Factor Augmented Dynamic Nelson–Siegel Model

• The model's main contribution is extending the Nelson–Siegel framework with a Wishart stochastic volatility process to capture time-varying multivariate volatility.
• It employs a Gaussian vector autoregression for latent factors and uses a collapsed-Gibbs sampler for efficient Bayesian estimation.
• Empirical results demonstrate improved in-sample fit and out-of-sample forecast performance in commodity futures risk management compared to baseline models.

The Factor Augmented Dynamic Nelson–Siegel (FADNS) model with Wishart stochastic volatility is a Bayesian state-space framework for modeling and forecasting the term structure of commodity futures, extending the classic Nelson–Siegel (NS) representation to incorporate time-varying multivariate volatility and multiple latent factors governing the dynamics of forward curves. The model builds on the dynamic 3-factor Nelson–Siegel model and its 4-factor Svensson extension, employing a Gaussian vector autoregression structure for the latent factors and a Wishart process to capture stochastic volatility. Its development enables robust estimation and uncertainty quantification for large cross-sections and long time series, yielding competitive performance for in-sample fit, out-of-sample prediction, and risk management in empirical applications such as WTI crude oil futures (Kleppe et al., 2019).

1. Model Structure: Measurement Equation

The observed data consist of daily log-prices $y_{i,t}$, where $i=1,\dots,N$ enumerates futures contracts with times-to-maturity $\tau_{i,t}$ at time $t$. The core observation model follows the dynamic Nelson–Siegel (NS) form: $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$ where $\beta_{1,t}$, $\beta_{2,t}$, $\beta_{3,t}$ denote the time-varying level, slope, and curvature factors.

The 4-factor Svensson extension augments this with a second curvature component: $$y_{i,t} = \beta_{1,t} + \beta_{2,t}\frac{1-e^{-\lambda_1\tau_{i,t}}}{\lambda_1\tau_{i,t}} + \beta_{3,t}\left( \frac{1-e^{-\lambda_1\tau_{i,t}}}{\lambda_1\tau_{i,t}} - e^{-\lambda_1\tau_{i,t}} \right) + \beta_{4,t}\left( \frac{1-e^{-\lambda_2\tau_{i,t}}}{\lambda_2\tau_{i,t}} - e^{-\lambda_2\tau_{i,t}} \right) + \varepsilon_{i,t}$$

Stacked notation gives: $$y_t = Z_t \beta_t + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \Sigma_y)$$ where $i=1,\dots,N$0 is the design matrix of NS or Svensson loadings.

2. Latent Factor Dynamics: Vector Autoregression

The evolution of latent factors $i=1,\dots,N$1 for $i=1,\dots,N$2 (NS) or $i=1,\dots,N$3 (Svensson) is governed by a first-order vector autoregression (VAR), typically set to a random walk: $i=1,\dots,N$4 where $i=1,\dots,N$5, $i=1,\dots,N$6 (commonly $i=1,\dots,N$7), and $i=1,\dots,N$8 is the time-varying precision matrix encoding the stochastic volatility. The generic transition density is

$i=1,\dots,N$9

which allows for parsimonious yet flexible modeling of factor persistence and interdependence.

3. Wishart Stochastic Volatility Specification

Multivariate time-varying volatility is introduced via a Wishart–Beta process (Uhlig 1994, 1997; Windle & Carvalho 2014), driving the evolution of the factor innovation precision matrix $\tau_{i,t}$0. Its transition equation in scaled-Beta form is

$\tau_{i,t}$1

where $\tau_{i,t}$2 (degrees of freedom) and $\tau_{i,t}$3 (scale) with the identification constraint $\tau_{i,t}$4 ensuring $\tau_{i,t}$5.

Initial precision $\tau_{i,t}$6 follows

$\tau_{i,t}$7

with user-selected positive-definite matrix $\tau_{i,t}$8.

4. Bayesian Posterior Inference and MCMC Estimation

The model is estimated in a fully Bayesian framework. The joint posterior for all parameters ($\tau_{i,t}$9), latent factors $t$0, and stochastic volatilities $t$1 is proportional to the product of Gaussian densities (likelihood and factor transition), the Wishart/Beta prior, and priors on remaining parameters: $t$2

Key conjugacies allow:

• Given $t$3, $t$4 form a linear-Gaussian state-space model with tractable Gaussian updates.
• Given $t$5, the Wishart specification produces a closed-form backward sampler for $t$6.
• Parameters $t$7 (Normal) and $t$8 (inverse gamma) have conjugate updates.

