# Wavelet Time-Scale Modeling of Brand Sales and Prices

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Wavelets Toolkit

- The j-level detail space ${W}_{j}=span({\psi}_{j,k},\phantom{\rule{0.166667em}{0ex}}k)$;
- The j-level approximation space ${V}_{j}=span({\phi}_{j,k},\phantom{\rule{0.166667em}{0ex}}k)$.

## 4. Development of the Mathematical Modeling System

- ${S}_{t}$ stands for the sales value at time t;
- ${P}_{t}$ stands for the price at time t;
- $C{P}_{t}$ stands for the competitor’s price at time t;
- $P{R}_{t}$ is the PROMO variable at time t;
- ${D}_{t}$ refers to percentage distribution of the main brand;
- ${\u03f5}_{t}$ is an error term.

- ${S}_{t,j}$ reflects the sales value at time t and the level or horizon j;
- ${P}_{t,j}$ stands for the price at level j and time t;
- $C{P}_{t,j}$ stands for the competitor’s price at level j and time t;
- $P{R}_{t,j}$ is the PROMO variable at horizon j and time t;
- ${D}_{t,j}$ stands for the percentage distribution at horizon j and time t;
- ${\u03f5}_{t,j}$ is an error term already relative to level j and time t.

## 5. Results and Discussion

**Figure 13.**The wavelet approximation ${A}_{6}$ for ${B}_{1}$ and $CP{B}_{1}$ prices and sales at level 6.

**Figure 14.**The wavelet approximations ${A}_{6}$ for ${B}_{2}$ and $CP{B}_{2}$ prices and sales at level 6.

**Figure 15.**The wavelet approximation ${A}_{6}$ for ${B}_{3}$ and $CP{B}_{3}$ prices and sales at level 6.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Armstrong, G.; Adam, S.; Denize, S.; Kotler, P. Principles of Marketing; Pearson Australia: Frenchs Forest, Australia, 2014. [Google Scholar]
- Jones, J.P. What’s in a Name? Advertising and the Concept of Brands; Lexington Books: Lexington, MA, USA, 1986. [Google Scholar]
- Farquhar, P.H. Managing Brand Equity. Mark. Res.
**1989**, 1, 24–33. [Google Scholar] - Marfatia, H.A. A fresh look at integration of risks in the international stock markets: A wavelet approach. Rev. Financ. Econ.
**2017**, 34, 33–49. [Google Scholar] [CrossRef] - Marfatia, H.A. Wavelet Linkages of Global Housing Markets and Macroeconomy. 2017. Available online: https://ssrn.com/abstract=3169424 (accessed on 20 July 2021).
- Abramovich, F.; Bailey, T.; Sapatinas, T. Wavelet Analysis and its Statistical Applications. Statistician
**2000**, 49, 1–29. [Google Scholar] [CrossRef] - Arfaoui, S.; Rezgui, I.; Ben Mabrouk, A. Wavelet Analysis on the Sphere, Spheroidal Wavelets; Degryuter: Berlin, Germany, 2017; ISBN 978-3-11-048188-4. [Google Scholar]
- Ben Mabrouk, A.; Kortass, H.; Ammou, S.B. Wavelet Estimators for Long Memory in Stock Markets. Int. J. Theor. Appl. Financ.
