Modelling and Estimating Interest Rate: A Comparative ... · best estimate accuracy. Our methodology consisted of a three-step sequential process: 1) Box-Jenkins technique was employed

European Journal of Scientific Research

ISSN 1450-216X / 1450-202X Vol. 155 No 4 March, 2020, pp.440 - 454

http://www. europeanjournalofscientificresearch.com

Modelling and Estimating Interest Rate: A Comparative Study

of ARIMA, and ARIMA Kalman Model

Re-Mi S. Hage

Department of Mathematics and Statistics

Notre Dame University Louaize, Zouk Mosbeh, 72, Lebanon

Tel: + 961 9218950; Fax: +961 9 225 164

E-mail: rhage@ ndu.edu.lb).

Sarah J. Mghames

Mathematics and Statistics, Notre Dame University

Zouk Mosbeh, 72, Lebanon

Tel: + 961 9218950; Fax: +961 9 225 164

E-mail: [email protected]

Abstract

In the past years, economic focus was centered on building a model to establish

interest rate as an important financial key and estimate. In this paper, currently available

different interest rate modeling approaches were discussed to conclude the model with the

best estimate accuracy. Our methodology consisted of a three-step sequential process: 1)

Box-Jenkins technique was employed to derive an Autoregressive Integrated Moving

Average (ARIMA) where the parameters were estimated using the Maximum likelihood

estimation technique. 2) the ARIMA model was considered as a state space in the Kalman

Filter algorithm where the estimation of its parameters was calculated at each time

increment using the Yule-Walker methodology. 3) Lastly, the estimated values were

compared to those of the ARIMA model alone. After comparing both models, ARIMA

combined with the Kalman Filter Algorithm offered an increased accuracy. Our results

show that the mean absolute percentage error for ARIMA standalone was close to 100%,

while it was close to 0% using the Kalman Filter.

Keywords: Box-Jenkins ARIMA models, interest rate, Kalman filter, Yule-Walker.

I. Introduction

Interest rate is a decisive financial element taken into serious consideration in decision-making by

many corporate entities including financial organizations, policymakers, and investors. An accurate

estimation and forecasting of interest rates can therefore provide valuable information for the financial

market by reducing interest rate risks; individuals and firms can take the appropriate measures to avoid

undesirable financial consequences especially for short-rate policies [1, 2, 3].

The first model for estimating and forecasting interest rates was formulated by Merton in 1973

[4], following which many other models were proposed including and not limited to: Brennan and

Schwartz [5,6], Vasicek [7], Dothan [8], Cox Ingersoll and Ross [9], Longstaff [10], Hull and White

[11], and Black and Karasinski [12].

However, this wide array of models differ in properly capturing the stochastic behavior of the

short term interest rate. For example, the Merton and Vasicek models are two stochastic models for

discount bond prices, which imply that the conditional volatility of changes in interest rate remains

Modelling and Estimating Interest Rate: A Comparative Study of ARIMA, and

ARIMA Kalman Model 441

constant. In contrast, the Cox Ingersoll and Ross model considers this change isproportional to r. Other

authors[13-15] applied Markov switching in order to model interest rates. To capture the volatility, a

few researchers [16-18] combined Markov switching with Garch or stochastic volatility; Lanne and

Saikkonen [19] used a mixture of the autoregressive process with time-varying transition probability

and Garch. While, Maheu and Yang [20] included the Markov switching of infinite dimension; a

Bayesian nonparametric model that allows changes in the unknown conditional distribution over time.

Chua [21] developed the Bayesian approach further by predicting a short term interest where posterior

model probabilities used in this model averaging were restricted to more recent predictive densities,

instead of those based on marginal likelihoods of the data set.

ARIMA is one of the most common and important time series model. Generally, any set of

financial data that changes over time is ascribed as time series data and can fit in an auto-regressive

moving average ARMA or ARIMA model (S&P 500 index volatility [22], forecasted short-term

interest rates [23] and in exchange rate ([24] – [27]). Pagan and Schwert [28] found evidence that

Autoregressive Integrated Moving Average (ARIMA) models give the best results when compared to

nonparametric models. Lanne and Saikkonen [29] took it further by combining a mixture

autoregressive process with time-varying transition probabilities and GARCH.

