Buys-Ballot Technique for the Analysis of Time Series with a Cubic-Trend Component

Time series, especially those with the cubic trend component, are encountered in many data analysis situations. The decomposition of such series into various components requires a method that can adequately estimate the cubic trend as well as other components of the series. In this study, the chain base, fixed base and classical methods of decomposition of time series with the cubic trend component are discussed with emphasis on the additive model. Chain base and fixed base estimators of the additive model parameters are derived. Basic properties of these two classes of estimators are equally determined. The derived chain base variables have the autocorrelation structure of an invertible third-order moving average model. The chain base estimators are found to be pairwise-negatively correlated estimators. Though the classical decomposition method and chain base method are both used for time series decomposition, the chain base method is recommended when a case of multicollinearity has been established.


Introduction
One of the tasks frequently performed by time series analysts is the decomposition of a given time series into its various components. The classical decomposition method is the first known method of decomposing time series. Its application is often predicated on the additive and multiplicative models. The objectives of the classical decomposition method have been mentioned in numerous studies. It helps us to investigate the presence of trend, seasonal and cyclical effects in a time series. Estimates of the four components of time series which include trend, seasonal, cyclical and irregular components are found with the help of this method . Classical decomposition models are also used for short term forecasting.
Inspite of its uses, the classical decomposition method has some limitations. Notable among the demerits is the tedious nature of the method since the components are estimated one after the other. Another disadvantage of this method is its frequent poor forecasting performance (Chatfield 1975;DeLurgio 1998). The least squares estimation procedure, which is used to estimate the trend component of a series in the classical decomposition approach, may not be reliable in the presence of multicollinearity. The high level of multicollinearity between two or more powers of the time variable (t) in a polynomial trend model often results in wrong inferences and model selection based on the least squares estimates of the concerned parameters (Chatterjee and Greenwood 1990; Schacam and Brauner 1997). As a consequence, the Buys-Ballot estimation procedure proposed in Iwueze and Nwogu (2004), which is capable of yielding estimates that are robust to multicollinearity, is considered when the multicollinearity problem exists (Nduka et al. 2017). The Buys-Ballot approach is primarily used for the decomposition of a relatively short series such that the trend and cyclical components are jointly estimated. The additive and multiplicative Buys-Ballot decomposition models are stated in Eqs. (1) and (2) respectively: (1) where X t is the observed value of the time series at time t, M t is the trend-cycle component at time t, S t is the seasonal component at t and e t is the irregular component or the error term at time t. In (1), e t ∼ N (0, σ 2 1 ) while in (2), e t ∼ N (1, σ 2 2 ). Apart from the work of Iwueze and Nwogu (2004) in which the chain base and fixed base estimation techniques were used in accordance with the linear trend-cycle component, the additive and multiplicative models, several studies have been subsequently undertaken within the context of the Buys-Ballot method of analysing time series data. In this regard, Iwueze and Ohakwe (2004) developed the Buys-Ballot procedure of analysing time series data with the quadratic trend-cycle component. Their specific contributions include the derivation of the chain base and fixed base estimators of the parameters of each of the additive and multiplicative models. The Buys-Ballot estimates for exponential and s-shaped curves were derived in Iwueze and Nwogu (2005). Certain properties of the chain base and fixed estimators have been discussed with respect to the linear trend-cycle. The unbiasedness and consistency of both esti-mators were established in Iwueze et al. (2010). According to Iwueze et al. (2011), if the trend-cycle component is linear, only the derived chain base variables are stationary with the autocorrelation structure of the moving average process of order one. These authors equally found the best linear unbiased estimate of the slope parameter using the variables associated with the chain base estimation. Works carried out in other areas of research interest in the Buys-Ballot Method include the development of the procedures through which one can determine when each of the additive and multiplicative models should be considered (Iwueze and Nwogu 2014), test for the presence of seasonal variations (Nwogu et al. 2016) and the influence of the mis-specification of error distribution on the prediction accuracy of the fitted Buy-Ballot model Nduka et al. (2017).
Though the trends in many time series can be represented by the linear , quadratic trend or exponential trend models, there are cases where the cubic time trend model is inevitably applicable (Chatfield 1975, Anna andAcquah 2011). In particular, the cubic trend model has found applications in Agronomy (Michael andMakowski 2013), Computational Statistics (Hargreaves andMcWilliams 2010), Epidermology (Ndong et al. 2014), Fishery (Fabrizio et al. 2000, Meteorology (Frankze 2012) and Psychology (Snook et al. 2009, Jebb et al. 2015.
Motivated by the wide applicability of the cubic trend model and robustness of the Buys-Ballot method to multicollinearity, in this study, we discuss the Buys-Ballot method of decomposing a time series with a cubic trend-cycle component. The chain base and fixed base estimators of the parameters of the cubic trend-cyle component and seasonal component of the additive model are derived. We also pay attention to the properties of the derived estimators of the parameters of the cubic trend-cycle component. The subsequent parts of this work are arranged in the following manner. Section 2 deals with the overview of the least squares estimation of the parameters of the cubic trend model. Some theoretical results are presented in Sects. 3 and 4 deals with the properties of the estimators derived in Sect. 3. In Sect. 5, we apply the results in Sect. 3 to real life time series data. Also considered in this section, is the comparison of the prediction performances of the fitted additive Buys-Ballot model and the classical decomposition model based on the least squares approach. The conclusion of this work is given in Sect. 5.

