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Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries

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This paper empirically investigates the Fisher effect in selected ECOWAS countries by employing annual data from 1961 to 2011. The inflation and interest rates for Burkina Faso, Cȏte d'Ivoire, Gambia, Ghana, Niger, Nigeria, Senegal and Togo are used in the study. Firstly, we investigate the order of integration of the 16 time series using the augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit root tests as a confirmatory test. Our empirical results indicate that inference based on the ADF and the Phillips-Perron test displays a considerable degree of robustness to the method of lag selection and the correction for heteroskedasticity and autocorrelation adopted, however, the robustness of the KPSS test to the method of computation of the long-run variance seems to be weak. On allowing for structural breaks, we found more evidence against the unit root hypothesis. Secondly, the Fisher equation is cast in the state space framework and the Kalman filter is applied to estimate the slope parameter. Our state space model results indicate that the strength of the Fisher effect does vary over time. For the ECOWAS countries; in some periods there appears to be a full Fisher effect, while in other periods, the relationship seems to be partial and non-existing at some other periods. The Harvey-Koopman procedure is also employed to detect the time of structural breaks and outliers in the state space model. We recommend that monetary authorities in the ECOWAS countries should aimed at making effective monetary policies and demonstrate strong commitments to monetary targets in order to strengthen the Fisher relation.
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International Journal of Statistics and Applications 2015, 5(5): 181-195
DOI: 10.5923/j.statistics.20150505.02
Fisher Effect, Structural Breaks and Outliers
Detection in ECOWAS Countries
Omorogbe J. Asemota1,*, Dahiru A. Bala2, Yahaya Haruna3
1Department of Statistics, University of Abuja, Nigeria
2Dept of Econ Engineering, Kyushu University, Japan
3Department of Statistics, University of Abuja
Abstract This paper empirically investigates the Fisher effect in selected ECOWAS countries by employing annual
data from 1961 to 2011. The inflation and interest rates for Burkina Faso, Cȏte d’Ivoire, Gambia, Ghana, Niger, Nigeria,
Senegal and Togo are used in the study. Firstly, we investigate the order of integration of the 16 time series using the
augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit root tests
as a confirmatory test. Our empirical results indicate that inference based on the ADF and the Phillips-Perron test displays a
considerable degree of robustness to the method of lag selection and the correction for heteroskedasticity and
autocorrelation adopted, however, the robustness of the KPSS test to the method of computation of the long-run variance
seems to be weak. On allowing for structural breaks, we found more evidence against the unit root hypothesis. Secondly,
the Fisher equation is cast in the state space framework and the Kalman filter is applied to estimate the slope parameter.
Our state space model results indicate that the strength of the Fisher effect does vary over time. For the ECOWAS countries;
in some periods there appears to be a full Fisher effect, while in other periods, the relationship seems to be partial and
non-existing at some other periods. The Harvey-Koopman procedure is also employed to detect the time of structural
breaks and outliers in the state space model. We recommend that monetary authorities in the ECOWAS countries should
aimed at making effective monetary policies and demonstrate strong commitments to monetary targets in order to
strengthen the Fisher relation.
Keywords Fisher effect, Inflation rates, Interest rates, Kalman filter, Outliers, Structural break
1. Introduction
The Fisher hypothesis postulated by Fisher [1] suggests
that when expected inflation rises, nominal interest rate will
rise with an equal amount without affecting the real interest
rate. This implies that a one-for-one relationship exist
between the nominal interest rate and the inflation rate.
Asemota and Bala [2] noted that this hypothesis has
important policy implications for the behavior of interest
rates, efficiency of financial markets and the conduct of
monetary policy. The Fisher effect is usually tested by
imposing rational expectations on inflation forecasts. This is
empirically tested by a regression of the nominal interest
rate on a constant and the realized inflation. The closer the
coefficient of the inflation rate is to one, the stronger the
Fisher effect, if the coefficient is substantially different
from one, any change in the nominal interest rates will also
change the interest rate and may consequently spread to
other macroeconomic variables.
* Corresponding author:
asemotaomos@yahoo.com (Omorogbe J. Asemota)
Published online at http://journal.sapub.org/statistics
Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved
Given the important policy implications of the Fisher
hypothesis, a large body of research has been devoted to
studying the Fisher relation; howbeit, there is no general
consensus among researchers on the Fisher hypothesis.
Asemota and Bala [2] using the cointegration and Kalman
filter approaches could not find evidence of a full Fisher
effect for the nominal interest and inflation rates in Nigeria.
Also, Engsted [3] and Hatemi-J [4] could not find evidence
of the Fisher hypothesis. However, Mishkin [5], Evans and
Lewis [6], Wallace and Warner [7], and Crowder and
Hoffman [8] found evidence in favour of long-run Fisher
effect. Badillo et al. [9] used the panel cointegration
approach to analyze the Fisher hypothesis for 15 European
Union (EU) countries. The empirical results showed that the
coefficient of inflation rate is significantly less than one,
which implies the existence of a Partial Fisher effect. Gul
and Acikalin [10] using monthly time series could only find
evidence of a partial Fisher effect for Turkey.
There are several reasons behind the inability to find
evidence of a full Fisher effect. Tobin [11] noted that
investors re-balance their portfolios in favour of real assets
during high inflationary periods. In addition, the different
types of interest rates and sample periods used in the
empirical analysis may also affect the results. It may also be
182 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
due to structural changes in the co-integrating vector.
Mishkin [5] noted that the relationship between interest rate
and inflation, shift with changes in monetary policy regimes.
Chuderewics [12] argued that the Central Bank behavior
plays an important role in understanding the varying
strengths and weakness associated with the Fisher relation.
