DYNAMICLINEAR Procedure
Discount Factors in Dynamic Linear Models
In accordance with the exposition of Petris, Petrone, and Campagnoli (2009), the discount factor 
signifies the proportion of the state evolution error variance to be accounted for during the transition from the posterior to the prior distribution. The state evolution is mathematically represented as
where 
is the state at time t, 
is the state transition matrix, 
is the process noise, 
is the covariance of the prior normal-gamma distribution at time t, and 
is the covariance of the posterior at time 
. This leads to an interpretation of 
as a fraction of the term 
:
which suggests that adjusts the covariance of the state evolution error to reflect the selected proportion of variance.
Petris, Petrone, and Campagnoli (2009) further note that, in practical settings, the discount factor is typically selected within the range of 0.9 to 0.99. A value closer to 1 indicates a higher confidence in the model’s stability over time, whereas a value closer to 0.9 suggests greater variability and less certainty in the persistence of the state’s dynamics.
The stochastic volatility discount factor, which is denoted by 
, plays a critical role in the evolution of the model by updating the precision 
of the observation error 
as you transition from the posterior to the prior distribution. To illustrate, if the gamma distribution of the posterior normal-gamma distribution at time is characterized by a shape parameter 
and a rate parameter 
, then the transition that is influenced by yields the parameters of the prior gamma distribution at time t as 
for the shape and 
for the rate. This reflects how modulates the degree of belief that is carried over from one time step to the next in the sequence of observations.