First draft (very preliminary): January 2004
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MODELLING CYCLICAL ASYMMETRIES IN EUROPEAN IMPORTS
José Ramón Cancelo
Estefanía Mourelle
(Universidade da Coruña, Spain)
Address: Dpto. Economía Aplicada II; Facultad Ciencias Económicas; Campus de Elviña; 15008 A Coruña (Spain). Fax: 34-981-167070. e-mail: Cancelo, ; Mourelle,
Abstract
This paper develops smooth transition autoregressions to investigate nonlinear behavior in the imports data of six major European economies: Belgium, France, Germany, Italy, Spain and the United Kingdom. Two classes of switch between regimes are considered: endogenously determined transition that assumes that nonlinearities are generated by idiosyncratic components specific to foreign trade, and exogenous transition based on GDP growth as a more direct indicator of the general cycle of the economy. Our results support the proposition that the dynamic properties of imports change over the business cycle: modelling the nonlinearity in the data allows to explain 25% to 30% of the residual variance of the best linear autoregression. In Belgium, France, Spain and the United Kingdom regimes are exogenously determined: in all cases the transition variable is current GDP growth, and the parameters of the model immediately adapt to changes in the economic conditions. In Germany and Italy the switch between regimes is endogenous. National characteristics play a role in defining the position of extreme regimes, the smoothness of the transition and local dynamics within each state.
JEL Classification: C32, E32, F15
Key words: smooth transition, nonlinear time series models, business cycle, European Union
1. INTRODUCTION
Modelling nonlinearities in economic time series related to business cycle asymmetries has long been of interest to applied economists. In the last fifteen years there has been an explosion of papers on the suitable statistical methods for summarizing and explaining cyclical behavior of macroeconomic data, most of them from an univariate point of view. Three types of models have most commonly been used (Potter, 1999): Markov switching models (MSM: Hamilton, 1989), Self-Excited Threshold Autoregressions (SETAR: Tong, 1990) and Smooth Transition Models (STM), which were developed by Teräsvirta and several co-authors (Luukkonen et al 1988a and 1988b, Granger and Teräsvirta 1993, Teräsvirta 1994 and 1998, Eitrheim and Teräsvirta 1996, van Dijk et al 2002, etc). In Markov switching models the relevant state of the economy is given by an unobserved, discrete variable. In threshold models there is a finite, usually small, number of regimes that are defined by the past values of the time series itself. Smooth transition models extend threshold models to allow the possibility that the variable may be in an intermediate state between extreme regimes.
In recent years many studies have used STMs to capture asymmetric cyclical behavior in macroeconomic variables like GDP, industrial production, unemployment, etc, including van Dijk and Franses (1999), Skalin and Teräsvirta (1999, 2002), Öcal and Osborn (2000) and Sensier et al (2002). Imports however have not received too much attention in the literature, probably because empirical work has concentrated mainly on the United States where the volume of international transactions is low compared to GDP. But international trade represents a significant proportion of economic activity in European countries, and the differences between domestic and imported goods are negligible with regard to that part of imports due to regional trade within the European Union. In short imports are expected to be highly sensitive to the state of the cycle in European countries, and constitute a natural candidate for unveiling some phase-dependent properties of the economy.
In this paper we investigate potential nonlinearities in the imports data of six major European economies: Belgium, France, Germany, Italy, Spain and the United Kingdom. We compare two variants of smooth transitions models. First we consider Smooth Transition AutoRegressions (STAR) with endogenous determination of regimes, following standard practice in the literature. Next we extend the basic univariate framework to allow for exogenous determination, so the switch between regimes is a function of a more direct indicator of the business cycle. This variant leads to what we call the smooth transition autoregression with exogenous transition (STAR-EXT) model, and on a priori grounds it is expected to provide a better description of the data than a strict univariate model.
The paper is organized as follows. Section 2 presents the two variants of smooth transition models. Section 3 reports the estimated models for the quarterly rate of growth of imports. Section 4 examines international evidence on cyclical asymmetries in the imports data. Section 5 concludes.
2. SMOOTH TRANSITION MODELS
Smooth transition models (STMs) are a special class of state-dependent, nonlinear time series models. In the most common case the variable is assumed to vary between two extreme regimes and the smoothness of the transition from one regime to the other is estimated from the data; in practice intermediate situations are frequently observed. The dependent variable is given by a linear combination of predetermined variables plus a random disturbance, where each coefficient is a function of a state variable. This parameterization permits a variety of dynamic behavior over the cycle and at the same time once the state is given the model is locally linear, which allows easy interpretation of the local dynamics. Granger and Teräsvirta (1993), Teräsvirta (1998) and van Dijk et al (2002) describe STMs in detail.
