S.-W. Chen / Economic Modelling 37 (2014) 1–15

Table 1 Studies adopt the univariate approach to test for the fiscal sustainability. Studies

Countries and samples covered

Methodology

Sustainability

Makrydakis et al. (1999) Sarno (2001)

Greece debt–GDP ratio (1958–1995) The US debt–GDP ratio (1916–1995) The US deficit–GDP ratio (1947–2001) The Spanish deficit-GDP ratio (1964–2001) The US deficit–GDP ratio (1947–2003)

Zivot and Andrews' (1992) unit root test with structural break

Hold

Teräsvirta's (1994) smooth transition autoregressive (STAR) model

Non-linear stationarity and hold

Caner and Hansen's (2001) TAR unit root test

Non-linear stationarity and hold

Caner and Hansen's (2001) TAR unit root test

Non-linear stationarity and hold

Perron's (1997) unit root test with structural break and Enders and Granger's (1998) TAR and MTAR unit root tests

Non-linear stationarity and hold

Arestis et al. (2004) Bajo-Rubio et al. (2004) Payne and Mohammadi (2006)

evidence from this line of research indicates that, by taking the nonlinear property into account, the US and the European Monetary Union (EMU) countries are no longer in violation of the intertemporal budget constraint. For the benefit of readers, we summarize the recent contributions to the government deficit sustainability after 2000 in Tables 1–2. There are at least two channels that make the budget deficit series become a non-linear process. The first rationale for incorporating possible asymmetry in the adjustment of the budget deficit stems in part from fiscal policy-makers that respond differently to a deviation of the deficit and/or surplus from its long-run trend. One would expect, for example, that the response would be more aggressive if the deficit is above its long-run trend than when it is below it (Bertola and Drazen, 1993). Second, the available empirical evidence suggests that various business cycle indicators exhibit asymmetric behavior (see, for example, Enders and Siklos, 2001). Given that the budget deficit is influenced by business cycle movements via automatic fiscal stabilizers and discretionary fiscal measures, it is reasonable to assume that the business cycle asymmetries could possibly translate into the budget deficit (Payne and Mohammadi, 2006). The aim of this study is to re-examine whether or not the government debt of the G-7 and some European countries is sustainable. An important implication of the standard unit root tests is the implicit assumption that the adjustment process is symmetric. Indeed, if the true adjustment process is asymmetric, then the restrictive symmetric adjustment implicitly assumed is indicative of model misspecification. In order to take account of the possibility of an asymmetric adjustment of the government deficits, in line with the literature, we adopt the threshold autoregressive (hereafter TAR) and the momentum threshold autoregressive (hereafter MTAR) unit root tests, proposed by Enders and Granger (1998), in this study. In addition, in order to take the possibility of non-linear trends into consideration, we also use the logistic smooth transition threshold autoregressive (hereafter LSTR-TAR) and the logistic smooth transition momentum threshold autoregressive (hereafter LSTR-MTAR) unit root tests, championed by Sollis (2004) and Cook and Vougas (2009), in this paper. These approaches permit structural breaks to occur gradually over time instead of instantaneously.3 The reason for adopting the LSTR-TAR and LSTR-MTAR models in this study is that non-linearity may occur in the form of structural changes in the deterministic components as emphasized by Bierens (1997) and Leybourne et al. (1998). In this study, we find that the debt–GDP ratios of the G-7 and European countries display smooth shifts in trend rather than sudden changes (see Figs. 1–11 for the details). As compared to the literature, the contributions of this study are threefold. First, the application of the threshold model overcomes the

weakness of the traditional linear unit root test in detecting the fiscal sustainability. It allows us to draw conclusions about the validity of the government intertemporal budget constraint in the long run. Furthermore, investigating the short-run dynamics of the threshold model provides a test concerning the importance of asymmetric adjustment in the government debt. Second, previous studies (e.g., Bajo-Rubio et al., 2010; Berenguer-Rico and Carrion-i-Silvestre, 2011) have shed light on the importance of recognizing the possibility of a structural shift in testing for the null hypothesis of a unit root for the debt–GDP ratio. We take this possibility into consideration by employing the Sollis (2004) and Cook and Vougas (2009) approaches. They combine the ideas of Enders and Granger (1998) and Leybourne et al. (1998) and develop a test for the null hypothesis of a unit root that under the alternative hypothesis allows for stationary asymmetric adjustment around a smooth transition between deterministic linear trends. Third, as compared with previous studies where the sample ends in the early 2000s (e.g., BajoRubio et al., 2004; Sarno, 2001), our period of analysis is extended to 2012, including the most recent developments in terms of the financial tsunami and the European sovereign debt crisis. The major findings from this study are as follows. First, by using standard unit root tests we find that the debt–GDP ratios of the G-7 and some European countries are non-stationary processes, implying that the fiscal sustainability does not hold. Second, the evidence from the TAR and MTAR unit root regressions suggests that the debt–GDP ratios are stationary series with asymmetric adjustment, indicating that the government intertemporal budget constraint is likely to be on a sustainable path in the long run. Third, the results from the LSTRTAR or LSTR-MTAR models show that the debt–GDP ratios of Canada, Germany, the US and Italy are stationarity after taking into account the non-linear trend. Nevertheless, it is model-dependent for the debt–GDP ratios of these countries to be asymmetrically adjusted after taking the non-linear trend into consideration. The remainder of this paper is organized as follows. Section 2 reviews the theoretical foundation of fiscal sustainability. Section 3 outlines the statistical methods used for testing non-linearity and the unit roots. Section 4 discusses the data used and the empirical results and compares our results with those of the extant literature. Finally, Section 5 concludes. 2. Fiscal sustainability Tests of fiscal sustainability are commonly based on the present value of the government's intertemporal budget constraint.4 Consider the following one-period government budget constraint: ΔBt ¼ Gt −Rt ;

ð1Þ

3

In the context of economic time series this has considerable intuitive appeal. Generally speaking, changes in economic aggregates are influenced by the changes in behavior of a very large number of agents. It is highly unlikely that all individual agents will react simultaneously to a given economic stimulus; while some may be able to (and want to) react instantaneously, others will be prone to different degrees of institutional inertia (dependent, for instance, on the efficiency of the markets in which they have to operate) and so will adjust with different time lags (Leybourne and Mizen, 1999, p 804).

where Bt is the real market value of government debt, Gt is real government expenditure inclusive of interest payments, Rt represents real tax 4 The content of this section draws heavily from Berenguer-Rico and Carrion-i-Silvestre (2011).

S.-W. Chen / Economic Modelling 37 (2014) 1–15

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Table 2 Studies adopt the cointegration approach to test for the fiscal sustainability. Studies

Countries and samples covered

Methodology

Sustainability

Cipollini (2001) Goyal et al. (2004)

The UK (1963–1997) India (1951–2000)

Non-linear stationarity and hold Hold

Bajo-Rubio et al. (2006)

Westerlund and Prohl (2008)

OECD countries (1977–2005)

Lusinyan and Thornton (2009)

South Africa (1895–2005)

Bajo-Rubio et al. (2010)

Spain (1850–2000)

Berenguer-Rico and Carrion-iSilvestre (2011) Mahdavi and Westerlund (2011)

The US (1947–2007)

Cointegration test with structural break and logistic smooth threshold error correction model Gregory and Hansen's (1996) cointegration test with structural break Shin's (1994) cointegration test Gregory and Hansen's (1996) cointegration test with structural break and Enders and Siklos, 2001) threshold cointegration test Westerlund's (2006) panel cointegration test with multiple breaks Gregory and Hansen's (1996) cointegration test with structural break Kejriwal and Perron's (2008, 2010) cointegration with multiple breaks and Hansen and Seo's (2002) threshold cointegration Multicointegration test with structural break

Non-linear stationarity and hold

Bajo-Rubio et al. (2008) Payne et al. (2008)

Spain (annual: 1964–2003) (quarterly: 1982–2004) Greece (1970–1998), Italy (1962–1997) Ireland and Netherlands (1957–1998) South Korea, Malaysia, Philippines Singapore, Thailand The US (1947–2005) Turkey (1968–2004)

The smooth transition error correction model Gregory and Hansen's (1996) cointegration test with structural break Hansen and Seo (2002) threshold cointegration

The US state and local governments (1961–2006)

Escario et al. (2012)

Spain (1857–2000)

Arghyrou and Luintel (2007) Baharumshah and Lau (2007)

