Application of Sarima Models in Modelling and Forecasting Nigeria s Inflation Rates

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1 American Journal of Alied Mathematics and Statistics, 4, Vol., No., 6-8 Available online at htt://ubs.scieub.com/ajams///4 Science and Education Publishing DOI:9/ajams---4 Alication of Sarima Models in Modelling and Forecasting Nigeria s Inflation Rates Otu Archibong Otu,*, Osuji George A., Oara Jude 3, Mbachu Hoe Ifeyinwa 3, Iheagwara Andrew I. 4 Deartment of Statistics, Central Bank of Nigeria, Owerri Deartment of Statistics, Nnamdi Azikiwe University, PMB 0, Awka Anambra State Nigeria 3 Deartment of Statistics, Imo State University, PMB 00, Owerri Nigeria 4 Deartment of Planning, Research and Statistics, Ministry of Petroleum and Environment Owerri Imo State Nigeria *Corresonding author: oaotu@yahoo.com Received December, 3; Revised December 7, 3; Acceted January 06, 3 Abstract This aer discussed the Alication of SARIMA Models in Modeling and Forecasting Nigeria s Inflation Rates. Time series analysis and forecasting is an efficient versatile tool in diverse alications such as in economics and finance, hydrology and environmental management fields just to mention a few. Among the most effective aroaches for analyzing time series data, the method roounded by Box and Jenkins, the Autoregressive Integrated Moving Average (ARIMA) was emloyed in this study. In this aer, we used Box-Jenkins methodology to build ARIMA model for Nigeria s monthly inflation rates for the eriod November 03 to October 3 with a total of data oints. In this research, ARIMA (,, ) (0, 0, ) model was develoed, and obtained as y ˆt+ = 0.387y t +43y t- 840e t e t- +68e t. This model is used to forecast Nigeria s monthly inflation for the ucoming year 4. The forecasted results will hel olicy makers gain insight into more aroriate economic and monetary olicy in other to combat the redicted rise in inflation rates beginning the first uarter of 4. Keywords: ARIMA Model, SARIMA Model, forecasting, ARMA Model, Box-Jenkins Methods, inflation rates, Akaike Information Criteria, Bayesian Information Criterion Cite This Article: Otu Archibong Otu, Osuji George A., Oara Jude, Mbachu Hoe Ifeyinwa, and Iheagwara Andrew I., Alication of Sarima Models in Modelling and Forecasting Nigeria s Inflation Rates. American Journal of Alied Mathematics and Statistics, no. (4): 6-8. doi: 9/ajams Introduction Inflation is the ercentage change in the value of the Wholesale Price Index (WPI) on a year-on year basis. It effectively measures the change in the rices of a basket of goods and services in a year. In India, inflation is calculated by taking the WPI as base. Thus, the formula for calculating Inflation is: WPI in month of current year WPI in same month of revious year 00 WPI in same month of revious year Inflation occurs due to an imbalance between demand and suly of money, changes in roduction and distribution cost or increase in taxes on roducts. When economy exeriences inflation, i.e. when the rice level of goods and services rises, the value of currency reduces. This means now each unit of currency buys fewer goods and services. It has its worst imact on consumers. High rices of day-to-day goods make it difficult for consumers to afford even the basic commodities in life. This leaves them with no choice but to ask for higher incomes. Hence the government tries to kee inflation under control. Contrary to its negative effects, a moderate level of inflation characterizes a good economy. An inflation rate of or 3% is beneficial for an economy as it encourages eole to buy more and borrow more, because during times of lower inflation, the level of interest rates also remains low. Hence the government as well as the central bank always strives to achieve a limited level of inflation. In Nigeria hardly does the day go by without government officials, oliticians and economist talking about inflation. In some cases we are told inflation is high for a articular month and low for another month. There are several imortant variables that hel to describe the state of an economy. These include inflation, unemloyment, the budget balance, the interest rates, and the balance of ayments. Inflation may be defined as a rise in the average level of a grou of rices in a country. The term is sometimes restricted to rolonged or sustained rises. Inflation creates a roblem because the urchasing ower of money falls as the rice level rises. It imoses an oortunity cost on holders of money. Thus inflation [] will reduce the real value of money wage, and savings accounts making holders of these instruments to lose. Inflation also encourages wasteful increase in the volume and freuency of transactions eole undertake and because it is difficult to foresee, it adds to the uncertainties of economic life. In real terms,