The basic univariate version of STMs is the Smooth Transition Autoregression (STAR): all predetermined variables are lags of the dependent variable and regimes are endogenously generated by the recent history of the time series itself. The STAR model of order p for a stationary and ergodic process yt is defined as
where F(yt-d) is a transition function that satisfies 0F1 and d is the transition lag.
The key feature about the transition function F(yt-d) is whether it is odd or even; in the first case F(-)=0 and F()=1 while in the second F()=1 and F(c)=0 for some finite c. In the empirical literature the odd case is usually represented by the logistic function
and the resulting model is known as the logistic STAR or LSTAR. The slope parameter determines the smoothness of the transition: the higher it is, the more abrupt the change from one extreme regime to the other. In the limit, as , the LSTAR model collapses into a two-regime SETAR. The location parameter c is such that F(c)=0.5 and separates the half-intervals of low and high values of the transition function.
For the even case the exponential function is used
so that (2.1) and (2.3) define the exponential STAR (ESTAR) model. The parameter has the same interpretation as in the logistic model and c now defines the middle extreme regime such that F(c)=0.
The fact that F(yt-d) is even or odd has interesting implications for the analysis of business cycle asymmetries. At first sight LSTARs seem more adequate, as the two extreme regimes correspond to very high and very low values of the variable and hence can be identified with strong recoveries and severe contractions. Nevertheless both specifications will be very similar in fitting the data when the estimated location parameter c is either very high or very low, a situation that arises quite frequently in empirical applications (Teräsvirta and Anderson 1992, Skalin and Teräsvirta 1999). Öcal and Osborn (2000) argue that in this case the preference for an exponential transition together with a low value of c may be explained because this combination is best suited than LSTARs for capturing transitions that are sharper near business cycle troughs than at peaks, see their paper for the details.
In the basic STAR (2.1) to (2.3) the state variable that determines the cyclical regime at each t is a lag of the dependent variable. While this may be a valid starting point in a strict univariate framework, it seems that a better alternative is to define a more direct indicator of the state of the cycle and to specify the switch as a function of that indicator. This gives rise to a second class of STMs that is midway between STARs and general Smooth Transition Regressions, as it combines univariate dynamic dependence and exogenous regime determination. We shall refer to this model as STAR-EXT and it is given by
with xt-d instead of yt-d in (2.2) and (2.3). The natural candidate for xt is GDP growth, and in the next Sections we compare the relative performance of (2.1) and (2.4) in order to assess the main force driving nonlinear behavior in European imports.
3. EMPIRICAL ANALYSIS
3.1 Data
We consider quarterly, seasonally adjusted data for imports of goods and services and GDP for Belgium, France, Germany, Italy, Spain and the United Kingdom. The source is the OECD Statistical Compendium and the final sample goes from 1981:1 to 2003:2. German data have been adjusted because of the unification. Standard unit root tests indicate that all imports and GDP series are I(1), so from now on all the variables are the first difference of the logarithms.
3.2 Testing linearity
The first step towards building STAR models is to test whether the data display the type of nonlinear behavior generated by smooth transition autoregressions. The usual approach is to compare the nested linear model against all variants of the general nonlinear model. The literature has not considered the case for testing against STAR-EXT in an explicit way, but it is straightforward to adapt the tests derived for general smooth transition regressions to cover this particular situation.
The tests are based on a sequence of auxiliary regressions and a review would demand some technical discussion that goes well beyond the scope of this paper, see Teräsvirta (1994, 1998) for the details. There is however a point that deserves some attention for interpreting Table 1. In the standard testing strategy first the lag order p is estimated by using Akaike Information Criterion (AIC) to select the proper number of lags in a linear autoregression, and next linearity tests are carried out conditional on this value. The transition lag d is determined either by varying it and choosing the value minimizing the p-value of his linearity test (Teräsvirta, 1994: 211), or by assuming that the transition variable is the linear combination , where '=(0 ... 1 ... 0)' is a selection vector with the only unit element corresponding to the unknown transition lag (Teräsvirta, 1998: 517) and zt is the transition variable. The first approach is called the conditional approach, while the second is known as the unconditional procedure.
In our analysis p varies from 1 to 8 and d from 1 to max(p,6) in the endogenous transition models and from 0 to max(p,6) when the switch depends on GDP growth. We took eight as the maximum lag order because eighth-order dynamics seem to be general enough for quarterly, seasonally adjusted data. Moreover the sample size is 89 observations and we wanted to retain an adequate number of degrees of freedom to avoid size distortions in carrying out hypothesis testing. Following standard practice in the literature d was assumed to be small and not greater than p.
