Testing linearity in term structures

Abstract

This article uses robust nonparametric techniques to investigate both crosssectional and dynamic properties of affine models, a popular framework to analyse Term Structures (TSs) of interest rates. The analysis shows the strong nonlinearity in the relationship of yields to the US and UK short rate. The nonlinear pattern is concave in the state variable, and increasing with respect to the maturity, for both countries. Linear and nonlinear specifications are then compared by means of a formal statistical criterion, the Generalized Likelihood-Ratio (GLR) test statistics, which confirms evidence against the linear specification.

Keywords:

• interest rates
• term structure
• affine models
• nonparametric methods

JEL Classification::

• C14
• E43
• G12

Acknowledgements

The views and opinions expressed in this article are not those of the Statec. Thanks are due to Peter Spencer, my PhD supervisor, to my advisors Karim Abadir and Gabriel Talmain and to my PhD awarding committee (Mike Wickens and John Knight). Thanks are also due to participants at ECOMOD2010, Istanbul, and the Conference on Computational Economics and Finance 2010, London, where this article has been presented.

Notes

¹ This is formally proven in Duffie and Kan (1996). One can also see Bolder (2001), who provides a clear illustration of discount bonds pricing computation from the diffusion specification.

² The conditional probability density of the interest rate is given by a Gaussian distribution in Vasicek's model, and by a noncentral χ² in CIR.

³ The extension of ATSs to multi-factor models does not alter the functional form and parametric specification of Equations (1) and (2).

⁴ Aït-Sahalia (this extended 1996a, who proposed a semi-parametric procedure for the variance based on a linear mean-reverting drift specification).

⁵ Under certain condition, the process in Equation (5) has a unique solution, and it is fully characterized by the two coefficients:

This shows that, with any specification for either drift or diffusion term, the other term will be specified, given the marginal density p of the diffusion process.

⁶ In contrast, the approach proposed by by Aït-Sahalia (1996a) requires the sample size to increase by prolonging the observation period, by adding more and more observations.

⁷ The update of the long end of the original McCulloch–Kwon dataset has been compiled by Gong and Remolona (1997) and researchers at the Federal Reserve Bank of New York. This dataset has also been analysed, using a different framework, by Spencer (2008). I am grateful to Peter Spencer for supplying a copy of this dataset. Treasury bills series are available at http://research.stlouisfed.org/fred2/.

⁸ Gurkaynak et al. (2007) present US daily yield curve data since 1961 to date. I do not use this dataset here for two reasons: (i) it does not include the shortest maturities (the authors argue that short-end estimates based on overnight federal funds rate maybe preferable to those based on Treasury bills, due to market segmentation); (ii) it is compiled using a parametric methodology that differs from the one – spline based – used by McCulloch and Kwon. McCulloch–Kwon data are also a better match to the UK yield curve, compiled using a similar method (see also Anderson and Sleath, 1999).

⁹ The 3-month Treasury bill rate series is compiled by the Office of National Statistics (ONS), series code: AJRP. This is used here for the following reason: the Bank of England uses repo rates to compute the yield curve at the shortest maturities (as discussed in Anderson and Sleath, 1999), but these data are available only from 1997. Furthermore, a comparable series is not available for the US.

¹⁰ Yield data are available on the Bank of England website: http://www.bankofengland.co.uk/Statistics/yieldcurve/index.html For the UK, the 20-year yield is not reported as observations are sparse.

¹¹ US density estimates computed for the same time period as for the UK do not feature bimodality, although have accentuated skewness and a fatter right tail.

¹² Diffusion estimates are computed using a Gaussian kernel; the bandwidth choice is guided by the discussion in Arapis and Gao (2006), who looked at the same data. So, the bandwidth is proportional to the optimal rate T ^−1/5, where T is the sample size. (The use of this rule for time series data analysis is discussed in Hall et al., 1995b).

¹³ This monthly frequency also avoids the noise and the computational high costs that are involved in higher frequency samples. It is supported by empirical results in Pritsker (1998) and Chapman and Pearson (2000), who argue that the bandwidth choice is affected by the memory properties of the series rather than the frequency of the observations.

¹⁴ The similarity of results for the UK and US is not affected by the shorter time period covered by the UK sample. A similar exercise, conducted by cutting the US sample to cover the same period as the UK sample, delivers similar results.

¹⁵ It would be useful to report bootstrap variability bands together with the estimation, as suggested in Jiang (1998); however, this is difficult in the present context and is left for future research.

¹⁶ The local-linear estimator for the regression model y = m(x) + ϵ has the form:

where ω _i  = K _h (x _i  − x){(S _{n, 2} − (x _i  − x)S _{n, 1})}, with . The kernel K is Gaussian.

¹⁷ Altman (1990)'s algorithm is an adjusted cross-validation criterion based on a parametric estimate of the correlation function. This method has the considerable advantage of being simpler than its alternatives proposed in the statistical literature, such as Hall et al.'s (1995a) method. Furthermore, simulation experiments show that, even when the parametric part is misspecified, this method provides a substantial improvement over rules developed under the independence assumption.

¹⁸ For robustness, one should note that the same nonlinear results are obtained when using different measures of the short rate, such as the overnight Federal Fund rate, as suggested in Gurkaynak et al. (2007). The curves are not reported here for reasons of space, but can be obtained from the author.

¹⁹ A regression equation which incorporates a constant and lagged values of the variable tested is used; the ADF statistics is the t-statistic for a first-order autoregressive coefficient equal to 1.

²⁰ Diebold and Rudebusch (1991) gave an important result: they showed that DF type tests have low power against fractionally integrated alternatives.

²¹ An alternative interpretation can be found in Bandi (2002), who argue that the misspecification of parametric affine interest rate models is due to the martingale nature of the process.