Medical, Pharma, Engineering, Science, Technology and Business

Student, Mount Kenya University , Eldoret, Kenya

- *Corresponding Author:
- Rotich Titus Kipkoech

Student, Mount Kenya University

Eldoret-9305-30100, Kenya

**Tel:**(053) 2033712

**E-mail:**[email protected]

**Received date:** October 04, 2015 **Accepted date:** October 23, 2015 **Published date:** October 28,
2015

**Citation:** Kipkoech RT (2015) Effects of Causality and Error Correction on
Volatility Modeling: A Simulation Approach. Int J Econ Manag Sci 4:297.
doi:10.4172/2162-6359.1000297

**Copyright:** © 2015 Kipkoech RT. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** International Journal of Economics & Management Sciences

Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) models are usually used to analyse time series data with high volatility clustering. In this paper, we analyse the effects of Granger Causality Model (GCM) and Error Correction Model (ECM) in analysing a time series and accordingly, we simulate two series of data using the GARCH1 model which are used for the analysis. The choice of the simulation model is based on its ability to capture volatility and heteroskedasticity. GCM2 and ECM3 models’ parameters are investigated for adequacy. Results from Augmented Dickey Fuller (ADF), Phillips PerronPhillips Perron (PP) and Kwiatkowski Philips Schmidt Shin (KPSS) tests indicate stationarity in the data as expected. GCM is built to demonstrate all the long term relationships. The two series Granger Caused each other. A linear ECM is also fitted and there is evidence that a short-term relationship exists between these two series. A high threshold value exists at the second lag, an indication of simple smoothing in the data. The residual deviance was greater than the degrees of freedom asserting that the model perfectly fit the data, supported by high R2 value of 0.871. Residuals from the fitted linear model are also stationary. The study concludes that ECMs and GCMs are appropriate in analysing time series. It is recommended that a similar study be undertaken but with a combination of ARMA Auto Regressive Moving Average (ARMA) Process and GARCH models. Further study should also be conducted on tail clustering analysis.

Granger causality, Error correction model, Simulation, Economics, Volatility

60K35, Granger causality, Error correction model, **Simulation**, Economics, Volatility

Escalation of interest rates and exchange rates in Kenya have been
a common phenomenon. Its random inter-data movements leads to
the subject of volatility. It has formed the basis of most researches in
time series analysis, with several scholars building various models in
the attempt to exhaustively examine the sources of these volatilities
and its predictions. Unfortunately, no particular scholar has been
able to exhaustively determine the sources of volatilities, neither have
they settled on any particular optimum model. Nevertheless, Leykam
[1], Zhongjian [2], Musyoki et al. [3], amongst others, concur that
modelling of these volatilities are vital for the health of an economy. It
has been pointed out that the most important aspect in any **volatility** analysis, including others, are to be able to; predict the future volatility
behaviour^{4}, and atleast establish the risk attached to these movements.

Volatility can be defined as the periodic displacement of a time
series from its long-term mean-level. Forces that displace these time
series from its mean-level is of great importance. The displacements as
well occur in phases, as suggested by the definition; the short term and
the long term. Most scholars have not yet done concurrent short term
and long term **analysis**. However, there is a great need to analyze these
shocks in two phases, the short and long-term, in order to capture the
movements exhaustively; where imputation of the two relationships
can be analyzed. Granger [4] first proposed a procedure, called granger
causality, which analyses the long-term movements. On the other
hand, an ECM, introduced by Granger in his work which analyses
a short-term relation in existence. He was analyzing a balance in an ECM where he realized that there was an imbalance in I(0) Integration
of Order 0 or stationarity^{5} and I(1) Integration of Order 1^{6} series,
(Granger).