Parameter estimation is an ongoing basic challenge in statistical analysis. In the context of time

series modeling, the common parameters estimation methods are the ordinary least squares and the

maximum likelihood method (MLE). In addition, there are two main benefits to represent a dynamic

system in a state space form; it allows unobserved variables (identified as the state variables) to be

incorporated into an observable model which consequently can be analysed using a powerful recursive

algorithm such as the Kalman filter (KF).

The Kalman Filtering algorithm is employed in this study. KF is an algorithm that estimates a

dynamic system’s state based on a series of noisy measurements. KF has been widely used in several

application areas including engineering, aeronautics, and recently in Financial Mathematics.

The objective of this paper is to find the best fit ARMA time series model for a set of financial

data; we chose US interest rates as our experimental data. Then, this data is processed through the KF

algorithm to confirm that the model ARMA-KF offers more accurate estimations compared to

standalone ARMA time series models (MLE technique); the parameters of the ARMA models are

estimated each time a new observation is entered using the Yule-walker [30-31] methodology.

This paper is divided into three main sections. Section 1 introduces the Kalman Filter, then

Section 2 overviews the time series analysis, elaborates the Box-Jenkin [32-34] approach commonly

used in finding the best fit ARMA model for a financial time series data, to finally write an ARMA

time series model in state space form. Lasly, the Kalman filtering algorithm is applied in Section 3 to a

set of interest rates data that follows a certain ARMA model, and resulting estimates are compared

with the estimates of the standalone counterpart.

II. Methodology A. Kalman Filter

The Kalman Filter (KF) [35-37] is a mathematical algorithm that provides an estimation of the

parameters of interest; often not fully observable mainly due to noise corruption. In theory, the KF

entails a recursive solution that estimates the state of a linear discrete-time dynamic systembased on a

series of noisy measurements. The process of finding the state estimate is divided into two steps: the

predicted and the updating step. During the prediction step, the filter produces an estimate of the

parameter of interest based on an explicit statistical model describing its evolution over time. Once the

new observation is available, the prediction is updated using the so-called Kalman gain factor weight.

Under the assumption of Gaussian error statistics, the Kalman gain is chosen in a way to ensure that

442 Re-Mi S. Hage and Sarah J. Mghames

the resulting estimate is an optimal estimate that minimizes the mean squared error function (i.e.

unbiased estimator with minimum variance).

The below calculations will use the following listed notations: x�: State vector at time k z�: Observation vector at time k u�: Input control vector at time k F�: State transition matrix G�: Input transition matrix H�: Observation transition matrix v�: Measurement noise vector w�: Process noise vector Q�: Process noise covariance matrix R�: Measurement noise covariance matrix x��|�: Estimation of x at time k based on time i; with k ≥ i P�: Covariance Matrix K�: Kalman Gain Matrix

A.1 State Space Model

The state space model is described by the following two equations:

State transition equation of the form (1) or (2)

x�� = F�x� + G�u� + w� (1)

x� = F�x�� + G�u� + w� (2)

or

The measurement equation of the form (3)

z� = H�x� + v� (3)

where the matrices F�, G� and H� are deterministic and known quantities

A.2 Assumptions

We start by assuming that the state equation and the measurement equation stated above are both linear

with discrete times.

The process and measurement noise vectors w� and v� are assumed to be two uncorrelated

white noise processes, i.e. E�w�v� ! = E"w�#E�v� ! = $ ∀ k, l having zero-mean, i.e. E"w�# =E"v�# = $ ∀ k and variance-covariance matrices respectively as follows:

Q� = E"w�w� # = (Var"w��# 0 … 00 Var"w�-# … 00 0 … 00 … 0 Var"w�.#/

with w� = (w��w�-⋮w�./ and



R� = E�v�v� ! = (Var"v��# 0 … 00 Var"v�-# … 00 0 … 00 … 0 Var"v�1#/

with v� = ( v��v�-⋮v�1/

Furthermore, the system's state x is uncorrelated with both error terms w� and v�.

