The return series, R t , can be obtained given that P t is the price of a unit share at time, t, while P t − 1 is the share price at time, t − 1.

R t = ∇ ln ( P t ) = ( 1 − B ) ln ( P t ) = ln ( P t ) − ln ( P t − 1 ) (1)

here, R t is regarded as a transformed series of the share price, P t , meant to attain stationarity while B is the backshift operator. Thus, both the mean and the variance of the series are stable [18] [19].

In [3] the extension of ARMA model to deal with homogenous non-stationary time series in which X t , itself is non-stationary but its d t h difference is a stationary ARMA model. Denoting the d t h difference of X t by

φ ( B ) = ϕ ( B ) ∇ d X t = θ ( B ) ε t (2)

where φ ( B ) is the nonstationary autoregressive operator such that d of the roots of φ ( B ) = 0 are unity and the remainder lie outside the unit circle. ϕ ( B ) is a stationary autoregressive operator (see also, [20] [21]).

The foundation of time series analysis is stationarity. Consider a finite set of return variables { R t 1 , R t 2 , ⋯ , R t n } from a time series process, { R ( t ) : t = 0 , + 1 , + 2 , ⋯ } . The k-dimensional distribution function is defined as

F R t 1 ⋯ R t k ( r 1 , r 2 , ⋯ , r k ) = P { R t 1 ≤ r 1 , R t 2 ≤ r 2 , ⋯ , R t k ≤ r k } , (3)

where r j , j = 1 , 2 , ⋯ , k are any real numbers.

A process is said to be:

1) first-order stationary in distribution if its one-dimensional distribution is time invariant. That is, if

F R t 1 ( r 1 ) = F R t 1 + k ( r 1 ) , (4)

for any integers t 1 , k and t 1 + k .

2) second-order stationary in distribution if

F R t 1 , R t 2 ( r 1 , r 2 ) = F R t 1 + k , R t 2 + k ( r 1 , r 2 ) , (5)

for any integers t 1 , t 2 , k , t 1 + k and t 2 + k .

3) the n t h -order stationary in distribution if

F R t 1 , R t 2 , ⋯ , R t n ( r 1 , ⋯ , r n ) = F R t 1 + k , R t 2 + k , R t n + k ( r 1 , ⋯ , r n ) , (6)

for any ( t 1 , ⋯ , t n ) and k integers.

A process is said to be strictly stationary if (3.6) is true for any n, that is, n = 1 , 2 , ⋯

According to [3], a process { R t } is weakly stationary if the mean E ( R t ) = μ is a fixed constant for all t and the autocovariances C o v ( R t , R t + k ) = γ k depends only on the time difference or time lag k for all t.

Stationary in the wide sense or covariance stationary is also referred to as second-order stationary process.

A process { a t } is called a white noise process if it is a sequence of uncorrelated random variables from a fixed distribution with constant mean, E ( a t ) = μ a , usually assumed to be zero, constant variance, V a r ( a t ) = σ a 2 and γ k = C o v ( a t , a t + k ) = 0 , for all k ≠ 0 . It is denoted by a t ~ WN ( 0 , σ a 2 ) , where WN stands for white noise [5]. By definition, a white noise process { a t } is stationary with autocovariance function,

γ k = { σ a 2 ,       k = 0 , 0 ,           k ≠ 0. (7)

The autocorrelation function is given as:

ρ k = { 1 ,           k = 0 , 0 ,           k ≠ 0. (8)

while the partial autocorrelation function is

φ k k = { 1 ,           k = 0 , 0 ,           k ≠ 0. (9)

Thus, the implication of a white noise specification is that the ACF and PACF are identically equal to zero.

According to [5], covariance between R t and R t + k denoted by C o v ( R t , R t + k ) , which is a function of the time difference, k, is called the autocovariance function { γ k } of the stochastic process. As a function of k, γ k is called the autocovariance function in the time series analysis since it represents the covariance between R t and R t + k from the same process. It is defined as

γ k = C o v ( R t , R t + k ) = E ( R t − μ ) ( R t + k − μ ) . (10)

The sample estimate of γ k is C k given by

C k = 1 n ∑ t = 1 n − k ( R t − R ¯ ) ( R t + k − R ¯ ) (11)

Similarly, the correlation between R t and R t + k denoted by C o r r ( R t , R t + k ) , which is a function of the time difference, k, is called the autocorrelation function { ρ k } of the stochastic process. As function of k, ρ k is called the autocorrelation function in time series analysis since it represents the correlation between R t and R t + k from the same process. It is defined as

