 Research
 Open Access
Performance evaluation of series and parallel strategies for financial time series forecasting
 Mehdi Khashei^{1} and
 Zahra Hajirahimi^{1}Email author
Received: 15 March 2017
Accepted: 20 October 2017
Published: 6 November 2017
Abstract
Background
Improving financial time series forecasting is one of the most challenging and vital issues facing numerous financial analysts and decision makers. Given its direct impact on related decisions, various attempts have been made to achieve more accurate and reliable forecasting results, of which the combining of individual models remains a widely applied approach. In general, individual models are combined under two main strategies: series and parallel. While it has been proven that these strategies can improve overall forecasting accuracy, the literature on time series forecasting remains vague on the choice of an appropriate strategy to generate a more accurate hybrid model.
Methods
Therefore, this study’s key aim is to evaluate the performance of series and parallel strategies to determine a more accurate one.
Results
Accordingly, the predictive capabilities of five hybrid models are constructed on the basis of series and parallel strategies compared with each other and with their base models to forecast stock price. To do so, autoregressive integrated moving average (ARIMA) and multilayer perceptrons (MLPs) are used to construct two series hybrid models, ARIMAMLP and MLPARIMA, and three parallel hybrid models, simple average, linear regression, and genetic algorithm models.
Conclusion
The empirical forecasting results for two benchmark datasets, that is, the closing of the Shenzhen Integrated Index (SZII) and that of Standard and Poor’s 500 (S&P 500), indicate that although all hybrid models perform better than at least one of their individual components, the series combination strategy produces more accurate hybrid models for financial time series forecasting.
Keywords
Background
Real time series forecasting with a high degree of accuracy is gaining increasing importance in many domains, particularly the financial markets, and thus, various attempts have been made to develop more accurate techniques. The objective of financial time series forecasting is to provide financial analysts and investors with reliable guidance on asset management. Thus, improving forecasting accuracy and introducing reliable forecasting methods can facilitate more profitable financial market investments by lead investors and financiers. To this effect, choosing a method that performs well in financial time series forecasting is imperative. To provide more accurate results, studies on time series forecasting and modeling widely use a combination of different models and metaheuristic optimization approaches. Considerable research has adopted optimization methods such as genetic algorithm (GA; Aghay Kaboli et al. 2016a, 2016b, Kaboli et al. 2016), particle swarm optimization (PSO; Aghay Kaboli et al. 2016a, 2016b), and gene expression programming (GEP; Aghay Kaboli et al. 2017a, 2017b; Aghay Kaboli et al. 2016a, 2016b). Kaboli et al. (2016) proposed the artificial cooperative search (ACS) algorithm to forecast longterm electricity energy consumption and numerically confirmed the effectiveness of the algorithm using other metaheuristic algorithms including the GA, PSO, and cuckoo search. ModiriDelshad et al. (2016) presented a backtracking search algorithm (BSA) and verified the reliability of the method in solving and modeling the economic dispatch (ED) problem. Aghay Kaboli et al. (2016a, 2016b) estimated electricity demand using GEP, a geneticbased method, as an expressiondriven approach and showed that GEP outperforms the multilayer perceptron neural (MLP) network and multiple linear regression models. Recent studies on time series forecasting largely focus on combination methods given the distinguishing features of hybrid models (e.g., unique modeling capability of each model), drawbacks in using single models, and the resultant improvements in forecasting accuracy. The key concept of combination theory is employing the unique merits of individual models to extract different data patterns. Importantly, the literature confirms that no individual model can universally determine datageneration processes. In other words, all characteristics of underlying data cannot be fully modeled by one model and therefore, combining different models or using hybrid ones helps analyze complex patterns in data more accurately and completely. Further, combining various models simplifies the selection of a model that is appropriate to process different forms of relationships in the data and reduces the risk of choosing an inefficient one.
Several approaches have been proposed to combine linear and nonlinear models. These combination methods are generally divided into two primary classes: series and parallel. In a series combination method, a time series is decomposed into linear and nonlinear parts. Accordingly, in the first stage, the model is used to process one time series component and then, the obtained values are used as inputs for the second model to analyze another component. On the other hand, in the parallel combination method, the original data are simultaneously considered to be inputs for different models and then, the linear combination of the forecasted results facilitates final hybrid forecasting.
Literature on series linear or nonlinear models for time series forecasting
Author(s)  Linear model  Nonlinear model  Year  Field 

