A REMODELLING OF MONTHLY RAINFALL FREQUENCY IN UMUAHIA, NIGERIA, BY A BOX-JENKINS
APPROACH
ABSTRACT
Monthly rainfall frequency in Umuahia, Imo State, Nigeria, has recently been modelled as a
SARIMA(0, 0, 0)x(0, 1, 1)12. This model was demonstrated to be better than the competing
SARIMA(0, 0, 0)x(1, 1, 1)12 on all counts; that is, in terms of minimum AIC and maximum R2.
Moreover in the latter model, the seasonal autoregressive, SAR, coefficient was observed to be
statistically non-significant. This automatically made the former model suggestive. The statistical
package used was R. However in this write-up by the use of Eviews 7 it is shown that the competing
models are the same but the latter model is the better model on all counts including residual normal
distribution. Therefore forecasting of the series may be done using the SARIMA(0, 0, 0)x(1, 1, 1)12
model.
KEY WORDS: Rainfall, Frequency, Time Series, Stationarity, SARIMA, R, Eviews 7.
INTRODUCTION AND LITERATURE REVIEW
Akpanta et al. (2015) have fitted a SARIMA(0, 0, 0)x(0, 1, 1)12 model to monthly frequency of rainfall
in Umuahia, Abia State, Nigeria. The model was adjudged the better of two suggestive ones, the
other one being a SARIMA(0, 0, 0)x(1, 1, 1)12 model. This work is another attempt to model the series
which was published by the authors using the same SARIMA approach with the two models as the
focus. The authors used the R software for analysis. Here, the Eviews 7 software is used. Results
obtained herein are contradictory to their results; the SARIMA(0, 0, 0)x(1, 1, 1)12 model is observed
to be the better model. This study might therefore be seen also as a comparative study of the
softwares R and Eviews 7 for SARIMA modelling. Etuk and Ojekudo (2014) have shown that SARIMA
modelling based on the two packages could produce contradictory results.
SARIMA modelling has been applied in recent times to model seasonal time series. For instance,
Abdul-Aziz et al. (2013) have modelled monthly rainfall pattern in the Ashanti region of Ghana by a
SARIMA(0, 0, 0)x(2, 1, 0)12 model. Zhang et al. (2015) fitted a SARIMA(0, 1, 0)x(0, 0, 1)4 model to
quarterly road traffic mortalities in China. Iraqi monthly oil production has been modelled as a
SARIMA(0, 1, 0)x(0, 0, 1)12 by Mostafaei and Sakhabakhsh (2011). Pelagic fish catch in Malaysia has
been modelled; for the trichiurus lepturus species the model was a SARIMA(1, 1, 0)x(0, 0, 1)12 and
for the amblygaster leiogaster species it was a SARIMA(0, 1, 1)x(0, 0, 1)12 (See Bako et al., 2013).
Urrutia et al. (2014) modelled foreign trade of the Philippines: the imports as SARIMA(5, 0, 8)x(0, 1,
1)12 and the exports as SARIMA(7, 2, 3)x(0, 1, 1)12. Mahsin et al. (2012) proposed a SARIMA(0, 0,
1)x(0, 1, 1)12 for the explanation of the variation in monthly rainfall in the Dhaka Division of
Bangladesh. Etuk and Natamba (2015) have modelled Daily Uganda shilling/US dollar exchange rates
by a SARIMA(0, 1, 1)x(0, 1, 1)7. Wang et al. (2015) have used a SARIMA(0, 1, 1)x(0, 1, 1)12 to model air
quality in Lanzhou, China. This is just to mention a few.
MATERIALS AND METHODS
Data: As mentioned earlier, the data for this work are 192 monthly rainfall frequencies from 1996 to
2011 in Umuahia, Nigeria, analysed and published by Akpanta et al. (2015). According to the authors
their source is the National Root Crop Research Institute, Umudike, Umuahia, Abia State of Nigeria.
Sarima modelling: An autoregressive moving average model of order p and q denoted by ARMA(p, q)
is defined by
Xt - 1Xt-1 - 2Xt-2 - ... - pXt-p = t +  1t-1 + 2t-2 + ... + qt-q
(1)
where {Xt} is a stationary time series, {t} is a white noise process and the ’s and ’s are constants
such that model (1) is stationary and invertible. Equivalently (1) may be put as
A(L)Xt = B(L)Xt
(2)
where A(L) is the autoregressive (AR) operator defined by A(L) = 1 - 1L - 2L2 - ... - pLp and B(L) is
the moving average (MA) operator defined by B(L) = 1 + 1L + 2L2 + ... + qLq. The symbol L is the
backshift operator defined by LkXt = Xt-k.
