Modelling Of The Short Term Interest Rate Biology Essay

Lin Huiping

U092944J

Supervisor:

Professor Xia Yingcun

Contents

Summary

Modelling of the short-term interest rate is important in the pricing of fixed income securities and determining the risk associated with holding portfolios of these securities. The key contribution of this thesis is the inclusion of a new term into various diffusion models that has been recommended in many well-known papers. A summary of these various diffusion models is shown in Table Summary of alternate models of short-term interest rate with different parameter restrictions of Section 3 and the models that we work on are presented in Table of Section 6.

We proposed that the various diffusion models should include the lagged difference of the short-term interest rate,, in the conditional mean and conditional variance of the short-term interest rate model. The new proposed model is shown in Section 3 as Equation Suggested model. A generalised model is shown as Equation in Section 3 which contains higher lagged difference of short-term interest rate in the model.

We used weekly observations of the yearly yield on U.S. Treasury Bills with three months to maturity. The data statistics is outline in section 4.

In this thesis, we estimate the new proposed model through various statistical methods namely (i) Ordinarily Least Squares Estimation, (ii) Weighted Least Squares Estimation and (iii) Maximisation Likelihood Function method. We test the significance of the new parameters through constructing confidence intervals. Bootstrap confidence intervals for the estimated parameters are constructed for estimation based on maximising of likelihood function. Wild bootstrap by Wu (1986) is implemented and I came up with the Matlab code [1] that is customised to our proposed models.

Conclusion on the significance of the newly proposed new variable is drawn in the last section.

All data processing was performed off-line using a commercial software package MATLAB® R2012b.

Introduction

The short-term riskless interest rate is important as it is one of the most basic and crucial in pricing fixed income securities and for the measurement of the interest rate risk that is associated with holding portfolios of these securities.

Vast efforts had being put in modelling and approximating short rate dynamics in recent years. The inherent problems of modelling short-term interest rate become more pronounced in the course of finding a model that has both a worthy theoretical justification and adequate practical application. The top theoretical models stipulate a continuous-time processes for the interest rate, stem from the model of Merton (Merton R. C., 1973) which is a depiction of arithmetic Brownian motion. This condition of the model is handy however it gives unfavourable negative interest rate and imprecise estimation of the real process. Alterations were made later on to improve the empirical structures. These include Vasicek’s model (Vasicek, 1977) which integrates a mean-reversion component to make the model stationary and reduces the probability of getting negative interest rates. The Cox, Ingersoll, and Ross (CIR) (Cox, Ingersoll, Jr, & Ross, 1985) model has volatility of the short-term interest rate varies proportionally to the square root of the interest rates level which inhibits the chance of getting negativity interest rates completely. Both Vasicek and CIR models can be extended to multi-factor models to give closed form outcomes for empirical work.

Chan, Karolyi, Longstaff, and Sanders (CKLS) (Chan, Karolyi, Longstaff, & Sanders, 1992) show that a more flexible and practical condition on the power of interest rate is needed to get a satisfactory depiction of the real short-rate process. CKLS eased the power restriction on the interest rate level in CIR model and resulted in single approximation of the power of as oppose to in the CIR. CIR and CKLS models parameterise short-term interest rate’s volatility as a function only on interest rate levels. They show that there exists a monotonic relationship between performance of the model and the degree to which interest rate levels are allowed to affect the volatility. However, a study by Brenner, Harjes, & Kroner (Brenner, Harjes, & Kroner, 1996) shows if shocks are allowed to affect volatility, this relationship converses and models with low sensitivity to levels will be preferred over those with high sensitivity to levels. Evidence from the estimation of (generalised) autoregressive conditionally heteroskedastic, (G)ARCH, models developed by Engle (Engle R. F., 1982) and Bollerslev (Bollerslev, 1986), directs towards exceptionally excessive volatility persistence in the interest rate process. In GARCH models, the model’s volatility depends on last period’s unexpected news,, as shown in Equation . However they are not without any intrinsic problems. In many practical uses of the model on interest rates and the model ended up with. Refer to papers such as Engle, D, & R (1987), Engle, V, & M (1990) and Flannery, A, & R (1992). This implies that half-life of a shock to volatility is infinite and volatility persists endlessly. Also (G)ARCH model allows interest rates to be negative. Lastly, (G)ARCH restricts its volatility to be a function lack of the interest rate levels which opposes to the theoretical literature as discuss earlier.

Equation GARCH Model’s volatily

These studies on volatility persistence influenced Brenner et al. (Brenner, Harjes, & Kroner, 1996) and Koedijk et al. (Koedijk, F.G.J.A., P.C., & C.C.P, 1994) to combine GARCH and CKLS models. Similarly, Longstaff and Schwartz (Longstaff & E.S, 1992) try a CIR term structure in a two-factor model that includes a stochastic volatility factor. These models are all approximate under discrete-time specification. There is now a universal agreement in the literature that short-rate models which account for both interest rate level effect and serial correlations in the volatility processes perform better than models that just parameterise serial dependence or level effect in the conditional variances. (Bali, 2000)

This thesis suggests an alternative way of modelling of the short-term interest rate’s conditional mean and conditional variances. We propose to parameterise the lagged difference of the short-term interest rate, in addition to the interest rate level effect in the conditional mean and conditional variances.

