Boundedness and nonlinearities in public debt dynamics: A TAR assessment

ECMODE-02818; No of Pages 7 Economic Modelling xxx (2013) xxx–xxx

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Boundedness and nonlinearities in public debt dynamics: A TAR assessment Yacouba Gnegne a, Fredj Jawadi b,⁎ a b

France Business School, Campus Amiens & University of Auvergne (CERDI-CNRS), France University of Evry & France Business School, Campus Amiens, France

a r t i c l e Available online xxxx JEL classification: C22 H6 Keywords: Public debt Nonlinearity Threshold models

i n f o

a b s t r a c t This study aims to investigate the dynamics of public debts over more than four decades for two of the main developed countries: the USA and the UK. To do this, we apply nonlinearity tests and threshold models. While the first tests enable us to check for further changes in the data, threshold models are required to assess the switching-regime hypothesis and to apprehend the main changes in public debts through different regimes. Our results provide several interesting findings. First, for both countries, we noted several structural breaks associated with well-known economic downturns, oil shocks, debt crises and financial crashes. Second, public debt dynamics seem to be characterized by various threshold effects that can improve the modeling and forecasting of public debt evolution. It is important to note that public debts vary significantly according to the regime and that a regime can be induced by specific macroeconomic factors. Keeping a close eye on such factors may help economists and policymakers to better control future public debt evolutions. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Public debt (PD) encompasses all the liabilities that are debt instruments owed by governments and public administrations, companies and organisms. We identify domestic PD detained by resident economic agents and external PD that refers to foreign holders (China, Japan, oil exporting countries, Brazil, etc., for US PD for example), as well as short-term (less than one year), mid-term (up to 10 years) and long-term (over 10 years) PD. PD constitutes a crucial instrument for governments to finance public expenses, especially when it is difficult to increase taxes and/or reduce expenditure. Accordingly, PD is considered as a crucial issue for consumers, investors and policymakers, since a high PD-to-GDP ratio drags the whole economy down. It operates through different channels: public investment, private savings, total factor productivity, and sovereign long-term nominal and real interest rates (Checherita and Rother, 2010). The state of the international economic system after the recent financial downturn is now a well-known 21st-century example. Indeed, the subprime crisis that began in the United States in August 2007 induced a global financial crisis (2008–2009) that was marked by a considerable liquidity crunch and bank losses. In order to save their banking systems and limit the risk of a new Great Depression, the US and European countries decided to stimulate banking and financial market liquidity through credit channels. Consequently, governments borrowed large amounts from resident and non-resident economic agents. Government deficit ratio and debt thus increased ⁎ Corresponding author at: Université d'Evry Val d’Essonne, UFR Sciences de Gestion et Sciences Sociales, 2, rue Facteur Cheval, 91000, Evry, France. Tel.: +33 3 22 82 24 41. E-mail addresses: [email protected] (Y. Gnegne), [email protected] (F. Jawadi).

rapidly in the United States and the United Kingdom, as in many other developed and emerging countries. This led PD to reach exceptional levels for several developed and emerging countries and sovereign default risks also exceeded historic levels (e.g. the Greek crisis). In addition, as illustrated in Fig. 1, PD in most countries has been increasing for several years. In particular, there was a considerable increase in PD after the 1980s and after 1990 for most countries (especially Europe) due to a fall in economic growth and to high spread between interest and growth rates. In France, according to the INSEE (National Institute for Statistics and Economic Studies), PD increased by 43.2 billion euros in the second quarter of 2012, reaching 1832.6 billion euros, in other words, 91% of GDP and 90.2% in the end of 2012. 1 In the United Kingdom, PD also reached extremely high levels due to the increase in public expenses and the nationalization of several banks since 2008. PD stood at 82.49% of GDP in 2010 and 85.3% in 2011, while it had not exceeded 42.7% before 2006. In the USA, the situation is even more dramatic as PD was about 102.93% of GDP in 2010, against 65.6% in 2006, due to tax reductions, military expenses, etc.2 A common feature is the abrupt increase that marked the entire PD of several developed countries following the subprime crisis. However, it is important to note that PD holders vary from one country to another. In addition, PD is not always expressed symmetrically in domestic or

1 The PD/GDP ratio moved from 66.7% (2006) to 83.5% (2009), 86.26% (2010) and 86.10% (2011), source: CIA World Factbook. 2 In the US, a critical threshold is fixed for PD, and the country is automatically considered as bankrupt if this threshold is exceeded. The threshold was passed in the US on December 26, 2012 (the fiscal cliff) and an exceptional agreement was signed to raise the ceiling of debt.

