On Business Cycle Asymmetries in ‘G5’ Countries
Khurshid M. Kiani
Department of Economics, 302-D Waters Hall, Kansas State University
Manhattan Kansas, 66506-4001, USA
Tel: +1-785-532-4581; Fax: +1-785-532-6919; E-mail: mkkiani@ksu.edu
Prasad V. Bidarkota*
Department of Economics, 245 Waters Hall, Kansas State University
Manhattan Kansas, 66506-4001, USA
Tel: +1-785-532-4578; Fax: +1-785-532-6919; E-mail: pbidarko@ksu.edu
May 29, 2002
Word count: Main text – 4,658
*
Corresponding author.
Abstract
We investigate whether business cycle dynamics in five industrialized countries are
characterized by asymmetries in conditional mean. We provide evidence on this issue
using a variety of time series models. Our approach is fully parametric. Our testing
strategy is robust to any conditional heteroskedasticity, outliers, and / or long memory
that may be present.
Our results indicate fairly strong evidence of nonlinearities in the conditional mean
dynamics of the GDP growth rates for Canada, Japan, and the US. For France and the
UK, the conditional mean dynamics appear to be largely linear.
Our study shows that while the existence of conditional heteroskedasticity and long
memory does not have much affect on testing for linearity in the conditional mean,
accounting for outliers does reduce the evidence against linearity.
Key phrases: business cycles; asymmetries; nonlinearities; conditional
heteroskedasticity; long memory; outliers; real GDP; stable distributions
JEL codes:
B22, B23, C32, E32
2
1
Introduction
The possible existence of asymmetries in economic fluctuations is being tested
extensively using aggregate macroeconomic data. While studies such as Neftci (1984),
Brunner (1992, 1997), Beaudry and Koop (1993), Potter (1995), and Ramsey and
Rothman (1996) conclude that there are significant asymmetries, others (Falk (1986),
Sichel (1989), DeLong and Summers (1986), and Diebold and Rudebusch (1990)) have
either failed to confirm these findings or have found only weak evidence supporting
them.
Detecting any nonlinearities that may be present in business cycles is important for
several reasons. Nonlinearities imply that the effects of expansionary and contractionary
monetary policy shocks on output are not symmetric. Any nonlinearities would invalidate
measures of the persistence of monetary policy and other shocks on GNP that are based
on linear models, including those derived from vector autoregressions (VARs).
Nonlinearities would necessitate that in order to validate theories of business cycles, such
as the real business cycle (RBC) theories, one would need to go beyond merely matching
the first and second moments of data with the moments implied by the theories in
question.
Granger (1995) recommends testing for linearity using heteroskedasticity-robust tests.
French and Sichel (1993) and Brunner (1992, 1997) show the existence of conditional
heteroskedasticity in real GNP data. Scheinkman and LeBaron (1989) report a weakening
3
of evidence against linearity after accounting for conditional heteroskedasticity in this
series.
There is a growing perception that the evidence of nonlinearity reported in several studies
so far may be due to the presence of outliers. Tsay (1988) demonstrates that linearity
could be rejected by the presence of outliers. Blanchard and Watson (1986) demonstrate
the existence of outliers in GNP data. Balke and Fomby (1994) and Scheinkman and
LeBaron (1989) report weakened evidence against linearity in US real GNP data once
outliers are taken into account.
Several macroeconomic time series data have been characterized as fractionally
integrated processes in a number of studies (Sowell, 1992b). While investigating the
possible existence of asymmetries in economic fluctuations it is important to use time
series models that describe both the long and short run properties of the data accurately.
Most of the studies on business cycles test for asymmetries without taking into account
conditional heteroskedasticity, outliers, and / or long memory. An exception is Bidarkota
(2000). This study finds robust evidence of non-linearities in the conditional mean
dynamics of the chain-weighted quarterly US GNP growth rates.
The present study seeks to extend the analysis in Bidarkota (2000) to an examination of
several developed countries. There are compelling reasons why such an extension is a
worthwhile undertaking. For instance, it is of interest to know whether business cycles in
4
different countries are all alike. If they are, then this provides a serious challenge to
macroeconomic theorists to develop theories of business cycles that can explain
fluctuations in economic activity in different countries without relying on country
specific institutional features. Documenting business cycle characteristics in different
countries is a first step in this task.
Also, in understanding spillover and contagion effects in international business cycles,
macroeconometricians traditionally rely on linear vector autoregressions (VARs). These
are convenient, simple to work with, well understood, and widely used in the literature.
However, there are some new studies that attempt to build nonlinear multivariate models
for studying business cycle linkages across countries (Anderson and Vahid (1998),
Anderson and Ramsey (2002)). Since multivariate nonlinear modeling is complex, it is
important that we document compelling evidence against linearity in international data
first before venturing to build these more complicated models.
This study is organized as follows. Section 2 outlines the various empirical models that
are used in our investigations. Section 3 discusses some important issues related to the
estimation of the models. Section 4 provides details on the data sources and some useful
summary statistics on the raw data. Section 5 reports detailed empirical results on
specification search and parameter estimates. Statistical tests for asymmetries and other
hypotheses of interest are reported in detail in Section 6. Important conclusions that can
be drawn from the results in the study are summarized in Section 7.
5
2
Non-Linear Time Series Models
In this study we use three classes of models to detect the possible presence of
asymmetries in real output series for the countries under investigation. These are termed
CDR-Augmented, CDR-Switching, and SETAR-Switching models. Within each of these
three classes of models, we entertain four different versions of the models. Model 1
incorporates stable distributions, conditional heteroskedasticity, and fractional
differencing. Imposing no fractional differencing (i.e., only integer differencing) on
Model 1 yields Model 2. Imposing homoskedasticity on Model 2 gives Model 3. Finally,
restricting the errors in Model 3 to come from Gaussian distributions gives Model 4.
Model 1 is the most general and it nests all other Models 2 through 4.
Each of the three classes of models listed above is described in detail in the following
three sub-sections.
2.1
CDR Augmented Models
Beaudry and Koop (1993) estimated a standard autoregressive moving average (ARMA)
model, augmented with an ad hoc nonlinear term for capturing asymmetries. This
nonlinear term labeled the current depth of a recession measures the gap between the
current level of output and the economy’s historical maximum level.
Bidarkota (1999, 2000) extended this model by incorporating stable distributions,
conditional heteroskedasticity, and long memory. In this study here we use this model.