The main objective of this paper is to identify the level of significance of the new term together with the interest rate level in the diffusion model. This involves estimating the new parameters by ordinary least squares estimation (OLSE), weight least squares estimation (WLSE) and lastly by maximisation of the log likelihood function (MLE) of our proposed model. Confidence intervals are constructed for all three methods of estimations. For MLE, in order to estimate the 95% confidence intervals for our estimated parameters, we carry out wild bootstrap (Wu, 1986) on the model. We then carry out the significance test on the new parameter base on the bootstrap samples we generated to test whether is significant in the short-term interest rate model.

The rest of the paper is arranged as follows: Section 3 provides an overview of the models used in various journals and a discussion on the new model we proposed. Section 4 described the data used in investigation and section 5 discuss about the method of approach to carry out the investigation of the significance of and in our new defined short-term interest rate model. Section 6 shows estimates of parameters from ordinary least squares estimator, weight least square estimator and maximisation of the log likelihood functions. The corresponding 95% confidence intervals of all the estimated parameters are shown too. Section 7 discuss about the fitting of the new model and evaluate all the approaches. Section 8 will be the conclusion for the paper.

The short-rate interest rate models and the Redefined diffusion model

The proposed generalized continuous time short rate specification by Chan et al. (Chan, Karolyi, Longstaff, & Sanders, 1992) is as follows:

,

Equation CKLS model

where denotes the short-term interest rate, is a measure on drift, is an assess of the degree of mean reversion in rate levels , measures how responsive volatility is to interest rate levels and is a Brownian motion. The drift component of short-term interest rate and variance of unexpected changes in short-term interest rate are represented by and respectively. The parameter scales the variance of unexpected interest rate deviations. With proper restrictions on the parameters, andin CKLS model many of the interest rate models discussed in the section 2 can be obtained. The set of models is summarised by Table Summary of alternate models of short-term interest rate with different parameter restrictions below.

Model

Merton (1973)

0

0

CEV

0

Vasicek (1977)

0

Brennan-Schwartz (1977)

1

Dothan (1978)

0

0

1

CIR VR (1980)

0

0

Rendleman–Bartter model (1980) GBM

0

1

CIR SR (1985)

Table Summary of alternate models of short-term interest rate with different parameter restrictions

The CEV model is the constant elasticity of variance model of Cox (1975); the CIR VR model is the Cox, Ingersoll and Ross (1980) variable-rate model; and the GBM is the geometric Brownian model.

It is common to reduce the continuous time model Equation to the following Euler-Maruyama discrete time approximation in Equation . Euler-Maruyama approximation is the simplest and clear-cut approximation and it converges to continuous time-process. (Kloeden & E, 1992) Though there are other better approximations, it is out of the scope of this paper and we will not discuss it.

,

; .

Equation Euler-Maruyama discrete time approximation of the continuous model

In Equation , denotes the set of information available at the time point, and represents the conditional variance of unexpected interest rate changes. As discussed in the Introduction, this model has conditional heteroskedasticity depends solely on the levels of the interest rate. If, a rise in interest rate inevitably leads to increase in volatility and vice versa. Such implication is not ideal and hence we proposed to generalise Equation by allowing to vary with the new proposed variable known as the last period difference interest rate. We also proposed to include in the conditional mean. This is shown in Equation Suggested model.

;

;

Equation Suggested model

We assumed, and are all positive since the conditional variance is theoretically positive. In this model, when the change in interest rates is large in the previous period, the increases and conditional volatility will increase. If the model reduces back to CKLS model and when, model reduces to an ARCH model.

Our generalisation focuses mainly on one-factor model. Note that Equation Suggested model is a one-factor model as it has only one source of uncertainty that is present in the mean equation and the volatility depends on the lagged difference of the interest rate which contains the same source of uncertainty. We could extend our models to two-factor models like the model in Longstaff and Schwartz (Longstaff & E.S, 1992) by adding another uncertainty in our volatility process. However, it is not our objective in evaluating a two-factor model in this thesis.

For a more general case, the proposed model in Equation Suggested model can be further expanded to include higher lagged terms of the difference interest rates as show in Equation . In Equation , the conditional mean and conditional variance is a function ofand respectively in additional to the interest rate level effect.

,

;

.

