The Interest Rate Risk

Published Date: 02 Nov 2017

Lin Huiping

U092944J

Supervisor:

Professor Xia Yingcun

Contents

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 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 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 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 triumph 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. These were shown in many practical uses of the model on interest rates and ended up with. Refer to paper 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.

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 function better than models that just parameterise serial dependence or level effect the conditional variances. (Bali, 2000)

This paper 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 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, weight least squares and lastly by maximising the log likelihood function (MLE). Confidence intervals are constructed for all three methods of estimation. For MLE, in order to estimate the confidence interval, 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 2 provides an overview of the models used in various journals and a discussion on the new model we proposed. Section 3 described the data used in investigation and section 4 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 5 shows estimates of parameters from ordinary least squares estimator, weight least square estimator and maximising of the log likelihood functions. The corresponding 95% confidence intervals of all the estimated parameters are shown too. Section 6 discuss about the fitting of the new model and evaluate the approach. Section 7 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 Brownian motion, is a measure on drift, is an assess of the degree of mean reversion in rate levels and measures how responsive volatility is to interest rate levels. 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 Introduction 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 to 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 above, 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 suggested 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 subsists in the mean equation and the volatility depends on the lagged difference 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 paper.

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

Year

First difference interest rateC:\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. 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 of two reasons. Firstly, 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. Secondly, weekly data set is of larger sample size and hence is a better set for study.

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 Fig.1 and Fig. 2 respectively and their summary statistics in Table .

From Fig. 1, 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 might signify that there are other terms on top of the level effects that might be equivalently crucial in explaining the volatility of short-term interest rates. Therefore we proposed to include the lagged term differenced interest rates which can be implied as the unexpected shock in explaining the volatility.

The kurtosis of the differenced data stated in Table is significantly greater than that of a standard normal distribution of 3. The negative Skewness coefficient is significantly different from zero a value indicating that the data set does not have a symmetric normal distribution. This is associated with the "asymmetric effect" where volatility increases with the fall in interest rates as compared to an increased in interest rate of the same magnitude. 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

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 4, concepts on how we use ordinary least squares estimator, weighted least squares estimator and maximum likelihood estimator will be reviewed. 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 be mentioned.

Estimation of parameters in the diffusion model

Maximum likelihood estimator

As reviewed earlier, the generalised proposed model of Equation can be expressed in matrix form as

Equation Matitx form of Equation

where and .

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

where and

The log likelihood function of is

The joint log likelihood function of the whole time series i.e. is

The maximisation likelihood estimation is then

In order to estimate the coefficient of and i.e. 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.

FMINCON tries to obtain a restricted minimum of the scalar function with numerous variables beginning at an initial estimate. This is normally 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 to first obtain initial estimates for the FMINCON to make it more likely to achieve the correct global maximum of our log likelihood function.

Ordinary least square (OLS) estimator

Our model for short-term interest rate has conditional heteroskedasticity. The estimated regression parameters through OLS approach are still unbiased however they no longer have minimum variance and may be rather inefficient. Mean square error of the OLS approach will underestimate the variance of the error terms and the standard deviation of the estimated parameters may extremely be underestimated too. Confidence intervals and test using t and F distributions for OLS will no longer strictly valid. (Kutner, Nachtsheim, Neter, & Li, 2005)

Since OLS estimator still provides an unbiased estimator, can be used as an initial value needed for the FMINCON to carry out maximisation of log likelihood function value.

Consider the transformation of Equation as follows:

,

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

After shifting the parameters around, we can say that denotes 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 are 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 we obtain we go on to regress on. The is obtained by minimising RSS, i.e. .

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

Weighted least square (WLS) estimator

The weighted least square estimator is a more suitable regression method for model with heteroskedasticity compared to the ordinary least square estimator. WLSE maximize the efficiency of parameter estimation. 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 using 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 then modified the weights. Typically, we will rerun the process at most twice 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.

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. Also, weighted least squares ignore the correlation between error terms and where s denotes number of periods apart and s=1,…,t-1. It simply assumes error terms are uncorrelated and set all correlation to zeros.

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

Bootstrap for evaluation of the MLE parameters

Diffusion models do not have constant error variances as the variances are correlated to 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.

General procedure

As mentioned in Section 4.1, 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 we use bootstrap method. In summary, bootstrap requires sampling from the observed sample data with replacement. Subsequently, bootstrap method requires calculating the estimated regression parameters from the bootstrap sample using the same fitting method for the original data. This leads to the first bootstrap estimate of and . This process is repeated 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 and (Kutner, Nachtsheim, Neter, & Li, 2005)

Bootstrap sampling

Wild bootstrap (Wu, 1986) is used for models which is heteroskedastic. This is suitable for our model as the volatility is not constant but varies with time. The idea is similar to residual bootstrap where we resample the residuals. When the regression model fitted is decent, the predictor variables can be considered unvarying, 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 4.1.1 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 tails percentiles. We generate 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. The null hypothesis is that the estimated parameters equal to zero, i.e. not significant. If the confidence interval contains zero, we do not reject the null hypothesis and if it does not include zero, we reject the null hypothesis. If the endpoints of the confidence intervals are near to zero, this might indicate that the estimates are in fact not that significant. More details on this will be discussed in Section 6 during the evaluation of the model from our estimation from the bootstrap samples.