Earlier, Fung and Hsieh [5] had used co-integration on their study on hedge funds. They criticized the conventional approaches of model constructions for asset-class indices to be applied in hedging. Seven factors were identified from which a model was built. On analysis of parameter stability, Fung and Hsieh applies the cumulative recursive residual method and plots on a time scale to investigate the reversion of the model parameter in the risk factor model. The factors are cointegrated and hence influence each others’ performance. Fung and Hsieh finally proposed a seven factor model to be applied for hedging.

Later, Leykam [1] in his work on cointegration and volatility in
the European natural gas spot markets tests the Granger causality in
volatily markets. Four markets were identified from which spot prices
were obtained. Granger causality tests were done for different pairs of
the markets. Of the four markets, one (Bunde) market indicated no
association with the other three markets. All the hypotheses that the
other three markets can be used in predicting volatility in Bunde were
rejected at 1% significance level. This meant that Bunde could not be
considered a price setter in the European gas spot **market**.

The study by Fung and Hsieh was further extrapolated by Zhongjian
who analyzed the same hedge funds in view of further examining the
validity of the method used in deriving the seven factors which had
been suggested by Fung and Hsieh for the inclusion in an hedging portfolio. In his research, he highlights that Fung and Hsieh did not
provide enough evidence to proof that the procedure used in choosing
the factors is quite different from the Sharpe and Fama-French which
only relies on one characteristics of the entire market. Contrary to Fung
and Hsieh, Zhongjian bases his parameter stability on the adjusted R2
statistic. Zhongjian does not mention the reason for his selection of R2
statistic instead of the cumulative recursive residual. He identifies nine
hedge indices which can be included in the hedging strategy. A full
rank co-integration in the **industry** was as well established, and an eight
factor model to be used for hedging strategies as the most powerful
model, is proposed.

Initially, Rashid [6] had applied granger causality in agriculture in a study on spatial integration of maize markets undertaken in the post-liberalized Uganda. Different markets were identified. Causality tests were conducated on different pairs of markets, and the results indicated that all pairs which included Kampala and Jinja failed to reject the causality null. This was an indication of a uni-directional causality, implying that the regional maize prices Granger caused the prices in these two large cities. Also, a two directional causality effect was established between Mbale and Hoima indicating a dependence behavior; that is, all deviances in one market affects the other.

Huang and Neftci [7] investigated co-integration relationship that
existed between the swap spreads and various rates such as the LIBOR
London Interbank Offered Rate^{7} rates, US corporate credit spreads
and the treasury **yield curve**; which found evidence of co-integration
existence. In their study, they showed that under the ECM framework,
the daily swap spreads reacted to the corrective long-run forces except
from the short-term fluctuations in the variables. They concluded that
the swap spread had a negative effect only on one measure, the treasury
yield curve, but positive in all the other rates.

Later on, Petrov [8] applies ECM in evaluating a pairwise cointegration
strategy between the South African equity market and other
emerging and developed markets; using the price indices rather than
MSCI Morgan Stanley Capital International^{8} index, as used by Biekpe
and Adjasi. There are two reasons; one, that price index is raw and two,
it enabled comparisons across different markets. Petrov shows that all
the markets was responding slowly to any long term disequilibrium.
The integration proved to be high between most markets and hence
portfolio selection was the most sensitive task to undertake. Petrov
applies an ECM to analyze different portfolios of different sizes . He
finds out that USA dominated in all the portfolios in which it was
introduced. It was then recommended that such portfolios should be
considered the most favorable for investors.

Credit risk is one of the most important type of risk which a bank
will be keen to assess to ensure that it remains in business. On the
other hand, most of the bank advances are made on a collateral basis.
Karumba and Wafula [9] studies this collateral lending characteristic of
the lending institutions^{9} and their implication on the general financial
equilibrium. They investigated its implication on the level of credit
risk faced by the banks. Karumba and Wafula applies co-integration
and **error** correction techniques to investigate long-run relationship.
The study found over-reliance on collateral in institutional lending.
A negative ECM adjustment coefficient was found indicating that
advances in loans and collaterals had a short-term adjustment. With
the introduction of credit referencing, the study concludes a general
reduction in credit risk.