Equation Suggested Generalised Eqaution

The Data

Year

Treasury bill rates

C:\Users\huiping\Desktop\New folder\new data plot.jpg

Figure 3-month Treasury bill rates

First difference interest rate

Year

C:\Users\huiping\Desktop\New folder\dy.jpg

Figure First difference interest rate

The data used in this paper is the weekly observations of the yearly yield on U.S. Treasury Bills with three months to maturity. (H.15 Selected Interest Rates) This data set is commonly used in papers that discussed short-term interest rate. The rates are computed as unweighted means of closing bid rates priced by at least five dealers in the secondary market. Weekly data was used instead of monthly data because the higher frequency weekly data is preferred over monthly data as it reduces the possible shortcomings of the discrete time approximation for the continuous time process.

The rate is sampled from January 1954 to January 2013 with a total of 3080 data points. The plot of the short-term interest rate series and the differenced interest rate are presented in Figure and Figure respectively and their summary statistics in Error: Reference source not found

From Figure , it is noticeable that volatility in the interest rate series varies directly with the present interest rate levels. It is exceptionally obvious in the Volker (1979 to 1983) regime where both the interest rate and volatility are high. This substantiates the interest rate level effect in the volatility model.

After the Volker monetary regime, the level effect is not as apparent. These empirical traits correspond with those described by Brenner et al. (Brenner, Harjes, & Kroner, 1996). This time-varying characteristic of the volatility in the sample data may signify that there are other terms on top of the level effects that can be equivalently crucial in explaining the volatility of short-term interest rates. Therefore we proposed to include the lagged term differenced interest rates expecting it to explain the volatility of the short-term interest rate.

The distribution of is skewed to the left and exhibit excess kurtosis. The Ljung-Box test statistic strongly rejects the null hypothesis of no serial correlation in the differenced data with p-value zero. The Jarque–Bera test strongly rejects the null hypothesis of normality in the interest rate series. These show that the data set of T-bill exhibits serial correlation and is not random distributed.

Variable

Mean

4.74%

-0.00040598%

Standard deviation

2.9872%

0.21521%.

Skewness

0.87148

-1.2919

Kurtosis

4.346

23.42

Ljung–Box test

56506 (0.00)

143.36 (0.0)

Jarque–Bera test

622.36 (0.001)

54352 (0.001)

Table Summary statistics for data used

Methodology

In section 5, we will discuss on the concepts on how we use ordinary least squares estimator, weighted least squares estimator and maximum likelihood estimator. We will also discuss how we obtain the bootstrap confidence interval for the MLE to evaluate our model.

Subsequently, the functions we used in Matlab and the algorithm used for these functions will also be mentioned.

Estimation of parameters in the diffusion model

Ordinary least square (OLS) estimator

OLS estimator is an unbiased estimator and we also use and as an initial value for the FMINCON in MATLAB to carry out maximisation of the log likelihood function outline in Section 5.1.3.

The generalised proposed model of Equation can be expressed in matrix form as

where and .

Equation Matrix representation of Equation

Consider the transformation of Equation as follows:

,

Equation Transformation of the short-term interest rate model to fit a regression model

After shifting the parameters, we can say that represents the independent white noise and Equation will be a simple linear regression model with the new transformed dependent variable and independent variable. This means that ordinary least square methods have their optimum properties with this model.

A common approach to estimate the parameters is as follows. We first regress on to obtain the least square estimate of. In order to estimate by OLS, we minimised the residual sum of squares (RSS), i.e. . After obtaining, we go on to regress on. The is obtained through minimising the RSS, i.e. .

The function used in Matlab is the REGRESS function which can solve the linear regression.

OLS approach ignores the conditonal heteroskedascity in the diffusion model and hence the parameters estimated no longer have minimum variace and may be rather inefficient. OLS approach underestimate the the variance of the error terms and standard deviation of the estimated parameters. (Kutner, Nachtsheim, Neter, & Li, 2005) Even though the 95% confidence interval will not be precise, we still hope to see that all these confidence intervals do not include zero for our parameters of interest, i.e. and, to further substantiate our initial objective of adding in and in the conditional variance and conditional mean model.

Weighted least square (WLS) estimator

The weighted least squares estimator maximise the efficiency of the parameters estimations through the introduction of weights. The weights will be inversely proportional to the variance at each level of the explanatory variables.

Since our conditional variance has unknown parameters, we will have unknown weights. In order to estimate the weights, a common approach is to use the squared residual regression. The steps are as follows:

Referring back to Equation

Step 1: Fit the regression model of on and estimate the conditional mean by assuming the weight as constant (all are 1), i.e. regress using ordinary least squares method. Estimate the residual by.

Step 2: Estimate the conditional variance based on the estimated residual by regressing on to get then estimate. The estimated parameters in the conditional variance are constraint to be positive before estimating.

Step 3: Repeat step 1 but with the weight being replaced by those obtained in step 2, i.e. inverse of.