In this class of models the most general model (Model 1) can be described as follows:
6
Φ ( L)(1 − L)d ( ∆y t − µ ) = [ Ω( L) − 1]CDR t + ε t
ε t | I t − 1 ~ z tc t ,
(1a)
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b2 c αt −1 + b3 | ε t −1 |α
(1b)
Here, ∆y t ≡ 100 * ∆(ln GDPt ) is the growth rate of GDP, its unconditional mean is µ, d is
the differencing parameter that takes on real values, Ω (.) and Φ (.) are polynomials of
orders r and p respectively in the lag operator L , with Ω (0) = Φ ( 0) = 1 . The term
CDR t is the current depth of recession that permits recessions to be less or more
persistent than expansions depending on the parameter estimates. It is defined as
CDR t = max{y t − j } j≥0 − y t .
A random variable X is said to have a symmetric stable distribution Sα (δ,c) if its log
characteristic function can be expressed as ln E exp( iXt ) = i δt − | ct | α . δ ∈ [−∞ , ∞ ] is the
location parameter that shifts the distribution either to the left or the right along the real
line, c ∈ [0, ∞ ] is the scale parameter that expands or contracts the distribution about δ ,
and α ∈ [0,2] is the characteristic exponent governing tail behavior. Smaller values of
this exponent indicate thicker tails. When α = 2 we obtain the normal distribution.
Equation 1b shows the evolution of the scales of the conditional distribution. When we
set α to 2 in the model, we obtain a normal GARCH (1,1) process for the conditional
variance in the volatility specification. Our Model 1b is analogous to the power ARCH
model introduced by Ding, Granger, and Engle (1993), with the exception that in the
7
latter specification the distribution of z t does not depend on the characteristic exponent
α. Dependence on α emerges here naturally because we are allowing for disturbances to
be drawn from the stable family. Liu and Brorsen (1995) also modeled volatility of the
daily foreign currency returns using stable errors. Similarly, McCulloch (1985) fitted a
GARCH-stable model to bond returns using absolute values instead of α powers.
When d = 0 we get a unit root in y t , but with d = −1 we end up with y t being integrated
of order zero I (0). ARFIMA models with long memory are defined in terms of the rate of
decay of their autocovariances, so the extension of these models to infinite variance
stable shocks is not immediate. Kokoszka and Taqqu (1995) contributed significantly to
the theory of fractionally differenced ARMA models with infinite-variance stable
innovations.
According to Brockwell and Davis (1991), a stationary casual and invertible solution to
an ARFIMA model with Gaussian errors requires that | d |< 0.5 . Kokoszka and Taqqu
(1995) showed that the existence of a unique casual MA (∞) representation to an
ARFIMA model with stable shocks requires α (d − 1) < −1 . This implies then that d be
positive when α > 1 . Moreover, for the ARFIMA model to be a solution to an AR (∞)
process requires that α > 1 and | d |< (1 − 1 / α) . In order to force our estimated models to
possess casual and invertible representations, we restrict α and d in Equation 1 to satisfy
these constraints.
8
Although we have an ad hoc non-linear CDR t term within an otherwise standard AR
framework with fractional differencing, this model is simple and parsimonious. When
Ω (L) = 1 , Equation 1a reduces to an autoregressive (AR) model with non-integer
differencing. Since it nests AR models, we can use the standard t-statistic or the
likelihood ratio (LR) statistic to test the statistical significance of the non-linear term
governing the conditional mean dynamics.1 With Ω (L) ≡ 1 + ω1L + ω2 L2 + ... + ωr Lr , when
the autoregressive lag order p is 0 and r is 1, ω1 = 0 yields a random walk with drift.
However, a positive ω1 implies that negative shocks are less persistent whereas a
negative ω1 implies that positive shocks are less persistent.
The existence of asymmetries essentially means that either the innovations are
asymmetric but the impulse transmission mechanism is linear, or that the innovations are
symmetric but the impulse transmission mechanism is nonlinear, or that the innovations
are asymmetric and the impulse transmission mechanism is nonlinear. However, it would
be hard to disentangle the asymmetric innovations from the nonlinear propagation
mechanism, if they both exist in a data series.
Although asymmetric α -stable distributions exist and are well defined, to determine
whether asymmetries in the conditional mean dynamics of the real GDP growth rates are
1
However, the asymptotic distribution of the t-test for the significance of the non-linear
CDR t term in the model given by Equation (1a) is non standard (Hess and Iwata, 1997),
both when the dependent variable is non-stationary [i.e. integrated of order one I(1)], and
when it is stationary [I(0)].
9
caused by asymmetric impulses being propagated linearly or symmetric impulses being
propagated nonlinearly or asymmetric impulses being propagated nonlinearly is beyond
the scope of this study. Here, we are merely investigating whether asymmetries exist in
the conditional mean regardless of how they can best be characterized.
2.2
CDR-Switching Models
Autoregressive models with time varying parameters (Tucci, 1995), threshold
autoregressive models (TAR), regime-switching models (Hamilton, 1989), and many
other nonlinear models have been used to capture asymmetries. TAR models (Tong and
Lim, 1980) are piecewise linear autoregressions. They are not only capable of
approximating a general nonlinear time series model of the form y t = f ( y t −1 ) + ε t and
capturing jump phenomenon, but they also admit limit cycles. Tsay (1988) introduces a
procedure for building and testing TAR models.
Beaudry and Koop (1993) also estimated a switching autoregressive moving average
model with the switch governed by a restriction defined in term of the current depth of
recession CDR t . In this section we use this switching model as modified by Bidarkota
(2000).
The most general model estimated within this class of models is as follows:
In Regime 1:
(1− φ1L − φ2L2 )(1− L)d (∆yt − µ1) = εt
(2a)
10
ε t | I t − 1 ~ z tc t ,
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b2 c αt −1 + b3 | ε t −1 |α
(2b)
In Regime 2:
(1 − φ3L − φ4L2 )(1 − L)d (∆yt − µ2 ) = εt
ε t | I t−1 ~ z t γc t ,
(2c)
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b 2ctα−1 + b3 | ε t−1 / γ |α
(2d)
When CDR t −1 = 0 , we get regime 1 and when CDR t −1 > 0 we obtain regime 2. The
unconditional mean of the process and the AR coefficients in the two regimes are
different. The parameter γ in regime 2 shows that the model has different scales in the
two regimes as well.
2.3
SETAR-Switching Models
This is identical to the CDR-Switching model, except for what governs switching
between the regimes. When a switch is governed by restrictions that are defined in terms
of the observed series y t , these models are called self-exciting threshold autoregressive
(SETAR) models. Potter (1995) estimated this type of model with a single restriction for
the log real GNP. The restriction is defined in terms of whether, ∆y t− d > r , where yt is
log real GNP, d is the delay and r is the threshold parameter.
