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Design and Estimation of a Quadratic Term Structure Model with a Mixture of Normal Distributions


  • Kentaro Kikuchi

    (Deputy Director and Economist, Institute for Monetary and Economic Studies, Bank of Japan (E-mail:


To keep yields non-negative in a quadratic Gaussian term structure model (QGTM), the short rate is represented by the quadratic form of the Gaussian state variables. The QGTM is among the most attractive candidate tools for analyzing yield curves for countries with low interest rates. However, the model is unlikely to capture the fat- tailed feature of changes in yields observed in actual bond markets. This study extends the QGTM by introducing state variables whose future distributions follow a mixture of normal distributions. This extension allows our model to accommodate vast changes in non-negative yields. As an illustrative empirical study, we applied our model to Japanese government bond (JGB) yields using the unscented Kalman filter. We then used the parameters obtained to investigate market views on past JGB interest rates by simulating future interest rate probability distributions under the physical measure and by decomposing interest rates into expected short rates and term premia.

Suggested Citation

  • Kentaro Kikuchi, 2012. "Design and Estimation of a Quadratic Term Structure Model with a Mixture of Normal Distributions," IMES Discussion Paper Series 12-E-08, Institute for Monetary and Economic Studies, Bank of Japan.
  • Handle: RePEc:ime:imedps:12-e-08

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    References listed on IDEAS

    1. Markus Leippold & Liuren Wu, 2003. "Design and Estimation of Quadratic Term Structure Models," Review of Finance, European Finance Association, vol. 7(1), pages 47-73.
    2. Markus Leippold & Liuren Wu, 2002. "Design and Estimation of Quadratic Term Structure Models," Finance 0207014, EconWPA.
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    More about this item


    affine term structure model; quadratic Gaussian term structure model; mixture of normal distributions; unscented Kalman filter; maximum likelihood method;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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