## 5 Discussion
### Granger Causality Test
Analyzing the time series plots shown above, we do not see a cause-and-effect relationship between two variables (\"the volume of gas consumption per month\" and \"the natural gas cost of 1 m\({}^{3}\)\") visually. Therefore, we will use the Granger causality test to investigate the cause-and-effect relationship between the specified variables. First, let us determine the lag order for the Granger causality test. To do this, we will perform a graphical analysis of the ACF and PACF for \"the natural gas cost of 1 m\({}^{3}\)\", and \"the volume of gas consumption per month,\" as well as use the AIC and SC criteria. The time series of \"the natural gas cost of 1 m\({}^{3}\)\" is not stationary. By applying two different transformations, we convert the time series of \"the natural gas cost of 1 m\({}^{3}\)\" into a stationary time series. Since the time series of \"the volume of gas consumption per month\" has the property of seasonality, we decompose it into its seasonal, trend and residual components using techniques like the Seasonal Decomposition of Time Series, and analyze the remaining part to determine the appropriate lag order. Let us plot the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the new stationary time series of \"the natural gas cost of 1 m\({}^{3}\)\" (Figures 7 and 8) of the residual component of the time series of \"the volume of gas consumption per month\" (Figures 9 and 10) and look for significant lags beyond which autocorrelation becomes negligible. The number of lags at which the ACF and PACF drop to zero or become insignificant can provide a visual indication of the lag order. Figure 5: Time series plot of “the natural gas cost of 1 m\({}^{3}\)\" (source: authors’ development). Figure 6: Time series plot of “volume of gas consumption per month” (source: authors’ development). Figure 8: Partial autocorrelation function of the stationary time series of “the natural gas cost of 1 m\({}^{3}\)” (source: authors’ development). Figure 10: Partial autocorrelation functions of the residual component of “the volume of gas consumption per month” (source: authors’ development). Figure 7: Autocorrelation function of the stationary time series of “the natural gas cost of 1 m\({}^{3}\)” (source: authors’ development). Figure 9: Autocorrelation functions of the residual component of the “volume of gas consumption per month” (source: authors’ development). Taking into account Figures 7-10 and the results of AIC and SC (the model with the smallest AIC, or the smallest SC, is preferred), we concluded that the order lags take the values \((p,\ q)=(2,\ 2)\). Let us check the null hypothesis of the Granger causality test for a single consumer of data (Table 1). We obtain the next result (Figure 11). Interpreting Key Parameters:  
F-statistic: the test statistic for the Granger causality test. \(p\)-value: the \(p\)-value associated with the F-statistic. Degrees of freedom (df): the degrees of freedom associated with the F-statistic. The chosen level of significance is \(\alpha=0.05\). Since \(p-\text{value}=0.516\), \(p-\text{value}>\alpha\), therefore the null hypothesis cannot be rejected. This means that past values of variable X (\"the natural gas cost of 1 m\({}^{3\sigma}\)) do not indicate that X Granger-causes variable Y (\"the volume of gas consumption per month\"). Applying the described model to the data of each consumer, we have determined that past values of variable X (\"the natural gas cost of 1 m\({}^{3\sigma}\)) do not indicate that X Granger-causes variable Y (\"the volume of gas consumption per month\") in 97.53% of cases (22,211 cases at or above the significance level and 563 cases below the significance level, respectively). Below are summarized results of research using all applied models in the section of separate categories of consumers (Table 2).