Full text of DOI:10.1080/00036840121698

Applied Economics
ISSN: 0003-6846 (Print) 1466-4283 (Online) Journal homepage: http://www.tandfonline.com/loi/raec20
A new turning point signalling system using the
Markov switching model with application to Japan,
the USA and Australia
Allan P. Layton & Masaki Katsuura
To cite this article: Allan P. Layton & Masaki Katsuura (2001) A new turning point signalling
system using the Markov switching model with application to Japan, the USA and Australia,
Applied Economics, 33:1, 59-70
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Applied Economics, 2001, 33, 59±70
A new turning point signalling system using
the Markov switching model with
application to Japan, the USA and
Australia
ALLAN P. LAYTON* and MASAKI KATSUURA
{
School of Economics and Finance, Queensland University of Technology, GPO Box
2434, Brisbane, Queensland, 4001, Australia and Department of Economics Meijo
{
University 1-501 Shiogamaguchi, Tenpaku, Nagoya, Aichi 468-8502 Japan
E-mail: a.layton@qut.edu.au and katsuura@meijo-u.ac.jp
A new business cycle turning point signalling system is proposed and examined by
using Japanese, US and Australian composite indexes of economic activity. Time
varying transition probabilities in a Markov regime-switching model are used as the
basis of the signalling system. The performance of the system is satisfactory, though
its reliability varies between peaks and troughs and across countries. Based on data
up until May 1998, the system suggests the absence of turning points in any of the
three countries in 1998.
I. INTRODUCTION
the next section, the signalling system proposed here is
based on the Markov regime-switching model originally
developed by Hamilton (1989). This paper employs the
two phase, time varying transition probability, regime-
switching model developed by Diebold et al. (1994),
Filardo (1994) and Durland and McCurdy (1994).1
Suitable signalling rules for predicting turning points are
developed by calculating time varying transition probabil-
ities of contraction and expansion using leading indicators
as explanatory variables.
One of the most important issues for macroeconomic pol-
icy makers when making decisions about stabilization poli-
cies is to predict the most likely time of the next business
cycle turning point. In particular, in Japan at the time of
writing, the government is endeavouring to devise and
implement policies to assist the economy out of its present
deep recession. The timing of the Japanese economic recov-
ery is of utmost interest to the Japanese government, other
national governments, and international ®nancial markets.
Similarly in the USA, analysts are keenly interested in how
long the present long expansion will continue.
In the leading indicator approach to business cycle
analysis, developed originally by Burns and Mitchell
(1946) at the National Bureau of Economic Research
(NBER), instead of basing a business cycle chronology
on a single series, analysts prefer to use a diŒusion index
(DI) or composite index (CI). Relatedly, recent develop-
ments have occurred in dating and predicting turning
points using time series econometrics techniques to calcu-
late probabilities of being in particular business cycle
phases or the likelihood of future turning points. Wecker
The purpose of this paper is to propose a new empirical
signalling system of turning points in the economy, to ex-
amine the validity of the system using Japanese, US and
Australian coincident and leading indexes, and to provide
information on the likelihood of imminent future turning
points in the three countries. While some methods for pre-
dicting turning points have been proposed as described in
* Corresponding author.
1 For applications of the time varying transition probability model in the business cycle context, see Durland and McCurdy (1994),
Layton (1998) for the USA, and Layton (1996) for Australia.
59
#
Applied Economics ISSN 0003±6846 print/ISSN 1466±4283 online
http://www.tandf.co.uk/journals
2001 Taylor & Francis Ltd

60
A. P. Layton and M. Katsuura
(1979), Neftci (1982), Hamilton (1989)2 and Stock and
Watson (1991, 1993) are important relevant papers. This
empirical work combines the leading indicator and regime-
switching modelling approaches in that it applies a variable
transition probability, regime-switching model using cycli-
cal leading and coincident indicators. The transition prob-
ability calculated in the model is used to determine
empirical rules for predicting turning points.
along with a leading CI in combination with a three
month type of rule allows turning points to be dated and
predicted. See Zarnowitz and Boschan (1975a, b).
Another signalling system for predicting turning points
using CIs was proposed by Zarnowitz and Moore (1982).
They explored sequential signals empirically by using both
a leading index (L) and a coincident index (C). L and C are
appropriately smoothed growth rates of leading and coin-
cident CIs. Their signals for peaks (troughs) consist of
three stages as follows:
The next section presents some existing rules for predict-
ing turning points using the leading indicator approach.
Section III outlines the regime-switching model and pro-
poses the turning point signalling system. Section IV pro-
vides the empirical results for the USA, Japan and
Australia. Applying the empirical rules of Section III, it
is shown that historical turning points would have been
quite well predicted using the signalling system in the
three countries. This paper also comments upon the poss-
ibility of turning points occurring in Japan (trough), the
USA and Australia (peaks) over the horizon immediately
following the sample period available at the time of the
analysis (endpoint ± May 1998). Section V contains some
concluding remarks.
1. L 3 3 and C 0 (L 0 and C 0);
< :
>
>
<
2. L 0 and C 3 3 (L 3 3 and C 0);
< : > :
<
>
3. L 0 and C 0 (L 3 3 and C 3.3).
> :
<
<
>
where 3.3% is the long term trend percentage growth rate
inherent in the CIs.
