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Weekly UpdatesAbstract Traditional specifications about imports have their foundations on the symmetry of the cycle. However, the wide debate about the asymmetric character of the cycle has aroused much interest in nonlinear dynamics due to the cyclical state of the economy. Economic theory maintains a linear long-run relationship linking imports, GDP and relative prices. This paper analyzes whether short-run deviations from this equilibrium display any kind of nonlinear behavior related to the state of the cycle. Nonlinearities will be captured by an error correction smooth transition regression (STR). Empirical evidence, focused on Spanish imports of goods, supports that short-run deviations of this variable from its linear equilibrium state display a nonlinear behavior. As it is demonstrated, this evolution is caused by the cyclical state of the economy.
Keywords Nonlinear behavior * Smooth transition models * Imports * Business cycle
JEL C32 * E32
Introduction
Economic time series behavior has long assumed symmetry, so that the related model building has essentially leant on the linear framework. The assumption of symmetry was not questioned in general until middle eighties, partly due to the influential article of Neftci (1984); despite which several authors have supported the existence of cyclical asymmetries for a long time (Mitchell 1913, Hicks 1950). Since then, many empirical works have confirmed the asymmetry of the cycle, which leads to a nonlinear behavior in the observed variables; see for instance Ocal and Osborn (2000) and Skalin and Terasvirta (2002).
Faced to this situation, models able to capture the asymmetry of the cycle must be applied. This paper focuses on the Smooth Transition (ST) model as preferred to other habitual specifications for several reasons: their flexibility enables the description of a wide range of nonlinear behaviors; a continuum of intermediate regimes is permitted; there exists a modeling procedure in the literature; standard nonlinear estimation techniques can be used, etc.
Imports hardly received attention from a nonlinear approach, as investigations traditionally assumed symmetry. The literature on this variable has so far focused on building aggregate demand functions, measuring income and price elasticities and examining the role of foreign trade on economic activity, without regard to the effect of cycle asymmetries (Bajo and Montero 1995, Domenech and Taguas 1997, Buisan and Gordo 1997). As an exception, two articles that adopt a nonlinear approach to imports are highlighted: Cancelo and Mourelle (2005), who developed a modeling cycle for capturing nonlinearities in European imports, and Escribano (1999), who considered nonlinear models for forecasting purposes.
The aim of this paper is to characterize cyclical asymmetries in Spanish imports of goods. Two key issues lead this empirical work: to investigate whether there is empirical evidence of nonlinear behavior related to cyclical asymmetries in Spanish imports and, in the affirmative, to find out the underlying economic component in this nonlinear evolution. Nonlinearities are considered to either arise from idiosyncratic components specific to foreign trade or, given the relevance of foreign trade in economic activity, from the cyclical state of the economy.
Spanish economy shows a high tendency to import and the weight of Spanish imports of goods over GDP has been growing for the last three decades. Their evolution is especially influenced by the economic activity level, picked up by GDP, and by (domestic and import) prices. GDP and prices bring about the dynamics of imports and GDP might also be the cause of the asymmetry of imports; thus, GDP plays a double role in this investigation.
The cointegration analysis may evidence whether a linear long-run relation between imports and the variables determining their behavior exists. This paper discusses, to what extent, short-run deviations of imports from the equilibrium location have linear or nonlinear dynamics. Error Correction-Smooth Transition Regressions will explain the latter.
The plan of the paper is as follows. The following section presents the foundations and building procedures of smooth transition regressions. Next, the data analysis and the estimated model for Spanish imports of goods are reported. Then, a study to extract information about the short-run dynamics of imports is developed. Finally, the main conclusions are outlined.
The Smooth Transition Regression Model
Foundations
ST models are members of the family of state-dependent models, which are a local linearization of the general nonlinear specification. The data-generating process (DGP) is a linear one that switches between a certain number of regimes according to a rule; the regime is characterized as a continuous function of a predetermined variable, so that interactions between variables are permitted, as well as intermediate states between the extreme regimes.
This parameterization allows for capturing different types of behavior that a linear model cannot characterize in an appropriate way; moreover, once the state is given, the model is locally linear, involving an easy interpretation of the local dynamics. See Granger and Terasvirta (1993), Terasvirta (1998) and Sensier et al. (2002) for further details on STs.
