Modeling and Forecasting Gambia ’ s Inflation Rates

In this paper, we examine the most appropriate method for modeling and forecasting Gambia’s inflation rates. We investigate the statistical properties of the inflation data and specify two models namely seasonal autoregressive integrated moving average (SARIMA) and k-factor Gegenbauer Autoregressive Moving Average (k-factor GARMA). The first model seasonal ARIMA(1, 1, 1)(0, 0, 1)12 was selected using the H-K Algorithm developed by Hyndman and Khandakar (2008) and 3-factor GARM A from both the spectral density graph and further analysis of the residuals from the 3-factor Gegenbauer model. The in-sample characteristics such as the Akaike Criterion and Schwarz Criterion following estimation using the first data set show that the ARIMA(1, 1, 1)(0, 0, 1)12 outperforms the 3-factor GARM A model. However, the second data set that was preserved and used for out-of-sample forecasting suggest that the 3-factor GARM A model outperforms the seasonal ARIMA(1, 1, 1)(0, 0, 1)12 model in out-of -sample forecasting. Our results indicated that inflation in Gambia is stationary with long-memory behavior at three distinct frequencies. We also found that the k-factor GARMA outperforms the seasonal ARIMA in out-sample forecasting which may be ascribed to the forecast horizon been large and series strongly long-range dependent.


Introduction
Inflation as defined by Webster (2000) is the persistent increase in the level of consumer prices or a persistent decline in the purchasing power of money.Inflation causes global concern because it can distort economic patterns when not anticipated.Inflation as described by Aidoo (2010) can cause uncertainty about the future price, interest rate, and exchange rate etc which as a result might increase the risk among potential traders and partners of a country.Inflation in The Gambia exhibit evidence of cyclical persistence justifying the use of a seasonal model such as SARIMA (seasonal autoregressive integrated moving average) which is an extension of the ARIMA model when the series contains both seasonal and non-seasonal behavior and a long memory model such as k-factor Gegenbauer autoregressive moving average (k-factor GARMA) in modeling it.
A stationary process is said to be long memory if it exhibits a slowly decaying autocorrelation function (ACF), denoted ρ(n), approximated as follows: ρ(n) ≈ c p (n)n 2d-1 , as n tends to infinity, where c p (n) is a slowly varying function at infinity and d is the long memory parameter such that 0 < d < 1/2, and therefore the infinite sum Σ|ρ(n)| diverges.In an equivalent way, in terms of spectral analysis, a stationary process is said to be long memory if for some frequency λ ∈ [0, π] the spectral density becomes unbounded (Woodward et al., 2011).Forecasters need to take into account long-range dependence in modeling time series to obtain accurate results.In many economic and financial applications, time series often possess persistent cycles, especially when dealing with daily or monthly frequencies.To take into account this periodic and persistent behavior long memory models have been proposed which makes it possible to integrate a certain type of nonstationarity in modeling and therefore avoids a differentiation of the series which strongly reduces the available information set.Long-range dependence has been studied extensively and evidence of it can be found in many areas of applied statistics.For instance, the pioneer work of Hurst (1951) stems from problems dealing with hydrology, and later many authors analyzed river flows through long memory processes, see for instance Noakes et al. (1998) or Ooms and Franses (1998).More recently, researchers founded evidence of persistence in telecommunication data (Taqqu et al., 1997) or in urban transport data (Ferrara & Guegan, 1999).However, most long memory applications concern financial time series such as stock market prices, exchange rates, inflation, etc.