11
The most general model estimated within this class of models is as follows:
In Regime 1:
(1− φ1L − φ2L2 )(1− L)d (∆yt − µ1) = εt
ε t | I t −1 ~ zt ct
z ~ iid
(3a)
Sα (0,1)
c αt = b1 + b2 c αt −1 + b3 | ε t −1 |α
(3b)
In Regime 2:
(1 − φ3L − φ4L2 )(1 − L)d (∆yt − µ2 ) = εt
ε t | It −1 ~ zt γct
z ~ iid
(3c)
Sα (0,1)
c αt = b1 + b 2ctα−1 + b3 | ε t−1 / γ |α
(3d)
When ∆y t −2 > 0 , we get regime 1 and when ∆y t −2 ≤ 0 , we get regime 2.
3
Estimation Issues
As mentioned earlier, Beaudry and Koop (1993) simply included an additive nonlinear
term in a standard autoregressive moving average (ARMA) model to capture
asymmetries in business cycles with the assumption that the shocks are normally
distributed. Although addition of a nonlinear term in a standard autoregressive moving
average framework is ad hoc, it does not impose any estimation problems. Bidarkota
12
(1999, 2000) used this model without any moving average terms but with errors having a
more general stable distribution, conditional heteroskedasticity, and long memory.
As in Bidarkota (1999, 2000), we do not consider any moving average (MA) terms in the
specification of the model. Maximum likelihood estimation of mixed ARMA models
with stable errors poses a challenge, although the Whittle estimator (Mikosch et al, 1995)
and minimum dispersion estimators (Brockwell and Davis, 1991) have been used in this
context.
We restrict ourselves to symmetric stable distributions here for a technical reason. We
use the computational algorithm due to McCulloch (1996) to obtain stable densities for
maximum likelihood estimation of our models. This algorithm works only when errors
are symmetric. However, Nolan (undated) has developed computational routines for
maximum likelihood estimation of models with stable distributions.
For the estimation of ARFIMA models the exact full information maximum likelihood
(ML) method of Sowell (1992a) may be adopted if the errors are iid normal. But for the
more complicated non-normal conditionally heteroskedastic models here, we use the
conditional sum of squares (CSS) estimator.
Baillie et al (1996) also used the conditional sum of squares (CSS) method, originally
proposed in the context of AFRIMA models by Hosking (1984), to estimate their
ARFIMA-GARCH models, with normal and Student-t errors. The CSS procedure is
equivalent to the full information MLE asymptotically. Baillie et al (1996) discussed
13
some properties of the CSS estimator in the context of ARFIMA models, particularly
with respect to its bias. They noted that not only does the CSS estimator do well when
they compared with it Sowell’s (1992a) exact MLE but also that it is computationally
feasible for more complex models.
Essentially we fit an ARMA model to the series (1 − L) d ( ∆y t − µ) that is obtained by
expanding the differencing operator (1 − L)d with the binomial expansion, and truncating
the infinite series at the first available observation. The CSS estimator is discussed for
ARMA models in Box and Jenkins (1976).
4
Preliminary Data Analysis
4.1
Data Sources
We obtained quarterly GDP data from the International Financial Statistics (IFS) CDROM (September 2001) for Canada, France, Japan, the United Kingdom (UK), and the
United States of America (USA).2 The dataset spans the period from 1957:1 to 2000:4 for
all countries except France for which the data begins in 1970:1. Table 1.1 provides
2
Initially, we tried working with all the G7 countries that include, in addition to the
above five countries, Germany and Italy. However, Germany poses a unique problem on
account of its reunification. It’s GDP increased by 35.43 percent in 1991:1. Data for Italy
showed an inexplicable spike of 87.5 percent annualized quarterly growth rate in 1970:1.
Consequently we decided to exclude these two countries from our analysis.
14
further details on the dataset used for each country. Figure 1 plots the logarithmic GDP
series and Figure 2 plots the annualized quarterly growth rates.
4.2
Summary Statistics
Table 1.2 provides some useful statistics on the raw data and Table 1.3 summarizes
preliminary statistical inferences for some interesting hypotheses. The average annualized
quarterly GDP growth rates range from 2.44 to 8.84 percent, with the average growth
rates for all countries being significantly positive. UK has the lowest and Japan has the
highest growth rate. The quarterly (annualized) standard deviations range from 2.40 for
France to 7.80 for Japan.
Skewness measures range from –0.54 for France to 0.64 for Japan, being statistically
significant for all countries except Canada. Excess kurtosis measures range from 0.14 for
Canada to 2.85 for the UK, with significant fat tails found in UK, US, and Japan
(marginal). The Jarque-Bera test rejects normality for all countries except Canada. These
preliminary results are subject to some qualification because the tests are based on the
assumption that the growth rates are iid. As will be evident later, this assumption is not
appropriate.
The Augmented Dicky Fuller (ADF) test indicates unit roots in levels (with constant and
time trend) for all countries but not in growth rates (with constant only). The only
15
exception is Japan for which the test fails to reject unit roots in growth rates with a
constant term only but does reject with constant and time trend.
Goldfeld-Quandt test fails to reject homoskedasticity in all countries and the Lagrange
Multiplier (LM) test detects autoregressive conditional heteroskedasticity (ARCH) only
in Japan.
5
Empirical Results
5.1
Specification Search
An extensive specification search for each country was conducted for the four versions
(Model 1 through Model 4) of each of the three classes of models described in section 2.
For the CDR-Augmented class of models, the specification search was done over all
parameterizations with lag orders for the autoregressive and CDR t terms of three or less
for parsimony. For the two classes of switching models, namely the CDR-Switching and
SETAR-Switching models, the search was done with the autoregressive lag polynomials
in the two regimes restricted to be of orders (3,3), (2,2), (1,1), or (0,0).
The best parameterizations for each version within each class of models are selected for
each country by the minimum Schwarz Bayesian criterion. The chosen parameterizations
are reported in Tables 2.1, 2.2, and 2.3 for CDR-Augmented, CDR-Switching, and
SETAR-Switching models, respectively. The table also reports the number of parameters
16
in the models, the maximized log-likelihood values, and the values of the Akaike
information (AIC) and Schwarz Bayesian (SBC) criteria, respectively.
Since the AIC is well known to yield relatively over parameterized specifications of
models in general, we choose the versions selected by the minimum SBC criterion in
what follows. These selected versions of the models are reported in Table 2.4.
