## Wednesday, January 15, 2014

### DNS/AFNS Yield Curve Modeling FAQ's

It's hard to believe that I haven't yet said anything about yield-curve modeling and forecasting in the dynamic Nelson-Siegel (DNS) tradition, whether the original Diebold-Li (2006) DNS version or the Christensen-Diebold-Rudebusch (2011) arbitrage-free version (AFNS). Here are a few thoughts about where we are and where we're going, expressed as answers to FAQ's, drawn in part from the epilogue of a recent book, Diebold and Rudebusch (2012).

1. What's wrong with unrestricted affine equilibrium models?

The classic affine equilibrium models, although beautiful theoretical constructs, perform poorly in empirical practice. In particular, the maximally-flexible canonical $$A_0(N)$$ models have notoriously recalcitrant likelihood surfaces. (Notation: $$A_x(N)$$ means a model with $$N$$ factors, $$x$$ of which have stochastic volatility.) See Hamilton-Wu (2012) et al.

2. What's right with DNS/AFNS?

DNS/AFNS just puts a bit of structure on factor loadings while still maintaining significant flexibility. That gets us to a very good place, involving both theoretical rigor (via imposition of no-arb in AFNS) and empirical tractability. That's all. It really is that simple.

3. Is AFNS the only tractable $$A_0(3)$$ model?

Not any longer, as recent important work has opened new doors. In particular, Joslin-Singleton-Zhu (2011) develop a well-behaved (among other things, identified!) family of Gaussian term structure models, for which trustworthy estimation is very simple, just as with AFNS. Moreover, it turns out that AFNS is nested within their canonical form, corresponding to three extra constraints relative to the maximally-flexible model.

4. If AFNS is no longer the only tractable $$A_0(3)$$ model, is it nevertheless still of special interest?

Yes! AFNS's structure conveys several important and useful characteristics, which are presently difficult or impossible to achieve in competing frameworks. First, as regards specializations, AFNS parametric simplicity makes it easy to impose restrictions. Second, as regards extensions, AFNS simplicity makes it similarly easy to increase the number of AFNS latent factors if desired or necessary, as for example with the five-factor model of Christensen-Diebold-Rudebusch (2009). Third, as regards varied uses, the flexible AFNS continuous basis functions facilitate relative pricing, curve interpolation between observed yields, and risk measurement for arbitrary bond portfolios.

And there's more. Fascinating recent work studying AFNS from an approximation-theoretic perspective shows that the Nelson-Siegel form is a low-ordered Taylor-series approximation to an arbitrary $$A_0(N)$$ model. See Krippner (in press).

5. What next?

Job 1 is flexible incorporation of stochastic volatility, moving from $$A_0(N)$$ to $$A_x(N)$$ for $$x>0$$, as bond yields are most definitely conditionally heteroskedastic. Doing so is important for everything from estimating time-varying risk premia to forming correctly-calibrated interval and density forecasts. Work along those lines is starting to appear. Christensen-Lopez-Rudebusch (2010)Creal-Wu (2013) and Mauabbi (2013) are good recent examples.