B. Heterogeneity and Dynamics
We characterize the dynamics of volatility parameters,
i,t 2
, using a discrete nonpara- metric approach from Jensen and Shore 2011. In such a model, the variable of inter-
est— here, the pair
i,t 2
≡
,i,t 2
,
,i,t 2
—can take one of L possible values where the number of these values and the values themselves are estimated from the data. The
probability that
i,t 2
takes a given value is a function of a the distribution of values in the population, b the distribution of values for each individual i, and c the number
of consecutive years with the most recent value. In other words,
i,t 2
has a given prob- ability of changing from one year to the next; when it changes, it changes to a value
drawn from the individual’s distribution, which in turn consists of values drawn from the population distribution.
We add structure and get tractability by adding a prior commonly used in Bayesian analysis of such discrete nonparametric problems: the Dirichlet process DP prior. In
a standard DP model, there is a “tuning parameter” that implicitly places a prior
on the total number of unique parameter values in the sample, L. In a hierarchical DP HDP model recently developed by Teh, Jordan, Beal, and Blei 2007, the usual DP
model is extended by adding a second tuning parameter,
i
, which implicitly places a prior on the total number of unique parameter values for any given individual, L
i
. Jensen and Shore 2011 extend this approach further to address panel data by includ-
ing a Markovian structure on the hierarchical DP giving us a Markovian hierarchical DP MHDP model. In this Markovian approach, the prior probability that the param-
eter is unchanged from the previous period depends on the number of consecutive years, Q
i ,t
, with that value. We add a third tuning parameter, , to place a prior on the
probability of changing the parameter value, p
i,t 2
=
i,t −1
2
| i, t = Q
i,t
+ Q
i,t
.
13
Given our research question, a key advantage of this setup is that it does not restrict the shape or the evolution of the shape of the cross- sectional volatility distribution.
We view our discrete nonparametric model and the structure placed on it by our MHDP prior as providing a sensible middle ground between tractability and fl ex-
ibility.
C. Estimation