Modeling cyclical asymmetries in GDP: international evidence.

Introduction

There is strong support for the business cycle asymmetry hypothesis. According to this view, business cycles consist of a sequence of phases, or regimes, and macroeconomic variables display distinct characteristics within each phase. A variety of theoretical explanations

Several testing procedures have been proposed to search for asymmetries in the data. Some authors advocate for using nonparametric tests that do not assume any specific model for the series under the alternative. Within this group, one may find tests for the equality of transition probabilities in a Markov chain [Neftqi, 1984], tests for the significance of indicator variables in multiple regressions and in seemingly unrelated regression models [Kontolemis, 1997], tests against unspecified nonlinearities like the BDS test [Brock et al., 1991], etc.

Another approach is to set up tests against a particular nonlinear time series model. If the stochastic properties of the variable are phase-dependent, then the true data generating process is nonlinear, and different types of phase dependence lead to different families of nonlinear models. A number of candidates have been proposed in the literature, see for instance the review articles by Mittnik and Niu [1994] and Potter [1999]. Carrying out the test of symmetric behavior against a well-defined alternative helps to pose the analysis of asymmetries in a structured framework when the null is rejected.

Cyclical asymmetries may arise because of asymmetric disturbances, an asymmetric propagation mechanism, or both. There is not a well-established procedure to determine the actual source of asymmetry, but most empirical work that has addressed on the issue opts for assuming symmetric noise and asymmetric transmission. As a consequence, the best way to characterize output asymmetry is to model how the impulse response function varies over different stages of the cycle. For doing so, an explicit representation of the data generating process is needed, and the modeler has to choose among several types of nonlinear models.

The choice is not straightforward. Although there is a great deal of empirical evidence for cyclical asymmetries, there is not a general pattern that fits all situations. On the contrary, asymmetries seem to be both variable-specific and country-specific. Centering on GDP, first one has to determine how many phases are relevant. The classical division into two phases--expansions and recessions--is too simple, and the recent literature advocates for considering three [Sichel, 1994], four [Emery and Koenig, 1992] and even six [Kontolemis, 1997] phases. Secondly, the number of relevant phases depends heavily on whether classical (level) cycles or growth cycles are considered [Zarnowitz, 1992; Kontolemis, 1997]. In the third place, the major economies have not evolved in the same way. Until recently, business cycles were shorter and more numerous in the United States than in Europe or Japan [Zarnowitz, 1992], and the relevant states of the cycle can not be assumed to be the same anywhere.

This paper aims to model and explain asymmetric behaviour in GDP growth in the USA, Europe, and Japan. Germany and France were chosen to represent the European economies, both because of their size in absolute terms and their role in promoting the European Union. The actual relevance of cyclical asymmetries is set forth by showing how GDP reacts to an exogenous shock, and to what extent the reaction depends on the state of the cycle at the time the shock occurs [Potter, 1994]. By comparing the responses at different countries the common facts are identified and separated from features that are country-specific.

It will be assumed that GDP growth is generated by a smooth transition autoregression (STAR) model [Granger and Terasvirta, 1993]. STARs are preferred to other popular alternatives, like the Markov switching model [Hamilton, 1989] or self-excited threshold autoregressions (SETAR) [Tong, 1990], for several reasons: they allow for a continuum of intermediate regimes; they nest the SETAR model as a special case; a modeling cycle, consisting of specification, estimation and diagnosis, exists; standard nonlinear inference techniques can be used; the models are locally linear and easy to interpret; etc. Besides, they display good performance in capturing cyclical behavior in macroeconomic variables like industrial production, unemployment, consumption, imports, etc, see for instance Terasvirta and Anderson [1992], Skalin and Terasvirta [1999; 2002], Ocal and Osborn [2000] and Cancelo and Mourelle [2005].

STARs are usually interpreted as consisting of two extreme regimes and a continuum of intermediate situations. On the one side, this is an advantage over competing representations of the data, as it does not pre-impose a discrete, fixed number of states. But it is also a shortcoming for our purposes, since there is not a clear-cut rule for defining qualitative cyclical states unless the transition is very rapid. For that reason in this paper, a procedure is proposed for grouping intermediate states that are equivalent in terms of their economic implications for GDP growth, in order to interpret the estimated model in terms of a small number of cyclical phases.