Finally, the initial system's state has a known mean and variance-covariance matrix given by

(4)

x�2|2 = E"x2# and P2|2 = E[�x2 − x�2|2!�x2 − x�2|2! ] (4)

A.3. Algorithm

The second step is to implement the following KF derivation (Fig.1):

Figure 1: Kalman filter algorithm

B. Time Series

The purpose behind studying time series is to find a mathematical model that can approximately

generate the historical data of the time series in addition to forecasting future observations.

A zero-mean moving average process of order q MA(q) is a linear combination of

independently and identically distributed (i.i.d.) white noise random variables ϑ_k's with mean zero

and variance δ^2 given by (5):The purpose behind studying time series is to find a mathematical model

Initial Estimates:

Fix 6�$|$ and 7$|$

6�8+9|8 = :86�8|8 + ;8<8 78+9|8 = :878|8:8= + >8

Predictive Estimates:

?8+9 = 78+9|8@8+9=�@8+978+9|8@8+9= + A8+9!−9 6�8+9|8+9 = 6�8+9|8 + ?8+9�B8+9 − @8+96�8+9|8! 78+9|8+9 = "C − ?8+9@8+9#78+9|8

Updated Estimates:

Measure zk+1

K = 0

k = k+1

444

that can approximately generate the historical data of the time series in addition to forecasting future

observ

independently and identically distributed (i.i.d.) white noise random variables

variance

where B is the Back

(ACF) that cuts off after lag q and partial autocorrelation function (PACF) that decays exponentially

after lag q.

Πexponentially after lag p and PACF that cuts off after lag p.

AR(p) given by (7)

⋯exponentially decaying functions AR(p) starting after lag q and PACF which is a mixture of

exponentially decaying functio

model within the class of auto

444


observations.

A zero


variance δ- given by (5):

where B is the Back

MA(q) processes are always stationary. They are characterized by its autocorrelation function


after lag q.

A zero

AR(p) processes are stationary under the condition that the roots of

ΠHBH � ⋯ Πexponentially after lag p and PACF that cuts off after lag p.

A zero

AR(p) given by (7)

ARMA(p,q) are stationary under the condition that the roots of

⋯ ΠJBJ � 0exponentially decaying functions AR(p) starting after lag q and PACF which is a mixture of


Box-Jenkins methodology is to a set of procedures for finding the best fit for a time series


zk �zk �

zk � Πϑk �

zk � Li�


A zero-mean moving average process of order q MA(q) is a linear combination of


given by (5):

where B is the Back-Shift



A zero-mean auto


ΠJBJ � 0 lie outside the unit circle. They are characterized by their

exponentially after lag p and PACF that cuts off after lag p.

A zero-mean auto

AR(p) given by (7)


0 lie outside the unit circle. ARMA(p,q) is characterized by its ACF which is a mixture of

exponentially decaying functions AR(p) starting after lag q and PACF which is a mixture of


Jenkins methodology is to a set of procedures for finding the best fit for a time series


� �1 � Ψ1B� Ψ(B) ϑk

Π1zk−1 � Π2z� �1 − Π1B −

L Πi zk−i

p

�1� L

i


mean moving average process of order q MA(q) is a linear combination of


Shift operator defined as B^j



mean auto-regressive process of order p AR(p) can be written as (6):


lie outside the unit circle. They are characterized by their


mean auto-regressive moving average ARMA(p,q) process is a mixture of MA(q) and


lie outside the unit circle. ARMA(p,q) is characterized by its ACF which is a mixture of


exponentially decaying functions AR(q) starting after lag p.


model within the class of auto-regressive moving average ARMA or ARIMA models (Fig.2):

Figure 2

� Ψ2B2 �

zk−2 � Π3zk−− Π2B2 − Π

ϑk � Π(B) z

L Ψi ϑk−i

q

i�1�




operator defined as B^j



regressive process of order p AR(p) can be written as (6):




regressive moving average ARMA(p,q) process is a mixture of MA(q) and




ns AR(q) starting after lag p.


regressive moving average ARMA or ARIMA models (Fig.2):

Figure 2: Box and Jenkins methodology

Ψ3B3 � ⋯

−3 � ⋯ Πp zk−Π3B3 − ⋯ Πpzk

ϑk




operator defined as B^j ϑ_k=ϑ_(k-j),j











ns AR(q) starting after lag p.



