Forecasting performance of logistic STAR model: An alternative version to the original LSTAR models

This paper proposes an alternative representation of the original version of the Logistic Smooth Transition Auto-Regressive (LSTAR) model. The Logistic Smooth Transition Auto-Regressive (LSTAR) and Exponential Smooth Transition Auto-Regressive (FSTAR) models are frequently used in empirical research. The LSTAR model describes asymmetrical nonlinear adjustment process, while the ESTAR model describes symmetrical nonlinear adjustment process (Sarantis, 1999), in Liews (2002). To Liews et a]., the theoretical assumption that exchange rate adjustment is symmetric makes the LSTAR model inappropriate for modelling exchange rate movements, for this reason, the LSTAR model has being neglected in the modelling of exchange rate in the past. They proposed the Absolute Logistic Smooth Transition Auto-Regressive (ALSTAR) model. This version they noted allows a V-shape symmetric adjustment in exchange rate behaviour. In this paper, we propose the Square Logistic Smooth Transition Auto-Regressive (SLSTAR) model which out-performs both the LSTAR and the ALSTAR model in many instances and has the inverted bell-shape of the exponential model which allows a symmetric adjustment in exchange rate behaviour. We have used Monte-Carlo studies and life data to show our claim. Introduction The investigation of non-linearities and asymmetries in macroeconomic variables is no doubt a popular area of empirical research. Among various non-linear models reported in (he literature are: the exponential auto-regressive (EAR) model, the Markov-switching model (Hamilton, 1989), where changes in regimes are assumed to be governed by the outcomes of an observed Markov chain, the threshold auto-regressive (TAR) model introduced in the time series literature by Howell Tong (see his 1983 and 1990 monographs), here, regimes are defined by the past values of the time series itself, where as in the Markov switching case regimes are defined by exogenous state of the Markov chain. The smooth transition model was first suggested by Chan and Tong (1986), to model a smooth transition between regimes and was subsequently developed by Timo Terasvirta and his various co-authors (see his 1993 monograph with Clive Granger). The former models are characterised by abrupt and sudden change from one regime to another about a particular threshold value. The abrupt regime changes in the threshold model coupled with the difficulties of the non-standard likelihood/least squares functions are unrealistic to many authors (Potter, 1999). This according to Yi-Nung Yang (2002) may not be consistent with the real world observations. Economic variables continue to receive shocks due to decision agents changing behaviour at a given point over time based on many alternative actions facing them. The economy consists of a great number of agents whose behaviour may switch sharply but not simultaneously. The STAR model is a STR model in which the transition variable is an endogenous variable becomes a useful tool. The introduction of smooth transition between regimes allows standard non-linear estimation techniques to be used. The main advantage in favour of STAR models (Nektarios, 2002), is that changes in economic aggregates are influenced by changes in the behaviour of many different agents and it is highly unlikely that all agents react simultaneously to a given economic signals. A STAR model allows that exchange rates alternates smoothly between two regimes. Two STAR models considered by Timo (1994), are the logistic STAR (LSTAR) model, and the exponential STAR (ESTAR) model. Anderson (1992), have successfully applied smooth transition models to a wide range of industrial production series. Yi-N.ung op cit applied STR to model the behaviour of the Dollar/Yen exchange rates. The STAR methodology according to Mark (2002), allows for the possibility that economies do not necessarily jump suddenly from one real exchange rate regime to another, for example between low and high real exchange rates on the basis of a single real exchange rate shock. Inugunn (2002) applied smooth transition regression model to investigate possible instability and non-linearity in their model for residential consumption of electricity in Norway. Mark op cil applied STAR models to US dollar real exchange rates of thirteen Latin American countries and found non-linear behaviour in seven countries with the LSTAR model capturing the non-linearities in six. The LSTAR model has been successfully applied by Terasvirta and Anderson (1992) and Terasvirta, Tyostheim and Granger (1994) to characterise the different dynamics of industrial production indexes in a number of OECD countries during expansions and recessions. The ESTAR model has been applied by Michael, Nobay and Peel (1997) and Taylor, Peel and Sarno (2000) to model real exchange rates It has also been applied to real effective rates by Sarantis (1999). The issue of a choice between the alternative specifications: LSTAR and ESTAR models in place of a linear model is beginning to generate ripples especially as it relates to the behaviour of the logistic smooth transition autoregressive model. According to Liew (2002), exponential smooth transition autoregressive (ESTAR) model is widely adopted in the exchange rate study as its symmetrical distribution matches that of the symmetrical exchange rate adjustment behaviour. In contrast the logistic smooth transition autoregressive model is discarded by most researchers in priori in their exchange rate modelling exercises due to its undesired property of being asymmetric. They investigated the validity of the claim by researchers that the ESTAR exchange rate model is superior to the LSTAR exchange rate model on the basis of forecast accuracy. They found that this claim is merely theoretical, as they could not find any empirical evidence'to support it. They however are of the opinion that LSTAR model should not be dropped just like that and so gave an alternative reparameterised version of the LSTAR model known as the absolute logistic smooth transition (ALSTAR) model. It is the purpose of this study to also give an alternative re-parameterised version of the original LSTAR model that performs better than the ALSTAR model proposed by Liews et al in some cases. We have used simulation and life data (monthly data on Nigerian Nominal Exchange Rate and the Seasonally Adjusted average Monthly Money Supply billion $ (ml)) for this study. The Models A smooth transition autoregressive model of order/? as given by Terasvirta (1994) is defined as: where y, is a stationary series, F(.) is a continuous transition function which is monotonicalfy increasing, twice differentiate and bounded by 0 and 1, y . is