Aguilar and West (2000) Bayesian Dynamic Factor Models and Portfolio Allocation

Summary

Develops a class of $k$-factor dynamic models for $q$-dimensional financial time series in which latent factor variances and series-specific idiosyncratic variances each follow univariate stochastic volatility (SV) processes, with log factor volatilities jointly governed by a first-order vector autoregression (VAR(1)) to capture cross-factor volatility dependencies. Bayesian inference proceeds via a multi-block Markov chain Monte Carlo (MCMC) sampler combining the Kim-Shephard-Chib (KSC, 1998) log-SV mixture-of-normals approximation with forward filtering, backward sampling (FFBS) for joint state sampling; sequential analysis uses the Pitt-Shephard (1999a) auxiliary particle filter (APF) with parameters fixed at MCMC posterior means from a training window. Applied to daily returns on six international exchange rates (Jan 1992–Sep 1998, $n = 1{,}827$), the factor model demonstrates markedly better out-of-sample portfolio performance than the variance discounting baseline.

Key Claims

• Factor model structure: For $q$-dimensional returns $y_t$, the $k$-factor representation is $y_t = \theta + X f_t + v_t$ where $f_t \sim N(0, H_t)$, $v_t \sim N(0, \Psi_t)$, $H_t = \text{diag}(h_{1t}, \ldots, h_{kt})$ is the diagonal matrix of time-varying factor variances, and $\Psi_t = \text{diag}(\psi_{1t}, \ldots, \psi_{qt})$ is diagonal with series-specific idiosyncratic variances. The implied conditional covariance is $\Sigma_t = X H_t X' + \Psi_t$. Factor loadings matrix $X$ is $q \times k$ lower-triangular with unit diagonal (Geweke-Zhou 1996 identification constraint).

• Factor SV as VAR(1): Log factor volatilities $\lambda_t = (\lambda_{1t}, \ldots, \lambda_{kt})'$ with $\lambda_{jt} = \log h_{jt}$ follow a stationary VAR(1): $\lambda_t = \mu + \Phi(\lambda_{t-1} - \mu) + w_t$, $w_t \sim N(0, U)$, where $\Phi = \text{diag}(\phi_1, \ldots, \phi_k)$ and $U$ is an unrestricted positive-definite innovations covariance matrix. Off-diagonal entries of $U$ and the implied stationary covariance $W$ (with $W_{ij} = U_{ij}/((1-\phi_i)(1-\phi_j))$) capture co-movement in factor volatility processes. Idiosyncratic log-variances $\gamma_{it} = \log \psi_{it}$ follow independent first-order autoregressive (AR(1)) models: $\gamma_{it} = \alpha_i + \rho_i(\gamma_{i,t-1} - \alpha_i) + s_{it}$.

• MCMC sampler: Multi-block Gibbs / Metropolis-within-Gibbs cycle. (1) Factor process $\{f_t\}$ via FFBS (Carter-Kohn 1994) treating $H_t$ as known. (2) Log factor volatilities $\{\lambda_t\}$ via KSC (1998) mixture-of-normals approximation — defining $z_t = \log(f_t^2)$ gives a multivariate dynamic linear model (DLM) with $\log\chi^2$ errors approximated by a finite mixture of normals — then FFBS for the full vector path. (3) Factor loadings $X$ sampled equation-by-equation exploiting the lower-triangular constraint. (4) Idiosyncratic log-variances $\{\gamma_{it}\}$ via univariate KSC smoother for each $i$. (5) Factor volatility mean $\mu$ and persistence $\Phi$ via normal conjugate updates. (6) Innovations covariance $U$ via Metropolis-Hastings with inverse-Wishart proposal $W_r^{-1}(U|R)$ where $r = r_0 + n - 1$ and $R = r_0 R_0 + (n-1)G$; acceptance probability $\min\{1, \alpha(U^*)/\alpha(U)\}$ with target $\alpha(U) \propto |W|^{-1/2}\exp(-\tfrac{1}{2}\text{tr}(W^{-1}V))$. (7) Mean return $\theta$ and idiosyncratic SV parameters $(\alpha_i, \rho_i, s_i^2)$ via standard conjugates.