: Box and Jenkins methodology

ΨqBq!ϑk

−p + ϑk Bp!zk

Re-Mi S. Hage




j),j≥0












: Box and Jenkins methodology

Mi S. Hage and



independently and identically distributed (i.i.d.) white noise random variables ϑ�'s with mean zero and




AR(p) processes are stationary under the condition that the roots of 1 �lie outside the unit circle. They are characterized by their


ARMA(p,q) are stationary under the condition that the roots of 1 � Π�B �lie outside the unit circle. ARMA(p,q) is characterized by its ACF which is a mixture of




and Sarah J. Mghames



's with mean zero and



� Π�B � Π ACF that decays


� Π-B- � Πlie outside the unit circle. ARMA(p,q) is characterized by its ACF which is a mixture of




Sarah J. Mghames



's with mean zero and

(5)



(6)

Π-B- �ACF that decays


(7)

ΠHBH �lie outside the unit circle. ARMA(p,q) is characterized by its ACF which is a mixture of





1. The stationarity condition can be tested using the ACF and PACF plots. The sampled ACF and

PACF of a stationary process cuts off completely or decays gradually after a few lags. In

contrast, a very slow decay in ACF and/or PACF plots indicates a non-stationarity trend.

2. The preliminary tool for model identification is the use of both sample ACF and PACF plots

with 95% confidence; the significant lags can be obtained from the correlogram

(autocorrelation plot) where their corresponding auto-correlation coefficients lie outside the

band given by: ± -√T.

3. The step right after identifying the time series model is to estimate its parameters based on the

Least squares method and/or the Maximum likelihood estimation technique.

4. The t-test is used to test whether or not the selected model fits the data best. In other words,

each chosen parameter chosen is tested for significance. The null hypothesis to be tested is: H2: Parameter� = 0; i = 1,2, . . , max (p, q)

C. ARMA Models in State Space Form

The ARMA(p,q) model given by (7) can be put in state space form with b=max(p, q+1). The state

vector is defined in (8)

(8)

The measurement equation is given by (9)

(9)

The state transition equation given by (10)

(10)

where

6k = ( zk Π2zk−1 � ⋯ � Πpzk−b�1 � Ψ1ϑk � ⋯ � Ψb−1ϑk−b�2⋮ Πbzk−1 � Ψbϑk/

Bk = 31 $′ b−15 6k⋮ Πbzk−1 � ΨbϑkBk = 31 $′ b−15 6k

1 0 … 00 1 … 0⎤ Bk = 31 $′ b−15 6k

6k =⎣⎢⎢⎢⎡ Π1 1 0 … 0 Π2 0 1 … 0⋮ ⋮ ⋮ ⋱ ⋮ Πb−1 0 0 … 1 Πb 0 0 … 0⎦⎥⎥

⎥⎤ 6k−1 �⎣⎢⎢⎢⎡ 1 Ψ1⋮ Ψb−2 Ψb−1⎦⎥⎥

⎥⎤ ϑk

Fk =⎣⎢⎢⎢⎡ Π1 1 0 … 0 Π2 0 1 … 0⋮ ⋮ ⋮ ⋱ ⋮ Πb−1 0 0 … 1 Πb 0 0 … 0⎦⎥⎥

⎥⎤ ; Gk = 0; <k = 0; dk =⎣⎢⎢⎢⎡ 1 Ψ1⋮ Ψb−2 Ψb−1⎦⎥⎥

⎥⎤ ϑk ; Hk = 31 $′ b−15 ; ek = 0; Ak = 0;