Overview of the Cubic Trend Estimation by Least Squares Method
Let {X t } and {T t } denote a given time series and the corresponding cubic trend component such that Suppose the least squares estimator of T t iŝ Then the least squares estimatorsâ 0 ,â 1 ,â 2 andâ 3 of a 0 , a 1 , a 2 and a 3 respectively, minimise the sum of squared deviations (S) of X t fromT t . For S = n t=1 (X t −T t ) 2 , we evaluate ∂ S ∂â r = 0 to obtain the normal equations: In matrix form, the system of linear equations in (5), (6), (7) and (8) becomes n t=1 t 2 n t=1 t 3 n t=1 t 4 n t=1 t 5 n t=1 t 3 n t=1 t 4 n t=1 t 5 n t=1 t 6 ⎞ ⎟ ⎟ ⎠ and The estimated variance-covarinace matrix for the estimators is Here,σ 2 = S n−4 . The decision to include any of the parameter estimates in the cubic trend model is made in line with the test of significance of the parameter. While a t-test may be appropriate for testing the significance of a single parameter, the significance of the overall cubic trend model can be investigated through the analysis of variance technique (Draper and Smith 1981). It is noteworthy that the regression outputs from many statistical packages contain the requisite summary of the test results and conclusions may be drawn on the basis of the computed p-values.
Once the cubic trend model parameters have been estimated, the classical decomposition method (CDM) may be employed in estimating the seasonal indices. Before the estimation of seasonal indices by this approach, the original series has to be detrended. Assuming the additive model, the process of detrending the series deals with the subtraction of the trend component from the series. If the detrended series is arranged in accordance with the seasons (months or quarters as the case may be), the seasonal averages can be obtained. These averages are used to find the seasonal indices.

Preliminary Results Based on the Buys-Ballot Table
An important aspect of the time series analysis using the Buys-Ballot approach is the arrangement of the observed seasonal time series data in a Buys-Ballot table as shown in Table 1. All the derivations made in this section are based on the deterministic component of the additive Buys-Ballot decomposition model.
If we make the substitution M t = a + bt + ct 2 + dt 3 into Eq.
(1) and consider only the deterministic part of Eq. (1), we have From Table 1, the ith row total is given as Table 1 The Buys-Ballot tabular arrangement of time series data Source: Iwueze and Nwogu (2004) Period Season In deriving Eq. (12), we made use of the assumption s j=1 S j = 0. Now, the ith row average is Next, we derive an expression for each of the jth column total and mean. With m i=1 S j = mS j , the jth column total becomes Dividing both sides of Eq. (14) by m, the jth column mean is obtained to bē Furthermore, the grand total of the observations is given as For ms = n, we have The grand mean is obtained by dividing Eq. (17) by n. Thus,

