## 1. INTRODUCTION

Nonlinear models play an increasingly important role in the analysis and forecasting of economic and financial time series. An attractive feature of these models is their state dependence, since it is often plausible that the nature of responses may vary with underlying conditions, such as the state of the business cycle, the central bank's monetary policy stance or conditions in financial markets. Smooth transition regression (STR) models are popular in this context because they provide an economic explanation for the regime, by making it a continuous function of an observed variable. Teräsvirta (1994) provides a coherent modelling strategy in the univariate smooth transition autoregression (STAR) context, which is generalized in Teräsvirta (1998) to the STR case. Recent applications include Anderson *et al.* (2007), Sensier *et al.* (2002) and Teräsvirta *et al.* (2005).

A crucial step in STR modelling is the specification of the transition variable that determines the regime. In practice, this is almost invariably taken to be a single lag of an observed variable, with researchers typically following the recommendation of Teräsvirta (1994) to specify the appropriate lag using linearity test statistics computed over a range of plausible values. However, the choice is often not clear-cut, since linearity may be rejected at multiple lags, implying that more than one lagged value contains information about the current regime. Although Medeiros and Veiga (2003, 2005) allow the transition variable to be an unknown linear function of multiple lags, the resulting procedure is fairly complicated and does not always deliver a transition variable that is a weighted average (with positive weights) of lagged values.

The present paper proposes a new procedure for transition variable lag specification, by defining it to be a parsimonious weighted average over potential lags. This WSTR (weighted STR) model requires estimation of only one additional parameter compared to procedures based on a single lag, which is a minimal cost in relation to the added flexibility it delivers. Although the Medeiros and Veiga (2003, 2005) specification is more general, our approach is preferable for the many situations in which regimes are anticipated to vary smoothly over time. The WSTR model is closely related to the STMIDAS specification of Galvão (2009), who exploits the mixed data frequency MIDAS model of Ghysels *et al.* (2005, 2006) to examine the use of high-frequency data for forecasting a lower-frequency variable in an STR context. Although the WSTR model can also be used with mixed-frequency data, our focus differs from Galvão (2009) in being more concerned with issues of model specification and nonlinearity testing.

As usual with models in the STR class, testing for nonlinearity has to confront the issue that transition function parameters are not identified under the null hypothesis. Our approach follows much of the literature by applying a Taylor series approximation. However, the implementation differs in that we propose searching over a plausible set of values for the WSTR weighting function parameters and applying the bootstrap approach of Hansen (1996). Further, we advocate the use of the wild bootstrap to account for possible heteroskedasticity of unknown form. In line with Becker and Hurn (2009), our results indicate that the wild bootstrap approach performs very well, delivering reliable finite sample size and power comparable to that achieved by tests that assume homoskedasticity when the true data-generating process (DGP) is homoskedastic.

The structure of the paper is as follows. Section 2 outlines the WSTR model, focusing on the transition variable. Inference is discussed in Section 3, which develops the proposed WSTR nonlinearity test and studies its properties through a Monte Carlo analysis. Section 4 then examines forecast performance in comparison with linear and other STR specifications. A concluding section completes the paper, with model estimation discussed in an Appendix.