In addition to the above rules they also used subjective
judgement based on their assessment of actual CI growth
rates. Another point to note is that they proposed three
stages in signalling turning points. Occasionally, only the
®rst and/or the second signals occurred, but the third did
not; i.e. no turning point eventuated. Also the ®rst signal
sometimes had a long lead (particularly for peaks) to the
turning point. The signals proposed here are similar to
their approach though the growth rate of CI as the basis
of the system is not used. Instead, time-varying transition
probabilities arising from the regime-switching econometric
model are used with only two stages of signals utilized.
II. SOME RULES FOR PREDICTING
TURNING POINTS
Indicator approach
In the indicator approach, leading and coincident indexes
are important inputs into the dating and prediction of turn-
ing points. When a DI is used, the point at which the
coincident DI crosses the 50% line from above (below)
determines a peak (trough). The point at which the leading
DI crosses the 50% line determines the number of months
lead in respect of the turning point in question. Cumulative
DI is also often used for dating and predicting turning
points.3 The number of months between turning points in
the leading indicators and coincident indicators at business
cycle turning points are analysed in so-called lead-lag
tables. Average or median leads and their standard devi-
ation are utilized for assessing the reliability and stability of
the leading indicators. As a DI often displays considerable
irregularities in monthly movements, a three month rule is
often adopted for dating and predicting turning points; i.e.
a judgement that a trough (peak) will occur is made only if
the DI is below (above) the 50% line for three successive
months after crossing the line.
A probabilistic approach to turning point prediction
The probabilistic approach evaluates the possibility of a
turning point by computing, in some way, its probability
of occurrence. In Wecker (1979), two binary variables
representing peaks and troughs were de®ned as functions
of some variable thought to be useful in predicting turning
points. Based on these two binary variables, the variable
wt, `time until the next turning point’, was introduced and
the empirical distribution of wt calculated by simulation. In
this method, the signal of the next turning point is the time,
wt, with the highest probability attached.
Neftci (1982) proposed sequential probability recursion
methods for calculating the probability, , of the occur-
ºt
rence of turning points.
following formula:
is calculated recursively by the
ºt
‡P Z t ‡1 Z
t
1 ¡ p1
ºt† gŠ
ˆ
j
‰ºt
…
> †f…
‡
1
t
ˆ
ºt‡
;
1
‡P Z t ‡1 Z t 1 ¡ p1
ˆ
j
‰ºt
…
> †…
ºt†
‡
j1
t
For CIs, turning points in the series itself represent turn-
ing points in the business cycle. Using a coincident CI
‡ 1 ¡ p0 1 ¡ P Z t ‡1 Z
t
ˆ
…
ºt† 1 f
…
> †gŠ
‡
t
2 The Neftci (1982) approach can be regarded as a special case of Hamilton’s model.
3 When cumulative DI is used, its own turning point represents the turning point of the business cycle. Tahara (1983) also proposed that
turning points of DI themselves could be utilized for predicting turning points because of their longer leads than the timing of crossing
the 50% line.

A new turning point signalling system
61
where Z denotes the time of the next turning point, and p0
and p1 denote the conditional densities of the variable dur-
ing normal and contractionary regimes, respectively. The
turning point signal occurs when pt exceeds some high
probability, say 0.90 or 0.95. Niemira (1991) applied this
method to leading indicators for the USA, the UK, Japan
and West Germany, and using 95% as the threshold for
predicting turning points, obtained useful results.
denote the business cycle indicator and st denote the
ˆ
phase (st 1 2). The probability density of y is assumed
;
t
to be
"
#
y ¡
…
·s †
1
t
t
ˆ
j
f y s
p exp ¡
2
… †
…
t; ³†
t
2
2
¼
2
º¼
st
si
2
st
and P,
methods.
and
are estimated by maximum likelihood
·
¼
st
Hamilton’s Markov, constant transition parameter,
regime-switching model ± as described in the next section
± provides the probability that the economy is in one phase
or another in any period.4 Using these probabilities,
Hamilton (1989) tried to establish a business cycle chron-
ology for the USA using GDP growth rate. The rule he
used was that: whenever the probability of contraction
(expansion) was over (less than) 0.5, the economy was con-
sidered to have transitioned into contraction (expansion).