Ghasemi et al. (2016)  ARIMA  SVM  2016  Electricity price and load forecasting 
Barrow (2016)  SMA  MLP  2016  Intraday call arrivals forecasting 
Katris and Daskalaki (2015)  FARIMA  MLP  2015  Internet traffic forecasting 
Chaâbane (2014)  FARIMA  MLP  2014  Electricity price forecasting 
Adhikari and Agrawal (2013)  RWM  MLP  2013  Financial time series forecasting energy 
Wang and Meng (2012)  ARIMA  MLP  2012  Consumption forecasting 
Khashei et al. (2012)  ARIMA  PNNs  2012  Time series forecasting 
Nourani et al. (2011)  SARIMAX  MLP  2011  Rainfall–runoff process modeling 
Wu and Chan (2011)  ARIMA  Time delay neural network (TDNN)  2011  Hourly solar radiation forecasting 
Aladag et al. (2009)  ARIMA  Elman’s recurrent neural networks  2009  Time series forecasting 
Bates and Granger (1969) introduced the concept of a parallel combination, which was subsequently used by many researchers such as Makridakis et al. (1982), Granger and Ramanathan (1984), Bunn (1989), and De Menezes and Bunn (1993). Wedding and Cios (1996) proposed a parallel combination model using radial basis function networks and the Box–Jenkins ARIMA model. More recently, several parallel hybrid forecasting models have been proposed to combine linear and nonlinear models. For instance, Wang et al. (2012) presented a parallel hybrid model using GA and by employing ARIMA, exponential smoothing (ES), and back propagation neural network (BPNN) models. Their numerical results showed that the proposed model outperforms all traditional models, including the ESM, ARIMA, BPNN, equal weight hybrid (EWH) model, and random walk (RWM) model. Forecasting stock returns, Rather et al. (2015) proposed a novel hybrid model that merges predictions by three individual models: ES, recurrent neural network (RNN), and ARIMA; the optimum weights of each model are identified using GA. Yang et al. (2016) presented a combined forecasting model using BPNN, adaptive networkbased fuzzy inference system (ANFIS), and SARIMA models, and thus, used a differential evolution metaheuristic algorithm to optimize the weights of a hybrid model. Their experimental case study showed that their proposed method performed better than the three individual methods and had higher accuracy.
In sum, several general conclusions can be drawn from the literature using hybrid models to explore time series forecasting. First, in recent years, there has a growing number of studies investigating the impact of using combination theory on forecasting accuracy; their objective is to enhance forecasting accuracy by combining different models. The numerical results of the reviewed papers evidence that the predictive capability and accuracy of hybrid models are better than those of single models. Moreover, hybrid models have recently become a dominated tool for time series forecasting. Second, scholars have introduced series and parallel combination methodologies to connect the components of hybrid models. However, the question of how to combine single models, that is, which combination yields more accurate results, remains unanswered. In other words, the literature has neglected to compare the two types of hybrid methods to introduce a more accurate one and focused on improving forecasting accuracy by employing hybrid models rather than their constituents. Third, the literature review revealed that among the linear and nonlinear models, ARIMA and MLPs have attracted overwhelming attention and perform well when part of hybrid models given their unique features. ARIMA models are one of the most important forecasting models that have been successfully applied in modeling and forecasting. The popularity of the ARIMA model can be attributed to its statistical properties and the wellknown Box–Jenkins (Box and Jenkins 1976) methodology in the modelbuilding process. The model assumes a linear correlation between the values of a time series and thus, performs well in linear modeling. MLP is the most wellknown artificial neural network that processes nonlinear patterns in data without any assumption and does not require the determination of a model’s form. MLPs are flexible computing frameworks and universal approximators with a high degree of accuracy and can be applied to a wide range of forecasting problems. The key advantage of the neural networks is their flexible nonlinear modeling.
Given that the literature on time series forecasting remains ambiguous on the choice of combination strategy, the core objective of this study is to introduce an effective combination methodology and elucidate how individual models can be combined to improve financial time series forecasting. Accordingly, this study presents a comprehensive discussion on series and parallel combination methods and then, constructs a model using both techniques to combine MLP as a nonlinear model and ARIMA as a linear model. Then, using two combination strategies, ARIMAMLP and MLPARIMA, the series and parallel hybrid models, comprising simple average (SA), linear regression (LR) and genetic algorithm (GA), are compared with their individual components. To evaluate the effectiveness of the hybrid models and introduce a more accurate and reliable hybrid method, two benchmark datasets, the closing of Shenzhen Integrated Index (SZII) and that of Standard and Poor’s 500 (S&P 500), are selected for the forecasting and modeling.
The remainder of this paper is organized as follows. Section Methods presents the basic concepts and modeling procedures of the ARIMA and MLP modes for time series forecasting. Section Series combination method of ARIMA and MLP models and Parallel combination of ARIMA and MLP describe the series and parallel combination techniques and the hybrid models constructed using these methods. Section Results and discussion reports the empirical results of the hybrid series and parallel models for a forecasting benchmark dataset. Section Comparison of forecasting results compares the performance of the models for the forecasting benchmark dataset. Section Conclusions concludes.
Methods
This section introduces the basic concepts and modeling procedures of the ARIMA and MLP models and series and the parallel hybrid methods for time series forecasting.
ARIMA model
Several diagnostic statistics and residual plots can be used to examine the goodness of fit of a tentatively adopted model to the historical data. If the model is deemed inadequate, a new tentative model is identified, which is also subjected to parameter estimation and model verification. Diagnostic information can help determine alternative model(s). This threestep modelbuilding process is typically repeated several times until a satisfactory model is identified. The final model is then used for the prediction.
MLP model
The choice of q is data dependent and there is no systematic rule in deciding this parameter. In addition to choosing an appropriate number of hidden nodes, selecting the number of lagged observations, p, and dimensions of the input vector is an important task in the ANN modeling of a time series. This is, perhaps, the most important parameter to be estimated in an ANN model because it plays a major role in determining the (nonlinear) autocorrelation structure of the time series.
Series combination method of ARIMA and MLP models
ARIMAMLP model
MLPARIMA model
Parallel combination of ARIMA and MLP
 I.
Modeling linear and nonlinear parts of time series using ARIMA and MLP models.
 II.
Calculating weights of obtained values from the previous stage.
 III.
Multiplying two desired weights coefficients to obtain forecasts from stage I and then summing them up.
Assigning the weights of each forecasting model is key to obtaining accurate forecasts using parallel methods because the weights indicate the importance and effectiveness of each individual component in a combined model. In addition, the forecasting results of a combined model with inappropriate weights may be less reliable than those of single models. The next section applies three wellknown weighting approaches, SA, LR and GA, to develop three possible hybrid models.
Simple averagebased hybrid model
Genetic algorithmbased hybrid model
A genetic algorithm is generally applied to solve optimization problems on the basis of a natural selection process that mimics biological evolution. The algorithm repeatedly modifies a population of individual solutions. At each step, the GA randomly selects individuals from the current population and uses them as parents to produce children for the next generation over successive generations; the population evolves toward an optimal solution. Given their ability to solve optimization problems, GAs are frequently used to determine optimum weights when using hybrid models.
Linear regressionbased hybrid model
Results and discussion
Shenzhen integrated index (SZII) dataset
ARIMAMLP hybrid model