The paper is organized as follows. The following section sketches the basics of smooth transition autoregressions. Next, the estimated models for the rate of growth of GDP are reported. Then, the underlying qualitative regimes are identified, and generalized impulse response functions are computed for each regime. The results are examined to determine whether the reaction is phase-dependent and to what extent there is a common pattern across countries. The final section concludes the paper.

The Model

A smooth transition autoregression assumes that the dependent variable [y.sub.t] is a linear function of its own past plus a random disturbance. The parameters of the autoregression depend on the state of the economy, which, in the strict univariate framework, is approximated by some lag [y.sub.t-d]. Different values of [y.sub.t-d] entail different underlying states, the response of [y.sub.t] to a given shock adapting to the specific state that prevails at each moment. Granger and Terasvirta [1993], Terasvirta [1994; 1998], and van Dijk et al. [2002] describe STARs with full particulars.

Let {[y.sub.t]} a stationary, ergodic process. The STAR model of order p and transition lag d is defined as:

[y.sub.t] = [[pi].sub.0] + [p.summation over (i=1)] [[pi].sub.i][y.sub.t-i] + F [y.sub.t-d] [[[theta].sub.0] + [p.summation over (i=1)] [[theta].sub.i][y.sub.t-i]] + [u.sub.t] (1)

where F(.) is a transition function that satisfies 0 [greater than or equal to] F [greater than or equal to] 1, and [u.sub.t] is a martingale difference sequence with respect to the history of the time series up to t - 1. For [y.sub.t-d] = [y.sup.*] then F([y.sup.*]) = [F.sup.*], and equation (1) collapses into the linear autoregression:

[y.sub.t] = [[pi].sub.0] + [F.sup.*] [[theta].sub.0] + [p.summation over (i=1)] ([[pi].sub.i] + [F.sup.*] [[theta].sub.i]) [y.sub.t-1] + [u.sub.t] (2)

A natural way of interpreting the STAR model is to think about two extreme regimes, defined by F = 0 and F = 1, and a set of intermediate situations that characterize the transition. The meaning of the extreme regimes depends on the transition function, and especially on whether it is odd or even. The odd case is usually represented by the logistic function:

F([y.sub.t-d]) = 1/1 + exp[- [gamma]([y.sub.t-d] - c)], [gamma] > 0 (3)

and equations (1)-(3) define the logistic STAR model or LSTAR. In equation (3) F(- [infinity]) = 0 and F(+ [infinity]) = 1, so the extreme regimes are defined for very high and very low values of [y.sub.t-d]; c is a location parameter such that F(c) = 0.5; [gamma] is a slope parameter that determines how rapid the transition is.

LSTAR models are expected to offer a suitable framework for analyzing nonlinear dynamics related to business cycles. The economy displays two plain states, one 'bad' (F = 0) and one 'good' (F = 1). The precise meaning of 'bad' and 'good' depends on the parameters c and [gamma] and can be adapted to each application. For example, one may find that in country A F [approximately equal to] 0 for GDP growing below 1.5 percent in annual terms, while in country B, F is close to 0 for negative rates of growth. In the first case the economy is in the 'bad' state either when it is in recession or in a slowdown, while in the second the 'bad' state is limited to strict recessions.

The exponential function is commonly considered in the case of an even transition:

F([y.sub.t-d]) = 1 - exp[- [gamma][([y.sub.t-d] - c).sup.2]], [gamma] > 0 (4)

and the pair of equations (1) and (4) define the exponential STAR model or ESTAR. In the exponential formulation the transition function is U-shaped, and the extreme regimes are defined for F(c) = 0 and F(-[infinity]) = F(+[infinity]) = 1.

At a first sight, ESTARs does not seem to be adequate for modeling cyclical asymmetries, since the two extreme regimes do not correspond to the classical states of expansion and recession. In empirical work, however, it is often found that ESTARs are preferred to LSTARs. The two formulations will be very similar in fitting the data when the location parameter c of an ESTAR model is very low, as almost all the observations will lie to the right of c. Ocal and Osborn [2000, p. 29] claim that, in this case, the basic difference between the two specifications is how the autoregressive coefficients [[pi].sub.i] + F[[theta].sub.i] change. In LSTARs, the coefficients vary at the same rate, both when F is close to zero and when it is close to one. In ESTARs, the change is sharper when [y.sub.t-d] begins to move from c to the right (i.e., when the economy is getting out of the recession) than when it is moving towards c from a much higher value.

