IJRRAS 14 (3) ● March 2013 www.arpapress.com/Volumes/Vol14Issue3/IJRRAS_14_3_14.pdf 596 A SEASONAL TIME SERIES MODEL FOR NIGERIAN MONTHLY AIR TRAFFIC DATA Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria ABSTRACT Nigerian Monthly Air Traffic Data (NAP) is analysed as a time series. The non-seasonal difference of its seasonal (i.e. 12-month) difference (DSDNAP) is observed to show some seasonality. The autocorrelation function of DSDNAP reveals a 12-month seasonality, the involvement of a seasonal moving average component of order 1 and the product of two autoregressive components: one non-seasonal and the other seasonal, both of order one. Therefore, a (1, 1, 0)x(1, 1, 1) 12 is proposed and fitted to the data. This model has been demonstrated to be adequate. Keywords: Air Traffic Data, Seasonal Time Series, ARIMA models, Nigeria. 1. INTRODUCTION A time series may be defined as data collected sequentially in time, the time points often equally spaced. A property of such a series is that neighbouring values are correlated. This correlation is called autocorrelation. Put as a function of the lag separating the correlated values, it is called autocorrelation function (ACF). The graph of the ACF is called the correlogram. A stationary time series refers to a time series with a constant mean, a constant variance and autocorrelation that is a function of the lag separating the correlated values. A stationary time series {X t } is said to follow an autoregressive moving average model of order p and q, denoted by ARMA(p, q) if it satisfies the following difference equation q t q t t t p t p t t t X X X X                          ... ... 2 2 1 1 2 2 1 1 (1) where { t } is called a white noise process and defined as a sequence of uncorrelated zero mean random variables with constant variance. The model (1) may be alternatively put as A(L)X t = B(L) t (2) where A(L) = 1 -  1 L -  2 L 2 - ... -  p L p and B(L) = 1 +  1 L +  2 L 2 + ... +  q L q and L is the backward shift operator defined by L k X t = X t-k . Besides stationarity, another necessary property for a time series is invertibility, which may be defined as the situation whereby the model is associated with a unique autocorrelation structure (Priestley[1]). For the model above to be stationary the equation A(L) = 0 must have roots all outside the unit circle and for it to be invertible, the equation B(L) = 0 must have all roots outside the unit circle. If p = 0, the model (1) or (2) becomes a moving average model of order q, designated MA(q). If, however, q = 0, the model (1) or (2) an autoregressive model of order p, designated AR(p). An autoregressive model of order p may be more specifically written as t p t pp t p t p t X X X X             ... 2 2 1 1 (3) The sequence of the last coefficients { ii } is called the partial autocorrelation (PACF) of {X t }. The PACF of an AR(p) cuts off at lag p, whereas that of an MA model dies off slowly. The ACF of an MA(q) model cuts off at lag q