**2008**, 12, 297–317. [Google Scholar] [CrossRef] - Ben Mabrouk, A.; Kahloul, I.; Hallara, S.E. Wavelet-Based Prediction for Governance, Diversification and Value Creation Variables. Int. Res. J. Financ. Econ.
**2010**, 60, 15–28. [Google Scholar] - Ben Mabrouk, A.; Ben Abdallah, N.; Hamrita, M.E. A wavelet method coupled with quasi self similar stochastic processes for time series approximation. Int. J. Wavelets Multiresolut. Inf. Process.
**2011**, 9, 685–711. [Google Scholar] - Ben Mabrouk, A.; Zaafrane, O. Wavelet Fuzzy Hybrid Model for Physico Financial Signals. J. Appl. Stat.
**2013**, 40, 1453–1463. [Google Scholar] [CrossRef] - Percival, D.B.; Walden, A.T. Wavelet Methods for Time Series Analysis; Camridge University Press: New York, NY, USA, 2000. [Google Scholar]
- Sarraj, M.; Ben Mabrouk, A. The Systematic Risk at the Crisis—A Multifractal Non-Uniform Wavelet Systematic Risk Estimation. Fractal Fract.
**2021**, 5, 135. [Google Scholar] [CrossRef] - Selcuk, F. Wavelets: A new analysis method. Bilkent J.
**2005**, 3, 12–14. (In Turkish) [Google Scholar] - Dekimpe, M.G.; Hanssens, D.M. Time-series Models in Marketing: Past, Present and Future. Int. J. Res. Mark.
**2000**, 17, 183–193. [Google Scholar] [CrossRef][Green Version] - Michis, A.A. Regression Analysis of Marketing Time Series: A Wavelet Approach with Some Frequency Domain Insights. Rev. Mark. Sci.
**2009**, 7, 45. [Google Scholar] [CrossRef] - Leeflang, P.S.; Wittink, D.R.; Wedel, M.; Naert, P. Building Models for Marketing Decisions; Kluwer Academic Publishers: Boston, MA, USA, 2000. [Google Scholar]
- Michis, A.A. Wavelet Analysis of Marketing Time Series; Berkeley Electronic Press: Berkeley, CA, USA, 2009. [Google Scholar]
- Leone, R.P. Generalizing What Is Known About Temporal Aggregation and Advertising Carryover. Mark. Sci.
**1995**, 14, 141–150. [Google Scholar] [CrossRef] - Burksdale, H.C.; Guffey, H.J., Jr. An Illustration of Cross-Spectral Analysis in Marketing. J. Mark. Res.
**1972**, 9, 271–278. [Google Scholar] [CrossRef] - Chatfield, C. Some Comments on Spectral Analysis in Marketing. J. Mark. Res.
**1974**, 11, 97–101. [Google Scholar] [CrossRef] - Bronnenberg, B.J.; Mela, C.F.; Boulding, W. The Periodicity of Pricing. J. Mark. Res.
**2006**, 43, 477–493. [Google Scholar] [CrossRef] - Dekimpe, M.G.; Hanssens, D.M. Empirical Generalizations about Market Evolution and Stationarity. Mark. Sci.
**1995**, 14, G109–G121. [Google Scholar] [CrossRef][Green Version] - Deleersnyder, B.; Dekimpe, M.G.; Sarvary, M.; Parker, P.M. Weathering Tight Economic Times: The Sales Evolution of Consumer Durables Over the Business Cycle. Quant. Mark. Econ.
**2004**, 2, 347–383. [Google Scholar] [CrossRef][Green Version] - Kaiser, R.; Maravall, A. Estimation of the Business Cycle: A modified Hodrick-Prescott Filter. Span. Econ. Rev.
**1999**, 1, 175–206. [Google Scholar] [CrossRef] - Kaiser, R.; Maravall, A. Measuring Business Cycles in Economic Time Series; Springer: New York, NY, USA, 2001. [Google Scholar]
- Pauwels, K.; Currim, I.; Dekimpe, M.G.; Hanssens, D.M.; Mizik, N.; Ghysels, E.; Naik, P. Modeling Marketing Dynamics by Time Series Econometrics. Mark. Lett.