2 American Journal of Alied Mathematics and Statistics 7 inflation means your money cannot buy as much as what it could have bought yesterday. Inflation retards [] economic growth because the economy needs a certain level of savings to finance investments which boosts economic growth. Inflation causes global concerns because it can distort economic atterns and can result in the redistribution of wealth when not anticiated. Inflation can also discourage investors within and without the country by reducing their confidence level in investments. This is because investors exect high ossibility of returns so that they can make good financial decisions. There are two main tyes of inflation: These are creeing or moderates inflation and hyer inflation. The creeing inflation, also known as mild inflation, is the tye in which the rates of rice change is not so severe. Examle of creeing inflation is the one Ghana exerienced in 99, 999, 0; and 0. A rate of inflation of about 0% annually can be described as creeing inflation. The hyer inflation is the tye in which the rates of change in rices are so severe. A tyical examle of hyer inflation is what has haened in Zimbabwe from 07 to 08. This country had a rates of inflation of about 8000%.This means that if you buy an item today in the morning, the rice of the item will change by the time you go there in the evening. The hyer inflation is the worse economic roblem any country will exerience. The effect of inflation is highly considered as a crucial issue for a country. The inflation roblems make a lot of eole living in a country much harder. Peole who are living on fixed income suffer most as when rices of commodities rise, since these eole cannot buy as much as they could reviously. In this study, the roblem is to forecast Nigeria s monthly inflation rates using time series Seasonal Autoregressive Integrated Moving Average (SARIMA) models. When it comes to forecasting, there are extensive number of methods and aroaches available and their relative success or failure to outerform each other is in general conditional to the roblem at hand. The rational for choosing this tye of model is contingent on the behaviour of the time series data. Also in the history of inflation forecasting, this model has roved to erform better than other models.. Review of Related Literature A research work [3] was carried out on SARFIMA model to study and redict the Iran s oil suly. The results of their analysis showed that the best model was SARFIMA (0,, )(0, -0.99, 0) which was used to redict the uantity of oil suly in Iran till the end of. Research was carried out [4] to evaluate the erformance of VAR and ARIMA models to forecast Austrian HICP inflation. Additionally, they investigate whether disaggregate modeling of five subcomonents of inflation is suerior to secifications of headline HICP inflation. Their modeling rocedure is to find adeuate VAR and ARIMA secifications that minimize the months out-of-samle forecasting error. The main findings are twofold. First, VAR models outerform the ARIMA models in terms of forecasting accuracy over the longer rojection horizon (8 to months ahead). Second, a disaggregated aroach imroves forecasting accuracy substantially for ARIMA models. In case of the VAR aroach the sueriority of modelling the five subcomonents instead of just considering headline HICP inflation is demonstrated only over the longer eriod (0 to months ahead). Two researchers [] also used a unified aroach to automatic modelling for univariate series. First, ARIMA models and the classical methods for fitting these models to a given time series were reviewed. Second, some objective methods for model identification were considered and some algorithmically rocedures for automatic model identification were described. Third, outliers are incororated into the model and an algorithm, for automatic model identification in the resence of outliers was roosed. Researchers [6] carried out an emirical study of the usefulness of SARFIMA models in energy science. The results indicate the aroriate model is SARFIMA (,,0)(0,73,0) was used to redict the consumtion rates of etroleum roducts till the end of 3. Having reviewed some related literatures, we shall now in this aer examine the alication of SARIMA models in modeling and forecasting Nigeria s inflation rates. 