hypothesis. Kwiatkowski et al. [17] propose the following
where 𝑦𝑡 is the original series, 𝛼�0 and 𝛼�1 are the drift and trend parameters respectively obtained by regressing 𝑦� on
𝑥
test statistics
𝐾𝑃𝑆𝑆 = 1 ∑𝑇
International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 1342
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Fractional Cointegration of Fisher Hypothesis in
Regulated and Deregulated Periods in Nigeria
Moses Abanyam Chiawa & Victoria Egbe Amali
This study investigates the validity of Fisher hypothesis in regulated period (January 1961 to September1986) and the deregulated (October 1986 to December 2010) using monthly nominal interest rate and inflation. Tests for unit root and fractional root were conducted on these variables for the two periods. Fractional cointegration analysis was used to test the long run relationship between nominal interest rate and inflation. The results indicate
fractional integration in the deregulated period for the variables. The variables were found to be fractionally cointegrated which implied that they are mean reverting and have long run relationship in the deregulated period. In the regulated period on the other hand, the variables were not fractionally cointegrated. The cointegration test did not support long run relationship between the variables in the deregulation period. This research work confirms the validity of Fisher hypothesis with deregulated period in Nigeria and there is no evidence of Fisher hypothesis in the regulated period. This is because in the regulated period interest rate is fixed and inflation is controlled, therefore not allowing the two economic variables to interplay. The study recommends that the Nigerian government should adopt policies that reduce inflation and interest rate to the minima points since Fisher hypothesis does exist in the current dispensation.
Nigeria’s financial system has gone through noticeable changes in term of ownership structure. Several policies, with a number of institutions established to provide the necessary economic environment and regulatory framework. Prior to 1986 when the structural adjustment programme (SAP) was introduced, the lending rate practiced by banks in Nigeria were strictly regulated under the surveillance of the supervisory Central Bank of Nigeria. The deregulation period which brought about the Structural Adjustment Programme (SAP) era came with stringent banking rules ([1]).
Financial institutions are regulated in some countries (that is, interest rate control, compulsory public debt placement and control on external capital flows) fixing nominal interest rate and raising fiscal deficits to lower inflation. This was done in Nigeria which resulted in repressed (even negative) real interest rates as observed in [2]. As a reversal policy however, the government in September 1986 introduced some measure of deregulation into interest rate management due to wide variations and unnecessarily high interest rate under the complete regulation policy ([1]).
The relationship between interest rate and inflation has
————————————————
• Dr. Moses Chiawa is currently an Associate Professor of Statistics at the Department of Mathematics and Computer Science, Benue State University, Makurdi, Nigeria. PH - +2348066715321.
E-mail: abanymoses@yahoo.co.uk
• Ms. Victoria Egbe Amali is currently pursuing masters degree program in
Statistics at University of Agriculture, Makurdi, Nigeria., PH-
+2348039735852. E-mail:vamali822@gmail.com.
been frequently investigated in both theoretical and
empirical economics. The one-to-one long-run relationship
between interest rates and inflation known as the Fisher hypothesis, has been tested extensively in the last two decades. The study of this relationship initially tested by [3] known as Fisher hypothesis or the Fisher effect, examines
the effect of nominal interest rate and inflation on the minimum required returns ([4]). According to [4], the Fisher effect or hypothesis states that a permanent change in the inflation rate will cause nominal interest rate to move in a one to one relation with inflation. Thus, the real interest rate will remain unchanged in response to a monetary shock if the Fisher effect holds. The real interest rate is determined entirely by the real factor in the economy, such as productivity of capital and investment time. This evidence of Fisher effect was found to be strong in some countries at certain period than others. For instance, United Kingdom, United States of America and Canada had strong evidence of Fisher effect in the post second world war period ([5]).