In Basel III accord, the main challenge is addressing rates volatility. Its evolution over time makes credit risk analysis more complex. In this research, we contribute to the bank of literature by investigating existence of short-term and long-term relation between lending and interest rates. This contributes to mitigation of credit risk analyzed by Karumba and Wafula. A procedure for modelling interbank lending rates can be seen as a milestone to the mitigation strategy. It will make the work proposed in Basel III accord much easier.

In this section, we review some important definitions on the various tests which will be used in section (5) for analyses. We start by reviewing definitions on granger causality in sub-section (4.1); followed by a review on ECM in sub-section (2.2), and finally defining some tests in sub-section (4.3), which are fundamental in the study.

**Granger causality**

In this section, we review basic concepts on granger causality. We define what is granger causality in sub-section (4.1.1), and give its mathematical representation in sub-section (4.1.2).

**Definition of granger causality: **Granger proposed a procedure
of investigating causality using lagged series and residuals. Suppose
we have a series or vector y_{t} from which we want to obtain K ahead
predictions , y_{t+k} from an information matrix ᴧ . Let ᴧ be a vector of
random variables/series (a_{t}, b_{t}, a_{t-1}, b_{t-1}…., a_{1}b_{1}). Obtaining the y_{t+k} using
least squares involves calculation of the conditional mean E[y_{y}/ᴧ]. In
time series, it involves regressing y_{t} on ᴧ where ᴧ in this case has the
variables (y_{t}, y_{t-1}, y_{t-2}….y_{1}). This is rather complex. An easier procedure
is to consider its causality.

**Theoretical representation of granger causality: **Let x_{t} and y_{t} be
two series. x_{t} is said to Granger cause y_{t} if the lagged values of x_{t} has
statistically important information about the future values of y_{t}. It is
calculated for stationary series. An appropriate procedure is chosen to
determine the lag to be used, to obtain optimum results. Regression is
used for estimation. t-tests are used to retain the significant variables
in the regression and f-test determines jointly significant variables to
be retained.

The procedure involves fitting a regression of lagged values of y_{t} such that;

y_{t}=λ_{0} + λ_{1}y_{t-1} + λ_{2}y_{t-2} +…λ_{k}y_{t-k} + ε_{t} (1)

where ε_{t} are the residuals. The k statistically significant lags of y_{t} is
augmented with lagged values of x_{t} such that;

y_{t}=λ_{0} + λ_{1}y_{t-1} + λ_{2}y_{t-2} +…λ_{k}y_{t-k} + μ_{a}x_{t-a} +…+ μ_{b}x_{t-b} (2)

The lagged values of x_{t} in equation are retained if it adds an
explanatory power to the regression equation. F-tests are used to
determine the retained lagged values of x_{t}. The shortest possible
regression has a values where longest has b values. The null hypothesis
of no Granger causality is rejected iff 3 at least one lagged value of x_{t} retained in equation.

**Error correction model**

When estimating a granger causality relationship, the requirement
is to ensure the series is I (1). Making a series I (1) implies differencing.
According to time series theory, differencing means removal of trend;
hence loss of important information about the time series behaviour
in the short-run. Also, a cointegration relationship assumes a linear relationship; which might not be always the case due to random shocks.
A displacement from the equilibrium relation implies a response from
one of the variables to attain the equilibrium. The rate at which either
variables re-attains equilibrium is modelled by an ECM. Simply,
an ECM is a **model** which gives an estimated response behavior of a
variable upon dis-equilibrium. An ECM can be estimated as

ΔA_{t} + λΔB_{t} = Ω + β (ᴧ) + ε_{t} (3)

Where λ and β are just coefficients, Ω an intercept which may
or may not be included; ε_{t} random noise; and ᴧ an ‘error **correction** component’.

**Empirical unit root tests**

In this section we review definitions of some tests which are fundamental to the empirical analysis section.