Westerlund and Edgerton's (2007) panel bootstrap cointegration test Multicointegration test

revenues and Δ = (1 − L) is the first difference operator. By denoting it as the real interest rate and assuming it to be stationary around a mean i, we can define Gt ¼ GEt þ it Bt−1 ;

ð2Þ

Non-linear stationarity and hold

Hold Weakly sustainable Non-linear stationarity and hold

Hold Hold Non-linear stationarity and hold

Hold Hold Hold

where EXP t ¼ GEt þ ðit −iÞBt−1 ; or, alternatively, as


where GEt is the real expenditure exclusive of interest payments, and the second term on the right-hand side of Eq. (2) represents interest payments on the level of debt accumulated at the end of the previous period. Then, we can express the debt as Bt ¼ ð1 þ iÞBt−1 þ EXP t −Rt ;

ð3Þ

Bt ¼


  1  1 Rtþ1 −EXP tþ1 þ B ; 1þi 1 þ i tþ1

via forward substitution we obtain Bt ¼

 jþ1 
 jþ1 ∞
 X 1 1 Rtþ jþ1 −EXP tþ jþ1 þ lim Btþ jþ1 : 1þi j→∞ 1 þ i j¼0

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4 1980

1983

1986

1989

1992 DEBT

1995

1998

2001

2004

2007

2010

DEBT_sm_C

Fig. 1. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Canada.

ð4Þ

4

S.-W. Chen / Economic Modelling 37 (2014) 1–15

1.2

1.0

0.8

0.6

0.4

0.2

0.0 1980

1983

1986

1989

1992

1995

1998

DEBT

2001

2004

2007

2010

DEBT_sm_C

Fig. 2. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: France.

If the government is satisfying its budget constraint intertemporally, it cannot asymptotically leave a debt with positive expected present value. Hence, the intertemporal budget balance or deficit sustainability holds if and only if the current value of outstanding government debt is equal to the present value of future budget surpluses if and only if it is held that
lim Et

j→∞

 jþ1 1 Btþ jþ1 ¼ 0; 1þi

ð5Þ

where Et denotes the expectation that is conditional on information at time t. If condition (5), also known as the transversality condition or no-Ponzi game rule, is satisfied, then the deficit is sustainable, since the stock of debt held by the public is expected to grow no faster, on average, than the growth rate of the economy, which is proxied by the real interest rate.

The empirical literature has followed two different routes when assessing fiscal sustainability. The first group of studies — hereafter, univariate-based approach — has concentrated on the analysis of the univariate properties of Bt (see, for example, Davig, 2005; Hamilton and Falvin, 1986). The second group of studies — henceforth, cointegrationbased approach — assesses whether Rt and Gt are cointegrated (see, among others, Hakkio and Rush, 1991; Martin, 2000; Quintos, 1995; Trehan and Walsh, 1988). The cointegration framework for testing the intertemporal budget constraint will follow once first differences are taken in Eq. (4):

ΔBt ¼

 jþ1  ∞
 X 1 ΔRtþ jþ1 −ΔEXP tþ jþ1 1þi j¼0
 jþ1 1 þ lim ΔBtþ jþ1 : j→∞ 1 þ i

0.8

0.7

0.6

0.5

0.4

0.3

0.2

1980

1983

1986

1989

1992 DEBT

1995

1998

2001

2004

2007

2010

DEBT_sm_C

Fig. 3. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Germany.

ð6Þ

S.-W. Chen / Economic Modelling 37 (2014) 1–15

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2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 1980

1983

1986

1989

1992

1995

1998

DEBT

2001

2004

2007

2010

DEBT_sm_C

Fig. 4. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Japan.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3 1980

1983

1986

1989

1992

1995

DEBT

1998

2001

2004

2007

2010

DEBT_sm_C

Fig. 5. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: the UK.

As before, the sustainability hypothesis holds if the expected value of the second term on the right-hand side of Eq. (6) is zero. Hence sustainability is associated with the condition
lim Et

j→∞

 jþ1 1 ΔBtþ jþ1 ¼ 0: 1þi

ð7Þ

In order to test for condition (7), the usual procedure consists of testing for the stationarity of ΔBt = Gt − Rt, provided that both Gt and Rt are I(1), with a cointegration relationship (1, −1). Note that even in the case where the cointegrating vector is set to (1, − 1), the univariate-based and cointegration-based approaches

are focusing on different variables, since the former concentrates on the debt while the latter concentrates on the deficit. If condition (5) is satisfied, then the budget deficit will be sustainable through the debt. If condition (7) is satisfied, then the budget deficit will be sustainable through the deficit. It is important to notice that sustainability through the debt implies sustainability through the deficit, but the reverse is not true. Hence, sustainability through the debt can be understood as a deeper concept of sustainability in which the government is equilibrating not only the flows, but also the stocks in the long run. In this paper, we analyze the possibility that the debt Bt is itself I(0) using univariate-based approach. Furthermore, Bt ~ I(0) implies ΔBt ~ I(−1), and sustainability is achieved both through the deficit and debt. Moreover, the key for the fiscal deficit ‘sustainability’ to hold is that the debt–GDP ratio follows a stationary process. For example, Berenguer-Rico and Carrion-i-Silvestre (2011) consider a linear

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S.-W. Chen / Economic Modelling 37 (2014) 1–15

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1980

1983

1986

1989

1992

1995

1998

DEBT

2001

2004

2007

2010

DEBT_sm_C

Fig. 6. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: the US.

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 1980

1983

1986

1989

1992

1995

1998

DEBT

2001

2004

2007

2010

DEBT_sm_C

Fig. 7. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Greece.

broken trend in the cointegrated regression and test multicointegration for the government expenditure and revenue to examine the fiscal sustainability for the US.5 They consider the following empirical model:

a way that they are also controlling, to some extent, the stock of debt. By letting β21 = 1 in Eq. (8) leads to Bt ¼

t X j¼1

′

R j ¼ β 0 ct þ β21

t X

t X j¼1

G j þ β11 Gt þ ut ;

t X ′ R j− G j ¼ β 0 ct þ β11 Gt þ ut ;

ð9Þ

j¼1

ð8Þ

j¼1

where ct collects the deterministic regressors. Eq. (8) includes the level of expenditure among the stochastic regressors in order to cover the feature that multicointegration can be present in stock-flow systems. The specification given in Eq. (8) implies working within a stock-flow setup, which allows us to consider whether governments are taking corrective measures on flows — revenues and expenditures — in such 5 Leachman et al. (2005), Kia (2008) and Escario et al. (2012) also tackle the assessment of fiscal solvency by testing multicointegration between public spending and revenues.

where Bt is the stock of debt. Imposing β21 = 1 in Eq. (9) implies working with the strong sustainability definition given in Quintos (1995), while leaving β21 to be free is in accordance with the weak sustainability concept in Quintos (1995). Berenguer-Rico and Carrion-i-Silvestre (2011) consider that the structural break only affects the deterministic component (Model 4) and the structural break affects both the deterministic component and the parameters of the stochastic regressors (Models 6 and 8).6 The idea of the LSTR-TAR and LSTR-MTAR models (see Section 3.2 for details) used in this paper are in line with that of 6

Readers are referred to Table 1 in their paper for detailed specifications of Models 1–8.

S.-W. Chen / Economic Modelling 37 (2014) 1–15

7

1.4

1.2

1.0

0.8

0.6

0.4

0.2 1980

1983

1986

1989

1992

1995

1998

DEBT

2001

2004

2007

2010

DEBT_sm_C

Fig. 8. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Ireland.

Berenguer-Rico and Carrion-i-Silvestre (2011) which allow structural break in the deterministic component. However, these models permit structural shifts to occur gradually over time instead of instantaneously.

where the indicator variable is defined as:  It ¼

1; 1;

if if

Bt−1 ≥τ; Bt−1 b τ

ð11Þ

1; 1;

if if

ΔBt−1 ≥τ; ΔBt−1 b τ

ð12Þ

3. Methodology or

3.1. TAR and MTAR unit root tests

 The well-known Dickey–Fuller test and its extensions assume a unit root as the null hypothesis and a symmetric adjustment process under the alternative. These tests are misspecified if the adjustment dynamics are non-linear or asymmetric. A formal way to quantify an asymmetric adjustment process as a generalization of the Dickey–Fuller test is given by the TAR and MTAR models proposed by Enders and Granger (1998) and Enders and Siklos (2001). Let Bt denote the debt-to-GDP ratio and consider the following regression: ΔBt ¼ I t ρ1 Bt−1 þ ð1−It Þρ2 Bt−1 þ εt

ð10Þ

It ¼

and τ denotes the value of the threshold which is derived by minimizing the residual sum of squares. Eqs. (10)–(11) and (10) and (12) are called the TAR model and the MTAR model, respectively. The MTAR model allows the speed and direction of adjustment, represented by ρ1 and ρ2, to depend upon the previous period's change in Bt − 1, which is especially valuable when the adjustment is believed to exhibit more momentum in one direction than the other.