To uncover this long run relationship between nominal interest rate and inflation, most literature applied unit root developed by [6] and the Johansen cointegration test by [7]. The standard Johansen cointegration analysis assumes that all variables are integrated of order one I(1) and hence restricts the error correction term to be a stationary process I(0). This method was used by [8] to investigate the Fisher effect in Nigeria. The research discovered Partial Fisher effect in the long run between interest rate and inflation rate in Nigeria. Asemota and Bala [9] employed Kalman filter and cointegration approach to investigate Fisher effect in Nigeria using quarterly data. Their study did not find evidence of full Fisher effect. Noor and Shamim [10] used
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Johansen cointegration test to determine the nature of relationship between nominal interest rate and inflation. Variance decomposition was used to explain the error
Hence, the excepted inflation is estimated by the following equation.
correction model and Granger causality to determine the
direction of relationship. The results found the presence of
𝜋 𝑒 = 𝜋𝑡
+ 𝜀𝑡
(3)
Fisher effect in Pakistan for the period 1980-2010.
There is a more generalized model whose order of integration is continuum of real number known as fractional cointegration. The model does not only nest the unit-root behavior within it, but also display stationary and non-stationary, mean-reverting dynamics, long-memory and anti-persistent dependencies ([11]). The concept of long memory and fractional integrated models in time series was introduced by [12], [13] and [14]. Specifically, in Fisher effect, fractional integrated series may imply the existence of an equilibrium long term relationship between nominal interest rate and inflation. Consequently, the error correction term is fractionally cointegrated thus there exist fractional cointegration between nominal interest rate and inflation in line with Fisher hypothesis.
This study is to find out whether there is any relationship between nominal interest rate and expected inflation in Nigeria in the regulated and deregulated periods. It is also to determine evidence of Fisher hypothesis in both periods.
This study employs Dickey-Fuller Generalised Least Square (DF-GLS) and Kwiatkowski, Phillips Schmidt and Shin (KPSS) to test for unit root and stationarity. Geweke and Porter Hudak (GPH), Robinson Log-periodogram (Roblpr) and the modified log-periodogram regression test for fractional differencing parameter of the long memory process.
The Fisher equation is of the form
𝑒
where 𝜀𝑡 is the error component. Thus the composite error
is equal to 𝛽𝜀𝑡 = 𝑍𝑡 . In particular, an estimate of 𝛽 is not
significantly different from one in the cointegration
regression equation (2), which indicate the presence of a
full Fisher effect. The term (𝐼𝑡 − 𝜋𝑡 ) is stationary if the estimate of 𝛽 is significantly lower than one. That is, there
will be a partial Fisher effect. Therefore the changes in the
expected inflation rate would be transmitted in a
proportion to (𝛽 < 1) the nominal interest rate.
By definition 𝒓𝒕 = 𝑰𝒕 − 𝝅𝒕 where 𝒓𝒕 is the real interest rate.
In other word, 𝒓𝒕 is given by the assumption of rational
expectation. The real interest rate would only differ by a
random stationary term, so that stationarity of the former
(real interest rate) implies stationarity of the later (𝑰𝒕 − 𝝅𝒕 ).
However it could happen that in equation (2). The nominal
interest rate and inflation rate would be cointegrated. If I t
and 𝝅𝒕 , are characterized by units root or are I(1) then
Fisher hypothesis may be tested by examining whether
these two variables form a stationary linear combination.
All conventional unit root tests are based on ARIMA (p,d,q) model. In a traditional ARIMA model, d is restricted to be a non negative integer. When d=1, the process is non- stationary, if d = 0, the process is stationary. ([15]) observed that a standard unit-root test such as Dickey-Fuller test may have low power against fractional alternative. For that reason this study employ DF-GLS test developed by [16] and KPSS by [17] to test for unit root and stationarity. These approaches are described below.
The Dickey-Fuller generalized least squares unit root test by [16] is a simple modification of the Augmented-Dickey
𝐼𝑡 = 𝑟𝑡 + 𝜋𝑡
(1)
Fuller test. Exploiting the result in [18] that uniformly powerful test do not exist for unit root tests, [16] modified
where I t is the nominal rate, rt is the ex-ante real interest
𝑒
the ADF test and showed that DF-GLS test has the limiting
rate and 𝜋𝑡
is the expected inflation rate. The Fisher
power function close to the point optimal test. In the
equation can also be written in the form
I t = 𝛼 + 𝛽𝜋𝑡 + 𝜂𝑡 (2)
where 𝛼 and 𝛽 are all parameter to be estimated. 𝜋𝑡 is the
actual inflation rate, 𝜂𝑡 is a long memory process, such as
ARFIMA. The expected inflation rate coincide with actually
recorded inflation rate (𝜋𝑡 ) plus the random error (𝜀𝑡 ).
