Univariate Nonlinear Prediction Models

Linear models have many useful properties and are a natural starting point for forecasting analysis given that they are generally easy to estimate and analyze as we saw in chapter 7. However, unless the data are jointly Gaussian, it could well be that there are better-performing nonlinear forecasting models. Indeed, we would expect that linear prediction models are misspecified for many economic variables. For example, recessions tend to be shorter than expansions, and economic recoveries can have very different dynamics from recessions; see, e.g., Pesaran and Potter (1997). Large increases in oil prices have been found to affect GDP growth differently from declines of a similar magnitude (Hamilton, 1996). Asset prices such as exchange rates are a third example in which crashes and recoveries can have very different dynamics as a result of the unwinding of carry trades in periods with waning appetite for risk among investors (e.g., Brunnermeier, Nagel, and Pedersen (2008)). See also Clements, Franses, and Swanson (2004) for further discussion and examples.

If economic (or scientific) theory does not provide guidance on the functional form of the forecasting model, we could posit the forecasting problem in its most general form,

$$f^{\ast }(z)=\arg \underset{f(\cdot )\in \mathcal{F}}{\min }{E}_Y\left[L(f(Z),Y)\right],\text{\hspace{1em}(8.1)}$$

where $\mathcal{F}$ is the set of all possible functions of the conditioning variables, Z. Faced with such a general model it is natural to use a nonparametric estimation approach. Nonparametric forecasting models are covered in more depth in chapter 11.

In practice, we often restrict $\mathcal{F}$ to a subset of all possible models. Linear models restrict the models to be linear functions of $Z$ with an additive error term. Nonlinear models could take the form

$$y_{t+1}=f(z_t)+{\epsilon }_{t+1},\text{\hspace{1em}(8.2)}$$

where the linear additive term captures the unpredictable component and $f(z_t)$ could be limited to a set of models, $\mathcal{F},$ , where the set of models is restricted to a parametric subset of all possible models that could be considered, ${\mathcal{F}}^R$ . Limiting $Z$ to past values of $Y$ suggests models of the form $y_{t+1}=f(y_t)+{\epsilon }_{t+1}$ We next examine such models, noting that different methods make different assumptions on ${\mathcal{F}}^R$ . We focus on parametric models, where ${\mathcal{F}}^R$ is limited to sets of nonlinear models that are known up to a set of parameters.

A large literature seeks to improve on ARMA models by allowing for parametric deviations from linearity. Typically the models employed have heuristic rather than theoretical motivations—e.g., there might be two or more regimes rather than a single one, which leads to switching regressions or smooth transition models for the data. Alternatively, the linear relation between current and past values of a series may break down as the data varies, which leads to threshold autoregression models. The models differ in their choice of exact functional form but all nest the linear model as a special case. While nonlinear least squares can be used to estimate model parameters, tests for nonlinear effects are difficult to interpret.

One limitation of these models is that they provide an inflexible approximation to unknown forms of nonlinearity. This should be contrasted with more general approaches to capture nonlinearity (such as series expansions) which provide more flexible approximations. These approaches are covered in chapter 11.

We first review some of the more popular models that have been applied to forecasting problems, including threshold autoregressive models (section 8.1), smooth transition autoregressive models (section 8.2), regime switching models (section 8.3), before briefly discussing tests for nonlinearities in section 8.4 and covering general procedures for forecasting with nonlinear models in section 8.5. Section 8.6 concludes.