>k =⎣⎢⎢⎢⎢⎡

δ2 Ψ1δ2 … Ψb−2δ2 Ψb−1δ2 Ψ1δ2 Ψ12δ2 Ψ1 Ψ2δ2 ⋯ Ψ1 Ψb−1δ2⋮ Ψ1 Ψ2δ2 ⋱ ⋱ ⋮ Ψb−2δ2 ⋮ ⋱ Ψb−22δ2 Ψb−1 Ψb−2δ2 Ψb−1δ2 Ψ1 Ψb−1δ2 … Ψb−1 Ψb−2δ2 Ψb−12δ2 ⎦⎥⎥⎥⎥⎤


D. Performance Test

After fitting a selected model to the time series data using Box-Jenkins approach, its performance can

be tested using the below statistical indicators:

Error (E) is the difference between the estimated value (EV) and the actual value (AV) as in

(11):

(11)

Absolute Error (AE) is the absolute value of the difference between the estimated value and the

actual value as in (12):

(12)

Relative Absolute Error (RAE) is the absolute value of the ratio of the difference between the

estimated value and the actual value as in (13):

(14)

Mean Absolute Percentage Error (MAPE) is the mean of the relative absolute error of the n

estimations as in (14):

(14)

III. Application The selected data consists of 110 quarterly interest rates (quarter 1 = Q1, quarter 2 = Q2, quarter 3 =

Q3, and quarter 4 = Q4) of the United States for a duration that extends from 1964_Q1 to 1991_Q2

(k=1 to k=110) where Interest rate (r) denotes the three-month Treasury bill rate. This data was

extracted from the Citibase Economic database, University of California, San Diego (Fig. 3).

Figure 3: Interest Rates from 1964_Q1 to 1991_Q2

As indicated by the plot above, the studied data has a sample mean of 7.18% and a sample

variance of 7.65 with maximum r = 15.904% in 1981_Q3 and minimum r = 3.514% in 1972_Q1.

fg = fhg − ihg

ifg = |fhg − ihg |fg = fhg − ihgifg = |fhg − ihg | jifg = | fhg −ihgihifg = |fhg − ihg |jifg = | fhg −ihgihg |

kilf = m∑ jifgog=1 p ∗jifg = | ihgkilf = m∑ jifgog=1o p ∗ 100



Fig. 3 also shows that the pattern of the interest rate per quarter is approximately the same.

Therefore, there is no need to divide the data into quarters while building its model.

A. Box & Jenkins Model Identification

In this section, the Box & Jenkins approach is applied to determine its corresponding time-series model

that will be used as a state space model in the Kalman Filter.

1. Stationarity: Fig. 3 shows that the original data series is non-stationary and non-seasonal.

Furthermore, the corresponding sample ACF (Fig. 4) shows to be non-stationarity as well

because it dies down extremely slowly. Thus, the first transformation of the original data series

should be performed and checked for stationarity.

Figure 4: Sample ACF of Original Data Series

The transformed data consists of 109 values (k=2 to k=110) with a mean of 0.02% and a

variance of 1.02. In addition, Fig. 5 shows that the pattern per quarter is approximately the same.

Therefore, there is no need to divide the transformed data into quarters. From the patterns of the

sample ACF and PACF plots of the transformed data (Fig. 6 and 7), one can recognize that the data is

now stationary.

Figure 5: Data Series after one differencing


Figure 6: Sample ACF of Differenced Data

Figure 7: Sample PACF of Differenced Data

2. Model Identification: Now that the transformed data appears to be stationary, its sample ACF and PACF

are analyzed to identify an appropriate model for it (i.e. orders p and q) from the large family of

available auto-regressive moving average models. The sample ACF decays fairly quickly and the

sample PACF cuts off after a lag c which turns out to be 4 or 5. This behavior is consistent with an

autoregressive model which is either of order 4 (cut off after lag 4) or 5 (cut off after lag 5) i.e. AR(4) or

AR(5).

3. Parameter Estimation: The parameters of an ARMA(4,1,0) or ARMA(5,1,0) are estimated based on the

maximum likelihood method which gives in our case the following results (Fig. 8 and 9 respectively).

