Generalized Dynamic Factor Models for Mixed-Measurement Time Series

Abstract

In this article, we propose generalized Bayesian dynamic factor models for jointly modeling mixed-measurement time series. The framework allows mixed-scale measurements associated with each time series, with different measurements having different distributions in the exponential family conditionally on time-varying latent factor(s). Efficient Bayesian computational algorithms are developed for posterior inference on both the latent factors and model parameters, based on a Metropolis Hastings algorithm with adaptive proposals. The algorithm relies on a Greedy Density Kernel Approximation (GDKA) and parameter expansion with latent factor normalization. We tested the framework and algorithms in simulated studies and applied them to the analysis of intertwined credit and recovery risk for Moody’s rated firms from 1982–2008, illustrating the importance of jointly modeling mixed-measurement time series. The article has supplemental materials available online.

Measurement invariance within and between individuals: a distinct problem in testing the equivalence of intra- and inter-individual model structures

“…First, we based our modeling on the linear, time-invariant Kalman filter and ML estimation which led to time-invariant time series models. Time-varying model parameters can—to some extent—be accommodated using the extended Kalman filter (e.g., Chow et al, 2011 ; Chow and Zhang, 2013 ) or a Bayesian approach (e.g., Del Negro and Otrok, 2008 ). Second, we employed a multi-group approach, i.e., a two-step procedure to address inter-individual differences in intra-individual dynamics.…”

Serial correlation structures in latent linear mixed models for analysis of multivariate longitudinal ordinal responses

“…The detailed derivations of these properties are given in Section A in Appendix S1. In summary, the proposed LLMM consists of specifications (1), (4), (5), (6), and (10), where Ω 1 and Γ are parameters satisfying the constraints (7), (8), and (9). Main parameters of interest include ∶= vec( T ) for the mean structure and the transition matrix e −ΓΔt for the dynamic structure.…”

“…• Fit the proposed model without any random effects and serial structure but with fixed covariates (to remove the effects of covariates 9,32 ) at level 2, that is, the proposed model now consists of (1), (4), and (6) where B i z ij ≡ 0 in (4) and ij(1) ≡ 0 in (6). In other words, the model at level 2 is given by:…”

“…1 While these impairments over time are not directly observed, they could be incorporated in any model developed for observed longitudinal data. [2][3][4][5][6][7][8] Specifically, Wang and Luo 5 fit a random intercept model with independent within-subject residuals to an ALS dataset to examine the treatment effect on the latent neurological functions. However, the model makes a strong and probably unrealistic assumption, that is, the correlations between latent repeated measurements is constant, regardless of time interval.…”

Bayesian Factorized Cointegration Analysis

“…Thus, the maximum autoregressive order for both latent factors are 8. For efficient MCMC sampling, priors are placed on parameters of the Parameter-eXpanded (PX) model as described by [7] and [8]. 30,000 posterior samples are drawn for model parameters, latent initial values and latent factors via MCMC after a 5000 iteration burn-in.…”