6 March, 2006 Reading Baye, Ch 3. pp 73-86
Collect Problem #7
Problem #8 due Wednesday
Reading: Ch 5. pp. 156-168
Lecture 19
REVIEW______:
III. Quantitative Demand Analysis
. Estimating Demand: Regression Analysis.
3. Doing regressions. Examples and an analysis of regression output.
Preview______
4. Interpreting the Significance of Individual Parameter Estimates.
5. Forecasting.
IV. Chapter 5. The Production Process and Costs
A. Introduction:
B. The Production Function.
LECTURE______
4. Interpreting the significance of individual parameter estimates.
Example #1. If I give you the estimated equation
Ù
Q = 24.207 – 2.069 P + 2.62 Px + 0.197 Adv. R2 = .82 , n=22
(10.1) (0.97) (0.49) (0.29) SER = 2.2
I might ask whether or not price was a significant explainer of sales. You could evaluate this by adding and subtracting 2 times the std. error of the regression to the parameter estimate. If the interval doesn’t include zero then at a 95% level of confidence, an inverse relationship exists between price and quantity.
-2.069 + 2(.97) = -.129
- 2.069 – 2(.97) = -4.009
Result: Yes, price is inversely related to quantity. On the other hand, consider advertising
.197+ 2(.29) = .1125
.197 – 2(.29) =-.383
This interval includes zero, so we cannot conclude that there is a direct relationship between advertising and quantity at a 95% level of confidence.
Comments on Multivariate Regression. Obviously, when constructing a demand relationship, you have some choice as to which variables to include. Increasing the number of independent variables always improves your estimate in the sense that you get a better "fit." (Intuitively, by adding terms you gain extra latitude in trying to minimize the squared differences between observed and predicted data.) Nevertheless, it is generally not a good idea to add variables to "maximize the fit," for you can easily add in too many things, disguising possible significant relationships.
Question: In the above, if Advertising does not significantly affect sales, should we delete it from the analysis? No, because advertising may still be important, and deleting imwe might introduce bias.
In general, the appropriate approach is to include all variables for which you have a straightforward reason for including.
5. Forecasting. One can also get a feel for the precision of a forecast by using the SER
Given independent variable values one can construct an approximate 95% forecast interval by adding and subtracting 2* the MSE of the regression to the point estimate.
For example, example 1 above, suppose P = 2, Px = 1 and Adv =10.
Then
Q = 24.207 – 2.069(2) + 2.62 (1) + 0.197 (10)
= 24.659
An approximate 95% confidence interval about the estimate would be
24.659 + 2(2.2) = 29.059
24.659 - 2(2.2) = 20.259
Your estimates get worse the further away you get from the mean of the sample. Thus, observations out of the range of the sample are very speculative. (Example: Can you forecast future sales, given price and advertising expenses? It depends on the relationship between your proposed price and advertising expenditures, and those you've observed in the past.)
Also, your forecasts will be worse, to the extent that there is any expectation that the future may be different from the past (For example, a forecast of cigarette sales would be very considerably more variable than a 95% confidence interval would suggest were there some substantial possibility that cigarette sales would be outlawed next year.
Some additional examples:
Example #2: Suppose you conduct a regression with n=9 data points, and generate the following results:
Qi= 10 - .5Pi + .1 Ai
(5) (.03) (.2)
n = 9, R2 = .83 SER = .5
- Interpret R2,
- Does price explain movement in sales?
- IF p=4 and A=10, make a 95% confidence interval for sales
Example #2. Consider a regression with the log of data.
lnQi= 50 - .2 ln Pi + .12ln Ai
(20) (.3) (.08)
n = 12, R2 = .68,
- Interpret R2
-. Does price alone explain movement in sales, Qi?
Notice finally, that we can make one further interesting insight with a log linear regression. Observe that h=-.2 Notice that two standard deviations about -.2 are not necessarily greater than zero. However, this interval does not include -1. Thus, we can conclude that we are on the inelastic portion of the demand curve.