8.1 THRESHOLD AUTOREGRESSIVE MODELS

8.1 THRESHOLD AUTOREGRESSIVE MODELS

Consider an extension of the $\mathrm{AR}(p)$ model,

$$y_{t+1}={\varphi }_1{y}_t+\cdot \cdot \cdot +{\varphi }_p{y}_{t-p+1}+\sigma {\epsilon }_{t+1},{\epsilon }_{t+1}\sim (0,1),$$

where the coefficients, including the variance of the residuals, can vary across different ranges of $y_{t-d}$ , for some $d\ge 0;$

$$y_{t+1}=\{\begin{array}{ll}{\varphi }_{11}{y}_t+\cdot \cdot \cdot +{\varphi }_{1p}{y}_{t-p+1}+{\sigma }_1{\epsilon }_{t+1}& \mathrm{if}{y}_{t-d}\in S_1,\\ ⋮& ⋮\\ {\varphi }_{k1}{y}_t+\cdot \cdot \cdot +{\varphi }_{kp}{y}_{t-p+1}+{\sigma }_k{\epsilon }_{t+1}& \mathrm{if}{y}_{t-d}\in S_k.\end{array}\text{\hspace{1em}(8.3)}$$

Here the states $S_i$ cover all possible values for ${\gamma }_{t-d}$ , the endpoints of which are known as thresholds. For example, for $k=2$ we might have $S_1=(-\infty ,s]$ and $S_2=(s,\infty )$ so the threshold is s. Models such as (8.3) are called threshold autoregressive (TAR) or “self-exciting ${\mathrm{TAR}}^{\mathfrak{n}}$ (SETAR) since the threshold depends on ${\gamma }_{t-d}$ as opposed to some exogenous variable.

When the thresholds, $S_1,\dots ,S_k$ , and $d$ are known, the extension over the standard linear model is to simply split the sample into $k$ fully defined groups. Parameter estimation is then straightforward, since least squares estimation can be applied separately to each group. Alternatively, if the errors are assumed to be normally distributed, one can use maximum likelihood estimation, which would lead to efficiency gains especially if there are parameter restrictions across the states, $e.g.$ if ${\sigma }_i=\sigma$ for $i=1,\dots ,k$

If the thresholds are unknown, this widens the class of models to search over. However, the thresholds can be and often are estimated. In a model with $k=2$ and $b{p}_1=b{p}_2$ , Chan (1993) showed that the threshold value, s , and the coefficients in each state can be consistently estimated when ${\gamma }_t$ is ergodic and strictly stationary. Moreover, the threshold estimate converges at rate $T_{\delta }$

One-step-ahead forecasts are easily computed from the TAR model. Suppose $k=2,p=1$ , and $d=0$ . Then, from (8.3) we have

$$f_{t+1|t}=[({\varphi }_{11}-{\varphi }_{21})1(y_t\le s)+{\varphi }_2]y_t,\text{\hspace{1em}(8.4)}$$

where 1 $(y_t\le s)$ is an indicator variable that is known at time $t,$ and so the forecast is conditionally linear in ${\gamma }_t$

Multistep forecasts are not as easily computed since they depend on future values taken by Y. Continuing with our earlier example, the two-step-ahead forecast depends on whether $y_{t+1}\le s:$

$$y_{t+2}=\{\begin{array}{ll}{\varphi }_{11}{y}_{t+1}+{\sigma }_1{\epsilon }_{t+2}& \mathrm{if}{y}_{t+1}\le s,\\ {\varphi }_{21}{y}_{t+1}+{\sigma }_2{\epsilon }_{t+2}& \mathrm{if}{y}_{t+1}>s.\end{array}\text{\hspace{1em}(8.5)}$$

This means that

$$f_{t+2|t}=E\left[\left[\left({\varphi }_{11}-{\varphi }_{21}\right)1\left(y_{t+1}\le s\right)+{\varphi }_{21}\right]y_{t+1}|Z_t\right],\text{\hspace{1em}(8.6)}$$

where $Z_t=\{y_1,\dots ,y_t\}$ . This expression depends on the future innovation, ${\epsilon }_{t+1}.$ which affects both the indicator function $1(y_{t+1}\le s)$ and $y_{t+1}$ and so $f_{t+2|t}$ is a nonlinear function of $y_{t+1}$ . To evaluate the expectation in (8.6) we need the distribution of the innovation, ${\epsilon }_t$ . Popular methods include Monte Carlo simulation or the bootstrap. The former can be used if the distribution of $\epsilon$ is known up to a set of estimated parameters, while the latter involves redrawing with replacement from the sample of residuals, $\{{\hat{\epsilon }}_1,\dots ,{\hat{\epsilon }}_t\}$ . Either method seeks to numerically evaluate the integral

$$\int [({\varphi }_{11}-{\varphi }_{21})1(y_{t+1}\le s)+{\varphi }_{21}]y_{t+1}P(y_{t+1}|Z_t)d{y}_{t+1}\text{\hspace{1em}(8.7)}$$

through some average

$$\frac{1}{B}\sum _{b=1}^B[({\varphi }_{11}-{\varphi }_{21})1(y_{t+1}^b\le s)+{\varphi }_{21}]y_{t+1}^b,\text{\hspace{1em}(8.8)}$$

where $y_{t+1}^b$ is the bth draw of $y_{t+1}$ generated by drawing a new innovation, ${\mathbf{\epsilon }}_{t+1}^b$ , and computing

$$y_{t+1}^b=[({\varphi }_{11}-{\varphi }_{21})1(y_t\le s)+{\varphi }_{21}]y_t+[({\sigma }_1-{\sigma }_2)1(y_t\le s)+{\sigma }_2]{\epsilon }_{t+1}^b.$$

Similarly, if a sequence of draws $\{{\epsilon }_{t+1}^b,{\epsilon }_{,t+2}^b,\dots ,{\epsilon }_{t+h-1}^b{\}}_{b=1}^B$ are available, we can recursively generate values of $y_{t+h}^b$ from $y_{t+h-1}^b.$

$$y_{t+h}^b=[({\varphi }_{11}-{\varphi }_{21})1(y_{t+h-1}^b\le s)+{\varphi }_{21}]y_{t+h-1}^b+[({\sigma }_1-{\sigma }_2)1(y_{t+h-1}^b\le s)+{\sigma }_2]{\epsilon }_{t+h-1}^b.$$

Finally, the forecast of $y_{t+h}$ can be computed as

$${\bar{f}}_{t+h|t}=\frac{1}{B}\sum _{b=1}^B{y}_{t+h}^b.\text{\hspace{1em}(8.9)}$$

Setting the number of draws, $B$, to some large number should generally give a good approximation to the forecast.