He developed a theoretical model that incorporates Central
Bank behavior and demonstrated that the strength of the
Fisher relation depends explicitly on its behavior. Recently,
more attention has been given to investigations of possible
structural breaks in macroeconomic time series. Hatemi-J [4]
argued that the inability to find a full Fisher effect may be
due to parameter instability. Structural breaks may occur
due to financial crises, policy changes, changes in
consumers’ preferences and behavior, technological
changes and political instability, among others.
The aim of this paper is to empirically investigate the
Fisher effect in selected ECOWAS 1 countries by
employing annual data from 1961 to 2011. The inflation
and interest rates for Burkina Faso, Cȏte d’Ivoire, Gambia,
Ghana, Niger, Nigeria, Senegal and Togo are used in the
study. First, we investigate the order of integration of the 16
time series using the augmented Dickey-Fuller (ADF),
Phillips-Perron (PP) and the Kwiatkowski-Phillips-
Schmidt-Shin (KPSS) unit root tests. We apply different lag
selection criteria for the ADF tests and different methods
are used for the computation of the long-run variance for PP
and KPSS tests. These were conducted to ascertain if
inferences based on unit root tests are affected by the
method of lag selection and method of constructing
heteroskedasticity and autocorrelation consistent (HAC)
estimators. In addition, we also conducted unit root tests
allowing for one and two endogenous structural breaks. To
examine the dynamic relationship between inflation and
nominal interest rate, the time varying parameter model is
constructed and the Kalman filter estimation method is
utilized to estimate and show the varying strength of the
response of the interest rates to changes in inflation. Finally,
the Harvey-Koopman technique is employed to detect the
time of structural breaks and outliers in the constructed
models. This is the first attempt in the literature to examine
the Fisher effect using Kalman filter methodology in the
ECOWAS region and it is aimed at filling an important gap
in the empirical literature on developing countries. In
addition, the paper is one of the few studies that empirically
compare and provide evidence on the performance of the
PP and KPSS test when different methods are used in the
computation of the long-run variance. The rest of the paper
is organized as follows: section 2 describes the
methodology. Section 3 presents the data and the empirical
findings, and the last section concludes.
1 ECOWAS is an acronym for Economic Community of West African States.
It is a regional group of fifteen countries founded in May 28, 1975 to promote
cooperation and integration, with a view to establishing an economic and
monetary union as a means of stimulating economic growth and development in
West Africa.
2. Econometric Methodology
2.1. Unit Root Tests without Structural Breaks
It is now a tradition in the field of econometrics to
investigate the order of integration of macroeconomic
variables prior to modeling the series. This is necessary in
order to avoid the problem of spurious regression as first
pointed out by Granger and Newbold [13]. Mishkin [5]
argued that both inflation and interest rates contain unit
roots, hence, the “traditional” forecasting equation suffers
from the spurious regression problem, except, if the
variables are cointegrated. The augmented Dickey-Fuller
(ADF), the Philips-Perron (PP) and the Kwiatkowski
PhillipsSchmidt–Shin (KPSS) tests will be used in this
study to examine the order of integration of the 16
macroeconomic variables. The augmented DickeyFuller
(ADF) test (Dickey and Fuller 1979 [14]) is based on the
following model:
11
−−
=
∆=+ + + +
k
t t i ti t
i
y ty y
αβ ρ θ ε
(1)
where
t
ε
is a well behaved error tem2 ;
ti
y
is the
lagged first difference added to correct for serial correlation
in the error and the maximum lag
k
is selected using the
Schwartz information criterion (SIC) and the ‘t sig’
approach proposed by Hall [15] and
α
,
β
,
ρ
and
θ
are the parameters to be estimated. Equation (1) tests the null
hypothesis of a unit root against a trend stationary
alternative.
The PhilipsPerron [16] method estimates the test
equation below:
'
1
,
∆= + +
t t tt
yyX
ρ δε
(2)
The PP test is based on the statistic
1/2 1/2
00 0 0 0
ˆ
( / ) ( )( ( )) / 2 ,= −−
t t f T f se f s
αα
γ γα
which modifies the DickeyFuller test. Where
ˆ
α
is the
estimate,
the t–ratio,
ˆ
()se
α
is coefficient standard
error, and
s
is the standard error of the test regression.
Further,
is a consistent estimate of the error variance in
(2), while
(0)f
ε
is an estimator of the residual spectrum
at frequency zero. The lag window or bandwidth use in the
study is estimated by two criteria; the Newey-West criterion
(Newey and West, [17]) using the Bartlett kernel and the
Andrews criterion (Andrews [18]) using the quadratic
spectral kernel.
However, Kwiatkowski et al. [19] argued that the
classical method of hypothesis testing is biased towards the
null hypothesis. Hence, it ensures that the null hypothesis is
accepted unless there is strong evidence against it. They
pointed out that the standard unit root tests are not very
2 The error term is said to be wellbehaved if it is independently and identically
normally distributed.
International Journal of Statistics and Applications 2015, 5(5): 181-195 183
powerful against relevant alternatives; see (Kwiatkowski
et al. [19], pg. 160). To circumvent this problem, KPSS [19]
proposed a test of the null hypothesis that a series is
stationary around a deterministic trend. They concluded that
by testing both the unit root null hypothesis and the
stationarity null hypothesis, researchers can distinguish
series that appear to be stationary, those that appear to have
a unit root, and those that the data (or the tests) are not
sufficiently informative to decide whether they are
stationary or integrated. The KPSS test is given by the
following equations:
= ++
t tt
y tr
δε
(3)
where
t
ε
is a stationary error and
t
r
is a random walk
given by:
1
= +
tt t
rr u
,
2
(0, )
tu
u iid
σ
(4)
The initial value
0
r
is treated as fixed and serves as the
intercept in the model and the null hypothesis of stationarity
is formulated as
2
0
: 0 =
u
H
σ
or
t
r
is constant. The LM
statistic is given by:
2
2
1
= ˆ
=
Tt
e
t
S
LM
σ
(5)
where
t
e
are residuals from the regression of
t
y
on an
intercept and time trend,
2
ˆe
σ
is the estimate of the error
variance of the regression and
t
S
is the partial sum of
t
e
defined by:
1
= , 1,2,..., .
