Identifying yield stability over time via time-varying GARCH processes and multivariate Horseshoe priors

Little is known about how grassland yield stability changes over time, where yield stability reflects the variance in yield, with higher-yielding grassland typically being less stable. Although Time-Varying Generalised Autoregressive Conditional Heterogeneity (TV-GARCH) processes are used heavily in economics and the medical sciences, presented is an application of these methods to better understand how conditional heterogeneous terms change over time for grassland ecosystems using data from the Park Grass Experiment, Hertfordshire.

Time varying processes can be modelled as a smooth non-parametric function, where Horseshoe priors (Carvalho et al. 2010) have been developed to prohibit overfitting of smooth functions through localised shrinkage of smooth coefficients through a series of zero-centred Normal distributions. Although Horseshoe priors provide good estimates for smoothed objects by localised shrinkage, we investigate the effects of covariate shrinkage through multivariate Horseshoe priors where the covariance matrix of a zero-centred multivariate Normal prior is decomposed into a vector of standard deviations and a matrix of correlations (Barnard et al. 2000). Covariate shrinkage allows for pairs of smooth coefficients to assume larger or smaller values depending on the correlation between coefficients and the shape of the
overall smooth function. We develop a Bayesian TV-GARCH model using the variance-parameterised Gamma likelihood function (Addy et al. 2022). With this formalised modelling procedure we were able to identify time-varying changes in yield stability, providing novel insight into the underlying process and confirming previous findings of yield stability changing during the 1990s on Park Grass.