The Smooth Transition Regression (STR) is the most general ST model: it contains an endogenous structure and exogenous variables. Suppose {[y.sub.t]} a stationary, ergodic process, and, without loss of generality, only one exogenous variable [x.sub.t]. The model is given by:
[y.sub.t] = [w'.sub.t][pi] + ([w'.sub.t][theta]) F ([s.sub.t];[gamma],c) + [u.sub.t] (1)
where [w'.sub.t] = (1, [y.sub.[t-1]], .., [y.sub.[t-p1]]; [x.sub.t], [x.sub.[t-1]], .., [x.sub.[t-p2]] is a vector of regressors, [pi] = ([[pi].sub.0], [[pi].sub.1], .., [[pi].sub.p])' and [theta] = ([[theta].sub.0], [[theta].sub.1], .., [[theta].sub.p])' are parameter vectors (p = p1 + p2 + 1), and [u.sub.t] is an error process, [u.sub.t]~Niid (0, [[sigma].sup.2]).
In equation (1), F([s.sub.i], [gamma], c) is the transition function, which is customarily bounded between 0 and 1; this makes the STR coefficients vary between [[pi].sub.j] and [[pi].sub.j] + [[theta].sub.j] (j=0, ..., p), respectively. The regime at each t is determined by the transition variable, [s.sub.t], and the associated value of F([s.sub.t]). The state variable can be: a lagged endogenous variable ([s.sub.t] = [y.sub.[t-d]]) for certain integer d > 0, d transition lag), an exogenous variable ([s.sub.t],=[x.sub.[t-d]]) or just another variable.
The STR model is a regime-switching specification that considers two distinct regimes (the basic version of STs), corresponding to F = 0 and F = 1. The transition from one regime to the other is smooth over time, meaning that parameters in (1) gradually change with the state variable. This model links two linear components through F([s.sub.t]), so that connection features depend on the formulation for F, especially on whether it is odd or even. On this matter, two specifications are highlighted. First, the logistic function, which has the form
F([s.sub.t]) = 1/1 + exp[-[gamma]([s.sub.t] - c)], [gamma] > 0 (2)
and the resulting model is the Logistic STR or LSTR. This function usually represents the odd case in the literature, which means that F(-[infinity])=0 and F([infinity])= 1. The slope parameter [gamma] defines the smoothness of the transition from one regime to the other: the greater it is, the more rapid the change. The location parameter c indicates the threshold between the two regimes; here, F(c)=0.5, so the regimes are associated with low and high values of [s.sub.t] relative to c.
Second, the exponential function
F([s.sub.t]) = 1 - exp [-[gamma]([s.sub.t] - c).sup.2], [gamma] > 0 (3)
provides the Exponential STR (ESTR) model. This even function implies F([+ or -][infinity])=1 and F(c)=0 for some finite c, defining the outer and the inner extreme regime, respectively.
The selection of the transition function is a key point when investigating nonlinearities, since LSTR and ESTR models describe quite different types of behavior. In the logistic model, the extreme regimes are associated with [s.sub.t] values far above or below c, where dynamics may be different; the ESTR model suggests a rather similar dynamics in the extreme regimes, related to low and high [s.sub.t] absolute values, while it can be different in the transition period.
The logistic transition seems to be the most adequate to capture cyclical asymmetries. However, Terasvirta (1994) indicates that the ESTR model can have similar implications to those of the LSTR model in practice when the vast majority of the data lie to the right of c.
For the purpose of this paper, error correction models with STR structure are considered in order to capture short-run and long-run effects in the data.
Modeling Procedure
Traditionally, the STR modeling cycle has relied on developing the iterative methodology proposed by Terasvirta (1994) for the univariate case. It involves three stages: search for specification, estimation and evaluation of the model. There exists a well-established STR modeling strategy in the literature (Granger and Terasvirta 1993, Terasvirta 1994, 1998).
The starting point consists of finding out the linear model that is characterizing the behavior of the series under study. Once this specification is obtained, its adequation to the relation being analyzed is tested: if the null hypothesis of the series being generated by a linear process against the alternative of a STR one is rejected, a preliminary specification of the nonlinear model is defined. This stage is centered on the selection of the appropriate transition lag and the form of the transition function. In the next step, the parameters of the STR specification are estimated by nonlinear least squares (NLS).
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