For cases with estimated coefficients that diverge considerably from the OLSE, we repeat the weighted least squares process by using the residuals from the weighted least squares fitting to estimate the conditional variance function again and revised the weights. Typically, we will repeat the process at twice or thrice to stabilise the estimated regression coefficient. This is called iteratively reweighted least squares. This is commonly used in heteroskedasticity models, one of which is outline in Abdelhakim ( 2012) paper.

For WLSE, Matlab has function LSCOV which weights can be specified explicitly at each level of the explanatory variables.

Note that because we need to estimate the unknown conditional variances and then the weights, we introduce another source of variability. Thus this means the confidence intervals calculated here are only approximations.

Maximum likelihood estimator

Recall the matrix representation from Equation ,

.The probability density function (p.d.f) of is given by the multivariate normal density in Equation .

where and

Equation Probability density function (p.d.f) of

The log likelihood function of is then as follows:

Equation Log likelihood function of

The joint log likelihood function of the difference short-term interest rate series i.e. is:

Equation The joint log likelihood function of

The maximisation likelihood estimation is then obtained by maximising the log likelihood function with respect to and as follows:

In order to estimate the parameters and, the log likelihood is maximized using FMINCON function in MATLAB. Since FMINCON is a minimising algorithm we will minimiseinstead. FMINCON is used as it can constraint our parameters in the conditional variance to be positive.

The OLS estimates are then used in FMINCON for MLE approach with positive constraint on the parameters in the conditional variance model i.e. We simply set parameters intialisation value for negative estimate to zero in FMINCON.

FMINCON tries to obtain a minimum of the scalar function with numerous variables beginning at an initial estimate under the set of restriction. This is called the constrained nonlinear optimization. (Mathworks Documentation Center, 2012) The initial values for the algorithm must not be far off from the values that give the global minimum else the algorithm may end up with inaccurate MLE. Therefore we used the unbiased least square approach as the initial estimates for the FMINCON to make it more likely to achieve the correct global maximum of our log likelihood function.

Bootstrap for evaluation of the MLE parameters

Diffusion models do not have constant error variances as the variances are dependent on the values from the past. Hence the methods used for standard fitted regression models in evaluating the precision and confidence interval of the estimation cannot be applied to our diffusion model estimated by MLE. The bootstrap method developed by (Efron, 1976) will be used to estimate the precision and confidence interval of sample estimates by MLE. Bootstrap method required computer intensive calculations and in this paper this is carried out by Matlab. The code that is required to carry out the bootstrapping is included in the attached CD.

General procedure

As mentioned in Section 5.1, we have fitted multiple regression model for short term interest rate to obtained the initial estimates of and. and are then used as the initial values to obtain MLE for and . In order to evaluate the precision and to construct the confidence interval for these maximisation of likelihood estimates of the parameters we use bootstrap method. In summary, bootstrap requires sampling from the observed sample data with replacement. Subsequently, bootstrap method requires calculating the estimated parameters from the bootstrap sample using the same fitting method for the original data. This leads to the first bootstrap estimate and. This process is reiterated for a large number of times. The estimated standard deviation of the entire bootstrap estimateand, represented by s*{ and s*{}, is an estimation of the variability of the sampling distribution of and and thus a quantification for the precision of for (Kutner, Nachtsheim, Neter, & Li, 2005)

Bootstrap sampling

Wild bootstrap (Wu, 1986) is an extension to Efron’s bootstrapping and is used for models which exhibit heteroskedasticity. The idea is analogous to residual bootstrap where we resample the residuals. When the model fitted is adequate, the predictor variables can be considered unvarying and fixed X sampling is apt. For each replicated value, the new y is computed using the following equation:

Equation Bootstrap Sampling

The residuals are randomly multiplied by a random variable  with mean 0 and variance 1. The difference between wild bootstrap and simple residual sampling is that it assumes a symmetric distribution for the residuals. Wild bootstrap allows bootstrapping to perform better even for small sample sizes. can be of different forms and in this paper we use the standard normal distribution.

These Y* values are then regressed on the original X variables by the same procedure used for the real sample which is mention in Section 5.1.3 to obtain the bootstrap estimates.

Bootstrap Confidence Interval

Bootstrapping can be used to set up confidence intervals. First, we order the bootstrap estimates, ( and) from the smallest to the biggest. The confidence interval can be found simply by Efron's percentile method (Efron, 1976) which takes the percentile and the percentile from these ordered bootstrap estimates. These will then be the endpoints of the confidence interval of the bootstrap estimates. The function used in Matlab is PRCTILE. In this paper we will let to be 0.05 and hence we will be finding the 95% confidence interval of the bootstraps estimates.

For construction of bootstrap confidence interval, it requires a much larger bootstrap sample sizes than bootstrap estimate of precision as now we required the tails percentiles. We generated 5000 bootstrap samples for each estimated model for the estimation of bootstrap confidence interval.

