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Cyclical asymmetries in unemployment rates: international evidence.

Abstract This paper investigates to what extent the observed nonlinearities in the unemployment rates of six major developed economies are the response to cyclical asymmetries. Two classes of models are compared: strict smooth transition autoregressions and models where the transition variable is GDP growth, which is considered a more direct indicator of the business cycle. The empirical evidence points out that nonlinearities in unemployment rates are induced by cyclical asymmetries. It is also found that in most countries the unemployment rate looks stationary and reverts to a long-run equilibrium rate in periods of normal growth, while in extreme cyclical situations it tends to become nonstationary as if each extreme cyclical episode had its own path of equilibrium.

Keywords Nonlinearity * STAR models * Business cycle * Unemployment * Unit roots

JEL Classification E32 * E24 * C52

Introduction

Nonlinear behavior in unemployment has been extensively documented in the literature. To mention just a few examples that include a variety of approaches, Neftci (1984) uses Markov chains to model asymmetric behavior. Montgomery et al. (1998) and Rothman (1998) estimate different classes of nonlinear time series models and compare their fit and forecasting performance. Krolzig et al. (2002) analyze the labor market in the United Kingdom with a regime-switching vector error correction model. Skalin and Terasvirta (2002) use smooth transition autoregressions to explain cyclical asymmetries and moving equilibria in unemployment rates for several OECD countries. Chauvet et al. (2002) set up a dynamic factor model with regime switching to extract the cyclical component of the unemployment rate in the USA.

This paper centers on the source of nonlinearities. It investigates to what extent nonlinear behavior is the response to cyclical asymmetries, or should be interpreted as being caused by idiosyncratic factors specific to the labor market. For doing so, it is assumed that nonlinearities can be captured by a smooth transition autoregression (STAR) model. The STAR model is a special case of an autoregressive process where the parameters depend on a transition variable [s.sub.t]. Given the value of [s.sub.t] the model simplifies into a linear autoregression, so nonlinearities arise because of the changes in [s.sub.t]. Each value of [s.sub.t] defines a regime, which is characterized by the properties of the linear autoregression that stems from it.

In strict univariate STARs the transition variable is a lag of the dependent variable and regimes are endogenously determined. If nonlinear behavior is generated by cyclical asymmetry, however, one would expect to achieve a better model by allowing the changes in the parameters to depend on a direct indicator of the business cycle. This extension leads to a smooth transition autoregression with exogenous transition, see for instance Cancelo and Mourelle (2005a). Such a parameterization can be seen as a special case of the general smooth transition regression model, and seems to be a natural extension of univariate STARs to explain nonlinearities induced by the business cycle. It should be noted that the models with exogenous transition are no longer an univariate representation of the Data Generating Process (DGP) of the dependent variable.

The relative performance of different alternatives for the transition variable in explaining the data provides an indication on whether nonlinearities are generated by cyclical asymmetries. In the affirmative, one would expect that the model with exogenous transition displays the best performance. On the contrary, if nonlinearities are due to idiosyncratic components specific to the labor market, then one should find that standard STARs are better in terms of their adequacy to the data.

The paper is organized as follows. The following section reviews the foundations of STAR models. Next, the modeling strategy is summarized and the final models are reported. Then, the estimated nonlinearities as captured by the models are interpreted in terms of cyclical asymmetries. The final section concludes.

Smooth Transition Autoregressions

In a Smooth Transition Autoregression (STAR) all predetermined variables are lags of the dependent variable and the autoregressive coefficients change with the transition variable [s.sub.t]. There are two extreme regimes, and the degree of smoothness of the transition from one extreme regime to the other is estimated from the data. Granger and Terasvirta (1993), Terasvirta (1998) and van Dijk et al. (2002) describe smooth transition autoregressions in detail.

The model of order p for a stationary and ergodic process [u.sub.t] is defined as

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

where F([s.sub.t]) is a transition function that satisfies 0[less than or equal to]F[less than or equal to]1. This paper centers on the so-called Logistic STAR or LSTAR, where F([s.sub.t]) is the logistic function

F([s.sub.t]) = 1/[1 + exp[-[gamma]([s.sub.t] - c)]], [gamma] > 0 (2)

In Eq. 2 F(-[infinity])=0 and F(+[infinity]) = 1, so the extreme regimes are defined for very high and very low values of [s.sub.t]; c is a location parameter such that F(c)=0.5; and [gamma] is a slope parameter that determines how rapid the transition between extreme regimes is.

LSTAR models are expected to offer a suitable framework for capturing the effects of cyclical asymmetries, in the sense that the extreme "good" and "bad" states correspond to very high and very low values of the transition variable. Other transition functions, notably the exponential function, have been considered in the literature of cyclical asymmetries, see for instance Terasvirta and Anderson (1992) or Cancelo and Mourelle (2005b). This paper focuses on an even transition because it has a more direct interpretation.

