EC 320
Regression analysis refers to a host of techniques used in statistics to explain relationships between variables in a data set. He helps us understand and make predictions of how dependent variables would change when independent variables are altered. Scott Armstrong (2012) says that “the estimation target is a function is a function of the independent variables called the regression function” (p. 9).It is this regression function that gives the mathematical relationship between the various variables in the data set. He says too that characterizing the disparity of the dependent variable around the regression meaning can also be described by the probability function.
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Regression analysis has many functions, one of which will be used later on in this paper but just to mention a few common ones. These include predicting and forecasting changed in dependent variables in a data set when independent variables change. It can also be used to tell which specific independent variables, in a list, are related to the dependent variables and to make out the exact character of the relationship.
This article accounts the findings of a study to determine the impact that the Florida Department of Citrus (FDOC) coupon promotion program has on the market demand for frozen concentrated orange juice (FCOJ).
The researchers are Jonq-Ying Lee and Mark G. Brown used a switching regression model to undertake this study. The core finding of the study is that “discounts and information provided by coupon were effective in increasing demand for FCOJ, and that the FDOC’s coupon programs had positive impacts on the probability of using coupons for purchasing FCOJ” (Lee & Brown, 1985, p. 647). It also reports that depending on the coupon promotion there was a different impact (ibid).
The article starts out by acknowledging a historical trend – that in the few preceding years to the conducting of this research “product promotion and advertising related to the cents-off coupons” (ibid) had grown very fast and steady. Manufacturers’ distributed coupons stood at 16 billion in 1970 but this number rose to a little over 100 billion in 1981 and that 58%, 65% and 76% of households used coupons in 1973, 1975 and 1980 respectively (ibid).
Lee and Brown then go into an explanation of how coupon programs work. They explain that the first stage of the implementation process is a dropping or mass distribution of coupons using a number of media. Coupons then redeem the coupons over the life of the program. The ground has been laid and so the researchers make the argument that, “Because redemptions represent individual uses of coupons, changes in individual consumption behavior can be related to the coupons redeemed (ibid). This is the argument around which their whole case and research is built. The researchers use the switching regression model to anylyze a possible couple redemption and household consumption.
Information used for this research is “monthly panel data from 9,552 households from July 1981 through June 1982” (ibid) obtained from the National Panel Data Research, Inc., an institution that keeps a diary of consumers’ purchases on a weekly basis. The data used here is a sample representing the consumer demographics in the United States and all relate to purchases of FCOJ, as reported by the panel.
Data is categorized into coupon users and nonusers and “demand equations estimated for each group” (p. 648). The data include quantity purchased by households – whether redeeming or not redeeming cents-off coupons.
The news media’s accuracy in reporting the finding
Among the news media that reported the findings of this research is the American Journal of Agricultural Economics in the 3rd issue of their 67th Volume in 1985. Their report on the overall findings of the research is fairly accurate. That report, however, does not say anything about the actual process – where the data samples came from, assumptions made before settling on the model to use, demerits of the model et al – other than mention that a switching regression was used to perform the analysis. The information provided in the news article is not very revealing but the little that it says is accurate.
The Switching Regression
The switching regression used by the researchers is defined by the equations 1, 2, 3, 5, and 6 below. These equations were used by the researchers to formulate the joint determination of whether to redeem a coupon and how much to purchase:
1) Q1t = X1tβ1 + Є1t iff It* ≥ 0
2) Q2t = X2tβ2 + Є2t iff It* < 0
3) It* = Ztγ - Єt
where:
Q1t is the FCOJ purchase rate during the month t for household redeeming coupons.
Q2t is the FCOJ purchase rate for the month t for a household not redeeming coupons.
I* is an unobservable index which determines the choice.
X1t, X2t and Zt are vectors of exogenous variables.
β1, β2 and γ are parameter vectors to be estimated and Є1t, Є2t and Єt are trivariately normally distributed disturbance terms with zero means.
Contemporaneous correlation between the disturbances is possible. In other words,
[Q1t, Q2t, It*]t ~ N3(µt, ∑)
t = 1, 2,, T
and ∑ is a positive definite matrix
4) ∑ = σ1t σ12 σk0
σ 21 σ22 σk1
σ 1* σ2* 1
Since Q1t and Q2t are assumed to be mutually exclusive, they cannot be observed simultaneously for any one household. What one does observe is the binary index It, which indicates whether or not coupons were redeemed and the quantity of the commodity purchased, Qt i.e
5) Qt = Q1t, It = 0 if I* ≥ 0
6) Qt = Q1t, It = 0 otherwise
(Lee & Brown, 1985, p. 4)
A classical assumption that might be violated by the switch aggression model is the assumption that independent variables are measured with no error. In this situation that is not the case and it is in cognizance of that danger that the model used by the researchers is built with errors-in-variables techniques. Even still, that danger is not wiped completely.
Endogeneity would still be possible. When σk0 = σk1 = 0 does not hold, the model will be said to involve endogenous switching.
Regression analysis can also play a part in studying non-experimental data too but the accuracy if those techniques can be reduced by various factors which may in turn result in a false sense of confidence and wrong conclusions from a study (Armstrong, 2012). Researchers have, therefore, over the years included in their work a check on their use of regression to ensure that they have taken steps to avoid illusions. Some of them have also modified their techniques and models to include features that would minimize possible arising from the analysis. That is one of the big strengths of regression as a modeling tool – it can be tailored to fit a situation and it can also be modified to include special features to take care of issues.
Armstrong, J. S. (2012). Illusions in Regression Analysis. International Journal of Forecasting, 1-11.
Lee, J.-Y., & Brown, M. G. (1985, August). Coupon Redemption and the Demand for Frozen Concentrated Orange Juice: A Switching Regression Analysis.
American Journal of Agricultural Economics, 67(3), 647-653. Retrieved from http://www.jstor.org/stable/1241088