**Review of augmented dickey fuller test:** This is a generalized form
of the Dickey Fuller test, (Dickey and Fuller. It relies on the assumption
that the residuals are independent and identically distributed. For a
series y_{t}, ADF uses the model

Δ y_{t} = α + λ_{t} + ηy_{t-1} + δ_{1}Δy_{t-1} +…..+δ_{p-1}Δy_{t-p+1} + ε_{t} (4)

which reduces to a random walk when α=0 and λ=0; and a random
walk with a drift when λ=0. The ADFAugmented Dickey Fuller test^{10} test thus detrends the series before testing for unit root. It uses lagged
difference terms to address serial correlation. The ADF test clearly
depends on differenced series. This thus possess a need for another
validating test.

An inspection of the p-value also determines whether the null
hypothesis of non-stationarity will be accepted. A small p-value^{11} leads
to the acceptance of the null hypothesis. An inspection of the Dickey-
Fuller value is as well important as this indicates the mean-reverting
property. It is normally a negative value. The larger its absolute value,
the lower the chance of occurrence of mean-reverting property.

**Review of Kwiatkowski philips schmidt shin test: **Contrary to
ADF test, the KPSS Kwiatkowski Philips Schmidt Shin test^{12} tests,
Kwiatkowski et al. [10] for the null hypothesis of level or trend
stationarity. It gives a way to specify whether to test with a trend or
without, in its test statistic. A regression model with linear combination
of a deterministic trend^{13}, a random walk and a stationary residual
series.

(5)

is used where δ_{t} is stationary, β_{t} is the trend component while is the random walk. β_{t=0} if we assume a without-trend regression. The
series in equation will be stationary if λ=0. Regression is used to obtain
the estimate of δ_{t}, that is , from which we compute

(6)

The test statistic for KPSS test is then calculated as

(7)

where the spectral density function estimator

(8)

is a linear combination of the variance estimator and covariance estimator

(9)

The test turns to a prudential choice of T in equation above.

**Review of phillip perron test:** The Phillips Perron approach
[11] applies a nonparametric correction to the standard ADF test
statistic, allowing for more general dependence in the errors, including
conditional heteroskedasticity. If there were strong concerns over
heteroskedasticity in the ADF residuals this might influence an analyst
to go for PP Phillip Perron test^{14}. If the addition of lagged differences
in ADF [12] did not remove serial correlation then this again might
suggest PP as an alternative.

Daily data on exchange rate of the Kenyan shilling against the dollar and the interbank lending rates are analysed in this section. The main aim is to establish if there exists any causality between the two rates. Granger causality is investigated in sub-section (5.1) while an ECM is built in sub-section (5.2) as follows:

**Granger causality**

The first step in granger causality analysis is to establish stationarity
of the time series. A basic investigation of this property is by visual
inspection of time plot. A time plot is plotted, represented in **Figure 1** below, and by inspection the series is non stationary.

ACF Auto Correlation Function^{15} is conventionally used in time
series analysis to inspect for stationarity. If the spikes tend to be
constantly high close to a value of 1, the series is non stationary. **Figure
2 **below represents the respective ACFs of the two rates.

An inspection of the ACFs suggest non stationarity. The respective
squares of the two series helps in the analysis of heteroskedasticity,
which will determine the method used to investigate for causality.
Nevertheless, mathematical tests such as KPSS, ADF and PP tests are
necessary to ascertain non stationarity. ADF test output of the two
series is presented in **Table 1** below.