3. Materials and Methods In this aer, the methodology and the theorems roounded by Box and Jenkins called the Autoregressive Integrated Moving Average (ARIMA) was extensively exlored. This is an advance forecasting techniue that takes into account historical data and decomoses it into an Autoregressive (AR) rocess, where there is a memory of ast values, an Integrated (I) rocess, which accounts for stabilizing or making the data stationary lus a Moving-Average (MA) rocess, which accounts for revious error terms making it easier to forecast. 3.. Autoregressive Moving Average Process (ARMA) or Mixed Process According to [7], autocorrelation atterns may reuire more comlex models. A more General model is a mixture of the AR() and MA() models and is called autoregressive moving-average model, ARMA(, ) model. He exlained further that this model forecasts Y as both a linear combination of actual ast values and a linear combination of ast errors. The general ARMA (, ) model is given by Y = µ + α Y + α Y + + α Y i t t t + e θ e θ e θ e i i i () Yi = αkyt k θ ket k + µ + et () k= k= Like the AR () model, the ARMA (, ), has autocorrelation that diminish as the distance between residuals increases. 3.. The Autoregressive Integrated Moving Average Model (ARIMA)

3 8 American Journal of Alied Mathematics and Statistics The order of the autoregressive comonent is, the order of differencing needed to achieve stationarity is d, and the order of the moving average comonent is. In general the ARIMA rocess (8) is of the form Zt = αkzt k θ ket k + µ + et k= k= (3) 3.3. The Backshift and Difference Oerators for ARIMA Reresentation To exress and understand differenced ARIMA models the concet of the backshift (lag) oerator, B, and difference oerator,, is used, These has no mathematical meaning other than to facilitate the writing of different tye of models that would otherwise be extremely difficult m to exress. The backshift is defined as B Y t = Y t m. For examle BYt = Yt. BYt = Yt, and B Yt = Yt. The difference oerator d d takes the form = ( B), when d differences are taken to achieve stationarity in the time series data. Using these notations,. The general AR() model Yt = α kyt k + µ + et is k= exressed as Yt αyt αyt αyt = α ( BY ) t= e+ µ, where α ( B) is the autoregressive oerator of order, defined by α( B) = α B α B α B (4). The general MA () model Yt = θ ket k + µ + et k= Yt= et θtet θet θet is exressed as = θ ( Be ) t + µ where (B) is the moving average oerator of order, defined by θ( B) = θ B θ B θ B () 3. The general ARMA (, ) Yt = µ + αyt + αyt + + αyt + et model,, is θet θet θet exressed as Yt = αyt αyt αyt = et θet θet θ et + µ ( αb αb αb ) Yt = ( θb θb θ B ) e+ µ (6) α( BY ) t = θ ( Be ) t + µ 4. Stationary series Z t obtained after d differencing of Y t T is given by d d Zt = Yt = ( B) Yt (7). A general ARIMA (, d, ) model is exressed d ( B) ( αb αb αb ) Yt = ( θb θb θb ) et (8) d ( B) α( BY ) t = θ( Be ) t Table. General Time Series Models MODEL STATIONARITY CONDITION INVERTIBILITY CONDITION ACF COEFFICIENTS PACF COEFFICIENTS AR() Yes No Die down Cuts off after lag MA() No Yes Cuts off after lag Die down ARM(,) Yes Yes Die down Die down Table gives the summary of the general non seasonal time series models and their statistical roerties. The table summarizes discussions on general AR, MA, and mixed ARMA [8] models Seasonal Autoregressive Models A urely seasonal time series is the one that has only seasonal AR or MA arameters. Seasonal autoregressive models are built with arameter called seasonal autoregressive (SAR) arameters. The SAR arameters reresent the autoregressive relationshis that exist between time series data searated by multiles of the number of eriods er season. A general AR model with P SAR arameters is given by Yt = αisyt is where Yt s i= is of order s, Yt s is of order s and Yt s, is of order s. A model with one SAR arameter is written as Yt= αsyt s+ et (9) Seasonal moving Average (SMA) models are built with seasonal moving average (SMA) arameters, and the general SMA model with Q arameters is given by: Q Yt = θiset is + et (0) i= The general mixed SAR and SMA model is given by Q Yt = αisyt is + θiset is + et i= i= () The order the seasonal ARMA rocess is given in terms of both Ps and Qs Table gives the summary of the stationarity and invertibility conditions of some secific seasonal time series models and the behaviour of their theoretical ACF and PACF.