Literature Review
Empirical researches have been carried out in the area of forecasting using Autoregressive Integrated Moving Average (ARIMA) models popularised by Box and Jenkins (1976).Junttila (2001), applied the Box and Jenkins (1976) approach to model and forecast Finnish inflation.Pufnik and Kunovac (2006) applied a similar approach to forecast short term inflation in Croatia.Alnaa and Ferdinand (2011) used ARIMA approach to predict inflation in Ghana.In their study, they used monthly data from June, 2000 to December, 2010 and found that ARIMA (6, 1, 6) is best for forecasting inflation in Ghana.An extended version of the Seasonal ARIMA, known as the Driftless Extended Seasonal ARIMA (DESARIMA) was introduced in a study by Pincheira and Medel (2012) to forecast inflation across 12 countries.Also, Barros and Gil-Alana (2012) employed a fractional approach (Autoregressive Fractionally Integrated Moving Average) to forecast inflation in Angola.Adjepong et al. (2013) considered the most appropriate short-term forecasting method for Ghana's inflation: Seasonal ARIMA vs. Holt-Winters.They concluded by proposing the Seasonal-ARIMA process as the most appropriate short-term forecasting method for Ghana's inflation.Recently Long-memory models have then become increasingly popular (Hassler and Wolters, 1995).(see, e.g., Chung & Baillie, 1993;Franses & Ooms, 1997).Much of the evidence supports the view that inflation is fractionally integrated with a differencing parameter that is significantly different from zero or unity (Sutcliffe, 1994).For instance, using US monthly data, Backus and Zin (1993) found a fractional degree of integration.They argued that aggregation across agents with heterogeneous beliefs results in long memory in the inflation process.Hasser (1993) and Delgado and Robinson (1994) provided strong evidence of long memory in the Swiss and Spanish inflation rates respectively.A new class of Fractionally Integrated Generalized AutoRegressive Conditionally Heteroskedastic (FIGARCH) processes was introduced by Baillie et al. (1996).Empirical evidence of long memory has also been found in monthly river flows (Ooms & Franses, 1998), stock market prices (Cheung & Lai, 1995), (Barkoulas & Baum, 1996), Willinger et al. (1999) or exchange rates (Cheung, 1993), Bisaglia and Guégan (1998), Velasco (1999).Caporale and Gil-Alana ( 2011) modeled European inflation rates using Multi-Factor Gegenbauer Processes.Their findings show that inflation in France and Italy is nonstationary, but in the former country this applies to both the long-run and the seasonal frequencies, whilst for the latter the nonstationarity concerns exclusively the long-run or zero frequency, and the contribution of the long-range dependence in the seasonal structure is relatively small.For the UK, inflation seems to be stationary, though with a large component of long-memory behaviour, especially at the zero frequency.In the present study we model Gambia inflation rates which to the best of our knowledge have not been done before.We specify a seasonal autoregressive integrated moving average (SARIMA) model and k-factor Gegenbauer Autoregressive Moving Average (k-factor GARMA) model for the Gambia inflation rates and compare their in-sample characteristics and out-sample forecasting performances.Our results indicated that inflation in Gambia is stationary with long-memory behavior at three distinct frequencies.We also found that the k-factor GARMA outperforms the seasonal ARIMA in out-sample forecasting which may be ascribed to the forecast horizon been large and series strongly long-range dependent.
The outline of this paper is as follows: section III describes the inflation data from the Central Bank of The Gambia.In section IV we present the two models namely seasonal autoregressive integrated moving average (SARIMA) model and k-factor Gegenbauer Autoregressive Moving Average (k-factor GARMA) model, section V deals with applications, and section VI gives a summary of our findings.

Data Presentation
The data was obtained from the Central Bank of the Gambia spanning the period January 1987 to June 2013 and is measured in percentages based on the consumer price index (CPI) of The Gambia. Figure 1 shows the graphical representations of realizations, distribution function, periodogram and the autocorrelation function.The spectral density is unbounded at the low frequencies and the autocorrelation function decay very slowly which suggest that the inflation series seems to be long memory process with fractional integration behavior.The periodogram exhibits three distinct spikes at the low frequencies and smaller peaks at other seasonal frequencies which are all indications that the inflation process in Gambia is possibly fractionally integrated with 0 < d < 1, i = 1, …, k where k is a finite integer indicating the number of seasonal structures.Patterns of seasonality are evident in the plot of the realizations and the autocorrelation function.

