17.2 Connection with ARCH and GARCH

The scale model is not a new invention. The literature knows models that focus on modelling the dynamics of the scale of distribution or, more specifically, of the second moment (e.g. variance). Engle (1982) proposed an Autoregressive Conditional Heteroscedasticity (ARCH) model to capture the time-varying variance using lagged values of the squared error term. Bollerslev (1986) expanded the idea by also including lagged values of variance, introducing a Generalised ARCH (GARCH), which can be formulated as: \[\begin{equation} \sigma_t^2 = a_0 + a_1 \sigma_{t-1}^2 + \dots + a_q \sigma_{t-q}^2 + b_1 \epsilon_{t-1}^2 + \dots + b_p \epsilon_{t-p}^2 , \tag{17.9} \end{equation}\] where \(\epsilon_t\sim \mathcal{N}(0,\sigma_t^2)\) and \(a_j\) and \(b_j\) are the parameters of the model. Bollerslev (1986) argue that GARCH, being equivalent to ARMA, will be a stationary process if its parameters are restricted so that \(\sum_{j=1}^p a_j + \sum_{j=1}^q b_j < 0\) and \(a_j, b_j \in [0,1)\) for all \(j\). The restriction on parameters guarantees that the resulting values of \(\sigma_t^2\) are positive, but as Pantula (1986) noted such parameter space might be too restrictive. To make sure that the predicted variance is always positive, Geweke (1986) and Pantula (1986) suggested to build GARCH in logarithms, leading to the log-GARCH model: \[\begin{equation} \log \sigma_t^2 = a_0 + a_1 \log \sigma_{t-1}^2 + \dots + a_q \log \sigma_{t-q}^2 + b_1 \log \epsilon_{t-1}^2 + \dots + b_p \log \epsilon_{t-p}^2 . \tag{17.10} \end{equation}\] Pantula (1986) pointed out that the model (17.10) is equivalent to ARMA(p,q) applied to logarithms of squared error \(\epsilon_t^2\). In our notations it can be written as: \[\begin{equation} \log \epsilon_t^2 = a_0 + b_1^\prime \log \epsilon_{t-1}^2 + \dots + b_p^\prime \log \epsilon_{t-p}^2 + a_1^\prime \log \eta_{t-1}^2 + \dots + a_q^\prime \log \eta_{t-q}^2 + \log \eta_t^2, \tag{17.11} \end{equation}\] which can be obtained by substituting \(\log \sigma_{t-j}^2 = \log \epsilon_{t-j}^2 -\log \eta_{t-j}^2\) for all \(j\). In this case \(b_j^\prime = b_j + a_j\) and \(a_j^\prime=-a_j\) for all \(j\). Given the connection of log-ARMA with log-GARCH and the discussion in Subsection 9.1.4, the model (17.11) is just a special case of the scale model for ADAM, being a special case of (17.8). ADAM Scale Model not only supports the ARMA elements, but it also allows for explicit time series components modelling in the variance and the usage of explanatory variables.

Remark. GARCH and log-GARCH typically assume that the error term follows Normal distribution: \(\epsilon_t \sim \mathcal{N}(0, \sigma_t^2)\) – and model the variance. At the same time, ADAM Scale Model works with several other distributions, discussed in Subsection 17.1.3 and is focused on modelling scale (variance is a special case of scale).

Furthermore, it can be shown that ETS(M,N,N) applied to the \(\epsilon_t^2\) can be considered as a special case of GARCH(1,1) with the restriction on parameter \(a_1=1-b_1\), \(b_1= \alpha\) and \(a_0=0\), because the latter becomes equivalent to SES (Geweke, 1986): \[\begin{equation} \sigma_t^2 = (1-\alpha) \sigma_{t-1}^2 + \alpha \epsilon_{t-1}^2 . \tag{17.12} \end{equation}\] The connection of SES and ETS(M,N,N) has been discussed in Subsection 4.3.2.

References

• Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics. 31, 307–327. https://doi.org/10.1016/0304-4076(86)90063-1

• Engle, R.F., 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica. 50, 987. https://doi.org/10.2307/1912773

• Geweke, J., 1986. Comment. Econometric Reviews. 5, 57–61. https://doi.org/10.1080/07474938608800097

• Pantula, S.G., 1986. Comment. Econometric Reviews. 5, 71–74. https://doi.org/10.1080/07474938608800099