Box-Jenkins Modelling of Nigerian Stock Prices Data

Nigerian stock prices data is modelled by Box-Jenkins approach and the use of automatic model selection criteria: Akaike Information criterion (AIC), Schwarz Information Criterion (SIC), R 2 . It is inferred that the most adequate model is autoregressive integrated moving average of orders 2, 1 and 3(ARIMA (2 ,1 ,3)). Forecasts are obtained on the basis of the model.


INTRODUCTION
A time series is defined as a set of data collected sequentially in time. It has the property that neighbouring values are correlated. This tendency is called autocorrelation. A time series is said to be stationary if it has a constant mean and variance. Moreover the autocorrelation is a function of the lag separating the correlated values called the autocorrelation function (ACF).
A stationary time series {X t } 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 (1) Or α α α α(B)X t = β β β β(B)ε t where {ε t } is a sequence of random variables with zero mean and constant variance, called a white noise process, and the α i 's and β j 's constants; α α α α(B) = 1 + α 1 B + α 2 B 2 + ... + α p B p and β β β β(B) = 1 + β 1 B + β 2 B 2 + ... + β q B q and B is the backward shift operator defined by B k X t = X t-k . If p=0, model (1) becomes a moving average model of order q (designated MA(q)). If, however, q=0 it becomes an autoregressive process of order p (designated AR(p)). An AR(p) model of order p may be defined as a model whereby a current value of the time series X t depends on the immediate past p values: X t-1 , X t-2, ..., X t-p . On the other hand an MA(q) model of order q is such that the current value X t is a linear combination of immediate past values of the white noise process: ε 1 , ε 2 , ..., ε q . Apart from stationarity, invertibility is another important requirement for a time series. It refers to the property whereby the covariance structure of the eries is unique (Priestley, 1981). Moreover it allows for meaningful association of current events with the past history of the series (Box and Jenkins, 1976).
An AR(p) model may be more specifically written as X t + α p1 X t-1 + α p2 X t-2 + ... + α pp X t-p = ε t Then the sequence of the last coefficients{α ii } is called the partial autocorrelation function(PACF) of {X t }. The ACF of an MA(q) model cuts off after lag q whereas that of an AR(p) model is a combination of sinusoidals dying off slowly. On the other hand the PACF of an MA(q) model dies off slowly whereas that of an AR(p) model cuts off after lag p. AR and MA models are known to have some duality properties. 5. An AR model is always invertible but is stationary if α α α α(B) = 0 has zeros outside the unit circle. 6. An MA model is always stationary but is invertible if β β β β(B) = 0 has zeros outside the unit circle.
Parametric parsimony consideration in model building entails preference for the mixed ARMA fit to either the pure AR or the pure MA fit. Stationarity and invertibility conditions for model (1) or (2) are that the equations α α α α(B) = 0 and β β β β(B) = 0 should have roots outside the unit circle respectively. Often, in practice, a time series is non-stationary. Box and Jenkins [2] proposed that differencing of an appropriate data could render a non-stationary series {X t } stationary. Let degree of differencing necessary for stationarity be d. Such a series {X t } may be modelled as where ∇ = 1 -B and in which case α α α α(B) = = 0 shall have unit roots d times. Then differencing to degree d renders the series stationary. The model (3) is said to be an autoregressive integrated moving average model of orders p, d and q and designated ARIMA(p, d, q). The purpose of this paper is to fit an ARIMA model to Nigerian stock prices.

MATERIALS AND METHODS
The data for this work are monthly stock prices data from January1987 to December 2006 obtained from Nigerian Stock Exchange Office, Port Harcourt, Nigeria.

Determination of the differencing order d:
Preliminary analysis of time series involves the time-plot and the correlogram. A stationary time series exhibits no trend and the degree of variability is invariant with time. In addition the covariance is a function of the time lag. The time plot of a stationary time series shows no change in the mean level as well as the variance over time. The autocorrelation function should decay fast to zero.

Test for stationarity:
The ACF of a non-stationary time series starts high and declines slowly. Moreover to test for stationarity we shall be using the Augmented Dickey-Fuller (ADF) test. This involves testing for b=1 against b < 1 in X t = a + bX t-1 + ε t. The software Eviews 3.1 that we shall use has facility for the ADF test also.

Determination of the orders p and q:
As already mentioned above, an AR(p) model has a PACF that truncates at lag p and an MA(q)) has an ACF that truncates at lag q. In practice are the nonsignificance limits for both functions. We shall explore the range of models ARMA(a,b), 0 ≤ a ≤ p, 0 ≤ b ≤ q for an optimum one. To do this we shall use the automatic model determination criteria AIC and SIC ( e.g. Akaike(1970), Etuk (1987Etuk ( , 1988, and Schwarz (1978) is the maximum likelihood estimate of the residual variance when the model has k parameters.The optimum model corresponds to the minimum of the criteria within the explored range.

Model Estimation:
The involvement of the white noise terms in an ARIMA model entails a nonlinear iterative process in the estimation of the parameters, α i 's and β j 's. An optimization criterion like least error of sum of squares, maximum likelihood or maximum entropy is used. An initial estimate is usually used. Each iteration is expected to be an improvement of the last one until the estimate converges to an optimal one. However,for pure AR and 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 for example, Etuk(1987Etuk( , 1988Etuk( , 1996).

Diagnostic Checking:
The model that is fitted to the data should be tested for goodness-of-fit. The automatic order determination criteria AIC and SIC are themselves diagnostic checking tools. Further checking can be done by the analysis of the residuals of the model. If the model is correct, the residuals would be uncorrelated and would follow a normal distribution with mean zero and constant variance.

RESULTS AND DISCUSSION
The time plot of the original series NSP in Figure1, the correlogram of Figure 2 and the ADF test of Table 1 clearly depict non-stationarity. Differencing the series once yields a stationary process, DNSP; the time plot is in Figure 3, the correlogram in Figure 4 and the ADF test in Table 2. We note that in this table the dependent variable is the second difference SNSP of the original series. From fig. 4, the ACF cuts off at lag 5 and PACF at lag 4. Exploring the range of models {ARMA(p,q): 0 ≤ p ≤ 4, 0 ≤ q ≤ 5} which are stationary and invertible with positive R 2 for the optimal on the basis of AIC and SIC yields an ARMA(2, 3). The model estimation is The chosen model as summarized in Table 3 is ARIMA(2, 1, 3) and is given by DNSP t = -0.061736DNSP t-1 -0.991882DNSP t-2 + 0.046505ε t-1 + 0.970226ε t-2 + 0.035801ε t-3 + ε t (±0.019321) (±0.020070) (±0.070918) (±0.014925) (±0.067853) Clearly non-linear techniques used by Eviews 3.1 involved an iterative process that converged after thirty one iterations. We observe that only the first and third MA coefficients are not significant, each being less than twice its standard error. The roots of α α α α(B) = 0 and β β β β(B) = 0 all lie outside the unit circle indicating stationarity and invertibility respectively. Besides the residual plot of Fig. 5 confirms that the residuals follow the normal distribution with zero (actually 0.1) mean.

FIGURE 5: HISTOGRAM OF MODEL RESIDUALS
Forecasting: An ARIMA(2, 1, 3) is of the form X X X ε ε β ε β ε β α α Taking conditional expectations at time t, is the k-point ahead forecast. That is the forecast of X t+k given the series up to X t .

CONCLUSION:
We have fitted an adequate ARIMA (2,1,3) model to Nigerian Stock Prices. That means that the first differences DNSP follow an ARMA (2,3) model. On the basis of the model, we have made some forecasts.