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Please cite this article as: Gnegne, Y., Jawadi, F., Boundedness and nonlinearities in public debt dynamics: A TAR assessment, Economic Modelling (2013), http://dx.doi.org/10.1016/j.econmod.2013.04.006

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90 80 70 60 50 40 30 80 82 84 86 88 90 92 94 96 98 00 02 04

OECD

USA

ZE

Source: Schalck (2007a) Note : ZE refers to the Euro Area. Fig. 1. Public debt evolution. Source: Schalck (2007a). Note: ZE refers to the Euro Area.

foreign currencies (various exchange risks), and central bank policies toward government debts differ. 3 Furthermore, economists are far from unanimous about the approaches adopted to measure PD. 4 PD repartition between governments and households varies according to the country. Overall, we can distinguish two models: the Anglo-Saxon model and the European one. In the first model, household debt is significant (around 100% of GDP in the UK), while the government is less ‘indebted’ but plays a limited role. The second model involves more debt however and a need for government reforms, with more savings capacity and less household debt. For example, French household debt in 2008 was under 25% of GDP. We should remember that PD includes two main components: budgetary balance and interest rates. Thus, the frequent revisions to country and debt notations by rating agencies affect interest rates and hence the dynamic and level of PD. Accordingly, with a high level of prior debt, high interest rates, absence of strong productive investment and moderate economic growth rates, it becomes very difficult for governments to stabilize PD. This is especially true in the Euro area where, in theory, debt monetization is not allowed (Central Bank Independence principle with reference to the Lisbon Treaty, article 101) and where the ECB only recently moved to acquire Treasury securities. This can indirectly justify the “Snowball Effect” observed in Fig. 1, which the current US and European fiscal directives are attempting to stop through the imposition of some degree of fiscal discipline. Of course, all these conditions and parameters result in a number of difficulties when treating PD data and dynamics. In practice, several international and European measures and directives exist to try to limit government deficit and public debt, such as the Maastricht Treaty, the Stability and Growth Pact and the new Fiscal Compact in the Economic and Monetary Union (EMU), the creation of fiscal policy committees, or the Balanced Budget Amendment in the US. However, it appears that none of them is actually fully enforced due to the severity of the recent financial turmoil and economic recession. In order to put some discipline back into fiscal policy, and to reduce government deficit and expenses and public

3 The ECB is less exposed to governent debt (6%) than the Fed (12%) and the Bank of England (25%). This can be linked to the quantitative easing effect that stimulates the public debt financed by central banks. 4 Official PD measures are not anonymously accepted as they reflect gross PD rather than net PD and explicit government contracts. Moreover, the pre-cited PD figures can vary according to how PD is defined (Maastricht, OCDE).

debt, several governments, notably in Europe, have shifted to severe austerity programs. Concerning the US and the UK, as Jawadi and Sousa (2013) argued, fiscal austerity can be detrimental to growth in the short term. This therefore provides an interesting challenge regarding public debt and the different measures required and adopted to manage its evolution. The main objective of this study is to clarify the challenges while investigating the dynamic of PDs and analyzing their main properties and the different cycles associated with their dynamics. In theory, neo-Keynesian economists recommend counter-cyclical fiscal policies in order to smooth production variations induced by changes to the economic cycle. Such measures can of course lead to decreases in public debt. However, the failure of expansionist fiscal measures in developed countries in the 1980s was accompanied by higher indebtedness, forcing these countries to call these policies into question. 5 Furthermore, in practice, policymakers in developed countries often favor increases in public expenditure around election time (i.e., the election effect), potentially inducing an increase in PD. The alternation of different policy regimes is also a source of sometimes conflicting reforms that impact on fiscal policy and PD dynamics due to the difficulty in finding a consensus on fiscal consolidation (Alesina and Perotti, 1995). That is, PD dynamics are expected to be not only cyclical but also asymmetrical. This asymmetry is due to further asymmetry between economic cycle phases. In order to better characterize PD dynamics, we focus on two major countries, the US and the UK, and attempt to clarify their debt dynamics (Fig. 2). In other words, rather than focus on research on PD factors that have been investigated in several previous studies, or on the effects of fiscal rules and PD on economic systems (Afonso and Sousa, 2012; Agnello and Sousa, 2011; Agnello et al., forthcoming; Sousa, 2012), we focus on the specifications of PD dynamics while attempting to develop appropriate econometric specifications to capture the time-varying aspect of PD. To do this, we check for further structural changes and nonlinearity in public debt dynamics. Our main contribution consists of proposing a dynamic on/off adjustment specification to model PD dynamics and apprehend their main changes. This specification has the advantage of being robust to abrupt multiple structural breaks in PD, which also can capture asymmetry and nonlinearity in PD data. To our knowledge, this is the first essay that focuses on modeling nonlinearity in PD data. In the nonlinear literature, authors either focus on nonlinearity in fiscal rules (Jawadi et al., forthcoming; Schalck, 2007b) or on nonlinearity in PD effects on economic growth (Egert, 2012). Only two previous studies have focused on detecting structural breaks in PD data: Uctum et al. (2006) and Jawadi and Sousa (2013), but neither of them test or model nonlinearity in PD. The present study fills this gap. 6 The paper is structured as follows. Section 2 dresses a literature review on PD dynamics. The econometric methodology is presented in Section 3. Section 4 discusses the main empirical results. Our concluding remarks are summarized in the last section. 2. Overview of the literature review The literature on fiscal policies and PDs has been very rich for several decades, with excellent theoretical and empirical papers (e.g. Barro, 1974) that rigorously discuss PD determinants, mechanisms and effects. The recent and sharp deterioration in fiscal balances and 5 Such measures are contrary to the ideas of several liberal and keynesians economists. Indeed, Barro (1974), with reference to the principle of rational expectations, renewed the famous “Ricardian Equivalence” hypothesis, also called the Barro–Ricardo Effect, while suggesting that expansive fiscal policy (that generates an increase in PD) implies householders'savings increase so as to pay expected future taxes. Accordingly, there is some equivalence between debts and taxes, thus making any economic policies to relaunch the economy ineffective. 6 This study can be considered as an extension to the paper by Jawadi and Sousa (2013) as it uses the same data,allowing us to make some interesting comparisions.