Another explanation for the empirical success of ESTARs is the need for considering three phases: recession, low growth or slowdown, and expansion. Sometimes slowdown periods are seen as situations where the uncertainty is high, while recessions and expansions are more stable states. In this case, it may be that the response to an exogenous shock depends on the degree of uncertainty, and the exponential function will perform better than the logistic transition.

Estimated Models

Quarterly, seasonally adjusted GDP series are considered for the United States, Germany, France, and Japan [Organization for Economic Cooperation and Development]. The sample goes from 1970:Q1 to 2002:Q2. German data have been adjusted because of the unification. Two outliers in Japan, dated at 1974:Q1 and 1997:Q2, were smoothed. Unit root tests indicate that all the series are I (1), so from now on the variables are the first difference of the logarithms.

It is commonplace to begin by computing linearity tests. It should be noted, however, that the authors do not consider the conclusions of such tests to be a relevant tool for guiding the modeling process, see also Potter [1999, p. 14] and Skalin and Terasvirta [2002, pp. 212-3]. On the one side, the test is usually computed conditional on the lag length p of the best linear autoregression. But if the process is nonlinear, then the linear approximation may underestimate the true order of the dynamics, and linearity tests are misspecified. Secondly, to carry out the tests, one has to assume that the transition lag is known or to compute a general test for unspecified d. In the former case, the usual practice is to calculate the test separately for several values of d, so the modeler does not control the overall significance level. In the latter case, the test displays low power, as there are many parameters that are non-significant even if the series is actually generated by a STAR model.

The tests reported in Table 1 have been computed for several values of the lag order p. The transition lag was assumed unknown, and linearity is tested against unspecified d in the interval [1, max(p,6)]. For the details of the auxiliary regressions and the test statistics, see Granger and Terasvirta [1993, pp. 70-5]. The empirical evidence supporting nonlinearity is strong for the USA, weak for France and Japan, and linearity is never rejected for Germany at the 5 percent significance level.

The next step was to specify and estimate STARs for the four series. Model building was based on an extensive search. A large number of potential models were specified by considering several combinations of p, d, and F. The lag order p varies from 1 to 8, as eighth-order dynamics seems to be general enough for quarterly, seasonally adjusted data. The transition lag goes from 1 to max(p, 6), and the transition function was allowed to be either logistic or exponential. All 66 specifications were estimated by nonlinear least squares, and the best models were selected for further refinement. Cross-parameter restrictions were evaluated, and non-significant coefficients were dropped to conserve degrees of freedom. Standard F-tests and AIC were used to check that the restrictions embedded in the final model were supported by the data. Several misspecification tests were computed to validate the final specifications, see below for further description of the diagnostic tests. All the computations were done in RATS. As a whole, the model-building methodology entails a lot of data mining, but any possible inadequacy of the model is expected to be detected at the evaluation stage.

The final selected models are presented in Table 2, together with some descriptive statistics and diagnostic tests. In regard to the latter, LJB is the Lomnicki-Jarque-Bera test of normality; ARCH denotes the statistic of no autoregressive conditional heteroskedasticity with four lags; BCH is Ocal and Osborn's [2000] test of business cycle heteroskedasticity computed by regressing the squared residuals on the values of the transition function. Three tests specially derived for smooth transition models in Eitrheim and Terasvirta [1996] are also displayed. AUTO tests serial independence against a process of order 12. NL is a test of no remaining nonlinearity in the residuals. The test is computed for several values of the transition lag under the alternative, and Table 2 reports the value minimizing the p-value of the tests; the p-value in parenthesis is computed from the standard F distribution and understates the actual value that would be obtained by considering the true, unknown distribution of the ordered statistic. PC is a general test of parameter constancy that allows for monotonically and nonmonotonically changing parameters under the alternative.

There are no indications of misspecification in the models, so one may conclude that the proposed STARs are adequate. According to the variance ratio, the nonlinear model explains 25 percent of the residual variance of the best linear autoregression in the USA, 19 percent in Japan, 13 percent in Germany, and 11 percent in France. It should be noted that there is no formal procedure to determine whether the models reported in Table 2 actually outperform linear autoregressions, as the test statistics do not consider the degrees of freedom that are lost in searching for d and p. Normality is rejected in two countries because a single outlier distorts the sample kurtosis coefficient (USA, 1978:Q2; France, 1974:Q4).