**2004**, 15, 167–183. [Google Scholar] [CrossRef][Green Version] - Michis, A.A.; Sapatinas, T. Wavelet Instruments for Efficiency Gains in Generalized Method of Moment Models. Stud. Nonlinear Dyn. Econom.
**2007**, 11, 4. [Google Scholar] [CrossRef] - Arfaoui, S.; Ben Mabrouk, A.; Cattani, C. Wavelet Analysis Basic Concepts and Applications; Chapman and Hall/CRC: New York, NY, USA, 2021. [Google Scholar]
- Daubechies, I. Ten Lectures on Wavelets; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1992. [Google Scholar]
- Franses, P.H.; Paap, R. Quantitative Models in Marketing Research; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Gençay, R.; Selçuk, F.; Whitcher, B. An Introduction to Wavelets and Other Filtering Methods in Finance and Economics; Academic Press: San Diego, CA, USA, 2002. [Google Scholar]
- Cifter, A.; Ozun, A. Multiscale Systematic Risk: An Application on ISE 30; MPRA Paper 2484; University Library of Munich: Munich, Germany, 2007. [Google Scholar]
- Cifter, A.; Ozun, A. A signal processing model for time series analysis: The effect of international F/X markets on domestic currencies using wavelet networks. Int. Rev. Electr. Eng.
**2008**, 3, 580–591. [Google Scholar] - Conlon, T.; Crane, M.; Ruskin, H.J. Wavelet multiscale analysis for hedge funds: Scaling and strategies. Physica A
**2008**, 387, 5197–5204. [Google Scholar] [CrossRef][Green Version] - DiSario, R.; Saraoglu, H.; McCarthy, J.; Li, H. Long memory in the volatility of an emerging equity market: The case of Turkey. Int. Mark. Inst. Money
**2008**, 18, 305–312. [Google Scholar] [CrossRef] - Fernandez, V. The CAPM and value at risk at different time-scales. Int. Rev. Financ. Anal.
**2006**, 15, 203–219. [Google Scholar] [CrossRef] - Gençay, R.; Whitcher, B.; Selçuk, F. Systematic Risk and Time Scales. Quant. Financ.
**2003**, 3, 108–116. [Google Scholar] [CrossRef] - Gençay, R.; Whitcher, B.; Selçuk, F. Multiscale systematic risk. J. Int. Money Financ.
**2005**, 24, 55–70. [Google Scholar] [CrossRef][Green Version] - In, F.; Kim, S. The hedge ratio and the empirical relationship between the stock and futures markets: A new approach using wavelet analysis. J. Bus.
**2006**, 79, 799–820. [Google Scholar] [CrossRef] - In, F.; Kim, S.; Marisetty, V.; Faff, R. Analysing the performance of managed funds using the wavelet multiscaling method. Rev. Quant. Financ. Account.
**2008**, 31, 55–70. [Google Scholar] [CrossRef] - Xiong, X.; Zhang, X.T.; Zhang, W.; Li, C.Y. Wavelet-based beta estimation of China stock market. In Proceedings of the 4th International Conference on Machine Learning and Cybernetic, Guangzhou, China, 18–21 August 2005; ISBN 0-7803-9091-1. [Google Scholar]
- Yamada, H. Wavelet-based beta estimation and Japanese industrial stock prices. Appl. Econ. Lett.
**2005**, 12, 85–88. [Google Scholar] [CrossRef] - Hanssens, D.M.; Leonard, J.P.; Randall, L.S. Market Response Models, Econometric and Time Series Analysis; Kluwer Academic Publishers: Boston, MA, USA, 2001. [Google Scholar]