Stage I  (Linear modeling): In the first stage of the ARIMAMLP model, Eviews software is used, which identifies ARIMA(1, 0, 0)as the best fit. The stationary test (augmented Dickey–Fuller [ADF] Test) is applied to the SZII time series to test whether a unit root test exists in the ARIMA model. According to the obtained results, ADF test statistic is −3.676528 and the critical value of −3.432 is significant at the 5% level, the null hypothesis that a unit root test exists in the SZII time series is rejected.

Stage II (Nonlinear modelling): To analyze the obtained residuals from the previous stage and based on the concepts of MLP models, in MATLAB software, the best fitted model composed of four inputs, four hidden and one output neurons (in abbreviated form N ^{(4, 4, 1)}), is designed.

Stage III  (Combination): In the final stage, the results obtained from stages I and II are combined. The estimated values of the ARIMAMLP model against the actual values for all data are plotted in Fig. 5.
MLPARIMA hybrid model

Stage I  (Nonlinear modeling): In the first stage of the MLPARIMA model, to capture the nonlinear patterns of a time series, an MLP with three input, two hidden, and one output neuron (abbreviated form: (N ^{ (3,2,1) } )), is designed.

Stage II  (Linear modeling): In the second stage of the MLPARIMA model, the residuals obtained from the previous stage are treated as the linear model. Thus, considering the lags of the MLP residuals as input variables of the ARIMA model, the bestfitted model is ARIMA(2, 0, 2).