Please cite this article as: Gnegne, Y., Jawadi, F., Boundedness and nonlinearities in public debt dynamics: A TAR assessment, Economic Modelling (2013), http://dx.doi.org/10.1016/j.econmod.2013.04.006

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3

US Public Debt Graphics 8.75 PDUS

8.50 8.25 8.00 7.75 7.50 7.25 7.00 6.75 1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009

UK Public Debt Graphics 13.50 PDUK

13.25

13.00

12.75

12.50

12.25 1965

1970

1975

1980

1985

1990

1995

2000

2005

Note: PDUS and PDUK denote the logarithm of public debts for the US and the UK respectively. Fig. 2. Evolution of PD in the US and the UK. Note: PDUS and PDUK denote the logarithm of public debts for the US and the UK respectively.

the historic level of PD in many countries have renewed the interest of academics, central banks and policymakers on this topic. However, there are still not many papers on PD dynamics in the literature. Here, we review some, mostly recent, studies on PD dynamics and evolutions. At the same time, we attempt to justify further nonlinearity in PD dynamics. Using two different approaches (unit root tests in the presence of structural breaks and a reaction function based on Barro's (1979) tax-smoothing model), Uctum et al. (2006) investigate the presence of the unit root and structural breaks in PD data for the G7 and several Latin American and Asian countries over the period 1970–2002. The authors do not reject stationarity and conclude in favor of sustainable fiscal policy for most countries. However, obviously, their results are considered sample-period dependent. A first practical note by Escolano (2010, Fiscal Affairs Department, IMF) develops a practice guide to better understanding the dynamic of PDs. The author, while recalling PD components (inflation, interest rates and fiscal adjustment), highlights the complexity of PD dynamics associated with the interaction between these factors. As in Bartolini and Cottarelli (1994) and Blanchard and Weil (1992), the author suggests that the adjustment of some fiscal rules can help us to identify different states and regimes for PD that are activated according to the threshold imposed on the spread between interest and economic growth rates. The underlying idea is that the fiscal rule framework of several countries consists of keeping the debt ratio to GDP below an upper bound in order to safeguard householders. If fiscal discipline

(i.e. the Golden Rule) is strictly imposed, it should imply only one trajectory for debt ratio to GDP, if not and with turbulent times, there are multiple feasible trajectories for PD. In a recent paper, Hasko (2010) associates the high PD volatility of OECD countries with extremely persistent inflation and high interest rate levels, exogenous economic shocks (i.e. demographic developments) and fiscal authorities. Using a reduced form of the VAR model, the author identifies a significant response of public debt to shocks on economic growth and monetary and fiscal policy since the mid-1970s. In particular, it seems that inflationary shocks were crucial factors in triggering the public debt problem. This recently led the Bank of England to reactivate monetary policy, for instance, in order to service budgetary policy and PD management after being independent for many years (Goodhart, 2012). Also using a VAR framework with debt feedback, Cherif and Hasanov (2012) test the effects of macroeconomic shocks (inflation, austerity, economic growth) on US PD dynamics, and show that austerity does not necessarily reduce debts. Interestingly, the authors point out that the reaction toward such shocks in normal times differs than that of a weak economic environment. This suggests that choosing the optimal timing for a shock can improve the management and reduction of public debt. As for the investigation of nonlinearity in PD and economic growth relationships, a recent working paper by Egert (2012) applies a threshold model, extends Reinhart and Rogoff (2010)'s approach and confirms the presence of a negative nonlinear relationship between public debt