Table 1 displays a summary of p-values of the linearity tests. To save space we only report the results against STAR models with exogenous transition for the lag order providing the minimum AIC in a linear autoregression. Both conditional and unconditional approaches are shown. Unconditional tests against STAR-EXT with exponential transition for the United Kingdom were computed for d ranging from 0 to 3, as the estimated p is high and there are not enough degrees of freedom for carrying out a joint test against unspecified d in the interval [0,6].
INSERT TABLE 1 HERE
The tests find evidence of nonlinear behavior in Belgium, France, Spain and the United Kingdom, but not for Germany or Italy. The number of rejections is higher against STAR-EXT models than against pure STARs (detailed results are available from the authors), a first indication that nonlinearities seem to be related to a general measure of the state of the economy. In fact should the basic testing strategy be taken literally one would not reject linearity against STARs with endogenous transition. Recent literature however advocates for attempting to build a valid nonlinear model even in that case, as it is expected that a false rejection of linearity will be discovered at some later stage (van Dijk et al 2002, Sensier et al 2002).
3.3 Estimated models
Model building was based on an extensive search. We specified a large number of potential models by considering all the possible combinations of p and d for both logistic and exponential transitions. This amounts to 66 models with endogenous regime determination and 82 models with exogenous transition for each country. All 148 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.
We were able to build valid STAR and STAR-EXT models for every country but Germany. We saw in Table 1 that for German imports the null of linearity was not rejected against a STAR-EXT model, and the modelling process confirmed that it is not possible to achieve an adequate STAR-EXT representation for that series. In Table 2 we report a summary of the estimated models.
INSERT TABLE 2 HERE
The final step is to assess the relative performance of the two approaches for determining the relevant regime, endogenous transition based on the recent history of imports and exogenous determination through GDP growth. This distinction plays a central role in explaining the main factor driving nonlinearities in the imports series: opting for the model with exogenous transition means that nonlinear behavior is related to general cyclical conditions, while the preference for the model with endogenous transition may be taken as an indication that nonlinearities are generated by idiosyncratic components specific to foreign trade.
To our knowledge formal tests for comparing non-nested, nonlinear time series models like those considered in Table 2 have not been developed, and the only feasible way to select between such competing specifications is to use information criteria. In Table 2 it can be seen that AIC is lower for the exogenous transition model in Belgium, France, Spain and the UK, while the strict univariate specification is preferred for Italy.
The selected models are presented in full detail in Table 3, 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 LM statistic of no ARCH with four lags; BCH is Öcal and Osborn's (2000) test of business cycle heteroskedasticity computed by regressing the squared residuals on the values of the transition function. We also consider three tests that were specially derived for smooth transition models in Eitrheim and Teräsvirta (1996). AUTO tests serial independence against a fourth-order process. 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 3 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.
INSERT TABLE 3 HERE
4. AN INTERNATIONAL PERSPECTIVE ON CYCLICAL ASYMMETRIES IN IMPORTS
In the previous section we showed that European imports display distinct characteristics associated with different states; in some cases the relevant state depends on the general cycle of the economy, while in the other countries it seems to be driven by more idiosyncratic factors specific to foreign trade. Nonetheless whatever the transition variable is a closer look at Table 2 reveals an interesting stylized fact of European imports. For every country the variance ratio, defined as the residual variance of the nonlinear model over the residual variance of the best linear autoregression selected with AIC, lies in the interval [0.71, 0.76]: the final nonlinear model explains 25% to 30% of the residual variance of the best linear autoregression in all six countries, although the way that bound is attained is not the same in all cases.
In Belgium and Spain the STAR model already does most of the job, as the variance ratios of STAR and STAR-EXT models are very similar. In the two countries the rates of growth of imports and GDP are closely related and lagged imports might be approximating GDP behavior in the model with endogenous transition. Thus considering GDP in an explicit way does not lead to a better fit but to a more efficient way of achieving the same explanatory power, both in the sense of requiring less parameters and by shortening the delay of response to a change in underlying economic conditions.
In France and the United Kingdom we have a different situation: the explanatory power of the univariate model is not much lower compared to the linear autoregression (variance ratios around 0.85), but it improves when exogenous transition is allowed for. The number of parameters of STAR-EXT models are the same (France) or higher (United Kingdom) than in the univariate specification, and the reduction in AIC is due to a better fit: we may conclude that in both countries regimes are determined by general economic conditions and that lagged rates of growth of imports are not a good approximation. The main shortcoming of the Italian model with exogenous transition is that it displays low explanatory power compared to the univariate specification. The fact that we were not able to set up a valid STAR-EXT model for Germany had no consequences on the reduction in the final variance ratio, which is similar to those observed in the five other countries.