Measuring the Dependence of the Responses to Shocks Over Different Cyclical Regimes

The previous section showed that the dynamics of GDP growth display nonlinear components that can be explained within the smooth transition framework. Now such nonlinearities are interpreted in terms of cyclical asymmetries. The economics of STAR models are easy to derive when the transition function is logistic and [gamma] [right arrow] [infinity], as the model becomes a two-state, self-excited threshold autoregression. But the interpretation is not so simple when the transition between the extreme regimes is slow and most observations lie halfway. By construction, a STAR model considers an infinite number of underlying states, one for each value of [y.sub.t-d], and the literature does not provide a clear-cut rule for grouping the values of the transition lag that are equivalent in terms of their economic implications on the behavior of [y.sub.t]. Such grouping should define a small number of qualitative cyclical regimes that are estimated from the data, instead of being determined on a priori grounds, and is essential for reinterpreting the nonlinearities captured by the model in terms of cyclical asymmetries.

The proposal of this paper relies on the local dynamics that arise for each [y.sub.t-d]. It was seen in equation (2) that for [y.sub.t-d] = [y.sup.*] the model becomes a linear autoregression with constant parameters. Small changes in [y.sub.t-d] will exert a minor influence on the roots of its characteristic polynomial and, hence, on the underlying dynamics of the model. In the medium term, the dynamics are ruled by the dominant root, i.e., the root with the highest modulus, and the effects of changes in [y.sub.t-d] on the dynamics can be summarized by looking at how the dominant root evolves. The values of the transition lag that render the same dominant features of the dynamics are put together, and the groups that are formed define the qualitative regimes that will be considered in analyzing the cyclical properties of the model.

Figure 1 depicts the estimated transition functions of the models reported in Table 2. Each dot represents (at least) one observation in the sample. The figure also displays the modulus of the dominant root for each [y.sub.t-d] in the sample. Although the models are globally stationary they may be locally unstable, unit and explosive roots arising for specific values of the transition lag.

In the United States, the transition is logistic and quite rapid, and the delay is four quarters. According to our interpretation, two qualitative regimes arise from the model, expansion ([y.sub.t-4] > 0.0036 or 0.36 percent in percentage terms; all the figures are quarterly rates of growth) and slowdown/recession ([y.sub.t-4] < 0.0036). The model is stable, as long as GDP growth remains above that bound. An exogenous shock may launch a period of slowdown (low, but still positive, growth) or recession (negative growth), with no remarkable changes in the model in the short run because of the recognition lag. But the economy will not remain indefinitely in this state: once [y.sub.t-4] is below 0.0036 the model becomes locally unstable and is dominated by an explosive cycle that sends GDP back to recovery and expansion.

In Germany, the transition function is exponential, the transition lag is two quarters and the model defines three states of the cycle: severe recession ([y.sub.t-2] < -0.01), recession (-0.01 < y.sub.t-2] < -0.0015) and slowdown/expansion ([y.sub.t-2] > -0.0015). There are only six observations in the first regime, so the proposition that the German economy behaves in the same way for positive rates of growth and in severe recessions should be considered a highly tentative approximation to the actual dynamics within each regime. The German economy is stable both for positive rates of growth and in extreme recessions, but unstable in recessions. Since the results for the severe recession regime should be taken with care because of the lack of data, one may think that if an exogenous shock launches a recession the internal forces of the economy are expected to start the recovery.

In the French model, the delay is two quarters, and the transition is exponential, although in practice it behaves as if it were logistic as there is only one observation at the left tail of the curve. The model is always stable, and the dominant root remains rather constant for every value of the transition lag. Given that the STAR model does not provide a clear indication of the underlying cyclical phases, in what follows the two classical regimes will be considered, one state of positive growth (expansion) and one of actual declines in the output (recession).

In Japan, the selected specification is a typical ESTAR model, as there is a significant number of data points at the left tall of the exponential function. The transition lag is four quarters and there are three regimes: recession ([y.sub.t-4] < 0.0010), slowdown (0.0010 < [y.sub.t-4] < 0.0055) and high-growth expansion ([y.sub.t-4] > 0.0055). The model is stable in recessions and in high-growth expansions, so once the economy falls in recession an exogenous shock is needed to launch the recovery. However, it is unstable for moderate rates of growth, and after a short time, a slowdown either translates into an actual decline of the output or proceeds towards a regime of rapid, steady growth.

