Estimates along with their standard errors and the equation’s R2are typically reported in equation format as: 2118.917.9081.8630.448 N=75() (6.35)*** (1.096)*** (0.683)***SALESPRICEADVERTRse(3.6) How to interpretations of the results? The negative coefficient on PRICE suggests that demand is price elastic; we estimate that, with advertising held constant, an increase in price of $1 will lead to a fall in monthly revenue of $7,908. The coefficient on advertising is positive; we estimate that with price held constant, an increase in advertising expenditure of $1,000 will lead to an increase in sales revenue of $1,863. The estimated intercept implies that if both price and advertising expenditure were zero the sales revenue would be $118,914. Clearly, this outcome is not possible; a zero price implies zero sales revenue. In this model, as in many others, it is important to recognize that the model is an approximation to reality in the region for which we have data; including an intercept improves this approximation even when it is not directly interpretable. Now use the model to predict sales revenue if price is $5.50 and advertising expenditure is $1,200: 5.5,1.2118.91 - 7.9081.863118.914-7.90795.51.86261.277.656PRICEADVERTSALESPRICEADVERT(3.6.1) The predicted value of sales revenue is $77,656 if the price is set at $5.50 and advertising expenditure is $1,200. Question: The negative sign attached to price implies that reducing the price will increase sales revenue. If taken literally, why should we not keep reducing the price to zero? Obviously that would not keep increasing total revenue. This makes the following important point: a)Estimated regression models describe the relationship between the economic variables for values similar to those found in the sample data. b)Extrapolating the results to extreme values is generally not a good idea and Predicting the value of the dependent variable for values of the explanatory variables far from the sample values invites disaster. 3.2.3 Estimation of the Error Variance σ2
6 We need to estimate the error variance, 22var()iieE e.But, the squared errors are unobservable, so we develop an estimator for σ2based on the squares of the least squares residuals: 12233ˆˆiiiiiieyyybb xb xAn estimator for σ2that uses the information from 2iˆeand has good statistical properties is: 221ˆˆNiieNK(3.7) where K is the number of β parameters being estimated in the multiple regression model. For the hamburger chain example: 75221ˆ1718.943ˆ23.874753iieMSENKNote that: 21ˆ1718.943NiiSSEeand ˆroot MSE 23.8744.8861. Both quantities typically appear in the output from your computer software, SAS, and different software refer to it in different ways.
- Fall '19