=
=
t
ti
i
S et T
(6)
When the errors are
iid
the estimator
2
ˆ
e
σ
converges to
2
σ
, however, when the errors are not
iid
, a consistent
estimator of the long-run variance
2
σ
is given by:
2 12 1
1 11
2
−−
= = = +
= +
∑∑
T lT
Tl t l t t
tt
T e T w ee
ττ
ττ
σ
(7)
where
l
w
τ
is an optimal weighting function that
corresponds to the choice of a spectral window. KPSS use
the Bartlett window suggested by Newey and West [17].
The modification is given by:
= 1 1
l
wl
τ
τ
(8)
However, Leybourne and McCabe [20] argued that
inference from the KPSS test can be very sensitive to the
value of the lag
l
that is used in the computation of the
2
Tl
σ
. Hence, in this paper, we also consider Andrews[18]
quadratic spectral kernel in the computation of the long-run
variance to ascertain the degree of sensitivity of the KPSS
test to the different method of computation of
heteroskedasticity and autocorrelation consistent long-run
variance.
2.2. Unit Root Tests with Structural Break
2.2.1. Zivot-Andrews Unit Root with one Structural Break
Test
Starting with the pioneer work of Perron [21], it has been
proved that conventional unit root tests are biased towards
the non-rejection of the unit root null hypothesis even if the
data are stationary with structural break(s). Perron [21]
incorporated dummy variables in the ADF test to account
for one exogenous structural break. However, this
exogenous imposition of break date has been criticized by
Christiano [22] and Zivot-Andrews [23]. Zivot-Andrews
[23] proposed a data dependent algorithm to determine the
breakpoint. The Zivot-Andrews test is carried out using the
following regression equations:
Model A (Crash Model):
11
−−
=
=++ + + +
k
t t t i ti t
i
y t DU y c y
µβ θ α ε
(9)
Model B (Changing Growth Model):
11
−−
=
=++ + + +
k
t t t i ti t
i
y t DT y c y
µβ γ α ε
(10)
Model C (Mixed Model):
11
−−
=
=++ + + + +
k
t t t t i ti t
i
y t DU DT y c y
µβ θ γ α ε
(11)
where
1=
t
DU
if
,>t TB
0 otherwise;
=
t
DT t TB
if
,>t TB
0 otherwise,
TB
is the date
of the endogenously determined break. Model A, referred to
as the “crash model” allows for a one-time change in the
intercept of the trend function, model B, referred to as the
“changing growth model” allows for a single change in the
slope of the trend function without any change in the level;
and model C, the “mixed model” allows for both effects to
take place simultaneously, i.e., a sudden change in the level
followed by a different growth path.3 The null hypothesis
for the three models is that the series is integrated (unit root)
without structural breaks = 1). The test statistic is the
minimum
t
over all possible break dates in the sample.
Zivot-Andrews [23] suggested using a trimming region of
(0.10T, 0.90T) to eliminate endpoints.
3 Perron [21] suggest that most macroeconomic time series can be adequately
modeled using either model A or model C. In addition, Sen [24] argued that if
one assumes that the location of the break is unknown, it is most likely that the
form of the break will be unknown as well. Sen [24] demonstrated that the loss
in power is quite negligible if the mixed model specification is used when in
fact that the break occurs according to the crash model or changing growth
model, and concluded that practitioners should specify the mixed model in
empirical applications. Hence, we used model C in our empirical analysis.
184 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
2.2.2. Lagrange Multiplier (LM) unit Root Test with
Structural Breaks
Lee and Strazicich [25] argued that the Zivot-Andrews
test omit the possibility of unit root with break. Hence,
researchers may incorrectly conclude that a time series is
stationary with breaks when in fact the series is
nonstationary with break(s). To circumvent this problem,
they proposed the minimum LM unit root test in which the
alternative hypothesis unambiguously implies trend
stationarity. The LM unit root test can be explained using
the following data-generating process (DGP):
'
= +
t tt
y Ze
δ
,
1
= +
ttt
ee
βε
(12)
where
is the observed time series,
δ
is a vector of
coefficients,
is a vector of exogenous variables and
t
ε
is a well-behaved error term. Corresponding to the
two-break equivalent of Perron’s [21] Model C, with two
changes in the level and the trend,
is defined by
[ ]
12 1 2
1, , , , , =
t tt t t
Z t D D DT DT
to allow for a
constant term, linear time trend, and two structural breaks in
level and trend.4 Under the alternative hypothesis, the
jt
D
terms describe an intercept shift in the deterministic trend,
where
jt
D
= 1 for
1≥+
Bj
tT
,
j
= 1, 2, and 0
otherwise;
Bj
T
denotes the time period when a break
occurs and
jt
DT
describes a change in slope of the
deterministic trend, where
=
jt Bj
DT t T
for
1
≥+
Bj
tT
,
j
= 1, 2, and 0 otherwise. Note that the
DGP includes breaks under the null
( 1)=
β
and
alternative
( 1)<
β
hypothesis in a consistent manner.
Lee and Strazicich [25] used the following regression to
obtain the LM unit root test statistic:
11
'
−−
=
∆= + + +

k
t t t i ti t
i
y Z S Su
δφ λ
(13)
where
=−−

t t xt
Sy Z
ψδ
,
t
= 2, ,
T
is a
de-trended series;
δ
are coefficients in the regression of
t
y
on
t
Z
;
x
ψ
is given by
11
yZ
δ
;
and
1
Z
denote the first observations of
t
y
and
respectively.