Layton (1996) extended this basic approach to using a
USA composite coincident index and the rule that: given
the series was currently in an expansionary (contraction-
ary) phase, a contractionary phase shift was identi®ed as
having occurred if a run of at least `®ve’ data point regime
probabilities in a row exceeded (were less than) 0.5.5
Layton (1997a) applied the same rule in dating and pre-
dicting Australian business cycles.6
Another version of the regime-switching model is the
time-varying transition parameter model. In this model,
explanatory variables are introduced as putative determi-
nants of the transition probabilities which are now
assumed to be time-varying. To ensure the probabilities
are bounded between zero and one they are usually mod-
elled as logistic functions of the list of putative explanatory
variables. In this paper, yt denotes the coincident index,
and (short and/or long) leading indexes are introduced as
possible explanatory variables. Probability matrix
Equation 1 becomes time dependent as
p11t p12t
ˆ
Pt
3
… †
p21t p22t
ˆ
ˆ
where p12t 1 ¡ p11t and p21t 1 ¡ p22t. These time-
varying transition probabilities are modelled as:
1
ˆ
piit
4
… †
1 ‡exp ¡X
…
t¡1- i†
II. A VARIABLE TRANSITION
PROBABILITY, REGIME SWITCHING
MODEL AND A DATING/ PREDICTION
RULE
ˆ
where Xt¡1
1 x
… ;
x
x
denotes the vector
of explanatory variables which are believed to in¯ uence the
;
2;t¡1 ;. . . ;
†
1;t¡1
k;t¡1
transition probabilities, and is a parameter vector to be
estimated
-
i
ˆ
i
…
1 2 . Diebold et al. (1991) used expected
; †
maximum likelihood (EML) method to estimate Equations
2 and 4 as a natural extension of Hamilton (1990). However,
as there sometimes exists the so-called `singularity’ problem,
Hamilton (1991) suggested a form of quasi-Bayesian estima-
tion using prior estimates of the parameters. As it was felt
that quasi-Bayesian estimation can sometimes be overly
aŒected by necessarily imprecise prior information, this
paper adopts constrained maximum likelihood estimation
by using selected, reasonable, parameter constraints such
Regime switching model
Hamilton (1989, 1990) has proposed the constant transi-
tion probability, Markov regime-switching model to allow
for non-linear shifts in a time series from one phase into
another. For each phase, the distribution of the variable
under study is assumed to be normal with diŒerent means
and variances, and the probability of switching from one
phase into the other is characterized by a ®rst order
Markov-type probability rule.
as
0, 0 and so on.
·1 < ·2 > ¼ >
0,
i
The matrix of Markov transition probabilities is given
by:
Rules for dating and predicting turning points using
transition probabilities
p11 p12
ˆ
P
1
… †
p21 p22
Using the estimated transition probabilities obtained
from the regime switching model in combination with
appropriately selected rules allows for the development of
ˆ
ˆ
where p12 1 ¡ p11 and p21 1 ¡ p22, and subscripts 1 and
2 denote contraction and expansion, respectively. Let yt
4 The method is not limited to two phases, but can be generalized to three or more phases. See Sichel (1994).
5 Layton (1996) p. 421.
6 Stock and Watson (1991, 1993) also calculated the probability of recession and expansion. However, their approach was based on
de®ning a business cycle indicator from some individual economic variables, and the approach is diŒerent from the other probabilistic
approaches referred to above.

62
A. P. Layton and M. Katsuura
a method for predicting the dates of turning points. This
contrasts with other earlier approaches to turning point
prediction which have used the estimated regime state
probabilities arising from a constant transition parameter
regime switching model. This paper attempts to extract use-
ful information from the estimated time-varying transition
probabilities. The switching probability from contraction
into expansion, p12t, isused as the basis for signalsof troughs,
and p21t, the switching probability from expansion into con-
traction, is used as the basis for the signals of peaks.
In previous papers using regime state probabilities (as
opposed to the transition probabilities used here) the rule
adopted for dating peaks and troughs related to the state
probability being above or below 0.5 (see Hamilton, 1989;
Layton, 1996, 1997a). In this context, the problem is to
determine a similar cut-oΠvalue for the relevant transition
probability which may reasonably be regarded as high
enough to be treated as a trigger signal of an imminent
phase change. A natural selection is the overall long term
mean value of the transition parameter.7 This then is one
aspect of the signal.
where r ¶ 5. The constraint r ¶ 5 is selected to be analo-
gous to a similar requirement in the long standing and
widely-used Bry±Bochan method8 for determining turning
points and which incorporates the requirement that no
phase will be recognized if it has a duration less than ®ve
months. By de®nition, the occurrence of the local minimum
leads the signal expressed by Equations 5 and 6. Therefore,
it may be regarded as the ®rst tentative signal of an
impending turning point. Of course, a signal is not formally
regarded as occurring until Equations 5 and 6 also are
satis®ed. In real time, the formal signal cannot be recog-
nized for at least ®ve months after the occurrence of the
de®ned local minimum.
In order to reduce the possibility of false signals one
needs to add a caveat to the above signalling system, viz.
that a potential local minimum is disregarded if it is fol-
lowed by a local maximum within ®ve months. This is also
analogous to the Bry±Boschan algorithm. A local maxi-
mum at time t is de®ned as:
pij;t max p
>
p
p
…
;
2 ;. . . ;
‡
†
‡
1
‡
5
ij;t
ij;t
ij;t
and
(8)
A second component of the signal addresses the issue of
whether the transition probability is su ciently large and
whether it has been su ciently large for a su ciently long
enough period. After all, given the transition probability
may be expected to ¯ uctuate around the mean over time, it
does not seem sensible to use the mean as the sole signalling
criterion. Thus, it is also desirable to compare the prob-
ability in any current period with that observed in some
appropriate recent time period. This comparison is made
by monitoring the ratio of the current transition probabil-
ity to the most recent local minimum value and judge that
the probability has risen enough if the ratio exceeds two.
Although the threshold value of two seems to be somewhat
arbitrary, some alternative values were also tested, viz. 1.5,
3.0, etc., and 2 gives the best signal for turning points.