When [s.sub.t] is some transformation of the dependent variable, like [u.sub.t-d] or [DELTA][u.sub.t-d], Eq. 1 becomes the standard Smooth Transition Autoregression (STAR) with endogenous transition, where regimes are endogenously generated by the recent history of the time series itself

[u.sub.t] = [[pi].sub.0] + [p.summation over (i=l)][[pi].sub.i][u.sub.t-i] + F([DELTA][u.sub.t-d])[[[theta].sub.0] + [p.summation over (i=l)][[theta].sub.i][u.sub.t-i]] + [e.sub.t] (3)

The other possibility is to assume that the autoregressive parameters depend on some exogenous variable [x.sub.t-d]. In this case the resulting model is midway between strict univariate STARs and general Smooth Transition Regressions, as it combines local univariate dynamic dependence and exogenous regime determination

[u.sub.t] = [[pi].sub.0] + [p.summation over (i=l)][[pi].sub.i][u.sub.t-i] + F([x.sub.t-d])[[[theta].sub.0] + [p.summation over (i=1)][[theta].sub.i][u.sub.t-i]] + [e.sub.t] (4)

In this paper [x.sub.t-d]=[DELTA]ln[GDP.sub.t-d], as GDP growth is a natural indicator for tracking the cycle. The actual interpretation of the extreme regimes given by F=0 and F=1 varies with the transition variable: when it is GDP growth the bad state corresponds to low values of [s.sub.t], while if it is unemployment growth the bad state is related to high values of [s.sub.t].

There are two major differences between Eqs. 3 and 4 for the purposes of this paper. First, there is an asymmetric propagation mechanism, in the sense that the response to a given shock varies as the coefficients of the impulse response function depend on the transition variable. But while a function that changes with some lagged transformation of unemployment would indicate that the propagation mechanism is driven by idiosyncratic forces of the labor market, in the model with exogenous transition the reaction of the unemployment rate to an exogenous disturbance will depend on the state of the cycle at the time the shock occurs.

The second difference refers to the interpretation of local dynamics and especially of local nonstationarity. Valid STAR models with endogenous transition are globally stable, and [u.sub.t] reverts to the same long-run equilibrium value in all situations. Such reversion is more or less rapid depending on the local dynamics. In particular, it is quite rapid when there is a root on or outside the unit circle, as [u.sub.t] passes the region of nonstationarity quickly on its way to values closer to the equilibrium.

When the transition is exogenous, the model explains the behavior of [u.sub.t] conditional on each value of [x.sub.t-d]. For most values of [x.sub.t-d] the model will be stable; in this case Eq. 4 gives the long-run equilibrium value of [u.sub.t] that corresponds to each [x.sub.t-d], and explains how the observed unemployment reverts to the equilibrium if an exogenous shock [e.sub.t] occurs and GDP growth remains constant. But it may happen that for some values of [x.sub.t-d] the roots of Eq. 4 are on or outside the unit circle, so that the absence of mean reversion indicates that there is not a unique long-run equilibrium rate of unemployment for those values of GDP growth. If there is a unit root, it would be possible to observe two different historical episodes with GDP growth constant at the same figure for a long period, and unemployment converging to a different value in each episode. As a consequence, if Eq. 4 is not locally stationary everywhere the long-run relationship between GDP growth and the unemployment rate is not uniquely defined for every value of the rate of growth of GDP.

Empirical Analysis

Data

Quarterly, seasonally adjusted data for the unemployment rates and GDP are considered for Canada, France, Germany, Italy, Japan and the United Kingdom (Organization for Economic Cooperation and Development). The sample goes from 1970:Q1 to 2004:Q4 in all countries but Italy, where it begins in 1974:Q1. German data have been adjusted because of the unification. The Italian data were also adjusted because of a break in the series.

There is some controversy about whether the unemployment rate follows a stationary process. It is known that standard unit root tests are not adequate when the underlying process is nonlinear. Although there are some results for specific nonlinear processes (Kapetanios et al. 2003) to the author's knowledge there is no formal procedure to test for unit roots in general STAR models with unknown p and d. Moreover, there seems to exist some differences across countries, and unemployment rates are more likely to be stationary in some countries than in others. As a consequence, the following specification of the STAR model will be considered

[DELTA][u.sub.t] = [[pi].sub.0] + [alpha][u.sub.t-1] + [p.summation over (i=1)][[pi].sub.i][DELTA][u.sub.t-i] + F([s.sub.t])[[[theta].sub.0] + [beta][u.sub.t-l] + [p.summation over (i=1)][[theta].sub.i][DELTA][u.sub.t-i]] + [e.sub.t] (5)

This expression is well behaved both when [u.sub.t] is I(0) and when it is I(1). The distributions of [t.sub.[^.[alpha]]] and [t.sub.[^.[beta]]] are unknown when [u.sub.t] is I(1), so the t-ratios can not be properly used to carry out formal tests on whether [u.sub.t] displays a unit root.