conventional ADF test for non-stationarity, the alternative hypothesis is ρ<1 which shows that the test is not conducted against any value of ρ. Under these circumstances a power envelope was constructed covering the continuous set of each possible value of ρ under the alternative. Elliot et al. [16] then proposed a family of tests whose power functions is tangent to the power envelope at one point and never lying below it. These tests were denoted by PT (0.5), which shows that they are optimal at the
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power of 50%. They went on to show that DF-GLS has the limiting power function close to P T (0.5).
In the DF-GLS test, the data are detrended (depending on whether the model is with drift only or has a linear trend) so that deterministic variables are taken out of the data prior to running the regression. A generalized least squares
𝑑
where 𝑥𝑡 is a random walk
𝑘𝑡 = 𝑘𝑡−1 + 𝑣𝑡 , 𝑣𝑡 ~𝑖𝑖𝑑(0, 𝜎2 𝑣) (10)
where 𝑧𝑡 is a stationary process. In this framework the
foregoing pair of hypothesis is equivalent to the pair.
2 = 0 𝑎𝑔𝑎𝑖𝑛𝑠𝑡 𝐻 : 𝜎2 ≠ 0 (11)
regression equation of detrended data, 𝑦𝑡
estimate associated with 𝑎� is given as
using the
𝐻𝑜 : 𝜎𝜐
𝑎 𝜐
𝑑 = 𝑦𝑡 − 𝛼�0 − 𝛼�1 t (4)
where 𝐻𝑜 is the null hypothesis, and 𝐻𝑎 is the alternative
hypothesis. Kwiatkowski et al. [17] propose the following
where 𝑦𝑡 is the original series, 𝛼�0 and 𝛼�1 are the drift and trend parameters respectively obtained by regressing 𝑦� on
𝑥
test statistics
𝐾𝑃𝑆𝑆 = 1 ∑𝑇
𝑇
2
𝑡
𝜎� 2
(12)
y = [ y , (1 − aL) y
,, (1 − aL) y ]′
(5)
where 𝑆
= ∑𝑡
𝑤� 2
x = [ x , (1 − aL) x
,, (1 − aL) x ]′
(6)
𝑡
given as
𝑗=1
𝑗 , with 𝑤𝑗 = 𝑦𝑡 − 𝑦� and 𝜎�∞ is an estimator
𝜎2 = lim
𝑇→∞
𝑇 −1 𝑉𝑎𝑟 ( ∑𝑇
𝑍𝑡
) (13)
with
x = (1,T )′ , α = 1 +
=c
, c is 7 if the data
that is 𝜎∞ is an estimator of the long run variance of the
𝑣
T
process zt , 𝑤𝑗 = 1 −
𝑗𝑞+1
is a Bartlett window with truncated
generating process has a drift only and c is 13.5 if the data
generating process has a drift with trend.
The DF-GLS test therefore involves estimating the equation
𝑑 𝑑 𝑑 𝑑
lag 𝑗𝑔 . KPSS uses Kernel estimators to correct serial for
correction, and Bartlett kernel is one of the best estimators.
The null hypothesis is rejected if the test statistics is greater
that the asymptotic critical values.
∆𝑦𝑡
= 𝛽0 𝑦𝑡−1 + 𝛽1 ∆𝑦𝑡−1 + … + 𝛽𝑝 ∆𝑦𝑡−𝑝 + 𝑣𝑡 (7)
The existence of fractional cointegration implies that the
As with the ADF test, the null hypothesis is H0 : 𝛽0 = 1. This
is considered with the t-ratio for 𝛽̂0 from equation (7). The
DF-GLS t-ratio follows a Dickey-Fuller distribution in the
case of constant only. The asymptotic distribution differs
when you include both constant and trend. ([16]) simulate the critical values of the test statistics in this later case. By
setting 𝑇 = {50, 100, 200, ∞}, the null hypothesis is rejected
if the test statistic falls below the critical value.