5.2
Parameter Estimates
Tables 3.1, 3.2, and 3.3 contain parameter estimates for the best parameterizations of the
most general versions (Model 1) within each of the three classes of models, namely, the
CDR-Augmented, CDR-Switching, and SETAR-Switching models. The estimates are
presented for all countries analyzed in this study.
In Table 3.1 the estimates of the unconditional mean µ range from –0.028 percent per
quarter for Japan to 0.905 percent per quarter for Canada. The value of the characteristic
exponent α ranges from virtually 2 for Canada and Japan to a low of 1.734 for the UK.
This is understandable given that fat tails were found in Table 1.2 to be most prominent
in the UK and absent in Canada. The volatility persistence parameter b 2 is estimated to
be between 0.827 for the US to 0.936 for Canada. The ARCH parameter b 3 takes values
from 0.026 for Canada to 0.055 for the UK. The sum of the GARCH parameter estimates
b 2 + b 3 takes values of 0.962 for Canada to a low of 0.877 for the US. The fractional
17
differencing parameter d is estimated to be between -0.464 for the US and 0.499 for
Japan.
The estimates of the autoregressive (AR) parameters range from 0.784 for the US to a
low of –0.389 for Japan. The best models selected within this class of models for Canada
and the UK have no AR terms.
In Table 3.2, the parameter estimates of the unconditional mean in regime 1 µ1 are close
to the values of µ reported in Table 3.1 for the CDR-Augmented model, except for
Japan. The estimates of the unconditional mean in regime 2 µ 2 vary between –2.65 for
Japan to 2.76 for the US. For all countries, excepting Japan, regime 1 is associated with
lower mean growth rates than regime 2. For Japan, this is reversed. The estimates of the
characteristic exponent α are very similar to those reported in Table 3.1. The estimates
of the volatility persistence parameter b 2 are uniformly higher and the estimates of the
ARCH parameter b 3 are uniformly lower in this case than the values of the
corresponding parameters reported earlier in Table 3.1.
For all countries excepting Japan, the scale ratio γ in regime 2 is estimated to be
between 0.914 (for the US) and 0.961 (for France). For Japan, this parameter is estimated
to be 1.08. Given that for all countries (excepting Japan) regime 1 is associated with
lower growth rates, this means that for all countries including Japan lower mean GDP
growth rate regimes are associated with relatively higher volatility.
18
Only Japan has AR terms (of first order only) appearing in the best parameterization
within this class of models. The estimates of the AR parameters in both regimes are
negative.
For the SETAR switching model estimates reported in Table 3.3, no single regime is
necessarily associated uniformly across all countries with lower growth rates.
Furthermore, the parameter estimates also suggest that low mean growth rate regimes are
no longer necessarily associated with relatively higher volatility. All other parameter
estimates, however, are generally similar to the values reported earlier for the CDRAugmented and CDR-Switching models.
6
Hypotheses Tests
We performed four types of hypotheses tests on the estimated models. The first is a test
for normality. The second is a test for homoskedasticity. The third tests for the possible
presence of long memory. The last is a test for linearity in the conditional mean. The
following sub-section describes the various tests and sub-section 6.2 provides empirical
results on the hypotheses tests.
6.1 Description of Various Tests
6.1.1
Test for Normality
19
We performed a normality test based on the value of α . If α equals 2, normality results.
If the value of a is less than 2, then the model is non-normal stable. This test compares
the likelihood ratio (LR) statistic for two models with identical parameterizations.
Since the null hypothesis for this test lies on the boundary of admissible values for α , the
LR test statistic does not have the usual χ 2 distribution asymptotically. Therefore, the
test here is based on small sample critical values generated by Monte Carlo simulations
reported in McCulloch (1997, Table 4, panel b).
6.1.2
Test for Homoskedasticity
The second test is a test for homoskedasticity. Under the null of homoskedasticity, the
GARCH parameters b 2 = b 3 = 0 . The test is based on the likelihood ratio test statistic.
Since b1 and c0 end up being trivial transformations of one another under the null
hypothesis, it is not clear whether the LR test statistic asymptotically has 2 or 3 degrees
of freedom. We base our statistical inference on critical values derived conservatively
from the χ32 distribution.
6.1.3
Test for Long Memory
The null hypothesis is d = 0 implying that the logarithm of GDP has a unit root. The
alternative of d ≠ 0 implies that the logarithm of GDP is described as a fractionally
differenced series. Depending on the estimates of d that are obtained, the logarithm of
GDP could be a stationary or a non-stationary series, with or without long memory. See
Baillie (1996) for an extensive survey on fractionally differenced processes. The test is
20
once again based on the likelihood ratio. Model 1 is the unrestricted model and Model 2
is the restricted model with d restricted to zero. The test statistic is distributed as the χ12
since we have only one restriction here.
6.1.4
Test for Linearity in Conditional Mean
The last is a test for linearity in the conditional mean. While the three tests above are
performed alternatively using the various versions within all the three classes of models,
the test for linearity in the conditional mean is done using different versions of the
models within only the two classes of switching models. This is because the minimum
SBC criterion ends up selecting among the four different versions of the CDRAugmented class of models a version and a parameterization that does not include the
additive CDR term for any of the countries except France (see Tables 2.4 and 2.1).
Footnote 1 reports some added difficulty in testing for the significance of the CDR term
with a standard t-test or an LR test.
Depending on the versions of the two classes of switching models used, we end up testing
for linearity successively using homoskedastic Gaussian models, homoskedastic stable
models, GARCH stable models, and long memory GARCH stable models. Under the null
hypothesis of linearity in conditional mean, the unconditional means in the two regimes
µ 1 and µ2 are equal, the scale ratio γ is equal to one, and the corresponding
autoregressive coefficients in the two regimes, if present in the specific parameterizations
of the model versions used in the test, are equal. If the null hypothesis is not rejected then
21
we only have one regime (linear conditional mean dynamics). If not, then we have two
distinct regimes describing the GDP growth rates.
6.2 Empirical Results on Hypotheses Tests
Hypotheses tests for normality, homoskedasticity, long memory, and linearity in
conditional mean were performed as elaborated in the earlier sub-section. The empirical
results of the various hypotheses tests listed above are reported in Tables 4.1, 4.2, and
4.3, respectively, for the CDR-Augmented, CDR-Switching and SETAR Switching
models. All tests are based on the likelihood ratio (LR) test statistic. For each of the three
tables, a different test is reported in the various rows of the first column. For each such
test, the numbers in the five rows in the other columns for that test are the LR test
statistics for the five countries arranged alphabetically in ascending order. P-values are
reported in parentheses. All statistical inferences are drawn at the five percent
significance level.
