The Estimation And The Bootstrap Deviation Correction Of The Dynamic Term Structure Model Of Interest Rates

The term structure of interest rates is a central issue of macro-economy and financial market,and it is also the important basis for making economic policies and managing financial risks and investments.Studies have shown that the yield curve contains a lot of macro-economic information,such as economic growth,inflation,monetary policy and so on.So as to speed up the reform of market-oriented interest rates and to conduct pertinently regulations on interest rates and macro-economy,it is important theoretically and practically for China to deeply analyze the shocks on the term structure of China's interest rates from supply and demand and to accurately grasp its dynamic characteristics.Affine dynamic term structure model(DTSM)is a standard empirical financial model of the yield curve which is also the most popular and widely used one.The estimation of the term structure of interest rates,however,has been faced by many challenges,such as the models to be estimated existing the initial value dependence and the maximum likelihood estimates not being the global optimal solution(local optimal solution).At home and abroad in recent years,the studies have shown that the DTSM estimation may have the deviations due to small sample,and people may have not been aware of the existence of bias in part.In this paper,we focus on two aspects,i.e.,the estimates of gaussian affine DTSM,and the corrections of its small sample bias.The content is divided into eight chapters,including:The first chapter is about the introduction,reviews and comments given to the present situation of this problem.In chapter 2,we review several estimate methods of traditional dynamic term structure models of interest rates,mainly including the NS model's two-step method,one-step kalman filter estimation method,and three factor gaussian affine DTSM Chen-Scott estimation method.The NS model and the gaussian affine DTSM all belong to affine models.The NS model is simple with obvious meanings of factors and then adopted by many scholars.The kalman filter estimation of three factors gaussian affine DTSM has the best fitting effect and can be completed by one step kalman filter estimation,so it is also widely used in traditional estimation method.The maximum likelihood estimation is popularly used among them.However,solutions of maximum likelihood estimation are not often globally optimal and lead to wrong conclusions when they are used to analyze specific issues.The traditional estimation methods have another problem of dependence on initial values.If variables needed to be estimated by traditional methods are too much,the models will not be solved with unconvergence due to unproper initial values.The theory part in Chapter 3 discusses the Bootstrap deviation correction method of small sample.Because the sample size of China's bond yields is small,and the traditional models of term structure of interest rates have a problem of small sample deviation that has long been ignored.The estimation of the DTSM may cause errors due to the small sample bias.Serious bias result in misleading estimates of expected future short-term interest rates and term premium,which do not accord with empirical research.So it is urgently to find a deviation correction method.The research in Chapter 3 have shown that the deviation of analytical formula method and the Bootstrap method are all smaller than OLS estimation method.Particularly,using the time series data of strongly sustainable yield,the Bootstrap method behaved better than an analytical formula for deviation correct.So the Bootstrap deviation correction method is selected in this paper.In the case of non-stationary appeared in the process of correction,we adopt the method of Kilian,because Kilian deviation correction method is not twisted on the finite sample properties of deviation correct method,while it does not increase a lot of variance,but decrease a lot of deviation,and so does not affect the mean square error.In order to solve the problem of flat surface of likelihood mentioned in chapter 2 and small sample deviation mentioned in chapter 3,this paper introduced in chapter 4 Joshlin,Singleton and Zhu(2011)estimation method(hereinafter referred to as JSZ).JSZ estimation methods are divided into two steps:the first step is a simple OLS estimation instead of maximum likelihood estimation,the second step is still maximum likelihood estimation,but JSZ describe the risk neutral distribution with a constant and a characteristic value of a drift matrix,so the parameters to be estimate in the traditional three factors DTSM are reduced from 22 to 4,which avoids the problem of overparametrization and also reduces the probability of local optimal solution.In this chapter based on the JSZ estimation method using to solve the problem of likelihood surface smooth,the small sample deviation in JSZ estimation must be corrected.Because the first step of JSZ estimation method is the OLS estimates,there may be a small sample deviation in this phase which can be made use of the Bootstrap method for larger correction,the second step of the JSZ method is unchanged,which is not to be corrected.The empirical analysis on small sample deviation correction is conducted to the term structure of China's interbank fixed interest rates bonds and the forward rate is decomposed into risk neutral rate and the forward premium.The results show that with deviation correction,the persistence of system increases significantly,and the adjusted risk neutral rate fluctuates significantly.The term premium is pro-cyclical before revision,but after small sample