An improved grey Markov chain model with ANN error correction and its application in gross domestic product forecasting

Abstract

Gross domestic product (GDP) is the most widely-used tool for measuring the overall situation of a country’s economic activity within a specified period of time. A more accurate forecasting of GDP based on standardized procedures with known samples available is conducive to guide decision making of government, enterprises and individuals. This study devotes to enhance the accuracy regarding GDP forecasting with given sample of historical data. To achieve this purpose, the study incorporates artificial neural network (ANN) into grey Markov chain model to modify the residual error, thus develops a novel hybrid model called grey Markov chain with ANN error correction (abbreviated as GMCM_ANN), which assembles the advantages of three components to fit nonlinear forecasting with limited sample sizes. The new model has been tested by adopting the historical data, which includes the original GDP data of the United States, Japan, China and India from 2000 to 2019, and also provides predications on four countries’ GDP up to 2022. Four models including autoregressive integrated moving average model, back-propagation neural network, the traditional GM(1,1) and grey Markov chain model are as benchmarks for comparison of the predicted accuracy and application scope. The obtained results are satisfactory and indicate superior forecasting performance of the proposed approach in terms of accuracy and universality.

Keywords

Gross domestic product grey Markov chain artificial neural network residual correction forecasting

1 Introduction

The Gross Domestic Product (GDP), which measures aggregate monetary value of all finished goods and services produced by an economy during a given period, serves as one of the most widely acknowledged and extremely important indicator to reflect market economic activity of a nation or region, as well as getting insight into the state of an economy. Hence, GDP forecasting in a timely manner which is very meaningful in guiding government policymakers, enterprises administrators and individuals to capture the development of economy and make the correct economic decision-making, has spurred widespread concerns by many researchers and practitioners in the last few decades. During recent years, with facing various downside risks for the global economics and aggravation of international trade disputes, GDP forecasting of an economy has become increasingly complex and unprecedented urgent.

Study on GDP forecasting has experienced explosive growth in recent years. Up to the present, methodologies used in these studies can roughly be summarized into four types: the traditional statistic model, neural network-based model, grey forecasting model and the combination model. As a set of mature procedures, statistics forecast methods including autoregressive integrated moving average (ARIMA) model, regression, exponential smoothing method etc. Box and Jenkins (1970) [1] initiated ARIMA model which as a typical time series method, involves three main stages namely identification, estimation and diagnostic checking. Gujarati and Porter (2004) [2] emphasized that logarithm of the raw GDP data series is adjusted before it is analyzed by ARIMA method. Through some empirical researches, ARIMA model has been proven to be reliable and accurate model in predicting macroeconomic variable GDP [3 –9]. Besides, the other common statistical prediction methods like multifactor regression [10], quadratic exponential smoothing [11], are also successfully applied into GDP forecasting. Due to the characteristics of nonlinear categorization, along with capacity of fast and accurate modeling, neural network, especially one of the most popular algorithm, i.e. back propagation (BP) neural network received much attention and achieved great results [12], including some concrete application of GDP forecasting [13 –15]. However, it should be noted that both statistical methods and neural network not only require a large sample of historical data but also typical distributions [16], thereby limiting the scope of their practical use.

Grey system theory originated by Deng (1982) [17] is a powerful mathematical tool to settle grey system with partial information known and partial information unknown. A significant advantage of this method embodies it only requires limited discrete data to estimate the behavior of a system with incomplete information and also achieves a certain degree of fitting effect to evaluate the reliability [18, 19]. Cause this strength there emerges a series of studies that applied the grey model to predict GDP in various nations or regions [20 –23]. Inadequacy lies in that the single grey model mainly focuses on the linear distributed data, and noise in the nonlinear data may give rise to significant errors in the estimate. Liu et al. (2020) [24] established a generalized grey forecasting technique named the optimized grey prediction model with time power term (abbreviated as OPTGM (1,1,α)) to predict China’s GDP, which shows higher forecasting accuracy compared with the traditional grey prediction model. But an inherent difficulties of this model lies in the determination of parameter α. Subsequently, great endeavor on merging grey model with other techniques has been devoted to extend the application scopes of nonlinear analysis together with increasing the predicting accuracy. An improved grey prediction model and its variants based on grey model and ANN are put forward in succession, arise in a variety of applications, such as transpacific air passenger market [25], flood and wind speed forecasting [26, 27], industrial control systems [28], air pollutant emission [29]. Some studies extended linear GM(1,1) to nonlinear grey Bernoulli model termed NGBM and applied it in predicting the foreign exchange rates [30], main economic indices of high technology enterprises [31], electricity power generation [32] and GDP forecasting [33, 34]. Another mathematical technique known as Markov chain (1997) [35], that due to its advantage of dealing with randomness, has been incorporated into grey model to enhance the predicted accuracy in recent studies. Wang and Meng (2008) [36] investigated forecasting electricity demand by using Grey-Markov model. Kumar and Jain (2010) [37] employed Grey-Markov model to predict energy consumption in India. Chen (2011)[38] put forward a Grey-Markov model to forecast financial crises for an enterprise. Other studies applied Grey-Markov model in prediction of global ICT development [39], fire accidents [40], aircraft accidental damage [41], equipment failure [42], urban medical services demand [43], developments of GDP, population and energy consumption [44], and demonstrated which can be as a viable alternative with a high prediction precision in various fields. The same inherent defect of Grey-Markov model with GM(1,1) lies in that which is hardly able to deal with nonlinearity cases that widely exist in reality.

