DYNAMICLINEAR Procedure

Gruber and West (2016) posit that in practical scenarios, models are likely to have small parental sets, denoted as , leading to a sparse matrix. This sparsity in often suggests a correspondingly sparse structure in the precision matrices . The connection between zero entries in and conditional independencies within the Gaussian graphical models is discussed by Carvalho and West (2007). This foundational concept informs the nomenclature of this model class as simultaneous graphical dynamic linear models. In cases where the number of variables m is even moderate, the parental sets tend to be relatively limited in scope. If k represents the maximum size of any parental set, there would be mk nonzero entries in . Therefore, a condition where ensures that avoids overparameterization.

The SGDLM, as implemented in PROC DYNAMICLINEAR, is specifically designed for adaptability and flexibility. Unlike fixed models, it allows the parental sets to vary over time, a feature that is crucial for evolving in response to data. This means that candidates of parental sets can differ across various time points and even for different variables. Such flexibility enables the model to effectively adapt to data, capturing the dynamics of the system under study.

Gruber and West (2017) provide an extensive discussion on the adaptability of SGDLMs. They find that smaller parental sets are generally preferable, provided that they are supported by data and predictive performance, thereby favoring increased sparsity. In larger-scale problems, a set of, for example, 20 parents for one series over a short time period might be practically indistinguishable from other candidate sets in which some parents are replaced by strongly collinear alternatives. The objective is to identify compact parental sets and adaptively revise them over time. This approach is essential for capturing and characterizing the structure and dynamics of resulting volatility matrices in terms of precision and variance. For example, in the study by Gruber and West (2017), a portfolio optimization problem that involves 401 assets is analyzed, using 20 parents for each series.