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An evaluation of alternative methods used in the estimation of Gaussian term structure models

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Abstract

This paper provides an evaluation of five methods, proposed in the literature, for extracting factors used in the estimation of Gaussian affine term structure models. We assert that irrespective of the method used for extracting state variables, cross-sectional and serial correlations exist in measurement errors. However, using a simulation design which is consistent with the data, we show that parameter estimation using the Kalman filter and the model-free method are quite precise in the presence of serial and cross-sectional correlations in the error term, and, in the presence of different measurement errors, the Kalman filter demonstrates strong empirical tractability.

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Notes

  1. Please see Yeh and Lin (2003) or Lundtofte (2013) for studies which implement interest rate models using a framework that is different from Duffie and Kan (1996). Lundtofte (2013) study interest rates using a general equilibrium model set up that enables consumers to learn within financial markets, while Yeh and Lin (2003) employ general equilibrium models and curve-fitting techniques (e.g. spline methods) to study the Taiwanese government bond market.

  2. Please see Chen and Yang (2002) for a theoretical application of the class of affine term structure models of Duffie and Kan (1996) to the derivation of a model of the term structure of interest rates.

  3. For this reason, we only provide an overview of the normalization of Dai and Singleton (2000) and refer readers to CDGJ (2008) or JSZ (2011) for more details on their respective methods. The overview is provided in Sect. 3.

  4. See Sect. 3 for more details on the CDGJ method.

  5. Such restrictions (and the implied volatility specification) are consistent with the Vasicek (1977) term structure model. Ritchken and Sankarasubramaniam (1996) demonstrate that the Vasicek (1977) model is one of the only Markovian models which uses a volatility specification that allows for an analytical relationship between bond prices and interest rates.

  6. Smaller values of λ produce slower decays and thus fit long-term maturities better, while bigger choices of λ produce faster decays and hence fit short-term rates better. We follow Diebold and Li (2006) in their selection of λ to match medium-term maturities.

  7. Comparing the JSZ-NS technique against the JSZ-PC technique resulted in a Chi squared test statistic value of 112.796, comparing the JSZ-PC technique against the JSZ-B technique led to a chi-squared test statistic value of 136.962, and finally, comparing the JSZ-NS technique against the JSZ-B technique led to a chi-squared test statistic value of 121.248. In each case, we have 66 degrees of freedom and the resulting p-values were, respectively, 0.0003, 0.0000, and 0.0000.

  8. \(\Updelta t\) is the time between observations and \(\Updelta z\) is a standard normal shock term.

  9. The intuition behind this simulation design rests on the fact that within the class of affine models and as was explained in Sect. 3, one would construct yield data by constructing the factor (i.e., state variable) and then using the factor to generate yields. Hence, any persistence observed in the data arises from persistence in the factor. The remaining steps compute the factor persistence.

  10. Since correlation is just a scaled version of covariance, the effect of this is to remove the necessity of a scaling factor.

  11. We also run unpaired t-tests in which we assume that each simulated dataset is obtained from independent samples of normal distributions. However, while the main conclusions were essentially the same, we feel that reporting a paired-sample t test is more appropriate since we are examining parameter estimates before and after the addition of noise.

  12. Employing maturities of up to 10 years for estimation is the norm in the literature (e.g., Dejong 2000; Duffee 2002; Tang and Xia 2007; Collin-Dufresne et al. 2008; Dempster and Tang 2011; Duffee 2011b; Duffee and Stanton 2012; Hamilton and Wu 2012). The reason for it is given by Dejong (2000), who notes that, “for the maturities over 10 years, the bond data are quite scarce, so the interpolation is not very accurate.” To overcome the belief that using up to 10 years of maturity may be ignoring the convexity risk in the term structure, and could be the reason why Gaussian models perform well, we investigated this issue further. We collected additional data corresponding to swaps with maturities of 15 and 20 years from Bloomberg and repeated every step of our data interpolation to construct additional zero-coupon yields with maturities ranging from 11 years through 20 years in increments of 1 year. Then, we re-estimated the JSZ normalization using the Kalman filter and the benchmark setting assuming that the 6-month, 2-year, and the 10-year are measured without error and all remaining bonds are measured with error and computed RMSEs corresponding to the difference between model-implied yields and actual yields. The results showed that for bonds with maturities greater than 10 years, the JSZ normalization of the Gaussian model still performed well.

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Acknowledgments

I would like to thank Chris Lamoureux, A. D. Amar, and seminar participants at the 2011 Financial Management Association Annual Meetings held in Denver, Colorado and the San Diego State University Workshop on Finance which also took place in 2011. I would also like to especially thank Cheng Few Lee, the associate editors, and the anonymous referees who are all affiliated with the Review of Quantitative Finance and Accounting for very helpful comments and insights that greatly improved the paper. I feel deeply indebted to them. The Data for the robustness check described in footnote 12 was obtained from the Bloomberg database via the Wells Fargo Financial Markets Laboratory at San Diego State University.

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  1. College of Business Administration, San Diego State University, San Diego, CA, 92182, USA

    Januj Juneja

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Juneja, J. An evaluation of alternative methods used in the estimation of Gaussian term structure models. Rev Quant Finan Acc 44, 1–24 (2015). https://doi.org/10.1007/s11156-013-0396-2

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