deviation correction term premiums shows the inverse cycle characteristics.Chapter 5 is an extension of chapter 4.Hamilton&Wu(2012)(hereinafter referred to as HW)described affine term structure model as "reduced-form" expression,which has not been used in previous affine term structure model.The purpose of introducing HW estimation method is to replace the maximum likelihood estimation with minimum chi-square estimation which is asymptotically equivalent to the maximum likelihood estimation which can judge quickly whether the optimal result is global or local.Before this,researchers tend to exert some restrictions to obtain the estimate or in some cases they didn't really get the global optimal solution of the likelihood function.After introducing the HW estimation method,small sample deviation in HW model is also corrected based on the Bootstrap method in this chapter.The first step of HW estimation method is through the OLS to estimate reduced-form VAR,therefore the Bootstrap deviation correction estimation can be used here;the structure model parameters in the second step are estimated by the least chi-square estimates which are unbiased in this step and don't need to be modified.The corrected HW estimation method is used to explore the change of risk neutral forward rate and term premium after exerted over identified restrictions.The empirical results show that the system is not obviously affected by the excessive identification restrictions without small sample deviation correction and has the significant persistent characteristics after small sample deviation correction with risk neutral forward rate fluctuations increased obviously as well.This shows that the small sample bias can lead to be underestimated of the sustainability of variables,and policy expected effect on the influence of the forward rate is undervalued.In chapter 6 of this paper,the dynamic relationships between macroeconomic variables and three factors of yield curve are studied by combining with time domain and frequency domain analysis method.The term structure of interest rates can be finally expressed by three potential factors,and to study the relationships between the yield curve and macro variables,and the relationships between the three factors and the macro variables are particularly important.The models composed of three factors and macro variable are often referred to as macro financial model.The VAR model is used in previous studies to explore the relationships between macroeconomic factors and the yield curve factor,which has been made great progress,but there is no consistent conclusion,the reason for which is that most studies focus on the analysis of time series properties of macro factor and potential factor ignoring the domain they are correlated,that is to say,the frequency domain characteristics,which they do not take into account.So the time-frequency analysis framework can be built up to get a comprehensive study of the relationship between them.The wavelet analysis is an effective approach to study of time-frequency domain analysis,and is adopted in this paper to study the relationships between three factors of the yield curve and macro variables.Three macro factors are selected in this chapter:industrial production(IP),the consumer price index(CPI)and broad money(M2).In order to obtain potential factors three methods are adopted to obtain three potential factor:NS model method,the traditional principal component analysis,and the Bootstrap deviation correction principal component analysis.Wavelet analysis shows that there is not significant and continuous wavelet coherence area in the level factor from NS model and the three macroeconomic variables,and the level factor obtained from the principal component analysis has significant wavelet coherency with the industrial production added value and consumer price index,and sustained time is longer,usually located in a long period.The slope factor of NS model fluctuate basically identical with that of the traditional principal component analysis and both have significant and lasting coherency with M2,located in a long period,but after the Bootstrap small sample deviation correction,the slope factor has significant and persistent wavelet coherency with all the three macroeconomic variables,located in a long period.This suggests that the small sample bias correction has remarkable influence on the slope factors that represent spreads,which is consistent with the small sample bias analysis to the risk neutral rate and term premiums in chapter 4;there are continuous and significant wavelet coherence between the curvature factor of the NS model and all three macro variables,but there are not continuous and significant wavelet coherence between the curvature factor obtained from the principal component analysis and any one of the three macro variables whether conducting Bootstrap small sample deviation correction or not.The research also finds in this chapter that some factors and macro variables exist wavelet coherence area for 1-2 years(in general for the 2008-2009),and in a short cycle,the traditional methods maybe arrive at the conclusions that these factors and macro variables are correlated,but the wavelet analysis show that the coherence time is not long,and the traditional methods are also unable to determine which cycle the correlations exist in and which leads or lags between the potential factors and macro variables.Therefore,when the central bank analysis on macro-economy with the three potential factors,it is necessary to know the relationship between the three factors and the macro factors and select the appropriate cycle to judge the changing direction of the potential factors