Exchange Rate | Interbank Lending Rate | |
---|---|---|

Dickey-Fuller Value |
-2.1937 | -3.6382 |

Lag Order |
14 | 14 |

P-value |
0.4963 | 0.02897 |

**Table 1:** ADF Test output for exchange and interbank lending rates.

The ADF test tests the null of non stationarity. From the ADF test output above, an inspection of the p-values indicates that the null hypothesis is not rejected at 5% level of significance for the Dollar exchange rate. We reject the null hypothesis at 5% level of significance for the Interbank lending rate. Conventionally, p-value indicates the amount of evidence we have against the null. It is therefore concluded that the exchange rate is non stationary while the interbank lending rate might be stationary. However, the ADF test has two weaknesses, namely;

1. The model for an ADF test uses the differenced series.

2. It assumes that the residuals are independent and identically distributed.

These weakness call for the use of KPSS test. The KPSS test uses the series to test for non stationarity without differencing. The assumption on the distribution of the residuals is not required in this test. It is therefore a tentative alternative for stationarity test.

Contrary to the ADF test, the KPSS test tests for the null of
stationarity. Therefore, from the output presented in **Table 2** above, an
inspection of the p-value calls for the rejection of the null hypothesis
at 5% level of significance and conclude that the two series are not
stationary. This is not in line with the results obtained from the
ADF output in **Table 1** above, where the interbank lending rate was
established to be stationary. To ascertain these results, we apply the PP
test. The results of this test is presented in the **Table 3** below.

Exchange Rate | Interbank Lending Rate | |
---|---|---|

KPSS Level |
5.0182 | 1.3153 |

Truncation Lag Parameter |
13 | 13 |

P-value |
0.01 | 0.01 |

**Table 2:** ADF Test output for exchange and interbank lending rates.

Exchange Rate | Interbank Lending Rate | |
---|---|---|

Dickey Fuller (Z) _{α} |
-7.3937 | -48.5564 |

Truncation Lag Parameter |
9 | 9 |

P-value |
0.6974 | 0.01 |

**Table 3:** ADF Test output for exchange and interbank lending rates.

Just like the ADF, the PP test tests the null of non stationarity. An
inspection of the p-values indicates that the Interbank lending rate
is stationary while the exchange rate is non stationary. This result is
contrary to the results from KPSS test in **Table 2** above, but in line with
the results obtained from **Table 1.** This therefore calls for an informed
judgement on whether to assume stationarity of the interbank lending
rates. In this study, we will assume that the interbank lending rates are
stationary and its only the exchange rate which is non stationary. This
is because two of the three tests performed supports this judgement
[13].

Following the above results, it is necessary to inspect whether the two series might be having a causal relationship. This can be investigated by superimposing the two series onto each other and checking whether their movements are similar.

Clearly from **Figure 3** above, the two series have causal relationship.

The exchange rate series is thus differenced and checked for stationarity. The same tests are applied.

The null hypothesis is rejected at 5% level of significance. The
exchange rate series is now stationary. Because of the weaknesses of the
ADF test discussed above in **Table 4**, KPSS test is done and the results
presented in **Table 5** below.

Augmented Dickey-Fuller Test | |
---|---|

Data:diff(Exchange) | |

Dickey-Fuller |
-14.2119 |

Lag order |
14 |

p-value |
0.01 |

Alternative hypothesis |
stationary |

**Table 4:** ADF Test output for the differenced series of the exchange rate.

KPSS Test for Level Stationarity | |
---|---|

data: diff(Exchange) | |

KPSS Level |
0.0755 |

Truncation lag parameter |
13 |

p-value |
0.1 |

**Table 5:** KPSS Test output for the differenced series of exchange rate.

Phillips-Perron Unit Root Test | |
---|---|

data: diff(Exchange) | |

Dickey-Fuller Z(alpha) |
-2765.71 |

Truncation lag parameter |
9 |

p-value |
0.01 |

alternative hypothesis |
stationary |

**Table 6:** PP Test Output for the Differenced Series of Exchange Rate.

The results are same as for ADF test. We fail to reject the null
hypothesis at 5% level of significance and conclude that the series is
stationary. To ascertain these results, the PP test is done and resluts
presented in **Table 6** below.

This tests wraps up the unit root tests and we infer that the differenced series is stationary. It is therefore concluded that the exchange rate is I(1).

