The probability distribution of a smooth time series process is independent of the displacement in time. If an arbitrary set of random variables is taken from the sequence and the sequence is shifted forward h times, its joint probability distribution remains unchanged ( Yu, 2022 ).

The first-order AR model, AR (1), is modeled as follows:

$r_t={\alpha }_1{r}_{t-1}+w_t$(1)

When ${\alpha }_1<1$, the effect of the previous moment’s fluctuation on the present moment’s fluctuation will be smaller and smaller. Based on this, the covariance as well as the autocorrelation coefficient of this model calculated in this paper it is a fixed value, i.e., smooth at this point.

For higher-order AR models, this paper still uses a similar analytical approach ( Guan, 2021 ), e.g., an AR (P) order model is as follows:

$r_t={\displaystyle {\sum }_{i=1}^p{\alpha }_i{r}_{t-1}}+w_t$ (2)

According to the above model formula, when ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_i$ are all less than 1, then this sequence is considered to be smooth in this paper, however, if any of ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_i$ is greater than or equal to 1, then this sequence is considered to be not smooth.

To determine whether in the model is smooth, that is, to determine whether all are ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_i$ less than 1, this time to use the characteristic equation of the AR model:

$1-{\alpha }_{1}x-{\alpha }_2{x}^2-\cdots -a_p{x}^p=0$(3)

There are p roots in this equation.

Going to test whether a particular AR series is smooth means testing whether one of the roots of that equation is greater than or equal to 1.

The equation form of the ARCH model is as follows:

Mean value equations: $y_t={\beta }_0+{\beta }_1{x}_{1t}+\cdots +{\beta }_k{x}_{kt}+{\epsilon }_t$.

Variance model: ${\sigma }_t^2={\alpha }_0+{\alpha }_1{\epsilon }_{t-1}^2$.

The variance of ${\epsilon }_t$ in the above equation, which depends only on the interference term in the immediately preceding moments, is also the process of ARCH (1), and if the interference term to which it is subjected is generalized, the process of ARCH (q) is obtained as ( Ji & Yang, 2021 ):

${\sigma }_t^2={\alpha }_0+{\alpha }_1{\epsilon }_{t-1}^2+{\alpha }_2{\epsilon }_{t-2}^2+\cdots +{\alpha }_q{\epsilon }_{t-q}^2$(4)

In this equation, $y_t$ represents the explained variable at moment t. ${\epsilon }_t$ is the momentary disturbance term, $I_t$ is the set of information at moment t, and ${\sigma }_t^2$ is the conditional variance of ${\epsilon }_t$. It is generally accepted that when $\omega$ is greater than 0, ${\alpha }_i$ is greater than 0, and ${\displaystyle \sum {\alpha }_i}$ is less than 1 ( $i=1,2,\cdots ,q$), this represents a smooth process of ARCH. In this model, ${\epsilon }_t$ obeys ARCH (q) process. And the ARCH model can reflect the volatility aggregation of the stock market ( Tang & Ju, 2020 ).

The generalized autoregressive conditional heteroskedasticity model, i.e., GARCH model, which introduces its own lagged value in the decision model of the conditional variance ${\sigma }_t^2$ of the current period of ${\epsilon }_t$, is defined as follows:

${\epsilon }_t=e_t{\sigma }_t$(5)

${\sigma }_t^2=\omega +{\alpha }_1{\epsilon }_{t-1}^2+{\alpha }_2{\epsilon }_{t-2}^2+\cdots +{\alpha }_q{\epsilon }_{t-q}^2+{\beta }_1{\sigma }_{t-1}^2+\cdots +{\beta }_p{\sigma }_{t-p}^2$(6)

The closer the value of the model coefficients in this model, i.e., ${\displaystyle {\sum }_{i=1}^q{\alpha }_i}+{\displaystyle {\sum }_{j=1}^p{\beta }_j}$, is to 1, the greater the volatility of the entire series. When the value is greater than 1, it indicates that the effects of this shock will persist and spread. When the sum of the values is less than 1, it means that the impact of the shock will gradually dissipate ( Hu & Xing, 2022 ). The simplest GARCH model is the standardized GARCH (1, 1) model:

$y_t={\beta }_0+{\beta }_1{x}_{1t}+\cdots +{\beta }_k{x}_{kt}+{\epsilon }_t$(7)

${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$(8)

The data selected for this paper is the daily closing price of as a sample, and the source of the data is Invesco.com. CSI 300 for the CSI 300 index from January 4, 2010-December 31, 2020 after constant logarithmic difference processing of daily closing prices：

$R_{t_i}=\ln y_{t_i}-\ln y_{t_i-1}$ (9)

where $R_{t_i}$ represents CSI the return of with 3,002,676 data and $y_{t_i}$ is the corresponding closing price data.

