1. Developing Country Studies www.iiste.orgISSN 2224-607X (Paper) ISSN 2225-0565 (Online)Vol 2, No.3, 2012A Seasonal Arima Model for Nigerian Gross Domestic Product Ette Harrison Etuk* Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria * E-mail: ettetuk@yahoo.comAbstractTime series analysis of Nigerian Gross Domestic Product series is done. A seasonal difference and then anon-seasonal one were obtained. The correlogram of the differenced series revealed seasonality of order 4.It also reveals an autocorrelation structure of a known seasonal model involving a seasonal autoregressivecomponent of order one and a non-seasonal moving average component of order one. The model has beenshown to be adequate.Keywords: Gross Domestic Product, ARIMA modelling, Seasonal models, Nigeria.1.Introduction1.1.ARIMA Models.A time series is defined as a set of data collected sequentially in time. It has the property that neighbouringvalues are correlated. This tendency is called autocorrelation. A time series is said to be stationary if it has aconstant mean and variance. Moreover the autocorrelation is a function of the lag separating the correlatedvalues and called the autocorrelation function (ACF).A stationary time series {Xt} is said to follow an autoregressive moving average model of orders p and q(designated ARMA(p,q) ) if it satisfies the following difference equation X t + α 1 X t −1 + α 2 X t − 2 + ... + α p X t − p = ε t + β 1ε t −1 + β 2 ε t − 2 + ... + β q ε t − q (1)or Α (B)Xt = Β (B)εt (2)where {εt} is a sequence of random variables with zero mean and constant variance, called a white noiseprocess, and the αi’s and βj’s constants; Α (B) = 1 + α1B + α2B2 + ... + αpBp and Β (B) = 1 + β1B + β2B2+ ... + βqBq and B is the backward shift operator defined by BkXt = Xt-k.If p=0, model (1) becomes a moving average model of order q (designated MA(q)). If, however, q=0 itbecomes an autoregressive process of order p (designated AR(p)). An AR(p) model of order p may bedefined as a model whereby a current value of the time series Xt depends on the immediate past p values:Xt-1, Xt-2, ..., Xt-p . On the other hand an MA(q) model of order q is such that the current value Xt is alinear combination of immediate past values of the white noise process: εt-1, εt-2, ..., εt-q. Apart fromstationarity, invertibility is another important requirement for a time series. It refers to the property wherebythe covariance structure of the series is unique (Priestley, 1981). Moreover it allows for meaningfulassociation of current events with the past history of the series (Box and Jenkins, 1976). 1
2. Developing Country Studies www.iiste.orgISSN 2224-607X (Paper) ISSN 2225-0565 (Online)Vol 2, No.3, 2012An AR(p) model may be more specifically written as Xt + αp1Xt-1 + αp2Xt-2 + ... + αppXt-p = εtThen the sequence of the last coefficients{αii} is called the partial autocorrelation function(PACF) of {Xt}.The ACF of an MA(q) model cuts off after lag q whereas that of an AR(p) model is a combination ofsinusoidals dying off slowly. On the other hand the PACF of an MA(q) model dies off slowly whereasthat of an AR(p) model cuts off after lag p. AR and MA models are known to have some dualityproperties. These include: 1. A finite order AR model is equivalent to an infinite order MA model. 2. A finite order MA model is equivalent to an infinite order AR model. 3. The ACF of an AR model exhibits the same behaviour as the PACF of an MA model. 4. The PACF of an AR model exhibits the same behaviour as the ACF of an MA model. 5. Α (B) = 0 has zeros outside the unit circle. An AR model is always invertible but is stationary if 6. An MA model is always stationary but is invertible if Β (B) = 0 has zeros outside the unit circle.Parametric parsimony consideration in model building entails preference for the mixed ARMA fit to eitherthe pure AR or the pure MA fit. Stationarity and invertibility conditions for model (1) or (2) are that theequations Α (B) = 0 and Β (B) = 0 should have roots outside the unit circle respectively. Often, inpractice, a time series is non-stationary. Box and Jenkins (1976) proposed that differencing of anappropriate data could render a non-stationary series {Xt} stationary. Let degree of differencing necessaryfor stationarity be d. Such a series {Xt} may be modelled as (1 + )∇dXt = Β (B)εt (3)where ∇ = 1 – B and in which case Α (B) = = 0 shall have unit roots d times. Thendifferencing to degree d renders the series stationary. The model (3) is said to be an autoregressiveintegrated moving average model of orders p, d and q and designated ARIMA(p, d, q).1.2 Seasonal ARIMA models.A time series is said to be seasonal of order d if there exists a tendency for the series to exhibit periodicbehaviour after every time interval d. Traditional time series methods involve the identification ,unscrambling and estimation of the traditional components: secular trend, seasonal component, cyclicalcomponent and the irregular movement. For forecasting purpose, they are reintegrated. Such techniquescould be quite misleading.The time series {Xt} is said to follow a multiplicative (p, d, q)x(P, D, Q)s seasonal ARIMA model if Α( B )Φ ( B s )∇ d ∇ s X t = Β( B )Θ( B s )ε t D (4)where Φ and Θ are polynomials of order P and Q respectively and s is the seasonal period. That is, Φ ( B s ) = 1 + φ1 B s + ... + φ P B sP , (5) 2