Thus, the two signal system proposed here consists of
two components as follows:
pij;t max p
>
p
p
…
;
ij;t¡2 ;. . . ;
†
ij;t¡1
ij;t¡5
Furthermore, a rule is required for determining if and
when a formal signal, having been given, should sub-
sequently be regarded as false. This would be the case if
a formal signal were given and a local maximum occurred
more than ®ve months later without a turning point being
in evidence before the occurrence of the next local mini-
mum. Finally, a turning point is regarded as missed if there
is no local minimum in evidence prior to the occurrence of
the turning point.
As an aside, it is recognized that, in general, Equation 8
represents a more natural de®nition of an extremum than is
Equation 7. However, the reason for adopting Equation 7
as the de®nition for a local minimum, rather than an ana-
logous version of Equation 8, is that Equation 7 results in
more frequent and more sensitive turning point signals
than Equation 8. It is also the case that one is only inter-
ested in monitoring for the required increases in the transi-
tion probability as described in the signalling system
represented by Equations 5 and 6.
-
1 p
:
p
5
>
… †
‡
ij;t0
s
ij
=
2 p
:
pij;t
2
6
… †
>
‡
s
ij;t0
0
-
where pij is the overall (long term) mean, t0 corresponds to
the most recent local minimum value of the relevant transi-
tion parameter, t0 ‡s expresses the time when the signal is
given, and s ¶ 5 (explained below).
IV. EMPIRICAL RESULTS
The time period, t0, is de®ned as the time at which the
relevant transition probability reached a local minimum.
The local minimum is formally de®ned as
Data
The analysis used monthly data on coincident composite
indexes (CC), short leading (LD), and/or long leading (LL)
pij;t
min p
…
p
p
2 ;. . . ;
ij;t0 r
7
… †
<
;
†
‡
1
‡
‡
ij;t0
ij;t0
0
7 As alternative rules, the mean plus one or two standard deviations were also considered. However, these alternatives were found to be
less successful in dating and predicting turning points.
8 See Bry and Boschan (1971).

A new turning point signalling system
63
composite indexes for Japan, the USA and Australia. The
indexes compiled by the Economic Cycle Research
Institute (ECRI) in New York are used, except in the
case of the short leading indexes for Japan and Australia.
As the short leading index for Japan, the leading composite
index published by the Economic Planning Agency (EPA)
of Japan is used and for Australia, the Westpac/IAESR
leading composite index is used.
(model 4) are used. The short (long) leading indexes are
transformed to moving six (eight) month growth rate
P
6
ˆ
cumulants, i.e. for LD, xt¡1
1 LDt¡i
The span of the moving sum is
, for LL,
ˆ
i
P
8
9
.
ˆ
xt¡1
1 LLt¡i
ˆ
i
expected to be the mean (or median) lead time of leading
indexes.
The indexes used are seasonally adjusted, but not trend-
adjusted series, and are transformed to month-to-month
growth rates. The sample period and descriptive statistics
of CC are shown in Table 1. The means of negative growth
rates and positive growth rates are quite diŒerent (as
one would expect), and the variances in negative growth
periods (regarded as broadly indicative of contractionary
regime periods) are larger than in positive growth periods
for all countries. The durations of expansions and contrac-
tions for the three countries over the sample period are
shown in Table 2. In all countries, the duration of expan-
sions is longer than that of contractions, particularly in the
case of the USA and Australia. See Tables 4, 6 and 8 for
the dates of peaks and troughs in each country.
Results for the USA
Parameter estimates for the USA are shown in Table 3.
Model 1 is the constant transition probability model, and
is included to compare with the results of the time varying
transition probability models.
Expected signs of parameters
ˆ
,
,
,
,
,
-
21
…- 10
-
-
-
11
12
20
¡ ¡ ‡ ‡ and estimates in model 2 and
- 22† …?;
;
model 3 are consistent with these expectations. In model
;?; ; †
^
4, the sign of 21 is not expected and some estimates are not
-
signi®cant. For this model, these results are probably the
result of multicollinearity between LD and LL. All esti-
mates in model 2 and model 3 are signi®cant and, in
terms of the maximized likelihood value, model 3, i.e.
using LL only, is preferred. Calculated time varying transi-
Alternative versions of a time varying transitional prob-
ability model were examined, viz. as explanatory variables
LD only (model 2), LL only (model 3) and both indexes
^
tion probabilities using model 3 are shown in Fig. 1 for p21t
and Fig. 2 for p^ .
12t
Before applying the turning point signalling system
described above, the system components are summarized
here again. The ®rst-stage signal is a local minimum
de®ned by Equation 7, and the second-stage signal is in
evidence when both Equations 5 and 6 are satis®ed given
the prior occurrence of the ®rst signal. If the second-stage
signal does not occur ± meaning that the next local maxi-
mum de®ned by Equation 8 is attained before the second
signal ± the signal is disregarded. The number of months
between the ®rst and the second signal is required to be at
least ®ve months, and the number of months between the
local minimum and the local maximum is also required to
be at least ®ve months.
Table 1. Descriptive statistics of coincident indexes
USA
Japan
Australia
Whole sample
Mean
Variance
St. dev.
