Slides
Equation Balance
The article by Pickup and Kellstedt forms the basis for summary remarks on time series and a useful transition into the study of multiple time series formally for week two.
The lingering issue that does not get adequate treatment owing to time is fractional integration methods. Indeed, the article by Lebo and Grant (2016) summarising the issues in the 2016 special issue is worth digesting.
A Note on ARCH and GARCH
ARCH and GARCH Models
- Models of the conditional volatility (variance) in the errors
- Widely used in financial econometrics with volatility in an assets being taken as an indicator of risk
ARCH Model of order \(q\)
\[ e_t = \sigma_t Z_t \] \(Z_t\) = white noise \(\sigma_t\) = standard deviation
\[ \sigma^{2}_t = a_0 + a_1e^{2}_{t-1} + \ldots + a_{q}e^{2}_{t-q} \]
GARCH Model
\(p\) is order of the GARCH terms \(\sigma^2\)
\(q\) is order of ARCH terms \(e^2\)
\[\sigma^2_{t} = w + \underbrace{\Sigma_{i=1}^{q} a_i e^{2}_{t-i}}_{(ARCH components)} + \underbrace{\Sigma_{i=1}^{p} B_i \sigma^{2}_{t-i}}_{(GARCH components)} \]