Limit Cycles in Japanese Macroeconomic and Financial Data
Yasuyuki NISHIGAKI*
Faculty of Economics, Ryukoku University
67 Fukakusa, Fushimi-ku, Kyoto 612-8577, JAPANE-mail: nisigaki@econ.ryukoku.ac.jp
Mitsuhiko SATAKE
Faculty of Economics, Doshisha University
Kamigyo-ku, Kyoto 602-8580, JAPANE-mail: msatake@mail.doshisha.ac.jp
Daiki MAKI
Faculty of Economics, Ryukoku University
67 Fukakusa, Fushimi-ku, Kyoto 612-8577, JAPANE-mail: maki@econ.ryukoku.ac.jp
Introduction
This paper examines the existence of limit cycles in Japanese macroeconomic variables
using a threshold and vector autoregressive (TAR and VAR) model. Recent business cycle
theories are classified into two main categories: (1) real business cycle and (2) endogenousbusiness cycle. In real business cycle theory, a stochastic business cycle model, which includes
stochastic exogenous influences for causing cyclical deviations from a growth trend, and alinear dynamical system, are used to describe the actual complicated economic fluctuations. An
autoregressive (AR) model, therefore, is utilized mostly to estimate a model based on thistheory.
Endogenous business cycle theory, on the other hand, is based on a nonlinear dynamicsystem that has a mechanism that induces complicated economic fluctuations endogenously.
Based on this theory, limit cycles characterized by a TAR model or VAR model, or chaotic
dynamical systems, are utilized to describe the actual complicated economic fluctuations. For
the estimation of the chaotic dynamical systems, see Nishigaki, Ikeda, Satake (2007).
Which theory is appropriate has significant implications for policy makers. Accordingly, it isnecessary to investigate whether economic fluctuations depend on an endogenous mechanism
or on exogenous shocks. To investigate this, we test for linearity in Japanese macroeconomic
and financial variables. The linearity test in the paper distinguishes the TAR model from theAR model. We find that most of them have TAR processes, i.e. the two-regime or three-regime
TAR processes. The results indicate that most Japanese macroeconomic data have limit cycles.
This implies that the government could control the business cycle in Japan, even though its
cyclical behaviour is extremely complicated, because the process follows not the AR model, butthe limit cycle expressed by the TAR model.

12. Limit Cycles and Business Cycle Theories
For decades, the aim of business cycle theory has been to model the basic underlying
dynamics of economic fluctuations. The Samuelson-Hicks-type multiplier and accelerator
mechanism explains the regular forces in oscillating time series, by using linear dynamic
systems of difference equations, with appropriate assumptions in the economic structures.
Although they succeeded in modelling the business cycles generated by the internal dynamicsof the economy, they failed to describe all the fluctuations of actual business cycles, which werecharacterized by many more irregularities, not only in the context of the monotonicity of thecycle, but also in its amplitude and frequency. Accordingly, a stochastic exogenous influence
had been thought to be included in the theoretically expressed linear systems.
The rational expectations theory of business cycles constitutes a stochastic business cycle
model that describes the actual complicated cycles by including stochastic exogenous influences,in addition to the implicit regularity with the linear difference and differential systems. Typicalexamples of such models are Lucas (1975) and Prescott (1986).
However, in endogenous business cycle theory, many papers have utilized non-linear
economic dynamics, such as chaotic dynamics or limit cycle theory, to model business cycles,
and they have succeeded in explaining the seemingly irregular fluctuations in actual timeseries. The basic idea of utilizing limit cycles in business cycle theory was first demonstrated byKaldor (1940), and mathematically formulated by Chang and Smith (1971) and Varian (1979).
Since then, the theoretical interest of economists has expanded this literature, e.g. Lorentz(1993).
To see the basic dynamics of limit cycle theory, consider the following two dimensional
ordinary differential equation systems.
x1  f1 x1 , x2 (2.1) ,x 2  f 2 x1 , x2 (2.2) ,or, in vector notation,
x  f x (2.3)

where x  ( x1 , x2 ) and f i : U  R 2 are smooth functions defined in U  R 2 . The dot over avariable denotes a time-derivative.
A solution curve, trajectory, or orbit is defined as  (t , x 0 ) : i.e. when a certain initial pointx 0 is given,  (t , x 0 ) provides the value of x at time t . For the existence and uniqueness ofa solution curve, the function f is assumed to be C 2 .

2A solution curve is called a periodic solution of period T, if there exists p such thatp   (T , p) for some T. In a phase space of dimension two, a periodic solution constitutes aclosed orbit, which is called a limit cycle.
A periodic solution curve  is stable for any   0 , if there exists a neighbourhood U of and d ( (t , x), )   , for any x U and t  0 . For asymptotic stability or asymptotic
convergence of a periodic solution  , the followings should be satisfied for all x that iscontained in the neighbourhood U of  .

lim d ( (t , x), )  0t (2.4)

The Poincare-Bendixson theorem is most often used to establish the sufficient conditions forthe existence of closed orbits in the dynamical system of (2.3), although it is of limited use fortwo-dimensional systems (see Lorentz (1993)).
So far, the discussion has been limited to continuous time nonlinear dynamical systems, but
this is only for a reference frame of the non-linear time series model in discrete time. A limit
cycle in discrete time is defined as follows (see Tong and Lim (1980)). Let xn denote a k  1
dimensional state that satisfies the equationxn  f xn1 (2.5) .

