In the GARCH(1,1) model,

\[\sigma_{t+1}^2 = \alpha_0 + \alpha_1 r_t^2 + \beta_1\sigma_{t}^2\]

and the one step forecast will be

\[\sigma_{t,1}^2 = \alpha_0 + \alpha_1r_t^2 + \beta_1\sigma_t^2\]

and two step forecast will be

\[\sigma_{t+2}^2 = \alpha_0 + (\alpha_1+\beta_1) \sigma_{t+1}^2+\alpha_1\sigma_{t+1}^2(\epsilon_{t+1}^2 -1) = \alpha_0+ (\alpha_1+\beta_1)\widehat{\sigma_{t,1}^2}\]

Recursively, we have

\[\begin{aligned} \widehat{\sigma_{t,l}^2} &= \alpha_0 + (\alpha_1 + \beta_1) \widehat{\sigma_{t,l-1}^2} \\ &= \alpha_0 \frac{1-(\alpha_1+\beta_1)^{l-1}}{1-(\alpha_1+\beta_1)} + (\alpha_1+\beta_1) \widehat{\sigma_{t,1}^2} \end{aligned}\]

as $l\rightarrow \infty$,

\[\widehat{\sigma_{t,l}^2} \rightarrow \frac{\alpha_0}{1-(\alpha_1+\beta_1)} = \mathbb{E}[r_t^2]\]

Nonlinear time series:

The general form is \(x_t = f(\epsilon_t, \epsilon_{t-1},\dots)\)

where $\epsilon_t$ iid. This is the Bernoulli shift process.

The form is too general. Define $\mathcal{F}t = \sigma(\epsilon_t, \epsilon{t-1},\dots)$ , then define

\(\mu_t = \mathbb{E}[x_t|\mathcal{F}_{t-1}] = g(\mathcal{F}_{t-1})\) we can also define

\[\sigma_t^2 = \text{Var}[x_t|\mathcal{F}_{t-1}] = h(\mathcal{F}_{t-1})\]

Then, we are interested in the time series that is of the form:

\[x_t = g(\mathcal{F}_{t-1}) + h(\mathcal{F}_{t-1}) \epsilon_t\]

This include ARMA, ARCH, GARCH.

If $x_t$ is weakly stationary with zero mean, then we can write $x_t$ as

\[x_t = \sum_{i=0}^\infty \psi_i \gamma_t\]

where $\gamma_t$ are uncorrelated but dependent. This is called the Wold decomposition. (time domain decomposition, corresponding to the frequency domain decomposition)

Threshold AR(TAR) model:

THe model is \(x_t = \phi^{(j)}_0 + \phi^{(j)}_1 x_{t-1} + \dots + \phi^{(j)}_p x_{t-p} + \epsilon_t\) with $j = 1,\dots, k$ regimes if $r_{j-1}\leq x_{t-d} < r_j $

Issue : $\mu_t = \mathbb{E}(x_t \mathcal{F}{t-1})$ is not a continuous function at the boundaries $(r_t){1}^k$.

For example

\[x_t = \left\{ \begin{aligned} &-1.5x_{t-1} + \epsilon_t \quad x_{t-1}<1 \\ &0.5x_{t-1} + \epsilon_t \quad x_{t-1}\geq 1 \end{aligned} \right.\]

Thus, the conditional mean is

\[\begin{aligned} \mu_t &= \mathbb{E}[(-1.5 x_{t-1} + \epsilon_t) I(x_{t-1}<1) + (0.5 x_{t-1}+\epsilon_t)I(x_{t-1}\geq 1) | \mathcal{F}_{t-1}] \\ &= -1.5 x_{t-1} I(x_{t-1} < 1) + 0.5 x_{t-1} I(x_{t-1}\geq 1) \end{aligned}\]

at $x_{t-1} = 1$, the exepcted mean jumps from -1.5 to 0.5 and makes the estimation at the boundary really hard.

In order to address the issue, we could add some smoothness in the jump and we introduce the STAR(smoothe TAR) model.

k-regime SETAR model:

\(x_t = \phi_0^{(i)} + \phi_1^{(j)} x_{t-1} + \dots + \phi^{(p)} x_{t-p} + \epsilon_t\) and STAR model: \(x_t = \epsilon_t + \left(c_0 + \sum_{i=1}^p\phi_{0,i} \epsilon_{t-i} \right)\)

as the baseline model, then we add transition

\[x_t = \epsilon_t + \left(c_0 +\sum_{i=1}^p \phi_{0,j} \epsilon_{t-i}\right) + K\left(\frac{x_{t-d-\Delta}}{s}\right) \left(c_1 + \sum_{i=1}^p \phi_{j,1} \epsilon_{t-j}\right)\]

and $d$ is the delay parameter, and $\Delta$ is the model transition parameter. $s$ is the scale parameter and $K$ is the kernel. This is the $k=2$ example.


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