Please cite this article as: Gnegne, Y., Jawadi, F., Boundedness and nonlinearities in public debt dynamics: A TAR assessment, Economic Modelling (2013), http://dx.doi.org/10.1016/j.econmod.2013.04.006

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and economic growth. Interestingly, the author points to further evidence of instability in this relationship, possibly associated with the nonlinear effect that varies not only within data frequency and timing but also according to the country under consideration. Finally, in a recent and short note about PD dynamics, Jawadi and Sousa (2013) provide further evidence of multiple structural breaks in US and British public debts. Indeed, the application of Bai and Perron tests (1998, 2003) enabled them to detect several significant breaks in PD dynamics. Interestingly, these structural breaks appropriately reproduced observed changes in PD ratios induced by well-known exogenous shocks and crashes (1973–79 oil shocks, debt crisis in 1982, equity market crash in 1987, Internet bubble in 2000, and subprime crisis in 2007). Retaining the same data, but with more focus on nonlinearity in PD, our study extends this work in several ways. On the one hand, we develop economic justification for nonlinearity and we address an overview of the literature on PD dynamics. On the other hand, not only do we check for nonlinearity in the data, but we also model this nonlinearity and capture different regimes to characterize the evolution of PD for the UK and the US over time. 3. Econometric methodology 3.1. Overview of threshold models In this paper we focus on a particular class of nonlinear models, namely, threshold models. These have recently been widely applied in economics and finance and have gained a considerable amount of attention. Such modeling has extended the linear model while enabling relationships to be nonlinear, which is of particular interest as it allows further asymmetry, structural breaks and nonlinearity to be captured in time series dynamics. 7 As financial data often exhibit abrupt changes, threshold specifications are therefore more realistic representations of financial and macroeconomic data generation processes. In our specific case, the study focuses on PD dynamics while checking for abrupt changes and asymmetry in PD data. Thus, threshold models would appear to be useful in dealing with time-varying and asymmetrical PD changes. Threshold models include Markov Switching models (Hamilton, 1989), Smooth Transition Autoregressive (STAR) models (Granger and Teräsvirta, 1993), and Threshold Autoregressive (TAR) models (Tong and Lim, 1980). While the first models, also called probabilistic processes, imply the presence of different relationships whose realizations are determined by an unobserved conditional probability, the transition between regimes for the latter is determined by a known and deterministic rule (the transition variable). STAR models can also be considered as a generalization of TAR models as the transition is abrupt for the latter while it is smooth for STAR models. Among the threshold models, we focus on TAR models to investigate structural breaks in PD dynamics as TAR models are required to represent price dynamics in the short term through different regimes. 3.2. TAR modeling 3.2.1. TAR models TAR models were introduced by Tong and Lim (1980) and extensively discussed in Tong (1990). They are particularly appropriate for reproducing asymmetry in business cycles through the specification of different regimes that are activated according to a certain threshold. Thus, the TAR model implies a relationship that is nonlinear over the

7 See Zapata and Gauthier (2003) for a brief note on threshold models and their applications and Guégan (1994) for more details about continous threshold models.

whole period but is linear per regime. It is a piecewise linear process as it defines a linear autoregressive model in each regime. Formally, a simple two-regime TAR model, noted TAR(2,p,St) corresponds to: p

p

p

Y t ¼ α 10 þ ∑i¼1 α 1i Y t−i þ ∑j¼1 β 1j X 1t−j þ ∑k¼1 δ1k X 2t−k þ ε1t if St ≤ c p p p Y t ¼ α 20 þ ∑i¼1 α 2i Y t−i þ ∑j¼1 β 2j X 1t−j þ ∑k¼1 δ2k X 2t−k þ ε 2t if St > c