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 here 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 3.

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 Introduction, we considered adding term in the CKLS model. We do it step by step by first considering putting in the conditional mean, then in the conditional variance and then and in both conditional mean and conditional variance.

Initially, we will use (power of interest rate level term in conditional variance) which is suggested by the square root models such as the CIR SR model mentioned in Table Summary of alternate models of short-term interest rate with different parameter restrictions. There is a study by Brenner, Harjes, & Kroner (1996) which studied alternative models of the short-term interest rate suggested nesting GARCH and CKLS model and their estimated is 0.459 which is close to 0.5 suggested by CIR SR model. Their model was quite analogous to ours here and hence we followed their findings on and use.

Different powers of the interest rate level, i.e., proposed previously in other journals (refer to Table Summary of alternate models of short-term interest rate with different parameter restrictions) are considered subsequently. The values of considered and used here are 0.5, 1, 0.75 and 0.25.

The estimated values 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

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

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

Log likelihood function value: 3817.466

(b) in conditional variance =0.5

 

MLE

1.44E-03

2.24E-04

1.80E-11

5.38E-03

1.134

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

2.060

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

Log likelihood function value: 4471.270

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

 

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

2.18E+00

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

Log likelihood function value: 4472.319

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

MLE

7.35E-12

5.38E-03

1.134

Lower CI

1.73E-10

6.47E-11

2.06E-05

Upper CI

3.70E-02

5.85E-02

2.618

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

Log likelihood function value: 4470.635

Models with =1

(a) in conditional mean =1

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

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

Log likelihood function value: 4059.360

(b) in conditional variance =1

 

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

3.2935

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

Log likelihood function value: 4637.211

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

 

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

3.4389

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

Log likelihood function value: 4638.279

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

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

3.9790

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

Log likelihood function value: 4636.538992

Models with =0.25

(a) in conditional mean =0.25

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

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

Log likelihood function value: 3543.610

(b) in conditional variance =0.25

 

MLE

9.55E-04

4.24E-04

3.47E-12

7.27E-03

1.3428

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

1.8835

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

Log likelihood function value: 4323.982

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

 

MLE

1.01E-03

3.91E-04

4.61E-02

1.75E-11

7.27E-03

1.3417

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

1.7749

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

Log likelihood function value: 4324.946

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

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

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

Log likelihood function value: 4323.251

Models with =0.75

(a) in conditional mean =0.75

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

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

Log likelihood function value: 3996.160

(b) in conditional variance =0.75

 

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

2.5350

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

Log likelihood function value: 4581.766

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

 

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

2.537

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

Log likelihood function value: 4582.840

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

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

2.866

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

Log likelihood function value: 4580.968

Evaluation/discussion of the new model

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 insignificant in all these models and should not be included in the model. Moreover, the interest rate level,, in the conditional mean model is also not significant proven by the bootstrap 95% confidence intervals. This corresponds well to the model in Brenner 1993 paper (Brenner, Harjes, & Kroner, 1996) where the all the parameters estimates for the conditional mean model are insignificant.

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.

Looking at the log likelihood function value across the different values of, it was the biggest when. This suggests that models with is a better fit for our suggested model. Also, from the MLE of it was of a reasonable with value less than 1. Considering the models with, model (c) with in both conditional variance and conditional mean and (d) with in both conditional variance and zero conditional mean has the largest log likelihood function value. Since both log likelihood function value differ from each other with merely a value of 2, we may say model (d) with lesser parameters is a better fit for short-term interest rate.

Ordinary Least squares estimates

As discussed in Methodology, we discussed how OLSE will not give accurate evaluations of the estimates’ standard deviation and so do the confidence interval. In fact we will underestimate the parameters’ standard deviation and will obtain a much smaller 95% confidence interval. This is clearly reflected in our results where the 95% confidence intervals of our OLSE parameters are much closer to the estimated value.

Even though the 95% confidence interval is not 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. Indeed, with the underestimated estimates’ standard deviation, the tighter confidence intervals for estimated and no longer consist of zeros. Lower bounds of the estimated are much further away from zero as compared to that of MLE. It is now of order -1.

Weighted least squares estimates

Compared to OLSE, WLSE is a slightly more efficient estimator. Nonetheless, the variability cause by the estimation of the conditional variance will affect the accuracy of the confidence intervals. Note that for model type (d) that estimates the conditional mean to be zero has exactly the same estimate as those in OLSE.

In summary, the conclusion drawn from the WLSE confidence intervals does not differ much from that of OLSE. The confidence intervals for estimated and does not consist of zeros for all different models.

Conclusion

Comparing the three methods of estimation of our suggested short-term interest rate model, the most accurate way of estimating confidence intervals will be through bootstrapping. Hence conclusion should be drawn using the results of MLE and its confidence interval to a larger extent.

For parameter, 95% bootstrap confidence intervals include zero for all models which evidently show that should not be included in the conditional mean model.

All three approaches conclude that the estimated is significantly different from zeros. This justifies the adding of in the conditional variance model.

Acknowledgement

I will like to thank the following people which had made this project possible.

To Professor Xia Yingcun: Thank you 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.

To my family and friends: Thank you for all the encouragement and moral support when I need it the most.