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Some Pitfalls in Smooth Transition Models Estimation: A Monte Carlo Study

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Abstract

Nonlinear regime switching models are becoming increasingly popular in recent applied literature, as they allow capturing state-dependent behaviors which would be otherwise impossible to model. However, despite their popularity, the specification and estimation of these type of models is computationally complex and it is far from being a univocally solved issue.This paper aims at contributing to this debate. In particular, we use Monte Carlo experiments to assess whether employing the standard trick of ‘concentrating the sum of squares’ by Leybourne et al. (Journal of Time Series Analysis, 19(1): 83–97, 1998) in the application of nonlinear least squares to smooth transition models yields estimates with desirable asymptotic properties. Our results confirm that this procedure needs to be used with caution as it may yield biased and inconsistent estimates, especially when faced with small samples.

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Notes

  1. The STECM model is a flexible econometric specification extending the basic ECM in order to capture a nonlinear-asymmetric adjustment process towards the equilibrium. In a nutshell, while in the standard ECM à laEngle and Granger (1987) the adjustment mechanism is linear (a constant parameter \(\alpha <0\)), in the STECM the short-run deviations are linked to the long-run equilibrium by means of a nonlinear function, depending on past values of a so called transition variable. Section 2 describes STECMs in more details.

  2. The term nonlinear cointegration may be ambiguous, as in the literature it can indicate both the presence of a nonlinear cointegrating long-run relationship, and a nonlinear short-run adjustment process. To avoid misunderstandings, when we use the term ‘nonlinear cointegration’ in this article, we are always referring to a nonlinear short-run adjustment process.

  3. A complete discussion of the stationary conditions for ST models is beyond the scope of this paper. For a more thorough treatment of this issue please refer to Dijk and Franses (2000).

  4. Van Dijk and Franses (2000, p. 90) state that beyond the choice of the starting values, two other issues deserve particular attention in the optimization procedure. The first one is exactly the one on which is article is focused, namely the CSQ procedure. The other one concerns the significance of the estimated smoothness parameter \(\gamma \). The issue is that in ST models the standard deviation of the smoothness parameter tends to grow with the size of the parameter itself, hence a precise estimate is always difficult to obtain. More precisely, Teräsvirta (1994) notes that when \(\gamma \) is large and at the same time the \(c\) parameters are sufficiently close to zero, a negative definite Hessian matrix is difficult to obtain for mere numerical reasons, even when convergence is achieved. That is the reason behind the apparent low significance of the estimate of \(\gamma \), which should then be evaluated with diagnostic different from the standard ones.

  5. See for example the Rats letter (2005). Among papers employing the CSQ trick we mention (Van Dijk et al. 2002), Lundbergh et al. (2003) and Shittu and Yaya (2010).

  6. For notational simplicity we will describe the procedure referring to model (3), extensions to more complicated models as (7) are straightforward.

  7. For a more detailed proof see (Wooldridge 2002, Chap. 12, p. 376).

  8. Notice that the preliminary step (Step 0) is to find starting values for both the NLS–CSQ and the ML estimation procedures.

  9. Different experiments where both the endogenous and exogenous were structurally generated have been carried out. However, their results have proven to be too erratic and we chose not to present them here.

  10. To be more precise it is a second-order logistic conditional STECM specification. It is conditional, in the sense that it estimates a ST-ECM model for \(Dy_{t}\) conditioning on \(Dx_{t}\) (The paper by VDF (2000) provides the rationale for this specification, here we take it for granted since we use the model as a benchmark). For notational simplicity we briefly refer to the model as a STECM(1).

  11. These values for the variance of the simulation error are chosen with the explicit objective to go as close as possible to the estimation results of VDF (2000). In particular, \(\sigma _{\varepsilon }^{2}=0.131\) is exactly the residual sum of squares error in VDF’s estimation when they employ weekly data (their lowest), while \(\sigma _{\varepsilon }^{2}=0.225\) is the one resulting from their monthly data estimation. Clearly, \(\sigma _{\varepsilon }^{2}=0.022\) is then chosen to be the more‘optimistic’ value for such variance, being 10 times smaller than the one obtained from VDF’s monthly estimation.