Testing Nonlinearity

It is usual to begin by testing that the data display the type of nonlinear behavior induced by smooth transitions. First, a linear AR is specified by selecting p with Akaike's Information Criterion (AIC) and checking that there is no autocorrelation remaining in the residuals. Note that the lag order of the process is p+1, see Eq. 5, unless [alpha]=[beta]=0 so [u.sub.t] is I(1). Next, linearity is tested against smooth transition nonlinearity. To circumvent the Davies' problem that arises because some parameters of the nonlinear model are not identified under the null hypothesis, a first-order Taylor approximation of the logistic function is considered. An auxiliary regression based on this approximation is estimated, and the resulting LM test has the standard distribution under the null of linearity.

Two strategies have been considered to deal with the fact that the transition lag d is unknown. The first consists of assuming that d is a free, unknown parameter that lies in the interval [[d.sub.min], [d.sub.max]], and setting up the test to consider all the possible values at the same time (Granger and Terasvirta 1993). In this way the size of the test is known, although its power may be low, as the auxiliary regression includes many terms that are non-significant even under the alternative. This procedure will be referred to as the unconditional approach, and the auxiliary regression is given by

[e.sub.LIN,t] = [[delta].sub.0] + [phi][u.sub.t-1] + [p.summation over (i=1)][[delta].sub.i][DELTA][u.sub.t-i] + [d max.summation over (j=d min)][[lambda].sub.j][s.sub.t-j] + [d max.summation over (j=d min)][[tau].sub.j][u.sub.t-1][s.sub.t-j] + [p.summation over (i=1)][d max summation over (j=d min)][[kappa].sub.ij][DELTA][u.sub.t-i][s.sub.t-j] + [v.sub.t] (6)

where [e.sub.LIN,t] stands for the residuals of the linear model, and the null hypothesis is [[lambda].sub.j] = [T.sub.j] = [[kappa].sub.ij] = 0 for all i=1,..., p and j=[d.sub.min],..., [d.sub.max].

A second approach is to compute a sequence of auxiliary regressions conditional on each value of d and to select the value minimizing the p-value of the linearity test, so Eq. 6 is simplified to

[e.sub.LIN,t] = [[delta].sub.0] + [phi][u.sub.t-l] + [p.summation over (i=1)][[delta].sub.i][DELTA][u.sub.t-i] + [lambda][s.sub.t-j] + [tau][u.sub.t-j][s.sub.t-j]+[p.summation over (i=1)][[kappa].sub.i][DELTA][u.sub.t-i][s.sub.t-j] + [v.sub.t,j] = [d.sub.min],..., [d.sub.max] (7)

and the null in each regression is [lambda] = [tau] = [[kappa].sub.1] = ... = [[kappa].sub.p] = 0. If the smallest p-value is below the critical level, then linearity is rejected, and the related d is assumed to be the true, unknown transition lag (Terasvirta 1994). The major shortcoming of this procedure is that it overemphasizes the need for considering nonlinear models, as the actual size of the test will be higher than the nominal size if the standard F distribution is used to compute the critical values. In empirical applications, linearity testing is usually based on the conditional approach when the number of possible values of d is large and the sample size is moderate (Terasvirta and Anderson 1992).

It should be noted that when the transition variable is an exogenous variable, then expressions (6) and (7) have power to detect linear dependence of [u.sub.t] on [s.sub.t]. Thus, rejecting the null in Eqs. 6-7 does not entail that the true DGP is a nonlinear model with exogenous transition, as the rejection may be due to omitted explanatory variables in a linear specification.

Table 1 reports the results of the linearity tests. Both the unconditional and conditional approaches are shown. The value of p comes from the linear model and was selected with AIC after searching in the interval [0, 8]. In Germany and Japan, two minima were found, so the tests were computed for both values of p. The transition lag ranged from 1 to 4. On the whole, linearity is not rejected against models with endogenous transition: at the 5% significance level, two rejections are found, in Canada and Japan. On the other side, it is widely rejected against models with exogenous determination of regimes.