6.2.1
Results on Normality
The test for normality easily rejects for the UK and the US. The evidence is somewhat
weaker for France and Japan. For Canada, however, the test fails to reject normality
overwhelmingly. The test results are largely unchanged when we account for conditional
heteroskedasticity and long memory.
6.2.2
Results on Homoskedasticity
22
From the three tables, it is clear that when we test for homoskedasticity using Model 2,
all countries except France show strong evidence against homoskedasticity. France
strongly fails to reject homoskedasticity. The statistical inferences remain unchanged
when we go from 5 to 10 percent significance level. Also, there are no changes in the
inferences when we do the tests after accounting for long memory with Model 1.
6.2.3
Results on Long Memory
The test for d = 0 strongly rejects in all countries with all three classes of models except
for the UK with SETAR Switching models. The statistical inferences are the same at the
5 and 10 percent significance level. However, it is important to note that Sowell’s
(1992b) Monte Carlo simulations suggest that the true small sample p-values may be
higher than the asymptotic χ12 p-values. Typically, GDP data tend to be largely
uninformative about the nature of their long run properties and it is generally hard to rule
out trend stationarity, unit roots, as well as fractional differencing from these data series.
6.2.4
Results on Linearity in Conditional Mean
Canada, Japan, and the US show fairly strong evidence against linearity in the conditional
mean dynamics of the GDP growth rates. Accounting for conditional heteroskedasticity
and long memory seems to be relatively unimportant when testing for linearity. However,
accounting for outliers decreases the evidence against linearity. Statistical inferences are
not affected much when we change the significance level of the test from 10 to 5 percent.
6.3 Discussion of Results on Hypotheses Tests
23
Our results on nonlinearity in the conditional mean for the US are in line with Bidarkota
(2000). This shows that the evidence against linearity in mean for the US is robust to
changes in the sample period.
Koop and Potter (2001) investigate whether nonlinearities could arise from structural
instability. Blanchard and Simon (2001) show a slowdown in the variance of US
economic activity suggesting a possible structural change in the early 1980s. We do not
account for this possibility in this study.
Diebold and Inoue (2001) demonstrate that spurious evidence of long memory may be
found in a series that undergoes occasional structural change. This is analogous to the
spurious evidence of unit roots in a series with occasional breaks (Perron, 1989). Our
results on long memory are subject to this caveat.
7
Conclusions
We used three classes of models, namely, the CDR-Augmented models, CDR-Switching
models, and SETAR-Switching models for testing asymmetries in the conditional mean
dynamics of GDP data in five industrialized countries. Our approach is fully parametric.
Our time series models account for the possibility of conditional heteroskedasticity,
outliers, and long memory in the data series.
24
Our results indicate fairly strong evidence of nonlinearities in the conditional mean
dynamics of the GDP growth rates for Canada, Japan, and the US. For France and the
UK, the conditional mean dynamics seem to be largely linear.
Our study shows that the Switching models capture non-linearity better than the CDRAugmented models. Accounting for conditional heteroskedasticity and long memory is
relatively unimportant when testing for linearity. However, accounting for outliers
decreases the evidence against linearity.
Our results also indicate that the GDP series for all countries excepting France reject
homoskedasticity. Long memory appears to be pervasive. While normality is strongly
rejected for the US and the UK, the evidence is somewhat weaker for France and Japan,
and overwhelmingly in favor of normality for Canada.
25
Table 1.1: Data Description
Canada
France
Japan
UK
USA
Quarterly
Real GDP
Quarterly
Real GDP
Quarterly
Nominal GDP
Quarterly
Real GDP
Quarterly
Real GDP
Sample Period
1957:12000:4
1970:12000:4
1957:12000:4
1957:12000:4
1957:12000:4
Sample Length
176
124
176
176
176
Data Series
Notes on Table 1.1
1. We obtained the quarterly seasonally adjusted GDP data for all countries from the
September 2001 edition of the International Financial Statistics (IFS) CD-ROM.
2. We used nominal GDP for Japan because seasonally adjusted data was only available
for the nominal series and not for the real series on the IFS CD-ROM.
26
Table 1.2: Summary Statistics
Canada
France
Japan
UK
USA
0.91
0.63
2.21
0.61
0.83
(p-value for mean = 0)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
Standard Deviation
0.97
0.60
1.95
1.05
0.95
Skewness
0.17
-0.54
0.64
0.36
-0.44
(0.49)
(0.01)
(0.00)
(0.03)
(0.01)
0.14
0.48
0.57
2.85
1.26
(0.32)
(0.14)
(0.06)
(0.00)
(0.00)
0.21
7.16
14.15
62.8
17.16
(0.89)
(0.03)
(0.00)
(0.00)
(0.00)
Levels (constant + trend)
-2.24
-2.25
-1.83
-1.39
0.14
First Differences (constant only)
-4.59
-3.99
-1.80
-4.64
-4.83
Orders of Significant
Autocorrelations (at 5% level)
Orders of Significant
Partial Autocorrelations
(at 5% level)
Goldfeld-Quandt Test
1-3
1,2
3
1,2
1,11
1,2
3,8,16,31
1,2
24.81
20.1
36.21
14.72
13.67
(p-value for homoskedasticity)
(1.00)
(1.00)
(0.99)
(1.00)
(1.00)
0.01
0.00
24.51
0.01
0.03
(0.91)
(0.95)
(0.00)
(0.91)
(0.85)
Mean
(p-value for skewness = 0)
Excess Kurtosis
(p-value for excess kurtosis = 0)
Jarque-Bera Test
(p-value for normality)
ADF Test for Unit Roots
LM Test for ARCH
(p-value for no ARCH effects)
1-10, 12,
13, 16
1-5,8,9
27
Notes on Table 1.2
1.
All statistics reported are for quarterly growth rates defined as 100∆ log y t ,
except the ADF test in levels for which the test statistics are for log levels
log y t .
2.
P-values for statistics pertaining to skewness, excess kurtosis, Jarque-Bera,
Goldfeld-Quandt, and the LM test for ARCH are reported in parentheses
below the test statistics.
3.
Optimal lag order for the ADF tests is chosen using the minimum SBC
criterion. Asymptotic critical values at the 10% significance level for the
model with constant, and with constant and time trend, are -2.57 and -3.13,
respectively.
4.
Optimal lag order for the LM test is chosen using the minimum SBC criterion.