Stage III  (Combination): In the final stage, the results obtained from stages I and II are combined. The estimated values of the MLPARIMA model against the actual values for all data are plotted in Fig. 6.
Parallel hybrid models

Stage I  (Linear and nonlinear modeling): Given the basic concepts of the ARIMA and MLP models in forecasting, the bestfit ARIMA and MLP models designed in Eviews and MATLAB software are ARIMA(1, 0, 0)and a onelayer neural network comprising three input, two hidden, and one output neuron (abbreviated form: (N ^{ (3,2,1) } )). Note that different network structures are examined to compare MLP’s performance, and the structure hat reported the best forecasting accuracy for the test data is selected.

Stage II  (Initializing weights): In this step, the optimum weights of the predicted values obtained from the previous stage are determined. Two weights are estimated by the LR model using the OLS approach in Eviews software, GA in MATLAB, and SA weighting approaches.

Stage III  (Combination): In this stage, the final combined forecast is calculated by multiplying two optimal weight coefficients on the forecasts obtained from stage I and then, summing them up. The estimated values of the SA, GA, and LRbased hybrid models against the actual values are plotted in Figs. 7, 8, 9, respectively. The performance of the hybrid models and their components in the train and test datasets to forecast SZII are reported in Table 2. The table shows that, in both datasets, the MLPARIMA series model achieved higher prediction accuracy than the parallel and individual base models.
Performance of models for SZII using train and test datasets
Model  Train  Test  

MAE  MAE  MAPE  RMSE  MAE  MSE  MAPE  RMSE  
ARIMAMLP  212.39  91,955  7.22%  303.24  1082.88  1,915,716  9.52%  1384.09 
MLPARIMA  210.91  86,221  7.23%  293.63  1064.91  1,915,422  10.16%  1383.98 
SAHM  217.78  95,399  7.12%  308.86  1123.87  1,997,323  9.91%  1388.69 
GAHM  215.38  94,577  7.08%  307.53  1102.75  1,969,593  9.79%  1403.42 
LRHM  215.54  94,573  7.09%  307.52  1074.87  1,928,479  9.64%  1388.69 
ARIMA  224.45  99,255  7.30%  315.04  1166.17  2,221,776  10.26%  1490.56 
MLP  215.38  94,577  7.30%  307.53  1102.33  1,974,479  9.80%  1405.16 
Standard and Poor’s 500 dataset
ARIMAMLP series hybrid model

Stage I  (Linear modeling): Similar to the linear modeling phase, ARIMA(1,0,0) is designed and the residuals of this step are used in the next step.

Stage II  (Nonlinear modeling): In this stage, the residuals of the previous step are used as input for the MLP model and a network with three input, five hidden, and one output neuron is fitted to extract the remaining nonlinear structures.

Stage III  (Combination): Here, the forecasted values of previous two stages are combined to generate the final combined forecast. The estimated values for the ARIMAMLP model against the actual values for all data are plotted in Fig. 11.
MLPARIMA series hybrid model

Stage I  (Nonlinear modeling): In the nonlinear modeling phase, a network with three input, three hidden, and one output neurons is designed to capture the nonlinear relationships in the time series generated and the generated residuals are used in the next step.

Stage II: (Linear modeling): In this step, an ARIMA (3, 0, 3) model is fit to process the linear structures that are not modeled by the MLP model.

Stage III: (Combination): In the final step, the forecasted values from stages I and II are combined. The estimated values of the MLPARIMA model against the actual values for all data are plotted in Fig. 12.
Parallel hybrid models

Stage I  (Linear and nonlinear modeling): Similar to the previous section, to capture the linear and nonlinear patterns in the data for the S&P time series, the ARIMA (1, 0, 0) and MLP models with three input, three hidden, and one output neuron are designed.

Stage II  (Initializing weights): In this state, the optimum weights are derived by applying the LR, GA, and SA weighting methods. Note that the OLS approach and GA are designed using Eviwes and MATLAB software.

Stage III  (Combination): According to the modeling procedure for the parallel hybrid models, the combined forecast is made using the values obtained from the previous two stages. The estimated values of the hybrid models based on parallel SA, GA, and LR against the actual values for all data are plotted in Figs. 13, 14, 15, respectively.
Performance of models for S&P 500 using train and test datasets
Model  Train  Test  

