Question

An investment analyst is interested in examining the long run relationship between the
3-month (91 days) Treasury bill (TB) rate and 6-month (182) Treasury bill rate. Data on
these series were obtained from the Bank of Ghana website, spanning from January, 2006,
to November, 2017.
Some preliminary tests were performed on the data and their outputs are given below:
CASE 1:
Augmented Dickey-Fuller Test on the 3-month TBill rate series.
Original (level) data
data: tb91
Dickey-Fuller = -2.3288, Lag order = 5, p-value = 0.4396
First Difference data
data: diff(tb91)
Dickey-Fuller = -3.7109, Lag order = 5, p-value = 0.02575
Augmented Dickey-Fuller Test on the 6-month TBill rate series.
Original (level) data
data: tb182
Dickey-Fuller = -2.0624, Lag order = 5, p-value = 0.5504
first difference data
data: diff(tb182)
Dickey-Fuller = -4.3089, Lag order = 5, p-value = 0.01
CASE 2:
Trace Test
######################
# Johansen-Procedure #
######################
Values of test statistic and critical values of test:
test 10pct 5pct 1pct
r <= 1 | 6.21 6.50 8.18 11.65
r = 0 | 30.23 15.66 17.95 23.52
S. Twumasi-Ankrah
Max Eigen Test
######################
# Johansen-Procedure #
######################
Values of teststatistic and critical values of test:
test 10pct 5pct 1pct
r <= 1 | 6.21 6.50 8.18 11.65
r = 0 | 24.03 12.91 14.90 19.19
a) Why do you think the investment analyst performed the above tests in CASE 1 and
CASE 2?
b) Interpret the test(s) in both CASE 1 and CASE 2.
c) Based on the information available, what time series model would you advise the
investment analyst to fit to the data? Justify

Answer 1

Modelling the Rate of Treasury Bills in Ghana

Ida Anuwoje Logubayom*, Suleman Nasiru, Albert Luguterah

Department of Statistics, University for Development Studies, P. O. Box 24 Navrongo, Ghana

*: Corresponding Author’s Email: [email protected]

Abstract

Treasury bills rate is a preeminent default-risk free rate asset in Ghana’s money market whose existence can

affect the purchasing power of other assets in the security market. Bank of Ghana sells its Bills to mop up excess

liquidity and buys Bank of Ghana Bills to inject liquidity into the system. This paper empirically models the

monthly Treasury bill rate of two short term Treasury bills (91 day and 182 day) from the year 2017

from the BoG using ARIMA models. From the results, it was realized that ARIMA _3, 1, 1_ model is appropriate

for modelling the 91-day Treasury bill rate with a log likelihood value of -328.58, and least AIC value of 667.17,

AICc value of 667.52 and BIC value of 683.05. Also, ARIMA _1, 1, 0_ best models the 182-day Treasury bill

rates with a log likelihood value of -356.50, and AIC value of 717.00, AICc value of 717.06 and least BIC value

of 723.35. An ARCH-LM test and Ljung-Box test on the residuals of the models revealed that the residuals are

free from heteroscedasticity and serial correlation respectively.

Keywords: Treasury bills, Ghana, Asset, Empirical, Short term.

1. Introduction

The acceptance of financial risk is inherent to the business of banking and insurance roles as financial

intermediaries. To meet the demands of customers and communities and to execute business strategies, financial

institutions make loans, purchase securities and deposit with different maturities and interest rates. A treasury

bill which is one of the common securities being purchased in many societies is a default-risk free short-term

bonds that matures within one year or less from their time of issuance. Treasury bills are sold with maturities of

four weeks (1month), 13 weeks (3 months-91-day), 26 weeks (6 months-182-day), and 52 weeks (12 months-

360-days) weeks, which are more commonly referred to as the one-, three-, six-, and 12-month T-bills,

respectively. Like a zero-coupon bond, Treasury bills are sold at a discount to par. “Par” is the value at which all

T-bills mature. Treasury Bills are issued to finance government deficits and Bank of Ghana sells Bank of Ghana

Bills to mop up excess liquidity and buys Bank of Ghana Bills to inject liquidity in the system (Brilliant, 2011).

Treasury Bills have so gained a high appeal among the population as securities with high returns and virtually no

default risk.

