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Differentiation, Integration, and Solutions of Ordinary Differential Equations
Published in Jamal T. Manassah, Elementary Mathematical and Computational Tools For Electrical and Computer Engineers Using Matlab®, 2017
The Root Test stipulates that for an > 0, the series ∑n=1∞an is convergent iflimn→∞(an)1/n<1
Infinite Series
Published in John Srdjan Petrovic, Advanced Calculus, 2020
In most applications, the Ratio Test is easier to use, and we will always try it first. However, the Root Test is more powerful. That means that whenever we are able to decide whether a series is convergent or divergent by using the Ratio Test, the Root Test would have been applicable. (Although, perhaps, harder to use.) On the other hand, there are situations when the Ratio Test is inconclusive (the limit of Dn is either 1 or does not exist), but the Root Test can provide the answer. Here is a more precise statement.
Comparative analysis of deep learning and classical time series methods to forecast natural gas demand during COVID-19 pandemic
Published in Energy Sources, Part B: Economics, Planning, and Policy, 2023
In this section, different time series forecasting models were used to examine the predictability of the natural gas demand of Turkey during the COVID-19 pandemic. Firstly, Augmented Dickey-Fuller (ADF) unit root test was performed to investigate the stability of the series. The results of the ADF unit root test with the value of different confidence intervals (1%, 5%, and 10%) are shown in Table 3. The maximum lag order for the ADF test was calculated as 12 from the sample size using the formula, provided by Schwert (1989). The analysis was carried out using RStudio software. Here, Q represents the original sequence, and Q* is the sequence after the first-order difference (). The ADF test statistic for the original series was −1.255, and the critical values for 1%, 5%, and 10% significance levels were calculated as −3.494, −2.890, and −2.582. Besides, the calculated p-value (0.648) is higher than the significance alpha level (α = 0.05). Therefore, it was concluded that there is a unit root in the natural gas consumption series. This means that at least the first difference is required to make the time series stationary. Thus, the first trend difference was applied to eliminate instability in the time series. When the ADF test results of the first-order difference sequence (Q*) is analyzed, it shows that the natural gas consumption series does not contain a unit root, that is, it becomes stationary (p-value <0.05)
Economic complexity–carbonization nexus in the European Union: A heterogeneous panel data analysis
Published in Energy Sources, Part B: Economics, Planning, and Policy, 2022
Mehmet Demiral, Emrah Eray Akça
The methodological steps of this study are as follows: The analysis begins by investigating the stationarity of the panel series. There are two generations of panel unit root tests to explore stationarity in the literature. The first group techniques assume cross-sectional independence, while the second-generation tests take possible cross-sectional dependence (hereafter CD) into consideration (Baltagi and Pesaran 2007). Thus, firstly, the panel series of the variables are checked for CD to determine an appropriate unit root test. After CD is confirmed, stationarity is controlled by the second-generation panel unit root tests. As the variables are not level-stationary but first-difference stationary, the analysis proceeds within the cointegration framework. To determine an appropriate cointegration setting, the models constructed in Eq. 2 and Eq. 3 are controlled for heterogeneity and CD which matter for the further unbiasedness of the results. Because of the rejection of both the homogeneity and cross-sectional independence, the cointegration analysis is conducted by accounting for heterogeneity and CD. After the cointegration analysis confirms the presence of long-run relationships among the variables in the models, we estimate these relationships using a second-generation method that considers both heterogeneity and CD. In the final step, we conduct a causality test, again using a second-generation approach.
Technology import modes, environmental regulation types and total factor energy efficiency
Published in Energy Sources, Part B: Economics, Planning, and Policy, 2022
Shuangshuang Li, Xin Miao, Enhui Feng, Yiqun Liu, Yanhong Tang
To avoid spurious regression, all variables used in this paper are tested for their stationary or integration of the same order by using LLCs (Levin, Lin, and Chu 2002), IPSs (Im, Pesaran, and Shin 2003), and ADF-Fisher tests (Maddala and Wu 1999). The results in levels and the first difference are reported in Table 3. There are variables that did not pass the significance test in levels; hence, the first difference test was conducted. The results of the unit root test in the first difference are statistically significant at the 1% level for all the time series. Following that, the long-term equilibrium between variables is tested using the panel cointegration test, which shows that the null hypothesis of no cointegration has been rejected. Therefore, there is strong evidence that the regression estimations presented in this paper are free of spurious regressions.