Padé Approximation Modelling of an Advertising-Sales Relationship
- 1 Department of Applied Economics, Faculty of Economics and Business Administration, University of La Laguna (ULL), Campus of Guajara, Tenerife, Spain
- 2 Department of Applied Economics, Faculty of Economics and Business Administration, University of La Laguna (ULL), Campus of Guajara, Tenerife, Spain
- 3 Department of Applied Economics, Faculty of Economics and Business Administration, University of La Laguna (ULL), Campus of Guajara, Tenerife, Spain
Abstract
Forecasting reliable estimates on the future evolution of relevant variables is a main concern if decision makers in a variety of fields are to act with greater assurances. This paper considers a time series modelling method to predict relevant variables taking VARMA and Transfer Function models as its starting point. We make use of the rational Padé-Laurent Approximation, a relevant type of rational approximation in function theory that allows the decision maker to take part in the building of estimates by providing the available information and expectations for the decision variables. This method enhances the study of the dynamic relationship between variables in non-causal terms and allows for an ex ante sensibility analysis, an interesting matter in applied studies. The alternative proposed, however, must adhere to a type of model whose properties are of an asymptotic nature, meaning large chronological data series are required for its efficient application. The method is illustrated through the well-known data series on advertising and sales for the Lydia Pinkham Medicine Company, which has been used by various authors to illustrate their own proposals.
- J. M. Beguin, C. Gourieroux and A. Monfort, “Identification of a mixed autoregressive-moving average process: The corner method,” In Time Series, O. D. Anderson, Ed., Amsterdam, North-Holland, pp. 423–436, 1980.
- G. M. Jenkins and A. S. Alavi, “Some aspects of modelling and forecasting multivariate time series,” Journal of Time Series Analysis, Vol. 2, pp. 1–47, 1981.
- K. Lii, “Transfer function model order and parameter estimation,” Journal of Time Series Analysis, Vol. 6, No. 3, pp. 153–169, 1985.
- P. Claverie, D. Szpiro and R. Topol, “Identification des modèles à fonction de transfert: La méthode Padé-transformée en z,” Annales D'Economie et de Statistique, Vol. 17, pp. 145–161, 1990.
- A. Berlinet and C. Francq, “Identification of a univariate ARMA model,” Computational Statistics, Vol. 9, pp. 117–133, 1994.
- G. E. P. Box, G. M. Jenkins and G. C. Reinsel, “Time series analysis: Forecasting and control,” (3rd ed.), Englewood Cliffs, Prentice-Hall, New Jersey, 1994.
- A. Bultheel, “Laurent series and their Padé approximations,” Birkhaüser, Basel/Boston, 1987.
- C. González-Concepción and M. C. Gil-Fari?a, “Padé approximation in economics,” Numerical Algorithms, Vol. 33, No. 1–4, pp. 277–292, 2003.
- C. González-Concepción, M. C. Gil-Fari?a and C. Pes- tano-Gabino, “Some algorithms to identify rational structures in stochastic processes with expectations,” Journal of Mathematics and Statistics, Vol. 3, No. 4, pp. 268–276, 2007.
- R. M. Helmer and J. K. Johansson, “An exposition of the Box-Jenkins transfer function analysis with an application to the advertising-sales relationship,” Journal of Marketing Research, Vol. 14, pp. 227–239, 1977.
- D. M. Hanssens, “Bivariate time-series analysis of the relationship between advertising and sales,” Applied Eco- nomics, Vol. 12, pp. 329–339, 1980.
- A. Aznar and F. J. Trivez, “Métodos de predicción en economía. Análisis de series temporales,” Editorial Ariel, Barcelona, Vol. 1–2, 1993.
- G. C. Tiao and R. S. Tsay, “Multiple time series modeling and extended sample cross-correlations,” Journal of Business and Economics Statistics, Vol. 1, pp. 43–56, 1983.
- J. F. Heyse and W. W. Wei, “Modelling the Advertising-sales relationship through use of multiple time series techniques,” Journal of Forecasting, Vol. 4, pp. 165–181, 1985.