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7 1980

1983

1986

1989

1992 DEBT

1995

1998

2001

2004

2007

2010

DEBT_sm_C

Fig. 9. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Italy.

8

S.-W. Chen / Economic Modelling 37 (2014) 1–15

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 1980

1983

1986

1989

1992

1995

1998

DEBT

2001

2004

2007

2010

DEBT_sm_C

Fig. 10. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Portugal.

If the system is convergent, ΔBt = τ is the long-run equilibrium value. In the case where ΔBt is above its long-run equilibrium value, the adjustment is ρ1Bt − 1, and if it is below its equilibrium value, the adjustment is ρ2Bt − 1. The Dickey–Fuller test is a special case of the MTAR Models (10) and (12) in case of a symmetry in the error correction process where ρ1 = ρ2. The MTAR model sets up the null hypothesis of a unit root in the debt–GDP ratio, that is, H0 : ρ1 = ρ2 = 0. The distribution for this statistic is non-standard and, therefore, the critical values provided in Enders and Granger (1998), and Enders and Siklos (2001), are used. We denote the statistics testing the null hypothesis of a unit root (or no cointegration) as FC. If this null hypothesis is rejected, the null hypothesis of symmetric adjustment, H0 : ρ1 = ρ2, can be tested using the usual F-statistics denoted as FA. In the case where the null hypothesis

H0 :ρ1 =ρ2 is not rejected, the result is in favor of a linear and symmetric adjustment in the debt–GDP ratio.

3.2. LSTR-TAR and LSTR-MTAR unit root tests Sollis (2004) and Cook and Vougas (2009) combine the ideas of Enders and Granger (1998) and Leybourne et al. (1998) and develop a test for the null hypothesis of a unit root, that under the alternative hypothesis allows for a stationary asymmetric adjustment around a smooth transition between deterministic linear trends. Leybourne et al. (1998) consider three models: Model A Bt ¼ α 1 þ α 2 St ðγ; cÞ þ ν t ;

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1980

1983

1986

1989

1992 DEBT

1995

1998

2001

2004

2007

2010

DEBT_sm_C

Fig. 11. Debt–GDP ratio (black line) and the fitted logistic smooth transition function (blue line) for Model C: Spain.

ð13Þ

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Table 3 Summary statistics.

Mean S.D. SK EK JB LB(24) ARCH(4)

Canada

France

Germany

Japan

UK

US

Greece

Ireland

Italy

Portugal

Spain

0.002 0.014 0.812 3.072 65.960** 138.839** 3.564**

0.006 0.015 0.831 1.802 32.839** 87.493** 9.199**

0.003 0.007 2.448 12.501 983.869** 52.552** 0.280

0.013 0.018 0.549 1.240 14.990** 174.150** 13.791**

0.003 0.012 0.778 0.918 17.835** 415.462** 89.571**

0.004 0.013 1.417 5.022 181.543** 176.706** 2.431**

0.011 0.022 0.462 1.787 22.125** 90.183** 19.327**

0.003 0.049 2.140 11.520 824.430** 113.651** 22.772**

0.002 0.013 -0.562 1.477 18.833** 205.268** 39.489**

0.009 0.012 0.287 −0.228 2.094 555.229** 83.878**

0.004 0.013 0.654 −0.101 9.418** 437.460** 54.175**

(1) ** denotes significance at the 5% level. (2) Mean and S.D. refer to the mean and standard deviation, respectively. (3) SK is the skewness coefficient. (4) EK is the excess kurtosis coefficient. (5) JB is the Jarque–Bera statistic. (6) LB(24) is the Ljung–Box Q statistic calculated with twenty-four lags. (7) ARCH(4) is the ARCH test calculated with four lags on raw returns.

Model B Bt ¼ α 1 þ β1 t þ α 2 St ðγ; cÞ þ ν t ;

ð14Þ

Model C Bt ¼ α 1 þ β1 t þ α 2 St ðγ; cÞ þ β2 tSt ðγ; cÞ þ νt ;

ð15Þ

where vt is a zero mean I(0) process, St(γ,c) is the logistic smooth transition function: St ðγ; cÞ ¼ ½1 þ expf∞−γðt−cT Þg

−1

:

ð16Þ

The parameter c determines the timing of the transition midpoint. Since γ N 0, we have S−∞(γ,c) = 0, S+∞(γ,c) = 1, and ScT(γ,c) = 0.5. The speed of transition is determined by the parameter γ. If vt is a zero-mean I(0) process, then in Model A Bt is stationary around a mean which changes from an initial value α1 to a final value α1 + α2. Model B is similar, with the intercept changing from α1 to α1 + α2, but allowing for a fixed slope term. In Model C, in addition to the change in intercept from α1 to α1 + α2, the slope also changes simultaneously,

and at the same speed of transition, from β1 to β1 + β2. The models (13), (14) and (15) can be used to test the following hypotheses; H 0 : Bt ¼ μ t ; μ t ¼ μ t−1 þ εt ; H 1 : ð13Þ; ð14Þorð15Þ; H 0 : Bt ¼ μ t ; μ t ¼ κ þ μ t−1 þ εt ; H 1 : ð14Þorð15Þ; where εt is assumed to be a stationary process with a mean of zero. Cook and Vougas (2009) combine Eqs. (13)–(15), (17)–(18) and propose a logistic smooth transition-momentum TAR (LSTR-MTAR) model as follows: ^ t ¼ Mt ρ ^ t−1 þ ð1−M t Þρ ^ t−1 þ ^1 ν ^2 ν Δν

k X

^δ Δν ^ t−i þ η ^t ; i

ð17Þ

i¼1

where Mt is the Heaviside indicator function,  Mt ¼

Table 4 Results of the linear unit root tests. Linear trend

Canada France Germany Japan UK US Greece Ireland Italy Portugal Spain

ADF

SP(1)

DF-GLS

−2.333 −3.442** −2.622 −1.329 −0.818 −1.754 −2.579 −1.671 −1.850 −2.240 −2.892

−0.886 −1.724 −1.689 −0.808 −1.052 −1.895 −1.561 −0.804 −1.530 −0.783 −0.975

−0.391 −0.671 −0.938 0.473 1.572 0.216 −0.879 −0.281 −0.782 0.242 −0.560

Quadratic trend and breaks tests

Canada France Germany Japan UK US Greece Ireland Italy Portugal Spain

SP(2)

ZA, Model C

LP, Model C

−1.224 −1.592 −1.759 −1.317 −1.542 −1.248 −2.211 −1.227 −1.708 −1.210 −1.437

−3.060 −3.130 −4.886 −3.798 −3.692 −3.497 −2.517 −3.124 −3.804 −3.171 −2.466

−6.127 −5.832 −6.352 −5.334 −5.176 −5.612 −6.229 −4.250 −5.010 −5.695 −5.308

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) ADF, SP(1) and DFGLS denote the augmented Dickey–Fuller test, Schmidt–Phillips τ test with linear trend and Elliott et al.'s (1996) DF-GLS test, respectively. (3) SP(2), ZA and LP denote the Schmidt–Phillips τ test with quadratic trend, Zivot and Andrews' (1992) and Lumsdaine and Papell's (1997) tests, respectively. (4) The 5% critical values for the ADF, SP(1) and DF-GLS tests are −3.43, −3.04 and −2.89, respectively. (5) The 5% critical values for the SP(2), ZA and LP tests are −3.55, −5.08 and −6.75, respectively.