ρ k = C o v ( R t , R t + k ) V a r ( R t ) V a r ( R t + k ) = γ k γ 0 (12)

The corresponding sample estimate is given by

ρ ^ k = C k C 0 , k = 0 , 1 , 2 , ⋯ (13)

The conditional correlation between R t and R t + k after their mutual linear dependency on the intervening variables ( R t + 1 , R t + 2 , ⋯ , R t + k − 1 ) has been removed, given by C o r r ( R t , R t + k / R t + 1 , R t + 2 , ⋯ , R t + k − 1 ) , is usually referred to as the partial autocorrelation in time series analysis ([5]).

Partial autocorrelation can be derived from the regression model, where the dependent variable, R t + k , from a zero-mean stationary process is regressed on k-lagged variables R t + k − 1 , R t + k − 2 , ⋯ and R t , that is

R t + k = φ k 1 R t + k − 1 + φ k 2 R t + k − 2 + ⋯ + φ k k R t + α t + k , (14)

where φ k i denotes the i t h regression parameter and α t + k is an error term with mean zero and uncorrelated with R t + k − j , for j = 1 , 2 , ⋯ , k . Multiplying R t + k − j on both sides of the above regression equation and taking the expectation, we get

γ j = φ k 1 γ j − 1 + φ k 2 γ j − 2 + ⋯ + φ k k γ j − k . (15)

Hence,

ρ j = φ k 1 ρ j − 1 + φ k 2 ρ j − 2 + ⋯ + φ k k ρ j − k . (16)

Diagnostic checking is applied with an objective of uncovering a possible lack-of-fit of the tentative model and possibly unraveling the cause of such a case. If no lack-of-fit is indicated, the model is ready for use. In other words, if any inadequacy is found, the iterative cycle of identification, estimation and diagnostic checking is repeated until a suitable and appropriate representation is obtained.

Once the parameters of the tentative models have been estimated, we check whether or not the residuals obtained from the estimated equation are approximately white noise. This is done by examining the ACF and PACF of the residuals to see whether they are statistically insignificant, that is, within two standard deviations at 5% level of significance. If the residuals are approximately white noise, the model may be entertained provided the parameters are significantly different from zero.

The Portmanteau lack-of-fit test uses the residual sample ACFs as a unit to check the joint null hypothesis test, which requires that several autocorrelations of a t are zero. [17] proposed the Portmanteau statistics given as:

Q ∗ ( m ) = T ∑ l = 1 m ρ ^ l 2 , (17)

where T is the number of observations.

A test statistic for the null hypothesis, H 0 : ρ 1 = ⋯ = ρ m = 0 , against the alternative, H a : ρ i ≠ 0 , for some i ∈ { 1 , ⋯ , m } under the assumption that { a t } is an i.i.d. sequence with certain moment conditions while Q ∗ ( m ) is asymptotically a Chi-square random variable with m degrees of freedom.

[22] modified the Q ∗ ( m ) statistic to increase the power of the test in finite samples as follows:

Q ( m ) = T ( T + 2 ) ∑ l = 1 m ρ ^ l 2 T − l , (18)

where T is the number of observations.

The decision rule is to reject H 0 if Q ( m ) > χ α 2 , where χ α 2 denotes the 100(1 – α)th percentile of a Chi-squared distribution with m – (p + q) degrees of freedom. The decision rule can also reject H 0 if the p-value is less than or equal to α, the significance level.

In practice, the selection of m may affect the performance of the Q(m) statistic. The choice m ≈ ln ( T ) provides better power performance [4].

Data collection was based on secondary source from the records of Nigerian Stock Exchange. The data on the daily closing share prices of the sampled banks (Union bank, Unity bank and Wema bank) from January 3, 2006 to November 24, 2016 were obtained from the Nigerian Stock Exchange [23] and delivered through contactcentre@nigerianstockexchange.com.

Figures 1-3 represent the share price series for the three banks. The share prices of all the banks do not fluctuate around a common mean, which clearly indicates the presence of a stochastic trend in the share prices, and is also an indication of non-stationarity. Since the share price series are found to be non-stationary, the first difference of the natural logarithm of share price series is taken to obtain stationary (returns) series. The inclusion of the log transformation is to stabilize the variance. Figures 4-6 show that the returns series appear to be stationary.