## 1. INTRODUCTION

To investigate yield-curve dynamics, researchers have produced a vast literature with a wide variety of models. Many of these models assume that at observed bond prices there are no remaining unexploited opportunities for riskless arbitrage. This theoretical assumption is consistent with the observation that bonds of various maturities all trade simultaneously in deep and liquid markets. Rational traders in such markets should enforce a consistency in the yields of various bonds across different maturities—the yield curve at any point in time—and the expected path of those yields over time—the dynamic evolution of the yield curve. Indeed, the assumption that there are no remaining arbitrage opportunities is central to the enormous finance literature devoted to the empirical analysis of bond pricing. Unfortunately, as noted by Duffee (2002), the associated arbitrage-free (AF) models can demonstrate disappointing empirical performance, especially with regard to out-of-sample forecasting. In addition, the estimation of these models is problematic, in large part because of the existence of numerous model likelihood maxima that have essentially identical fit to the data but very different implications for economic behavior (Kim and Orphanides, 2005).^{1}

In contrast to the popular finance AF models, many other researchers have employed representations that are empirically appealing but not well grounded in theory. Most notably, the Nelson and Siegel (1987) curve provides a remarkably good fit to the cross section of yields in many countries and has become a widely used specification among financial market practitioners and central banks. Moreover, Diebold and Li (2006) develop a dynamic model based on this curve and show that it corresponds exactly to a modern factor model, with yields that are affine in three latent factors, which have a standard interpretation of level, slope and curvature. Such a dynamic Nelson–Siegel (DNS) model is easy to estimate and forecasts the yield curve quite well. Despite its good empirical performance, however, the DNS model does not impose the presumably desirable theoretical restriction of absence of arbitrage (e.g. Filipović, 1999, and Diebold et al., 2005).

In Christensen et al. (2007), henceforth CDR, we show how to reconcile the Nelson–Siegel model with the absence of arbitrage by deriving an affine AF model that maintains the Nelson–Siegel factor loading structure for the yield curve. This arbitrage-free Nelson–Siegel (AFNS) model combines the best of both yield-curve modeling traditions. Although it maintains the theoretical restrictions of the affine AF modeling tradition, the Nelson–Siegel structure helps identify the latent yield-curve factors, so the AFNS model can be easily and robustly estimated. Furthermore, our results show that the AFNS model exhibits superior empirical forecasting performance.

In this paper, we consider some important generalizations of the Nelson–Siegel yield curve that are also widely used in central banks and industry (e.g. De Pooter, 2007).^{2} Foremost among these is the Svensson (1995) extension to the Nelson–Siegel curve, which is used at the Federal Reserve Board (see Gürkaynak et al., 2007, 2008), the European Central Bank (see Coroneo et al., 2008) and many other central banks (see Söderlind and Svensson, 1997, and Bank for International Settlements, 2005). The Svensson extension adds a second curvature term, which allows for a better fit at long maturities. Following Diebold and Li (2006), we first introduce a dynamic version of this model, which corresponds to a modern four-factor term structure model. Unfortunately, we show that it is not possible to obtain an arbitrage-free ‘approximation’ to this model in the sense of obtaining analytically identical factor loadings for the four factors. Intuitively, such an approximation requires that each curvature factor must be paired with a slope factor that has the same mean-reversion rate. This pairing is simply not possible for the Svensson extension, which has one slope factor and two curvature factors. Therefore, to obtain an arbitrage-free generalization of the Nelson–Siegel curve, we add a second slope factor to pair with the second curvature factor. The simple dynamic version of this model is a generalized version of the DNS model. We also show that the result in CDR can be extended to obtain an arbitrage-free approximation to that five-factor model, which we refer to as the arbitrage-free generalized Nelson–Siegel (AFGNS) model.

Finally, we show that this new AFGNS model of the yield curve not only displays theoretical consistency but also retains the important properties of empirical tractability and fit. We estimate the independent-factor versions of the four-factor and five-factor non-AF models and the independent-factor version of the five-factor arbitrage-free AFGNS model. We compare the results to those obtained by CDR for the DNS and AFNS models and find good in-sample fit for the AFGNS model.

The remainder of the paper is structured as follows. Section 2 briefly describes the DNS model and its arbitrage-free equivalent as derived in CDR. Section 3 contains the description of the AFGNS model. Section 4 describes the five specific models that we analyze, while Section 5 describes the data, estimation method and estimation results. Section 6 concludes the paper, and an Appendix contains some additional technical details.