Vougas [27] has shown that the LM type test using the
above optimal de-trending device is more powerful and
finds more evidence in favor of trend stationary than the
ADF type test.
ti
S
,
i
= 1, …,
k
terms are included as
necessary to correct for serial correction. Note that the test
4 Lee et al. [26] noted that there are technical difficulties in obtaining relevant
asymptotic distributions and the corresponding critical values of endogenous
test with three or more breaks.
regression (12) involves
t
Z
instead of
t
Z
so that
t
Z
becomes
[ ]
12 1 2
1, , , ,
∆∆
tt t t
D D DT DT
. The unit
root null hypothesis is described by
0=
φ
, and the LM test
statistics are given by:
=
T
ρφ
, (13a)
τ
=
t
-statistic for the null hypothesis
0=
φ
. (13b)
To endogenously determine the break points (
Bj
T
), the
minimum LM unit root test uses a grid search as follows:
inf ( )=
LM
ρλ
ρλ
, (14a)
inf ( )=
LM
τλ
τλ
. (14b)
Where
/=
b
TT
λ
, and
T
is the sample size. Vougas
[27] indicated that in the application of LM test, the
studentized version (
τ
) takes into account the variability of
the estimated coefficients and is more powerful than the
coefficient test (
ρ
). The breakpoints are determined to be
where the test statistic is minimized. As is typical in
endogenous break test, we use a trimming region of (0.15T,
0.85T) to eliminate endpoints. Critical values are tabulated
in Lee and Strazicich [25].
2.3. The State Space Model
The Fisher effect is usually investigated by regressing the
nominal interest rate on a constant and the realized inflation
rate. The value of the coefficient on the inflation rate
provides a measure of the Fisher effect. The closer the value
of the coefficient is to one, the stronger the Fisher effect.
The Fisher effect is usually represented by the following
equation:
+ =e
t tt
i rf
(15)
where
t
i
is the nominal interest rate,
t
r
is the ex-ante real
rate of interest and
e
t
f
is the expected inflation rate.
Assuming rational expectations on the inflation forecasts
implies that;
= +
e
t tt
ff
ε
(16)
where
t
ε
the forecast error is white noise. Hence, the
regression equation for Fisher effect is:
=++
t t tt
ir f
αε
(17)
The strength of the Fisher effect is usually assessed by
comparing the coefficient of inflation
α
to one. If
1=
α
in equation (17), we have the full Fisher effect, however, if
1<
α
it is known as the partial Fisher effect. The
specification in (17) above assumes that the
α
coefficient
is constant throughout the time under investigation. This
specification may be spurious especially in economic and
International Journal of Statistics and Applications 2015, 5(5): 181-195 185
business applications where the level of randomness is high,
and also where the constancy of patterns or parameters
cannot be guaranteed. In addition, the monetary policy
behavior of the central banks is not constant through time.
Thus, to capture the dynamic economic environment and the
evolving policy behavior, a more flexible model that allows
the parameter to vary randomly over time is adopted. This
flexible model is popularly referred to as the time varying
parameter model. The state space representation of (17) as a
time varying parameter model is given as:
1
=++
= +
t t tt t
tt t
ir f
αε
αα ξ
(18)
The first equation in (18) is called the measurement
equation while the second is the state or transition equation.
The measurement equation relates the observed variables
(data) and the unobserved state variable (
t
α
), while the
transition equation describes the evolution of the state
variable. The observation error
t
ε
and state error
t
ξ
are
assumed to be Gaussian white noise (GWN) sequences. The
overall objective of state space analysis is to study the
evolution of the state (
) over time using observed data.
When a model is cast in a state space form, the Kalman filter
is applied to make statistical inference about the model. The
Kalman filter (hereafter, KF) is simply a recursive statistical
algorithm for carrying out computations in a state space
model. A more accurate estimate of the state vector or slope
coefficient can be obtained via Kalman Smoothing (K.S).
The unknown variance parameters (
2
ε
σ
and
2
ξ
σ
) in
model 12 are estimated by the maximum likelihood
estimation via the Kalman filter prediction error
decomposition initialized with the exact initial Kalman filter.
Harvey and Koopman [28] demonstrated that the auxiliary
residuals in the state space model can be very informative in
detecting outliers and structural change in the model. For a
complete exposition of the state space model and Kalman
filter, see Durbin and Koopman [29] and Hamilton [30].
3. Data, Results and Discussion
3.1. The Data
The data used in this study are obtained from the
International Financial Statistics (IFS) database CD-ROM
(June, 2012). We utilize annual inflation and nominal
interest rates data of ECOWAS countries over the time
period of 1961-2011 depending on data availability.5 A time
series plot of the series for the different countries is depicted
in figure 1.
5 The inflation and interest rates for Burkina Faso, Cȏte d’Ivoire, Gambia,
Ghana, Niger, Nigeria, Senegal and Togo are used in the study. The other
ECOWAS countries are omitted from the study because the data available on
their inflation and interest rates are too scanty to be included in the analysis.
3.2. Unit Root with and without Structural Breaks Tests
Results
It has now become standard practice in time series
analysis to investigate the order of integration of the series
prior to its statistical modeling. In the econometric literature,
the augmented Dickey-Fuller test is most commonly used to
ascertain the order of integration of the time series data. In
addition, we also apply the Phillips-Perron [16] and the
Kwaitkowski et al. [19] tests to the series while varying the
method of lag selection (for the ADF test) and the method
of correction for autocorrelation and heteroskedasticity (PP
and KPSS tests).
The conventional unit root tests (ADF, PP and KPSS)
that do not allow for the possibility of structural breaks have
been proven to be biased towards the non-rejection of the
null hypothesis. Hence, we also apply the one-break and
two-break unit root tests. The results are provided in the
table 1.