The confidence interval created can be used as an alternative form for hypothesis testing with the null hypothesis denoted as the estimated parameters equal to zero. More details on this will be discussed in Section 7 during the evaluation of the model.

Empirical Results

In this section, results from the ordinary least square estimation (OLSE), weighted least squares estimation (WLSE) and maximum likelihood estimation (MLE) of the time-varying parameters of the model of short-term interest rate will be presented. The corresponding 95% confidence interval for each estimate will be shown. For MLE, we will be looking at the 95% bootstrap confidence interval. We looked into 4 different models with and without and in the conditional mean and conditional variance model and evaluate their importance in these different models. Different values are also considered.

Our aim is to see the significance of the parameters of in the conditional mean model and in the conditional variance in the various models considered.

Model considered

The models considered for studies are summarised below in Table . Table shows the set of variables that are included or not included in the various models considered.

Recall from Equation Suggested model:

;

;

in

Conditional mean model

Conditional variance model

(a) Mean model

0

(b) Variance model

0

(c) Both

(d) Variance model

0

0

0

Table Summary on the models considered for studies

As mention in section 2, we considered adding term in the CKLS model. We do it step by step by first considering the model with in the conditional mean, then in the conditional variance and finally and in both conditional mean and conditional variance. Different powers of the interest rate level, i.e., are considered subsequently. The values of considered and used here are 0.5, 1, 0.75 and 0.25.

Note that for model (a) to (d), they are related to previously proposed models in other journals mentioned in Table Summary of alternate models of short-term interest rate with different parameter restrictions. For example model (d) is analogous to that of Dothan (1978) and CIR VR (1980) where value is 1 and 3/2 respectively.

The estimated values in section 6.2 are all corrected to 3 significant figures.

The ordinary least squares estimates, weighted least squares estimates and the maximum likelihood estimates

Models with =0.5

(a) in conditional mean =0.5

OLSE

1.25E-02

-2.70E-03

8.93E-02

-1.02E-01

7.26E-02

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-1.23E-01

6.32E-02

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-8.17E-02

8.21E-02

WLSE

1.08E-02

-2.27E-03

7.79E-02

-1.02E-01

7.27E-02

Lower CI

1.04E-02

-3.61E-03

4.30E-02

-1.23E-01

6.32E-02

Upper CI

1.11E-02

-9.17E-04

1.13E-01

-8.18E-02

8.22E-02

MLE

1.07E-02

-1.74E-03

6.55E-02

5.21E-16

1.70E-02

Lower CI

5.33E-05

-9.57E-03

-3.03E-01

2.25E-06

8.07E-11

Upper CI

3.40E-02

4.67E-04

3.12E-02

3.57E-02

3.87E-02

Log likelihood function value: 3817.466

(b) in conditional variance =0.5

 

OLSE

1.13E-02

-2.45E-03

-8.04E-02

5.66E-02

2.40E-01

Lower CI

-2.94E-03

-5.00E-03

-1.01E-01

4.69E-02

2.06E-01

Upper CI

2.56E-02

9.38E-05

-5.95E-02

6.64E-02

2.74E-01

WLSE

7.83E-03

-1.49E-03

-8.05E-02

5.67E-02

2.41E-01

Lower CI

3.92E-03

-2.93E-03

-1.01E-01

4.69E-02

2.07E-01

Upper CI

1.17E-02

-4.05E-05

-5.96E-02

6.65E-02

2.75E-01

MLE

1.44E-03

2.24E-04

1.80E-11

5.38E-03

1.13

Lower CI

3.77E-05

-1.93E-02

5.29E-10

2.52E-11

3.60E-08

Upper CI

2.71E-02

3.85E-04

4.11E-02

4.60E-02

2.06

Log likelihood function value: 4471.270

(c) in both conditional mean and conditional variance =0.5

 