4 American Journal of Alied Mathematics and Statistics 9 ARMA MODEL STATIONARITY CONDITION Table. Secific Pure Seasonal Time Series Models INVERTIBILITY ACF COEFFICIENTS CONDITION PACF COEFFICIENTS (,D,0) s - < α s < None Die down Cuts off after one seasonal lag (,D,0) s α, + α s < None Die down Cuts off after one seasonal lag (0,D,) s None - < θ s < Cuts off after one seasonal lag Die down (0,D,) s None θ s + θ s < θ s - θ s < θ s < Cuts off after two seasonal lag Die down (,D,) s` - < α s < - < θ s < Die down Die down 4. Data on Nigeria s Inflation Looking at Table 9 in the Aendix, it shows the data of Nigeria s monthly inflation ratess from November 03 to October 3, totaling one hundred and twenty () monthly observations. The data were obtained from the National Bureau of Statistics. Figure and Figure show the lot of Nigeria s monthly inflation and the trend analysis lot resectively. Figure 3 and Figure 4 also describe the features of the data that is the autocorrelation lot and the artial autocorrelation lot resectively. TIME SERIES PLOT OF THE ORIGINAL INFLATION DATA Inflation Index Figure. TIME SERIES PLOT OF NIGERIAN S MONTHLY INFLATION Inflation 0 TREND ANALYSIS PLOT OF NIGERIAN'S MONTHLY INFLATION Linear Trend Model Yt = *t Variable Actual Fits Accuracy Measures MAPE MAD MSD Index Figure. TREND ANALYSIS PLOT OF NIGERIAN S MONTHLY INFLATION

5 American Journal of Alied Mathematics and Statistics AUTOCORRELATION PLOT OF NIGERIA'S INFLATION.0 Autocorrelation Figure 3. AUTOCORRELATION PLOT OF NIGERIA S INFLATION PARTIAL AUTOCORRELATION PLOT OF NIGERIAN'S INFLATION.0 Partial Autocorrelation Figure 4. PARTIAL AUTOCORRELATION PLOT OF NIGERIAN S INFLATION ACF of Residuals for Inflation (with % significance limits for the autocorrelations) Autocorrelation Figure. ACF PLOT OF RESIDUALS OF NIGERIAN S INFLATION

6 American Journal of Alied Mathematics and Statistics A look at the time series lot of the original data in Figure imlies that the series is non-stationary. More so, the trend analysis as shown in Figure shows a decreasing trend. However, the ACF lot as shown in Figure 3 dies down in a sinewave fashion and the PACF lot as shown in Figure 4 tails off at lag (though there is a sike at lag 8, it is considered surious and therefore neglected). With these results above, an AR [] model is susected. The result of estimates of arameters, the ACF and the PACF of the residuals obtained using MINITAB version.0 are shown below, Figure and Figure 6 resectively. Table 3. ESTIMATES OF PARAMETERS FOR AR() MODEL Tye Coef SE Coef T P AR AR Constant Mean.4.86 Number of observations: Residuals: SS = (back forecasts excluded) MS = DF = 7 Partial Autocorrelation Function for Inflation (with % significance limits for the artial autocorrelations).0 Partial Autocorrelation Figure 6. PACF PLOT OF RESIDUALS OF NIGERIAN S INFLATION Time Series Plot for Inflation (with forecasts and their 9% confidence limits) Inflation Time Figure 7. TIME SERIES PLOT FOR FORCAST USING AR ()

7 American Journal of Alied Mathematics and Statistics 4.. Modified Box-Pierce (Ljung-Box) Chi- Suare Statistic Chi-Suare DF P-Value A look at Figure 6 and Figure 7 show some significant number of sikes outside the limit. kk kk kk ie αϕ ϕ αϕ ϕkk which euals 3 ϕ kk 3 suggesting that the residuals are not random. Figure also shows that the P-values for the Ljung-Box statistics are significant. The forecast as shown by Figure 7 do not seem to be consistent with the forecast of inflation figures. We try differencing the data to bring about stationarity in mean. TIME SERIES PLOT OF FIRST DIFFERENCE OF THE ORIGINAL DATA 7. ST DIFF OF ORIGINAL DATA Index Figure 8. TIME SERIES PLOT OF ST DIFFERENCE OF THE ORIGINAL DATA Figure 8 