^ t−1 ≥ 0; 1; if Δν ^ t−1 b 0; 0; if Δν

ð18Þ

and vt is the residual from the first step by using the non-linear least squares approach for Eqs. (13)–(15).7 If the null hypothesis of H0 : ρ1 = ρ2 = 0 cannot be rejected in ^ t and therefore Bt contain a unit root. The statistics are Eq. (17), then ν ∗ ∗ referred to as Fα∗, Fα(β) and Fαβ corresponding to Models A, B and C, respectively. If the null hypothesis of H0 : ρ1 = ρ2 = 0 is rejected and ^ t (Bt) is a stationary LSTR-MTAR process with ρ1 = ρ2 b 0 holds, then ν symmetric adjustment. If H0 : ρ1 = ρ2 = 0 is rejected and ρ1 b 0, ρ2 b 0, ^ t (Bt) is a stationary LSTR-MTAR process and ρ1 ≠ ρ2 hold, then ν displaying asymmetric adjustment. Critical values must be tabulated via Monte Carlo simulations. The TAR model allows the degree of autoregressive decay to depend upon the state of the debt–GDP ratio, measuring the “deep” cycles. For instance, if the autoregressive decay take place rapidly when the imbalance is above the trend and slow when the imbalance is below the trend, troughs will be more persistent than peaks. Likewise, if the autoregressive decay is slow when the imbalance is above the trend and fast when the deficit is above the trend, peaks will be more persistent than troughs. On the other hand, the MTAR model allows the debt– GDP ratio to display differing amounts of autoregressive decay depending on whether the imbalance is increasing or decreasing, measuring the “sharpness” of cycles (Payne and Mohammadi, 2006). 7

If we replace Eq. (18) with 

Mt ¼

1; 0;

if if

^ t−1 ≥0; ν ^ t−1 b 0; ν

then it is called the logistic smooth transition TAR (LSTR-TAR) model proposed by Sollis ^t (2004). If the null hypothesis of H0: ρ1 = ρ2 = 0 cannot be rejected in Eq. (17), then ν nt and therefore Bt contain a unit root. The statistics are referred to as Fα, Fα(β) and Fαβ corresponding to Models A, B and C, respectively.

10

S.-W. Chen / Economic Modelling 37 (2014) 1–15

Table 5 Results of the TAR unit root test — G-6. Demeaned data (1) Canada Attractor 0.963 FC 3.902 FA 3.078** [0.081] ρ1 −0.130* (0.068) −0.009** ρ2 (0.004)

Demeaned and detrended data

(2) France

(3) Germany

(4) Japan

(5) UK

(6) US

(7) Canada

(8) France

(9) Germany

(10) Japan

(11) UK

(12) US

0.838 0.992 0.313 [0.576] 0.007 (0.022) −0.005 (0.004)

0.674 3.2539 2.449 [0.120] −0.201 (0.124) −0.006** (0.003)

0.487 1.929 0.511 [0.476] 0.006** (0.003) −0.046 (0.072)

0.410 2.307 3.832* [0.052] −0.004 (0.006) −0.058** (0.027)

0.681 1.667 1.012 [0.316] −0.001 (0.012) −0.017 (0.009)

0.370 + 0.002t 10.518** 20.834** [0.000] −0.001 (0.003) −4.783** (1.047)

0.248 + 0.006t 8.641** 6.069** [0.015] −0.866** (0.337) −0.031** (0.010)

0.202 + 0.003t 6.811** 13.608** [0.000] 0.002 (0.017) −48.202** (13.066)

−0.135 + 0.01t 8.395** 10.483** [0.001] 0.014** (0.005) −0.904** (0.592)

0.173 + 0.002t 3.314 5.984** [0.015] −0.002 (0.003) −0.322** (0.130)

0.312 + 0.003t 6.364** 11.287** [0.001] −0.003 (0.007) −0.107** (0.030)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) FC and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the FC statistic of demeaned data are 3.74, 4.56 and 6.47, respectively. (4) The 10%, 5% and 1% critical values for the FC statistic of demeaned and detrended data are 5.18, 6.12 and 8.23, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values. Table 6 Results of the MTAR unit root test — G-6. Demeaned data (1) Canada Attractor 0.697 FC 4.271 FA 5.092** [0.025] ρ1 −0.023** (0.009) ρ2 0.009 (0.010)

Demeaned and detrended data

(2) France

(3) Germany

(4) Japan

(5) UK

(6) US

(7) Canada

(8) France

(9) Germany

(10) Japan

(11) UK

(12) US

0.495 2.530 4.880** [0.029] −0.008 (0.007) 0.020 (0.009)

0.674 2.014 0.049 [0.825] −0.007 (0.004) −0.005** (0.007)

0.487 4.298 0.438** [0.015] 0.010** (0.004) 0.007 (0.006)

0.464 5.301** 8.357** [0.004] −0.005 (0.007) −0.051** (0.015)

0.725 1.599 0.522 [0.471] −0.012** (0.007) −0.001 (0.013)

0.691 + 0.0002t 4.154 4.619** [0.033] −0.026** (0.009) 0.005 (0.010)

0.197 + 0.006t 7.104** 1.227 [0.270] −0.072** (0.020) −0.034 (0.026)

0.309 + 0.003t 6.887** 4.242** [0.041] −0.072** (0.020) −0.001 (0.021)

0.303 + 0.015t 6.736** 1.652 [0.201] −0.019** (0.005) −0.006 (0.008)

0.391 + 0.001t 6.889** 8.631** [0.004] −0.003 (0.007) −0.041** (0.011)

0.489 + 0.003t 3.725 0.107 [0.743] −0.027** (0.011) −0.020 (0.016)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) FC and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the FC statistic of demeaned data are 4.05, 4.95 and 6.91, respectively. (4) The 10%, 5% and 1% critical values for the FC statistic of demeaned and detrended data are 5.64, 6.65 and 8.85, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

The data include quarterly observations of the debt imbalance as percentages of GDP. We consider the G-7 industrial countries, i.e., Canada, France, Germany, Italy, Japan, the UK and the US, as well as some European countries, i.e., Greece, Ireland, Portugal and Spain, in our empirical study. The sample period was determined primarily based on the availability of the data. The sample period is 1980:Q1–2012:Q4, and provides a total of 132 observations. All data are obtained from Datastream. Some descriptive statistics of changes of the debt–GDP series are outlined in Table 3, which details the first four moments of each series and presents tests for normality and serial correlation. Several interesting facts are observed from Table 3. First, with the exception of Italy, the coefficients of skewness of all of the series are positive, implying that the changes in the debt–GDP ratios are flatter to the right compared to the normal distribution. The coefficients of excess kurtosis for the raw changes are much higher than 0, indicating that the empirical distributions of these samples have fat tails. The coefficients of skewness and excess kurtosis reveal non-normality in the data with the exception of Portugal. This is confirmed by the Jarque–Bera normality test as shown in Table 3. Second, the Ljung–Box Q-statistics, LB(24), denote significant autocorrelations for all of the series. We also report a standard ARCH test for the changes in the debt–GDP ratios. The test results indicate that a significant ARCH effect exists for most of the debt–GDP ratios, with the exception of Germany. As a preliminary analysis, we apply a battery of linear unit root tests to determine the order of integration of the debt–GDP ratio.8 We consider the Augmented Dickey–Fuller (ADF) test, as well as the ADF-GLS

test of Elliott et al. (1996) in this study. Vougas (2007) highlights the usefulness of the Schmidt and Phillips (1992) (SP hereafter) unit root test in practice. Therefore, we also employ it in this study. These authors propose some modifications of existing linear unit root tests in order to improve their power and size. For the ADF and ADF-GLS tests, an auxiliary regression is run with an intercept and a time trend. To select the lag length (k) we use the ‘t-sig’ approach proposed by Hall (1994). That is, the number of lags is chosen for which the last included lag has a marginal significance level less than the 10% level. The results of applying these tests are reported in Table 4. We find that, with the exception of France, the null hypothesis of a unit root cannot be rejected at the 5% significance level for the ADF statistics. Based on the well-known low power problem of the ADF test, we turn our attention to other statistics. The SP test (see Schmidt and Phillips, 1992), with parametric correction, cannot reject the unit root hypothesis with both linear and quadratic trends at the 5% significance level.9 The results from the DF-GLS (see Elliott et al., 1996) indicate that the null hypothesis of a unit root cannot be rejected for all of the debt–GDP ratios, suggesting that the debt–GDP ratios for these countries are nonstationary processes. Based on the linear unit root test results, the presence of a unit root in the debt–GDP ratio is not in accordance with fiscal debt sustainability. As Perron (1989) pointed out, in the presence of a structural break, the power to reject a unit root decreases if the stationary alternative is true and the structural break is ignored. To address this, we use Zivot and Andrews' (1992) sequential one trend break model and Lumsdaine and Papell's (1997) two trend breaks model to investigate the order of the empirical variables. We use the ‘t-sig’ approach proposed by Hall (1994) to select the lag length (k). We set kmax = 12 and

8 An important advantage of the stationarity test is that it does not involve the estimation of an unknown cointegrating parameter.