3. Developing Country Studies www.iiste.orgISSN 2224-607X (Paper) ISSN 2225-0565 (Online)Vol 2, No.3, 2012 Θ( B s ) = 1 + θ1 B s + ... + θ Q B sQ , (6)where the φ i and θ j are constants such that the zeros of the equations (5) and (6) are all outside the unitcircle for stationarity and invertibility respectively. Equation (5) represents the autoregressive operatorwhereas (6) represents the moving average operator.Existence of a seasonal nature is often evident from the time plot. Moreover for a seasonal series the ACFor correlogram exhibits a spike at the seasonal lag. Box and Jenkins (1976) and Madsen (2008) are a fewauthors that have written extensively on such models. A knowledge of the theoretical properties of themodels provides basis for their identification and estimation. The purpose of this paper is to fit a seasonalARIMA model to Nigerian Gross Domestic Product Series (NGDP).2.Materials and Methods:The data for this work are quarterly Gross Domestic Products(NGDP) from 1980 to 2007 obtainable fromAbstracts of the National Bureau of Statistics of Nigeria.2.1. Determination of the orders d, D, p, P, q and Q:Seasonal differencing is necessary to remove the seasonal trend. If there is secular trend non-seasonaldifferencing will be necessary. To avoid undue model complexity it has been advised that orders ofdifferencing d and D should add up to at most 2 (i.e. d + D < 3). If the ACF of the differenced series has apositive spike at the seasonal lag then a seasonal AR component is suggestive; if it has a negative spikethen a seasonal MA component is suggestive. Box and Jenkins(1976) and Madsen(2008) have given acatalogue of seasonal models and their covariance structures for possible use for modelling.As already mentioned above, an AR(p) model has a PACF that truncates at lag p and an MA(q)) has anACF that truncates at leg q. In practice ±2/√n where n is the sample size are the non-significance limits forboth functions.2.2. Model Estimation:The involvement of the white noise terms in an ARIMA model entails a nonlinear iterative process in theestimation of the parameters. An optimization criterion like least error of sum of squares, maximumlikelihood or maximum entropy is used. An initial estimate is usually used. Each iteration is expected to bean improvement of the previous one until the estimate converges to an optimal one. However,for pure ARand pure MA models linear optimization techniques exist (See for example Box and Jenkins (1976),Oyetunji (1985)). There are attempts to adopt linear methods to estimate ARMA models (See forexample, Etuk(1996). We shall use Eviews software which employs the least squares approach involvingnonlinear iterative techniques.2.3. Diagnostic Checking:The model that is fitted to the data should be tested for goodness-of-fit. We shall do some analysis of the 3
4. Developing Country Studies www.iiste.orgISSN 2224-607X (Paper) ISSN 2225-0565 (Online)Vol 2, No.3, 2012residuals of the model. If the model is adequate, the autocorrelations of the residuals should not besignificantly different from zero.3.Results and Discussions:The time plot of the original series NGDP in Figure 1 shows secular trend and marked seasonality thatgrows with time, which is an indication of a multiplicative seasonal model. Seasonal (i.e. 4-month)differencing of the series produces a series SDNGDP with much lesser trend and seasonality. See Figure 2for the time plot. Non-seasonal differencing yields a series DSDNGDP with some seasonality but no trend(see Figure 3). Its ACF as shown in Figure 4 has the following features: 1. There are significant (positive) spikes at lags 4, 8 and 12 indicating seasonality of order 4 as well as a seasonal AR component. 2. The spikes immediately before and after each seasonal lag are of the same sign and equal. 3. As the lag increases autocorrelation generally decreases.These features fit the description of the (0, 0, 1)x(1, 0, 0)s model (5.131) of page 132 of Madsen(2008)which is given by (1 - ΦBs) Xt = (1 - θB)εt With s=4, this translates into Xt = α4 X t-4 + β1εt-1 + εt (7)where X represents DSDNGDP. The estimation of the model summarized in Table 1 yields the model Xt = 0.2356Xt-4 - 0.9043εt-1+ εt (8)The estimation involved 7 iterations. We note that both coefficients are significantly different from zero,each being larger than twice its standard error. Moreover, there is considerable agreement between theactual and the fitted models as shown in Figure 5. The correlogram of the residuals in Figure 6 also depictsthe adequacy of the model since all the autocorrelations are not significantly different from zero.4.Conclusion:The DSDNGDP series has been shown to follow the seasonal model (8). This model has been shown to beadequate.References: 4
5. Developing Country Studies www.iiste.orgISSN 2224-607X (Paper) ISSN 2225-0565 (Online)Vol 2, No.3, 2012Box, G. E. P. And Jenkins, G. M. (1976). Time Series Analysis, Forecasting and Control. San Francisco:Holden-Day.Etuk, E. H. (1996). An Autoregressive Integrated Moving Average (ARIMA) Model: A Case Study.Discovery and Innovation, 10 ( 1 & 2): 23 – 26.Madsen, H. (2008). Time Series Analysis, London: Chapman & Hall/CRC.Oyetunji, O. B. (1985). Inverse Autocorrelations and Moving Average Time Series Modelling. Journal ofOfficial Statistics, 1: 315 – 322.Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic. London: Academic Press.Dr Ette Harrison Etuk (M’76-SM’81-F’87) is an Associate Professor of Statistics at Rivers StateUniversity of Science and Technology, Nigeria. Born in 1957 he had his B. Sc. in 1981, M. Sc. in 1984 andPh.D in 1987. These degrees were in Statistics and all obtained from University of Ibadan, Nigeria. Hisresearch interests are Time Series Analysis, Operations Research and Experimental designs. He is a Fellowof Institute of Corporate Administration of Nigeria and also that of Institute of Human and NaturalResources of Nigeria.Table 1: Model Estimation 5
11. Developing Country Studies www.iiste.orgISSN 2224-607X (Paper) ISSN 2225-0565 (Online)Vol 2, No.3, 2012FIGURE 6: CORRELOGRAM OF THE RESIDUALS 11
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