The results of the various unit root tests displayed in
Table 1 reveal some interesting findings. The ADF test that
utilizes the Schwarz information criterion rejects the unit
root null for 8 of the 16 series, while the Phillips-Perron test
that utilizes the Bartlett kernel in the computation of the
long-run variance also rejects the unit root null for 8 of the
16 series. With the KPSS test, we cannot reject the null of
stationarity at the usual critical levels for 10 series when the
Bartlett kernel is used as the method for heteroskedasticity
and autocorrelation correction. The overall conclusions of
the confirmatory analysis are quite contradictory: the
confirmatory analysis yield 8 cases of conflicting results; a
situation where the individual test arrives at different
conclusion, 6 cases of genuine-stationarity and 2 cases of
genuine-unit root process. We then alter the method of lag
selection for the ADF test to the t -statistic approach and the
method of computation of the long-run variance to the
quadratic spectral kernel for the Phillips-Perron and KPSS
tests. The results in Table 2 reveals that the ADF and the
Phillips-Perron tests also rejected the unit root hypothesis
for 8 of the 16 series, while the KPSS test cannot rejects the
null hypothesis of stationarity for 12 of the 16 series.
The overall confirmatory analysis using the three tests
indicated conflicting conclusions for 6 series, 7 series with
genuine-stationarity and 3 series with genuine-unit root
conclusion. Based on these findings, we can say that the
ADF and the Phillips-Perron test displays a considerable
degree of robustness to the method adopted in correction for
heteroskedasticity and autocorrelation while the robustness
of the KPSS test to the method of computation of the
long-run variance is weak.
186 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
Figure 1. Time Series Plots of Inflation and Interest Rates for the ECOWAS countries
International Journal of Statistics and Applications 2015, 5(5): 181-195 187
Figure 1. Continued
Table 1. Unit root tests without structural breaks
Countries Series ADF PP KPSS Conclusion
Null: Unit root Null: Unit root Null: Stationary
Burkinafaso
INT t = -0.580 t = -1.635 t = 0.225 Conflicting results
INF t = -7.297*** t = -7.353*** t = 0.143 Genuine-stationary
Cotedevoir
INT t = -0.580 t = -1.635 t = 0.225 Conflicting results
INF t = -4.183*** t = -4.277*** t = 0.188 Genuine-stationary
Gambia
INT t = -3.462 ** t = -1.965 t = 0.139* Conflicting results
INF t = -3.373 ** t =-3.331 ** t = 0.066 Genuine-stationary
Ghana
INT t = -0.600 t = -0.579 t = 0.168**T Conflicting results
INF t = -2.418 t = -4.286*** t = 0.194 Conflicting results
Niger
INT t = -0.581 t = -0.581 t = 0.210 C Conflicting results
INF t = -4.617*** t = -4.636 *** t = 0.192 Genuine-stationary
Nigeria
INT t = -2.050 t = -1.807 t = 0.142 * T Genuine-unit root
INF t =-3.316 ** t = -3.147 ** t = 0.158** Conflicting results
Senegal
INT t = -2.056 T t = -1.985 t = 0.189** T Conflicting results
INF t = -5.301*** t = -5.269 *** t = 0.073 Genuine-stationary
Togo
INT t = -1.931 T t =-1.855 T t = 0.195**T Genuine-unit roots
INF t = -4.783*** t = -4.772*** t = 0.162 Genuine-stationary
Notes: ADF test is conducted using the Bayesian information criterion as lag selection, while the PP and KPSS tests are conducted
using the Bartlett kernel (Newey-West correction). C and T denote model estimated with constant and trend respectively. ***, **
and * denote significant at the 1%, 5% and 10% levels respectively.
188 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
Table 2. Unit root without structural breaks
Countries Series ADF PP KPSS Conclusion
Null: Unit root Null: Unit root Null: Stationary
Burkinafaso
INT t = -0.459 t = -0.524 t = 0.161 Conflicting results
INF t = -3.673*** t = -7.293*** t = 0.199 Genuine-stationarity
Cote devoir
INT t = -0.667 t = -0.524 t = 0.161 Conflicting results
INF t = -4.183*** t = -4.180*** t = 0.168 Genuine stationarity
Gambia
INT t = -3.462** t = -2.358 t = 0.150** Conflicting results
INF t = -3.373** t = -3.366** t = 0.058 Genuine stationarity
Ghana
INT t = -0.600 t = -0.523 t = 0.264*** Genuine-unit root
INF t = -2.522 t = -4.046*** t = 0.184 Conflicting results
Niger
INT t = -0.445 t = -0.516 t = 0.143 Conflicting results
INF t = -4.617*** t = -4.607*** t = 0.177 Genuine stationarity
Nigeria
INT t = -1.233 t = -1.718 t = 0.094 Conflicting results
INF t = -3.775*** t = -3.455** t = 0.197 Genuine stationarity
Senegal
INT t = -2.056 t = -1.974 t = 0.123** Genuine-unit root
INF t = -5.301*** t = -5.303*** t = 0.054 Genuine stationarity
Togo
INT t = -1.931 t = -1.842 t = 0.129** Genuine-unit root
INF t = -4.783*** t = -4.785*** t = 0.155 Genuine stationarity
Notes: ADF test is conducted using the t-statistic method of lag selection, while the PP and KPSS tests are conducted using the
Quadratic spectral kernel (Andrews’s correction). C and T denote model estimated with constant and trend respectively. ***,
** and * denote significant at the 1%, 5% and 10% levels respectively.
We also conducted unit root test that account for the
possibility of one and two endogenous breaks in the
specification. The Zivot-Andrews and the minimum
Lagrange multiplier endogenous break results are reported
in Table 3 while the results of two-breaks are reported in
Table 4.
At the conventional significance level, the ZA test rejects
the unit root null for 12 of the 16 series; this implies that, 12
series are found to be stationary with one structural break.
The minimum LM test rejects the null hypothesis of unit
root with one break for 10 of the 16 series; therefore, we
find evidence of stationarity with one structural break for 10
series. The combined analysis of both results yield 8
conflicting conclusions, 7 cases of genuine cases of
stationarity with one structural break and 1 case of genuine
unit root. The different break dates selected by the ZA and
minimum LM one-break test and the 50% conflicting
conclusion in the combined analysis of both tests are
puzzling.