ð1Þ where: (α10, α1i, β1j, δ1k) and (α20, α2i, β2j, δ2k) refer to the coefficients in the first and second regime respectively, ∀i, j = 1,…, p. Yt refers to the endogenous variable (e.g. PD), and Yt − i, X1t − j and X2t − 2 denote the list of possible explanatory variables. The errors ε1t and ε2t are white noise processes. St and c are the transition variable and threshold parameter respectively,8 while p denotes the maximum lag number. 9 A TAR specification has several features (limit cycle, amplitude dependent frequencies, and jump phenomena). It thus enables us to reproduce asymmetry and periodic behavior in data. Accordingly, the TAR model implies an abrupt transition between regimes when the transition variable exceeds a certain threshold. Alternatively, a more general specification defining Smooth TAR (STAR) models (Teräsvirta and Anderson, 1992) makes the regime switching smooth rather than abrupt as it is carried out through a continuous function.10 A TAR model also requires a number of conditions to generate time series that are stationary (Caner and Hansen, 2001). As for TAR modeling, the lag number (p) can be determined using information criteria (Franses and Van Dijk, 2000). The estimation of TAR models requires the application of sequential conditional least squares only. In particular, according to Tong and Lim (1980), the implementation of TAR modeling is carried out in three main steps. First, we define arbitrary values for c and d, then determine p using information criteria and finally estimate both equations by the least square (LS) method. Second, we keep d fixed, vary the threshold value c and reestimate both equations by the LS method. The optimal value for c should minimize the information criteria. Third, we execute setups 1 and 2 to estimate the value of d, and its optimal value should also minimize the information criteria. This method is not simple as the values of c and d are not known. Also, it has a certain cost in terms of timing. Accordingly, an alternative procedure based on linearity tests was introduced to specify TAR models. This approach is somewhat sequential and is conditioned by estimated values for c and d. 3.2.2. Linearity tests These tests aim to specify the values of c and d while testing the null hypothesis of linearity against its alternative of nonlinearity. To this end, linearity tests are applied for several values of d: 1 ≤ d ≤ p. The optimal value should minimize the p-value of the linearity test. In practice, two main linearity test strategies were introduced: the Tsay (1989) test and the Hansen (1996) test. 11 Tsay (1989) proposed a linearity test that is related to the Portmanteau test of nonlinearity discussed by Petruccelli and Davies (1986), based on arranged regression and predictive residuals. Tsay (1989)'s test is considered as a combination of linearity tests by Keenan (1985), Tsay (1986) and Petruccelli and Davies (1986). The test is simple and widely applicable in four main steps. First, we select the autoregressive order p using a partial autocorrelation function

8 St can be a lagged endogenous variable (Yt − d) where d denotes the delay parameter (1 ≤ d ≤ p), or an explanatory variable, or another exogenous variable. Note that p may also differ from regime to regime, thus providing a Self-Exciting TAR (SETAR) model. 9 If p varies from one equation to another, we obtain a SETAR (Self-Exciting TAR) model. 10 For a recent comparison between TAR and STAR models based on simulation exercises for stock returns, see Gibson and Nur (2011). 11 We briefly discuss these methods. See Ben Salem and Perraudin (2001) for more details about these linearity tests.

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(PACF) of Yt (model 1) and information criteria 12 and retain possible values for the delay parameter d that defines the threshold variable. For example, possible threshold lags may correspond to: 1 ≤ d ≤ p. Second, we fit arranged autoregressions for a given p and apply a threshold nonlinearity test (CUSUM test) while ordering observations according to the increasing values of the threshold variable. This implies two regressions. The first corresponds to k observations associated with weak values of the threshold variable while the second is associated with its higher values. Accordingly, we obtained the following ordering model that corresponds to model (1) for which the threshold parameter is located between the k and (k + 1) observations: p

p

p

p

Y ðOÞ ¼ α 10 þ ∑i¼1 α 1i Y ðOÞt−i þ ∑j¼1 β1j X ðOÞt−j þε 1ðOÞ for the k first values of SO Y ðOÞ ¼ α 20 þ ∑i¼1 α 2i Y ðOÞt−i þ ∑j¼1 β2j X ðOÞt−j

ð2Þ

þε 2ðOÞ for the next values of SO

Q ðpÞ ¼

T X 2 ^ 2t e^t − u

t¼1

t¼1 T X

^ 2t u

T−k−2p−1 pþ1

Table 1 Robust unit root tests. ADF

PP Δ

Level US UK

a

a

−1.96 −7.88a

1.87 0.71a

KPSS

Level a

3.31 0.70a

Δ

Level a

−2.92 −8.41a

b

1.44 0.37b

t¼1

where: k = (T / 10) + p, êt denotes normalized error εt, and ût corresponds to residuals of the regression e(o) on (1, Y′(O)). 13 Under the null hypothesis of linearity, this statistic follows a Fisher test noted F (p + 1, T − k − 2p − 1). If linearity is rejected, the optimal value of d should maximize this statistic and we move on to the next step. In the third step, the threshold value (c) is determined graphically as the graph can provide useful information when locating the threshold. In particular, while plotting the t-ratio values of recursive estimates of the autoregressive coefficients of model (2) versus the threshold variable, the optimal threshold value should correspond to the first observed structural break. The t-ratios of various coefficients may be examined as long as they are statistically significant. Indeed, estimated AR coefficients and t-ratios begin changing when recursion reaches the threshold value. Furthermore, according to Tsay (1989), the estimated threshold value should normally belong to the interval [Min St, Max St]. Finally, after determining c and d, the TAR model is estimated in the last step by the usual LS method. The methodology developed by Hansen (1996) has the advantage of introducing a more global strategy while suggesting how to determine both c and d in line with the Tsay (1989) principle. Accordingly, his linearity test depends on these two parameters. Hence, we first estimate an AR model of p order and we recuperate its estimated residual êt and consider the possible values for d. Second, we apply 12 Tsay (1989) prefers PACF over information criteria as it imposes no penalty on highorder terms. Also, information criteria could be misleading with nonlinear processes. 13 See Tsay (1989) for more details on this test.

