Stability of Conditional Variance and Stability Forecasts for Natural Gas Contracts Using NAGARCH Models

The stochastic process is described as second order stationary that is meant a variance and mean in the process are fixed is not reliant on time , the majority of significant condition for random error all the time model series ought to be the white noise process with constant variance and zero mean and uncorrelated. This model was discussed by a number of researchers and suggested some nonlinear time series model referred to as autoregressive conditional, heteroscedastic avoidance volatility in data, that causes dependency mean and variance on time t. The basic model suggestion in 1982 by R. Engle abbreviated as ARCH [1].

Quite a few researchers have studied and extended the stability of these extension ARCH and GARCH models. Glosten et al. [2] proposed GJR-GARCH.

Non-linear Asymmetric Generalized Autoregressive Conditional Heteroscedasticity Variance model in 1992 which was short (NAGARCH model) proposed the NAGARCH model as an expansion of GARCH model [3].

Motivation for a study:

• Natural gas price volatility is of strategic importance to global financial markets and energy sectors, directly affecting investment decisions and economic policies. These volatility makes it necessary to develop accurate models to analyze and predict them. In this context, GARCH models are the most widely used to study conditional variance. However, traditional models (ARCH and GARCH) lack the flexibility to deal with asymmetry in the impact of positive and negative stocks, with led to the development of models such as EGARCH, and GJR-GARCH.
• The NAGARCH model was chosen because it shows advanced ability to capture asymmetric variance and complex market dynamics, making it more suitable for analyzing natural gas prices, which are highly affected by unexpected events such as geopolitical crises and weather.

The research aims to develop and analyze the stability of the NAGARCH model to determine the optimal conditions for stability and apply the NAGARCH model to natural gas price data to verify the predictive performance in addition to integrating the NAGARCH model with ARIMA to create a hybrid model and analyze its performance compared to individual models.

The importance of the research is in providing accurate tools for analyzing natural gas price fluctuations, which enables decision-makers in the energy and finance sectors to predict fluctuations and make informed decisions. It also contributes to bridging the research gap by expanding the uses of NAGARCH models and analyzing their stability, as research combines NAGARCH model with ARIMA model, opening new horizons in hybrid models for analyzing time data. NAGARCH was chosen compared to other models because: dealing with asymmetry: it’s characterized by its ability to deal with the effects of different shocks on the conditional variance, provides more advanced techniques for analyzing the dynamics of volatile markets, and previous studies have proven its effectiveness in analyzing commodities and markets that change in a manner similar to natural gas.

NAGARCH(Q,P) models will a new stability conditions by using the local linear approximation technique and converting the nonlinear model into a difference equation with fixed coefficients proposed by Ozaki [4] when he detect the exponential autoregressive model's stability condition (EXPAR). Ali and Mohammad [5] found stability conditions of limit cycle for Gompertz Autoregressive model, Mohammad and Mudhir [6] found the stability of the EGARCH model, stability of GJR-GARCH model by Noori and Mohammad [7], stability conditions for a nonlinear time series model [8], N-SIR by Hussien and Azher [9], and stability exponential double autoregressive model [10].

This study including three sections: Section 1 including introduction, Section 2 including some primaries and theorems with its proving to find a stability of NAGARCH model, while Section 3 including application to apply the conditions which was finds in Section 2.

3.1 Data descriptions

The data used in our study represents monthly mean of historical data natural gas contracts from: MAY.1990-FEB.2022 by 381 views. Data availability the data used to support the finding of this study is available on the following website https://sa.investing.com/commodities/natural-gas-historical-data.

3.2 Modeling and building the NAGARCH model

The modeling and programming process NAGARCH’s model is applied to the data series. We note and verify that the predicted conditional variance value approaches the value of the unconditional variance. Parameters estimation by AIC, and BIC criteria, and then predict the value of the conditional variance, it is worth noting that to study the GARCH family models, we will follow the Box-Jenkins methodology, provided that the heterogeneity is detected; Because the architecture of GARCH models is very close in syntax to ARMA models.