Ohara [31] noted that just as the unit root tests that fail to
account for structural breaks are biased towards the
non-rejection of the null hypothesis, under specification of
the number of break may also lead to misleading inferences.
Hence, we also report the results of the minimum Lagrange
multiplier two-break unit root tests in table 4. Using the
two-break LM test, we find evidence of stationarity with
two-break points for 12 of the series. Kwiatkowski et al. [19]
noted that the test of the unit root null hypothesis should be
complemented by the test of the null hypothesis of
stationarity for confirmatory analysis, i.e., to ascertain our
conclusion about the unit roots. However, this assertion
has been regarded as an illusion by Burke [32] who through
a detailed Monte Carlo study reveals that the KPSS test has
the same poor power properties as the ADF test. He
concluded that even if confirmation occurs, it may be
incorrect. In addition, the different break dates in our results
and the issue of the actual number of breaks to be included
in the unit root test with structural breaks still remains
controversial in the literature. For example, the one-break
endogenous test results of Zivot-Andrews yields 12
stationarity with breaks series, while the minimum LM test
yields 10 stationarity with break series. Which is the more
International Journal of Statistics and Applications 2015, 5(5): 181-195 189
powerful of the two tests? Furthermore, applying the
two-break LM test also yields 12 stationary series with two
structural breaks. What if there are more than two structural
breaks? Hence, in the face of all these issues, why testing
for unit root in macroeconomic time series then? According
to Harvey [33], “why worry about testing for unit roots in
the first place’’? A more recent direction in econometrics is
the modeling of macroeconomic series in the framework of
state space models. In the state space framework, structural
change is accounted for in the model building methodology
and testing for non stationarity is not important. Hence, in
the next section, we assess the strength of the Fisher
relationship using the time varying parameter model
estimated by the Kalman filter methods.
Table 3. Unit root with one structural break
Countries Series Zivot-Andrews LM test Conclusion
t-statistic (k) Break date t-statistic (k) Break date
Burkinafaso
INT t = -7.783‡ (7) 1969:01 t = -3.347 (7) 1991:01 Conflicting results
INF t = -8.765‡ (0) 1984:01 t = -7.332‡ (0) 1973:01 Stationarity + break
Cotedevoir
INT t = -4.970 (7) 1970:01 t = -3.347 (7) 1991:01 Conflicting results
INF t = -6.314‡ (0) 1981:01 t = -4.660† (0) 1973:01 Stationarity + break
Gambia
INT t = -5.117† (1) 1985:01 t = -4.650† (1) 2007:01 Stationarity + break
INF t = -4.678 (0) 1988:01 t = -4.452† (0) 1988:01 Conflicting results
Ghana
INT t = -4.796 (0) 1976:01 t = -4.092 (7) 1990:01 Unit root
INF t = -4.672 (7) 1976:01 t = -6.613‡ (0) 1983:01 Conflicting results
Niger
INT t = -4.828* (4) 1969:01 t = -3.9633 (7) 1991:01 Conflicting results
INF t = -5.826‡ (0) 1983:01 t = -4.9540† (0) 1982:01 Stationarity + break
Nigeria
INT t = -4.676 (4) 1966:01 t = -4.5101† (4) 1992:01 Conflicting results
INF t = -6.053‡ (1) 1998:01 t = -4.776† (1) 1996:01 Stationarity + break
Senegal
INT t = -5.153† (0) 1993:01 t = -3.5674 (7) 1991:01 Conflicting results
INF t = -6.272‡(0) 1976:01 t = -5.268‡ (0) 1985:01 Stationarity + break
Togo
INT t = -5.201†(0) 1993:01 t = -3.5691 (7) 1991:01 Conflicting result
INF t = -5.598‡(0) 1994:01 t = -4.7416† (0) 1982:01 Stationarity + break
Notes: Critical values for the Zivot-Andrews test for model C are -5.570, -5.080 and -4.820 at the 1%, 5% and 10% significance level respectively.
The critical values for the one-break LM test depend on the location of the break (
/=
B
TT
λ
) and are symmetric around
λ
and (
1
λ
).
Generally, the values range between (-5.15 to -5.05) at 1%, (-4.51 to -4.45) at 5% and (-4.21 to -4.17) at 10% significance level. ‡, †, and denote
significance at 1%, 5% and 10% levels respectively.
190 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
Table 4. Two-break unit root test
Countries Series LM Two Breaks Unit Root Test
t
-statistic (
k
) Break-date 1 Break-date 2
λ
Conclusion
Burkinafaso
INT -5.2880 (3) 1978:01 1996:01 ( 0.4;0.7) 2-break unit root
INF -8.4582 (0) 1973:01 1983:01 (0.3;0.5) 2-break stationary
Cotedevoir
INT -5.2880 (3) 1978:01 1996:01 (0.4;0.7) 2-break unit root
INF -5.3202 (0) 1973:01 1980:01 (0.3;0.4) 2-break stationary
Gambia
INT -9.6944 (7) 1985:01 2001:01 (0.3; 0.7) 2-break stationary
INF -6.7709 (7) 1984:01 1994:01 (0.3;0.6) 2-break stationary
Ghana
INT -6.4364 (3) 1993:01 2001:01 (0.6; 0.8) 2-break stationary
INF -7.8549 (0) 1976:01 1983:01 (0.3;0.4) 2-break stationary
Niger
INT -5.3840 (3) 1979:01 1995:01 (0.3;0.7) 2-break stationary
INF -6.0529 (8) 1980:01 1992:01 (0.4;0.6) 2-break stationary
Nigeria
INT -6.9237 (6) 1978:01 1992:01 (0.4;0.6) 2-break stationary
INF -6.9019 (1) 1992:01 1997:01 (0.6;0.7) 2-break stationary
Senegal
INT -4.6754 (3) 1984:01 1995:01 (0.4;0.6) 2-break unit root
INF -6.7792 (0) 1986:01 1995:01 (0.4;0.6) 2-break stationary
Togo
INT -4.8399 (6) 1978:01 1995:01 (0.3;0.6) 2-break unit root
INF -5.9024(0) 1983:01 1995:01 (0.4;0.6) 2-break stationary
Note: ‡,and denote significance at 1%, 5% and 10% respectively.