b

0.11 0.53b

Level

Δ

−3.62 −1.87

−4.81 −4.96

a linearity test of Multiplier Lagrange (LM test) for each value of d and we compute LM(c) statistics for different values of c following this formula: 0

ð3Þ

Z&A Δ

Note: “Level” and “Δ” designate series in levels and in first-differences, respectively. (a): model with neither trend nor constant; (b): model with constant, but without trend. The critical values for the ADF and the PP tests at 5% statistical level are −1.95 for model (a), −2.89 for model (b) and −3.45 for model (c). The critical values for the KPSS tests at 5% significance level are 0.463 and 0.146 for model (b), respectively. The critical value for the Z&A test is −4.42 at 5% significance level.

LMðcÞ ¼ SðcÞ IðcÞSðcÞ

where O denotes the observation ranking according to increasing values of the threshold variable (St). The arranged autoregression (AR) has the advantage of grouping observations into two groups so that all of the observations in a group are described by the same linear AR model. Interestingly, this separation does not require the precise value of the threshold as only the number of observations in each group depends on it (Tsay, 1989). The estimation procedure would be simpler if the threshold value is known, but since it is unknown, then the estimation is carried out sequentially. Accordingly, the TAR model is estimated by a recursive method for each value of d and the linearity hypothesis consists of testing the equality between the AR coefficients of the two regimes under consideration (model (2)). From Tsay (1989), we note that the statistic of this test corresponds to: T X

5

ð4Þ

Where S(c) denotes the estimated model score under the null hypothesis, while I(c) refers to the Fisher matrix of information. In order to check the power of this test, Hansen (1996) suggests computing different statistics: sup LM(c), exp LM(c) and Mean LM(c). If linearity is rejected, the optimal value of d should maximize these statistics and we move on to next step. Third, the threshold parameter is estimated while minimizing the residual variance of estimated TAR models for different possible values of d. Finally, we estimate the TAR model using the LS method. Ben Salem and Perraudin (2001) compare these two approaches and suggest that it is difficult to conclude whether one strategy supplants another or not.14 As for the TAR estimation, the ordinary LS method is still useful because the TAR model is locally (per regime) linear. Overall, TAR modeling makes a great contribution through the decomposition of a stochastic system into subsystems and the definition of several states and regimes. This is particularly interesting as it enables us to capture time series dynamics within structural breaks, asymmetrical and threshold effects and switching regime hypothesis. This framework however requires stationarity series and is more appropriate for investigating short-term dynamics. 4. Data and empirical results 4.1. Preliminary analysis In order to explore PD dynamics for the USA and the UK, we use quarterly data according to data availability: 1970:1–2009:2 for the US, and 1962:4–2009:2 in the case of the UK. We start by looking at the data evolution through Fig. 1 that plots the (log) evolution of the level of public debts. We noted a significant increase in both US and UK PDs since the 1980s as suggested in the previous section. It would appear that the PD increase has been exponential over the last few years (8% for the US and 18% in the UK), suggesting further evidence of extreme volatility, several structural breaks and recurrent changes in PD data. Second, we checked for the unit root hypothesis in PD data as in Uctum et al. (2006) and Jawadi and Sousa (2012), using the usual unit root tests: Dickey–Fuller (ADF) test, Phillips–Perron (PP) and the KPSS test, The second test is preferred as it is robust to autocorrelation and heteroscedasticity. According to Table 1, not all of these tests reject the unit root hypothesis, indicating that both the US and the UK are I(1). To check the robustness of our findings for the presence of structural breaks as in Uctum et al. (2006), we apply the unit root test by Zivot and Andrews (1992), denoted as Z&A. Our conclusions remain unchanged. 14

We checked linearity in this study using both tests.

Please cite this article as: Gnegne, Y., Jawadi, F., Boundedness and nonlinearities in public debt dynamics: A TAR assessment, Economic Modelling (2013), http://dx.doi.org/10.1016/j.econmod.2013.04.006

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Table 2 Descriptive statistics.

YUS YUK

Mean

St. dev.