Analysis Steps:

• Identify the NAGARCH model, which including some stages firstly using Box-Jenkins Methodology, secondly examine the time series and its properties using ACF and PACF, third determine the best model order using AIC and BIC [19, 20], forth test the fit model using series residual analysis.
• Build the hybrid model (ARIMA-NAGARCH): use stage-1 with the ARIMA model to capture seasonal and stochastic trends, and stage-2 apply the NAGARCH model to the ARIMA residuals to analyze conditional variance, finally combine the results of two models to predict future values.
• Test stability: apply the local linear approximation to analyze the stability of fixed points in the model, or test for the presence of stationary limit cycle and confirm their stability.
• Evaluate the performance of models: compare the performance using AIC and BIC, with test the forecasts accuracy using the validation set (12 months of excluded data), and measure MSE [21-23], to compare actual values to predictions.
• Use the selection models to predict 250 future time steps, and verify the predicts values are within the confidence interval $(\pm 1.96 \sqrt{MSE})$.

(1) We enter the data into the program used to create the data sequence, where the Figure 2 is represented by monthly mean of historical data natural gas contracts.

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Figure 2. Represented monthly mean of historical data natural gas contracts

It is noticeable when the time series is drawn that it follows irregular components, that is, the series of result of unexpected and irregular effects, and also does not repeat a specific pattern, the diagram helps visualize the process of identifying fixed points and checking their stability. Then follows the step of drawing the time series to convert the series to a return series and we do this conversion by using the mathematical formula.

$r_t=\log \left(\frac{x_t}{x_{t-1}}\right)=\log x_t-\log x_{t-1}$

where, $r_t$ is returns series and $x_t, x_{t-1}$ are observation at $t, t-1$ respectively. Figure 3 expresses the series of returns.

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Figure 3. The series of returns

Figure 3 shows the general trends and fluctuations in natural gas prices over the study period. The series shows random fluctuations and no clear seasonal pattern.

Although the time series possessed stability, it still suffers from some correlations that we have to remove. This is evident by drawing the ACF and PACF functions as in Figure 4 where, the correlations appear outside the confidence intervals specified $\pm \frac{1.96}{\sqrt{n}}$ where $n$ is the sample size [24].

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Figure 4. ACF and PACF functions for the time series

We can notice from the previous figure that there are a number of lags outside the boundaries of the confidence interval that need to be addressed before starting the modeling and forecasting process [25].

(2) To find out whether there is an effect of variance heterogeneity by finding the error-squared series of a return’s series out of the rapport $e_t=\left(r_t-\bar{r}_t\right)^2$, where $\bar{r}_t$ is the mean of the returns.

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Figure 5. The error-squared series of the return’s series

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Figure 6. ACF and PACF functions for the error-squared series of the return’s series

Figure 5 shows the squared errors of the time series of returns. The series exhibits non-stationary variance, indicating heteroscedasticity, Figure 6 shows values outside the confidence intervals, indicating conditional heteroscedasticity, we notice that some values still fall outside the confidence interval in the ACF and PACF functions at the time differences (Lags) for $k=1,2$.

(3) To detect attacks in return series, we use the Ljung-Box test to measure the effect of heteroscedasticity where the “Table 1” represents the test results.

From Table 1, it is noted that at most lags, the p-value is less than 0.05, indicating that there is autocorrelation in the series. This means that the data needs further processing to remove autocorrelation before modeling.

Table 1. Results of the Ljung-Box test for return series

(4) Now we fit parameters estimation for the model in order to choose the best rank for the model, we will use AIC, CAIC and BIC information criteria and calculate the values. Where Table 2 gives the values of AIC, and BIC criteria for different ranks of the NAGARCH’s model.

Table 2. The values of AIC, CAIC and BIC criteria for different ranks of the NAGARCH

Table 2 shows a comparison between the performance of different NAGARCH models based on AIC and BIC criteria. From this, we get that the best model is NAGARCH(1,1), as it achieved the lowest value for both AIC and BIC. Also, increasing the model order (P,Q) led to a decrease in the quality of the model due to the increase in complexity, so the model formula is $x_t=\psi_t e_t$ where $e_t \sim iid\ N(0,1)$

$\begin{gathered}\psi_t^2=0.016519  +0.043359\left(x_{t-i}-0.37924 \psi_{t-i}\right)^2-0.18201 \psi_{t-j}^2\end{gathered}$          (21)

And by implementation When we examine the model's stability conditions, we see that it is stable according to the condition in Eq. (10), then:

$\begin{aligned} \eta_1\left(1+\gamma_1^2\right)+\delta_1 & =0.043359\left[1+(0.37924)^2\right]-0.18201  =-0.1324149795<1\end{aligned}$

The value of the NAGARCH model's unconditional variance is given in Eq. (15) as follows:

$\psi^2=0.0247$

(5) Now checking the adequacy of the model this stage is to check the suitability of the chosen model in order to verify the adequacy of the NAGARCH(1,1) model in interpreting the data used in calculating future predictions, but the standard series must be checked. Perform a Ljung-Box test and plot the ACF and PACF functions, but on the residual series.