To sum up, three major limitations existing in these approaches as mentioned above. First, the traditional regression procedures, neural network technique and their variants requires a large database to guarantee forecasting accuracy. Second, GDP is highly influenced by various external shocks and thus performs instability of growth, especially for some emerging economies. Single grey model is incapable for acquiring accurate predication in the situation of random vibration and high noise. Employing Markov chain model highly requires the calculation of GDP transition probability. GDP transaction features vary across categories and nations, and probability matrix of GDP transaction exhibits difference from one nation to another. Third, both traditional grey model, Markov chain and their integrations focus on (approximate) linear predictions, but them are not good at handling of nonlinearity.

This paper aims to address the limitations outlined above and enhance the precision accuracy in GDP forecasting by developing a new hybrid model which is based on GM (1,1) model, combines Markov chain model and artificial neural network (ANN), named Grey-Markov chain model with ANN error correction. In essence the combined model can be seen as a Grey-Markov model with residual modification by ANN sign estimation. It takes advantage of small sample forecasting of GM(1,1), modelling the randomness on the discretized states of Markov chain, and takes advantage of nonlinear treatment capability of ANN. In order to test prediction accuracy and better adaptability of the presented model, we utilize it to GDP forecasting which can be approximate quantified as a nonlinear forecasting under limited data with random vibration and noise. Four countries consisting of two developed economies, i.e. the United States and Japan, two emerging economies, i.e. China and India, are specialized selected which is due to the feature of GDP trend for each alternative in the period 2000-2019 differs significantly. The details are displayed in Fig. 1 and Table 1, so that these sample data possess typicality with sufficient proof to verify the effectiveness of our proposed model.

Fig. 1

Comparison of the actual real GDP growth rate in the United States, Japan, China and India, 2000-2019. (Original data is collected from open database of the World Bank.)

Table 1

Mean, variance and variance coefficient of GDP growth rate in 2000-2019 of the four countries

Country	Mean	Variance	Variance coefficient	Typical features of economic growth
US	2.1%	2.13e - 04	0.70	medium to low speed with slight fluctuation
JP	0.9%	3.53e - 04	2.08	low speed with significant fluctuation
CN	9.0%	4.47e - 04	0.23	steadily growth at a high speed
IN	6.5%	3.29e - 04	0.28	steadily growth at a medium to high speed

The remainder of the paper is organized as follows. Section 2 contains background material related to grey model and Markov chain, and describes the procedure of the improved grey Markov chain model with ANN error correction. Section 3 states applications to GDP forecasting, presents empirical results and compares with prediction performance of methods included. The final comments and conclusions are provided in Section 4 and 5.

2 Methodology

The improved Grey Markov Chain model with ANN error correction (GMCM_ANN) is assembly in three methods, including GM(1,1) model, Markov chain and Artificial Neural Network (ANN). The purpose of GM (1,1) is to initially fit and predicts GDP in the linear manner by adopting historic data. The technique of ANN is applied to make residual modification in GM(1,1) for adapting nonlinear characteristics of reality. In order to further increase the prediction precision, Markov chain is incorporated in the GM(1,1) with ANN error correction model that revises the estimates by considering the transferring states of previous forecasting sequences.

2.1 The GM(1,1) Model

The GM(1,1) as a mature technique is generally developed as the following procedure.

