Slide 1  2005 South-Western Publishing A chief uncertainty for managers is the future. They fear what will happen to their product? »Managers use forecasting, prediction & estimation to reduce their uncertainty. »The methods that they use vary from consumer surveys or experiments at test stores to statistical procedures on past data such as regression analysis. Objective of the Chapter : Learn how to interpret the results of regression analysis from demand data. Estimation of Demand Chapter 4

Slide 2 Demand Estimation Using Marketing Research Techniques Consumer Surveys »ask a sample of consumers their attitudes Consumer Focus Groups »experimental groups try to emulate a market (Hawthorne effect = behave differently in when observed) Market Experiments in Test Stores »get demand information by trying different prices Historical Data »what happened in the past is guide to the future Consumer Surveys »ask a sample of consumers their attitudes Consumer Focus Groups »experimental groups try to emulate a market (Hawthorne effect = behave differently in when observed) Market Experiments in Test Stores »get demand information by trying different prices Historical Data »what happened in the past is guide to the future

Slide 3 Statistical Estimation of Demand Functions: Plot Historical Data Look at the relationship of price and quantity over time Plot it »Is it a demand curve or a supply curve? »The problem is this does not hold other things equal or constant. quantity Price Is this curve demand or supply?

Slide 4 Steps to take: »Specification of the model -- formulate the demand model, select a Functional Form linearQ = a + bP + cY log Q = a + blog P + clog Ydouble loglog Q = a + blog P + clog Y quadraticQ = a + bP + cY+ dP 2 »Estimate the parameters -- determine which are statistically significant try other variables & other functional forms »Develop forecasts from the model Statistical Estimation of Demand Functions

Slide 5 Specifying the Variables Dependent Variable -- quantity in units, quantity in dollar value (as in sales revenues) Independent Variables -- variables thought to influence the quantity demanded Tastes Time Trend »Instrumental Variables -- proxy variables for the item wanted which tends to have a relatively high correlation with the desired variable: e.g., Tastes Time Trend

Slide 6 Functional Forms: Linear Linear Q = a + bP + cY »The effect of each variable is constant, as in  Q/  P = b and  Q/  Y = c, where P is price and Y is income. »The effect of each variable is independent of other variables »Price elasticity is: E D = (  Q/  P)(P/Q) = bP/Q »Income elasticity is: E Y = (  Q/  Y)(Y/Q)= cY/Q »The linear form is often a good approximation of the relationship in empirical work.

Slide 7 Functional Forms: Multiplicative or Double Log Multiplicative Q = A P b Y c »The effect of each variable depends on all the other variables and is not constant, as in  Q/  P = bAP b-1 Y c and  Q/  Y = cAP b Y c-1 »It is double log (log is the natural log, also written as ln) Log Q = a + bLog P + cLog Y »the price elasticity, E D = b »the income elasticity, E Y = c »This property of constant elasticity makes this approach easy to use and popular among economists.

Slide 8 Simple Linear Regression Y t = a + b X t +  t time subscripts & error term Find “best fitting” line  t = Y t - a - b X t  t 2 = [Y t - a - b X t ] 2. min  t 2 =  [Y t - a - b X t ] 2. Solution: slope b = Cov(Y,X)/Var(X) and intercept a = mean(Y) - bmean(X) _X_X Y _Y_Y a XX YY

Slide 9 Simple Linear Regression : Assumptions & Solution Methods 1.The dependent variable is random 2.A straight line relationship exists 3.error term has a mean of zero and a finite variance 4.the independent variables are indeed independent Spreadsheets - such as »Excel, Lotus 1-2-3, Quatro Pro, or Joe Spreadsheet Statistical calculators Statistical programs such as »Minitab »SAS »SPSS »ForeProfit »Mystat

Slide 10 Sherwin-Williams Case (p. 141) Ten regions with data on promotional expenditures (X) and sales (Y) Result: Y = X One use of a regression is to make predictions. If a region had promotional expenditures of 185, the prediction is Y = , by substituting 185 for X The regression output will tell us also the standard error of the estimate, s e. In this case, s e = Approximately 95% prediction interval is Y  2 s e. Hence, the predicted range is anywhere from to

Slide 11 T-tests Different samples would yield different coefficients Test the hypothesis that coefficient equals zero »H o : b = 0 »H a : b  0 RULE: If absolute value of the estimated t > Critical-t, then REJECT H o. »We say that it’s significant! The estimated t = (b - 0) /  b The critical t is: »Large Samples, critical t  2 N > 30 »Small Samples, critical t is on Student’s t Distribution, page B-3 at end of book, usually column D.F. = # observations, minus number of independent variables, minus one. N < 30

Slide 12 Sherwin-Williams Case In the simple linear regression: Y = X The standard error of the slope coefficient is (This is usually available from any regression program used.) Test the hypothesis that the slope is zero, b=0. The estimated t is: t = (.434 – 0 )/ = The critical t for a sample of 10, has only 8 degrees of freedom »D.F. = 10 – 1 independent variable – 1 for the constant. »Table B2 shows this to be at the.05 significance level Therefore, |2.939| > 2.306, so we reject the null hypothesis. We informally say, that promotional expenses (X) is “significant.”

Slide 13 Correlation Coefficient We would expect more promotional expenditures to be associated with more sales at Sherwin-Williams. A measure of that association is the correlation coefficient, r. If r = 0, there is no correlation. If r = 1, the correlation is perfect and positive. The other extreme is r = -1, which is negative. r = -1 r = +1 r = 0 Y Y Y X X X

Slide 14 Coefficient of Determination: R 2 R-square is the percentage of the variation in dependent variable that is explained ^ – R 2 =  [Y t -Y t ] 2 /  [Y t - Y t ] 2 = SSR / SST As more variables are included, R-square rises Adjusted R-square, however, can decline _X_X Y _Y_Y YtYt Y t predicted ^

Slide 15 Association and Causation Regressions indicate association, but beware of jumping to the conclusion of causation Suppose you collect data on the number of swimmers at a beach and the temperature and find: Temperature = Swimmers, and R 2 =.88. »Surely the temperature and the number of swimmers is positively related, but we do not believe that more swimmers CAUSED the temperature to rise. »Furthermore, there may be other factors that determine the relationship, for example the presence of rain or whether or not it is a weekend or weekday. Education may lead to more income, and also more income may lead to more education. The direction of causation is often unclear. But the association is very strong.

Slide 16 Multiple Linear Regression Most economic relationships involve several variables. We can include more independent variables into the regression. To do this, we must have more observations (N) than the number of independent variables, and no exact linear relationships among the independent variables. At Sherwin-Williams, besides promotional expenses, different regions charge different selling prices (SellPrice) and have different levels of disposable income (DispInc) The next slide gives the output of a multiple linear regression, multiple, because there are three independent variables

Slide 17 Figure 4.9 on page 151 Dep var: Sales (Y) N=10 R-squared =.790 Adjusted R 2 =.684 Standard Error of Estimate = VariableCoefficient Std error T P Constant Promotion SellPrice DispInc

Slide 18 Interpreting Multiple Regression Output Write the result as an equation: Sales = Promotion SellPrice DispInc Does the result make economic sense? »As promotion expense rises, so does sales. That makes sense. »As the selling price rises, so does sales. Yes, that’s reasonable. »As disposable income rises in a region, so does sales. Yup. That’s reasonable. Is the coefficient on the selling price statistically significant? »The estimated t value is given in Figure 4.9 to be »The critical t value, with 6 ( which is 10 – 3 – 1) degrees of freedom in table B2 is »Therefore |-2.663| > 2.447, so reject the null hypothesis, and assert that the selling price is significant!