Volatility modelling in time and space
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This thesis contributes to the scientific community in several aspects. We introduce both spatial- and spatio-temporal extensions to the family of GARCH and ARMA-GARCH models and present asymptotic statistics for the quasi maximum likelihood estimators [QMLE] for the GARCH extensions. An important property of these extensions are their spatial- and spatio-temporal stationarity, which is part of the model specifications. The models all exist on an equidistant d-dimensional grid, be it purely spatial or spatiotemporal. Volatility modelling is important in finance, but we also present applications from other fields of study, e.g. climate, meteorology and even cell biology. In stationary spatial statistics on infinite lattices, a boundary problem arises. This is dealt with, in two of the papers, by assuming a circular model. This means wrapping the spatial part of the grid of observation onto a torus surface by connecting opposing edges, and effectively removing the boundaries so that each site’s neighbours are observed. The torus space is good for visualization and the point is that we regard sites on opposite sides of the rectangle we observe as neighbours. Circulation changes the area of observation from infinite to being closed and finite, and proving asymptotic results becomes easier. Consistency and asymptotic normality of the QMLE is established in the circular situation for GARCH models. The circular model can be used as an approximation of an infinite grid model, in which the circular estimator will be biased. In this setting, we suggest a parametric bootstrap bias correction to compensate for the false links between boundary sites due to circulation. In simulation studies, this approach provides good results for both GARCH and ARMA-GARCH models. For ARMA-GARCH, it is not uncommon to fit an ARMA model to data and a GARCH model to its residuals, but simultaneously estimating all parameters is better. We show by a simulation experiment that the variance of the ARMA-part of the QMLE can be reduced by doing this. The second paper of this thesis is an application of non-stationary GARCH modelling in climate research. We investigate how volatility has developed in a daily temperature series at Svalbard Airport over the last 44 years. During this period the temperature there has increased intensively. We model the volatility using a GARCH model with a trend, where the slope depends on the day of the year. Except for the summer, we find a decreasing temperature variability, i.e. a negative trend. The temperature on Svalbard is getting higher and more stable at the same time and we believe this is due to the reduced sea ice extent in the region. Without the circulation, on an infinite grid and in a potentially purely spatial setting, we turn to half-space GARCH models in the final paper. These models use an ordering of the spatial locations, extending non-deterministic time series to space. The MLE used is based on a modified likelihood, and we show that it is consistent and asymptotically Gaussian. Instead of the standard Lyapunov condition for existence of a stationary solution, a generalization of Nelson’s criteria is used.
Has partsPaper A: Hølleland, S., & Karlsen, H. A. (2020). A Stationary Spatio‐Temporal GARCH Model. Journal of Time Series Analysis, 41(2), 177-209. The article is available in the thesis file. The article is also available at: https://doi.org/10.1111/jtsa.12498
Paper B: Hølleland, S., & Karlsen, H. A. (2020). Decline in temperature variability on Svalbard. Journal of Climate, 33(19), 8475-8486. The article is not available in BORA due to publisher restrictions. The published version is available at: https://doi.org/10.1175/JCLI-D-20-0174.1
Paper C: Hølleland, S., & Karlsen, H. A. Space-Time ARMA-GARCH models with applications. The article is not available in BORA.
Paper D: Karlsen, H. A. & Hølleland, S. Spatial GARCH processes. The article is not available in BORA.