A limit point in the vector space is defined as follows: A k  1 dimensional vector x * is calleda limit point, if there exists x0  x* such that, starting with n  0 , x n tends to x * as n*tends to infinity: lim xn  x .
n

A limit cycle in discrete time is defined as follows: Let C denote the set of k  1
dimensional vector ci , i  1,  , T , T being a positive integer   . 1) C is called a limit cycleof period T if there exists x0  C , such that as n   , x n will ultimately fall intox * : xn  x* ，2) ci  f ci1  i  2,3,, T ，cT i  ci i  1,2,, T ， and 3) T is the smallest
such positive integer.
On the effectiveness of the TAR model in inferring the existence of limit cycles in time series,
the following notation by Tong (1983) is instructive:“Physically, limit cycles represent the stationary state of sustained oscillations (nowdynamic) which do not depend on initial conditions but depend exclusively on theparameters of the system, i.e. they are intrinsic properties. In addition, there exist
limit cycles which have the properties of being robust, i.e. insensitive to small
perturbation of parameters of the system.”

33 Linearity Tests
In this section, the method of linearity testing, which checks whether there are limit cycles
in macroeconomic variables in Japan, is briefly explained. Following Hansen (1999), we show
the model that tests the linearity of time series data.
The threshold autoregressive (TAR) model is a nonlinear model that consists of several
regimes, each of which constructs an autoregressive model that is linear. The variable that
decides in which regime each observation is contained is named a threshold variable. Amongthe several classes of TAR models, the TAR model in which the threshold variable is its own lag
is named the ‘Self Exciting Threshold Autoregressive’ (SETAR) model.
A SETAR(m) model, in which there are m regimes, takes the formyt  1 X t 1I1t  , d      m X t 1I mt  , d   et

where X t 1  1, yt 1 , yt 2 , , yt  p(3.1) , , which has k 1 vector ( k  1  p ),    ,,   ,1m1(  1   2     m1 ), which is named a threshold value to group the observations into regimes.

I jt  , d   I  j 1  yt d   j  are dummy variables, and are 1 when  j 1  yt d   j , and 0otherwise. When the observations are in  j 1  yt d   j , the processes of the time series arein accordance with AR(p) process Yt   j X t 1 .
The parameters of (1) may be collected as   1 ,  2 , ,  m ,  , d  . The parameters are
estimated by least-squares (LS), given  , d  .  , d  are determined by minimizing the sumof the squared residuals of (1).
The testing for linearity is to test the null hypothesis of SETAR(1) against the alternative ofSETAR(2). The SETAR(1) model can be written asyt  1 X t 1  et(3.2) ,which is an AR model, and SETAR(2) asyt  1 X t 1I1t  , d    2 X t 1I 2t  , d   et(3.3) .

 S1  S 2  , where S m
 S2 The statistic to test the linearity against the TAR model is F12  n( m  1,2 ) is the sum of squared residuals of the m-regime TAR model. If the threshold value and delay parameter d are fixed, F12 is distributed to  2 k  under the null hypothesis of4

SETAR(1) which is actually an AR model. However, the asymptotic distribution of F12 is not 2 , unless the threshold value  or delay parameter d are identified. The distribution isestimated by the bootstrap method. If the value of F12 is larger than the critical value of  2 k  ,i.e. if the linearity is rejected, the SETAR(2) model is supported.
When the SETAR(2) model is adopted, we proceed to test whether the series are subject toSETAR(2) or SETAR(3) models. The SETAR(3) model can be written asyt  1 X t 1I1t  , d    2 X t 1I 2t  , d    3 X t 1I 3t  , d   et(3.4) ，where    1 ,  2  , because there are two threshold values in the three-regime TAR model. If
the threshold value and delay parameterdare fixed, the test statistic is S  S3  1
 , which is also distributed to  2 k  under the null hypothesis of theF23  n 2
 S3 SETAR(2) model. Otherwise, the distribution of F23 is not  2 . The distribution is estimatedby the bootstrap method. If the value of F23 is larger than the critical value of  2 k  , theSETAR(2) model is rejected, and the SETAR(3) model is adopted.

4. Empirical Results and the Implications
Satake, Maki and Nishigaki (2009) conducted the two-regime TAR model test and showed
the results as in Table 1.

Table 1: Linearity tests(1) H 0 :AR Model vs H 1 : Two-regime TAR test

P-valueP-value

CIDI

CPI

WPI

NIKKEI

0.223

0.042

0.018

0.001

0.233

TOPIX

M2

M1LR

0.016

0.041

0.046

0.554

(2) H 0 : Two-regime TAR vs H 1 : Three-regime TAR test

P-value
1DI

CPI

0.205

0.045

WPI
0.008

TOPIX
0.016

The notation of F23 is the same as that of F12 .
5

M1
0.112

M2
0.070

* Monthly data are used, and are transformed into log-difference, i.e. {ln( yt )  ln( yt 1 )}  100 . P-values arecalculated by bootstrap method, considering heteroscedasticity when it is observed in the error terms. The lag length isselected by BIC criteria.

Linearity is rejected for all variables except CI (Composite Index), NIKKEI (Aver. StockPrice) and LR (Rate of Return on Government Bond). It turns out that the processes of DI(Cumulated Diffusion Index), CPI (Consumer Price Index), WPI (Wholesale Price Index),TOPIX, M2 (M2+CD), and M1 are subject to the TAR model and, therefore, it is suggested thatthey have limit cycles.
In this study, we will conduct a multi-regime VAR model test as well as a TAR model test on
Japanese macroeconomic and financial data and will investigate the existence of
two-dimensional limit cycles in those data.
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