Estimated Models

The next step was to set up nonlinear models following the guidelines reported in Table 1. All the values of the transition lag d for which the null of linearity was rejected were considered, although the value rendering the minimum p-value was given more attention. The equations were estimated by nonlinear least squares. Following standard practice (Terasvirta 1994), the slope parameter [gamma] was standardized by dividing it by the standard deviation of the transition variable. Cross-parameter restrictions were evaluated and non-significant coefficients were dropped. Standard F-tests and AIC were used to check that the restrictions embedded in the final model were supported by the data. Inference on [alpha], [beta] and the intercepts [[pi].sub.0] and [[theta].sub.0] was rather ad hoc, as there are no formal results for carrying out hypothesis testing on these parameters when it is not known whether [u.sub.t] is I(1) or I(0). As the cost of dropping a significant parameter is much higher that the inconveniences of retaining a nonsignificant coefficient, it was decided to maintain these parameters in the model unless their point estimates and t-ratios were close to 0. All the computations were done in rats.

According to the results of the linearity tests, the actual source of the observed nonlinearities in unemployment rates is uncertain in Canada and Japan, as the null is rejected both for endogenous and exogenous transitions. For Japan it was not possible to achieve a valid STAR with endogenous transition, so no univariate STAR provides an appropiate alternative to explain nonlinear behavior. In Canada, both a strict STAR and a model with exogenous transition were successfully estimated, and the latter was much better in fitting the data: taking AIC (R-square) as a measure of fit, the results are -2.27971 (0.39) for the univariate model and -2.50521 (0.51) for the model with exogenous transition (the two models have the same number of parameters).

Valid models with exogenous transition were achieved for every country, so one may conclude that the observed nonlinearities of the unemployment rate are directly related to cyclical asymmetries. Table 2 shows the final models, 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; and BCH is Ocal and Osborn's (2000) test of business cycle heteroskedasticity.

Besides, some tests especially derived for smooth transition models in Eitrheim and Terasvirta (1996) are also displayed. AUTO is a test of serial independence against an eighth-order process. NLxx is a test of neglected nonlinearity to detect whether the proposed specification provides an adequate description of the nonlinearity in the data. The test is computed separately for transition lags varying from 1 to 4 under the alternative, and Table 2 reports the value minimizing the p-value of the tests. The p-value in parenthesis comes from the standard F distribution and understates the actual value that would be obtained by considering the true, unknown distribution of the ordered statistic. Two variants have been considered: NLEN is the test against endogenous transition (i.e., remaining nonlinearity driven by some lag of the unemployment rate), while NLEX tests against neglected nonlinearity ruled by lagged GDP growth.

Interpreting Nonlinearities in Terms of Cyclical Asymmetries

Table 3 summarizes local dynamics by displaying the roots of the characteristic polynomial at the extreme values of the transition function, as well as the roots of the linear AR. To save space only the roots with modulus greater than 0.8 are shown. It should be noted that such roots are point estimates of the true, unknown roots that were computed from the point estimates of the autoregressive parameters. It would be interesting to test whether the population roots are actually inside, on or outside the unit circle, but to the author's knowledge there is no formal procedure for carrying out such tests. For convenience of presentation in the comments that follow, the upper extreme regime is defined for F in the interval [0.9,1], while the lower extreme regime is for F[member of][0,0.1].

In Canada the transition lag is one and the dynamics of the unemployment rate react immediately to the changes in the cyclical conditions of the economy. There is a major improvement in fitting the data with respect to the linear AR, as the nonlinear model explains 25.8% of the residual variance of the former.

The unemployment rate looks stationary everywhere, and nonlinear behavior arises as a response to strong recessions in Canadian GDP. In the upper extreme regime (quarterly GDP growth higher than -0.3%), the long-run rate of unemployment is 7.4%, while in severe recessions ([DELTA]ln[GDP.sub.t-1]<-0.8%) it increases to 10.4%.

In France, the transition lag is one. The nonlinear model explains 12% of the residual variance of the linear AR and behaves like a threshold autoregression (TAR). The threshold is 0.3% (1.2% in annual terms), so the two cyclical regimes derived from the model may be referred to as expansion and slowdown/recession. It should be noted, however, that GDP growth is rather volatile and unemployment changes from one regime to the other quite frequently.

When the economy is expanding the unemployment rate looks stationary and its long-run expected value is 7.4%. In slowdowns/recessions, the unemployment rate displays a unit root, and there is no reversion to the mean.

In Germany, the unemployment rate depends on the fourth lag of GDP growth, so the reaction to changes in the cyclical conditions is slower than in other countries. The ratio of variances is 0.843 and the nonlinear model explains 15.7% of the residual variance of the linear AR. The transition between the extreme regimes is smooth and there is a continuum of intermediate situations. The upper extreme regime is defined for [DELTA]ln[GDP.sub.t-4]>0.6%, or 2.4% in annual terms, and the lower extreme regime for [DELTA]ln[GDP.sub.t-4]<-0.4%(-1.6%).

