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Likelihood-based estimation of a semiparametric time-dependent jump diffusion model of the short-term interest rate

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Abstract

This paper proposes a semiparametric time-dependent jump diffusion model in an effort to capture the dynamic behavior of short-term interest rates. The newly proposed model includes a wide variety of well-known interest rate models, incorporating the time-varying instantaneous return, volatility as well as jump component. The local likelihood density estimation technique together with pseudo likelihood estimation method is employed to estimate the parameters of the model. Some simulations are conducted to examine the statistical performance of our estimators. The proposed procedure is then applied to analyze daily federal funds rate.

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Acknowledgements

The research work is supported by the National Natural Science Foundation of China under Grants No. 11701286, the Natural Science Foundation of Jiangsu Province of China under Grants No. BK20171073, the University Natural Science Foundation of Education Department of Jiangsu Province of China under Grants No. 17KJB110006, the University Philosophy and Social Science Foundation of Education Department of Jiangsu Province of China under Grants No. 2017SJB0350.

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Correspondence to Tianshun Yan.

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Appendix: Technical details

Appendix: Technical details

The Bernoulli approximation here can be achieved as follows: Subdivide the interval (0, T) into n equal subintervals each of length \({{\Delta }_i}\), where \({{\Delta }_i} =T/n\), for a time interval of length T. Define the indicator variable \(Y_i=1\) if a jump occurs, else \(Y_i=0\) for some i.

$$\begin{aligned} \begin{aligned}&\qquad \mathrm{{Prob}}[{{Y}_{i}}=0]=1-\lambda {{{\Delta }_i }}+\mathrm{O}({\Delta }_i), \\&\qquad \mathrm{{Prob}}[{{Y}_{i}}=1]=\lambda {{{\Delta }_i}}+\mathrm{O}({\Delta }_i),\ \ \ \mathrm{{for}} \ \ \ i=1,2,\ldots ,n\\&\qquad \mathrm{{Prob}}[{{Y}_{i}}>1]=\mathrm{O}({\Delta }_i).\\ \end{aligned} \end{aligned}$$

Let \({M}={\sum \nolimits _{i=1}^{n}} Y_i\). M is distributed binomial being the sum of independent Bernoulli variables. For k occurrences,

$$\begin{aligned} {\mathrm{{Prob}}[M=k]}=C_{n}^{k}{{\left( \frac{\lambda T}{n} \right) }^{k}}{{\left( 1-\frac{\lambda T}{n} \right) }^{n-k}},\quad k=0,1,2,\ldots ,n. \end{aligned}$$

Then we have

$$\begin{aligned} \underset{n\rightarrow \infty }{\mathop {\lim }}\,\mathrm{{Prob}}[{M}=k]=\frac{{{e}^{-\lambda T}}{{(\lambda T)}^{k}}}{k!}.\ \ \ \ k=0, 1, 2,\ldots ,n \end{aligned}$$

This is a standard construction of the Poisson process (for further details, see Karlin and Taylor 1975).

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Yan, T., Zhao, Y. & Wang, W. Likelihood-based estimation of a semiparametric time-dependent jump diffusion model of the short-term interest rate. Comput Stat 35, 539–557 (2020). https://doi.org/10.1007/s00180-019-00875-1

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