Modeling The Term Structure Of Interest Rates Based On Functional Data Analysis

This paper models the term structure of interest rates using functional data analysis.We first construct an FPCA-K model based on the functional principal component analysis(FPCA)method.Then,we propose a DNCS model based on the natural cubic spline(DCS)function.Finally,we extend the FPCA-K model and the DNCS model by considering the long memory characteristics of the yield rates.The empirical results show that both the FPCA-K model and the DNCS model have advantages in the out-of-sample forecasting of the term structure of interest rates.Specifically,we conduct three studies in this paper as follows.First,we propose a functional principal component analysis(FPCA)method to model the term structure of interest rates.We construct an FPCAK model using the leading principal components and the corresponding component scores.The yield curve forecasting is based on the prediction of the principal component scores.We also introduce the functional autoregressive(FAR)model,which is widely studied in the statistics,and we deal with the ill-inversed problem of the covariance operator in a finite Hilbert subspace.We conduct the empirical study using monthly yield data from January 2002 to December 2017.We adopt the FAR model,the dynamic Nelson-Siegel(NS)models and the random walk model as benchmarks,and we evaluate the forecasting performance of different models using the root mean squared error,the mean absolute error and the Diebold-Mariano statistic.The results show that in the 1-month-ahead forecasting of the yield curve,our FPCA-K model outperforms the FAR model and dynamic NS models,but it cannot outperform the random walk model.In the 3-,6-and 12-month-ahead forecasting,our FPCA-K model outperforms all its competitors,including the random walk model.We provide robustness checks by conducting rolling forecasts,changing the width of the rolling window,checking the subsample's robustness,and comparing the correct trend rate statistic.We find that the out-of-sample forecast performance of the FPCA-K model is robust under different settings.Second,we construct a two-step dynamic natural cubic spline(DNCS)model based on the natural cubic spline(NCS)function.The DNCS model can be related to the preferred habitat theory of the term structure of interest rates.We first estimate the interpolation matrix and the knot yields of the DNCS model,then we obtain the forecasts of the yield rates based on the prediction of the knot yields.We also introduce the widely used functional signal plus noise(FSN)model,which is expressed in a linear state-space model.We select the number of knots based on the explained variance criterion and choose the positions of knots using the in-sample crosssectional regression.We adopt the FSN model,the dynamic NS models and the random walk model as benchmarks,and we evaluate the forecasting performance of different models using the root mean squared error,the mean absolute error and the Diebold-Mariano statistic.The out-of-sample forecasting results are similar with the FPCA-K model.In the 1-monthahead forecasting,the DNCS model outperforms the FSN model and the dynamic NS models,but it cannot outperform the random walk model.In the 3-,6-and 12-month-ahead forecasting,the DNCS model outperforms all its competitors,including the random walk model.We also provide robustness checks by conducting rolling forecasts,changing the width of the rolling window,checking the subsample's robustness,and comparing the correct trend rate statistic.We find that the out-of-sample forecast performance of the DNCS model is robust under different settings.Finally,we compare the forecasting performance of the four functional models,namely the FPCA-K model,the FAR model,the DNCS model and the FSN model.The results show that the FPCA-K model and the DNCS model outperform the FAR model and the FSN model.In the 1-and 3-month-ahead forecasts,the FPCA-K model performs best and in the 6-and 12-month ahead forecasts,the DNCS model performs best.Third,we extend the FPCA-K model and the DNCS model by considering the long memory characteristics of the yield rates.we analyze the long memory feature of the yield rates,the long-run expectations of the short rate and the term premiums,and the out-of-sample forecasting of the yield rates using the long memory models.As the observed US yields are extremely persistent,we employ both the Chinese and US yield rates to conduct the empirical analysis.The results show that the US yields are more persistent compared with the Chinese yields.We show that stationary I(0)models imply an unrealistically low degree of volatility in the long-run expectations of the short rate due to fast mean reversion.The long memory models imply more volatile long-distance expectations of the short rate due to the persistent long-memory latent factors.The long-distance expectations of the US short rate implied by the long memory models are most volatile.The term premium estimates implied by the long memory modes differ from those of the I(0)model and are more volatile and more realistic.Finally,we compare the out-of-sample forecasting performance of the long memory models and the I(0)models using the root mean squared error,the mean absolute error and the Diebold-Mariano statistic.The results show that the long memory models can substantially improve the out-of-sample forecasts of the US yield rates and thus more suitable for the modeling of the US yields.