Dynamic factor models and PCA on returns

Dynamic factor models reduce a high-dimensional panel of time series to a small number of unobserved factors. They are the time-series workhorse for risk modelling, term-structure decomposition, and macro forecasting — combining the dimension-reduction power of PCA with the temporal coherence of state-space models.

The static factor model

Y_t = B f_t + e_t
Y_t ∈ Rⁿ (observed),  f_t ∈ Rᵏ (factors),  B ∈ Rⁿˣᵏ (loadings),  e_t (idiosyncratic)

With k << n, the n × n covariance Σ_Y = B Σ_f Bᵀ + Σ_e is parsimoniously described. This is the static skeleton; the dynamic version adds time-series structure on f_t and possibly on e_t.

Dynamic factor model

f_t = A f_{t-1} + η_t,        η_t ~ N(0, Q)
Y_t = B f_t + e_t,            e_t ~ N(0, Σ_e)

Factors follow a VAR(1) (extensible to VAR(p)). Idiosyncratic errors are uncorrelated across components. Estimable by Kalman filter EM, by approximate principal-components extraction (Stock-Watson), or by Bayesian methods. The Stock-Watson approach scales to hundreds of series and is the basis of the Federal Reserve Bank of Philadelphia's coincident index.

Yield-curve factor models

Nelson-Siegel

y(τ) = β_1 + β_2 (1 - e^(-τ/λ)) / (τ/λ) + β_3 [(1 - e^(-τ/λ)) / (τ/λ) - e^(-τ/λ)]

Three factors with shape functions corresponding to level (β_1), slope (β_2), and curvature (β_3). The λ parameter is the curvature peak location. Fits curves remarkably well; widely used by central banks for curve estimation.

Dynamic Nelson-Siegel (Diebold-Li 2006)

Treat (β_1, β_2, β_3) as time-varying state vectors, each following an AR(1). Get short-rate-like forecasts: project β's forward, reconstruct the yield curve. Outperforms naive random-walk forecasts for the yield curve at most horizons.

Affine term-structure models

Theoretically-grounded counterpart to NS: assume no arbitrage and a small state vector x_t following Gaussian dynamics. Yields are affine in x_t under risk-neutral measure. The Ang-Piazzesi-Wright and Gürkaynak-Sack-Wright frameworks decompose the US Treasury curve into expected-rate paths and risk-premium components — the standard analytical tool of macro fixed-income desks.

Equity factor models in time-series form

Fama-French 3-factor returns as a VAR: the factors (Mkt, SMB, HML) have their own dynamics; firm-level returns load on the factors plus idiosyncratic noise. Time-varying factor exposures (rolling betas, Kalman-filtered betas) account for slow drift in firm characteristics. The basis of conditional asset pricing (e.g., Lettau-Ludvigson 2001).

PCA vs DFM

• PCA: extracts static factors from the full panel; treats observations as i.i.d.
• DFM: imposes time-series structure on factors; estimates factor dynamics; uses Kalman smoothing.
• For nowcasting: DFM is preferred because it handles mixed frequencies and ragged-edge data.
• For risk modelling: PCA is often enough; DFM is more elegant but only marginally better empirically.

Number of factors

Bai-Ng (2002) information criteria: ICp1, ICp2, ICp3. Compute the residual variance for k = 1, 2, 3, ... factors and pick the k minimising IC. Onatski (2010) provides a more powerful test based on the gap between successive eigenvalues. In practice: for monthly macro panels of 100-200 series, k = 4-6 is typical; for daily equity returns, k = 3-5 explains > 50% of variance.