The Proposed Buys-Ballot Estimators of the Cubic Trend-Cycle and Seasonal Components of the Additive Buys-Ballot Decomposition Model
Two classes of estimators, namely chain base and fixed base estimators of the requisite parameters are derived following the procedure of Iwueze and Ohakwe (2004). For the purpose of deriving the estimators, we find the first, second and third differences of the ith row mean series. From Eq. (13), we obtain Given the forward difference operator , then the first forward difference ofX i . has the following representation Hence, From the foregoing, the following relationship between the estimators of b, c and d can be easily deduced Equation (25) simply reveals that unless the estimators of c and d are known, it is practically impossible to estimate b within the context of the Buys-Ballot method used in this research work. Consequently, one may wish to know if the estimator of c is a function of that of d. To obtain the estimator of c, we evaluate the second difference ofX i. . For this purpose, we have If we substitute Eqs. (13), (19) and (20) into (25) and simplify the resulting expression completely, the result in Eq. (26) is obtained It follows from Eq. (26) thatĉ The estimator of c in Eq. (27) certainly depends on that of d. To find the estimator of d, we first evaluate the third difference ofX i. . Let Simplifying Eq. (28) afterX i. ,X (i+1). ,X (i+2). andX (i+3). have been replaced by Eqs. (13), (19), (20) and (21) respectively, yields Using Eq. (29), the chain base estimator of d i iŝ Let W i. = Z (i+1). − Z 1. Then, W i. = 2cs 2 + 3ds 2 (2(i + 1)s + s + 1) − (2cs 2 + 3ds 2 (2s + s + 1)) For the fixed base estimator of d i , we get Again, the fixed base estimator of d iŝ In line with Eq. (18), an estimator of a is generally given aŝ By substituting n = ms into Eq. (15) and using Eq. (34), the estimator of S j is found to bê +d 2n 2 + n − 4 j 3 − 6 j 2 n + 6 j 2 s − 4 jn 2 + 6 jns − 2 js 2 + 2n 2 s − ns 2 4 Owing to the fact that the derived estimators of a, b, c and S j depend on that of d, those estimators are said to be chain base estimators if there are all functions of the chain base estimator of d. On the other hand, there are called fixed base estimators if there are determined using the fixed base estimator of d. Observe that in the absence of trend,â =b =ĉ =d = 0 andŜ

Properties of the Estimators of Parameters of the Cubic Trend-Cycle Component
When two or more estimators of a particular parameter exist, as it is the case in this study, efforts are made to determine the best among the competing estimators. To determine a better class of estimators between the chain base and fixed base families of estimators, we derive and compare properties of the estimators belonging to the two classes.
For an estimator to be unbiased for a particular parameter, its expectation must be equal to the given parameter. Taking expectation of both sides of Eq. (30) leads to We can deduce from the assumption of e t associated with Eq. (1) thatē i. ∼ N (0, Also, It is now evident that bothd C B E i andd C B E are unbiased for d. The variances of these estimators are obtained as follows: Acknowledging the facts that errors are uncorrelated, we get Having knowledge of the covariance betweend C B E i andd C B E h , where i = h helps us to determine if the chain base derived variables are independent or not. It also serves as a tool for deriving the autocorrelation function for the derived variables. With the autocorrelation function, one can figure out whether the variables are generated by a stationary series or a nonstationary series. Notationally, The following autocorrelation function ford C B E i , is derived using R C B E (k) It is now certain that the CBE derived variablesd C B E i , i = 1, 2, 3, ., ., ., m − 3 have the autocorrelation structure of a third-order moving average process, indicating that there are generated by a stationary process. We can also deduce from the work of Okereke et al. (2015) that the moving average process is invertible.
Next, we consider the properties ofĉ C B E .
In order to determine the nature of the relationship betweenĉ C B E andd C B E , we need to find the covariance (Cov(ĉ C B E ,d C B E )) ofĉ C B E andd C B E . This covariance will be needed for the derivation of the V ar(b C B E ) .
Expectation, variance and covariances which are predicated on each the chain base estimatorsb C B E andâ C B E of b and a respectively are given in the "Appendix I". The chain base estimator (Ŝ C B E j ) of the seasonal component stated in (35) is easily seen to be unbiased for (S C B E j ). Let V c be the variance-covariance matrix pertaining to the derived chain base estimators. Then Efficiency comparison based on determinants of variance-covariance matrices has been discussed by Hamilton (1994). Given two point estimation methods, then one of the methods is said to be more efficient than the other if its corresponding variancecovariance matrix has a smaller determinant. We may observe that V c depends on σ 2 1 . Since σ 2 1 is often not known, it is estimated by the mean squared error (MSE). If we replace σ 2 1 in Eq. (56) by MSE, we obtain the estimated variance-covariance matrix. So far, we have focused on the properties of the chain base estimators based on Eq. (1). In what follows, properties of fixed base estimators are discussed. Considering Eq. (32) and the associated error terms, we have After estimating parameters of the cubic trend-cycle model, it will be expedient to test hypothesis about the individual parameters. Notably, each of the concerned chain base estimators of the parameters of the cubic trend-cycle model is a linear function of the observed time series. Suppose we wish to test the null hypothesis H 0 : g = g 0 against H 1 : g = g 0 , where g is any of the parameters a, b, c and d. For a Guassian white noise process and unknown σ 2 1 , the t-test statistic may be used, where est.std(ĝ) is the estimated standard deviation of the given parameter. At α level of significance, we reject H 0 if |t| ≥ t α 2 ,n−4 .