The integrated properties of a series 𝑦𝑡 may also be
investigated by testing the null hypothesis
𝐻0 = 𝑦𝑡 = 𝐼(0) against the alternative hypothesis
𝐻𝐴 : 𝑦𝑡 = 𝐼(1) (8)
econometric data may be randomly drifting away from equilibrium for long episodes, but may finally return to equilibrium. Fractional cointegration enables us to distinguish between the case where the equilibrium errors are non-mean reverting and where they are actually mean- reverting but exhibiting a significant persistence in the short-run. If the equilibrium error is found to be integrated of order b with b>0 (which might not be necessarily I(0)), the series is fractionally cointegrated.
Conventional unit-root tests are based on ARIMA (p,d,q) model and d the differencing parameter is restricted to be an integer. The ARFIMA model was however, introduced by [12] and [14]. The differencing parameter d is a real
That is the hypothesis that the data generating process is stationary [I(0)] against a unit root [I(1)]. Kwiatkowski et al.
number. An ARFIMA ( p, d ′, q )
process that satisfies:
process is a stationary
[17] have derived a test for the pair of hypothesis in the
φ ( L ) (1 − L )d y
= c + θ
( L )ε ,
t=1,2,…,T (14)
absence of linear trend term. They start from a data generating process.
p t q t
where d is the parameter of fractional differentiation, c is a
constant and φp
and θq
are autoregressive and moving
𝑦𝑡 = 𝑥𝑡 + 𝑧𝑡
average polynomials of order p (a9n)d q, respectively. The
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autocorrelations,
ρk ,
for an ARFIMA process for large k
where xi is the regressor ( ln�4𝑠𝑖𝑛2 �𝜆𝑗⁄2��). If
and
d < 1 are given by the following approximation (see
limT →∞ [ g (T )] and limT →∞ g (T ) / T = 0
2 2
then
([12])).
p lim s = π
/ 6 where s2 is the sample variance of the
𝜌𝑘 ≈
Γ(1−𝑑)
Γ(𝑑 )
𝑘2𝑑−1 (15)
residuals from the regression equation (19).
which is a monotonic function. The autocorrelation
function of ARFIMA decreases slowly to zero while its spectral density is infinite (see [14]),
The test developed by [19] is a spectral regression-based method for estimating fractional difference (d) in the equilibrium error. The test was carried out using Monte Carlo simulation experiment performed by [19]. The test also provides a general way of testing for fractional difference which is not dependant on nuisance parameters of the ARMA process.
𝑓𝑥 (𝜆) = 𝜎2 ⁄2𝜋{4𝑠𝑖𝑛2(𝜆⁄2}2 𝑓𝑢 (𝜆) (16)
Assume a sample of 𝑋𝑡 of size T is available. Take
logarithms from both sides of equation (16), and evaluate it
2𝜋𝑗
The value of the power factor, µ , is the main determinant
of ordinates included in the regression. Traditionally the number of periodogram ordinates is chosen from the interval [T0.45, T0.55]. However, ([20]) recently showed that the optimal m is of order O(T0.8).
at harmonic frequencies 𝜆𝑗 =
, j= 0,1,2,…T-1. After
𝑇
slope to differ for each series. A choice must be made of the
adding 𝐼(𝜆𝑗 ) the periodogram at ordinate j, to both sides of
the log form of equation (16), becomes:
𝐼𝑛{𝐼𝜆𝑖 } =
𝐼𝑛{𝜎2 𝑓𝑢 (0)⁄2𝜋} − 𝑑𝐼𝑛{4𝑠𝑖𝑛2(𝜆⁄2} + 𝐼𝑛�𝐼(𝜆𝑗 )⁄𝑓𝑥 (𝜆𝑗 )� (17)
The last term on the right-hand side of equation (17)
becomes negligible when low-frequency ordinates 𝜆𝑗 are
near to 0. The following simple regression is hence
suggested.