Assume that the original data series is X⁽⁰⁾ ={ x⁽⁰⁾ (1) , x⁽⁰⁾ (2) , …, x⁽⁰⁾ (n) }, where x⁽⁰⁾ (i) >0, i = 1, 2, …, n. A sequence operator which can weaken the randomness of the source data called one time accumulating generation operation (1-AGO) accumulates the data series and gains a new series, denoted by X⁽¹⁾ ={ x⁽¹⁾ (1) , x⁽¹⁾ (2) , …, x⁽¹⁾ (n) }, satisfying $x^{(1)} (k) = \sum_{i = 1}^{k} x^{(0)} (i)$ , k = 1, 2, …, n.

Furthermore, the next mean generation sequence is defined as: Z⁽¹⁾ (n) ={ z⁽¹⁾ (2) , …, z⁽¹⁾ (n) }, where $z^{(1)} (k) = \frac{1}{2} [x^{(1)} (k - 1) + x^{(1)} (k)], k = 2, \dots, n$ .

Due to the fact that the above monotonically increasing sequence by using 1-AGO is similar to the solution curve of the first order linear differential equation. One can derive the whitenization differential equation in GM (1,1) model as follows: $\frac{d {\hat{x}}^{(1)}}{dt} + α {\hat{x}}^{(1)} = β .$ (1)

Another equivalent expression in GM(1,1) is: $x^{(0)} (k) + α z^{(1)} (k) = β .$ (2)

In the equations above, symbol ∧ represents the grey predicted value, α represents the developing coefficient, and β represents the grey action. Let u = [α, β] ^T be the parameter list, one can obtain its values by using least square method: $u = (α, β)^{T} = (B^{T} B)^{- 1} B^{T} Y .$ (3) Where $Y = [\begin{matrix} x^{(0)} (2) \\ x^{(0)} (3) \\ ⋮ \\ x^{(0)} (n) \end{matrix}]$ , and $B = [\begin{matrix} - z^{(1)} (2) & 1 \\ - z^{(1)} (3) & 1 \\ ⋮ & ⋮ \\ - z^{(1)} (n) & 1 \end{matrix}]$ .

Consequently, the prediction value at step k (k = 1, 2, …, n - 1) is determined by the grey prediction equations: $\begin{matrix} {\hat{x}}^{(1)} (k + 1) = [x^{(0)} (1) - \frac{β}{α}] e^{- k α} + \frac{β}{α} . \\ {\hat{x}}^{(0)} (k + 1) = {\hat{x}}^{(1)} (k + 1) - {\hat{x}}^{(1)} (k) \\ = [x^{(0)} (1) - \frac{β}{α}] (1 - e^{α}) e^{- k α} . \end{matrix}$ (4)

2.2 Grey Markov chain forecasting model

Practical prediction of GDP involves its composition, which is often affected by some unknown and random factors. Such as unexpected supply and demand shocks, seasonal variations etc. Thus, the traditional GM(1,1) model sometimes is inadequate of predicting accurate GDP with original data series. In order to improve the prediction precision of GDP, a brief reviews regarding GMCM in which Markov chain model combined with GM(1,1) model is introduced in the current section. The integrated model will be more competent for tracking the randomness and volatility of macroeconomic total quantity, and performs a higher accurate prediction on GDP compared to the single model GM(1,1). The main procedures of GMCM are outlined in the following.

2.2.1 The partition of transferring

Based on the actual GDP denoted by x⁽⁰⁾ (i) (i = 1, 2, …, n) drawing from the initial data series and the forecasting values ${\hat{x}}^{(0)} (i) (i = 1, 2, \dots, n)$ using GM(1,1). By calculating a ratio of actual value to predicted values, i.e. relative variation ratio, one obtains a non-stationary random series in Markov chain, denoted by s (i), $s (i) = x^{(0)} (i) / {\hat{x}}^{(0)} (i)$ . Therefore, this random series s (i) can be divided into different statuses by equidistant partition. We partition series s (i) into m segments using ${\hat{x}}^{(0)} (i)$ curve as center, denoted by {Q₁, …, Q_m}, and each segment Q_j represents a state j. Each s (i) can be located into one corresponding status j satisfying s (i) ∈ Q_j, which Q_j is defined as: $Q_{j} = {\begin{matrix} j = m . \end{matrix}$ (5) Where L_j, U_j indicates the low bound and upper bound of state in ratio sequence, respectively. Which are computed as: $\begin{matrix} L_{j} = min s (i) + \frac{j - 1}{m} [max s (i) - min s (i)], \\ U_{j} = min s (i) + \frac{j}{m} [max s (i) - min s (i)] . \end{matrix}$ (6)

2.2.2 The establishment of the state transition probability matrix

We use the symbol P_ij (k) to denote the probability of ratio transferring from state i to j in step k, in which P_ij (k) = M_ij (k)/M_i. Where M_ij (k) represents the sample size of the ratio transferring from state i to j in step k, and M_i is the sample size of the ratio state i.