Nomination | Brand | Description (Sector) |
---|---|---|

${B}_{1}$ | Jarir bookstore | Books and electronics |

${B}_{2}$ | Almarai | Dairy and poultry |

${B}_{3}$ | STC | Telecommunications |

${B}_{4}$ | Al Abdullatif | Household durables |

${B}_{5}$ | EIC | Electrical industries company |

${B}_{6}$ | Al Aseel | Textiles, apparel and luxury goods |

Brand | Mean | Median | Min | Max | Std | Skewness | Kurtosis | Jarque–Bera $(\mathit{h},\mathit{p})$ |
---|---|---|---|---|---|---|---|---|

The brand prices | ||||||||

${B}_{1}$ | 161.42 | 159.2 | 105.44 | 225 | 26.48 | 0.21 | 2.37 | (1,${10}^{-3}$) |

${B}_{2}$ | 52.91 | 53.3 | 36.95 | 63.7 | 4.07 | −0.08 | 3.15 | (0,0.34) |

${B}_{3}$ | 100.25 | 99.75 | 67.10 | 139.2 | 16.35 | 0.20 | 2.52 | (1,10${}^{-3}$) |

${B}_{4}$ | 15.88 | 12.66 | 8.15 | 40.35 | 8.12 | 1.76 | 4.64 | (1,10${}^{-3}$) |

${B}_{5}$ | 20 | 20.34 | 14.46 | 29.4 | 3.55 | 0.39 | 2.35 | (1,10${}^{-3}$) |

${B}_{6}$ | 35.52 | 31.35 | 14.88 | 69.75 | 16.44 | 0.65 | 2.01 | (1,10${}^{-3}$) |

The brands sales | ||||||||

${B}_{1}$ | 153.63 | 109.31 | 7.56 | 4020 | 258.03 | 10.70 | 140.71 | (1,10${}^{-3}$) |

${B}_{2}$ | 564.28 | 418.03 | 34.70 | 12,140 | 689.47 | 9.25 | 129.83 | (1,10${}^{-3}$) |

${B}_{3}$ | 890.12 | 512.49 | 33.30 | 1.21310 | 3967.75 | 27.64 | 831.52 | (1,10${}^{-3}$) |

${B}_{4}$ | 809.01 | 223.43 | 8.24 | 22,200 | 1935.15 | 6.08 | 51.00 | (1,10${}^{-3}$) |

${B}_{5}$ | 2577.69 | 1720 | 111.39 | 37,490 | 3161.13 | 4.89 | 40.35 | (1,10${}^{-3}$) |

${B}_{6}$ | 243.89 | 64.04 | 0.01 | 5250 | 570.35 | 4.70 | 29.97 | (1,10${}^{-3}$) |

Brand | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|

The brand prices | ||||||

${B}_{1}$ | −0.0002 | 0.0007 | 0.0016 | 0.0147 | 0.0147 | 0.0912 |

${B}_{2}$ | 0.0000 | 0.0000 | −0.0001 | −0.0054 | −0.0054 | −0.0041 |

${B}_{3}$ | −0.0000 | −0.0004 | 0.0001 | 0.0042 | 0.0042 | 0.0656 |

${B}_{4}$ | −0.0000 | −0.0001 | −0.0008 | 0.0001 | 0.0001 | −0.0106 |

${B}_{5}$ | −0.0000 | 0.0002 | −0.0007 | −0.0045 | −0.0045 | −0.0061 |

${B}_{6}$ | 0.0000 | 0.0002 | −0.0008 | 0.0054 | 0.0054 | −0.0395 |

The brand sales | ||||||

${B}_{1}$ | −0.0073 | 0.0196 | −0.0206 | 0.0079 | 0.0079 | 0.0034 |

${B}_{2}$ | −0.0645 | −0.0049 | 0.0065 | 0.2211 | 0.2211 | 0.7129 |

${B}_{3}$ | 0.0012 | −0.4976 | −0.9052 | 36.9157 | 36.9157 | 20.2465 |

${B}_{4}$ | −0.2039 | 0.2151 | −0.0401 | 0.3270 | 0.3270 | 1.0747 |

${B}_{5}$ | −0.2954 | 0.2021 | −1.5023 | 4.1040 | 4.1040 | 21.1156 |

${B}_{6}$ | −0.0060 | 0.0060 | −0.0210 | 0.4643 | 0.4643 | 0.5981 |

Brands | Top Competitor(s) |
---|---|

${B}_{1}$—Jarir Bookstore | Alobeikan ($CP{B}_{1}$) |

${B}_{2}$—Almarai | Nadec ($CP{B}_{2}$) |

${B}_{3}$—STC | Zain ($CP{B}_{3}$) |

**Table 5.**Model (9) coefficient estimations for brands ${B}_{i}$, $1\le i\le 6$, at different levels.

Coefficient | Model (8) | ${\mathit{A}}_{6}$ | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|---|---|

The brand ${B}_{1}$ | ||||||||

${\beta}_{0}$ | $-95.21$ | $-167.57$ | $-0.01$ | $0.01$ | $-0.01$ | $-0.02$ | $-0.09$ | $-0.24$ |

${\beta}_{1}$ | $1.93$ | $2.45$ | $1.27$ | $12.75$ | $4.55$ | $2.85$ | $-7.13$ | $-4.82$ |

${\beta}_{2}$ | $-6.05$ | $-7.21$ | $-10.78$ | $-8.27$ | $15.78$ | $-24.60$ | $-3.54$ | $-12.48$ |

The brand ${B}_{2}$ | ||||||||

${\beta}_{0}$ | $-42.76$ | $-131.92$ | $-0.06$ | $0.02$ | $-0.002$ | $-0.42$ | $0.21$ | $1.10$ |

${\beta}_{1}$ | $10.31$ | $12.98$ | $227.21$ | $166.02$ | $-10.61$ | $45.20$ | $-6.22$ | $-34.84$ |

${\beta}_{2}$ | $2.17$ | $0.31$ | $53.16$ | $-72.86$ | $-14.65$ | $60.14$ | $-6.97$ | $23.62$ |