OLSE

1.25E-02

-2.70E-03

8.93E-02

-8.02E-02

5.66E-02

2.30E-01

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-1.00E-01

4.71E-02

1.97E-01

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-5.99E-02

6.60E-02

2.63E-01

WLSE

8.51E-03

-1.69E-03

6.94E-02

-8.04E-02

5.67E-02

2.31E-01

Lower CI

4.63E-03

-3.14E-03

2.85E-02

-1.01E-01

4.71E-02

1.98E-01

Upper CI

1.24E-02

-2.46E-04

1.10E-01

-6.00E-02

6.62E-02

2.64E-01

MLE

1.49E-03

1.89E-04

4.57E-02

7.24E-12

5.38E-03

1.13

Lower CI

5.88E-05

-2.03E-02

-1.20E-01

4.41E-10

2.54E-11

2.95E-02

Upper CI

2.72E-02

3.65E-04

6.69E-02

4.03E-02

4.96E-02

2.18

Log likelihood function value: 4472.319

(d) in conditional variance model only (assume mean equal zero)=0.5

OLSE

-8.07E-02

5.68E-02

2.42E-01

Lower CI

-1.02E-01

4.69E-02

2.07E-01

Upper CI

-5.97E-02

6.66E-02

2.76E-01

WLSE

-1.14E-01

6.29E-02

-4.27E-01

Lower CI

-1.56E-01

3.40E-02

-5.76E-01

Upper CI

-7.17E-02

9.18E-02

-2.78E-01

MLE

7.35E-12

5.38E-03

1.13

Lower CI

1.73E-10

6.47E-11

2.06E-05

Upper CI

3.70E-02

5.85E-02

2.62

Log likelihood function value: 4470.635

Models with =1

(a) in conditional mean =1

OLSE

1.25E-02

-2.70E-03

8.93E-02

-6.61E-02

2.36E-02

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-7.93E-02

2.13E-02

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-5.29E-02

2.60E-02

WLSE

4.84E-03

-1.08E-03

3.43E-02

-6.67E-02

2.38E-02

Lower CI

2.22E-03

-2.65E-03

-8.98E-05

-8.02E-02

2.14E-02

Upper CI

7.45E-03

4.83E-04

6.87E-02

-5.32E-02

2.62E-02

MLE

3.69E-03

-8.39E-04

4.86E-02

2.34E-04

7.80E-03

Lower CI

1.37E-05

-1.69E-02

-3.75E-01

9.10E-07

3.79E-11

Upper CI

2.36E-02

4.16E-04

3.24E-02

3.73E-02

9.00E-02

Log likelihood function value: 4059.360

(b) in conditional variance =1

 

OLSE

1.13E-02

-2.45E-03

-5.43E-02

1.92E-02

2.08E-01

Lower CI

-2.94E-03

-5.00E-03

-6.78E-02

1.67E-02

1.74E-01

Upper CI

2.56E-02

9.38E-05

-4.08E-02

2.17E-02

2.42E-01

WLSE

4.33E-03

-8.57E-04

-5.45E-02

1.92E-02

2.09E-01

Lower CI

2.11E-03

-2.28E-03

-6.81E-02

1.67E-02

1.75E-01

Upper CI

6.56E-03

5.63E-04

-4.09E-02

2.17E-02

2.43E-01

MLE

1.23E-03

1.15E-04

4.37E-05

2.97E-03

8.64E-01

Lower CI

3.45E-05

-1.72E-02

1.36E-06

5.61E-13

1.08E-01

Upper CI

2.45E-02

5.22E-04

3.56E-02

5.01E-02

3.29

Log likelihood function value: 4637.211

(c) in both conditional mean and conditional variance =0.5

 

OLSE

1.25E-02

-2.70E-03

8.93E-02

-5.42E-02

1.92E-02

1.98E-01

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-6.72E-02

1.67E-02

1.65E-01

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-4.11E-02

2.16E-02

2.31E-01

WLSE

4.61E-03

-9.66E-04

5.30E-02

-5.45E-02

1.92E-02

2.01E-01

Lower CI

2.39E-03

-2.39E-03

1.29E-02

-6.78E-02

1.67E-02

1.67E-01

Upper CI

6.83E-03

4.56E-04

9.31E-02

-4.12E-02

2.17E-02

2.34E-01

MLE

1.22E-03

8.56E-05

4.26E-02

4.56E-05

2.96E-03

8.62E-01

Lower CI

2.50E-05

-1.78E-02

-1.42E-01

4.71E-07

8.13E-13

1.43E-01

Upper CI

2.45E-02

5.17E-04

9.14E-02

3.46E-02

4.92E-02

3.44

Log likelihood function value: 4638.279

(d) in conditional variance model only (assume mean equal zero)=1

OLSE

-5.46E-02

1.92E-02

2.09E-01

Lower CI

-6.82E-02

1.67E-02

1.75E-01

Upper CI

-4.10E-02

2.18E-02

2.44E-01

WLSE

8.95E-03

-1.27E-02

5.70E-01

Lower CI

8.74E-03

-1.69E-02

2.14E-01

Upper CI

9.17E-03

-8.48E-03

9.26E-01

MLE

4.95E-05

2.96E-03

8.66E-01

Lower CI

6.19E-10

1.24E-11

1.02E-01

Upper CI

3.33E-02

7.15E-02

3.98

Log likelihood function value: 4636.538992

Models with =0.25

(a) in conditional mean =0.25

OLSE

1.25E-02

-2.70E-03

8.93E-02

-1.55E-01

1.44E-01

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-1.88E-01

1.21E-01

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-1.22E-01

1.67E-01

WLSE

1.09E-02

-2.34E-03

8.51E-02

-1.55E-01

1.44E-01

Lower CI

1.05E-02

-3.69E-03

5.00E-02

-1.88E-01

1.21E-01

Upper CI

1.12E-02

-9.99E-04

1.20E-01

-1.22E-01

1.67E-01

MLE

1.06E-02

-2.27E-03

6.25E-02

5.10E-15

2.79E-02

Lower CI

4.48E-07

-2.75E-02

-4.01E-01

6.13E-08

9.79E-10

Upper CI

3.03E-02

3.24E-04

2.35E-02

3.92E-02

4.39E-02

Log likelihood function value: 3543.610

(b) in conditional variance =0.25

 