10/1948±5/1998
0.2538
0.3126
10/1965±5/1998
0.1183
0.2929
10/1952±5/1998
0.2466
0.2829
0.5591
0.5412
0.5319
Negative period*
Mean
Variance
¡0.4146
0.1754
0.4188
¡0.3748
0.2555
0.5055
¡0.3976
0.1871
0.4326
St. dev.
Positive period**
Mean
Variance
0.4846
0.1524
0.3904
0.3989
0.0970
0.3115
0.4593
0.1323
0.3638
St. dev.
Applying this signalling system, the leads and lags of the
signals to the reference dates (the lead-lag table) for each
Notes: * for data yt < 0; **for data yt > 0.
Table 2. Months of duration of expansion and contraction
USA
Japan
Australia
Expansion
Contraction
Expansion
Contraction
Expansion
Contraction
Number
Mean
Median
Max.
Min.
8
51.4
42
106
11
9
10.7
10
16
6
7
35.7
28
57
22
6
21.2
17
36
9
6
65.2
48
152
18
6
13.8
11.5
26
8
Note: see notes of Tables 4, 6 and 8 for reference dates in each country.
9 A number of diŒerent spans were examined for the moving sum for of the US LL data; and eight months appeared most suitable in
terms of maximizing likelihood. For the justi®cation for using moving sums, see Layton (1997b): 261±2.

64
A. P. Layton and M. Katsuura
Table 3. Parameter estimates for US coincident index
Parameter
Model 1
Model 2
Model 3
Model 4
p11
p22
0.8822 (0.0377)
0.9710 (0.0090)
¡0.3950 (0.0714)
0.4103 (0.0223)
·
¡0.3708 (0.0752)
¡0.3738 (0.0716)
¡0.3638 (0.0968)
1
·
0.4106 (0.0230)
0.4090 (0.0219)
0.4094 (0.0220)
2
2
¼
0.2839 (0.0456)
0.1935 (0.0133)
0.2954 (0.0495)
0.2950 (0.0521)
0.3056 (0.0796)
1
2
¼
0.1927 (0.0139)
0.9286 (0.5711)
¡0.1074 (0.0517)
0.1949 (0.0133)
1.8329 (0.3771)
0.1940 (0.0139)
1.1503 (0.5541)
¡0.0796 (0.0464)
¡0.0400 (0.0357)
3.9397 (0.6394)
¡0.0028 (0.0331)
0.1329 (0.0483)
2
-
-
-
-
-
-
10
11
¡0.0602 (0.0321)
12
3.1473 (0.3782)
0.0454 (0.0248)
4.0868 (0.5736)
20
21
0.1409 (0.0422)
22
Log likelihood
¡446.99
¡430.58
¡424.30
¡421.95
Notes: The value of parentheses are asymptotic standard errors by White (1982).
Model 1: Constant transition probability model.
Model 2: Varying transition probability model using LD only.
Model 3: Varying transition probability model using LL only.
Model 4: Varying transition probability model using both LD and LL.
pii, ·i, ¼2i : See (1) and (2).
ˆ
- ij: parameter for ith phase (i 1;2) and jth explanatory variable (j 0 for constant term, j 1 for LD and j 2 for LL) in (4).
ˆ
ˆ
ˆ
ˆ
Fig. 1. Transition probabilities from contraction to expansion by model 3: the US case (mean 0.1725)
ˆ
Fig. 2. Transition probabilities from expansion to contraction by model 3: the US case (mean 0.0436)

A new turning point signalling system
65
model is shown in Table 4. The business cycle chronology
(o cial peaks and troughs presented in the table) is estab-
lished by the business cycle dating panel convened under
the auspices of the NBER. When the local maximum inter-
venes between the second-stage signal and the o cial turn-
ing point, the signal is regarded as a `false signal’. A turning
point is regarded as `missed’ if the local minimum occurs
after the turning point.
All models perform well for all nine troughs. This is
because the duration of contraction in the USA is quite
short (as shown in Table 2) and prediction is therefore
easier than in the case for peaks. The shorter duration of
contractions is also re¯ ected in Figs 1 and 2; the mean
transition probability from contraction to expansion is
much larger than that from expansion to contraction.
For peaks, model 2 gives too many false signals. The per-
formance of each of models 3 and 4 is almost the same but,
given the aforementioned discussion of the parameter esti-
mates, model 3 is preferred.
Table 4. Leads and lags of signals to US business cycle chronology
Model 2
1st
Model 3
1st
Model 4
1st 2nd
Model
Signal
2nd
0
2nd
Peaks
Nov-1948
NA
NA
¡5
NA
‡
‡
Jul-1953
Aug-1957
Apr-1960
Dec-1969
Nov-1973
Jan-1980
Jul-1981
Jul-1990
¡5
4
¡5
¡1
(missed)
1
‡
‡
6
‡
‡
1
(missed)
¡1
2
4
(missed)
¡8
¡3
¡5
¡9
0
¡9
¡10
¡14
¡5
¡3
¡5
¡9
0
¡9
¡3
¡5
¡9
0
Now using model 3 for peaks, one peak was missed with
four false signals in evidence for the remaining eight peaks.
The leads are also longer than the case for troughs. This is
to be expected and corresponds to the usual observation
that leads at peaks are longer than those at troughs. In
both peaks and troughs, the ®rst signal always leads the
turning point, but the second signal sometimes lags it, espe-
cially at troughs. In general, given all these results as
described, the signalling system may be regarded as quite
successful and acceptable.