9 The terms SP(1) and SP(2) denote the Schmidt–Phillips τ tests with the linear and quadratic trend, respectively.

4. Data and results 4.1. Data description and basic statistics

S.-W. Chen / Economic Modelling 37 (2014) 1–15

11

Table 7 Results of the TAR unit root test — European countries. Demeaned data

Attractor FC FA ρ1 ρ2

Demeaned and detrended data

(1) Greece

(2) Ireland

(3) Italy

(4) Portugal

(5) Spain

(6) Greece

(7) Ireland

(8) Italy

(9) Portugal

(10) Spain

1.078 5.443** 9.999** [0.002] 0.031** (0.011) −0.012** (0.005)

0.294 2.386 3.591** [0.049] −0.004 (0.005) −0.487 (0.242)

1.256 1.674 1.152 [0.285] −0.061 (0.052) −0.004 (0.003)

0.637 6.352** 12.572** [0.001] 0.016** (0.005) −0.013** (0.004)

0.603 2.496 1.208 [0.273] 0.001 (0.008) −0.010** (0.004)

0.580 + 0.008t 22.872** 32.205 [0.000] −5.995** (1.052) −0.018** (0.005)

1.040 − 0.002t 2.314 1.203 [0.274] −0.045 (0.026) −0.014 (0.010)

0.742 + 0.003t 10.654** 21.288** [0.000] −0.001 (0.004) −63.343** (13.728)

0.084 + 0.006t 3.509 7.003** [0.009] −0.001 (0.003) −1.430** (0.540)

0.593 + 0.002t 6.115** 7.327** [0.007] −3.153** (1.162) −0.005** (0.002)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) FC and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the FC statistic of demeaned data are 3.74, 4.56 and 6.47, respectively. (4) The 10%, 5% and 1% critical values for the FC statistic of demeaned and detrended data are 5.18, 6.12 and 8.23, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

Table 8 Results of the MTAR unit root test — European countries. Demeaned data

Attractor FC FA ρ1 ρ2

Demeaned and detrended data

(1) Greece

(2) Ireland

(3) Italy

(4) Portugal

(5) Spain

(6) Greece

(7) Ireland

(8) Italy

(9) Portugal

(10) Spain

0.690 3.832 5.609** [0.019] −0.001 (0.005) 0.020** (0.007)

0.802 2.130 0.603 [0.438] −0.021** (0.010) −0.001 (0.024)

1.200 2.127 2.202 [0.140] −0.001 (0.004) −0.017 (0.009)

0.725 4.405 8.078** [0.005] −0.0001 (0.002) −0.027** (0.009)

0.723 2.907 1.592 [0.209] −0.008** (0.003) −0.0001 (0.005)

0.394 + 0.010t 6.229* 0.658 [0.418] −0.034** (0.011) −0.019** (0.013)

0.952 − 0.002t 2.213 0.588 [0.444] −0.025** (0.012) −0.001 (0.029)

0.985 + 0.002t 2.375 1.584 [0.210] 0.006 (0.008) −0.027** (0.013)

0.055 + 0.006t 5.106 10.211** [0.001] −0.011** (0.005) 0.013** (0.005)

0.459+0.003t 8.625** 11.382** [0.001] −0.024** (0.005) 0.004 (0.005)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) FC and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the FC statistic of demeaned data are 4.05, 4.95 and 6.91, respectively. (4) The 10%, 5% and 1% critical values for the FC statistic of demeaned and detrended data are 5.64, 6.65 and 8.85, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

use the approximate 10% asymptotic critical value of 1.60 to determine the significance of the t-statistic on the last lag. We use the ‘trimming region’ [0.10 T, 0.90 T] and select the break point endogenously by choosing the value of the break that maximizes the ADF t-statistic. We report the results in Table 4. The results suggest that the null hypothesis of a unit root cannot be rejected at the 5% significance level for all of the debt–GDP ratios. These findings fully echo those obtained from the SP and DF-GLS linear unit roots.

4.2. Results based on the TAR and MTAR approaches We report the results for the demeaned, as well as demeaned and detrended data for Bt based on the following reason: if there is a time trend in the data and the regression equation does not contain a trend term, then the test has low power. On the other hand, if the regression equation contains a trend term but a trend does not exist in the data, then the null hypothesis is rejected too often. The debt–GDP ratio is demeaned by regressing Bt on a constant, C, and, alternatively, demeaned and detrended, C, T, by regressing Bt on a constant, as well as a linear trend prior to estimation in the TAR and MTAR regression equations. Hence, we allow for a constant term and a linear trend as attractors. We perform the tests with a linear time trend included due to its possible impact on the properties of the tests. The threshold or attractor is consistently estimated via Chan's (1993) method. This involves sorting the estimated residuals in ascending order, excluding 15% of the largest and smallest values, and selecting from the remaining 70% the threshold parameter which yields the lowest residual sum of squares (Enders and Siklos, 2001). We employ the ‘t-sig’ approach proposed by Hall (1994) to select the lag length (k). We set kmax = 12 and use the approximate

10% asymptotic critical value of 1.60 to determine the significance of the t-statistic on the last lag. The results of applying the TAR tests for the demeaned as well as the demeaned and detrended data of the debt–GDP ratios of the G-6 countries are reported in the top and bottom panels of Table 5, respectively. For the demeaned data (columns (1)–(6) in Table 5), the FC statistics are not significant at the conventional level, indicating that the null hypothesis of a unit root (H0 : ρ1 = ρ2 = 0) in the debt–GDP ratio cannot be rejected. However, for the demeaned and detrended data (columns (7)–(12) in Table 5), the null hypothesis of a unit root (H0 : ρ1 = ρ2 = 0) in the debt–GDP ratios of Canada, France, Germany, Japan and the US (columns (7)–(10) and (12) in Table 5) must be rejected at the 5% significance level. This finding can be interpreted as evidence in favor of a cointegrating relationship between the stock of government expenditure and stock of government revenue with a [1, −1] cointegrating vector, i.e., that the debt–GDP ratio is stationarity. Thus, for those countries in which the null hypothesis of a unit root is rejected using the TAR specification, the null hypothesis of symmetry where ρ1 = ρ2 is tested. The FA statistics show that this hypothesis is rejected at the 5% significance level for Canada, France, Germany, Japan and the US (columns (7)–(10) and (12) in Table 5), therefore indicating the presence of an asymmetric adjustment phenomenon. Taking France as an example, the point estimates suggest that the debt–GDP ratio tends to decay at the rate of |ρ1| = 0.866 for Bt − 1 above the attractor 0.248 + 0.006(t − 1) and at the rate of |ρ2| = 0.031 for Bt − 1 below the attractor. In regard to the speeds of adjustment, it appears that |ρ1| is greater than |ρ2|. This is evidenced by the null hypothesis of symmetry (ρ1 = ρ2) which is rejected. Thus, the evidence suggests that the ‘deep’ cycles (adjustments) around the threshold value of the debt are asymmetric. Next, the MTAR specification which has favorable power and size properties relative to the alternative of symmetric adjustment (Enders

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S.-W. Chen / Economic Modelling 37 (2014) 1–15

Table 9 Results of the LSTR-TAR unit root test — G-6. Model B

Fα(β) FA ρ1 ρ2

Model C

(1) Canada

(2) France

(3) Germany

(4) Japan

(5) UK

(6) US

(7) Canada

(8) France

(9) Germany

(10) Japan

(11) UK

(12) US

5.069 8.002** [0.005] −0.013 (0.009) −0.517** (0.178)

5.648 0.436 [0.510] −0.058** (0.027) −0.084** (0.031)

13.012** 5.447** [0.021] −0.128** (0.040) −0.304** (0.069)

5.137 2.502 [0.116] −0.192** (0.080) −0.061** (0.026)

7.240 0.045 [0.830] −0.053** (0.020) −0.047** (0.016)

9.600 2.085 [0.151] −0.064** (0.022) −0.122** (0.035)