When F is close to 1, the unemployment rate seems to revert to its long-run mean, 8.9%. On the other side, for F=0 the unemployment rate is nonstationary; in fact, it could be generated by a I(2) process or even be locally explosive. It should be noticed that F=0 is an extreme situation that is seldom observed, and the dominant root is below the unity for values of F as low as 0.2 or [DELTA]ln[GDP.sub.t-4] = -0.25% (-1%).

In Italy, the parameters of the model are ruled by GDP growth lagged three quarters. The explanatory power of the nonlinear model is not much higher than that of the linear model, as the former explains 9.1% of the residual variance of the latter. The model separates the phase of very high growth of GDP from the rest of the phases of the cycle.

When quarterly GDP increases below 1.1% (4.5% in annual terms), the unemployment rate follows a stationary process with long-run mean 9.6%. As GDP growth proceeds beyond that bound, the unemployment rate comes closer to nonstationarity, and for [DELTA]ln[GDP.sub.t-3] > 1.5%, there is no mean reversion and hence no long-run equilibrium rate.

In Japan, the transition is driven by the third lag of GDP growth and the model explains 13.4% of the residual variance of the linear AR. The transition is slow, and three states can be distinguished: the upper extreme regime which corresponds to very rapid growth, for [DELTA]ln[GDP.sub.t-3] > 1.8 or 7.3% in annual terms; the lower extreme regime which comprises mild growth, slowdowns and recessions and is defined for [DELTA]ln[GDP.sub.t-3] < 0.7% (2.8%); and the intermediate interval 0.7% < [DELTA]ln[GDP.sub.t-3] < 1.8%.

In the upper extreme regime of high-growth expansions, the unemployment rate is not stationary and there is no reversion to a given long-run equilibrium value. The same conclusion seems to hold for quarterly GDP growth between 0.7 and 1.8%. In the lower extreme regime, however, the unemployment rate looks stationary with long-run expected value 3.9%. It should be noted that the estimated linear AR displays a unit root, see Table 3, so the major improvement of the nonlinear model is to point out that even though the unemployment rate appears to be nonstationary in the most part of the sample, there is mean reversion to a long-run rate of equilibrium for low rates of growth of GDP.

In the United Kingdom, the nonlinearities of the unemployment rate react to the second lag of GDP growth and explain 18% of the residual variance of the linear autoregression. The preferred model behaves like a TAR that allows for specific dynamics when the economy experiences a severe recession, [DELTA]ln[GDP.sub.t-2] < -0.58%. Within this regime, the dominant roots are a pair of explosive, complex roots with period 16.3 quarters. Otherwise, the unemployment rate is a stationary process, with a long-run mean equal to 3.6%.

The estimated models show that the position of extreme regimes and local behavior within each regime are country-specific. Nonetheless, some common features may be drawn. There is widespread empirical evidence that nonlinearities in the unemployment rate are induced by the cyclical states of the economy. Moreover, the changes in local dynamics seem to explain the failure of standard testing strategies to determine whether the unemployment rate displays a unit root.

According to the models developed in this paper, in five countries the dominant root is inside the unit circle in some phases of the cycle, and on or outside the unit circle in the others. There is not a general pattern that fits all situations: in some countries nonstationarity is related to periods of very rapid growth of GDP, while in others nonstationarity is a characteristic of severe recessions. One may suspect that the specific sample used in this paper explains some of the differences across countries, as different countries had different cyclical experiences along the time interval that was considered. Finally, it should also be noted that when the model is locally stable, there are a number of roots close to the unity, so shocks take a long time to fade away even if GDP growth remains constant at a normal rate.

Conclusions

This paper investigated to what extent the observed nonlinearities in the unemployment rates of six major developed economies are the response to cyclical asymmetries, or should be interpreted as being caused by idiosyncratic factors specific to the labor market. It was assumed that nonlinear behavior can be captured by smooth transition autoregressions (STAR), a special case of an autoregressive process where the parameters depend on a transition variable [s.sub.t]. Given the value of [s.sub.t], the model simplifies into a linear autoregression, so nonlinear behavior arises because of the changes in the transition variable.

Two classes of models were considered: strict univariate STARs, where the transition between regimes is endogenously determined, and models where the changes in the autoregressive coefficients depend on GDP growth, which is considered to be an adequate indicator of the business cycle. The empirical analysis showed consistent evidence across countries that the latter are much better in fitting the data than the former, a result that supports the proposition that the behavior of the unemployment rate changes with the phases of the cycle in a nonlinear way.