¡8
¡10
¡14
¡5
¡15
¡5
(missed)
¡6
¡1
¡6
¡1
False signals
Total signals
11
17
4*
11
4
11
Troughs
Oct-1949
May-1954
Apr-1958
Feb-1961
Nov-1970
Mar-1975
Jul-1980
¡5
¡5
¡1
¡1
¡1
¡1
¡1
¡2
¡1
0
¡7
¡3
¡1
¡2
¡9
¡2
¡1
¡11
¡1
0
¡5
¡5
¡1
¡1
¡6
¡2
¡1
¡2
¡1
0
‡
‡
‡
1
2
1
‡
‡
‡
4
4
4
‡
‡
‡
3
4
4
4
‡
‡
3
4
¡2
‡
‡
‡
4
3
‡
‡
‡
4
4
4
Results for Japan
‡
‡
4
Nov-1982
Mar-1991
3
¡5
3
A similar analysis was applied to Japanese data, and the
results are shown in Table 5 for the parameter estimates,
Table 6 for the lead-lag results, and Figures 3 and 4 for the
transition probabilities. Note that the sample periods are
diŒerent between model 3 and models 2 and 4 due to avail-
ability of data. Business cycle reference dates are estab-
lished by the EPA.
‡
‡
‡
4
4
False signals
Total signals
0
9
0
9
0
9
Notes: Models used are as in Table 3.
Figures are the number of months of signals to the turning points.
The datesofpeaksand troughsarebased on NBER referencedates.
Table 5. Parameter estimates for Japanese coincident index
Parameter
Model 1
Model 2
Model 3
Model 4
p11
p22
0.9722 (0.0217)
0.9825 (0.0127)
¡0.0109 (0.0770)
0.3745 (0.0406)
·
¡0.2975 (0.1264)
¡0.0161 (0.0769)
¡0.3047 (0.1132)
1
·
0.1760 (0.0281)
0.3875 (0.0405)
0.1824 (0.0275)
2
2
¼
0.3721 (0.0159)
0.2258 (0.0193)
0.4546 (0.0336)
0.4526 (0.0451)
0.4662 (0.0365)
1
2
¼
0.1356 (0.0141)
3.1486 (1.2847)
¡0.0980 (0.2678)
0.2282 (0.0277)
3.5553 (0.7139)
0.1270 (0.0131)
2.9863 (1.5382)
¡0.2414 (0.5076)
¡0.1113 (0.2236)
3.4340 (0.8098)
¡0.0818 (0.3461)
0.0374 (0.2094)
2
-
-
-
-
-
-
10
11
¡0.0577 (0.0954)
12
3.5693 (0.7359)
0.0319 (0.1639)
4.0254 (0.7786)
20
21
0.2695 (0.1088)
22
Log likelihood
¡325.62
¡177.20
¡318.89
¡175.14
Notes: sample period: 10/1965±5/1998 for model 1 and 3, 1/1975±5/1998 for model 2 and 4.
See notes of Table 3.

66
A. P. Layton and M. Katsuura
Table 6. Leads and lags of signals to Japanese business cycle
chronology
the other models. A possible reason for the insigni®cance
^
of
is because the duration of contractions in Japan is
-
12
substantially longer than is the case in the USA and the
explanatory power of the leading index is apparently rela-
tively much lower in contractions. This is re¯ ected in Table
6. All seven peaks are well predicted using model 3 with no
missing peaks, and only two false signals. Two o cial
troughs are missed with two false signals for the six
troughs. Unlike the USA, in the case of Japanese turning
points, forecasting peaks appears to be easier than is the
case for troughs. In particular, the missed troughs are the
more recent ones which correspond to periods of contrac-
tion in Japan which have been relatively longer. The diŒer-
ence in duration between expansion and contraction in
Japan is not as large as that in the other two countries
under study and the mean value of the transition probabil-
ities, p12t and p21t, are almost the same (see Figs 3 and 4).
Model 2
1st
Model 3
1st
Model 4
1st 2nd
Model
Signal
2nd
2nd
Peaks
Jul-1970
‡
NA
NA
¡9
¡10
¡10
¡9
5
NA
NA
Nov-1973
Jan-1977
Feb-1980
Jun-1985
Feb-1991
Mar-1997
¡4
‡
¡10
5
0
(missed)
(missed)
(missed)
(missed)
(missed)
‡
¡14
¡3
2
‡
(missed)
(missed)
(missed)
¡4
4
‡
¡4
1
‡
¡3
2
False signls
Total signals
0
2
2*
9
2
2
Troughs
Dec-1971
Mar-1975
Oct-1977
Feb-1983
Nov-1986
Oct-1993
NA
NA
¡9
¡9
¡7
¡3
¡1
NA
NA
‡
3
The statistical insigni®cance of ^ , the contractionary
·
regime mean growth rate, might be explained by the sample
1
‡
¡2
¡7
¡3
¡1
¡3
¡1
¡7
¡6
1
‡
3
‡
‡
4
5
‡
‡
3
(missed)
(missed)
¡11
4
period. From Table 5, it is apparent that the estimated
mean derived from using either models 1 or 3 is quite dif-
ferent from that resulting from model 2 or 4. It is specu-
lated that this diŒerence arises from the diŒerent sample
period used. The diŒerence between the two periods is the
period of the late 1960s and early 1970s. This period is well
known to have been a period of rapid growth in Japan. In
this period, even in contraction, some positive growth rates
‡
8
(missed)
False signals
Total signals
1
5
2**
6
1
4
Notes: See note of Table 4.
