Research on GDP Forecast of Ji’an City Based on ARIMA Model

In this paper, based on the GDP of Ji’an City from 1978 to 2018, we use the ARIMA(0, 2, 1) model to forecast the GDP of Ji’an City from 2019 to 2023. The prediction results can provide scientific reference for Ji’an City to formulate economic development goals.


Introduction
Since the reform and opening up, China has made remarkable achievements in economic development. The living standards of the people are constantly improving, and we are heading toward the goal of completing the building of a moderately prosperous society in all respects. We can feel the rapid development of China's economy in our daily life, but from the national level, we need specific data to analyze the overall state of the national economy. Gross domestic product (GDP) refers to the market value of all final products produced by all resident units of a country or region within a certain period of time. GDP not only plays an irreplaceable role in reflecting a country's national income, consumption capacity and economic development, but also helps people understand the economic condition of a country or region from a macro perspective. It is the key basis for the formulation of national or regional economic policies, as well as a key means to test whether economic policies are effective and scientific. Therefore, if we use appropriate statistical methods to reveal the law of GDP data changes and make high-precision prediction of short-term GDP, then it will be of great practical significance to the overall planning of macro-economy.
Ji'an City, located in central Jiangxi Province, is a famous Red City on the In recent years, more and more scholars have realized the importance of accurately forecasting GDP. They used a variety of methods to forecast GDP from multiple perspectives. The main forecasting methods could be divided into four categories: regression analysis (Jin, 2011;Chen, 2012), gray prediction (Li & Li, 2010;Tian & Liu, 2018;Zhang & Xie, 2019), artificial neural network (Huang, 2007;Zhao, 2017;He, Wu, & Xia, 2020) and ARIMA model (Li & Xue, 2013;Zhou, 2015;Yan, 2018). ARIMA model is popular because of its simple operation and high precision. It is one of the most commonly used methods to forecast GDP. Li and Xue (2013)  This paper is organized as follows. In Section 2, we give some definitions and ARIMA model modeling steps that will be used in this paper. We use the steps 2) The mean function does not change with time. , 0, 1, 2, 3) The autocovariance function only depends on the time interval, that is , , Autocorrelation function and autocorrelation function describe the degree of correlation between a random event at two different times. In other words, it describes the impact of one's historical behavior on the current situation. However, this kind of correlation is not pure, because when the degree of correlation between t X and t k X − is measured by calculating the lag k-step autocorrelation function a, the influence of k − 1 random variable actually doped. Therefore, we introduce the concept of partial autocorrelation function.
It can be seen that the partial autocorrelation characterizes the degree of correlation between t X and t k X − that is not affected by White noise is a sequence of uncorrelated random variables with equal zero mean variances. It is the simplest and most basic stationary time series. Its importance lies in the fact that white noise is the "generator" of many important models or sequences. Obviously, the autocovariance function k γ and the auto-
2) Data preprocessing. If the time series data studied is non-stationary, the data must be preprocessed, such as logarithm operation, difference operation and so on, so that the data can be transformed into stationary. Otherwise, move on to the next step.
3) Check whether a stationary time series is a white noise series. If it is, it means that the data does not have any analytical value and the analysis should be stopped; if it is not, then the next step of modeling can be carried out. 4) ARIMA model structure selection. The autocorrelation and partial autocorrelation function graphs of stationary time series are obtained by using statistical analysis software. The model structure is judged by observing the specific characteristics of these two images. The judgment basis is shown in Table 1.
As the observation of images is subjective, the judgment may not be so accurate. Therefore, when determining the structure of the model, it is appropriate to select several values around the determined p value and q value to construct the ARIMA model. The predicted value is obtained by substituting the known data into the relation established by the above model.
The visual operation process of the ARIMA model modeling steps is shown in Figure 1.

Data Selection and Preprocessing
In this article, we use the Ji'an Yearbook released by Ji'an City as the data source, and select the Ji'an City GDP data from 1978 to 2018 as a sample to establish an ARIMA model to predict the Ji'an City GDP from 2019 to 2021.

Stationarity Test
According to the data in Table 2, we use MATLAB software to draw the GDP time series diagram of Ji'an City from 1978 to 2018, as shown in Figure 2. It can be clearly seen that the series has an exponential increasing trend and is a non-stationary time series.
In order to eliminate this trend of exponential increase, we perform logarithmic operations with base e on the original data, and records the processed sample time series as lnGDP, and then use MATLAB again to make a sequence diagram of the new series obtained after taking the logarithm. As shown in Figure   3.
It can be seen from Figure 3 that the upward trend of time series lnGDP is more gradual, but still presents a linear upward trend. The first-order difference of the time series lnGDP is denoted as ∇lnGDP, and its sequence diagram is shown in Figure 4.     It can be seen from the image that the sequence obtained by the first difference always oscillates around a certain constant value. It follows that the new sequence is roughly stable. In order to avoid the interference of subjective consciousness and obtain more rational and accurate conclusions, it is necessary to continue to carry out the unit root test, which has become the most widely used  The results show that, ∇lnGDP is a non-stationary series. Therefore, in order to obtain the stationary sequence, we still need to do further difference operation. The second-order difference is made for the time series lnGDP, which is denoted as ∇ 2 lnGDP. The time sequence diagram is shown in Figure 5.
From Figure 5, it can be preliminarily judged that the second-order difference ∇ 2 lnGDP of the time series lnGDP is stationary. Through the ADF stationarity test of the sequence ∇ 2 lnGDP, the MATLAB program running result is which confirms the stationarity of ∇ 2 lnGDP.

Model Recognition and Order Determination
Through the above stationarity analysis, the logarithmic difference order d = 2 has been determined. In order to determine the p and q in the ARIMA(p, d, q) model, autocorrelation and partial autocorrelation analysis are performed on the second-order difference sequence ∇ 2 lnGDP. The results are shown in Figure 6 and Figure 7. It can be seen from the figure that the autocorrelation function graph is truncated after order 1, and the partial autocorrelation function graph is tailed. According to the characteristics of autocorrelation function and partial autocorrelation function, ARIMA(0, 2, 1) is considered for fitting.   It is not accurate to determine the values of P and Q only through Figure 6 and Figure 7. According to the BIC criterion, the BIC function values of a limited number of models are investigated within a certain range, and then the relative optimal order is selected. The corresponding BIC values of each model are shown in Table 3.   According to BIC criterion, BIC value is the smallest when p = 0 and q = 1, which is consistent with the above image analysis results. Therefore, the optimal fitting model of time series ∇ 2 lnGDP is ARIMA(0, 0, 1), that is, the optimal fitting model of time series lnGDP is ARIMA(0, 2, 1).