Hence the Markov state transition probability matrix, denoted by P (k), with its entries P_ij (k) can be expressed as: $P (k) = [\begin{matrix} P_{11} (k) & P_{12} (k) & \dots & P_{1 m} (k) \\ P_{21} (k) & P_{22} (k) & \dots & P_{2 m} (k) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ P_{m 1} (k) & P_{m 2} (k) & \dots & P_{mm} (k) \end{matrix}] .$ (7)

2.2.3 Identification of the forecasting value

Let π_j (t) be probability of state in period t, a state transaction vector with regard to sequence of periods is derived, denote by (π₁ (t) , π₂ (t) , …, π_m (t)). In addition, π_j (0) indicates the initial probability vector of the state transaction, utilizing of the initial probability vector and the state transaction probability matrix, to ascertain the probability vector of forecast objects in any period at any state. A hybrid model of prediction with mixture of GM(1,1) and Markov chain is given by: ${\tilde{x}}^{(0)} (t + 1) = \frac{1}{m} \sum_{j = 1}^{m} [π_{j} (t) \otimes Q_{j}] \cdot {\hat{x}}^{(0)} (t + 1) .$ (8)

Where π_j (t) = π_j-1 (t) P (1), π_j (t) ⊗ Q_j = π_j (t) · L_j + (1 - π_j (t)) U_j. In essence ${\tilde{x}}^{(0)} (t + 1)$ means product of the expected value of relative variation ratio and the predicted values by GM(1,1). Actually it further extracts information of data transfer so as to improve the precision of forecasting. More details can be refer to [45].

2.3 The improvement of grey Markov chain model with ANN error correction

The existing of error between raw data and forecasting is unavoidable. Hence, minimizing the error is a practicable way to increase forecasting precision. In this section we are going to propose a new method by using ANN corrects the error to develop the grey Markov chain prediction model. The original forecasting value is obtained by GM(1,1), followed by one calculates the estimates forecast error and updates predictive values the using ANN, and finally investigates information of states transfer by Markov chain to obtain the estimated forecast values. In order to implement the proposed model, the procedure of obtaining the forecasted values by GMCM_ANN is described in the following, showed as Fig. 2:

Collect the original series of data and divide into two groups, i.e. the training subset and testing subset. The former is used to train the forecasting model, and the latter is employed to test performance of forecasting model.

Use the grey forecasting model GM(1,1) to train the training samples, then detection effect is validated by testing data, and compute the predictive values.

Obtain the residual series. Let {e⁽⁰⁾ (1) , e⁽⁰⁾ (2) , …, e⁽⁰⁾ (j) , …, e⁽⁰⁾ (n)} be the residual series, where $e^{(0)} (j) = x^{(0)} (j) - {\hat{x}}^{(0)} (j)$ , x⁽⁰⁾ (j), ${\hat{x}}^{(0)} (j)$ denotes the real values and forecasted values in the period j, respectively. Then standardize the residual series and compute the standardized error denoted by ɛ⁽⁰⁾ (j) in the following: $ɛ^{(0)} (j) = \frac{e^{(0)} (j) - {min}_{j} e^{(0)} (j)}{{max}_{j} e^{(0)} (j) - {min}_{j} e^{(0)} (j)} .$ (9)

Training the ANN model based on the training subset of the standardized error sequence. Let Γ be the prediction order, let ɛ⁽⁰⁾ (j - 1) , …, ɛ⁽⁰⁾ (j - Γ) (j = Γ, …, n) be defined as input samples, ɛ⁽⁰⁾ (j) be defined as the expected value of forecast by ANN. In this study we set Γ = 3, set learning rate with 0.05 and maximum training times with 5000 for generating neural network. And run twenty experiments on the training data for ANN error correction and select the optimal fitting results as the outputs of ANN error correction.

Compute the forecasting values with ANN error correction. It can be expressed as: ${\tilde{x}}^{(0)} (j) = {\hat{x}}^{(0)} (j) + {\tilde{e}}^{(0)} (j) .$ (10) Where ${\tilde{x}}^{(0)} (j)$ , ${\tilde{e}}^{(0)} (j)$ denotes the predictive value of ${\hat{x}}^{(0)} (j)$ and fitting residual in period j by GM-ANN, respectively. And ${\tilde{e}}^{(0)} (j) = ɛ^{(0)} (j) \cdot [max_{j} e^{(0)} (j) - min_{j} e^{(0)} (j)] + min_{j} e^{(0)} (j)$ .