The brand ${B}_{3}$ | ||||||||

${\beta}_{0}$ | $139.14$ | $72.38$ | $-0.006$ | $-0.45$ | $-0.82$ | $-14.90$ | $35.88$ | $21.80$ |

${\beta}_{1}$ | $-5.91$ | $-7.47$ | $-590.90$ | $80.18$ | $-314.70$ | $41.34$ | $01.99$ | $16.75$ |

${\beta}_{2}$ | $122.77$ | $139.51$ | $2436.79$ | $-396.59$ | $351.10$ | $-394.74$ | $-980$ | $-167.01$ |

**Table 6.**Confidence intervals at $95\%$ for model (9) coefficient estimations for brands ${B}_{i}$, $1\le i\le 6$, at different levels.

Coeff | Model (8) | ${\mathit{A}}_{6}$ | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|---|---|

The brand ${B}_{1}$ | ||||||||

${\beta}_{0}$ | [−96.24; −95.18] | [−168.60; −165.53] | [−0.04; 0.02] | [−0.02; 0.04] | [−0.04; 0.02] | [−0.05; 0.02] | [−0.12; −0.08] | [−0.27; −0.21] |

${\beta}_{1}$ | [1.89; 1.96] | [2.41; 2.48] | [1.23; 1.30] | [12.71; 12.78] | [4.51; 4.58] | [2.81; 2.88] | [−7.16; −7.09] | [−4.85; −4.78] |

${\beta}_{2}$ | [−6.11; −6.00] | [−7.27; −7.16] | [−10.84; −10.73] | [−8.33; −8.22] | [15.71; 15.82] | [−24.66; −24.55] | [−3.60; −3.49] | [−12.54; −12.43] |

The brand ${B}_{2}$ | ||||||||

${\beta}_{0}$ | [−43.79; −42.77] | [−132.95; −130.88] | [−0.19; 0.02] | [−0.11; 0.05] | [−0.03; 0.02] | [−0.55; −0.30] | [0.10; 0.30] | [1.01; 1.23] |

${\beta}_{1}$ | [10.27; 10.34] | [12.94; 13.01] | [227.17; 227.24] | [165.98; 166.05] | [−10.64; −10.57] | [45.16; 45.23] | [−6.25; −6.18] | [−34.87; −34.80] |

${\beta}_{2}$ | [2.10; 2.22] | [0.24; 0.36] | [53.09; 53.21] | [−72.92; −72.80] | [−14.71; −14.59] | [60.07; 60.19] | [−7.03; −6.91] | [23.55; 23.67] |

The brand ${B}_{3}$ | ||||||||

${\beta}_{0}$ | [139.10; 139.17] | [72.34; 72.41] | [−0.03; 0.02] | [−0.48; −0.41] | [−0.85; −0.78] | [−14.93; −14.86] | [35.84; 35.91] | [21.76; 21.83] |

${\beta}_{1}$ | [−5.94; −5.87] | [−7.50; −7.43] | [−590.93; −590.86] | [80.14; 80.21] | [−314.73; −314.66] | [41.30; 41.37] | [1.95; 2.02] | [16.71; 16.78] |

${\beta}_{2}$ | [122.70; 122.81] | [139.44; 139.55] | [2436.72; 2436.83] | [−396.65; −396.54] | [351.03; 351.14] | [−394.80; −394.69] | [−980.06; −979.95] | [−167.07; −166.96] |

Brand | Model (8) | ${\mathit{A}}_{6}$ | Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | Level 6 |
---|---|---|---|---|---|---|---|---|

${B}_{1}$ | 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 |

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**MDPI and ACS Style**

Alanazi, T.M.; Ben Mabrouk, A. Wavelet Time-Scale Modeling of Brand Sales and Prices. *Appl. Sci.* **2022**, *12*, 6485.
https://doi.org/10.3390/app12136485

**AMA Style**

Alanazi TM, Ben Mabrouk A. Wavelet Time-Scale Modeling of Brand Sales and Prices. *Applied Sciences*. 2022; 12(13):6485.
https://doi.org/10.3390/app12136485

**Chicago/Turabian Style**

Alanazi, Tawfeeq M., and Anouar Ben Mabrouk. 2022. "Wavelet Time-Scale Modeling of Brand Sales and Prices" *Applied Sciences* 12, no. 13: 6485.
https://doi.org/10.3390/app12136485