shows the time series lot of the first difference of Nigerian s original inflation data. There is stationarity in mean and the existence of seasonality is evident. TREND ANALYSIS PLOT FOR ST DIFFERENCE OF THE ORIGINAL DATA Linear Trend Model Yt = *t ST DIFFERENCE OF ORIGINAL DATA Variable Actual Fits Accuracy Measures MAPE MAD.38 MSD Index Figure 9. TREND ANALYSIS FOR IST DIFFERENCE OF THE ORIGINAL DATA Figure 0 and Figure show the autocorrelation function and the artial autocorrelation function of the first difference of Nigerian s original inflation data resectively. The ACF and PACF show insignificant number of sikes dieing down in a sinewave fashion.

8 American Journal of Alied Mathematics and Statistics 3 ACF OF ST DIFFERENCE OF THE ORIGINAL DATA (with % significance limits for the autocorrelations) Autocorrelation Figure 0. ACF OF IST DIFFERENCE OF THE ORIGINAL DATA PACF PLOT OF ST DIFFERENCE OF ORIGINAL DATA (with % significance limits for the artial autocorrelations) Partial Autocorrelation Figure. PACF OF ST DIFFERENCE OF ORIGINAL DATA TIME SERIES PLOT OF THE SEASONAL DIFFERENCE OF THE ST DIFF. DATA 0 DIFFERENCE Time Figure. TIME SERIES PLOT OF THE SEASONAL DIFFERENCE OF THE ST DIFFERENCE DATA

9 4 American Journal of Alied Mathematics and Statistics Figure shows the time series lot of the seasonal difference of the first differenced of Nigerian s inflation data which shows stability in mean at both the seasonal and the non-seasonal levels. TREND ANALYSIS PLOT FOR ST DIFFERENCE OF THE ORIGINAL DATA Linear Trend Model Yt = *t DIFF Variable Actual Fits Accuracy Measures MAPE.8770E+4 MAD.064E+00 MSD E Index Figure 3. TREND ANALYSIS OF THE SEASONAL DIFFERENCE OF THE ST DIFFERENCED DATA Figure 3 shows the trend analysis of the seasonal difference of the first differenced of Nigerian s original inflation data. The trend is neither increasing nor decreasing which is indicative of stationarity in mean. Figure 4 shows the autocorrelation function of the seasonal difference of the first differenced of Nigerian s original inflation data. ACF OF THE SEASONAL DIFFERENCE OF THE ST DIFFERENCED DATA.0 Autocorrelation Figure 4. ACF OF THE SEASONAL DIFFERENCE OF THE ST DIFFERENCED DATA The time series lot of the st differenced data and the trend analysis as indicated in Figure 8 and Figure 9 show stationarity in mean and variance. There were significant sikes in the time series lot at lags, 4, and so on. This is indicating that seasonality is evident in the monthly inflation rates with a eriod of. This call for seasonal differencing of the st non-seasonal differenced data, as shown in Figure. Figure 0 and Figure are the lots of the autocorrelation function (ACF) and the artial autocorrelation function (PACF) of the st differenced data. The ACF dies down after lag and the PACF also tails off after lag, suggesting that = and = would be needed to describe these data as coming from a non-seasonal autoregressive and a moving average rocess resectively. Hence, the time series model that gives rise to these observations was an ARIMA (,, ) model, since the data was differenced once (i.e. d=) to attain stationarity. Figure and Figure 3 show the time series lot of the seasonal difference of the st differenced series and the trend analysis lot resectively. The trend analysis shows stationarity at the seasonal level. Figure 4 and Figure show the ACF and the PACF of the seasonal difference of the s differenced series resectively. A critical look at the seasonal lags show that both ACF and the PACF sikes at seasonal lag dies down to zero for other seasonal lags, suggesting that = and = would be needed to describe these data as coming from a seasonal autoregressive and moving average rocess.