Jacoby, et al., provided theoretical arguments to show how treasury bills impacts stock market prices.

Jones showed that stock prices predicted expected returns in the time-series. As more data has become

available, recent work has shifted focus on studying time-series properties of risk in equity markets as well as in

Treasury bills. Huang, et al.,related risk to return volatility, while Brandt studied the relationship

between liquidity, order flow and the yield curve.

Aboagye et al., in studying the performance of stocks in Ghana, using an investment of the same amount

in treasury bills and shares over a period of2017, found out that investors in stock exchange traded

shares earned on average 54% per annum, whereas treasury bill investors Earned 36.3%. The changing rate

relationship across the spectrum of maturities is analyzed by some researchers by a yield curve risk: A yield

curve is the graph of required interest rates for various maturity times.

Ghanaians are so impressed with the observable high rate of returns on treasury bills that many believe treasury

bills offer the chance to earn higher returns than can be earned on other financial securities. Due to the

impressive nature of treasury bills in the financial market, many researchers on the Ghanaian economy focus

much on comparative study on the performance of treasury bills and other stock investments but none in this

country has concentrated on modeling Treasury bills. This study therefore is focused on modeling the 91-day and

182-day Treasury bills, to determine an appropriate times series model for predicting these Treasury bills and

forecast future outcomes. The Modelling of Treasury bills in useful to investors in their choice of Treasury bill to

invest in. It will also be useful in the financial markets and the security agencies in the control of the different

degrees of liquidity in their securities.

2. Materials and Methods of Analysis

We obtained monthly data on the 91-day and 182-day Treasury bills rate from the Bank of Ghana (BoG) from

2017. The rates were modeled using Autoregressive Integrated Moving Average

(ARIMA) models. The ARIMA is a modified form of the Autoregressive Moving Average Model

(ARMA model where the times series variable is non-stationary. An ARMA model is the

combination of an Autoregressive process and a Moving average process into a compact form in order to reduce

the number of parameters. For an ARMA model, _ is the order of the Autoregressive process and                is the

order of the Moving average process. An ARMA is used only if a time series variable is weakly stationary. If the

times variable is non-stationary (that is has a unit root), the ARMA model is extended to an ARIMA

model where _ is the order of integration of the series (number of times the series is differenced to

make it stationary).

The general form of the ARMA model is:

whiles an ARIMA model is represented by the backward shift operator as:

where _1" is the non-seasonal difference filters.__ is a white noise series and p and q are positive integers. _ is determined from the

PACF plot whiles              is determined from the ACF plot.

The modelling of an ARIMA model as outlined by Box–Jenkins consist of Model identification,

Parameter Estimation and Diagnostic of selected model.

Model Identification: The order of the Autoregressive component , the Moving Average component_ and

the order of integration, were obtained by a model identification process. Generally, for an ARIMA_

model, is identified from the Autocorrelation function (ACF) plot and _ is taken from the Partial

Autocorrelation function (PACF) plot. For any ARIMA process, the theoretical PACF has non-zeros

partial autocorrelations at lag 1, 2,……. . , _, but zero partial autocorrelation at all lags, whiles the theoretical

ACF has non-zeros partial autocorrelations at lag1, 2,……. . ,          , but zero partial autocorrelation at all lags. In this

study, the non-zero lags were taken as the _ and                for the model estimation as expected. Before the orders were

identified the following test were carried out on the data.

Ljung Box test: We performed the Ljung Box test to test jointly whether or not several autocorrelations of

the time series variable measured were zero. The Ljung Box statistic is given by:

is approximately a chi-square distribution with ) degrees of freedom. We reject -. and conclude there is

serial correlation when the p − value < 0.05.

Augmented Dickey Fuller (ADF) Unit Root Test: We used the Augmented Dickey Fuller (ADF) Test to

determine whether the times series has a unit root (non-stationary) or is weakly stationary. This test is based on

the assumption that the series follows a random walk with model;

and hypothesis:

Where is the characteristic root of an AR polynomial and 7_ is a white noise series.

An Autocorrelation plot of the series shows no serial correlation and randomness if all sample autocorrelations

fall within the two standard error limits.

