Heterogeneity and Dynamics Statistical Model

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