Seasonal ARIMA Model
The extension of ARIMA model to the SARIMA model comes in when the series contains both seasonal and non-seasonal behavior.This behavior of the series makes the ARIMA model inefficient to be applied to the series.This is because it may not be able to capture the behavior along the seasonal part of the series and therefore mislead to a wrong order selection for non-seasonal component.
If d and D are nonnegative integers, the {x} is a seasonal ARIMA(p, d, q)(P, D, Q) S process with period s if the differenced series y t = (1-B) d (1-B) D x t is a casual ARMA process defined by: . In applications d is rarely more than one and P and Q are typically less than three (Brockwell & Davis, 2002).
To select the best SARIMA model we use an algorithm developed by Hyndman and Khandakar (2008) called the HK-algorith.

The k-factor GARMA Model
The k-factor Gegenbauer process (X t ) t ∈ Z is defined by the following quation: where k is a finite integer indicating the maximum number of seasonal structures, where |v j |≤1 for j = 1, …, k, where d j 's are fractional values for j = 1, …, k, where μ is the mean of the process and where (ε t ) t ∈ Z is covariance stationary with mean zero and variance σ ε 2 (Gray et al., 1989).
The k-factor GARMA model is defined by the following equation: (For stationarity and invertibility conditions refer to Gray et al. (1989)).
To estimate the parameters of the k-factor GARMA model, we use in this study the Whittle approach as in Diongue and Guegan (2008).
Where I X (ω j ) is the periodogram of the process (X t ) t=1 T and expressed as follows : with the frequencies ω j are such that T the integer part of 2 T , and i a complex is defined by www.ccse

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The prope We now d the autoco numbers Khandaka spike at l Figure 2( that q is adding an Khandaka of AIC, A    The ACF of the residuals displayed in Figure 3 depicts that the autocorrelation of the residuals are approximately all zero, that is to say they are uncorrelated and the p-value for the Ljung-Box statistic in Table 6 clearly exceed 5% at lag 12, indicating that there is no significant departure from white noise for the residuals.The ARCH-LM test is used to test for constant variance.From the ARCH-LM test results shown in Table 7, we fail to reject the null hypothesis of no ARCH effect (homoscedasticity) in the residuals of the selected model.The spectral density in Figure 1 is unbounded at the low frequencies which suggest that the inflation series seems to be a long memory process with fractional integration behavior (Woodward et al., 2011).The empirical ACF of the series in Figure 1 clearly shows a strong dependence between distant observations, as well as a cyclical behavior with a pretty long period.Moreover, the spectral density of the series clearly possesses three distinct peaks with the first peak having frequency located very close, but not necessarily equal, to zero which are all distinctive features of a weakly stationary Gegenbauer process.The properties of the data suggest using a 3-factor Gegenbauer process to model inflation in Gambia.
We assume the inflation series has the following specification: -0.
The following results were obtained from the estimation of the long memory model, the 3-factor Gegenbauer model.The parameter estimation is done using the parametric Whittles method.The three orders of integration are d 1 =0.117, d 2 =0.2744, d 3 =0.0831with LLF = 1.9098.(See Figure 4 for the residual correlogram).From the autocorrelation function in Figure 4 it is obvious that the residuals are not white noise.An ARCH-LM test performed on the residuals fails to reject evidence of ARCH effect.We adjust various ARMA(p, q) specifications on the residuals and found ARMA(0, 1) to be most adequate with the parameter θ 1 = −0.654.
From the autocorrelation function in Figure 4 it is obvious that the residuals are not white noise.It is important to note that the SARIMA methodology has certain limitations.For instance, it requires large number of observations for model identification and sometimes estimation and selection involves some form of art.Also, differencing the series in the case of nonstationarity may reduce the available information set.However, the model is parsimonious with respect to the coefficients and good in providing unconditional forecasts.
Note also that the number of peaks to be chosen in the spectral density for the Gegenbauer model remains unclear and must be investigated.
estimator based on the periodogram that involves the function

Figure 5 .
Figure 5. Behavior of ACF and PACF the distribution of the estimated residuals

Table 3 .
A

Table 6 .
Box-Ljung test of the null hypothesis of randomness for the residuals