OLSE

1.13E-02

-2.45E-03

-1.19E-01

1.10E-01

2.56E-01

Lower CI

-2.94E-03

-5.00E-03

-1.52E-01

8.63E-02

2.23E-01

Upper CI

2.56E-02

9.38E-05

-8.53E-02

1.33E-01

2.90E-01

WLSE

9.18E-03

-1.74E-03

-1.19E-01

1.10E-01

2.57E-01

Lower CI

5.07E-03

-3.18E-03

-1.52E-01

8.63E-02

2.23E-01

Upper CI

1.33E-02

-2.98E-04

-8.53E-02

1.34E-01

2.91E-01

MLE

9.55E-04

4.24E-04

3.47E-12

7.27E-03

1.34

Lower CI

6.29E-05

-2.33E-02

4.35E-10

1.09E-10

7.66E-09

Upper CI

2.88E-02

3.53E-04

4.55E-02

5.53E-02

1.89

Log likelihood function value: 4323.982

(c) in both conditional mean and conditional variance =0.25

 

OLSE

1.25E-02

-2.70E-03

8.93E-02

-1.18E-01

1.10E-01

2.46E-01

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-1.51E-01

8.69E-02

2.13E-01

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-8.61E-02

1.33E-01

2.79E-01

WLSE

9.92E-03

-1.96E-03

7.03E-02

-1.19E-01

1.10E-01

2.47E-01

Lower CI

5.85E-03

-3.40E-03

2.92E-02

-1.51E-01

8.68E-02

2.14E-01

Upper CI

1.40E-02

-5.16E-04

1.11E-01

-8.60E-02

1.33E-01

2.80E-01

MLE

1.01E-03

3.91E-04

4.61E-02

1.75E-11

7.27E-03

1.34

Lower CI

7.99E-05

-1.97E-02

-1.18E-01

4.67E-10

1.25E-10

1.95E-02

Upper CI

2.93E-02

3.34E-04

7.04E-02

4.49E-02

5.46E-02

1.77

Log likelihood function value: 4324.946

(d) in conditional variance model only (assume mean equal zero)=0.25

OLSE

-1.19E-01

1.10E-01

2.58E-01

Lower CI

-0.152722952

0.086372089

0.223870687

Upper CI

-0.085445622

0.133902095

0.292057424

WLSE

9.24E-02

-9.30E-02

2.6960

Lower CI

1.56E-02

-1.64E-01

2.5038

Upper CI

1.69E-01

-2.20E-02

2.8883

MLE

1.78E-11

7.28E-03

1.3409

Lower CI

1.55E-14

2.24E-10

7.11E-06

Upper CI

4.06E-02

6.56E-02

2.1563

Log likelihood function value: 4323.251

Models with =0.75

(a) in conditional mean =0.75

OLSE

1.25E-02

-2.70E-03

8.93E-02

-8.11E-02

4.13E-02

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-9.70E-02

3.67E-02

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-6.52E-02

4.59E-02

WLSE

1.06E-02

-2.23E-03

6.35E-02

-8.13E-02

4.13E-02

Lower CI

1.03E-02

-3.61E-03

2.86E-02

-9.74E-02

3.67E-02

Upper CI

1.10E-02

-8.43E-04

9.83E-02

-6.53E-02

4.60E-02

MLE

6.89E-03

-1.52E-03

6.43E-02

8.54E-06

1.13E-02

Lower CI

1.31E-06

-1.95E-02

-3.92E-01

1.26E-07

6.09E-11

Upper CI

2.59E-02

2.62E-04

2.79E-02

3.52E-02

5.95E-02

Log likelihood function value: 3996.160

(b) in conditional variance =0.75

 

OLSE

1.13E-02

-2.45E-03

-6.52E-02

3.29E-02

2.24E-01

Lower CI

-2.94E-03

-5.00E-03

-8.14E-02

2.80E-02

1.90E-01

Upper CI

2.56E-02

9.38E-05

-4.91E-02

3.77E-02

2.58E-01

WLSE

6.02E-03

-1.17E-03

-6.54E-02

3.29E-02

2.24E-01

Lower CI

2.84E-03

-2.59E-03

-8.17E-02

2.81E-02

1.90E-01

Upper CI

9.21E-03

2.62E-04

-4.92E-02

3.78E-02

2.59E-01

MLE

1.63E-03

9.78E-05

2.40E-11

4.03E-03

9.59E-01

Lower CI

1.46E-05

-1.91E-02

3.70E-09

7.05E-12

6.19E-04

Upper CI

2.55E-02

4.28E-04

3.66E-02

4.88E-02

2.54

Log likelihood function value: 4581.766

(c) in both conditional mean and conditional variance =0.75

 