Estimation of lag value to be used in the estimation of causal
relation between the two series follow. AIC is the commonly used
procedure in the estimation. The output of the estimation is presented
in the following **Table 7**.

Degrees of Freedom | Sum of Squares | RSS | AIC | |
---|---|---|---|---|

74156 | 10319 | |||

Diff(Exchange) |
1 | 247.4 | 74403 | 10328 |

**Table 7:** Results for the AIC lag Estimation.

From the output, the second lag is the most appropriate for
estimation. The model effects are therefore investigated and from **Figure 4**, it is clear that the model has a level effect except for tail values.

Since the model effects are level within the mean, the necessity of
lag inclusion in the model is investigated. The results of this test is as
shown in **Table 8** below.

Granger causality test | |
---|---|

Model 1 |
Y Lags(Y, 1:1) + Lags(X, 1:1) |

Model 2 |
Y Lags(Y, 1:1) |

Res.Df Df F Pr(>F) 1 3317 2 3318 -1 25.674 4.264e-07 *** |

**Table 8:** Statistical Test output on lag inclusion in the model.

Y^{16} and X^{17} represents the interbank lending rate and the exchange
rate, respectively. The null hypothesis of the saturated model is not
rejected. Therefore, 3 sufficient evidence that the inclusion of the
lagged values in causality estimation leads to the overall improvement
in the model predictive ability. The coefficients for the granger causality
model are presented in the **Table 9** below.

Intercept | X11 | X12 | X13 |
---|---|---|---|

-7.9150651 | 0.1942408 | 0.1574291 | -0.8701337 |

**Table 9:** Granger Causality Model Coefficients.

The third variable^{18} is found to be insignificant in the model, at 5%
level of significance. The variable is dropped. A linear model is thus fit
with the series itself and its second lagged value as follows (**Table 10**).

Intercept | X11 | X13 |
---|---|---|

-7.9123922 | 0.1942063 | -0.7914011 |

**Table 10:** A Linear Model with significant lagged values.

All the model parameters were found to be statistically significant at 5% level. From the output above, the granger causality model therefore becomes:

y_{t} + 7.91239 = 0.19421x_{t} -0.79140 x_{t-2} (10)

where y_{t} represents the interbank lending rate while x_{t} is the dollar
exchange rate.

It therefore remains to check on the direction of the causality. The
output is presented below **Table 11**.

F-statistic | p-value | |
---|---|---|

Exchange- Interbank |
6.473963 | 0.00156268 |

Interbank- Exchange |
2.992947 | 0.05027497 |

**Table 11:** Direction of the causality.

It is clear that exchange rates granger causes the interbanking lending rates. Therefore, movements in interbank lending rates are more likely to be caused by the movements in the exchange rates. However, interbank lending rate does not granger cause the exchange rate, as could have been expected.

**Error correction model**

Once the granger causality model discussed above has been built^{19}, an ECM is easily built by considering the residuals of the model in
equation above. ECM involves fitting a regression equation of the
differenced series and the residuals of the fitted granger causality
model. Due to the inclusion of the residuals, dynamic linear modeling
is used. The fitted model will involve two main parts; the residual part
which might be considered more stable than the differenced series part,
hence the use of a dynamic linear model. The output of an estimated
ECM is as shown in **Table 12** below.