and macro factors and potential factors lead or lag period through the phase difference in order to avoid the wrong conclusion.In chapter 7,the small sample bias is discussed in spanning hypothesis.Spanning hypothesis refers to all relevant information to forecast yield or excess profits can be generated by the level factor,slope factor and curvature factor.Whether the spanning hypothesis is valid or not is of vital importance in the financial and macroeconomic theory.If the hypothesis is right,then explaining the monetary policy expectations and bond risk premium does not need any other macroeconomic data,all the necessary information in the current is reflected in the shape of the yield curve.If the hypothesis is not right,then explain the monetary policy expectations and bond risk premium also need other macroeconomic information,only part of the state variables are not enough to fully explain the term structure of interest rates.The influence of the small sample to spanning hypothesis test is first discussed in this chapter,under the case of the prediction error series correlated and the lack of strictly exogenous constraints through the method of simulation,and then our Treasury yields and macroeconomic data and CPI(IP)are used again for empirically test.Simulation and the empirical results show that the small sample bias influence statistical test and regression model estimates,which is a cause of the generated assumptions may not be established.In the empirical test,test results in this paper is analyzed before and after the modification.The change of R square is calculated before and after joining macro variable.Results show that after the Bootstrap small sample deviation correction the change of R square is far smaller than correction before.The value of T test also shows that the CPI is significant in the regression mode before correction,but after the Bootstrap small sample deviation correction,CPI does not become significant in the model,so the explaining ability of macro variables is limited.Therefore,we did not find evidence that the spanning hypothesis in china's bond market is not established.The innovation of this paper is mainly reflected in three aspects:The first is the small sample bias correction of the term structure model of interest rate in our country.The two modern interest rate term structure model of JSZ estimation method and HW estimation method are introduced,and the quantitative method for the Bootstrap deviation is referenced from Bauer,Rudebusch and Wu(2014),effectively solving the problems of likelihood surface flat and the deviation of small sample in the traditional term structure of interest rates model.The model by deviation revised can not only get the global optimal solution,but the rate of returning to its mean decreases and the persistence of dynamic system increasing significantly.The ratio of the adjusted risk neutral rate in the forward rate obvious increases and its volatility increases significantly,which shows that the forward rate is strongly influenced by policy expectations.Before small sample deviation correction the term premium is pro-cyclical,but it is counter-cyclical after revised,which is in line with the macro economic reality.Second is the time-frequency analysis of the potential factors of term structure of interest rates in China and the macroeconomic factors.The traditional way is usually with the aid of the VAR model to analyze relations of potential factors and macro variables.Compared with the traditional methods,in this paper the relationships between potential factors and macro variables are analyzed from two aspects:time and frequency.At the same time,as to the extraction of potential factors,this paper considers the small sample bias correction.So,the conclusion in this paper has been greatly improved compared with that of traditional method without deviation correction.Empirical shows that the slope factor obtained from principal component analysis with the small sample bias correction have significant wavelet coherence and duration longer area with all the three macroeconomic variables,which is very different from before,and the revised slope factor and three macro factors are synthetic.The results can provide theoretical references for macro-economic policy and financial regulations for Chinese government and central Banks.The third is Spanning Hypothesis test in China's bond market.This paper deals with Chinese bond yield data to test the theory of Spanning hypothesis is established or not.In this paper,the empirical analysis shows that the excess returns can be predicted by the level factors,slope factor and curvature factor of the yield curve.After the Bootstrap deviation correction of small sample,the increment of R square in regression model when joining the macro variables is very small,and the macro variables in the regression model is not significant.So,after deviation correction,this paper did not find evidence of macro variables which can significantly increase the ability to predict.In this paper,the research also has some shortcomings:There is no empirical test to expectations hypothesis of term structure and also didn't discuss the inspection results after the deviation correction,which will be the next problem needed to be studied in this paper in the further.Although the affine term structure model can well fit the term structure of interest rates,the nonlinear,non-Gaussian term structure theory has been mature.For instance,SR model has been put forward by Andreasen(2014).Affine term structure model extended to nonlinear and non-Gaussian with investigating the correlated deviation correction is also a research direction in the future.