OLSE

1.25E-02

-2.70E-03

8.93E-02

-6.51E-02

3.28E-02

2.13E-01

Lower CI

-1.69E-03

-5.24E-03

5.41E-02

-8.07E-02

2.82E-02

1.80E-01

Upper CI

2.68E-02

-1.61E-04

1.25E-01

-4.94E-02

3.75E-02

2.46E-01

WLSE

6.54E-03

-1.33E-03

6.48E-02

-6.53E-02

3.29E-02

2.15E-01

Lower CI

3.37E-03

-2.76E-03

2.45E-02

-8.12E-02

2.82E-02

1.82E-01

Upper CI

9.71E-03

9.41E-05

1.05E-01

-4.95E-02

3.76E-02

2.49E-01

MLE

1.65E-03

6.44E-05

4.40E-02

2.43E-11

4.03E-03

9.57E-01

Lower CI

4.16E-05

-1.74E-02

-1.24E-01

5.99E-07

5.27E-12

7.39E-02

Upper CI

2.56E-02

4.14E-04

7.55E-02

3.72E-02

4.55E-02

2.54

Log likelihood function value: 4582.840

(d) in conditional variance model only (assume mean equal zero)=0.75

OLSE

-6.56E-02

3.30E-02

2.25E-01

Lower CI

-8.19E-02

2.81E-02

1.91E-01

Upper CI

-4.93E-02

3.78E-02

2.59E-01

WLSE

2.24E-01

-1.00E-01

1.46

Lower CI

-1.99E-03

-2.14E-01

4.25E-01

Upper CI

4.49E-01

1.34E-02

2.50

MLE

2.56E-11

4.03E-03

9.61E-01

Lower CI

2.62E-10

1.84E-11

9.51E-03

Upper CI

3.44E-02

5.72E-02

2.87

Log likelihood function value: 4580.968

Discussion on the new model

Ordinary Least squares estimates

Theand have 95% confidence intervals that does not contain zero. This indicates that both and are significantly different from zero at 5% level for all the suggested models discuss in Table .

Weighted least squares estimates

The 95% confidence intervals of the estimated parameter under WLSE, the only model that show is not significantly different from zero at 5% level is model (a) with =1. The 95% confidence interval for this model is [-8.98E-05, 6.87E-02] which consist of zero. The 95% confidence intervals of for the rest of the models with in the conditional mean model (i.e. model (a) and model(c)) do not consist zero and hence conclude that is significantly different from zero at 5% level for these models.

The 95 % confidence intervals for of all the models with in the conditional variance model do not consist of zero. This shows that is significantly different from zero at 5% level.

The conclusion drawn from the WLSE confidence intervals does not differ much from that of OLSE. With the exception for one model, the confidence intervals for and does not consist of zeros for all different models.

Maximising likelihood estimates

We can see that for all models with in the conditional mean model, the estimated parameter has 95% bootstrap confidence interval that contains zero. This indicates that is not significantly different from zero in all these models and should not be included in the model.

We will now focus on the significance of parameter in the conditional variance model. Since the parameters in the conditional variance are constraint to be positive, in order to determine if the parameter is significantly different from zero, we look at the lower bounds of the 95% confidence intervals to see if it contains zero.

For all the models, lower bounds of the confidence intervals do not include zero. Models with =0.5 has MLE of to be roughly around 1.13 and lower bound ranging from order -08 to -02. Models with =1 has MLE of to be roughly around 0.86 and lower bound order to of -01. Models with =0.25 has MLE of to be roughly around 1.34 and lower bound ranging from order -09 to -02. Models with =0.75 has MLE of to be roughly around 0.96 and lower bound ranging from order -04 to -01. Thus we can conclude that the parameter estimates are significantly different from zero.

Conclusion

For parameter, the 95% confidence intervals by OLSE and WLSE, majority of them suggest that the estimated is significantly different form zero at 5% level. The 95% bootstrap confidence intervals however include zeros for all models which evidently show that should not be included in the conditional mean model. Due to this conflicting conclusion drawn from different confidence intervals, we cannot strongly conclude the adding of variable in the conditional mean.

All the three methods of estimation of our suggested short-term interest rate models have 95% confidence intervals for the estimated that does not consist zeros. Therefore we conclude that the estimated is significantly different from zero and this justifies the adding of in the conditional variance model.

For future research we can consider the model with higher lagged difference of the short-term interest rate as mentioned in Equation to further evaluate the significance of these terms.

Acknowledgement

I will like to thank Professor Xia Yingcun for always being so patience and willing in supervising me for the project and teaching me all the techniques used in modelling short-term interest rate. Thank you for your valuable guidance and advices.