A collapsed-Gibbs sampler cycles through:

1. Jointly sampling $t$9: update $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$0 via random-walk Metropolis (marginalizing $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$1), update $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$2 with a sparse Gaussian precision sampler (as in Chan & Jeliazkov 2009).
2. Jointly sampling $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$3: update $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$4 via marginal likelihood, then backward-sample $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$5 from the shifted singular Wishart.
3. Standard updates for $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$6 and $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$7.

Posterior quantities are obtained by averaging across post-burn-in MCMC cycles.

5. Empirical Performance in Crude Oil Futures

An empirical application to 24 monthly WTI crude-oil log-futures (Jan 1996–May 2016, $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$8) compares several model variants: 3F (homoscedastic NS), 3F-SV (NS + Wishart SV), 4F (homoscedastic Svensson), and 4F-SV (Svensson + Wishart SV). Weakly-informative priors are specified.

Estimation and forecasting results include:

• Posterior means (4F-SV, second estimation window): $$y_{i,t} = \beta_{1,t} + \beta_{2,t} \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} + \beta_{3,t} \left( \frac{1-e^{-\lambda_1 \tau_{i,t}}}{\lambda_1 \tau_{i,t}} - e^{-\lambda_1 \tau_{i,t}} \right) + \varepsilon_{i,t}, \quad \varepsilon_{i,t} \sim N(0,\sigma_{y,i}^2)$$9, $\beta_{1,t}$0, $\beta_{1,t}$1, $\beta_{1,t}$2 (implying $\beta_{1,t}$3). The intercept $\beta_{1,t}$4 is close to zero, indicating factor processes are approximately unit-root random walks.
• Effective sample sizes for all parameters exceed 200.
• Deviance Information Criterion (DIC) ranks: 4F-SV best, followed by 4F, 3F-SV, then 3F.
• Out-of-sample log-predictive likelihoods (2008 window, 24 contracts, 576 dates): 3F–24,853; 3F-SV–25,019; 4F–26,034; 4F-SV–26,195. Thus, 4F-SV displays superior predictive density performance.
• One-day RMSE (2008 window): 3F–0.0290; 3F-SV–0.0290; 4F–0.0289; 4F-SV–0.0289 versus the random-walk benchmark of 0.0286.
• In VaR forecasting, 4F-SV produces the most accurate hit rates at 1%, 5%, and 10%, and passes unconditional/conditional coverage tests more consistently than alternatives.

Posterior smoothed volatility and cross-correlation estimates closely track rolling-window realized measures. Both curvature factors are empirically distinct: one captures longer-horizon term-structure curvature, while the second extracts a short-horizon “bump.”

6. Model Comparison and Interpretation

Model comparisons indicate that the 4-factor Svensson with Wishart stochastic volatility (4F-SV) outperforms lower-order or homoscedastic alternatives in both in-sample goodness-of-fit and out-of-sample density/point forecasting. The parsimonious stochastic volatility specification captures temporal variation and cross-sectional dependence among risk factors with high persistence ($\beta_{1,t}$5, $\beta_{1,t}$6), recovering both the level and complex curvature dynamics of the commodity forward curve.

A plausible implication is that incorporating Wishart SV enhances risk management and density forecasting capability compared to homoscedastic or classic factor models, notably improving Value-at-Risk assessment for portfolios including long–short (bull spread) positions.

Model Variant	Factors	SV Process	Best Use
3F	NS (level, slope, 1 curvature)	No	Baseline comparison
3F-SV	NS	Yes (Wishart)	Volatility modeling
4F	Svensson (2 curvature)	No	Enhanced fit
4F-SV	Svensson	Yes (Wishart)	Forecast/risk

7. Conclusions and Practical Value

The FADNS model with Wishart stochastic volatility, as formulated in Kleppe et al. (Kleppe et al., 2019), provides a flexible, computationally efficient, and Bayesian-coherent approach to modeling the term structure of commodity futures. The fully conjugate state-space and stochastic volatility structure enables robust inference even for large panels ($\beta_{1,t}$7, $\beta_{1,t}$8), with jointly estimated latent factors and time-varying covariance, and competitive or superior performance versus common benchmarks such as linear Gaussian state-space models and random walk processes. The empirical evidence suggests that the model is particularly effective for forecasting time-varying forward curves, volatility, cross-factor covariances, and portfolio risk in commodity markets.