Construct a data sequence {s (Γ) , … , s (n) } with its entries $s (j) = x^{(0)} (j) / {\tilde{x}}^{(0)} (j)$ , indicate the ratio of actual value to predictive value. And partition the sequence into m kinds of states using Equation 5–6. In this study we set m = 3.

Ascertain the state transition probability matrix, and train the forecasting model incorporated in Markov chain with respect to training subset of {s (Γ) , … , s (n) }. By applying Equation 8, one obtains the final predictive values of the improved GMCM_ANN.

Fig. 2

Algorithm flowing chart for GMCM_ANN.

Note that GMCM_ANN which we put forward primarily embodies its advantages in three aspects. First, it makes prediction without the restriction of sample size and is qualified to solve the problems of insufficient number of training samples. Second, the traditional GM(1,1) model is as a linear grey forecasting model, and Markov chain mainly adapts to stationary objects, while GMCM_ANN extends the linear grey forecasting to nonlinear forecasting, as well the novel model can deal with non-stationary objects. Third, GMCM_ANN can modify the volatility and noise in prediction. The new method fully plays their respective advantages GM, MC and ANN, can extend the applicable scopes to nonlinear dynamic information prediction and better improve performance of predicated accuracy. To quantitative verify the effectiveness of GMCM_ANN, the following section applies it with empirical data, and compares the performance with other forecasting models.

3 Empirical experiment

3.1 Data

We use time series with annal data which is with regard to two developed countries United States (US) and Japan (JP), two developing countries China (CN) and India (IN), spanning the years 2000 through 2019, to verify the proposed model. Note that GDP is the market value of all of the final goods and services produced within the country’s borders in a one-year period. It is commonly regarded as the most important indicator to measure the national economic development, and has the features that include approximate linear growth rate, multiple causes, and influence of other random factors. The data used in this paper is collected from open database of the World Bank. Data from 2000 to 2015 is used to train the forecasting models, and the rest periods as testing set that is employed to test the performance of different forecasting models. Original data of GDP from 2000 to 2019 are listed in the Appendix.

3.2 Procedure

To assessing forecast performance of the proposed method GMCM_ANN, four forecasting methods, i.e. autoregressive integrated moving average model (ARIMA), Back-Propagation neural network (BP), the traditional GM(1,1) and grey Markov chain prediction Model (GMCM) are as benchmarks for comparison. By importing the original data into these models as mentioned above separately, we successively investigate GDP forecasting of four selected countries. Matlab software package (2010) is employed to implement these models and derives the corresponding estimated results. Plots of the actual data and the predicted values of GDP are contained in Figs 3–6. These blue line reflect the actual GDP situations of various countries, the brown line, red line, yellow line, purple line and green line represents the predictions which are calculated by ARIMA, BP, GM, GMCM and GMCM_ANN, respectively. It is marked in each figure below that one takes 2000-2015 as models training segment, 2016-2019 as models testing segment to examine prediction accuracy of each model.

Fig. 3

Forecasting GDP of the United States by ARIMA, BP, GM(1,1), GMCM and GMCM_ANN.

Fig. 4

Forecasting GDP of Japan by ARIMA, BP, GM(1,1), GMCM and GMCM_ANN.

Fig. 5

Forecasting GDP of China by ARIMA, BP, GM(1,1), GMCM and GMCM_ANN.

Fig. 6

Forecasting GDP of India by ARIMA, BP, GM(1,1), GMCM and GMCM_ANN.

The test focuses on three common evaluation criteria, including relative percentage error (RPE), mean absolute percentage error (MAPE) and root mean square error (RMSE), to compare the five models’ prediction performance in these criteria. These formulas are defined as follows: $\begin{matrix} {RPE}_{i} = \frac{{\hat{x}}^{(0)} (i) - x^{(0)} (i)}{x^{(0)} (i)} \times 100 %, \\ MAPE = \frac{100 %}{n} \sum_{i = 1}^{n} | {RPE}_{i} |, \\ RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {({\hat{x}}^{(0)} (i) - x^{(0)} (i))}^{2}} . \end{matrix}$ (11)