The dates of peaks and troughs are based on EPA reference dates.
* False signals of peak by local minimum (turning point signal) in
model 3 are: Dec-1987(Nov-1988)and Sep-1994(Aug-1995).
** False signals of trough by local minimum (turning point signal)
in model 3 are: Apr-1980(May-1981) and Jun-1992(Mar-1993).
were experienced. Therefore, ^ in models 1 and 3, whose
·
1
sample includes the period of rapid growth rates, is larger
than that found from models 2 and 4, the estimation of
which was based on the sample covering the period of
lower overall growth rates. This result is not inconsistent
with Table 2.
From Table 5 some estimates in models 2, 3 and 4 are
^
insigni®cant and, in model 4, the sign of
is inconsistent
-
21
with expectations. In model 3, insigni®cant estimates are ^
·
1
^
and
though in model 2 both
for the contractionary regime (i.e. subscript 1)
The di culty in predicting troughs does not mean the
signal system is without value. After all, the eŒectiveness of
any signalling system derives from the nature of the under-
lying data upon which the model estimation is based.
-
12
^
^
are statistically insig-
21
and
-
-
11
ni®cant. This latter result implies that LD does not lead
CC. It would therefore appear that model 3 is better than
ˆ
Fig. 3. Transition probabilities from contraction to expansion by model 3: the Japanese case (mean 0.0394)

A new turning point signalling system
67
ˆ
Fig. 4. Transition probabilities from expansion to contraction by model 3: the Japanese case (mean 0.0704)
Results for Australia
somewhat self-ful®lling. However, the signals nonetheless
show as good a performance as in the USA. This is par-
ticularly the case for troughs. This is probably also because
Australian contractions are much shorter than expansions
as is shown in Table 2. The longer periods of expansion
provide for the possibility of more false signals in peaks.
These results for Australia provide further generality and
evidence of the eŒectiveness of the signalling system in
dating and predicting turning points, even though the tar-
geted turning points are not o cially recognized.
For the Australian case, relevant tables and ®gures are
Tables 7 and 8, and Figs 5 and 6.
The signs of all estimates are as expected. However,
some estimates in model 4 are not signi®cant, most prob-
ably because of multicollinearity. Though models 2 and 3
seem to be reasonable alternatives on the basis of estima-
tion (although model 2 has a slightly larger likelihood), the
lead-lag tables suggest model 3 is preferred for predicting
turning points. Note that the de®ned turning points used
here are not o cially recognized as in the USA and Japan,
because Australia does not have any o cial monthly busi-
ness cycle chronology. The peaks and troughs are deter-
mined by applying the Bry±Boschan turning point method
to the coincident series.10 Therefore, the results might be
Predicting the next turning points
As the purpose of the proposed signalling system is to
predict turning points, we now comment on the likelihood
Table 7. Parameter estimates for Australian coincident index
Parameter
Model 1
Model 2
Model 3
Model 4
p11
p22
0.8796 (0.0471)
0.9796 (0.0104)
¡0.2332 (0.0871)
0.3212 (0.0266)
·
¡0.2332 (0.0825)
¡0.2312 (0.0809)
¡0.2246 (0.0816)
1
·
0.3162 (0.0257)
0.3209 (0.0249)
0.3194 (0.0248)
2
2
¼
0.3888 (0.0429)
0.2025 (0.0119)
0.3884 (0.0456)
0.3877 (0.0414)
0.4033 (0.0467)
1
2
¼
0.2099 (0.0113)
2.1015 (0.6394)
¡0.1920 (0.1543)
0.2058 (0.0116)
2.5807 (0.7013)
0.2047 (0.0114)
2.4543 (0.6951)
¡0.0525 (0.2184)
¡0.0496 (0.0657)
4.6869 (1.3194)
0.0554 (0.3492)
0.0374 (0.1396)
2
-
-
-
-
-
-
10
11
¡0.0865 (0.0515)
12
4.3439 (0.8363)
0.6080 (0.2452)
5.0173 (0.3028)
20
21
0.1488 (0.0839)
22
Log likelihood
¡407.61
¡399.86
¡400.19
¡399.08
Notes: sample period: 10/1952±5/1998.
See note of Table 3.
10 See Bry-Boschan (1971) and Layton (1997a).

68
A. P. Layton and M. Katsuura
Table 8. Leads and lags of signals to Australian business cycle
chronology
the one hand, and Japan pulling itself out of contraction on
the other, was of worldwide concern and interest at the
time this analysis was done.
Model 2
1st
Model 3
1st
Model 4
1st 2nd
Table 9 shows the transition probabilities from January
1997 to the latest period available at the time the analysis
was undertaken. For the USA, there is no signal for a peak
in evidence. For Australia, the ®rst signal was given in June
1997. However, after a further 11 months, the second signal
is not yet in evidence given that the level of the transition
probability has been lower than the long-term mean value
throughout this period. If a local maximum is attained
before the second signal is given, the ®rst signal will be
disregarded. Thus, though there exists the possibility of a
peak occurring in the near future, the possibility is cur-
rently very unlikely.