Time series regression with "numeric" data: Start = 1, End = 3320 |
|||||||||

Call: dynlm(formula = Interbank3 Exchange4 + resid) | |||||||||

Residuals: |
|||||||||

Min |
1Q |
Median |
3Q |
Max |
|||||

-4.5456 | -0.6501 | 0.0874 | 0.5570 | 6.5569 | |||||

Coefficients: |
|||||||||

Estimate Std. |
Error |
t value |
Pr(>|t|) |
||||||

(Intercept) |
7.108730 | 0.021622 | 328.78 | <2e-16*** | |||||

Exchange4 |
-0.775746 | 0.055974 | -13.86 | <2e-16*** | |||||

resid |
1.000388 | 0.004744 | 210.85 | <2e-16 *** | |||||

--- | |||||||||

Signif. codes: 0 ‘ ***’ 0.001 ‘ **’ 0.01 ‘ *’ 0.05 ‘ .’ 0.1 ‘ ’ 1 | |||||||||

Residual standard error: 1.246 on 3317 degrees of freedom | |||||||||

Multiple R-squared: 0.9308, Adjusted R-squared: 0.9308 | |||||||||

F-statistic: 2.231e+04 on 2 and 3317 DF, p-value: < 2.2e-16 |

**Table 12:** An Estimation of an ECM for Exchange and Interbank Lending Rates.

All the parameters from the model are significant at 5% level. R^{2} value of 0.9308 shows that the overall model fits well to the data. The
fitted ECM therefore becomes;

y_{t} = 70108730 - 0.775746 Δ x_{t} + 1.000388ᴧ (11)

where y_{t} represents the interbank lending rate, x_{t} the exchange rate
while ᴧ denotes the error correction component.

Contrary to time series theory, tests on interbank lending rates return stationarity. Nevertheless, the exchange rates seem to be consistently non stationary. The stationarity of the interbank lending rate can be attributed to the fact that;

• The interbank lending rates are controlled locally and are mainly set following fluctuation of world wide economic performance. The stability of interbank lending rates is mainly determined by the central bank.

• The exchange rate is normally controlled by the overall world wide economic performance. Its fluctuation is thus not influence locally by any country, i.e, it is not controlled monopolistically.

From the above arguements, we expect the interbank lending rates to be more stable of the two.

Results of the granger causality indicate that the exchange rates
granger causes the interbank lending rates. Movements in exchange
rates can be used as an indication of the most probable movement in
interbank lending rate. The residual sum of squares from the granger
causality model is very low, an indication that the model optimally explains the variations in the data. From the superimposed plot of the
two series in **Figure 2,** it is expected that the exchange rate and the
interbank lending rate granger causes each other. However, contrary
to this intuition, the interbank lending rate does not granger cause the
exchange rate. This can as well be attributed to the local nature of the
interbank lending rate.

Finally, an ECM model is built and results presented in **Table 11** in the above section. The model is presented in equation. From the
output, the R2 value is 0.9308, an indication that the model well fits
to the data. Further, the adjusted R2 value is 0.9308, same as the R2
value, meaning the model can as well perfectly fit to any other data
with similar characteristics. Therefore, the model can be used for
predictive purposes. This arguement is ascertained by the very small
sum of squared residuals. The model can therefore be used to analyse
data with similar characterisitcs as those used in the study. It can as well
be adobted by institutions for their internal hedging and as a liquidity
guard.

It is recommended that the same study be conducted with interbank lending rates being differenced to investigate whether their will be any change in the overall model. Also, an investigation on the stationarity of the interbank lending rates should be done to establish its cause and why it occurs. Further, a study on the tail values should be done to examine the cause of the tail clustering and its impact on the overall model. Given the granger causality model, it is recommended that a cointegration model be built to examine whether the two models differ from each other; and if so, examine the reasons for the difference.

^{1}Generalized AutoRegressive Conditional Heteroskedasticity

^{2}Granger Causality Model

^{3}Error Correction Model

^{4}At some level of significance

^{5}Integration of order 0 or stationarity

^{6}Integration of order 1

^{7}London Interbank Offered Rate

^{8}Morgan Stanley Capital International

^{9}Banks

^{10}Augmented Dickey Fuller test

^{11}Less than 0.05 or 0.01 depending on the statistician

^{12}Kwiatkowski Philips Schmidt Shin test

^{13}If test statistic is with a trend

^{14}Phillip Perron test

^{15}Auto Correlation Function

^{16}Interbank lending rate

^{17}Exchange rate

^{18}First lag of the exchange rate

^{19}Equation

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