Break points Critical values
λ = (TB1/T, TB2/T) 1% 5% 10%
λ = (0.2 , 0.4) -6.16 -5.59 -5.27
λ = (0.2 , 0.6) -6.41 -5.74 -5.32
λ = (0.2 , 0.8) -6.33 -5.71 -5.33
λ = (0.4 , 0.6) -6.45 -5.67 -5.31
λ = (0.4 , 0.8) -6.42 -5.65 -5.32
λ = (0.6 , 0.8) -6.32 -5.73 -5.32
Source: Lee and Strazicich (2003)
3.3. The State Space Model Estimation Results
We employed the time varying parameter techniques to
capture the behavior of interest rates to changes in the
inflation rates over time. The relationship is examined in the
state space framework and the Kalman algorithm is apply to
assess if this relationship is stable through time. Prior to
Kalman filtering and smoothing, we estimate the unknown
variance parameter of the model using the maximum
likelihood method maximized by the BFGS
(Broyden-Fletcher-Goldfarb-Shannon) method. After
estimating the unknown variances, the Kalman filter
technique is applied to estimate the time path of the
parameter of interest (Fisher coefficient). The Kalman filter
estimate of the behavior of the parameter is depicted in
figure 2.
International Journal of Statistics and Applications 2015, 5(5): 181-195 191
Figure 2. Kalman Filter Estimate of the Fisher Coefficient for the ECOWAS Countries
192 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
The plots in Figure 2 indicate that the relationship
between interest and inflation rates vary over time and the
variation in the strength of the Fisher effect varies among
countries. The plots indicate that the strength of the Fisher
effect fluctuates around 0.01 to 0.06 for Burkina Faso
during the period from 1973 to 2011, while for Ghana, the
strength of the Fisher effect hovers around 0.2 to 1.1 during
the period from 1980 to 2011. In fact, in the case of Ghana,
we can say a full Fisher effect exists between the periods
from 1997 to 2000. However, this full Fisher effect
gradually decline to partial Fisher effect from 2001 until
the end of the sample. In the case of Gambia, the strength of
the Fisher effect was 0.20 to 1.25 during the period from
1985 to 2011. The strength of the Fisher relationship
gradually increased from the mid-1990s and attained a full
Fisher effect in the early 2000s and relatively maintained
the high strength through the sample period. The strength of
the Fisher relationship for Nigeria was around 0.20 in the
mid-1980s and attained its highest strength of 0.60 in the
early 2000s and gradually declines afterwards. There is no
noticeable period of full Fisher effect in the case of Nigeria
over the period under consideration. With the exception of
Cote d’ivoire that attained a relatively full Fisher effect in
the mid-1980s, for all the other Francophone ECOWAS
countries, there is no noticeable period of full Fisher effect
over the period of study. For example, the strength of the
Fisher relationship increased from 0.1 in 1980 to about 0.3
in 1985 and gradually decreased afterwards until it drifted
into the negative axis from 1994. Hence, a partial Fisher
effect exists for Senegal around 1974 to 1993. Similarly, in
the case of Togo, the strength of the relationship increased
from the early 1970s to about 0.4 in 1983 and gradually
decreased afterwards until it drifted into the negative axis in
1994. An interesting result about the Francophone
ECOWAS countries, is the sudden drift of the Fisher effect
parameter into negative territory in 1994 (with the
exception of Niger). Hence, a pondering question is; why
did the parameter of the Fisher effect drifts into the negative
axis for the Francophone ECOWAS countries in 1994? This
may be as a result of sharp and strong devaluation of the
CFA franc (a common currency for the ECOWAS
francophone) that occurred in January 1994. Prior to
devaluation, the value of CFA has been criticized as being
too high, and only favors the urban elites of the ECOWAS
francophone countries at the expense of the farmers who
cannot easily export their agricultural products. Hence, the
devaluation of the 1994 was aimed at addressing these
imbalances.
Our results indicate that the strength of the Fisher effect
does vary over time. Specifically, for the ECOWAS
countries; in some periods there appears to be a strong
relationship between the interest and inflation rates (full
Fisher effect), while in other periods, the relationship seems
to be weak (partial Fisher effect) and non-existing at some
other periods. The variation in the strength of the Fisher
effect may be attributed to the monetary policies adopted by
the central banks at different time periods. Chuderewicz [12]
has demonstrated using a theoretical model that the strength
of the Fisher relation depends explicitly on the behavior of
the central bank. He noted that the higher the degree of
commitment of central banks to monetary targets, the
stronger is the relationship between the interest and
inflation rates. Hence, we recommend that monetary
authorities in the ECOWAS countries should aim at making
more effective monetary policies and demonstrate strong
commitments to monetary targets in order to strengthen the
Fisher relations.
3.4. Structural Breaks and Outliers Detection Results
We exploit the capability of the state space model in
detecting the time of structural breaks and outliers. Harvey
and Koopman [28] demonstrated that auxiliary residuals in
state space models are potentially useful not only for
detecting outliers and structural breaks but for
distinguishing between them. The auxiliary residuals are
estimators of the disturbances associated with the
unobserved components. The detection strategy is to plot
the standardized residuals. In a Gaussian model, indicators
of outliers and structural breaks arise for values greater than
2 in absolute value. We apply this procedure to the
standardized observation equation residuals and the state
equation residuals in (18) to detect outliers and time of
structural breaks respectively. The plots are presented in
Figure 3.