28
Table 1.3: Summary of Statistical Inferences
Canada
France
Japan
UK
USA
Nonzero Mean
Y
Y
Y
Y
Y
Skewness
N
N
Y
Y
N
Fat Tails
N
N
N
Y
Y
Non-Normality
N
Y
Y
Y
Y
Levels (constant +
trend)
First Differences
(constant only)
Y
Y
Y
Y
Y
N
N
Y
N
N
Homoskedasticity
Y
Y
Y
Y
Y
ARCH Effects
N
N
Y
N
N
Unit Roots
29
Notes on Table 1.3
1.
Y indicates presence of, and N indicates absence of, the characteristic listed in
the first column of the corresponding row. For example, N for USA in the second
row indicates absence of skewness. These inferences are based on the statistics
and p-values reported in Table 1.2.
3.
All statistical inferences are done assuming a 5 percent level of significance.
30
TABLE 2.1: Specification Search - CDR-Augmented Models
Model 4
Model 3
Model 2
Model 1
(1,0)
(2,1)
(3,0)
(0,0)
(1,0)
(1,0)
(2,1)
(3,0)
(0,0)
(1,1)
(1,0)
(2,1)
(3,0)
(1,0)
(1,0)
(0,0)
(2,0)
(1,0)
(0,0)
(1,0)
3
5
5
2
3
4
6
6
3
5
7
9
9
7
7
7
9
8
7
8
Log likelihood
-231.10
-106.31
-308.26
-252.12
-224.23
-231.07
-105.28
-301.97
-240.44
-218.70
-219.01
-105.28
-281.87
-219.84
-209.63
-219.30
-107.27
-282.45
-219.73
-207.33
AIC
468.21
222.62
626.52
508.24
454.46
470.14
222.57
615.93
486.87
447.39
452.01
228.56
581.74
453.68
433.26
452.60
232.55
580.90
453.45
430.66
SBC
477.65
236.52
642.23
514.54
463.90
482.73
239.24
634.78
496.32
463.13
474.04
253.65
610.02
475.71
455.29
474.63
257.63
606.04
475.48
455.84
Model
parameterization
selected by minimum
SBC criterion
Number of parameters
31
Notes on Table 2.1
1. For each item listed in the first column, the five rows in the subsequent columns
for that item denote the corresponding statistics for the five countries Canada,
France, Japan, UK, and USA in that order.
2. The most general model is Model 1. We get Model 2 by imposing d = 0 .
Imposing homoskedasticity on Model 2, we get Model 3. Setting α = 2 in Model
3 yields Model 4.
32
TABLE 2.2: Specification Search – CDR-Switching Models
Model 4
Model 3
Model 2
Model 1
(1,1)
(2,2)
(2,2)
(0,0)
(1,1)
(1,1)
(1,1)
(3,3)
(0,0)
(1,1)
(1,1)
(1,1)
(3,3)
(0,0)
(1,1)
(0,0)
(0,0)
(1,1)
(0,0)
(0,0)
6
8
8
4
6
7
7
11
5
7
10
10
14
8
10
9
9
11
9
9
Log likelihood
-227.18
-108.92
-306.92
-249.50
-213.74
-226.95
-112.54
-295.19
-238.63
-210.32
-211.11
-112.30
-269.70
-218.04
-202.42
-212.22
-108.83
-274.89
-216.05
-201.03
AIC
466.35
233.85
629.83
507.00
439.48
467.90
239.07
612.38
487.26
434.64
442.21
244.61
567.4
452.08
424.84
442.44
235.67
571.79
450.09
420.06
SBC
485.20
256.08
654.92
519.57
458.33
489.89
258.53
646.87
502.97
456.63
473.63
272.40
611.30
477.21
456.25
470.72
260.68
606.28
478.37
448.34
Model
parameterization
selected by minimum
SBC criterion
Number of parameters
Notes on Table 2.2
1. As in Table 2.1.
33
TABLE 2.3: Specification Search – SETAR-Switching Models
Model 4
Model 3
Model 2
Model 1
(0,0)
(2,2)
(3,3)
(3,3)
(1,1)
(0,0)
(1,1)
(3,3)
(2,2)
(0,0)
(0,0)
(1,1)
(3,3)
(0,0)
--
(0,0)
(0,0)
(1,1)
(0,0)
(1,1)
4
8
10
10
6
5
7
11
9
5
8
10
14
8
--
9
9
11
9
11
Log likelihood
-231.08
-107.20
-303.93
-243.15
-214.09
-231.08
-110.97
-297.60
-235.93
-219.47
-228.60
-110.86
-273.23
-217.57
--
-217.49
-109.46
-275.13
-216.37
-199.64
AIC
470.17
230.38
627.86
506.30
440.19
472.17
235.95
617.20
489.85
448.94
473.20
241.71
574.45
451.14
--
452.98
236.92
572.27
450.75
421.28
SBC
482.20
252.62
659.22
537.71
459.04
487.88
255.40
651.69
518.12
464.65
498.33
269.51
618.35
476.27
--
481.25
261.93
606.76
479.02
455.83
Model
Parameterization
selected by minimum
SBC criterion
Number of parameters
Notes on Table 2.3
1. As in Table 2.1.
2. The symbol “--“ denotes missing numbers because the numerical algorithm for
maximization of the log likelihood function failed to converge.
34
TABLE 2.4: Model Selection
Canada
France
Japan
UK
USA
Minimum AIC
M2
M3
M1
M1
M1
Minimum SBC
M2
M4
M1
M1
M2
Minimum AIC
M1
M4
M2
M1
M1
Minimum SBC
M1
M4
M1
M2
M1
Minimum AIC
M1
M3
M2
M1
M1
Minimum SBC
M1
M4
M1
M2
M1
CDR-Augmented
CDR-Switching
SETAR-Switching
35
Notes on Table 2.4
1. For each country and within each class of models, the best versions among all
Models 1 through 4 (M1 to M4) are selected and reported in the table.
2. Model selection is done alternatively by two different criteria – the minimum
Akaike information criterion (AIC) and the Schwarz Bayesian criterion (SBC).
3. AIC = T ln( RSS) + 2n , where n in the number of estimated parameters and T is
the number of usable observations.
4. SBC = T ln( RSS ) + n ln( T) , where n in the number of estimated parameters and T
is the number of usable observations.
