8.2 SMOOTH TRANSITION AUTOREGRESSIVE MODELS

8.2 SMOOTH TRANSITION AUTOREGRESSIVE MODELS

The smooth transition autoregressive (STAR) model is similar to the threshold autoregressive model, the key difference being that it allows a smoother transition between the states. To see the relation between the two models, consider the twostate TAR model:

$$\begin{array}{rl}& y_{t+1}=\left({\varphi }_{11}{y}_t+\cdot \cdot \cdot +{\varphi }_{1p}{y}_{t-p+1}+{\sigma }_1{\epsilon }_{t+1}\right)1(y_{t-d}\in S_1)\\ & +\left({\varphi }_{21}{y}_t+\cdot \cdot \cdot +{\varphi }_{2p}{y}_{t-p+1}+{\sigma }_2{\epsilon }_{t+1}\right)(1-1(y_{t-d}\in S_1)).\end{array}\text{\hspace{1em}(8.10)}$$

The transition between the states can be smoothed by replacing the indicator function $1(\cdot )$ with a function $F(s_t;\gamma ,c)\in (0,1)$ that smoothly moves between 0 and 1, where $s_t$ is data dependent— $-e.g.,s_t=y_{t-d}-\mathrm{and}\gamma >0$ and c are parameters affecting the transition across states:

$$\begin{array}{rl}& y_{t+1}=\left({\varphi }_{11}{y}_t+\cdot \cdot \cdot +{\varphi }_{1p}{y}_{t-p+1}+{\sigma }_1{\epsilon }_{t+1}\right)F(s_t;\gamma ,c)\\ & +\left({\varphi }_{21}{y}_t+\cdot \cdot \cdot +{\varphi }_{2p}{y}_{t-p+1}+{\sigma }_2{\epsilon }_{t+1}\right)(1-F(s_t;\gamma ,c)).\end{array}\text{\hspace{1em}(8.11)}$$

The main STAR models are the logistic STAR (LSTAR) and exponential STAR (ESTAR) models, summarized below, where in each case, $\gamma >0$

(8.12)

For the LSTAR model, the weight on the first model exceeds 0.5 when $s_t>c$ and is otherwise smaller. When $\gamma \to \infty$ the LSTAR model approaches the TAR model, while for $\gamma 0$ it approaches a linear AR model.

The ESTAR model does not nest the TAR model. It is close to the TAR model in the second state, provided $s_t$ is near c and is otherwise close to the TAR model in the first state. The ESTAR model converges to a linear model when $\gamma \to \infty$ or when $\gamma 0$

When $s_t$ equals the time index, these models become time-varying STAR models, and are akin to a permanent break model with smooth transitions between breaks and no mean reversion. Multiple regimes can also be handled. Surveys of the models are provided by van Dijk, Teräsvirta, and Franses (2002) and Teräsvirta (2006). For a book length treatment, see Franses and van Dijk (2000).

Assuming quadratic loss, the model parameters can be obtained through M-estimation using the objective function

$${\mathrm{SSR}}_T(z,\beta )=\frac{1}{T}\sum _{t=1}^T{\left(y_t-{\varphi }_1(L)F\left(s_t;\gamma ,c\right)-{\varphi }_2(L)(1-F\left(s_t;\gamma ,c\right))\right)}^2.\text{\hspace{1em}(8.13)}$$

Here the M-estimator amounts to nonlinear least squares and the parameters are $\beta =({\varphi }_{11},\dots ,{\varphi }_{1p},{\varphi }_{21},\dots ,{\varphi }_{2p},\gamma ,c)$ . We could also assume that the errors are normal and employ maximum likelihood. The model is partially nonlinear with nonlinearities arising only through F . For given values of $(\gamma ,c)$ the optimization problem is linear, which enables the search problem to be reduced to a maximum of two dimensions.

Forecasts from STAR models can be computed in the same way as for the TAR models. For example, given a set of parameter estimates, $\hat{\mathbf{\beta }}$, the one-step-ahead plugin forecasts reduce to

$$\begin{array}{rl}& f_{t+1\mid t}(z)=\left({\hat{\varphi }}_{11}{y}_t+\cdot \cdot \cdot +{\hat{\varphi }}_{1p}{y}_{t-p+1}\right)F(s_t;\hat{\gamma },\hat{c})\\ & +\left({\hat{\varphi }}_{21}{y}_t+\cdot \cdot \cdot +{\hat{\varphi }}_{2p}{y}_{t-p+1}\right)(1-F(s_t;\hat{\gamma },\hat{c})).\end{array}$$

These forecasts are computed as a weighted average of the forecasts in each of the states, with weights dependent on $F\left(s_T;\hat{\gamma },\hat{c}\right)$ . Multistep forecasts must again be handled by numerical methods that account for the nonlinear effect of future innovations.

Smooth transition methods can also be used to model the local dynamics of a time series around exogenous variables such as a deterministic trend. For example, González, Hubrich, and Teräsvirta (2011) propose a local deterministic trend of the form

$$\mu (t)={\mu }_0+\sum _{i=1}^g{\delta }_{i}g({\gamma }_i,c_i,t/T),$$

where $t/T$ is the local point in time, measured as a fraction of the total sample, $T,{\mu }_i$ , and ${\gamma }_i>0$ are parameters, and $g$ is a logistic transition function,

$$g({\gamma }_i,c_i,t/T)=(1+\exp (-{\gamma }_i(t/T-c_i)){)}^{-1}.\text{\hspace{1em}(8.14)}$$

Here $c_1<\cdots <c_q$ are the points in time (as a fraction of the full sample) at which the deterministic component transitions from one level, ${\mu }_0+{\mu }_1+\cdot \cdot \cdot +{\mu }_{i-1}$ to the next, ${\mu }_0+{\mu }_1+\cdot \cdot \cdot +{\mu }_{i-1}+{\mu }_i,i\le q$ . When ${\gamma }_i$ is close to $0,$ the transition will be smooth and gradual, while a large value of ${\gamma }_i$ gives rise to a sudden shift, akin to a step function.