When {Xt} is non-stationary Box and Jenkins (1976) proposed that differencing it a sufficient number
of times could make it stationary. Suppose d is the number of times for which {Xt} is differenced to
ensure its stationarity. Suppose the differences, denoted by {dXt} with  = 1 - L, follow an ARMA(p,
q) model. Then the original series is said to follow an autoregressive integrated moving average
model of order p, d and q denoted by ARIMA(p, d, q).
For a seasonal series of period s, Box and Jenkins proposed that it could be modelled by
A(L)(Ls)dDsXt = B(L)(Ls)t
(3)
where  and  are the seasonal AR and MA operators respectively and Ds is the seasonal
difference operator equal to 1 – Ls. The symbol D is the order of seasonal differencing. Suppose the
seasonal operators are respectively polynomials of degrees P and Q, the model (3) is called a
seasonal autoregressive integrated moving average model of order p, d, q, P, D, Q and s denoted by
SARIMA(p, d, q)x(P, D, Q)s.
The estimation of (3) entails the determination of the orders p, d, q, P, D, Q and s and then the
estimation of the coefficients. The seasonal period may be determined from knowledge of the
seasonal nature of the series or from its initial inspection. The AR orders p and P may be determined
as the lags (non-seasonal and seasonal respectively) at which the autocorrelation function (ACF)
becomes non-significant. Similarly the MA orders q and Q may be estimated as the lags at which the
partial autocorrelation functions (PACF) becomes non-significant. The difference orders are mostly
such that it is enough that they sum up to at most 2 for stationarity of the series. Model parameters
are usually estimated using the maximum likelihood approach or the least squares technique. Before
and after each differencing stage stationarity may be ascertained using the augmented Dickey Fuller
(ADF) technique. Data analysis herein is done on the basis of the Eviews 7 package which is based on
the least squares approach to optimization.
RESULTS AND DISCUSSION
The time plot of the series called herein NFMR in Figure 1 shows clearly a seasonal pattern of a
yearly period. Its correlogram in Figure 2 is therefore as expected: sinusoidal-patterned with 12month periodicity. A 12-monthly differencing of the series yields SDNFMR with a time plot in Figure
3 showing a stationary pattern confirmed by the correlogram of Figure 4. Moreover the ACF
indicates the possibility of some models, two of which are as pointed out by Akpanta et al. (2015): a
SARIMA(0, 0, 0)x(1, 1, 1)12 and a SARIMA(0, 0, 0)x(0, 1, 1)12. The estimation table and the residual
histogram of the former model are in Table 1 and Figure 5 respectively. Those of the latter model are
in Table 2 and Figure 6 respectively.
The SARIMA(0, 0, 0)x(1, 1, 1)12 model as evident from Table 1 is given by
Xt + 0.1786Xt-12 = t – 0.9077t-12
(0.0791)
(0.0171)
(4)
It may be observed that contrary to the observation of Akpanta et al. (2015) both coefficients of the
model are significant; their observation was that seasonal AR coefficient is non-significant which
gave room for the proposal of the latter model.
The SARIMA(0, 0, 0)x(0, 1, 1)12 model as estimated in Table 2 is
Xt = t – 0.8985t-12
(0.0215)
(5)
Model (4) is better than model (5) on all counts: it has a lower AIC, Schwarz criterion and HannanQuinn criterion; it has a higher R2 and a lower residual variance. Moreover at a significance level of
5% its residuals are normally distributed whereas those of model (5) are not. See Figures 5 and 6 for
the Jarque-Bera tests.
CONCLUSION
It may be concluded that contrary to the observation of Akpanta et al. (2015) model (4) is more
adequate than model (5). The difference could also be between the software R and the software
Eviews 7. As earlier mentioned, Etuk and Ojekudo (2014) observed discrepancy of analytical results
between the two packages over the data set analyzed by Martinez et al. (2011). This calls for further
research.
REFERENCES
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FIGURE 2: CORRELOGRAM OF NFMR
FIGURE 4: CORRELOGRAM OF SDNFMR
TABLE 1: ESTIMATION OF THE SARIMA(0, 0, 0)X(1, 1, 1)12 MODEL
FIGURE 5: HISTOGRAM OF THE SARIMA(0, 0, 0)X(1, 1, 1)12 RESIDUALS
TABLE 2: ESTIMATION OF THE SARIMA(0, 0, 0)X(0, 1, 1)12 MODEL
FIGURE 6: HISTOGRAM OF THE SARIMA(0, 0, 0)X(0, 1, 1)12 RESIDUALS