12.362** 12.011** [0.001] −0.512** (0.126) −0.077** (0.023)

4.236 0.604 [0.438] −0.081** (0.033) −0.048 (0.028)

12.791** 4.429** [0.037] −0.133** (0.042) −0.288** (0.066)

8.060 0.406 [0.524] −0.192** (0.065) −0.143** (0.047)

8.163 0.672 [0.413] −0.067** (0.021) −0.046** (0.015)

19.872*** 2.041 [0.155] −0.237** (0.046) −0.160** (0.035)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) Fα(β) and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the Fα(β) statistic of Model B are 10.23, 11.56 and 14.26, respectively. (4) The 10%, 5% and 1% critical values for the Fαβ statistic of Model C are 11.34, 12.78 and 15.89, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

and Siklos, 2001, p. 166) is examined. The results of the MTAR unit root test for the debt–GDP ratios of the G-6 countries are reported in Table 6. It is shown that, with the exception of the UK (column (5) in Table 6), the null hypothesis of a unit root cannot be rejected at the 5% significance level for the demeaned data. For the demeaned and detrended data, the null hypothesis of non-stationarity is rejected at the 5% significance level for France, Germany, Japan and the UK (columns (8)–(11) in Table 6), implying that the debt–GDP ratios of these countries are stationary processes and thus sustainable. In addition, the asymmetric adjustment hypothesis is supported for Germany and the UK (columns (9) and (11) in Table 6) since the FA statistic is rejected at the conventional level. For France, the point estimates suggest that the debt–GDP ratio tends to decay at the rate of |ρ1| = 0.072 for ΔBt − 1 above the threshold 0.197 + 0.006(t − 1) and at the rate of |ρ2| = 0.034 for ΔBt − 1 below the threshold, while the size of |ρ1| is slightly greater than |ρ2|. The null hypothesis of symmetry where ρ1 = ρ2 is not rejected. Thus, it appears that the ‘sharpness’ cycles (adjustments) around the threshold value of the debt are symmetric. The same exercise is applied to the European countries, i.e., Greece, Ireland, Italy, Portugal and Spain. We summarize the results of the TAR and MTAR unit root tests in Tables 7 and 8, respectively. The TAR model for the demeaned data provides evidence that the null hypothesis of a unit root is rejected at the 5% significance level for the debt–GDP ratios of Greece and Portugal (columns (1) and (4) in Table 7). In addition, the null hypothesis of symmetric adjustment is rejected at the 5% level for Greece and Portugal, indicating that the debt–GDP ratios for the two countries are stationary processes with asymmetric adjustment. For the demeaned and detrended data, the FC statistics must be rejected at the conventional significance level for Greece, Italy and Spain (columns (6), (8) and (10) in Table 7). Only the debt–GDP ratios of Italy and Spain (columns (8) and (10) in Table 7) display asymmetric adjustment since the null hypothesis of ρ1 = ρ2 is rejected at the 5% significance level.

For the case of the MTAR model with the demeaned data, the null hypothesis of non-stationarity cannot be rejected for Greece, Ireland, Italy, Portugal and Spain (columns (1)–(5) in Table 8). For the case of the demeaned and detrended data, the test results show that the debt–GDP ratio of Greece (column (6) in Table 8) exhibits a stationary process with symmetric adjustment. For the case of Spain (column (10) in Table 8), the debt–GDP ratio follows a stationary process with asymmetric adjustment.

4.3. Results of the LSTR-TAR and LSTR-MTAR approaches Bierens (1997) and Leybourne et al. (1998) emphasize that nonlinearity may occur in the form of structural changes in the deterministic components. That is, a broken time trend is a particular case of a nonlinear time trend. In this study, for instance, there is an obvious shift in the trend of the debt–GDP ratio for Canada circa 1997 (see Fig. 1) and the US circa 2005 (see Fig. 6). Of particular importance, the shift seems to be smooth rather than abrupt. In order to take the possibility of the smooth transition non-linear trends into consideration, we use the logistic smooth transition TAR and MTAR models, championed by Sollis (2004) and Cook and Vougas (2009) respectively, in this study. These approaches permit structural shifts to occur gradually over time instead of instantaneously. The LSTR-TAR specification is examined by first testing the null hypothesis of a unit root, i.e., ρ1 = ρ2 = 0, in Eq. (17) and comparing the appropriate critical values from Sollis (2004). The results of applying the LSTR-TAR test for Model B (Eq. (14)) as well as Model C (Eq. (15)) of the debt–GDP ratios of the G-6 are reported in the top and bottom panels of Table 9, respectively. From the results of the Fα(β) and Fαβ as shown in Table 9, for Canada, Germany and the US (columns (3), (7), (9) and (12) in Table 9), the null hypothesis of a unit root is rejected at the 5% significance level or better. The results imply that the debt–

Table 10 Results of the LSTR-MTAR unit root test — G-6. Model B

∗ Fα(β) FA

ρ1 ρ2

Model C

(1) Canada

(2) France

(3) Germany

(4) Japan

(5) UK

(6) US

(7) Canada

(8) France

(9) Germany

(10) Japan

(11) UK

(12) US

5.289 1.431 [0.233] −0.068** (0.023) −0.032 (0.020)

7.446 3.831* [0.052] −0.108** (0.029) −0.032 (0.026)

12.877** 1.336 [0.250] −0.243 (0.055) −0.160** (0.053)

5.475 1.731 [0.190] −0.124 (0.039) −0.049 (0.044)

7.282 0.106 [0.744] −0.046** (0.017) −0.054** (0.018)

9.187 0.198 [0.656] −0.094** (0.028) −0.077** (0.026)

11.470 0.826 [0.365] −0.172** (0.043) −0.122** (0.038)

5.436 2.938* [0.089] −0.104** (0.033) −0.030 (0.028)

12.807 1.418 [0.236] −0.243** (0.055) −0.157** (0.052)

9.199 2.193 [0.141] −0.208 (0.050) −0.099** (0.059)

8.074 0.058 [0.809] −0.057** (0.018) −0.051** (0.018)

20.264*** 1.351 [0.247] −0.237** (0.049) −0.171** (0.036)

∗ (1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) Fα(β) and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the F∗α(β) statistic of Model B are 10.31, 11.52 and 14.18, respectively. (4) The 10%, 5% and 1% critical values for the F∗αβ statistic of Model C are 11.67, 13.05 and 16.15, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

S.-W. Chen / Economic Modelling 37 (2014) 1–15

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Table 11 Results of the LSTR-TAR unit root test — European countries. Model B

Fα(β) FA ρ1 ρ2

Model C

(1) Greece

(2) Ireland

(3) Italy

(4) Portugal

(5) Spain

(6) Greece

(7) Ireland

(8) Italy

(9) Portugal

(10) Spain

1.085 0.393 [0.531] −0.015 (0.032) −0.040 (0.027)

9.499 9.234** [0.003] −0.018 (0.031) −0.179** (0.041)

11.639** 7.668** [0.006] −0.161 (0.043) −0.036** (0.011)

4.017 0.799 [0.372] −0.024 (0.014) −0.011** (0.005)

3.228 2.585 [0.110] −0.005 (0.007) −0.026** (0.010)

1.099 0.429 [0.513] −0.015 (0.033) −0.041 (0.028)

10.832 0.388 [0.534] −0.206** (0.059) −0.259 (0.071)

13.077** 13.692** [0.000] −0.274** (0.046) −0.031** (0.010)

1.393 0.424 [0.515] −0.001 (0.013) −0.010 (0.006)

6.974 0.105 [0.745] −0.049** (0.017) −0.041** (0.016)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) Fα(β) and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the Fα(β) statistic of Model B are 10.23, 11.56 and 14.26, respectively. (4) The 10%, 5% and 1% critical values for the Fαβ statistic of Model C are 11.67, 13.05 and 16.15, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

Table 12 Results of the LSTR-MTAR unit root test — European countries. Model B

∗ Fα(β) FA

ρ1 ρ2

Model C

(1) Greece

(2) Ireland

(3) Italy

(4) Portugal

(5) Spain

(6) Greece

(7) Ireland

(8) Italy

(9) Portugal

(10) Spain

4.521 6.923** [0.009] −0.069** (0.023) 0.012 (0.021)

4.180 0.104 [0.746] −0.055** (0.021) −0.074 (0.055)

10.839* 0.187 [0.665] −0.070** (0.018) −0.058** (0.022)

5.365 2.448 [0.120] −0.023** (0.007) −0.004 (0.010)

2.933 1.384 [0.241] −0.020** (0.008) −0.006 (0.007)

4.701 7.423** [0.007] −0.071** (0.024) 0.012 (0.020)

10.798 0.146 [0.702] −0.214** (0.065) −0.248 (0.067)

10.550 0.254 [0.614] −0.075** (0.019) −0.059** (0.024)

1.285 1.496 [0.223] −0.001 (0.005) −0.011 (0.007)

10.219 7.513** [0.007] −0.080** (0.018) −0.015 (0.014)

(1) *, **, *** denote significance at the 10%, 5% and 1%, respectively. (2) F∗α(β) and FA denote the F-statistics for the null hypothesis of a unit root H0 : ρ1 = ρ2 = 0 and symmetry H0 : ρ1 = ρ2, respectively. (3) The 10%, 5% and 1% critical values for the F∗α(β) statistic of Model B are 10.31, 11.52 and 14.18, respectively (4) The 10%, 5% and 1% critical values for the F∗αβ statistic Model C are 11.67, 13.05 and 16.15, respectively. (5) The numbers in parentheses are standard errors. (6) The numbers in square brackets are p-values.