National characteristics play a role in determining how the unemployment rate evolves along the cycle. There is not a general pattern that explains all situations, as the position of extreme regimes, the smoothness of the transition, the local dynamics within each regime and the percentage reduction in the residual variance of the linear AR vary from country to country. In spite of this plenty of country-specific features, the following remarkable stylized fact was found.

It is known that standard unit root tests provide mixed evidence as to whether the unemployment rate follows a stationary process. This is a major issue in analyzing the labor market, as the unit root controversy is related to the existence of a long-run, equilibrium rate of unemployment. The models estimated in this paper point out that the existence of such equilibrium is phase-dependent, and that would explain the failure of univariate unit root testing. First, if the unit root exists only in specific periods, then the results of the tests are sensitive to a particular sample period. Second, the force driving the existence of the unit root is exogenous to the unemployment rate, and standard tests can not control for its influence as they are posed in a strict univariate framework.

The models indicate that it seems that in periods of normal, moderate growth, the unemployment rate is stationary and always reverts to the same long-run mean, albeit the local dynamics are such that it may take a long time for shocks to fade away. But in extreme situations (strong recoveries in some countries, severe recessions in others), the unemployment rate becomes nonstationary, as if each extreme cyclical episode had its own, characteristic path of equilibrium. It would be interesting to relate this interpretation of the unemployment rate behavior based on the business cycle, to the economic theories of the labor market aimed to explain the existence of a natural rate of unemployment and why it varies across countries. This issue will be considered in further research.

Acknowledgments A previous version was presented at the 61st International Atlantic Economic Conference, Berlin, Germany, March 15-19, 2006. The author thanks the conference participants and an anonymous referee for their comments and suggestions.

References

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Published online: 24 May 2007

[c] International Atlantic Economic Society 2007

J. R. Cancelo ([mailing address])

Facultad de Ciencias Economicas y Empresariales, Dpto. Economia Aplicada II, Universidade da Coruna, Campus de Elvina, 15071 Corunna, Spain

e-mail: jramon@udc.es
Table 1 Linearity tests against different transition variables: p-values

              Endogenous Transition, [s.sub.t]=[DELTA][u.sub.t-d]
                      Conditional         Unconditional
Country    p  1      2      3      4      1 to 4

Canada     1  0.035  0.982  0.750  0.562  0.043
France     2  0.069  0.179  0.264  0.265  0.285
Germany    1  0.122  0.484  0.390  0.796  0.597
           5  0.599  0.463  0.537  0.484  0.491
Italy      5  0.136  0.558  0.183  0.177  0.123
Japan      3  0.620  0.206  0.614  0.441  0.487
           8  0.415  0.043  0.175  0.185  0.427
United     5  0.400  0.709  0.523  0.263  0.215
  Kingdom

              Exogenous Transition, [s.sub.t]=[DELTA]ln [GDP.sub.t-d]
                      Conditional         Unconditional
Country    p  1      2      3      4      1 to 4

Canada     1  0.000  0.055  0.489  0.842  0.002
France     2  0.002  0.018  0.755  0.538  0.003
Germany    1  0.110  0.612  0.921  0.029  0.021
           5  0.333  0.568  0.824  0.010  0.185
Italy      5  0.767  0.560  0.018  0.946  0.274
Japan      3  0.001  0.462  0.012  0.046  0.000
           8  0.045  0.524  0.020  0.186  0.006
United     5  0.068  0.008  0.404  0.109  0.003
  Kingdom

Table 2 Estimated models with exogenous transition for the unemployment
rates

CANADA
[DELTA][u.sub.t] =[4.178.(5.90)] - [0.402.(5.83)][u.sub.t-1] +
  [0.85.(4.62)][DELTA][u.sub.t-1] + (-[3.897.(5.46)] +
  [0.36.(5.21)][u.sub.t-1] - [0.55.(2.77)][DELTA][u.sub.t-1]) x
  [1 + exp {-[9.51.(0.25)] x 121.23 ([DELTA]ln[GDP.sub.t-1] -
  [0.0061.(11.08)])}][.sup.-1] + [e.sub.t]
s=0.277, [R.sup.2]=0.51, AIC=-2.50521, [s.sup.2]/[s.sub.L.sup.2] =
0.742, LJB=7.46 (0.02), AUTO= 1.11 (0.36), ARCH=1.57 (0.19), BCH=4.28
(0.04), NLEN=0.99 (0.38), NLEX=5.79 (0.00).