For predicting GDP of the United States in Fig. 3, it is straightforward to conclude that BP has the worst fitting performance whether in training segment or testing segment due to the significant errors. Forecasting GDP of China in Fig. 5 and India in Fig. 6 also show the same features with accuracy. But forecasting GDP of Japan in Fig. 4 displays a slightly difference that among ARIMA, BP and GM have indistinguishable unsatisfactory performances. Compare prediction performances of the five models by taking into account the testing period, it can be seen that the result by GMCM_ANN has the slightest errors on the whole, subsequent to the GMCM, while the ARIMA, GM and BP have significant residuals. Due to the fact it indicates the hybrid models GMCM and GMCM_ANN have performed more accurate prediction than the rest three single models with remarkable improvements in precision. Especially GMCM_ANN further revising the forecasting results by GMCM to achieve the highest accuracy. The experiment results proves the validity of the proposed method. However, It can be seen roughly that the fitting degrees by GMCM_ANN in Figs. 3–4 are higher than the fitting degrees in Figs. 5–6. Other four models also encounter the common feature of lower accuracy in predicting GDP of China and India. This phenomenon shows that prediction accuracy of models depends on the samples, and more difficulties arise in predicting GDP of China and India rather than the United States and Japan.

3.2.1 Results

It is customary that testing data is used to compare the forecasting accuracy of different models. According to the corresponding procedures above for various models, these estimated values and RPE for GDP of four selected countries during the testing period (2016-2019) are presented. The concrete results are listed in Table 2 as below.

Table 2
The comparison of forecasting results obtained by five models.(Units: Billion US Dollars)