The final step is to assess how the response of the output to an exogenous shock changes over different cyclical states. Generalized Impulse Response Functions (GIRF) are computed to characterizing the propagation mechanisms implied by the models [Koop et al., 1996; Potter, 2000a]. The effect of a shock [u.sub.t] = [delta] on [y.sub.t+n] conditional on the cyclical regime i is given by

GIRF(n, [delta], [[OMEGA].sup.(i).sub.t-1]) = E([y.sub.t+n] | [u.sub.t] = [delta], [[OMEGA].sup.(i).sub.t-1]) - E([y.sub.t+n] | [[OMEGA].sup.(i).sub.t-1]) (5)

where [[OMEGA].sup.(i).sub.t-1]) is the set of histories that belong to regime i.

To derive the GIRFs for each country, all the histories in the sample were assigned to a cyclical regime. In the USA, for instance, for any t the relevant history consists of the vector [[omega].sub.t-1] = ([y.sub.t-1], ...., [y.sub.t-8]), and it is assigned either to the expansion regime or to the slowdown/recession regime according to whether [y.sub.t - 4] is higher or lower than 0.0036. Four types of initial shocks were considered, [+ or -] s and [+ or -] 2 s, where s stands for the residual standard error of the model. For each combination of history and initial shock ([u.sub.t] = s, -s, 2 s, and -2 s), 1000 replicates of a 21-step prediction sequence are generated; two paths are derived, one with the selected shock in the first step and noise onward, and the other with noise everywhere; random draws of the residuals are used as noise; the same sequence of random disturbances is used in each replication of the four types of initial shocks. For each horizon, the mean over the 1000 replicates of the difference between the response with and without the initial shock is computed, and that sequence of means is the estimated GIRF for that history. The final GIRF for the cyclical regime i is computed as the mean of the GIRFs of all the histories in that regime.

Figures 2 to 4 show the estimated responses. For convenience of representation, the figures plot the effect of the shock on the level of GDP in percentage terms. Figure 2 depicts the response to shocks of magnitude one standard error of the residuals (unit shocks), either positive or negative, to assess how the response changes over regimes and whether it is symmetric. Figures 3 and 4 focus on proportionality. For each regime, the figures represent the ratio between the GIRF for shocks of magnitude two standard errors over the GIRF for unit shocks; Figure 3 is for negative shocks and Figure 4 for positive shocks.

[FIGURES 2-4 OMITTED]

In the U.S., the response to positive unit shocks is the same in all regimes. When a negative unit shock hits the system, its effect depends on the state of the cycle: 20 quarters after the shock, the fall in GDP is higher if the economy was expanding (-1.1 percent), than if it was in the slowdown/recession regime (-0.6 percent). The response to changes in the size of the shock is proportional when the economy is expanding, but GDP reaction is more than proportional in the case of a negative shock that occurs when the economy is in the slowdown/recession regime.

In Germany, there is an asymmetric response to unit shocks. The German economy looks more sensitive in the recession regime, especially for negative shocks. The decline in GDP 20 quarters after the economy was hit by a negative unit shock is -0.9 percent if it was in severe recession, -0.8 percent in slowdown/expansion and -1.1 percent in recession. The response to positive shocks is proportional in every regime but is phase-dependent for negative shocks: although a shock of -2 s has twice the effect of a shock of size -s in the recession regime, the ratio increases to 2.5 in the regime of severe recession and to 3 in the slowdown/expansion state.

The French economy is more sensitive to negative shocks in both regimes, and the effect on GDP is higher when the economy is expanding whatever the sign of the shock. The two extremes are positive shocks in the recession state (GDP increases 0.8 percent) and negative shocks in the expansion state (GDP falls -1.1 percent). The response is rather proportional for negative shocks but seems to be less than proportional for positive shocks in either regime. Therefore, the STAR model detects some dependence of the response on the state of the cycle, despite the fact that regimes were exogenously determined.

In Japan, the propagation mechanism remains rather constant over different states of the cycle. The response to unit shocks is smaller when the economy is in the slowdown regime, but the difference may be due to sampling variability. The proportionality restriction seems to hold in general terms.