Empirical Results
A numerical example is given to illustrate the chain base method (CBM). We also compare the method with the classical decomposition method (CDM) through the decomposition of a real time series data set. The prediction accuracy measures employed in this work are the mean squared error (MSE), mean absolute error (MAE) and mean absolute percentage error (MAPE).

Real Life Example
As a practical application of CDM and CBM, we consider the time series decomposition of the monthly concentrations of atmospheric CO 2 . The monthly atmospheric concentrations of co 2 data set for the period January, 1959-December, 1997 is available in R package. Following Cleveland et al. (1983) and Bacastow et al. (1985), it can be deduced that the co 2 data exhibit trend and seasonal variation. In particular, the monthly atmospheric concentrations of co 2 data set for the period January, 1959-December, 1997 have been shown to have cubic trend and seasonal variations.
The estimated least squares cubic trend model for the time series isT t = 316.265 + 0.0290513t + 0.000292787t 2 − 2.90208 × 10 −7 t 3 . Having fitted a cubic trend model, we proceed to investigate the significance of the overall regression using the results in Table 2. With a p-value of 0.000, it is obvious that the the fitted cubic trend model is statistically significant at 5% significance level. This shows that the time series has a cubic trend component. Consequently, it is imperative to determine the coefficients that should be included in the fitted cubic trend model for the time series data. This calls for tests for significance of coefficients of the model parameters. Results based on these tests, are summarised in terms of p-values in Table 3. From all indications, it is necessary to consider a cubic trend model containing all the four coefficients.
Trend parameter estimates as well as seasonal indices are given in Table 4 for CDM and CBM. It is not surprising that minimum values of MSE, MAE and MAPE are associated with the classical decomposition method (CDM) since the chain base method (CBM) is basically meant for short series. The issue is then how short should the series be for CBM to be a good rival of CDM when assumptions of the latter hold. To address this issue, we examine the following figures. From Fig. 1, it is evident that the series has a cubic trend and seasonal components. Figure 2 shows that fits based on CDM are quite close to the actual observations for all time points. On the other hand, fits based on CBM are close to the actual observations from January of 1959-December, 1967. Within this period, fits based on the two methods are also close, as indicated in Fig. 2. The closeness of the two sets of fitted values, may be attributed to closeness of the corresponding least squares and chain base estimates in Table 4. Beyond the

Conclusion
Presented in this paper, are the Buys-Ballot and classical methods of decomposing a time series with a cubic trend component. Two sets of estimators, namely chain base and fixed base estimators have been derived and their properties investigated. While the two sets of estimators are generally unbiased, there is still a remarkable difference between their properties. The chain base estimators are functions of stationary variables with the third-order moving average model autocorrelation structure whereas the fixed base estimators depend on nonstationary variables. This is in agreement with the findings in Iwueze et al. (2010Iwueze et al. ( , 2011. One might think of differencing the nonstationary variables to obtain the modified fixed base estimators based on stationary variables. We may note that the first difference of the derived fixed base variables yields the derived chain base variables which are stationary. As a result, the modified fixed base estimators will not be different from the already derived estimators of chain base type.
Being a linear combination of the derived chain base variables, each of the chain base estimators, is normally distributed when the underlying time series is normally distributed. It follows that hypothesis testing about the significance of a cubic model parameter can be carried out using a t statistic and an appropriate chain base estimate. There is no doubt that the chain base estimators are not mutually independent. Undoubtedly, the estimators are pairwise-negatively correlated.
We have graphically illustrated that for the first 108 obervations on the time series, CBM competes favourably with CDM. However, if there is a case of multicollinearity, CBM may be considered.