Model
Signal
2nd
2nd
Peaks
‡
‡
Dec-1955
¡5
5
0
¡5
¡10
¡20
¡4
¡1
¡1
¡9
¡5
5
0
Dec-1960
Jul-1974
Sep-1976
Sep-1981
Apr-1990
¡9
¡19
¡15
¡15
¡6
¡7
‡
(missed)
2
(missed)
‡
‡
¡8
2
¡8
0
¡8
2
¡19
¡6
(missed)
(missed)
False signals
Total signals
5
10
6
11
6
10
Troughs
‡
‡
‡
3
Aug-1956
Sep-1961
Mar-1975
Nov-1977
May-1983
Jun-1992
¡1
¡4
¡6
6
‡
2
¡1
¡4
5
‡
4
¡1
¡4
5
‡
For Japan, December 1997 may be regarded as the ®rst
signal, but the second signal is not yet given and, if March
1998 turns out to be the local maximum (if the probabilities
in the subsequent three months are lower than the value of
that month), the ®rst signal will be disregarded. Therefore,
a reasonable conclusion at this point would be that a
trough in the Japanese economy is not imminent.
However, it should be noted that, as was described
above, the signals of troughs in Japan, in particular
for recent troughs, do not perform well. Of course, the
available data extend only up until May 1998. This
period does not include the recent economic policies ±
the so-called Emergency Economic Measures ± viz., the
permanent tax cut, the very large supplementary budget
spending, bringing the Long-term Credit Bank of Japan
and the Nippon Credit Bank under government control,
the provision of public funds to several big banks, and so
on.11
‡
‡
‡
1
¡6
2
¡6
2
‡
‡
(missed)
¡10
¡9
1
¡10
¡9
1
¡14
¡3
¡1
¡1
‡
(missed)
¡2
4
(missed)
False signals
Total signals
1
5
1
7
1
6
Notes: See note of Table 4.
The dates of peaks and troughs are determined by Bry±Boschan
methods for coincident index. See Layton (1977b).
of imminent turning points for the three countries under
study. As stated earlier, and shown by reference dates, the
USA and Australia are in expansion, and Japan is in con-
traction. Because of the size of the US and Japanese econo-
mies, the possibility of the USA falling into contraction on
ˆ
Fig. 5. Transition probabilities from contraction to expansion by model 3: the Australian case (mean 0.1195)
11 As it subsequently turned out, no peak was in evidence in either the U.S. or Australia throughoutthe rest of 1998 and 1999. The trough
in Japan apparently occurred in early 1999.

A new turning point signalling system
69
ˆ
Fig. 6. Transition probabilities from expansion to contraction by model 3: the Australian case (mean 0.0246)
minimum) of the transition probability. Applying this
system to three countries, it performs reasonably well.
However, the performance in each country is sensitive to
the estimation that in turn is sensitive to the duration of
phases in that country. The model which includes only the
ECRI long leading index (model 3) is suitable in all three
countries. This means LL has predictive power as an expla-
natory predictor variable for business cycle turning points.
And, as far as data up until May 1998 are concerned, there
is no sign that a turning point will occur during 1998 in any
of the three countries.
Table 9. Recent transition probabilities
USA
Japan
Australia
1/97
2/97
3/97
4/97
5/97
6/97
7/97
8/97
9/97
10/97
11/97
12/97
1/98
2/98
3/98
4/98
5/98
0.0121
0.0073
0.0056
0.0110
0.0091
0.0053
0.0045
0.0045
0.0039
0.0042
0.0048
0.0036
0.0032
0.0046
0.0046
0.0064
0.0045
0.0346
0.0236
0.0254
0.0194
0.0204
0.0177
0.0131
0.0161
0.0177
0.0212
0.0160
0.0130
0.0166
0.0193
0.0232
0.0188
0.0215
0.0054
0.0032
0.0037
0.0026
0.0010
0.0007
0.0013
0.0016
0.0008
0.0014
0.0014
0.0009
0.0020
0.0024
0.0024
0.0043
0.0048
In this paper, LD and LL are used as explanatory vari-
ables in the time varying transition probability model.
However, as stated earlier, the duration of phases seems
also to aŒect the results. The in¯ uence of duration could be
studied by introducing duration as an additional explana-
tory variable as in Durand and McCurdy (1994). And, in
relation to the estimate of the contractionary regime mean
growth rate in Japan, growth rates need not be classi®ed
simply into two phases, viz. contraction and expansion.
Using three phases, as in Sichel (1994) and Layton and
Smith (2000), is also a natural extension of this paper.
Future work could involve studying whether the signalling
system proposed here can be improved by utilizing such
appropriately speci®ed and estimated extended models.
mean
0.0436
0.0394
0.0246
Notes: For the USA and Australia, the ®gures are transition prob-
abilities from expansion into contraction. For Japan, the ®gures
are transition probabilities from contraction into expansion.
All probabilities are estimated by model 3 for each country.
V. CONCLUSION
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