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Most affine models of the term structure with stochastic volatility predict that the variance of the short rate should play a ‘dual role’ in that it should also equal a linear combination of yields. However, we find that estimation of a standard affine three-factor model results in a variance state variable that, while instrumental in explaining the shape of the yield curve, is essentially unrelated to GARCH estimates of the quadratic variation of the spot rate process or to implied variances from options. We then investigate four-factor affine models. Of the models tested, only the model that exhibits ‘unspanned stochastic volatility’ (USV) generates both realistic short rate volatility estimates and a good cross-sectional fit. Our findings suggest that short rate volatility cannot be extracted from the cross-section of bond prices. In particular, short rate volatility and convexity are only weakly correlated.

This paper investigates the relation between interest rate volatility and the cross section of bond yields. It is well-established that at least three factors are needed to capture bond yield dynamics: Litterman and Scheinkman (1991) interpret them as level, slope, and curvature. It is also well-established that interest rate volatility is stochastic.1 As such, this paper focuses on those three- and four-factor models of the term structure that capture stochastic volatility. By imposing only the condition of no-arbitrage, it can be shown that state variables that drive changes in interest rate volatility generally play a ‘dual role’ in that they also drive changes in bond yields.2 For example, theoretical considerations generally imply a strong link between changes in short-rate volatility and changes in curvature. Not surprisingly, this prediction is manifest in the specific models proposed in the literature. For example, affine models of the term structure predict that state variables driving interest rate volatility are also linear combinations of yields. As a special case, the ‘preferred’ A1(3)A1(3) model of Dai and Singleton (DS, 2000) predicts that the variance of the spot rate is also a linear combination of the level, slope, and curvature of the yield curve. The first goal of this paper is to investigate whether the variance state variable in the preferred A1(3)A1(3) model can simultaneously satisfy its dual roles as the source of time-varying yield volatility and a factor in the yield curve. We choose this model because Duffee (2002) concludes that it offers the best forecasting performance among three-factor models with stochastic volatility, while DS find that it offers the best characterization of unconditional yield volatilities and a sufficiently flexible correlation structure.3 On the other hand, there is evidence that the A1(3)A1(3) model is misspecified.4 Therefore, a more general goal of the paper is to investigate the joint dynamics of level, slope, curvature, and volatility factors within a more flexible four-factor affine framework that does not impose at the outset a deterministic link between them. We also investigate a subset of the affine class that displays ‘unspanned stochastic volatility’ (USV). These models impose strong parameter constraints in order to generate bond yields that are independent of some of the variables driving volatility. As such, these ‘unspanned’ variables do not play a dual role and thus are free to more accurately capture the time-series properties of interest rate volatility. This potential benefit is not without costs, however. First, the large number of parameter constraints may prove to be overly restrictive. For example, the A1(4)A1(4) USV model has only 11 risk-neutral parameters that affect bond yields, compared to 22 for its unrestricted counterpart. Second, limiting a variance state variable to have only a time-series role means that such a model will be less able to explain cross-sectional yield patterns. For example, the A1(4)A1(4) USV model has only three factors to capture the cross section of yields. Prior empirical work on USV has focused almost exclusively on the spanning relation between interest rate derivatives and bond prices. For example, Collin-Dufresne and Goldstein (CDG, 2002) find changes in swap rates are only weakly correlated with returns on at-the-money cap straddles, while Heidari and Wu (2003) obtain similar results using implied volatilities from swaptions. Li and Zhao (2006) investigate quadratic models of the term structure and find them incapable of explaining the returns on caps of various maturities and strike prices. In contrast, Fan, Gupta, and Ritchken (2003) report that swaption returns are in fact well-spanned by yield changes, while both Bikbov and Chernov (2009) and Joslin (2007) report that the USV restrictions are strongly rejected when bond and option data are used. Our paper differs from the previous work by focusing on the ability of USV models to explain bond prices themselves. Specifically, we are interested in determining whether USV models are able to simultaneously match both the cross-sectional and time-series properties of bond yields. While it seems clear that the USV model will match some aspects of the time series of volatility (i.e., conditional second moments of yields) relatively well, it is unclear how the restrictions imposed to generate USV will affect the model's ability to capture the cross section and time series of yields (i.e., conditional first moments). We note that many standard econometric methods used to estimate affine models are unsuitable for investigating models exhibiting USV. Indeed, a consequence of imposing USV restrictions is that the one-to-one mapping assumed by Duffie and Kan (DK, 1996) between yields and factors does not hold. This implies that standard estimation techniques that rely on the ‘invertibility’ of the term structure (e.g., Chen and Scott, 1993 and Pearson and Sun, 1994) with respect to the latent factors cannot be implemented. The Kalman filter-based approaches of Duan and Simonato (1999) and de Jong (2000) are also unsuitable for our purposes because USV restrictions make it impossible for a linear filter to properly update the distribution of the unknown volatility state variable. We therefore write term structure dynamics in a nonlinear state space form and estimate the parameters of the models using Bayesian Markov chain Monte Carlo (MCMC). The results from estimating the unrestricted essentially affine three-factor model are striking. Most significantly, we obtain the ‘self-inconsistent’ result that the volatility factor extracted from this model (i.e., the ‘term structure-implied volatility’) is basically unrelated to volatilities estimated using rolling windows, GARCH volatilities, or implied volatilities from