Estimation of Parameters: We selected the best model among all candidate models by the Akaike Information

criterion (AIC), Akaike Information Corrected criterion (AICc), Normalized Bayesian Information Criterion

(BIC) and the Log-likelihood values. The best model is the model with the maximum Log-likelihood value and

least AIC, AICc and BIC value.

Model Diagnostics: The selected model was checked to determine whether or not it appropriately represented

the data set. The diagnostic check on the residuals of the fitted model to check whether they are white noise

series was done: These include an ACF plot of the residuals, a Ljung Box test and an ARCH-LM test on the

residuals of the best model to determine whether they are random and their variance, homoscedastic (constant) or

Heteroscedastic respectively.

3. Results and Discussion

Modelling of 91-Day Treasury Bill

The time series plot of the 91-day Treasury bill rate in figure 1 shows that the series does not fluctuate about a

fixed point and thus gives an indication of non-stationary in the series. This is also seen from the ACF plot of the

series which shows a slow decay and also from the PACF plot which has a very significant spike at lag 1. The

Augmented Dickey-fuller test further confirms this assertion: The test is insignificant at the 0.05 significance

level showing that the series has a unit root and hence not stationary. The series was therefore first differenced

and tested for stationary with the Augmented Dickey-fuller test: The first difference was enough to make the

series stationary as shown by the test.

Table 2 shows the different models fitted to the series, ARIMA_3, 1, 1_ appears to be the best model as it has the

least AIC, AICc, BIC values and the maximum Log-likelihood. The estimates of the parameters of the model,

shown in table 3, indicates that AR_1_, AR_2_ and MA_1_ models are significant at the 0.05 significance level

whiles AR_3_ is significant at 0.10 level of significance. Our diagnostic checking of the ARIMA _3, 1, 1_ model

revealed that the model was adequate for the series. The ARCH-LM test showed that there was no ARCH effect;

hence the residuals have a constant variance. The Ljung-Box p–values (> 0.05) showed that there is no serial

correlation in the residuals of the model. The ACF plot of the residuals also shows that the residuals are white

noise series.

Modelling of 182-Day Treasury Bill

A plot of the 182-day Treasury bill rate gave an indication that the series was not weakly stationary as shown in

figure 6. This was also realized from the ACF and PACF plots in figure 7; the ACF plot of the series had a slow

decay and the PACF plot showed a very significant spike at lag 1. The Augmented Dickey-fuller test which was

also insignificant at the 0.05 level of significance further confirms the non-stationarity of the series. The series

was therefore stationarized after the first differencing; the Augmented Dickey-fuller test of the differenced series

was significant indicating that the series was stationary.

From table 7, ARIMA_1, 1, 0_ model was selected as the best model among the different ARIMA (_, 1,        _

models fitted for the 182-day series since it has the smallest BIC value. Even though the AIC and AICc value of

the ARIMA_1, 1, 0_ model was larger than other models fitted, the BIC criterion was used for selecting the best

model because the BIC criterion is a consistent estimator and tends to select models with less parameters as

compared to AIC criterion. The parameter estimate of ARIMA _1, 1, 0_ shown in table 8 indicates that AR_1_ is

significant at the 0.05 significance level. Our diagnostics of the ARIMA _1, 1,0_ model showed that the model

best fit the series. It was realized that there was no ARCH effect on the residuals of the selected model due to an

insignificant ARCH-LM test statistic hence the residuals are homoscedastic. Also the Ljung-Box statistic was

not significant thus gives an indication of no serial correlation among the residuals of the selected model at 0.05

level of significance. The ACF plot of the residuals further showed that the residuals were white noise series.

These tests revealed that ARIMA _1, 1, 0_ model was adequate in representing the 182-day Treasury bill rate.

4. Conclusion

This study used time series to model the treasury bills in Ghana using data from the Bank of Ghana (BoG) from

the year 1988 to 2012. The modeling of the treasury bills was done mainly by ARIMA model. The Study

revealed that, the 91-day Treasury bill rate is best modeled with ARIMA _3, 1, 1_ whiles the 182-day Treasury

bill rates is best modeled by ARIMA _1, 1, 0_. The diagnostics of these two models showed that these models

adequately fits the two series hence are adequate for the forecasting of Treasury bill rate in Ghana.

1, 0) model