𝐼𝑛�𝐼�𝜆𝑗 �� = 𝛽0 + 𝛽1 𝐼𝑛�4𝑠𝑖𝑛2 �𝜆𝑗 ⁄2�� + 𝜂𝑗 (18)
number of harmonic ordinates to be included in the
spectral regression. Robinson's estimator is not restricted to
using a small fraction of the ordinates of the empirical periodogram of the series. It also allows the removal of one or more initial ordinates, and the averaging of the periodogram over adjacent frequencies. Given the
assumption that 𝑋𝑡 represent a G-dimension vector with
𝑔𝑡ℎ element 𝑋𝑔𝑡 , 𝑔 = 1 , … 𝐺. If 𝑋𝑡 has a spectral density
matrix
𝜋
−𝜋
where (𝑔, ℎ) is an element in 𝑓𝑔ℎ (𝜆). The 𝑔𝑡ℎ diagonal
element 𝑓𝑔𝑔 (𝜆), is the power of the spectral density of 𝑋𝑔𝑡
for 0 < 𝐶𝑔 < ∞ 𝑎𝑛𝑑 −
< 𝑑𝑔 < .
where intercept 𝛽0 = 𝐼𝑛{𝜎2 𝑓𝑢 (0)⁄2𝜋}, parameter 𝛽1 = −𝑑,
Hence Robinson log-periodogram is given as
error term 𝜂𝑗 = 𝐼𝑛�𝐼(𝜆𝑗 )⁄𝑓𝑥 (𝜆𝑗 )� and j = 1,2,…,n. The
𝐼 (𝜆) = (2𝜋𝑛)−1�∑𝑛
𝑋 𝑒𝑖𝑡𝑋 2
𝐺
number of observations, that is, the number of ordinates to
𝑡=1
𝑔𝑡
� , 𝑔 = 1, … , 𝐺 (21)
be used in the estimation of the regression is n = g(T), where g(T) satisfy the following conditions: limT →∞ g (T ) = ∞ and limT →∞ g (T ) / T = 0 . The function g (T ) = T µ , with 0 < µ < 1, is the number of periodogram ordinates used to estimate d and satisfies both conditions and the estimator of d is consistent. The test of hypothesis for the parameter can be done based on the
The GPH estimation is inconsistent against 𝑑 > 1 of the
alternative hypothesis.
asymptotic distribution of dˆ, derived by [19].
Let 𝑋𝑡
be fractionally integrated process, it implies 𝑋𝑡
𝑑̂ → 𝑁(𝑑, 𝜋 2⁄6 ∑(𝑥𝑖 − 𝑥 2 )) (19)
satisfies a general model in equation (21) in the case where
the 𝑡 > 0 in the error term, 𝑓𝑢 (𝜆) is the spectral density.
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𝜇𝑡 is stationa6ry with zero mean. In the case where the
𝑡 > 0 in the error term
This section comprises of analysis of data for regulation period (1961M1 – 1986M9) and deregulation period (1986M10 – 2010M12). Consideration was given to period
𝜇𝑡 = 𝐶(𝑙)𝜀𝑡 = ∑∞
𝐶𝑗 𝜀𝑡−𝑗 , ∑∞ 0
𝑗�𝐶𝑗 � < ∞ 𝐶(1) ≠ 0
(22)
when interest rate was regulated by government and
period when government remove some restrictions to
with 𝜀𝑡 = 𝑖𝑖𝑑(0, 𝜎2 ) with Ε|𝜀𝑡 |𝑝 < ∞ and 𝑝 > 4. Equation
(22) implies that spectrum of 𝜇𝑡 is continuously
differentiable for all frequencies.
The fractional process is
(𝑑)𝑘
interest and introduce the Structural Adjustment Programme from 1986 - 20101.The section contain plots, unit root tests, fractional difference estimations and model fittings. The variables employed in this study are nominal
𝑋𝑡 = (1 − 𝑑)−𝑑 𝜇𝑡 = ∑𝑡
𝑘!
𝜇𝑡−𝑘 ,
interest rate (nint) and inflation (inf). Fractional difference
𝑤𝑖𝑡ℎ 𝜇𝑡 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ≤ 0 (23)
Γ(𝑑 + 𝑘)
series of nominal rate (fint) and inflation (finf) and
computed.