options. Furthermore, the strong in-sample fit of that model breaks down following the end of the estimation period, suggesting deep misspecification. We interpret these findings as evidence that three-factor models cannot simultaneously describe the yield curve's level, slope, curvature, and volatility. That is, volatility is unable to play the dual role that such a model predicts it does. The estimation of such a model therefore presents a tradeoff between choosing volatility dynamics that are more consistent with one role or the other. For the data set we investigate, and with no parameter restrictions imposed, that tradeoff is heavily tilted toward explaining the cross section.5 Both the A1(3)A1(3) and A1(4)A1(4) models exhibiting USV imply realistic behavior for the dynamics of short rate volatility, but the A1(3)A1(3) USV model fails on (at least) two dimensions. First, with just two factors affecting the cross section of yields, it cannot provide the same accuracy in fitting the cross section as does the unrestricted three-factor model or the four-factor USV model. Second, the unconditional yield volatilities implied by the model are inconsistent with the data, as they fail to reproduce the ‘hump shape’ relation between unconditional volatility and maturity found in the data. Therefore, in this paper, we report only the results for the A1(4)A1(4) USV model and compare them to the results for the unrestricted A1(3)A1(3) and A1(4)A1(4) models.6 Of the models investigated, only the A1(4)A1(4) USV model is able to generate both good cross-sectional and time-series fits of yields. Since it has three factors that affect yields, the model's in-sample fit is very tight. Furthermore, in contrast to the unconstrained A1(3)A1(3) model, this model is just as accurate (if not more so) in our three-year ‘hold-out’ sample. The model is able to simultaneously forecast volatilities of yields at all maturities, both in- and out-of-sample, and it generates the correct hump shape in the term structure of unconditional volatilities. Given our use of a relatively short sample period, we were unable to gauge the model's yield forecasting performance with much accuracy, but we note that the model has at least as much flexibility in its risk premia as the most general models considered by Duffee (2002). While the unconstrained A1(4)A1(4) model performs better than the unconstrained A1(3)A1(3) model in that its predicted volatility is at least positively correlated with volatility estimates such as GARCH, it is still grossly inadequate at capturing the volatility of short rates. In particular, for short maturity yields, the USV model outperforms the unconstrained A1(4)A1(4) model both in- and out-of-sample when it comes to volatility forecasts. Further, the USV model, fitted to weekly data, delivers forecasts that are equal to or better than GARCH forecasts based on daily data. However, for longer maturities, the USV and non-USV models seem to perform similarly with respect to volatility predictions, and both are inferior to GARCH. These results indicate that within the unconstrained four-factor model there is a tension between capturing the unconditional distribution of yields and the short rate volatility dynamics. These results also suggest that a better model would allow for different drivers for the volatilities of short and long maturity yields. An important implication of our findings is that any strategy that attempts to hedge the volatility risk inherent in fixed income derivatives (if feasible at all) must be substantially more complex than the convexity-based ‘butterfly’ positions discussed by Litterman, Scheinkman, and Weiss (1991) and implied in Brown and Schaefer, 1994a and Brown and Schaefer, 1994b. Indeed, our results suggest that implied spot rate volatility measures extracted from the cross section of the yield curve are likely to be bad estimates of actual volatility. The rest of the paper is as follows. In Section 2, we characterize maximal three- and four-factor models exhibiting USV. In Section 3, we describe an estimation methodology that remains valid under USV, while Section 4 includes all empirical results. We conclude in Section 5.

We investigate several affine models of the term structure that generate stochastic volatility in yields. We find that the unrestricted A1(3)A1(3) model implies a volatility time series that is essentially unrelated to the actual volatility of the short rate process. This surprising result is a consequence of the dual role played by the volatility state variable in the unrestricted affine model: it is both a linear combination of yields (i.e., it affects the cross section of the term structure) and the quadratic variation of the short rate (i.e., it impacts the time series of the term structure). Bayesian estimation results in more weight being placed on the first role at the expense of the second. While the in-sample fit of yields is excellent, a clear out-of-sample breakdown casts doubt on the model's adequacy in this role as well. We then investigate an A1(4)A1(4) model exhibiting USV. The USV specification allows the model to fit level, slope, and curvature while simultaneously producing a volatility process that is highly correlated with both GARCH and option-implied volatility series. It does so by explicitly introducing variation in curvature that is unrelated to volatility, a straightforward generalization within the representations introduced by CGJ. This model is also (mostly) successful in replicating the ‘hump shape’ in term structure volatility, and it performs as well out-of-sample as it does in-sample. We also consider an unrestricted A1(4)A1(4) model. In principle, this model could fit level, slope, curvature, as well as volatility perfectly. However, results indicate that the likelihood criterion is strongly tilted toward fitting the cross section of yields at the expense of the predicted volatility series, which, even in this unconstrained four-factor model, bears little resemblance to actual volatility time series estimated via GARCH or from implied option volatilities. However, the model performs well in fitting the unconditional volatility structure and longer-maturity yield conditional volatilities. We conclude that fitting short maturity volatilities is not necessary to obtain a good fit of yields and longer maturity conditional volatilities, as short- and long-dated volatilities appear to be driven by different components. While our results confirm the findings of Litterman and Scheinkman (1991) that at least three factors are needed to explain the cross sectional features of the yield curve, it further demonstrates that these factors are an inadequate description of the state space, as they are incapable of replicating observed patterns of conditional volatility.