MAE  MSE  MAPE  RMSE  MAE  MSE  MAPE  RMSE  
ARIMAMLP  9.94  176.76  0.01%  13.30  9.02  159.33  0.64%  12.62 
MLPARIMA  9.96  180.17  0.01%  13.42  9.20  157.69  0.65%  12.56 
SAHM  9.98  180.69  0.01%  13.44  9.22  164.48  0.65%  12.83 
GAHM  9.97  180.31  0.65%  13.42  9.23  159.33  0.85  12.62 
LSHM  9.97  180.28  0.01%  13.43  8.98  152.55  0.63%  12.35 
ARIMA  10.02  182.21  086%  13.50  9.36  173.85  0.66%  13.19 
MLP  9.97  180.31  0.01%  13.43  9.24  159.34  0.65%  12.62 
Comparison of forecasting results
This section compares the predictive capabilities of the hybrid models constructed by applying the series and parallel combination methods with either of their components, MLP, and ARIMA, using the two abovementioned datasets. The comparative analysis is conducted from two viewpoints: comparison of series and parallel hybrid models and analysis of average percentage improvement in the series and parallel hybrid models in comparison with their components. Two performance indicators, MAE and MSE, are employed to compare the forecasting performance of the hybrid models and their components.
Overall performance of series and parallel models for SZII
Model  Train  Test  

MAE  MSE  MAE  MSE  
Series models  211.65  89,088.05  1073.89  1,915,569.00 
Parallel models  216.23  94,849.02  1100.49  1,965,131.66 
ARIMA  224.45  99,255.51  1166.17  2,221,776.70 
MLP  215.38  94,577.00  1102.33  1,974,479.94 
Overall performance of series and parallel models for S&P 500
Model  Train  Test  

MAE  MSE  MAE  MSE  
Series models  9.95  178.46  9.11  158.51 
Parallel models  9.97  180.42  9.14  158.78 
ARIMA  10.02  182.21  9.36  173.85 
MLP  9.97  180.31  9.24  159.34 
Average improvement in series and parallel models over ARIMA and MLP models for SZII
Hybrid model  ARIMA  MLP  

Train (%)  Test (%)  Train (%)  Test (%)  
MAE  MSE  MAE  MSE  MAE  MSE  MAE  MSE  
Series models  5.70  10.24  7.91  13.78  1.72  5.80  2.57  2.98 
Parallel models  3.65  4.43  5.64  11.55  −1.11  −0.40  0.27  0.47 
Average improvement in series and parallel models over ARIMA and MLP models for S&P 500
Hybrid model  ARIMA  MLP  

Train (%)  Test (%)  Train (%)  Test (%)  
MAE  MSE  MAE  MSE  MAE  MSE  MAE  MSE  
Series models  0.69  2.05  2.66  8.82  0.3  1.02  1.40  1.25 
Parallel models  0.49  0.63  1.45  6.37  0.00  −0.21  1.04  0.52 
Conclusions
Forecasting realworld time series, particularly financial time series, is a critical task that has recently received overwhelming attention. Given the importance of accurate forecasting, several related methods have been proposed in the literature. In addition to single methods, studies have combined different methods to generate more accurate results and confirmed that combining different models enhances forecasting accuracy and accounts for the unique features of individual models. Although numerous studies have used series or parallel methods to construct hybrid models and confirm that combining different models reduces forecasting error and offer more accurate results, they remain vague on the precise combination that produces a more accurate hybrid model. Thus, this study proposed a more efficient technique to forecast financial time series and then, conducted a comprehensive comparison of the predictive capabilities of the series and parallel combination techniques that were combined with linear and nonlinear models, such as ARIMA and MLP, along with their individual components. First, the series and parallel hybrid models were compared, followed by a comparison of the hybrid models’ average percentage improvement with those of their base models. The empirical results for the two benchmark datasets, SZII and S&P 500, indicated that all hybrid models constructed using the two combination methods generated superior results than at least one of their individual components. The results also show that the series method generate more accurate hybrid models and has a higher improvement percentage than the parallel method. Therefore, the series combination method can be considered an efficient alternative to construct more accurate hybrid models in both analytical approaches to forecasting financial time series.
Future works should consider implementing the series and parallel hybrid methodologies to develop an approach with three or more individual models and accordingly, compare and analyze the obtained results. Researchers can also examine other statistical and intelligent models, such as GARCH and SVM models, to construct series and parallel hybrid models to forecast financial time series.
Declarations
Acknowledgements
The authors express their gratitude to Dr. Farimah Mokhatab Rafiei, associate professor of industrial engineering at the Tarbiat Modares University of Tehran, and Dr. Mehdi Bijari, professor of industrial engineering at Isfahan University of Technology, for their insightful and constructive comments, which have helped considerably improve this paper.
Funding
The authors have no funding to report.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors have equally contributed to this work and approve of this submits.
Competing interests
The authors declare that they have no competing interests.
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