The plots of the auxiliary residuals for Burkina Faso,
Gambia, Ghana and Nigeria (others are available on request)
are shown in figure 3. Outliers are detected in the Fisher
relations in 1992 and 1993 for Burkina Faso, in 1994 and
1996 for Gambia, 1992 and 1998 for Ghana and 1990 and
1999 for Nigeria. The plots of the other countries also
reveals the presence of outliers in the Fisher relation for
Togo in 1992, 1989, 1990 and 1992 for Cote d’ivoire, 1992
for Niger, and 1991 and 1992 for Senegal.
On the basis of the plots, we find evidence of structural
breaks in the Fisher relationship in 1980 for Burkina Faso,
1980, 1981, 1993-1994 for Cote d’ivoire, around 2005-2007
for Gambia and 1997 and 1999-2000 for Ghana. We also
detect structural breaks in 1994, 1997-1998 for Nigeria,
1985-1986 for Niger, 1972-1974 for Senegal and
1973-1975 for Togo.
International Journal of Statistics and Applications 2015, 5(5): 181-195 193
Figure 3. Detecting Outliers and Structural Breaks
194 Omorogbe J. Asemota et al.: Fisher Effect, Structural Breaks and Outliers Detection in ECOWAS Countries
4. Conclusions
This paper empirically investigates a fundamental
relationship in macroeconomics: the relationship between
interest and inflation rates. The fundamental issue addressed
in this paper is whether this relationship is stable over time
for ECOWAS countries. The inflation and interest rates for
Burkina Faso, Cȏte d’Ivoire, Gambia, Ghana, Niger,
Nigeria, Senegal and Togo are used in the study. First, we
investigate the order of integration of the 16 time series
using the augmented Dickey-Fuller (ADF), Phillips-Perron
(PP) and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS)
unit root tests as a confirmatory test. Using the BIC as the
lag selection criteria for the ADF test and the Bartlett kernel
for the computation of the covariance matrix for both the PP
and KPSS, the results yield 8 cases of conflicting results, 6
cases of genuine-stationarity and 2 cases of genuine-unit
roots. When the
t
-statistic is used for the selection of the
lag order for the ADF test and the Quadratic Spectral kernel
is used for the PP and KPSS, the test results indicate 6 cases
of conflicting results, 7 cases of genuine-stationarity and 3
cases of genuine-unit roots. This indicates that inference
based on unit root tests may be affected by the method of
lag selection and method of constructing heteroskedasticity
and autocorrelation consistent (HAC) estimators. We also
conduct unit root tests allowing for one and two structural
breaks. On allowing for structural breaks, the unit root
hypothesis is rejected for 12 out of the 16 series considered
in the study. Secondly, the Fisher equation is cast in the
state space form and the Kalman filter is apply to estimate
the slope parameter. Our results indicate that the strength of
the Fisher effect does vary over time. Specifically, for the
ECOWAS countries; in some periods there appears to be a
strong relationship between the interest and inflation rates
(full Fisher effect), while in other periods, the relationship
seems to be weak (partial Fisher effect) and non-existing at
some other periods. Using the Harvey-Koopman procedure,
we detect the time of structural breaks and outliers in our
model. Finally, we recommend that monetary authorities in
the ECOWAS countries should aimed at making effective
monetary policies and demonstrate strong commitments to
monetary targets in order to strengthen the Fisher relation.
ACKNOWLEDGEMENTS
We thank Professor Peter C. B. Phillips, Felix Chan and
other participants at the New Zealand Econometric Study
Group Meeting 2013 for their helpful comments when this
paper was first presented at the New Zealand Econometric
Society Conference. The usual disclaimer applies.
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In this study, was examined the validity of the Fisher Hypothesis in Turkey using the methods of analyzing econometric time series based on the proportions of profits paid to participation accounts on monthly data for the period 2004: 01-2021: 02 and the relationship between nominal interest rates and inflation. The analysis explored the stationarity of the series, primarily by applying the ADF Unit Root Test, the Zivot-Andrews Unit Root Test, and the Lee-Strazicich Unit Root Test. As a result of the unit root tests, the causation relationship between the series was analyzed with the Toda-Yamamoto Causality Test and the Breitung�Candelon Frequency Area Causality Test. As a result of the analysis, the Fisher hypothesis for Turkey is valid and the one-way and long-term causation relationship from inflation to interest has been established. In the context of the Fisher hypothesis, the analysis of the relationship between inflation and the profit sharing has established a one-way and a short-term causation relationship from inflation to the profit sharing. Keywords: Fisher Hypothesis, Inflation Rate, Interest Rate, Profit Share Rate, Turkish Economy, Frequency Area Causality Test Result.
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In this study, was examined the validity of the Fisher Hypothesis in Turkey using the methods of analyzing econometric time series based on the proportions of profits paid to participation accounts on monthly data for the period 2004: 01-2021: 02 and the relationship between nominal interest rates and inflation. The analysis explored the stationarity of the series, primarily by applying the ADF Unit Root Test, the Zivot-Andrews Unit Root Test, and the Lee-Strazicich Unit Root Test. As a result of the unit root tests, the causation relationship between the series was analyzed with the Toda-Yamamoto Causality Test and the Breitung�Candelon Frequency Area Causality Test. As a result of the analysis, the Fisher hypothesis for Turkey is valid and the one-way and long-term causation relationship from inflation to interest has been established. In the context of the Fisher hypothesis, the analysis of the relationship between inflation and the profit sharing has established a one-way and a short-term causation relationship from inflation to the profit sharing. Keywords: Fisher Hypothesis, Inflation Rate, Interest Rate, Profit Share Rate, Turkish Economy, Frequency Area Causality Test Result. JEL Classification: G21, G23, E31, E40, C22, C32.
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