• Variance discounting baseline: Exponential-smoothing estimate $S_t = (1 - a_t)S_{t-1} + a_t y_t y_t'$ with $a_t = 1/(1 + \delta n_{t-1})$ and discount factor $\delta \in (0.9, 1)$; backward-smoothed by $S_{t|n}^{-1} = (1-\delta)S_t^{-1} + S_{t+1|n}^{-1}$ (West-Harrison 1997, p. 608–609). Computationally trivial, over-smooths volatility peaks, and imposes no factor structure.

• Sequential portfolio allocation: Parameters $\{\theta, X; \mu, \Phi, U; \alpha_i, \rho_i, s_i\}$ fixed at MCMC posterior means from training observations $t = 1, \ldots, T_0 = 1{,}000$. The Pitt-Shephard (1999a) auxiliary particle filter then propagates a particle cloud through $\{H_t, \Psi_t\}$ at each new observation, delivering one-step-ahead $g_t = E(y_t|D_{t-1})$ and $G_t = V(y_t|D_{t-1})$ for mean-variance portfolio construction. The "honest" strategy is validated by the near-identical parameter estimates from MCMC on training data alone vs. the full series.

• Exchange-rate application ($q=6$ currencies DEM/GBP/JPY/FRF/CAD/ESP, $k=3$ factors): DEM and GBP dominate factor structure; JPY volatility is largely idiosyncratic; CAD is almost entirely idiosyncratic (low loadings on all factors); FRF and ESP loadings near unity on the DEM factor. Britain's Exchange Rate Mechanism (ERM) exit (late 1992) and Japan's 1995 rate changes produce clearly identified factor volatility spikes absent in smoothed discount trajectories. Over 827 out-of-sample days, the unconstrained factor-model portfolio outperforms the unconstrained discount portfolio; both beat equal-weight; unconstrained beats unit-sum-constrained by approximately 4$\times$.

• Factor number diagnostic: Running MCMC with too many factors (e.g., $k=3$ when data support $k=2$) causes poor convergence as the excess factor collapses toward zero — a practical signal for over-parameterisation.

Concepts Introduced or Extended

• Factor Model
• Dynamic Factor Model
• Stochastic Volatility
• Gibbs Sampler
• Metropolis-Hastings

Entities Mentioned

• Omar Aguilar
• Mike West
• Neil Shephard
• Kim-Shephard-Chib (1998)

Quotes

"The models are direct generalizations of univariate stochastic volatility models and represent specific varieties of models recently discussed in the growing multivariate stochastic volatility literature." (p. 338)

"The discount method produces more heavily smoothed trajectories, pointing to one of the major known drawbacks of that method and so obscures some of the evident peaks in volatility related to major economic changes and events." (p. 349)

"Simultaneous sequential inference on the factor-model parameters... as well as all the time-evolving volatility processes... is an open research problem." (p. 343)

My Take

The central contribution is making multivariate SV tractable by imposing a lower-triangular factor structure and routing the MCMC through the KSC mixture approximation — an elegant marriage of Geweke-Zhou factor identification with the Kim-Shephard-Chib univariate SV sampler, extended here to a VAR(1) on log factor volatilities. Chib-Nardari-Shephard (2006) directly extends this line with reduced blocking for higher dimensions ($p=50$). The APF-based sequential updating is pragmatic: fixing parameters at MCMC means is an honest strategy that works when parameters are stable, as confirmed empirically here, but provides no guarantees in general; Liu-West (2001) and later particle MCMC partially address the simultaneous learning problem. The portfolio comparison is suggestive but limited — one data set, one time window, no transaction costs, no statistical inference on the return differences. The factor number choice problem (fixed throughout) remains a genuine gap.