𝑤ℎ𝑒𝑟𝑒 (𝑑)𝑘 =
Γ(𝑑)
This is Pochammer’s symbol for the forward factorial
function. The modified estimator is expressed as
1
𝑤𝑢 (𝜆) = 𝑤𝑥 (𝜆)𝐷𝑛 �𝑒−𝑖𝜆 ; 𝑑� + �𝑋�
(𝑑) − 𝑒𝑖𝑛𝜆 𝑋�
(𝑑)� (24)
√2𝜋𝑛
𝜆0
𝜆𝑛
𝑤ℎ𝑒𝑟𝑒 𝑋�
𝑛−1
(𝑑) = 𝐷 �𝑒−𝑖𝜆 𝐿; 𝑑� = � 𝑑̃
𝑒−𝑖𝑝𝜆 , 𝐿𝑃
𝜆𝑛
𝑛−1
𝑋𝑛 = � 𝑑̃
𝑛
𝑒−𝑖𝑝𝜆 𝑋𝑛−𝑝 ,
𝑝=0
𝜆𝑝
5 -4.5257 0.00005 5 -4.6950 0.0000
4 -4.0210 0.00014 4 -4.6981 0.0000
𝑝=0
1
𝑤𝑥 (𝜆) = ∑𝑛
𝑋𝑡 𝑒𝑖+𝑡
(25)
3 -3.0532 0.00253 3 -5.6437 0.0025
√2𝜋𝑛
𝑡=1
2 -3.0532 0.00252 2 -15.5349 0.0000
The discrete Fourier transform of 𝑋𝑡 is 𝑊𝑥 (𝜆) which is
associated with periodogram co-ordinates 𝑤𝑢(𝜆). The
Geweke and Porter-Hudak test establish consistency and
asymptotic normality for 𝑑 < 0. Robinson [21] proved consistency and asymptotic normality for 0 < 𝑑 ≤ 1�2 in
the case of non-stationary ARMA while [23] showed that
𝑑>1 is consistency with 𝑑 exihibiting asymptotic bias
towards unity. The null hypothesis of consistency is tested
against both 𝑑 < 1 𝑎𝑛𝑑 𝑑 > 1.
negative and tend hyperbolically to zero in such case, the process is considered anti-persistent or with intermediate memory.
1 -3.0532 0.00251 1 -15.5249 0.0025
0 -3.0532 0.00250 0 -15.5249 0.0025
The critical values of the t-statistic are -3.4686, -2.9128 and -
2.6099 at 1%, 5%, and 10% level of significance respectively, for both level and first difference.
The test was carried out including the trend and intercept. The results show that the inflation series is stationarity at both level and first difference at 5% level of significance.
(ii) If 0 < 𝑑 < 0.5, then all the auto-correlation are
positive and also decline hyperbolically. Therefore
they are persistent and have a long memory process.
(iii) If 0.5 < 𝑑 < 1, the process is covariance non- stationary but mean reverting since an innovation
will have no permanent effect on its value. This is in contrast to an I(1) which will be both covariance non-stationary and non-mean reverting in which case the effect of an innovation will persist forever.
5 -2.1295 0.0341 5 -9.3311 0.0000
4 -1.3796 0.0000 4 -18.7312 0.0000
3 -1.3189 0.1883 3 -18.7312 0.0000
2 -1.3189 0.1883 2 -18.7312 0.0000
1 -1.3189 0.1883 1 -18.6927 0.0000
1 For the purpose of this study, we are limiting the period of government relaxation of rule on interest rates (deregulation period) to 2010
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0 -1.4680 0.0000 0 -18.6927 0.0000
The critical values of the t-statistic are -3.4683, -2.9134 and -
2.6110 at 1%, 5%, and 10% level of significance respectively, for both level and first difference.
The test is carried out including both trend and intercept. The results show that the series is stationary only at first difference of the series at 5% level of significance.
The critical values of the t-statistic are -3.4708, -2.9084 and -
2.6012 at 1%, 5%, and 10% level of significance respectively,
for both level and first difference.
In Table 4 the test was carried out including both trend and intercept. The results show that the interest rate series is stationary in level at 5% only at lag 0 and stationary for all lags in first difference at 5% level of significance.