GDP ratios are non-linear trend stationary processes for these countries. 10 In addition, the null hypothesis of symmetry (ρ1 = ρ2) is rejected at the conventional significance level for Canada and Germany (columns (7) and (9) in Table 9), but not for the US. Thus, the evidence suggests that the ‘deep’ cycles (adjustments) around the threshold value of the debt–GDP imbalances of Canada and Germany are asymmetric. Next, we turn our attention to the results of the LSTR-MTAR model. This model allows the adjustment to depend on the previous period's change in the government deficit. The results for the test statistics F∗α(β) and F∗αβ of the LSTR-MTAR model for Models B and C are reported in the top and bottom panels of Table 10, respectively. For Germany and the US (columns (3), (9) and (12) in Table 10), the null hypothesis of a unit root (ρ1 = ρ2 = 0) in Model C is rejected at the 10% or better significance level. The results indicate that the debt–GDP ratios for these countries once again exhibit non-linear trend stationarity. However, the null hypothesis of symmetry (ρ1 = ρ2) is not rejected at the 5% significance level. Thus, it appears that the ‘sharpness’ cycles (adjustments) around the threshold values of the debt–GDP ratios of these countries are symmetric. The results of the LSTR-TAR and LSTR-MTAR approaches for the debt–GDP ratios of Greece, Ireland, Italy, Portugal and Spain are outlined in Tables 11 and 12, respectively. The results from the LSTR-TAR test provide the evidence that the debt–GDP ratio of Italy (columns (3) and (8) in Table 11) exhibit a non-linear trend stationary process with asymmetric adjustment. From the LSTR-MTAR model, the debt–GDP ratios of Italy (column (3) in Table 12) is non-linear trend stationary processes and exhibits symmetric adjustment.

10 Strictly speaking, Fα(β), Fαβ and Fα(β)⁎, Fαβ⁎ are not tests for long-run IBC. The rejections obtained from Fα(β) and Fαβ Fαβ⁎ (Models B and C of the LSTR-TAR model) as well as Fα(β)⁎ and Fαβ⁎ (Models B and C of the LSTR-MTAR model) simply reveal that the hypothesis of stationarity around a non-linear trend is preferred to the hypothesis of a unit root.

All in all, the empirical results for the respective LSTR-TAR and LSTRMTAR models reveal that the debt–GDP ratios of Canada, Germany, the US and Italy are stationarity after taking into account the non-linear trend. Nevertheless, it is model-dependent for the debt–GDP ratios to be asymmetrically adjusted after taking the non-linear trend into consideration. Figs. 1 to 11 present the time series plots of the debt–GDP ratios (black line) and the estimated logistic smooth transition functions (blue line) for the G-6 and European countries, respectively.11 Intuitively, if the true data generating process follows the non-linear process of a logistic smooth transition function, then the estimated logistic smooth transition trend is close to that of the raw data. As such, it is highly possible to reject the null hypothesis of non-stationarity. If we take the US as an example (Fig. 6), the estimated logistic smooth transition trend of Model C is quite close to that of the raw data. This plot echoes the rejection of the null hypothesis of a unit root by the F∗α(β) and F∗αβ statistics as shown in Table 10. 5. Concluding remarks This paper attempts to reexamine the non-stationarity of the debt– GDP ratios of the G-7 and some European countries in order to test for fiscal sustainability. A variety of unit root tests ranging from univariate estimators to non-linear testing principles are employed in an effort to obtain inferences that are robust to problems associated with nonstationarity. We adopt the TAR, MTAR, LSTR-TAR and LSTR-MTAR unit root tests which help detect the non-linear debt–GDP relationship without specifying the threshold in advance. For the benefit of readers, we summarize our empirical results in Table 13. This study reaches the following key conclusions. First, by using a battery of univariate unit root tests, the debt–GDP ratios of the G-7 11 The detailed estimation results of the logistic smooth transition model, i.e., Eqs. (15)–(18), are available from the author upon request.

14

S.-W. Chen / Economic Modelling 37 (2014) 1–15

Table 13 Summary of a variety of non-linear unit root tests. Country

TAR (asymmetry)

Canada France Germany Japan UK US Greece Ireland Italy Portugal Spain

Yes (yes) Yes (yes) Yes (yes) Yes (yes) Yes (yes) Yes (yes) Yes (yes) Yes (yes) Yes (yes)

MTAR (asymmetry)

LSTR-TAR (asymmetry)

LSTR-MTAR (asymmetry)

Yes (yes) Yes Yes (yes) Yes Yes (yes)

Yes (yes)

Yes

Yes

Yes

Yes (yes)

Yes

Yes

Yes (yes)

(1) Term “yes” denotes that the null of a unit root is rejected and in favor of the stationary process with symmetric adjustment. (2) Term “yes (yes)” denotes that the null of a unit root is rejected and in favor of the stationary process with asymmetric adjustment. (3) TAR and MTAR denote the non-linear unit root tests proposed by Enders and Granger (1998). (4) LSTR-TAR and LSTR-MTAR denote the non-linear unit root tests proposed by Sollis (2004) and Cook and Vougas (2009), respectively.

and some European countries are characterized by a unit root process, which is in violation of fiscal sustainability. Second, the results from the TAR and MTAR unit root regressions suggest that the debt–GDP ratios are stationary series with asymmetric adjustment, implying that the government's intertemporal budget constraint is sustainable in the long run. Our results echo the findings of, to name a few, Sarno (2001), Arestis et al. (2004), Bajo-Rubio et al. (2004, 2008) and Payne and Mohammadi (2006). Third, empirical evidence from the LSTRTAR and LSTR-MTAR tests shows that the debt–GDP ratios of Canada, Germany, the US and Italy are stationarity after taking account of the non-linear trend. Nevertheless, it is model-dependent for the debt– GDP ratios to be asymmetrically adjusted after taking the non-linear trend into consideration. Acknowledgments I would like to thank the editor, Professor Stephen George Hall, and two anonymous referees of this journal for helpful comments and suggestions. The usual disclaimer applies. References Afonso, A., 2005. Fiscal sustainability: the unpleasant European case. FinanzArchiv 61, 19–44. Afonso, A., Rault, C., 2010. What do we really know about fiscal sustainability in the EU? A panel data diagnostic. Rev. World Econ. 125, 731–755. Arestis, P., Cippollini, A., Fattouh, B., 2004. Threshold effects in the US budget deficit. Econ. Inq. 42, 214–222. Arghyrou, M.G., Luintel, K.B., 2007. Government solvency: revisiting some EMU countries. J. Macroecon. 29, 387–410. Baharumshah, A.Z., Lau, E., 2007. Regime changes and the sustainability of fiscal imbalance in East Asian countries. Econ. Model. 24, 878–894. Bajo-Rubio, O., Diaz-Roldán, C., Esteve, V., 2004. Searching for threshold effects in the evolution of budget deficit: an application for the Spanish case. Econ. Lett. 82, 239–243. Bajo-Rubio, O., Diaz-Roldán, C., Esteve, V., 2006. Is the budget deficit sustainable when fiscal policy is non-linear? The case of Spain. J. Macroecon. 28, 596–608. Bajo-Rubio, O., Diaz-Roldán, C., Esteve, V., 2008. US deficit sustainability revisited: a multiple structural change approach. Appl. Econ. 40, 1609–1613. Bajo-Rubio, O., Diaz-Roldán, C., Esteve, V., 2010. On the sustainability of government deficits: some long-term evidence for Spain, 1850–2000. J. Appl. Econ. 13, 263–281. Berenguer-Rico, V., Carrion-i-Silvestre, J.L., 2011. Regime shifts in stock-flow I(2)-I(1) systems: the case of US fiscal sustainability. J. Appl. Econ. 26, 298–321. Bertola, G., Drazen, A., 1993. Trigger points and budget cuts: explaining the effects of fiscal austerity. Am. Econ. Rev. 83, 11–26. Bierens, H.J., 1997. Testing the unit root with drift against non-linear trend stationarity, with an application to the US price level and interest rate. J. Econ. 81, 29–64. Bohn, H., 1998. The behavior of US public debt and deficits. Q. J. Econ. 113, 949–963. Caner, M., Hansen, B.E., 2001. Threshold autoregression with a unit root. Econometrica 69, 1555–1596. Chan, K., 1993. Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Ann. Stat. 21, 520–533. Cipollini, A., 2001. Testing for government intertemporal solvency: a smooth transition error correction model approach. Manch. Sch. 69, 643–655.