From the Table 3, it's noted that most of the p-values are high 0.05, indicating that the model explains the time series well and there is no longer autocorrelation.

Table 3. Results of the Ljung-Box test for the standard residuals series

Figure 7 shows the distribution of the standardized residuals and the test for autocorrelation. It's clear that the series is stationary and does not suffer from signification autocorrelation.

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Figure 7. Standard residuals series, distribution curves, ACF and PACF for squared standard residuals series

Figure 8 shows the conditional variance of the model and how it approaches the unconditional variance as the model shows stability over time. Figure 9 shows how closely the conditional variance approaches the unconditional variance indicating that the model has reached good stability.

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Figure 8. Model-output conditional variance series and conditional variance forecasting

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Figure 9. Conditional variance, its stability, and its approach to unconditional variance

(6) After verifying the readiness of the NAGARCH(1,1) model in the previous five stages, we will perform in this last stage the forecasting and plot the resulting conditional variance series from the NAGARCH(1,1) model and the forecasting conditions in Figure 8 and Figure 9. The conditional variance and its stability and its approach to the unconditional variance.

3.3 Analysis using the ARIMA-NAGARCH hybrid model

1. The data is entered and the original series is converted into a return series. As mentioned previously, the ARMA-NAGARCH mixture is modeled, and verify the stability of the mixture by dividing and testing the NAGARCH models and the stability of the ARMA models separately.

From Table 4, the best hybrid model is ARIMA(1,0,1)-NAGARCH(1,1), which achieves the lowest AIC and BIC values with data stability.

2. After we have chosen the best order for the model and verified its stability, we find the standard remainder series of using the infer command in MATLAB and we draw the quantum function, which is a visual examination of whether the series has a normal distribution or not.

Table 4. Stability test and values of AIC and BIC

Figure 10 refers to plot standard remainder series and plot quantum function.

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Figure 10. The standard residual series and the quantum function of the normal distribution

It becomes clear that the series forms a straight line next to the normal distribution line, and this confirms that the series is normally distributed. Figure 11 shows the distribution plot of the standard residual series compared to the plot of the normal distribution curve and the shape of the frequency distribution of the residual series as follows:

3. Make a prediction for the chosen model with 250 future steps, and the value of the unconditional mean is proved and fallen within the confines of the confidence interval which is specified by y±1.96/√MSE, where the MSE is the values of the mean squares of the predicted errors, which are observed to get closer to zero as the prediction steps increase. Figure 11 shows the mean least square errors for predictions on the return series as well as the conditional standard deviation predictions.

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Figure 11. The normal distribution curve for the standard remainder series and the frequency distribution curve for the series

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Figure 12. The series of original and predicted returns and the series of calculated and predicted variances

From Figure 12, the model shows acceptable performance in short-term forecasting, but may need improvement to accommodate dynamic changes in the long-term. Also, the conditional variance matches the actual behavior, which enhances the stability of the model.

4. The final stage of our modeling examined the predictive performance of the model used as we left out a calendar period of 12 months by using MSE. The model was used to predict return values for 11 future months and then return the return forecast values to their original values using the directive $d_t={ret} 2 {price}\left(r_t, 2.771\right)$, where 2.771 represents the last value of the training set.

Table 5 compares the actual and forecast values for natural gas returns over the 12 months. The forecasts are relatively close to actual values, but there are some differences that indicate poor model performance in predicating accurately.

Table 5. Monthly rate forecasts

Figure 13 shows that the squared errors gradually approach zero in the long run, indicating that the model is trying to stabilize but may face challenges with high volatility. Low errors in the short run enhance the efficiency of hybrid ARIMA-NAGARCH model for near-term forecasts. However, there is a need to improve model strategies to address errors that appear in long- term forecasts.

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Figure 13. Prediction errors of the model ARMA(1,0,1)-NAGARCH(1,1)