10 American Journal of Alied Mathematics and Statistics PACF OF THE SEASONAL DIFF. OF THE ST DIFFERENCED DATA (with % significance limits for the artial autocorrelations) Partial Autocorrelation Figure. PACF OF THE SEASONAL DIFF. OF THE ST DIFFERENCED DATA 4.. Identification of the ARIMA Model Two goodness-of-fit statistics that are most commonly used for the model selection are; Akaike Information Criterion (AIC) and Schwarz Bayesian Information Criterion (BIC). The AIC and BIC are determined based on a likelihood function. The AIC [9] and BIC are k calculated using the formulas below: AIC = In( SSE) + n k and BIC = In( SSE) + In( n) where n is the total number n of observations, SSE is the sum of the suared errors, and k = ( + + P+ Q+ d + s). In this aer, n = data oints. Four tentative ARIMA models are tested for the data series and the corresonding AIC and BIC values for the models are resented in Table 4. Table 4. AIC and BIC values for four Tentative SARIMA Models ARIMA MODEL (,d,) AIC BIC ( ) ( 0 ) ( ) (0 0 ) ( ) ( 0 0) (0 ) ( 0 ) The models that have the lowest AIC and BIC are ARIMA ( ) (0 0 ) and ( ) ( 0 ). Since two models are identified, the most suitable model is selected by the rincile of arsimony. ARIMA ( ) (0 0 ) model has fewer arameters than ARIMA ( ) ( 0 ) model. Furthermore all the coefficients of ARIMA ( ) (0 0 ) model are significantly different from zero and the estimated values satisfy the stability as indicated in Table 3. We then roceed to the next stage of the Box- Jenkins aroach which is the estimation of arameters of the tentative model Parameter Estimation of SARIMA (,, ) (, 0, ) and SARIMA (,, ) (, 0, ) Models Immediately a suitable SARIMA (P, d, )(P, D,Q) structure is identified, the next ste is the arameter estimation or fitting stage. The arameters are estimated by the maximum likelihood method. The results of arameter estimations are reorted in Table and Table 6. Table (a). Estimates of arameters of SARIMA (,, ) (, 0, ) model Final Estimates of Parameters Tye Coef SE Coef T P AR SAR MA SMA Differencing: regular difference Number of observations: Original series, after differencing 9 Residuals: SS = (back forecasts excluded) MS =.98 DF = Table (b). Modified Box-Pierce (Ljung-Box) Chi-Suare Statistic Modified Box-Pierce (Ljung-Box) Chi Suare Statistic Chi-Suare DF P-Value Table 6(a). Estimates of arameters of the tentative SARIMA (,, ) (0, 0, ) model Final Estimates of Parameters Tye Coef SE Coef T P AR MA SMA Differencing: regular difference Number of observations: Original series, after differencing 9 Residuals: SS = (back forecasts excluded) MS =.937 DF = 6 Table 6(b). Modified Box-Pierce (Ljung-Box) Chi-Suare Statistic Modified Box-Pierce (Ljung-Box) Chi Suare Statistic Chi-Suare DF P-Value

11 6 American Journal of Alied Mathematics and Statistics We roceed in our analysis to check if the arameters contained in the models are significant. This ensures that there are no extra arameters resent in the model and the arameters used in the model have significant contribution, which can rovide the best forecast. The estimates of autoregressive, moving average and the seasonal moving average arameters are labeled AR.., MA.. and SMA.., which are 43, 68, and 840, resectively. Based on 9% confidence level, we conclude that all the coefficients of the ARIMA (,, ) (0, 0, ) model are significantly different from zero as shown in Table 3(a). Furthermore, the -vales for the Ljung-Box