Given that z� = r� − r��(k = 2, … ,5) and based on Fig. 9, fitting an AR(4) to tranformed data yields

(15)

(15) for k ≥ 6 and ϑ�~N(0, 0.885177)

Given that z� = r� − r��(k = 2, … ,5) and based on Fig. 10, fitting an AR(5) to transformed

data yields (16)

for k ≥ 7and ϑ�~N(0, 0.882565)

4. Testing: the t-test explicitly shows that the first four AR coefficients in AR(4) or AR(5) are significant

at α = 0.05. Applying the t-test on the fifth coefficient, namely Π{, results in the following hypotheses: H2: Π{ = 0 versus H}: Π{ ≠ 0

zk = −0.0143155 − 0.525325zk−1 − 0.875319zk−2 − 0.33017zk−3 − 0.397905zk−4 � ϑk

zk = −0.013971 − 0.503797zk−1 − 0.857341zk−2 − 0.282451zk−3 − 0.369112zk−4 � 0.0545495zk−5 � ϑk


ARIMA Kalman Model

Since the p

the fifth parameter is 0. Hence, the AR(4) given by (16) is the best suitable model for the transformed

data for k

B. Modeling Interest Rates using ARMA time

The Box & Jenkins approach theoreti

transformed data.

In this section, the AR(4) model is tested to see wheter it practically fits the transformed data.

Subsequently, the estimated values of the transformed data are caluclated u

determined and compared to 1.

Fig. 10 illustrates the AR(4) estimated values vs. the actual transformed data for a sample of 98

(k=6 to k=103). This graph shows that the AR(4)

for the interest rates.

zk


ARIMA Kalman Model

Since the p-value for the fifth parameter is larger than

the fifth parameter is 0. Hence, the AR(4) given by (16) is the best suitable model for the transformed ≥ 6:

Figure 8:

Figure 9



transformed data.






or the interest rates.

= −0.0143155


ARIMA Kalman Model

value for the fifth parameter is larger than


igure 8: Parameters Estimation when fitting AR(4) to transformed data

Figure 9: Parameters Estimation when fitting AR(5) to transformed data








0143155 − 0.525325


value for the fifth parameter is larger than


Parameters Estimation when fitting AR(4) to transformed data

: Parameters Estimation when fitting AR(5) to transformed data


The Box & Jenkins approach theoretically proved that the AR(4) is the best suitable model for the





525325zk−1 − 0


value for the fifth parameter is larger than α,




B. Modeling Interest Rates using ARMA time-series model (MLE technique)

cally proved that the AR(4) is the best suitable model for the




(k=6 to k=103). This graph shows that the AR(4) model does not give a practically accurate estimation

0.875319zk−2


we therefore fail to reject the hypothesis that




series model (MLE technique)





model does not give a practically accurate estimation

2 − 0.33017z






series model (MLE technique)



Subsequently, the estimated values of the transformed data are caluclated using AR(4), the MAPE is



zk−3 − 0.397905







sing AR(4), the MAPE is



397905zk−4 � ϑ

449



(16)



sing AR(4), the MAPE is



ϑk


Figure 10: AR(4) Estimated Values vs. Actual Transformed Data

Fig. 11 is the difference between the AR(4) estimated values and the actual transformed data;

i.e. the error curve. The MAPE is very close to 1 indicating a significant error between the estimate and

the actual data.

Figure 11: Error between AR(4) Estimated Values and Actual Transfromed Data

We can conclude from the above results that applying time-series alone to interest rates data

does not give a practicalaccurate estimation. We will consequently use time-series as a state space

model in the Kalman filter in order to check any changes in the accuracy.

C. Modeling Interest Rates using Kalman Filter with ARMA time series-model as state space

The transformed data fits an AR(4) which can be put in state space form as follows:

The state vector is defined in (17)

(17)

The measurement equation is given in (18)

(18) The transition equation is given in (19)

6k = ( zk Π2zk−1 � Π3zk−2 � Π4zk−3 Π3zk−1 � Π4zk−2 Π4zk−1/,

Bk = 31 0 0 05 6k Π4zk−1Bk = 31 0 0 05 6k

1 0 0 1



(19)

With

The parameters Π�, Π-, ΠH, Π�, μ, and δ- are estimated using Yule-Walker at the realization of

each new observation. Thus, F�, u�, and Q� change in respect to time, i.e. with the estimation of the

new parameters.