Min

Max

Skewness

Kurtosis

JB (p-value)

ARCH (p-value)

0.011 0.002

0.01 0.02

−0.04 (2000:2) −0.04 (2000:2)

0.08 (2008:4) 0.17 (2007:4)

0.80 2.20

5.54 15.9

0.00 0.00

0.00 0.34

Note: JB denotes the statistics of the Jarque–Bera test.

To better characterize these data, we analyze descriptive statistics for PD changes and report the main results in Table 2. We even noted some indication of volatility excess; the ARCH effect is only accepted in the US case. The departure from normality and the symmetry hypotheses are rejected however for both countries. These indications with the significant leptokurtic characteristic reveal some indication of further asymmetry and nonlinearity in US and UK PD distributions. Interestingly, as reported in Table 2, the highest level of PD changes for both countries was reached after the US subprime crisis, particularly in 2008:4 for the US and in 2007:4 for the UK. This result confirms our analysis in the previous sections. In order to explicitly check these indications, in the next step we apply two classes of linearity tests: the Tsay test and the Hansen test. 4.2. Linearity tests As discussed in the Econometric methodology section, these tests check both hypotheses of switching regimes and nonlinearity of US and UK PD data. To adopt Granger and Teräsvirta (1993)'s approach that consists of moving from nonlinear general specifications to specific nonlinear representations, we apply both the Tsay and the Hansen tests. We retain the TAR model as a nonlinear alternative and we test linearity against nonlinearity of the threshold autoregressive (TAR) type. This choice is justified by the abrupt increase inherent to US and UK PD over the last years. Under the nonlinearity hypothesis, as for the economic cycle, we assume the presence of two further PD regimes. In the first one, the economic growth is enough and perhaps the maintaining of fiscal discipline is possible, implying a low PD level. In the second regime, however, the occurrence of the economic crises is the source of a high level of fiscal deficit and PDs. Accordingly, both linearity tests were applied and reported in Table 3. We should recall that the Hansen (1996) test enables us to test for further threshold breaks in the data, while the application of Tsay (1989)'s test enables us to perform an arranged regression test for threshold autoregression. Interestingly, both tests converge to show significant evidence of nonlinearity in US and UK public debt. This result is interesting as it points to the presence of switching-regimes in the dynamics of public debt, which covers the boundedness effect and target expected by fiscal rules and authorities. Last but not least, this confirms our preliminary analysis as well as previous studies mentioned in the second section as it seems that the change between regimes occurred during the last decade, notably in 2003:3 for the US and 2002:3 for the UK. The last step consists of modeling PD dynamic regimes through the estimation of a two-regime TAR model. 4.3. Estimation of a two-regime TAR model After identifying nonlinearity and abrupt switching in US and UK PD changes and showing significant threshold effects in their trajectories,

Table 3 Results of nonlinearity tests (p-values).a

we attempted to specify PD per regime. This is particularly useful to check whether PD changes differ according to the regime under consideration or not. To this end, we re-parameter the following two-regime TAR specification for each country: p

St ¼ β10 þ ∑i¼1 β1i St−i þ ε1t if St−k ≤ c p St ¼ β20 þ ∑i¼1 β2i St−i þ ε2t if St−k > c

ð3Þ

where: β1i and β2i refer to coefficients in the first and second regimes and p denotes the lag number; c denotes the threshold parameter and (ε1t and ε2t) are the error-terms of the first and second regimes. St refers to PD change. In this specification, the transition between regimes is expected to be activated abruptly when previous PD changes exceed a threshold that is endogenously specified. This specification has three main advantages. It is suitable for depicting the PD dynamic that can differ according to the quality of the macroeconomic news: i.e. good or bad news. It is also appropriate for apprehending changes frequently observed within fiscal rule changes. Finally, it requires a simple estimation method. Our main findings, reported in Table 4 and 5, provide several interesting results. First, for each country we identify two different regimes describing the evolution of their PD. For example, for the US, the upper regime for high PD is described through the following AR(2): St = 1.3 ∗ 10−3 + 0.09 St − 1 + 0.67 St − 2. The low PD-level dynamic (lower regime) however corresponds to the following AR(2): St = 2 ∗ 10−3 + 0.692 St − 1 + 0.147 St − 2. For the UK (Table 5), we also identify two regimes but with a more persistent dependence structure, as for both regimes we retain an AR(3).15 Second, the optimal transition variable refers to two-quarter lagged PD for the US, whereas three-quarter lagged PD is used for the UK, implying further memory and inertia effects in PD dynamics, but which do exceed two quarters (resp. three quarters) for the USA (resp. the UK). Third, we noted significant and strong evidence of asymmetry between lower and upper PD regimes for both countries. Indeed, for the US, 84% of PD dynamics is captured by the first regime, while the second regime captures 16%. As for the UK, we capture 85% of PD change in the first regime and 15% is represented in the second regime. The transition between regimes is carried through when US PD change (respect. UK PD change) exceeds a threshold of 2.4% (respect. 1.75%). Interestingly, in dating the transition between regimes, it seems that for the US it occurred in the second quarter of 2003 after the intervention of the US army in Iraq, which corresponds to a request to the Congress for 87 billion $ by the president Georges Bush to finance the Wars in Iraq and Afghanistan. Concerning the UK, it seems that PD shifts occurred 15 months before the US shift in the first quarter of 2002. This date matches the 2002–2003 UK firefighter dispute that started when the UK firefighters union, the Fire Brigades Union (FBU), voted to take strike action in order to obtain a better salary, demanding a 39% pay increase. It was the first nationwide firefighters' strike organized in the UK since the 1970s. Two-regime TAR specification p