Conclusions

The paper investigated whether GDP displays the type of nonlinear behavior generated by cyclical asymmetries. Smooth transition autoregressive models were specified, estimated and evaluated for the rates of growth of GDP in the United States, Germany, France, and Japan. The actual relevance of cyclical asymmetries was set forth by showing how the response of GDP to a given shock changes with the state of the cycle at the time the shock occurs. For doing so, a procedure was proposed for defining a small number of cyclical regimes from the estimated STAR models by grouping the values of the transition lag that render the same dominant features of the local dynamics.

The analysis showed that there is not overwhelming evidence that STAR models outperform linear autoregressions. Nevertheless, they improve the sample fit by introducing nonlinear components that can be interpreted in terms of the business cycle asymmetry hypothesis. In the U.S., such improvement proceeds both in terms of reducing the residual variance and by showing how the propagation mechanism adapts to the cyclical state prevailing at the moment the shock hits the system. In Germany, the decrease in the residual variance is lower than in the United States, but the model provides a clear indication that the response function changes over the cycle. In the two countries, the effects of negative shocks are more heterogeneous than the response to positive shocks.

In France, the reduction in the residual variance is small, and the model fails to provide an empirical characterization of the relevant cyclical regimes. Despite that, the model shows that the response to a shock is not the same for expansions and recessions. In Japan, modeling the nonlinear components in GDP growth leads to a higher decrease in the residual variance than in the European countries, and three underlying cyclical regimes may be identified from the proposed specification. But there are only minor differences in the estimated propagation mechanisms, and the response of GDP to an exogenous shock is quite stable over the cycle.

References

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JOSE RAMON CANCELO, AND ESTEFANIA MOURELLE *

* Universidade da Coruna--Spain. The authors thank an anonymous referee for the comments to a previous version.

TABLE 1
Linearity Tests Against Smooth Transition Autoregressions (P-Values)

                     Country
Lag
Order    United States        Germany

P       LSTAR    ESTAR    LSTAR    ESTAR

4       0.5984   0.1517   0.9616   0.4476
5       0.3282   0.0060   0.5125   0.5676
6       0.1784   0.0057   0.5602   0.5321
7       0.0355   0.0303   0.3281   0.3994
8       0.0169   0.0333   0.1672   0.2278

                     Country
Lag
Order       France            Japan

P       LSTAR    ESTAR    LSTAR    ESTAR

4       0.3939   0.1281   0.8657   0.4375
5       0.0676   0.2558   0.6915   0.1573
6       0.0279   0.6071   0.7248   0.1374
7       0.0532   0.4555   0.7256   0.0154
8       0.1279   0.4298   0.7240   0.0231

TABLE 2
Estimated STAR Models for GDP Growth

UNITED STATES (LSTAR)

[y.sub.t] = 0.0141 + 0.27 [y.sub.t-1] - 0.12 [y.sub.t-3]
           (0.0026) (0,08)             (0.15)

+ 0.56[y.sub.t-4] - 0.25 [y.sub.t-5] - 0.25 [y.sub.t-6]
 (0.24)            (0.16)             (0.16)

- 0.11 [y.sub.t-7] - 0.49 [y.sub.t-8]
 (0.14)             (0.13)

+ (-0.0124 + 0.44 [y.sub.t-3] - 0.64 [y.sub.t-4] + 0.25 [y.sub.t-5]
   (0.0040) (0.21)             (0.27)             (0.16)

+ 0.39 [y.sub.t-6] + 0.11 [y.sub.t-7] + 0.49 [y.sub.t-8])
 (0.20)             (0.14)             (0.13)

x [[1 + exp {-9.73 x 113.41 ([y.sub.t-4] - 0.0052)}].sup.-1]
            (10.44)                       (0.0010)

+ [u.sub.t]

s = 0.0074, [R.sup.2] = 0.35, AIC = -9.7098, [s.sup.2] /
[s.sup.2.sub.L] = 0.75, LJB = 32.18 (1.0 x [10.sup.-7]),
ARCH = 1.12 (0.35), BCH = 0.21 (0.65), AUTO = 1.52 (0.13),
NL = 1.36 (0.15), PC = 0.46 (0.99)

GERMANY (ESTAR)

[y.sub.t] = 0.0237 + 0.50 [y.sub.t-1] + 4.92 [y.sub.t-2]
           (0.0119) (0.33)             (2.33)