The initial estimated parameters; state vector and covariance matrix, are found using the first

four values of the transformed data (k=2 to k=5) and are given by: Π� = −0.8095, Π- = 0.5847, ΠH =−0.2163, Π� = 0.029,

μ = 1, δ- = 0.0129, 62|2 = ( 0.2220.1027−0.03980.0054 / and 72|2 = (0.0129 0 0 00 0 0 00 0 0 00 0 0 0/

Running the KF algorithm 98 times recursively results in acheiving the estimated values of the

transformed data for k=6 to k=103.

Fig. 12 illustrates the AR(4) with KF estimated values vs. the actual transformed data (k=6 to

k=103). This graph shows that the AR(4) with the KF model gives an accurate estimation for the

interest rates.

Figure 12: AR(4) with KF Estimated Values vs. Actual Transformed Data

Fig. 13 is the difference between the AR(4) with KF estimated values and the actual

transformed data using the same scale of Fig. 10 ; i.e. the error curve. The MAPE is very close to 0

indicating a negligible error between the estimation and the actual data.

B = 31 0 0 05 66k = (Π1 1 0 0Π2 0 1 0Π3 0 0 1Π4 0 0 0/ 6k−1 � (1000/ ϑk � ��000�

1 0 0 0 � 10

(1

Fk = (Π1 1 0 0Π2 0 1 0Π3 0 0 1Π4 0 0 0/ ; Gk = (1 0 0 00 1 0 00 0 1 00 0 0 1/ ; <k = ��000� ; dk = (1000/ ϑk ; Hk = 31 0 0 05 ; ek= 0; Ak = 0;

>k = (δ2 0 0 00 0 0 00 0 0 00 0 0 0/


Figure 13: Error between AR(4) with KF Estimated Values and Actual Transformed Data

The error curve is now illustrated in Fig. 14 using a different scale to zoom in on the

differences. The average error and standard deviation are approximately 0.

Figure 14: Error between AR(4) with KF Estimated Values and Actual Transformed Data

We can conclude from the above results that applying time-series with KF to interest rates data

increases the accuracy of the estimation.

Conclusion Interest rate is an imperative factor in borrowing and lending. Therefore, accurate

estimation/forecasting of interest rates is one of the critical economic variables regarding decision-

making. Interest rate models with different parameter combinations are successful to some extent.

However, the primary drawback of the majority of the interest rate models is the instability of its

parameters.

To overcome these shortcomings, we used the Box-Jenkins approach to find the ARIMA time-

series model that best fits the interest rate data. The mean absolute percentage error (MAPE) of the

ARIMA model using the maximum likelihood estimation technique as a parameter estimation

technique was around 100% indicating a significant error between the estimation and the actual data.

Hence, we re-estimated the parameters of this time series-based model using the Yule-Walker method

and the Kalman Filter algorithm to get a better estimation of the quarterly interest rates where MAPE

was close to 0%.

The main contribution of this article is the efficiency of the Kalman filter stochastic approach

even to estimate interest rates by only using the past values of the difference between the log-levels.

Moreover, the Kalman filter offers the possibility to introduce an error model that enables the detection

and exclusion of outliers. Thus, the estimation is robustified by statistic tests, which was made possible



by the Kalman filter formalism. In addition, estimation of the parameters was determined at each

increment of time using the Yule-Walker methodology Finally, the key point or novelty in the

evolution of this model lies in the estimation of the parameters re-calculated at each time increment.

More research is needed to better understand the mechanism of the interest rate by comparing the

Kalman Filter having ARIMA time series model as a state space with multiple regression combined

with metaheuristic algorithm.

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Modelling and Estimating Interest Rate: A Comparative ... · best estimate accuracy. Our methodology consisted of a three-step sequential process: 1) Box-Jenkins technique was employed