Country Year GDP ARIMA RPE BP RPE GM RPE GMCM RPE GMCM_ANN RPE

US 2016 16972 17006 0.22% 17641 3.9% 16820 -0.9% 16989 0.1% 17075 0.6%

2017 17349 17229 -0.66% 17930 3.4% 17104 -1.4% 17275 -0.4% 17522 1.0%

2018 17856 17391 -2.52% 18179 1.8% 17394 -2.6% 17568 -1.6% 17843 -0.1%

2019 18300 17566 -4.0% 18416 0.6% 17688 -3.3% 17865 -2.4% 18070 -1.3%

JP 2016 6025 6010 -0.3% 5941 -1.4% 5979 -0.8% 5981 -0.7% 6021 -0.1%

2017 6141 6032 -1.8% 6020 -2.0% 6017 -2.0% 6019 -2.0% 6072 -1.1%

2018 6190 6049 -2.3% 6116 -1.2% 6055 -2.2% 6058 -2.1% 6111 -1.3%

2019 6211 6079 -2.1% 6182 -0.5% 6094 -1.9% 6096 -1.8% 6133 -1.3%

CN 2016 9491 8956 -5.6% 8800 -7.3% 10182 7.3% 10078 6.2% 9936 4.7%

2017 10132 9397 -7.3% 8810 -13.1% 11158 10.1% 11044 9.0% 10666 5.3%

2018 10797 9808 -9.2% 8814 -18.4% 12227 13.2% 12102 12.1% 11184 3.6%

2019 11520 10185 -11.6% 8816 -23.5% 13399 16.3% 13262 15.1% 12162 4.5%

IN 2016 2482 2437 -1.8% 2439 -1.7% 2434 -2.0% 2440 -1.7% 2446 -1.5%

2017 2660 2577 -3.1% 2502 -6.0% 2598 -2.3% 2604 -2.1% 2616 -1.7%

2018 2842 2738 -3.6% 2522 -11.3% 2773 -2.4% 2779 -2.2% 2789 -1.9%

2019 2940 2910 -1.0% 2527 -14.1% 2960 0.7% 2966 0.9% 2992 1.8%

Country	Year	GDP	ARIMA	RPE	BP	RPE	GM	RPE	GMCM	RPE	GMCM_ANN	RPE
US	2016	16972	17006	0.22%	17641	3.9%	16820	-0.9%	16989	0.1%	17075	0.6%
	2017	17349	17229	-0.66%	17930	3.4%	17104	-1.4%	17275	-0.4%	17522	1.0%
	2018	17856	17391	-2.52%	18179	1.8%	17394	-2.6%	17568	-1.6%	17843	-0.1%
	2019	18300	17566	-4.0%	18416	0.6%	17688	-3.3%	17865	-2.4%	18070	-1.3%
JP	2016	6025	6010	-0.3%	5941	-1.4%	5979	-0.8%	5981	-0.7%	6021	-0.1%
	2017	6141	6032	-1.8%	6020	-2.0%	6017	-2.0%	6019	-2.0%	6072	-1.1%
	2018	6190	6049	-2.3%	6116	-1.2%	6055	-2.2%	6058	-2.1%	6111	-1.3%
	2019	6211	6079	-2.1%	6182	-0.5%	6094	-1.9%	6096	-1.8%	6133	-1.3%
CN	2016	9491	8956	-5.6%	8800	-7.3%	10182	7.3%	10078	6.2%	9936	4.7%
	2017	10132	9397	-7.3%	8810	-13.1%	11158	10.1%	11044	9.0%	10666	5.3%
	2018	10797	9808	-9.2%	8814	-18.4%	12227	13.2%	12102	12.1%	11184	3.6%
	2019	11520	10185	-11.6%	8816	-23.5%	13399	16.3%	13262	15.1%	12162	4.5%
IN	2016	2482	2437	-1.8%	2439	-1.7%	2434	-2.0%	2440	-1.7%	2446	-1.5%
	2017	2660	2577	-3.1%	2502	-6.0%	2598	-2.3%	2604	-2.1%	2616	-1.7%
	2018	2842	2738	-3.6%	2522	-11.3%	2773	-2.4%	2779	-2.2%	2789	-1.9%
	2019	2940	2910	-1.0%	2527	-14.1%	2960	0.7%	2966	0.9%	2992	1.8%

Taking absolute RPE as a yardstick to measure prediction performance of different models, several stylized facts can be revealed from Table 2. First, in aggregate the mixture models are more accurate in contrast to the single component model. Second, the model fitting results indicates GMCM_ANN with error correction has lower prediction error than GMCM in most cases, such as in the GDP forecasting of Japan, China and India. Third, the estimated results by GMCM is casually closer to the real data than GMCM_ANN, like forecasting US GDP in 2016-2017. However, to forecast US GDP in 2018, the absolute RPE for GMCM suddenly rises from 0.4% to 1.6%, while the absolute RPE for GMCM_ANN significantly narrows from 1% to 0.1%, the latter performs more stability of prediction accuracy with less fluctuation of RPE.

For a better comparison, MAPE and RMSE as two crucial guidelines are on assessing forecasting accuracy of the four approaches to formally demonstrate the validity of GMCM_ANN model. The results of the comparison based on the testing period (2016-2019) are presented in Table 3 and Fig. 7.

Fig. 7

The comparison of fitting results by five models.

As shown in Table 3 and Fig. 7, GMCM_ANN has the highest imitation with the original testing data. Comparing with the three single models ARIMA, BP and GM(1,1), GMCM_ANN has sharply reduced 61.17%, 69.96% and 64.56% in MAPE for forecasting US GDP. Comparing with the mixture model GMCM, GMCM_ANN also has significant declined 35.40% in MAPE for forecasting US GDP. Similar patterns also appear in the GDP forecasting of other three countries. Investigation of RMSE also supports the same conclusions. These results are able to prove the validity of the proposed model, which has a better superiority of prediction precision by comparison among four remaining methods. Consequently, it is reasonable to expect that the predicted result of each country’s GDP in 2020-2022 calculated by GMCM_ANN are good simulations of the actual values with lower forecast error. The concrete results are displayed in Table 4 as below.

Table 3

The prediction precisions by five mentioned models

Model	US		JP		CN		IN
	MAPE	RMSE	MAPE	RMSE	MAPE	RMSE	MAPE	RMSE
ARIMA	1.88%	439.18	1.61%	111.03	8.41%	946.98	2.41%	71.79
BP	2.43%	474.99	1.26%	83.85	15.54%	1835.31	8.26%	274.03
GM(1,1)	2.06%	409.87	1.71%	110.93	11.74%	1332.85	1.85%	53.26
GMCM	1.13%	263.62	1.67%	108.86	10.60%	1216.08	1.74%	49.03
GMCM_ANN	0.73%	152.94	0.93%	65.34	4.51%	511.09	1.68%	46.68

Table 4

The prediction results for the four countries’ GDP by GMCM_ANN.(Units: Billion US Dollars)

Year∖Country	US	JP	CN	IN
2020	18656	6253	12162	3125
2021	19069	6294	13054	3424
2022	19441	6308	13949	3633

4 Discussion

The more accurate predicting of future GDP trend is obviously of significant importance to government, corporation and individual decision makers. In this study we employ the proposed model GMCM_ANN and the other four models ARIMA, BP, GM(1,1), GMCM as reference models in GDP forecasting of the United States, Japan, China and India. The empirical results show that GMCM_ANN has superior accuracies for GDP forecasting with different countries. Then we obtain the new predicted values of each country in 2020-2022 by using GMCM_ANN. From the foregoing, some suggestions are provided to be considered in further GDP forecasting.

