DYNAMICLINEAR Procedure
Simultaneous Graphical Dynamic Linear Models
The classical dynamic linear model (DLM) often falters when confronted with high-dimensional time series that are characterized by intricate correlations. Recognizing this limitation, Gruber and West (2016) and Gruber and West (2017) introduced an enhanced version known as a simultaneous graphical dynamic linear model (SGDLM). This advanced model is adept at efficiently managing large-scale, correlated time series.
Consider an m-dimensional time series that is denoted by 
. The SGDLM can be expressed as
In this representation, 
and 
. The expression 
denotes a vector that represents values of other series indexed by 
, where 
denotes the simultaneous parental set of series j. The vector 
denotes the vector of external predictors for series j at time t; it is important to note that 
might vary in dimensions across the m variables.
First, here are the definitions of certain matrices and vectors that are related to SGDLMs:
-
Let
represent the collection of 
values for every j, formally written as 
-
Define
as a diagonal matrix that contains 
values: 
-
is the vector given by 
where the individual components are defined as
.
The matrix 
captures the relationships between various time series. It is defined as
where 
for each 
.
Use these definitions to represent the distribution of 
as
where the matrices 
and 
are given by
and where 
is the precision matrix.
Finally, PROC DYNAMICLINEAR integrates the local-level dynamic linear model with the simultaneous graphical feature, as follows:
where 
for all j, and 
for all j and t.
Forward Filtering in SGDLMs
To understand the forward filtering process in an SGDLM, you need to understand the distributions at play. Here are the key distributions that this process uses. For more information about these distributions, see Gruber and West (2016) and Gruber and West (2017).
-
The prior at time t is given by
where
denotes the information set at time t; 
denotes the data set up to time 
; and 
denotes the density function of 
and
where
is a multivariate normal distribution that has the mean 
and the covariance matrix 
, and 
is a gamma distribution that has the shape parameter 
and the rate parameter 
. -
The naive posterior approximation at time t to the true posterior
is given by 
where
-
The true posterior at time t is
with the components