36
Table 3.1: Parameter Estimates - CDR-Augmented Models
Canada
(0,0)
France
(2,0)
Japan
(1,0)
UK
(0,0)
USA
(1,0)
0.905
(0.253)
0.652
(0.092)
-0.028
(1.029)
0.675
(0.099)
0.825
(0.028)
d
0.266
(0.058)
-0.379
(0.233)
0.499
(0.000)
0.145
(0.064)
-0.464
(0.209)
α
1.999
(0.000)
1.931
(0.074)
1.999
(0.000)
1.734
(0.109)
1.867
(0.095)
c0
1.016
(0.200)
0.319
(0.218)
2.313
(0.879)
1.102
(0.427)
1.252
(0.594)
b1
0.000
(0.000)
0.010
(0.017)
0.018
(0.009)
0.002
(0.004)
0.017
(0.015)
b2
0.936
(0.039)
0.856
(0.171)
0.889
(0.031)
0.861
(0.049)
0.827
(0.084)
b3
0.026
(0.020)
0.041
(0.051)
0.038
(0.017)
0.055
(0.024)
0.050
(0.029)
φ1
0.614
(0.236)
-0.389
0.063
φ2
0.285
(0.149)
-107.20
-282.15
Model
Parameterization
µ
Log Likelihood
-219.30
0.784
(0.163)
-219.93
-207.33
37
Notes on Table 3.1
1.
Parameter estimates are reported for the best model specification for version 1
within the class of CDR-Augmented models as determined by the minimum SBC
criterion.
2.
The most general CDR-Augmented model (Model 1) can be written as:
Φ ( L)(1 − L)d (∆y t − µ ) = [Ω ( L) − 1]CDR t + ε t
(1a)
ε t | I t−1 ~ z t c t , z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b2 c αt −1 + b3 | ε t −1 |α
(1b)
38
Table 3.2: Parameter Estimates - CDR-Switching Models
Canada
(0,0)
France
(0,0)
Japan
(1,1)
UK
(0,0)
USA
(0,0)
µ1
0.906
(0.280)
0.560
(0.255)
0.299
(1.072)
0.721
(0.130)
0.702
(0.314)
µ2
1.631
(0.746)
1.205
(0.599)
-2.653
(3.424)
0.830
(0.259)
2.760
(0.977)
d
0.279
(0.062)
0.281
(0.088)
0.499
(0.000)
0.157
(0.069)
0.292
(0.070)
α
1.999
(0.000)
1.941
(0.141)
1.999
(0.000)
1.781
(0.126)
1.908
(0.089)
c0
0.914
(0.103)
0.473
0.140
2.020
(0.461)
1.178
(0.402)
0.996
(0.337)
b1
0.000
(0.000)
2.1e-11
(0.000)
0.007
(0.005)
4.6e-9
(0.004)
1.4e-7
(0.006)
b2
1.014
(0.010)
1.014
(0.044)
0.965
(0.009)
0.930
(0.055)
0.963
(0.047)
b3
0.000
(0.000)
1.03e-11
(0.022)
0.000
(0.000)
0.049
(0.025)
0.032
(0.019)
Model
Parameterization
φ1
-0.355
(0.070)
-0.703
(0.209)
φ3
γ
Log Likelihood
0.943
(0.020)
0.961
(0.060)
1.080
(0.039)
0.938
(0.029)
0.914
(0.043)
-212.22
-108.83
-274.89
-216.04
-201.03
39
Notes on Table 3.2
1.
Parameter estimates are reported for the best model specification for version 1
within the class of CDR-Switching models as determined by the minimum SBC
criterion.
2
The most general CDR-Switching model (Model 1) can be written as follows.
In Regime 1 when CDR t −1 = 0 :
(1− φ1L − φ2L2 )(1− L)d (∆yt − µ1) = εt ,
ε t | I t − 1 ~ z tc t ,
(2a)
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b 2ctα−1 + b3 | ε t −1 |α
(2b)
In Regime 2 when CDR t −1 > 0 :
(1 − φ3L − φ4L2 )(1 − L)d (∆y t − µ2 ) = ε t ,
ε t | I t−1 ~ z t γc t ,
(2c)
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b 2ctα−1 + b3 | ε t−1 / γ |α
(2d)
40
Table 3.3: Parameter Estimates - SETAR-Switching Models
Parameters
Model
Parameterization
Canada
(0,0)
France
(0,0)
Japan
(1,1)
UK
(0,0)
USA
(1,1)
µ1
0.966
(0.247)
0.811
(0.207)
-0.233
(1.034)
0.743
(0.106)
0.830
(0.023)
µ2
1.201
(0.704)
0.281
(0.404)
5.872
(6.079)
0.503
(0.283)
0.685
(0.035)
d
0.254
(0.066)
0.228
(0.079)
0.499
(0.000)
0.125
(0.066)
-0.473
(0.000)
α
1.999
(0.000)
1.960
(0.140)
1.999
(0.000)
1.772
(0.122)
1.897
(0.086)
c0
1.160
(0.353)
0.419
(0.132)
1.554
(0.347)
1.052
(0.385)
0.895
(0.278)
b1
0.000
(0.000)
7.4e-86
(0.000)
0.005
(0.005)
3.1e-9
(0.027)
0.000
(0.000)
b2
0.910
(0.051)
1.040
(0.057)
0.970
(0.009)
0.916
(0.056)
0.964
(0.021)
b3
0.051
(0.037)
2.7e-9
(0.019)
0.000
(0.000)
0.053
(0.027)
0.022
(0.012)
φ1
-0.398
(-0.067)
-0.254
(0.444)
φ3
γ
Log Likelihood
0.940
(0.085)
-217.49
0.888
(0.082)
-109.46
1.161
(0.063)
-275.97
0.769
(0.060)
0.486
(0.158)
0.913
(0.066)
-216.37
0.929
(0.064)
-199.64
41
Notes on Table 3.3
1. Parameter estimates are reported for the best model specification for version 1
within the class of SETAR-Switching models as determined by the minimum
SBC criterion.