Cook, S., Vougas, D., 2009. Unit root testing against an ST-MTAR alternative: finitesample properties and an application to the UK housing market. Appl. Econ. 41, 1397–1404. Davig, T., 2005. Periodically expanding discounted debt: a threat to fiscal policy sustainability? J. Appl. Econ. 20, 829–840. Elliott, G., Rothenberg, T.J., Stock, J.H., 1996. Efficient tests for an autoregressive unit root. Econometrica 64, 813–836. Enders, W., Granger, C.W.J., 1998. Unit-roots tests and asymmetric adjustment with an example using the term structure of interest rates. J. Bus. Econ. Stat. 16, 304–311. Enders, W., Siklos, P.L., 2001. Cointegration and threshold adjustment. J. Bus. Econ. Stat. 19, 166–176. Escario, R., Gadea, M.D., Sabaté, M., 2012. Multicointegration, seigniorage and fiscal sustainability. Spain 1857–2000. J. Policy Model 34, 270–283. Feve, P., Henin, P., 2000. Assessing effective sustainability of fiscal policy within the G-7. Oxf. Bull. Econ. Res. 62, 175–195. Gabriel, V.J., Sangduan, P., 2011. Assessing fiscal sustainability subject to policy changes: a Markov switching cointegration approach. Empir. Econ. 41, 371–385. Getzner, M., Glatzer, E., Neck, R., 2001. On the sustainability of Austrian budgetary policies. Empirica 28, 21–40. Goyal, R., Khundrakpam, J.K., Ray, P., 2004. Is India's public finance unsustainable? Or, are the claims exaggerated? J. Policy Model 26, 401–420. Gregory, A.W., Hansen, B.E., 1996. Residual-based tests for cointegration in models with regime shifts. J. Econ. 70, 99–126. Hakkio, G., Rush, M., 1991. Is the budget deficit too large? Econ. Inq. 29, 429–445. Hall, A.D., 1994. Testing for a unit root in time series with pretest data based model selection. J. Bus. Econ. Stat. 12, 461–470. Hamilton, J.D., Falvin, M.A., 1986. On the limitations of government borrowing: a framework for empirical testing. Am. Econ. Rev. 76, 808–819. Hansen, B.E., Seo, B., 2002. Testing for two-regime threshold cointegration in vector errorcorrection models. J. Econ. 110, 293–318. Kalyoncu, H., 2005. Fiscal policy sustainability: test of intertemporal borrowing constraints. Appl. Econ. 12, 957–962. Kejriwal, M., Perron, P., 2008. The limit distribution of the estimates in cointegrated regression models with multiple structural changes. J. Econ. 146, 59–73. Kejriwal, M., Perron, P., 2010. Testing for multiple structural changes in cointegrated regression models. J. Bus. Econ. Stat. 28, 503–522. Kia, A., 2008. Fiscal sustainability in emerging countries: evidence from Iran and Turkey. J. Policy Model 30, 957–972. Leachman, L., Bester, A., Rosas, G., Lange, P., 2005. Multicointegration and sustainability of fiscal practices. Econ. Inq. 43 (2), 454–466. Leybourne, S., Mizen, P., 1999. Understanding the disinflations in Australia, Canada and New Zealand using evidence from smooth transition analysis. J. Int. Money Financ. 18, 799–816. Leybourne, S., Newbold, P., Vougas, D., 1998. Unit roots and smooth transitions. J. Time Ser. Anal. 19, 83–98. Lumsdaine, R.L., Papell, D.H., 1997. Multiple trend breaks and the unit root hypothesis. Rev. Econ. Stat. 79, 212–218. Lusinyan, L., Thornton, J., 2009. The sustainability of South African fiscal policy: an historical perspective. Appl. Econ. 41, 859–868. Maddala, G.S., Wu, S., 1999. A comparative study of unit root tests with panel data and a new simple test. Oxf. Bull. Econ. Stat. 61, 631–652. Mahdavi, S., Westerlund, J., 2011. Fiscal stringency and fiscal sustainability: panel evidence from the American state and local governments. J. Policy Model 33, 953–969. Makrydakis, S., Tzavalis, E., Balfoussias, A., 1999. Policy regime changes and the long-run sustainability of fiscal policy: an application to Greece. Econ. Model. 16, 71–86. Martin, G., 2000. U.S. deficit sustainability: a new approach based on multiple endogenous breaks. J. Appl. Econ. 15, 83–105. Payne, J.E., Mohammadi, H., 2006. Are adjustments in the U.S. Budget deficit asymmetric? Another look at sustainability. Atl. Econ. J. 34, 15–22. Payne, J.E., Mohammadi, H., Cak, M., 2008. Turkish budget deficit sustainability and the revenue-expenditure nexus. Appl. Econ. 40, 838-830. Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401. Perron, P., 1997. Further evidence on breaking trend functions in macroeconomic variables. J. Econ. 80, 355–385. Quintos, C., 1995. Sustainability of the deficit process with structural shifts. J. Bus. Econ. Stat. 13, 409–417. Sarno, L., 2001. The behavior of US public debt: a non-linear perspective. Econ. Lett. 74, 119–125. Schmidt, P., Phillips, P.C.B., 1992. LM test for a unit root in the presence of deterministic trends. Oxf. Bull. Econ. Stat. 54, 257–276. Shin, Y., 1994. A residual-based test of the null of cointegration against the alternative of no cointegration. Econ. Theory 10, 91–115. Sollis, R., 2004. Asymmetric adjustment and smooth transitions: a combination of some unit root tests. J. Time Ser. Anal. 25, 409–417. Teräsvirta, T., 1994. Specification, estimation and evaluation of smooth transition autoregressive models. J. Am. Stat. Assoc. 89, 208–218. Trehan, B., Walsh, C.E., 1988. Common trends, the government's budget balance, and revenue smoothing. J. Econ. Dyn. Control. 12, 425–444. Trehan, B., Walsh, C.E., 1991. Testing intertemporal budget constraints: theory and applications to US federal budget and current account deficits. J. Money Credit Bank. 23, 206–223. Uctum, M., Wickens, M., 2000. Debt and deficit ceilings, and sustainability of fiscal policies: an intertemporal analysis. Oxf. Bull. Econ. Res. 62, 197–222.

S.-W. Chen / Economic Modelling 37 (2014) 1–15 Vougas, D.V., 2007. Is the trend in post-WW II US real GDP uncertain or non-linear? Econ. Lett. 94, 348–355. Westerlund, J., 2006. Testing for panel cointegration with multiple structural breaks. Oxf. Bull. Econ. Stat. 68, 101–132. Westerlund, J., Edgerton, D.L., 2007. A panel bootstrap cointegration test. Econ. Lett. 97, 185–190.

15

Westerlund, J., Prohl, S., 2008. Panel cointegration tests of the sustainability hypothesis in rich OECD countries. Appl. Econ. 42, 1335–1364. Wilcox, D., 1989. The sustainability of government deficits: implications of the presentvalue borrowing constraint. J. Money Credit Bank. 21, 291–306. Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil-price shock and the unit root hypothesis. J. Bus. Econ. Stat. 10, 251–270.