Dynamic Factor Model

Definition

A dynamic factor model (DFM) represents a $q$-dimensional time series $y_t$ as a linear combination of $k < q$ latent factor processes plus idiosyncratic noise: $y_t = \theta + X f_t + v_t$, where $X$ is a $q \times k$ factor loadings matrix, $f_t \sim N(0, H_t)$ with time-varying diagonal $H_t$, and $v_t \sim N(0, \Psi_t)$ with diagonal idiosyncratic variance matrix $\Psi_t$. The conditional covariance decomposes as $\Sigma_t = X H_t X' + \Psi_t$ (low-rank-plus-diagonal). Dynamic models extend static factor analysis by allowing $H_t$ and $\Psi_t$ to evolve over time as stochastic processes.

Key Ideas

• Dimensionality reduction: Reduces free covariance parameters from $O(q^2)$ to $O(qk)$ loadings plus $q$ idiosyncratic variances plus $k$ factor variance processes — critical for inference when $q$ is large.
• Identification (Geweke-Zhou 1996): The loadings matrix $X$ is generically unidentified up to rotation ($X f_t = (XR)(R^{-1}f_t)$ for orthogonal $R$). The standard convention fixes $X$ to be lower-triangular with unit diagonal, requiring a meaningful ordering of the $q$ series (e.g., placing the "dominant" series first).
• Stochastic Volatility (SV) extension: Factor log-variances $\lambda_{jt} = \log h_{jt}$ follow a vector autoregression (VAR(1)): $\lambda_t = \mu + \Phi(\lambda_{t-1} - \mu) + w_t$, $w_t \sim N(0, U)$. Off-diagonal entries of $U$ capture co-movement in factor volatility processes. Idiosyncratic log-variances follow independent autoregressive (AR(1)) models (Aguilar-West 2000).
• GARCH extension: Factor variances can follow GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) processes (Engle-Ng-Rothschild 1992 factor-ARCH; Demos-Sentana 1998). Easier to estimate by Maximum Likelihood Estimation (MLE) but less flexible for Bayesian analysis.
• Relationship to state-space: DFMs are special cases of state-space models with $f_t$ as the state vector; Forward Filtering Backward Sampling (FFBS, Carter-Kohn 1994) applies directly to draw $\{f_t\}$ in one block given the variance paths.
• Connection to Cholesky discounting: The variance discounting baseline (West-Harrison 1997) recovers a $k=q$ factor structure via the Cholesky decomposition of the smoothed estimate $S_t$, but with no parsimonious factor reduction.

How It Works

1. Set $k$: Choose the number of factors guided by the Cholesky decomposition of the sample covariance or subject-matter knowledge. Excess factors cause Markov Chain Monte Carlo (MCMC) convergence failure (a useful diagnostic).
2. Order series: Place the "dominant" series first so that the lower-triangular constraint is substantively interpretable.
3. MCMC (Aguilar-West scheme):
   • Draw latent factor paths $\{f_t\}$ via FFBS treating $H_t, \Psi_t$ as given.
   • Draw log-volatility paths via Kim-Shephard-Chib (KSC, 1998) mixture-of-normals + FFBS for the multivariate dynamic linear model (DLM).
   • Draw loadings $X$ equation-by-equation exploiting the lower-triangular constraint.
   • Draw idiosyncratic SV parameters via univariate KSC smoother.
   • Draw $U$ via Metropolis-Hastings with inverse-Wishart proposal.
4. Sequential updating: Fix parameters at MCMC posterior means from a training window; use the auxiliary particle filter (Pitt-Shephard 1999) to update volatility states in real time.

Why It Matters

• Portfolio allocation: One-step-ahead $G_t = V(y_t|D_{t-1}) = X\hat{H}_t X' + \hat{\Psi}_t$ feeds directly into mean-variance optimization; the factor structure makes matrix inversion tractable via the Woodbury identity.
• Interpretability: Factors may correspond to economically meaningful latent drivers (e.g., a "European currency factor" dominating DEM/FRF/ESP co-movements).
• Selective intervention: Named factors allow targeted prior interventions when one component is expected to change (e.g., an anticipated central bank policy shift affects only the JPY factor, not DEM).
• Volatility peak sharpness: Unlike the exponential-smoothing (discounting) baseline, the SV factor model captures sharp volatility spikes (e.g., Britain's 1992 Exchange Rate Mechanism (ERM) exit) without over-smoothing.

Open Questions

• Number of factors: $k$ is typically fixed; Lopes-West (1998) developed MCMC for joint inference on $k$ in static models; extension to dynamic SV factors remains open.
• Sequential parameter learning: Fixing parameters at MCMC means is a pragmatic approximation; simultaneous sequential learning on both parameters and states is computationally hard (Liu-West 2001 partially addressed this; particle MCMC methods provide a fuller solution).
• Scaling: Chib-Nardari-Shephard (2006) extends to $p=50$ with a reduced blocking scheme; beyond $\sim 100$ series, full DFM MCMC becomes prohibitive.

Related

• Factor Model
• Bańbura-Giannone-Reichlin (2010) Large Bayesian VARs vs. FAVAR
• Stochastic Volatility
• Gibbs Sampler
• State-Space Representation
• Aguilar and West (2000)
• Mike West
• Omar Aguilar
• Chib-Nardari-Shephard (2006)