2. The most general SETAR-Switching model (Model 1) can be written as follows.
In Regime 1 when ∆y t − 2 > 0 :
(1− φ1L − φ2L2 )(1− L)d (∆yt − µ1) = εt ,
ε t | I t − 1 ~ z tc t ,
(3a)
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b 2ctα−1 + b3 | ε t −1 |α
(3b)
In Regime 2 when ∆y t − 2 ≤ 0 :
(1−φ3 L −φ4L2 )(1− L)d (∆yt −µ2 ) = ε t ,
ε t | I t−1 ~ z t γc t ,
(3c)
z t ~ i.i.d.Sα ( 0,1)
c αt = b1 + b 2ctα−1 + b3 | ε t−1 / γ |α
(3d)
42
TABLE 4.1: Hypotheses Tests – CDR-Augmented Models
Model 4
LR ( α = 2 )
LR (no GARCH)
LR ( d = 0 )
Model 3
Model 2
Model 1
0.06
2.06
12.58
23.36
8.26
0.00
0.54
6.72
20.34
4.96
0.00
0.00
16.28
18.62
4.38
24.3 (0.00)
0.02 (0.99)
40.20 (0.00)
41.16 (0.00)
83.74 (0.00)
21.16 (0.00)
1.32 (0.72)
35.78 (0.00)
41.24 (0.00)
15.96 (0.00)
17.88 (0.00)
34.10 (0.00)
55.66 (0.00)
17.22 (0.00)
4.60 (0.03)
43
Notes on Table 4.1
1.
The table presents likelihood ratio (LR) test statistics and their associated p-values
in parentheses.
2.
For each item reported in column 1, row one in column 2 and subsequent columns
for that item presents statistics for Canada, row two for France, row three for
Japan, row four for the UK, and row five for the USA.
3.
LR ( α = 2 ) is a test for normality. The distribution of the test statistic in this
instance is not standard χ 2 because the null hypothesis is on the admissible
boundary of α . However, critical values at the 10% and 5% significance level are
available through Monte Carlo simulations from McCulloch (1997, Table 4, panel
b). These are 0.243 and 1.120, respectively.
4.
LR (no GARCH) is a test for homoskedasticity. The null hypothesis is
b 2 = b3 = 0 . Under this null, b1 and c0 are trivial transformation of one another.
As such, it is not clear whether the LR statistic has asymptotic χ 2 distribution
with two or three degrees of freedom. We report conservative χ 23 p-values in
parentheses.
5.
LR ( d = 0 ) is a test for the absence of long memory.
44
TABLE 4.2: Hypotheses Tests - CDR-Switching Models
Model 4
LR ( α = 2 )
Model 3
Model 2
Model 1
0.46
0.16
15.44
21.74
6.84
4.12
2.56
0.78
58.18
4.80
0.00
5.22
0.00
22.52
11.44
31.68 (0.00)
0.48 (0.92)
50.98 (0.00)
40.66 (0.00)
15.80 (0.00)
12.66 (0.01)
0.00 (1.00)
55.72 (0.00)
44.92 (0.00)
--
LR (no GARCH)
LR ( d = 0 )
LR (one regime)
22.42 (0.00)
19.84 (0.00)
35.68 (0.00)
36.32 (0.00)
19.07 (0.00)
5.86 (0.00)
2.15 (0.14))
16.36 (0.00)
2.72 (0.26)
7.5 (0.06)
6.22 (0.10)
1.30 (0.72)
13.56 (0.00)
0.98 (0.61)
11.30 (0.01)
13.12 (0.00)
1.46 (0.69)
24.34 (0.00)
4.46 (0.11)
0.00 (1.00)
10.68 (0.00)
0.10 (0.95)
14.20 (0.00)
4.16 (0.00)
13.44 (0.00)
45
Notes on Table 4.2
1. The table presents likelihood ratio (LR) test statistics and their associated p-values
in parentheses.
2. For each item reported in column 1, row one in column 2 and subsequent columns
for that item presents statistics for Canada, row two for France, row three for
Japan, row four for the UK, and row five for the USA.
3. LR (one regime) is a test for linear conditional mean dynamics. The null
hypothesis is µ1 = µ2 , γ = 1, and the corresponding autoregressive coefficients in
the two regimes are equal.
4. See notes 3-5 in Table 4.1.
5. The symbol “--“ denotes missing numbers because the numerical algorithm for
maximization of the log likelihood function failed to converge.
46
TABLE 4.3: Hypotheses Tests – SETAR-Switching Models
Model 4
LR ( α = 2 )
Model 3
Model 2
Model 1
0.00
2.80
12.66
14.44
2.98
0.00
0.00
0.38
13.76
--
0.00
4.74
0.00
46.20
12.12
11.50 (0.17)
0.22 (0.97)
48.74 (0.00)
40.20 (0.00)
--
18.74 (0.00)
0.00 (1.00)
75.82 (0.00)
41.26 (0.00)
--
LR (no GARCH)
LR ( d = 0 )
LR (one regime)
22.22 (0.00)
12.96 (0.00)
56.38 (0.00)
2.40 (0.12)
-7.74 (0.01)
5.58 (0.23)
3.54 (0.62)
9.00 (0.00)
6.80 (0.08)
7.76 (0.02)
4.46 (0.22)
2.96 (0.71)
4.74 (0.32)
9.92 (0.01)
1.28 (0.53)
4.34 (0.23)
17.28 (0.00)
5.46 (0.14)
--
0.14 (0.93)
1.06 (0.59)
46.54 (0.00)
3.52 (0.32)
14.58 (0.00)
47
Notes on Table 4.3
1. The table presents likelihood ratio (LR) test statistics and their associated p-values
in parentheses.
2. For each item reported in column 1, row one in column 2 and subsequent columns
for that item presents statistics for Canada, row two for France, row three for
Japan, row four for the UK, and row five for the USA.
3. LR (one regime) is a test for linear conditional mean dynamics. The null
hypothesis is µ1 = µ 2 , γ = 1 , and the corresponding autoregressive coefficients in
the two regimes are equal.
4. See notes 3-5 in Table 4.1.
5. The symbol “--“ denotes missing numbers because the numerical algorithm for
maximization of the log likelihood function failed to converge.
48
8
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52
Figure 1 Plot of Logrithmic Real GDP over Time
Canada
France
5
5
4.5
4.5
4
4
3.5
3
1950
1975
2000
2025
3.5
1970
1980
1990
Japan
2000
2010
UK
14
5
13
4.5
12
11
4
10
9
1950
1975
2000
2025
2000
2025
USA
9.5
9
8.5
8
7.5
1950
1975
3.5
1950
1975
2000
2025
Figure 2
Plot of Real GDP Growth Rates Over Time
Canada
France
0.04
0.02
0.01
0.02
0
0
-0.01
-0.02
1950
1960
1970
1980
1990
2000
2010
-0.02
1970
1975
1980
Japan
1985
1990
1995
2000
2005
UK
0.15
0.06
0.04
0.1
0.02
0.05
0
0
-0.05
1950
-0.02
1960
1970
1980
1990
2000
2010
1990
2000
2010
-0.04
1950
1960
1970
1980
1990
2000
2010
USA
0.04
0.02
0
-0.02
-0.04
1950
1960
1970
1980
54