Inf 0.053 0.216 0.021 0.216 0.148 0.216 0.036 0.216
5 -3.1606 0.0017 5 -7.2758 0.0000
Int rate
0.063 0.110 0.100 0.110 0.258 0.110 0.064 0.110
4 -3.1606 0.0017 4 -7.2758 0.0000
3 -3.1606 0.0017 3 -7.2758 0.0000
2 -2.5292 0.0000 2 -7.2758 0.0000
1 -2.1530 0.0321 1 -9.0975 0.0000
0 -1.6249 0.1052 0 -13.3222 0.0000
The critical values of the t-statistic are -3.4708, -2.9084 and -
2.6012 at 1%, 5%, and 10% level of significance respectively,
for both level and first difference.
The test was carried out including both trend and intercept. The results show that inflation is stationary in level at 5% from lags 3 to 5 and stationary for all lags in first difference at 5% level of significance.
At 5% level of significance, the DF-GLS test reject that the null hypothesis of unit root for nominal interest rate and inflation rate. The KPSS tests uniformly reject trend stationarity for all series tested at the 5% level of significance. The combination of these evidences imply that both nominal interest rate and inflation rate are not stationary I (1) processes at level in the deregulated period but stationary in the regulated period.
These conventional unit root tests have frequently been employed to establish the non stationarity of time-series processes. Critiques of this strategy have been carried out by [24], [25] and [26]. Perron and Vogelsang [25] demonstrated in a simulation study that shifts in the intercept and/or slope of the trend function of a stationary time series biases these standard unit-root tests toward non rejection of the null hypothesis. This is the reason why an alternative measure of estimating order of integration has been employed in this study.
5 -2.4627 0.0000 5 -16.1295 0.0000
4 -2.2152 0.0275 4 -16.1295 0.0000
3 -2.2152 0.0000 3 -16.1295 0.0000
2 -2.2152 0.0275 2 -16.1295 0.0000
1 -2.4627 0.0143 1 -16.1295 0.0000
0 -3.7575 0.0002 0 -26.5841 0.0000
Conventional unit-root tests, generally fail to consider the
possibility that the order of integration of these series may
be fractional I(d), rather than integer (0, 1, 2, ...). The mean-
reverting properties of nominal interest rate and inflation series may not be detectable by standard integer order of differencing (unit-root tests), due to some of the short comings earlier mentioned (low power against fractional alternatives).
Fractional integration tests are employed to overcome this challenge. This study makes use of three nonparametric
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International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 1348
ISSN 2229-5518
tests for the order of fractional integration2. These are spectral based regression estimator [19], the log- periodogram regression estimator of [21] and the estimator
of [22]. The Phillips' estimator is a recently proposed
extension of [19] which addresses some of the weaknesses
of the test in [19].
0.70 0.7644 1.4284 -0.1999 0.4011
0.75 0.8380 1.3616 -0.1189 0.3292
0.80 0.7099 1.2256 -0.2189 0.2951
0.85 0.6280 1.1889 -0.2494 0.3132
0.90 0.5169 1.0252 -0.3236 0.2369
0.50 0.3335 1.0551 -0.3290 0.0694
0.55 0.6296 1.0486 -0.2915 0.0555
0.60 0.8690 1.3170 -0.1416 0.3227
0.65 0.9028 1.2447 -0.1361 0.2420
0.70 0.8026 1.3816 -0.2186 0.4506
Result in Table 8 shows fractional integration of interest rate at level, while inflation at level is non-stationary but is fractionally integrated at first difference.
0.75 0.8750 1.3506 -0.0913 0.3536
0.80 0.7498 1.2902 -0.2171 0.2704
From Table 6 GPH estimation reject the null hypothesis of stationarity in level of both series, but does not reject stationarity at first difference of both series, hence both series are fractional integrated at first difference.
0.70 0.4769 1.1065 0.0314 0.1
0.75 0.7200 0.9784 0.0301 0.0
0.80 0.8730 0.9156 0.0010 0.0
0.85 0.7393 0.9386 -0.0054 0.0
0.90 0.8065 0.8299 -0.0295 0.0
The two variables in Table 9 are fractionally integrated.
