# Wavelet Time-Scale Modeling of Brand Sales and Prices
## 5 Results and Discussion
This will have an important effect on the economy and on society, as it will provide jobs in the national labor market, thus affecting quality of life and well-being in society from the economic point of view. The promotional coefficient \(\beta_{3}\) also has a small value (almost null) which suggests that the promotion factor does not greatly influence the movements of prices and sales. This finding is also reasonable in KSA, as all these top brands apply promotional prices throughout the whole year. Every week, promotions of different products under the same brand are announced. This is also an important point, as it leads to a increase in sales and thus influences sales especially. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Coeff** & **Model (8)** & \(\mathbf{A_{6}}\) & **Level 1** & **Level 2** & **Level 3** & **Level 4** & **Level 5** & **Level 6** \\ \hline \multirow{3}{*}{\(\beta_{0}\)} & \multirow{3}{*}{[-96.2k\(-95.18\)]} & \multirow{3}{*}{[-166.530]} & \multirow{3}{*}{[-0.04; 0.02]} & \multirow{3}{*}{[-0.02; 0.04]} & \multirow{3}{*}{[-0.04; 0.02]} & \multirow{3}{*}{[-0.05; 0.02]} & \multirow{3}{*}{[-0.12; -0.08]} & \multirow{3}{*}{[-0.27; -0.21]} \\  & & & & & & & \\ \cline{1-1}  & [18.99; 16] & [21.42; 2.48] & [12.23; 1.30] & [12.27; 12.78] & [45.14; 5.84] & [2.81; 2.88] & [-7.16; -7.09] & [-4.458; -4.78] \\ \cline{1-1}  & \(\beta_{2}\) & [-61.11; -6.00] & [-7.27; -7.16] & [-8.38; -10.73] & [-8.38; -2.821] & [15.75; 15.82] & [-3.466; -3.49] & [-12.56; -12.43] \\ \cline{1-1}  & & & & & & & & \\ \cline{1-1}  & & & & & & & & \\ \cline{1-1}  & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Confidence intervals at 95% for model (9) coefficient estimations for brands \(B_{i}\), \(1\leq i\leq 6\), at different levels. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Brand** & **Model (8)** & \(\mathbf{A_{6}}\) & **Level 1** & **Level 2** & **Level 3** & **Level 4** & **Level 5** & **Level 6** \\ \hline \(B_{1}\) & 0.26 & 0.58 & 0.17 & 0.12 & 0.78 & 0.55 & 0.56 & 0.23 \\ \(B_{2}\) & 0.68 & 0.19 & 0.40 & 0.33 & 0.23 & 0.21 & 0.18 & 0.10 \\ \(B_{3}\) & 0.39 & 0.63 & 0.30 & 0.14 & 0.12 & 0.99 & 0.65 & 0.23 \\In Figures 13-15, comparisons between the brand prices and sales for \(B_{1}\), \(B_{2}\), and \(B_{3}\) and those of their competitor brands \(CPB_{1}\), \(CPB_{2}\), and \(CPB_{3}\), respectively, are illustrated graphically at the maximum level of wavelet decomposition \(J=6\), by means of the filtered prices. These last figures are in complete agreement with the results previously discussed, especially those in Table 3. Indeed, we notice from Figure 13 that the variations in the \(B_{1}\) prices and sales are ahead of those of its competitor \(CPB_{1}\). Prices and sales for both of them show a global increase, with higher \(B_{1}\) prices. However, the sales also show some oscillations, with higher competitor sales at the longer time scales. This may be explained by the fact that \(B_{1}\) sales are in fact imported products from outside the kingdom, except for a few types of products. However, the competitor, Alobeikan, mainly focuses on a national printing press concerned with printing books internally, or at least those imported and distributed within GCC countries. This meant that its sales increased during the COVID-19 period, which here represents the long-term horizons. The brand \(B_{2}\) and its competitor \(CPB_{2}\) are both concerned with national foods, and thus were not widely influenced by the COVID-19 pandemic as they already used internal products. Nevertheless, the dominance of the brand \(B_{1}\) sales and prices is clearly illustrated in Figure 14. This is also compatible with Table 3, where the corresponding coefficient \(\beta_{2}\) is globally positive, although low for \(A_{6}\), and perturbed (oscillating between positive and negative values) for some medium-to-long time scales. Figure 15 reflects the variation in sales/prices for the brand \(B_{3}\) and its competitor \(CPB_{3}\).