The mean value of the logarithmic return of CSI 300 index in the sample is 0.000145, the median is 0.000345, the maximum value is 0.064989, the minimum value is −0.091544, and the standard deviation is 0.014608. The skewness is −0.693074, and the kurtosis of left skewness of the series is 7.918043, which is bigger than the kurtosis of the normal distribution, which indicates that the distribution of the data shows the state of “sharp peak and thick tail” with left skewness. This indicates that the data distribution shows “sharp peaks and thick tails” with left skewness. At the same time, the JB normality test is conducted, and the final value of the statistic is 2910.015 and significant, which indicates that the distribution of CSI 300 index returns is significantly different from the normal distribution, according to which this paper carries out the next step of the smoothness test and so on.

The results of the unit root test for the variable m using Eviews 10 software are shown in the table below, and based on the graphs presented it can be observed that the original hypothesis is rejected at the 5% level of significance, i.e., the series is a smooth series with no time trend or random trend. The mean value modeling can be carried out ( Liao, 2023 ). The specific results are shown in Table 1 :

From Table 2 , it can be seen that a large portion of the p-values are less than 0.05, and this paper concludes that there is serial autocorrelation. In the later modeling analysis is in this test a condition. And ACF and PACF are not obvious truncation and partial tail phenomenon, this paper can be considered that it is consistent with the establishment of model characteristics. ARMA ( Liu & Qian, 2023 ).

In this paper, we constructed and characterised an ARMA(p, q) volatility model for the Shanghai Composite Index, employing the first-order difference of the logarithmic difference of the CSI 300 Index. In order to ensure the efficiency of data utilization, from eight models are fitted for this purpose. During the fitting process ARMA (2, 2), ARMA (2, 1), ARMA (11, 2), ARMA (1), AR (2), AR (1), MA (2), MA (1). This paper found that only after the significance test ARMA (1, 1) and ARMA (2, 2) models passed the significance test for all coefficients. This paper then compares the AIC, SC, and HQC values to determine the final optimal model.

As shown in Table 3 , it can be observed that in ARMA (1, 1) model the AIC value is −5.618596 and SIC value is −5.614191, and in model the ARMA (2, 2) AIC value is −5.616713 and SIC value is −5.607903, and based on the judgement of the information criterion, this paper selects the model with smaller value as the optimal model.

From the above table, it can be observed that the ARMA (1, 1) model satisfies among the three information criteria the minimum, so finally chosen AIC and SIC ARMA (1, 1) is as the mean model. The autocorrelation test is performed the residual series of the model as LM test ARMA (1, 1) shown in Table 4 :

From Table 4 , it can be seen that the developed ARMA (1, 1) model passes the LM test and does not reject the original hypothesis at the level of significance, and the lag 10% same conclusion is obtained at order, and is therefore identified as the final mean model.

The test for effect is to test whether the residual series are heteroskedastic and whether the residual squared series are correlated. The graphs of correlation coefficients and partial autocorrelation coefficients of the squared residual series show that there is a strong autocorrelation ARCH ( Liu, Yang, & Shen, 2019 ).

The ARCH-LM test method and the results below further carried out are shown in Table 5 .

From Table 5 , the original hypothesis is rejected at the 5% level of significance, thus the model has ARCH effect.

In this paper, we fit each order model, because p = 1, q = 1; the lag order should not be too large, so the highest lag order in this paper is 2^nd order. After several attempts of GARCH (p, q) model, respectively, to establish normal distribution, generalized error distribution and GARCH (1, 1), GARCH (2, 2), GARCH (GARCH (1, 2), 2, 1) model, and ultimately finally determined to choose the generalized error distribution under the GARCH under the T-distribution (1, 1) model.

The inspection process is as follows:

In the GARCH (1, 1) model:

In the GARCH (1, 1) model, all coefficients passed the significance test; however, higher-order ARCH effects were present and were therefore omitted.

In the T-GARCH (1, 1) model:

As shown in Table 7 , the coefficient portion is not significant and is therefore omitted.

Under conditions of heteroskedasticity, namely in the EGARCH (1, 1) model:

As shown in Table 8 , All coefficients are statistically significant and pass the heteroscedasticity test. This paper concludes that the model can be used to fit the return volatility of the CSI 300 Index.

The final result obtained is: ${\sigma }_t^2=0.00000124+0.05559u_{t-1}^2+0.940159{\sigma }_{t-1}^2$.

In order to test the fit of the model, as shown in Table 6 , the model is tested for the effects of ARCH.

As shown in Table 9 , based on the results of the test, as well as Table 6 , it is known that the model is no longer heteroskedastic.