  12. The Levenberg–Marquardt algorithm is a local maximization algorithm which interpolates between the Gauss–Newton algorithm and the method of gradient descent. It is thought to be more robust than the Gauss–Newton, since in many cases it finds a solution even if it starts very far off the final minimum. Even if, for well-behaved functions and reasonable starting parameters, it tends to be a bit slower than the Gauss–Newton. The Berndt, Hall, Hall, and Hausman algorithm follows Newton–Raphson, but replaces the negative of the Hessian by an approximation formed from the sum of the outer product of the gradient vectors for each observation’s contribution to the objective function. For least squares and log likelihood functions, this approximation is asymptotically equivalent to the actual Hessian when evaluated at the parameter values which maximize the function. When evaluated away from the maximum, this approximation may be quite poor.

  13. We described those changes in details in ‘Section 4’.

  14. Alternative measures of bias can be found in the tables reported in the appendix.

  15. For simplicity we reported here only the results pertaining with the more optimistic scenario with \(\varepsilon =0.022\); the other results are qualitatively not different though.

  16. It must be said that the pioneers of ST models [see for example Teräsvirta (1994) and Franses and Dijk (2000)] are always very clear on this point, and they suggest to perform a grid search to make sure to have appropriate starting values. Nevertheless, it is clear that the more the models gets complicated and the higher the number of parameters, the higher is the probability that the grid search does not give the expected results or that it takes too much time.

  17. Indeed the problem of near singularity of the root of the loglikelihood function happens quite frequently, especially in the estimation of complex models like ST. The problem can be handled with the choice of an appropriate algorithm (like Marquardt which has a ridge correction factor) and/or by modifying appropriately the starting values.

  18. Weise (2009) synthetically defines global optimization algorithms as the ones that “employ measures that prevent convergence to local optima and increase the probability of finding a global optimum.” For a thorough treatment of global optimization algorithms please refer to the above mentioned book.

  19. The name stands for for easy-to-use code based on the gradient method.

  20. More exactly the number of replications is 2,29,378. It is important to notice also that the algorithm works with a minimization program, hence the loglikelihood had to be specified with a negative sign. For this reason, all the figures pertaining to the application of EZGRAD will display the negative of the loglikelihood value on the vertical axis, and when we refer to the values of the loglikelihood function the minus sign is always implicit.

  21. From the theoretical viewpoint, the Rational Expectation Hypothesis provides one of the possible rationales for using the term spread as an indicator of market expectations about future economic conditions. In this framework, long-term interest rates are averages of appropriate expected future short-term interest rates, the link being summarized by the so called yield curve. A change in the term spread corresponds to a change of the slope of the yield curve, hence when the market anticipates a recession, a reduction in expected future short-term interest rates is anticipated by a flattening yield curve. Other possible theories that have been proposed to explain this relationship span from a purely monetary explanation, to a more theoretical one relating to with consumers’ expenditure smoothing behavior.

  22. On this topic see for example Galbraith and Tkacz (2000), Venetis et al. (2003), and more recently Brunetti and Torricelli (2009).

  23. See Wheelock and Wohar (2009) for a comprehensive survey.

  24. The grid search procedure employed here is exactly the same as the one used in Experiments I, II and III and it is described more in details at the end of Sect. 5.

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Acknowledgments

I would like to thank Marco Tucci for his guidance and support throughout my research. A special thanks goes to Giampiero Gallo for suggesting this title. The usual disclaimer applies. This version of the paper has greatly benefited from the comments of the two anonymous referees, all remaining errors are mine.

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  1. Robert Schuman Centre for Advanced Studies (RSCAS), European University Institute, San Domenico di Fiesole, FI, Italy

    Novella Maugeri

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Maugeri, N. Some Pitfalls in Smooth Transition Models Estimation: A Monte Carlo Study. Comput Econ 44, 339–378 (2014). https://doi.org/10.1007/s10614-013-9395-6

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