+ 0.14 [y.sub.t-3] - 0.06 [y.sub.t-5] + 1.33 [y.sub.t-6]
 (0.08)             (0.04)             (0.44)

+ (-0.0180 - 0.50 [y.sub.t-1] - 4.92 [y.sub.t-2] - 0.06 [y.sub.t-5]
   (0.0120) (0.33)             (2.33)             (0.04)

- 1.45 [y.sub.t-6])
 (0.45)

x [1 - exp{ -7.83 x 10536.83 [([y.sub.t-2] + 0.0060).sup.2]}]
            (3.88)                          (0.0007)

+ [u.sub.t]

s = 0.0090, [R.sup.2] = 0.17, AIC = -9.3453, [s.sup.2]/[s.sup.2.sub.L]
= 0.86, LJB = 1.84 (0.40), ARCH = 0.67 (0.61), BCH = 2.25 (0.14),
AUTO = 1.67 (0.09), NL = 1.09 (0.37), PC = 0.76 (0.83).

FRANCE (ESTAR)

[y.sub.t] = 0.0050 + 0.63 [y.sub.t-2] - 0.17 [y.sub.t-3]
           (0.0017) (0.44)             (0.18)

- 0.49 [y.sub.t-4] + 0.15 [y.sub.t-5]
 (0.22)             (0.09)

+ (-0.0021 + 0.53 [y.sub.t-1] - 0.90 [y.sub.t-2] + 0.55
  (0.0067)  (0.24)             (0.37)             (0.31)

[y.sub.t-3] + 0.49 [y.sub.t-4] - 0.13y [t.sub.t-6])
             (0.22)             (0.12)

x [1 - exp{-0.26 x 30959.52 [([y.sub.t-2] + 0.0041).sup.2]}]
           (0.27)                          (0.0046)

+ [u.sub.t]

s = 0.0048, [R.sup.2] = 0.30, AIC = -10.5839, [s.sup.2]/
[s.sup.2.sub.L] = 0.89, LJB = 48.49 (2.9 x [10.sup.-11]),
ARCH = 0.59 (0.67), BCH = 0.35 (0.56), AUTO = 1.32 (0.22),
NL = 1.38 (0.15), PC = 0.57 (0.97).

JAPAN (ESTAR)

[y.sub.t] = -0.0037 + 0.51 [y.sub.t-1] - 1.18 [y.sub.t-2]
            (0.0086) (0.36)             (0.45)

+ 0.43 [y.sub.t-3] + 1.71 [y.sub.t-4] + 1.15 [y.sub.t-5]
 (0.08)             (2.23)             (0.55)

- 0.51 [y.sub.t-6]
 (0.39)

+ (0.0085 - 0.51 [y.sub.t-1] + 1.61 [y.sub.t-2] - 1.71 [y.sub.t-4]
  (0.0088) (0.36)             (0.45)             (2.23)

- 1.57 [y.sub.t-5] + 0.51 [y.sub.t-6]
 (0.55)             (0.39)

x [1 - exp{-17.13 x 11643.92 [([y.sub.t-4] - 0.0034).sup.2]}]
            (7.26)                          (0.0003)

+ [u.sub.t]

s = 0.0074, [R.sup.2] = 0.42, AIC = -9.7206, [s.sup.2]/[s.sup.2.sub.L]
= 0.81, LJB = 0.67 (0.72), ARCH = 1.13 (0.35), BCH = 3.03 (0.08),
AUTO = 1.02 (0.44), NL = 1.30 (0.20), PC = 1.15 (0.30).

Notes: [y.sub.t] denotes the quarterly rate of growth of GDP.
Values under regression coefficients are standard errors of the
estimates; s is the residual standard error; [R.sup.2] the
determination coefficient; AIC the Akaike Information Criterion;
[s.sup.2]/[s.sup.2.sub.L] is the variance ratio of the residuals
from the nonlinear model and the best linear AR selected with AIC;
LJB is the Lomnicki-Jarque-Bera normality test; ARCH is the
statistic of no ARCH based on four lags; BCH is a business cycle
heteroskedasticity test; AUTO is the test for residual autocorrelation
of order 12; NL is the test for no remaining nonlinearity; PC is
the general parameter constancy test. Numbers in parentheses
after values of LJB, ARCH, BCH, AUTO, NL, and PC are P-values.