FRANCE
[DELTA][u.sub.t] =[0.096.(4.55)] + [0.84.(7.52)][DELTA][u.sub.t-1] -
  [0.26.(2.55)][DELTA][u.sub.t-2] + ([0.027.(0.69)] - [0.017.(4.49)]
  [u.sub.t-1] - [0.15.(1.11)][DELTA][u.sub.t-1] + [0.10.(0.80)][DELTA]
  [u.sub.t-2]) x [1 + exp {-[73.00.(0.65)] x 182.34([DELTA]
  ln[GDP.sub.t-1] - [0.0035.(3.81)])}][.sup.-1] + [e.sub.t]
s=0.139, [R.sup.2]=0.60, AIC=-3.87943, [s.sup.2]/[s.sub.L.sup.2] =
0.880, LJB=7.83 (0.02), AUTO= 1.92 (0.06), ARCH=4.92 (0.00), BCH=0.18
(0.67), NLEN=2.05 (0.09), NLEX=1.78 (0.14).

GERMANY
[DELTA][u.sub.t] =[1.68.(7.50)][DELTA][u.sub.t-1] - [0.78.(3.94)][DELTA]
  [u.sub.t-2] + [0.19.(1.63)][DELTA][u.sub.t-4] + ([0.161.(2.14)] -
  [0.018.(2.14)][u.sub.t-1] - [1.26.(4.62)][DELTA][u.sub.t-1] +
  [1.02.(3.75)][DELTA][u.sub.t-2] - [0.14.(0.86)][DELTA][u.sub.t-3] -
  [0.19.(1.30)][DELTA][u.sub.t-4]) x [1 + exp {-[4.36.(2.26)] x 103.43
  ([DELTA]ln[GDP.sub.t-4] - [0.0006.(0.50)])}][.sup.-1] + [e.sub.t]
s=0.143, [R.sup.2]=0.67, AIC=-3.80660, [s.sup.2]/[s.sub.L.sup.2] =
0.843, LJB=0.72 (0.70), AUTO=1.66 (0.12), ARCH=1.96 (0.10), BCH=0.10
(0.75), NLEN=0.68 (0.72), NLEX=0.91 (0.52).

ITALY
[DELTA][u.sub.t] = [0.131.(1.71)] - [0.014.(1.61)][u.sub.t-1] +
  [0.09.(1.25)][DELTA][u.sub.t-1] + [0.30.(2.62)][DELTA][u.sub.t-2] +
  [0.29.(3.20)][DELTA][u.sub.t-3] - [0.22.(1.86)][DELTA][u.sub.t-4] +
  [0.23.(2.45)][DELTA][u.sub.t-5] + ([0.301.(2.07)] - [0.039.(2.17)]
  [u.sub.t-1] + [0.14.(0.44)][DELTA][u.sub.t-1] + [0.22.(0.81)][DELTA]
  [u.sub.t-2] - [0.99.(3.33)][DELTA][u.sub.t-3] - [0.20.(0.79)][DELTA]
  [u.sub.t-4] - [0.20.(0.84)][DELTA][u.sub.t-5]) x [1 + exp
  {-[7.72.(1.10)] x 127.32 ([DELTA]ln[GDP.sub.t-3] - [0.0131.(9.63)])}]
  [.sup.-1] + [e.sub.t]
s=0.195, [R.sup.2]=0.40, AIC=-3.13781, [s.sup.2]/[s.sub.L.sup.2] =
0.909, LJB=1.24 (0.54), AUTO= 0.37 (0.93), ARCH=1.78 (0.14), BCH=1.37
(0.24), NLEN=2.37 (0.04), NLEX=1.22 (0.30).

JAPAN
[DELTA][u.sub.t] = [0.105.(2.57)] - [0.027.(2.10)][DELTA][u.sub.t-1] +
  [0.28.(2.41)][DELTA][u.sub.t-1] + [0.23.(1.63)][DELTA][u.sub.t-2] +
  [0.25.(2.22)][DELTA][u.sub.t-3] - [0.17.(1.28)][DELTA][u.sub.t-6] +
  (-[0.218.(2.62)] + [0.064.(2.19)][u.sub.t-1] - [0.65.(2.51)][DELTA]
  [u.sub.t-1] - [0.23.(1.63)][DELTA][u.sub.t-2] + [0.69.(2.22)][DELTA]
  [u.sub.t-6] - [0.32.(1.32)][DELTA][u.sub.t-7] - [0.44.(1.83)][DELTA]
  [u.sub.t-8]) x [1 + exp {-[3.48.(2.71)] x 112.11([DELTA]
  ln[GDP.sub.t-3] - [0.0122.(9.63)])}][.sup.-1] + [e.sub.t]
s=0.095, [R.sup.2]=0.33, AIC=-4.61168, [s.sup.2]/[s.sub.L.sup.2] =
0.866, LJB=3.54 (0.17), AUTO=1.14 (0.34), ARCH=1.27 (0.29), BCH=1.10
(0.30), NLEN=1.70 (0.07), NLEX=1.50 (0.13).

