PLC Inc. is a manufacturer in food industry. Its marketing department is seeking to develop an econometric model for forecasting the future sales of one of its products. An analyst feels that the most important factors affecting the sales (measured in Kgs) are the selling price X1 (measured in dollars), advertisement spending X2 (measured in thousand dollars), the purchasing power of its potential buyers X3 (represented by GDP per person which is measured in thousand dollars), and time T measure in quarter (valued 1, 30). Thirty (30) consecutive quarters of quarterly data for these variables were obtained. The following three regression equations were tried.
1. Y = ß0 + ß1X1 + ß2X2 + ß3X3 + ß4T + µ
2. Y = ß0 + ß1lnX1 + ß2lnX2 + ß3lnX + µ
3. lnY = ß0 + ß1X1 + ß2X2 + ß3X3 + ß4T + µ
The regression results of these three models are as follows (figures in the parentheses are the standard errors of the corresponding estimated regression coefficients):
Model 1: Y = -15.00 - 0.20X1+ 0.06X2 + 2.00X3 + 1.00T
(5.293) (0.20) (0.02) (1.00) (2.00)
R2=0.67 adj R2=0.55
Model 2: Y= -50.453 - 3.168lnX1+ 2.308lnX2+14.687lnX3
(13.728) (2.917) (0.110) (3.10)
R2=0.65 adj R2=0.58
Model 3: lnY= -3.131 - 0.45lnX1+ 8.04lnX2+ 0.250X3+ 0.02T
(1.216) (0. 015) (1.865) (0.121) (0.100)
R2=0.725 adj R2=0.68
By using mathematical manipulation, interpret and give specific meanings of the regression coefficients associated with the advertisement spending (X2) in the three models (i.e. 0.06, 2.308, and 8.04 respectively). Show your mathematic work step by step.