statistic clearly all exceed % for all lag orders, imlying that there is no significant dearture from white noise for the residuals. We then roceed to the next ste after arameter estimation which is the Diagnostic Checking or model validation. The Box and Jenkins (970) estimation rocess for seasonal ARIMA model is shown in Figure Diagnostic Checking and Model Validation The model verification is concerned with checking the residuals of the model to determine if the model contains any systematic attern which can be removed to imrove on the selected ARIMA model. It is obvious that the selected model may aear to be the best among a number of models considered; it becomes necessary to do diagnostic checking to verify that the model is adeuate. Verification of an ARIMA model is tested (i) by verifying the ACF of the residuals, (ii) by verifying the normal robability lot of the residuals. ACF of Residuals for SARIMA (,, ) (0, 0, ) MODEL (with % significance limits for the autocorrelations) Autocorrelation Looking at Figure 6, the autocorrelation checks of the residuals indicate that the model is good because they a white noise rocess. That is the residuals have zero mean, constant variance and also uncorrelated. Also, the -values for the Ljung-Box statistic from Table 3 as shown clearly exceed % for all lag orders, indicating that there is no significant dearture from white noise for the residuals. Since the model diagnostic tests show that all the arameter estimates are significant and the residual series are random, it can then be concluded that (,, ) (0, 0, ) model is adeuate for the inflation series. Therefore, (,, ) (0, 0, ) is used to forecast the inflation series of Nigeria. 4.. Point Forecast with SARIMA (,, ) (0, 0, ) Model The ARIMA (,, )(0, 0, ) is selected to forecast the inflation variable, where autoregressive term = (nonseasonal), P = 0(seasonal) [that is, ( - αb)( 0)]; differencing term d = (non-seasonal difference), Q = 0(seasonal difference) [that is ( - B)( 0)] and moving average term = (non-seasonal), Q = (seasonal) [that is ( - θ B)( - θ B ). For the dataset in Table 3, the fitted model is given by Figure 6. ACF of Residuals for SARIMA (,, ) (0, 0, ) model ( B)( αby ) t = ( θb)( θb ) et () yt αbyt Byt + αb yt = et θb et θbet + θθ B et 3 yt = et θb et θbet + θθ B et + αbyt + Byt αb yt (3) Transforming the back oerator, euation (3) becomes; yt = et θet θet + θθ et 3 + ( + α) yt αyt (4) 4.6. Forecast Results by SARIMA (,, )(0, 0,) Model In order to forecast one eriod ahead that is, y t+, the subscrit of the euation (4) is increased by one unit throughout as given by yt+ = ( + α) yt αyt + et+ () θet θet + θθ et

12 American Journal of Alied Mathematics and Statistics 7 The term e t+ is not known because the exected value of future random errors has been taken as zero. There are data oints from November 03 to October 3 used to build the ARIMA model. From the Table 3, using α = 43, θ = 68, θ = 840, we have θ θ = Thus, euation () is given as yt+ = 0.387yt + 43yt 840et 0.789et + 68et + et+ In order to forecast inflation for the eriod (that is, November 3), euation () is given by yˆ = 0.387y + 43y9 840eˆ eˆ eˆ + eˆ e ˆ = 0 eˆ 09 = y09 yˆ 09 =.3 8 =.87 eˆ 08 = y08 yˆ 08 =.7 =.648 eˆ = y yˆ = =.786 The forecast uantity for eriod can now be calculated as follows: y ˆ = 0.387(7.8) + 43(8.0) 840(.87) 0.789(.648) + 68(.786) + 0 = 3.