UNITED KINGDOM
[DELTA][u.sub.t] =[0.168.(1.85)] - [0.023.(1.09)][u.sub.t-1] +
  [1.71.(5.18)][DELTA][u.sub.t-1] - [0.66.(1.70)][DELTA][u.sub.t-2] -
  [0.29.(1.57)][DELTA][u.sub.t-4] + (-[0.147.(1.51)] + [0.017.(0.78)]
  [u.sub.t-1] - [0.80.(2.40)][DELTA][u.sub.t-1] + [0.66.(1.70)][DELTA]
  [u.sub.t-2] + [0.15.(0.72)][DELTA][u.sub.t-4] - [0.10.(1.08)][DELTA]
  [u.sub.t-5]) x [1 + exp {-[129.26.(0.59)] x 103.60([DELTA]
  ln[GDP.sub.t-2] + [0.0058.(81.09)])}][.sup.-1] + [e.sub.t]
s=0.106, [R.sup.2]=0.88, AIC=-4.40424, [s.sup.2]/[s.sub.L.sup.2] =
0.820, LJB=10.07 (0.01), AUTO= 0.35 (0.94), ARCH=2.90 (0.02), BCH=2.05
(0.15), NLEN=0.71 (0.68), NLEX=1.40 (0.20).

[u.sub.t] denotes the unemployment rate, and [DELTA]ln[GDP.sub.t] is the
quarterly rate of growth of GDP. Values under the regression
coefficients are t-ratios in absolute value; s is the residual standard
error; [R.sup.2] is the determination coefficient; AIC is the Akaike
Information Criterion; [s.sup.2]/[s.sub.L.sup.2] is the variance ratio
of the residuals from the nonlinear model and the linear AR; LJB is the
Lomnicki-Jarque-Bera normality test; AUTO is the test for residual
autocorrelation of order 8; ARCH is the statistic of no ARCH based on
four lags; BCH is a business cycle heteroskedasticity test; NLEN (NLEX)
is the test for no remaining nonlinearity when the transition is
endogenous (exogenous). Numbers in parentheses after values of LJB,
AUTO, ARCH, BCH, NLEN and NLEX are p-values. The true distributions of
the ordered statistics NLEN and NLEX are unknown and the p-values shown
in the table overreject the null that the model is correctly specified.

Table 3 Local dynamics in extreme regimes and in the linear AR: roots
with modulus greater than 0.8

Country         Extreme Values of F  Roots                Modulus

Canada          0                     0.72[+ or -]0.57 i  0.92
                1                     0.94                0.94
                Linear                0.90                0.90
France          0                     1.00                1.00
                1                     0.96                0.96
                Linear                0.98                0.98
Germany         0                     1.00                1.00
                                      1.12                1.12
                1                     0.96                0.96
                                     -0.83                0.83
                Linear                0.96                0.96
Italy           0                    -0.66[+ or -]0.56 i  0.87
                                      0.93[+ or -]0.03 i  0.93
                1                     0.96                0.96
                                      0.75[+ or -]0.70 i  1.02
                Linear                0.96                0.96
                                      0.81                0.81
                                     -0.65[+ or -]0.51 i  0.83
Japan           0                     0.93                0.93
                                      0.80[+ or -]0.22 i  0.82
                1                     1.03                1.03
                                     -0.56[+ or -]0.88 i  1.05
                                      0.32[+ or -]0.87 i  0.94
                                      0.82[+ or -]0.27 i  0.87
                Linear                1                   1
                                     -0.73[+ or -]0.37 i  0.82
                                     -0.36[+ or -]0.77 i  0.85
                                      0.84[+ or -]0.28 i  0.89
United Kingdom  0                     0.93                0.93
                                      1.08[+ or -]0.44 i  1.16
                1                     0.92[+ or -]0.01 i  0.92
                Linear                0.92[+ or -]0.06 i  0.92

Country         Extreme Values of F  Period

Canada          0                      9.4
                1                      -
                Linear                 -
France          0                      -
                1                      -
                Linear                 -
Germany         0                      -
                                       -
                1                      -
                                       -
                Linear                 -
Italy           0                      2.6
                                     225.5
                1                      -
                                       8.4
                Linear                 -
                                       -
                                       2.5
Japan           0                      -
                                      23.4
                1                      -
                                       2.9
                                       5.2
                                      19.6
                Linear                 -
                                       2.4
                                       3.1
                                      19.4
United Kingdom  0                      -
                                      16.3
                1                    576.5
                Linear                91.0
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Comment:Cyclical asymmetries in unemployment rates: international evidence.
Author:Cancelo, Jose Ramon
Publication:International Advances in Economic Research
Geographic Code:1USA
Date:Aug 1, 2007
Words:6903
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