Tests

US

UK

Hansen (1996) test Tsay (1989) test

0.01 0.00

0.00 0.00

a Results in Tables 1, 2 and 3 are in line with those of Jawadi and Sousa (2013), confirming asymmetry and nonlinear in PD data.

St ¼ β10 þ ∑i¼1 β1i St−i þ ε1t if St−k ≤ c p St ¼ β20 þ ∑i¼1 β2i St−i þ ε2t if St−k > c

15 For both, the lag number is retained with reference to information criteria and autocorrelation functions.

Please cite this article as: Gnegne, Y., Jawadi, F., Boundedness and nonlinearities in public debt dynamics: A TAR assessment, Economic Modelling (2013), http://dx.doi.org/10.1016/j.econmod.2013.04.006

Y. Gnegne, F. Jawadi / Economic Modelling xxx (2013) xxx–xxx Table 4 TAR estimation for US PDs. Regime 1

Estimators

Regime 2

Estimators

B^ 10

2.10−3 (1.2 ∗ 10−3) 0.692 (0.12) 0.147 (0.12) 0.024 131

B^ 20

1.3 ∗ 10−3 (7 ∗ 10−3) 0.09 (0.23) 0.67 (0.30) 0.024 24

B^ 11 B^ 12 C N

B^ 21 B^ 22 c n

Note: n denotes the number of observations per regime. Values between brackets correspond to robust estimator's standard deviation.

Table 5 TAR estimation for UK PD. Regime 1

Estimators

Regime 2

Estimators

^ 10 B

4.10−4 (2 ∗ 10−3) 0.407 (0.09) 0.229 (0.08) 0.077 (0.08) 0.0175 154

B^ 20

−0.012 (5 ∗ 10−3) 0.049 (0.08) 0.579 (0.10) 0.533 (0.05) 0.0175 29

^ 11 B ^ 12 B ^ 13 B C N

B^ 21 B^ 22 B^ 23 C N

Note: n denotes the number of observations per regime. Values between brackets correspond to robust estimator's standard deviation.

5. Conclusion This paper set out to investigate public debt adjustment dynamics for two major countries (the USA and the UK) over more than four decades. To this end, we developed nonlinear specification based on nonlinearity tests and threshold models. Our specification develops an on/off framework suitable to apprehend asymmetry and nonlinearity in PD changes. Our specification can also be used to capture structural breaks and shifts in fiscal rules and public debt dynamics. Accordingly, our empirical results provide interesting findings as, first, we point to further evidence of instability and time-variation in public debt changes, confirming previous studies that suggest further multiple trajectories for public debt that has been particularly active over the last decade. Second, we show that public debt dynamics for both the US and the UK exhibit asymmetry and nonlinearity. This confirms the complexity associated with PD computation and interactions with financial and macroeconomic factors such as interest rate, inflation, growth rate, budgetary balance, etc. It is also interesting because it implies that the introduction of nonlinearity while modeling public debt dynamic helps economists and policymakers to improve their comprehension of public debt evolutions and the forecasting of their future dynamics. Finally, the specification of threshold effects that delimit public debt regimes not only permits us to better characterize changes in public debt but also to date public debt shifts and to associate these shifts with stylized and realized policy and economic facts. Obviously, the conception of threshold effects is relevant and its application may be extended to investigate relationships with public debt and macroeconomic determinants. It would also be useful to better apprehend linkages between international public debts. References Afonso, A., Sousa, R.M., 2012. The macroeconomic effects of fiscal policy. Applied Economics 44, 4439–4454.

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Please cite this article as: Gnegne, Y., Jawadi, F., Boundedness and nonlinearities in public debt dynamics: A TAR assessment, Economic Modelling (2013), http://dx.doi.org/10.1016/j.econmod.2013.04.006