0.50 0.6935 0.4416 -0.2416 -0.0433
0.55 | 1.1928 | 0.3906 | 0.1882 | 0.0876 |
Level First difference |
0.60 | 1.2471 | 0.3386 | 0.2870 | 0.0240 | Power Interest Inflation Interest Inflation |
0.65 0.70 | 1.1757 1.1467 | 0.3840 0.4639 | 0.2131 0.1975 | 0.0343 0.0146 | rate rate 0.50 -0.1942 0.6042 -0.0555 0.0186 |
0.75 | 1.0089 | 0.7047 | 0.0344 | 0.0308 | 0.55 -0.0914 0.8955 -0.0705 0.3241 |
0.80 0.9622 0.8974 0.0237 0.0035
0.60 -0.1023 1.1236 0.1061 0.4270
0.65 0.0666 1.0473 0.0363 0.2680
The result of Table 7 show fractional integration of inflation | 0.70 | 0.2443 | 1.0727 | 0.0873 | 0.2272 |
at level hence inflation in regulation period is persistent and has a long memory process while interest rate of the | 0.75 | 0.5031 | 0.9115 | 0.0574 | 0.0610 |
same period is non-stationary but mean reverting. | 0.80 | 0.7536 | 0.8893 | 0.0166 | 0.0093 |
2 another methods of fractional integration estimation is the
Lo modified rescaled range(R/S) test.
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International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 1349
ISSN 2229-5518
0.50 0.5326 0.9150 -0.3798 0.0696
0.55 0.6277 0.9626 -0.3355 0.0566
0.60 0.8632 1.0772 0.1777 0.3285
0.65 0.8971 1.1154 -0.1665 0.2407
0.70 0.7994 1.1921 -0.2422 0.4367
0.75 0.8713 1.2284 -0.1100 0.3413
0.80 0.7492 1.1958 -0.2330 0.2590
Table 10 and Table 11 results agree with that of Geweke and Poter-Hudak and the log periodogram regression estimation.
The long memory parameter d estimated in Table 6 and 8 are used to generate fractionally difference series for inflation and interest rate for the deregulation period. The new set of data is now used to perform cointegration. The cointegration test is carried out using Johansen test for cointegration see Table 10.
Number of equation = 2, Lag interval = 1 to 4.
The result in Table 10 does not show a long run relationship between the two variables at 95% confidence interval.
Var | Coeff | Error | t-stat | p-value |
Const | 9.2135 | 0.1246 | 73.94 | 0.00001 |
Inf | 0.3819 | 0.0074 | 5.186 | 0.0516 |
Least square regression is given as
𝐼𝑡 = 9.21350 + 0.381939 × 𝑖𝑡 (26)
Results from Table 11 show a partial Fisher effect between
nominal interest rate and inflation in regulated period. The least square regression is given as
Variable | Coefficient | Standard error |
Constant | 21.4202 | 0.436624 |
Inflation | 0.0274743 | 0.140552 |
The least square regression is given as
𝐼𝑡 = 21.4202 + 0.0275 ×it
(27)
Equation (27) implies partial relationship between interest
rate hand inflation in deregulation period.
In this study, we provide further empirical evidence on the validity of the Fisher hypothesis, which proposes a positive relationship between nominal interest rates and inflation by applying the fractional cointegration techniques to a data of regulation and deregulation periods. The results show that, nominal interest rate and inflation are fractionally cointegrated in the deregulation period using the Geweke and Poter-Hudak, the Robinson log-periodogram test and the modified log-periodogram regression estimation. The residuals of regulation period are not are fractionally integrated.
Therefore, the results show that there is a stable long run relationship between nominal interest rate and inflation suggesting the validity of Fisher hypothesis in deregulation era. The results also imply that equilibrium errors display long memory or deviation from the long run relationship
Eigen-value | Trace- stat | 5% Critical value | P-value |
0.053025 | 23.31639 | 15.49471 | 0.0027 |
0.027521 | 7.897682 | 3.841460 | 0.0050 |
shared by nominal interest rates and inflation takes a long time to dissipate and return to their equilibrium relationship, but the same can't be said for the regulation period.
This study is in agreement with the findings of [11] who examined the Fisher hypothesis using fractional integration. This thus confirm the validity of Fisher’s hypothesis in the current dispensation of deregulation in Nigeria and found clear evidence that the research methodology used in several recent contribution to the Fisher equation literature using standard cointegration techniques is inappropriate. In conclusion, our results justify the increasing use of fractional cointegration analysis to validate the findings of Fisher hypothesis.
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ISSN 2229-5518
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