% Once our model has been obtained and its arameters have been estimated, we can use it to make our rediction. Table 7 summarizes months ufront inflation forecast from November 3 to October 4 with 9% confidence interval. Table 7. - Month Forecasted Inflation for November 3 to October 4 Month Period Forecast (%) Lower Uer November December January February March Aril May June July August Setember October Table 8. Basic Statistic of Nigeria s Monthly Inflation Data in Percentages No. of observation Mean St. Dev. Variance Min. Max Conclusion From Figure, it can be confirmed that inflation exhibit volatility starting from somewhere around 06. The volatility in Nigerian s inflation series can be attributed to several economic factors. Some of these factors are money suly, exchange rates dereciation, etroleum rice increases, and oor agricultural roduction. Box-Jenkins Seasonal Autoregressive Integrated Moving Average (SARIMA) was emloyed to analyze monthly inflation rates of Nigeria from October 03 to November 3. The study mainly intended to forecast the monthly inflation rates for the coming eriod of November, 3 to November 4. Series of tentative models were develoed to forecast Nigeria s monthly inflation, but based on minimum AIC and BIC values and after the estimation of arameters and series of diagnostic test were erformed, ARIMA(,,)(0,0,) model was judge to be the best model for forecasting after satisfying all model assumtions. The forecast results revealed a decreasing attern of inflation rates in the first uarter of 4 and turning oint at the beginning of the second uarter of 4, where the rates takes an increasing trend till the Setember. Definition of Terms The full meanings of the abbreviations used in this aer are: ACF Autocorrelation function PACF Partial Autocorrelation function AR Autoregressive MA Moving Average SMA Seasonal Moving Average SAR Seasonal Autoregressive ARIMA Autoregressive Integrated Moving Average SARIMA Seasonal Autoregressive Integrated Moving Average AIC Akaike Information Criterion BIC Bayesian Information Criterion References [] Rudiger Dornbusch and Stanley Fischer, (993). Moderates Inflation, The Bank Economic Review, Vol.7, Issue, P.-44. [] Jack H.R, Bond M.T., and Webb J.R, (989). The Inflation- Hedging Effectiveness of Real Estate. Journal of Real Estate Research, Vol.4. P [3] Hamidreza M. and Leila S. (). Using SARFIMA model to study and redict the Iran s oil suly. International Journal of Energy Economics and Policy. Vol., No.,, [4] Fritzer, F., Gabriel, M. and Johann, S. (0). "Forecasting Austrian HICP and its Comonents using VAR and ARIMA Models," Working Paers 73, Oesterreichische National bank (Austrian Central Bank). [] Gomez V., and Maravall A., (998.) "Automatic Modelling Methods for Univariate Series," Banco de Esaña Working Paers 9808, Banco de Esaña. [6] Leila S. and Masoud Y. (). An Emirical Study of the Usefulness of SARFIMA models in Energy Science. International Journal of Energy Science. IJES Vol. No.. [7] [Jeffrey J., (990). Business forecasting Methods. Atlantic Publishers. [8] Box, G. E. P and Jenkins, G.M., (976). Time series analysis: Forecasting and control, Holden-Day, San Francisco. [9] Akaike, H. (974). A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control 9 (6):

13 8 American Journal of Alied Mathematics and Statistics APPENDIX-A Table 9. Nigeria s Monthly Inflation Ratess from November 03 to October 3 Month Inflation Month Inflation Month Inflation Month Inflation Source: National Bureau of Statistics