As the world’s two biggest emerging economies, both China and India are growing at rates noticeably faster than the rest two mature economies the United States and Japan in recent years. And the two exhibit a very obvious nonlinear growth characteristics. The main driving forces could be attributed to these development phases also comparatively primary, rich human resources, huge domestic consumption market, and market-oriented economy system innovation etc. Partly because of that, predicting GDP of China and India are likely to be fraught with more difficulties, as well as a relatively low accuracy.

It should be noted that due to insufficient training model exists in the period 2000 - 2015 which is limited by sample size of database. This results in a larger fitting error in the fitting curve of GDP in the testing period by applying the single prediction models ARIMA, BP and GM(1,1). The prediction model based on BP is inappropriate for GDP forecast because data does not appear as S-type fluctuation characteristics. And the prediction ARIMA and GM(1,1) models also has poor performances because of limited sample size and non-linear economic growth for the four countries, especially for the two emerging economies.

In order to modify the grey forecasting, the GMCM model by considering the interrelationship between the states of time series through using Markov state transition probability matrix, which is tentatively fitted to the data and certainly improves the overall prediction accuracies in our experiment. But inadequate lies in that the model only captures (approximate) linear and stationary information.

The hybrid model GMCM_ANN overcomes the shortage of unsuitability for nonlinear sampled-data in the GM(1,1) model, and weakens the volatility and noisiness of data by using Markov chain, the artificial neural network (ANN) is incorporated in the GMCM error correction model to further achieve reliable and precise predicted results. Taking advantage of three involved components, the GMCM_ANN model has exhibited good adaptability for limited data and various volatilities. With using the same group of historical data of GDP, the proposed model has a better prediction than ARIMA, BP, GM(1,1) and GMCM models. In terms of the noisiness and volatility of databases in GDP forecasts, the GMCM_ANN model is more stable and reliable for predicting economic growth trends. This model enriches methodologies for forecasting that can be easily extended to settle down other prediction problems with limited and volatility data.

5 Conclusions

Grey predicting only requires limited sample size and has sufficient capacity to dealing with incomplete and uncertain information. Markov chain is accustomed used for extraction rule from data with randomness. ANN is an approximate simulation of biologic nerve system so that has great nonlinear processing capability to adapt to reality. Therefore, a combination prediction model consisted of GM(1,1), Markov chain and ANN is capable for GDP forecasting based on the raw data series with fluctuation and noise, as well as achieving high predicting accuracies. This paper presents a new hybrid prediction model abbreviated as GMCM_ANN with ANN error correction for GDP forecasting of the United States, Japan, China and India. The idea of correcting prediction error in advance makes the new technique flexible in enhancing predicting accuracy. The obtained results in empirical experiment validate the feasibility and effectiveness of the proposed method.

It is worth emphasizing that the development of GMCM_ANN as an assembly of three different models, has a relatively complex process so as to require a large amount of calculation and time consumption. And in reality economic aggregate of one country is affected by multiple influences, accordingly the data form change from single time series data into panel data, even mixed-frequency data. Hence it is often not enough to be convincing that GDP forecast only depends on historical GDP series. Further research includes developing the proposed approach to multi-factor prediction model with a variety of data types, and focusing on comparing predictive capability and precision in different economic scenarios, along with recommending some constructive policy suggestions for improving economic growth.

Footnotes

Acknowledgments

The work was supported by the Research Funds of Collaborative Innovation Center for Urban Industries Development in Chengdu-Chongqing Economic Zone, China (Grants No. 950218032). The authors are highly grateful to the editor and anonymous referees for their valuable comments and constructive suggestions which help to improve the present form of this paper.

Appendix

Table 5

Original data from 2000 to 2019. (Units: Billion US Dollars)

Year∖GDP	US	JP	CN	IN
2000	12620	5349	2232	873
2001	12746	5371	2418	915
2002	12968	5377	2639	950
2003	13339	5459	2904	1025
2004	13846	5580	3198	1106
2005	14332	5672	3562	1194
2006	14742	5753	4015	1290
2007	15018	5848	4586	1389
2008	14998	5784	5029	1432
2009	14617	5471	5502	1544
2010	14992	5700	6087	1676
2011	15225	5694	6669	1763
2012	15567	5779	7193	1860
2013	15854	5894	7751	1978
2014	16243	5916	8317	2125
2015	16710	5989	8892	2295
2016	16972	6025	9491	2482
2017	17349	6141	10132	2660
2018	17856	6190	10797	2842
2019	18300	6211	11520	2940

Source: The World Bank Open Data.

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