Chapter 4 - Basic estimation techniques. After completing this unit, you should be able to: Set up a regression equation that can be estimated using a computerized regression routine, interpret and understand how to use the computer output to investigate problems that are of interest to managers of a firm, specify a relation or model between a dependent variable and the appropriate independent variable(s) that can be estimated using regression techniques,...
Managerial Economics ninth edition Thomas Maurice Chapter Basic Estimation Techniques McGrawHill/Irwin McGrawưHill/Irwin ManagerialEconomics,9e ManagerialEconomics,9e Copyrightâ2008bytheMcGrawưHillCompanies,Inc.Allrightsreserved ManagerialEconomics Simple Linear Regression Simple linear regression model relates dependent variable Y to one independent (or explanatory) variable X Y a bX • I nt e r c e pt par am e t e r ( a) g ive s value o f Y wh e r e r e g r e s s io n line c r o s s e s Y ax is (value o f Y wh e n X is z e r o ) • Slope parameter (b) gives the change in Y associated with a oneunit change in X, b 42 Y/ X Managerial Economics Method of Least Squares • Parameter estimates are obtained by choosing values of a & b that minimize the sum of squared residuals • T h e r e s id ual is t h e d if f e r e nc e b e t we e n t h e ac t ual & f it t e d value s o f Y , Yi Yˆi • The sample regression line is an estimate of the true regression line Yˆ 43 ˆ aˆ bX Managerial Economics Sample Regression Line (Figure 4.2) S 70,000 60,000 ei 50,000 ) sr all od( s el a S 20,000 10,000 • • 40,000 30,000 60,000 Si • Sˆ i • • • 46,376 • 2,000 4,000 6,000 8,000 Advertising expenditures (dollars) 44 Sample regression line Sˆ i 11573 , 9719 A 10,000 A Managerial Economics Unbiased Estimators • The estimates of aˆ & bˆ not generally equal the true values of a & b • aˆ & bˆ ar e r and o m var iab le s c o m put e d us ing d at a f r o m a r and o m s am ple • The distribution of values the estimates might take is centered around the true value of the parameter • An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter 45 Managerial Economics Relative Frequency Distribution* (Figure 4.3) Relative Frequency Distribution* for bˆ when b = Relative frequency of bˆ 1 ˆ Least-squares estimate of b (b) 46 *Also called a probability density function (pdf) 10 Managerial Economics Statistical Significance • Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e., b 0) b = it is possible that the sample will produce an estimate bˆ • Even if that is different from zero • Test for statistical significance using t-tests or p-values 47 Managerial Economics Performing a t-Test • First determine the level of significance • Probability of finding a parameter estimate to be statistically different from zero when, in fact, it is zero • Probability of a Type I Error • – level of significance = level of confidence 48 Managerial Economics Performing a t-Test • t -ratio is computed as t bˆ Sbˆ where Sbˆ is the standard error of the estimate bˆ • Use t-table to choose critical t-value with n – k degrees of freedom for the chosen level of significance • n = number of observations • k = number of parameters estimated 49 Managerial Economics Performing a t-Test • If absolute value of t-ratio is greater than the critical t, the parameter estimate is statistically significant 4 Managerial Economics Using p-Values • Treat as statistically significant only those parameter estimates with p-values smaller than the maximum acceptable significance level • p-value gives exact level of significance 4 • Also the probability of finding significance when none exists Managerial Economics Coefficient of Determination • R2 measures the percentage of total variation in the dependent variable that is explained by the regression equation • Ranges from 0 to 1 • High R2 indicates Y and X are highly correlated 4 Managerial Economics F-Test • Used to test for significance of overall regression equation • Compare F-statistic to critical Fvalue from F-table • Two degrees of freedom, n – k & k – 1 • Level of significance • If F-statistic exceeds the critical F, the regression equation overall is statistically significant 4 Managerial Economics Multiple Regression • Uses more than one explanatory variable • Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable 4 Managerial Economics Quadratic Regression Models • Use when curve fitting scatter plot is U-shaped or -shaped U • 4 Y a bX cX • Fo r line ar t r ans f o r m at io n c o m put e ne w var iab le Z X • Es t im at e Y a bX cZ Managerial Economics Log-Linear Regression Models • Use when relation takes the form: Y b Pe r c e nt ag e c h ang e in Y Pe r c e nt ag e c h ang e in X • c Pe r c e nt ag e c h ang e in Y Pe r c e nt ag e c h ang e in Z • T r ans f o r m b y t ak ing nat ur al lo g ar it h m s : • b and c ar e e las t ic it ie s • 4 aX b Z c lnY lna b ln X c ln Z ... Yˆ 4 3 ˆ aˆ bX Managerial Economics Sample Regression Line (Figure 4. 2) S 70,000 60,000 ei 50,000 ) sr all od( s el a S 20,000 10,000 • • 40 ,000 30,000 60,000 Si • Sˆ i • • • 46 ,376 • 2,000 4, 000... parameter 4 5 Managerial Economics Relative Frequency Distribution* (Figure 4. 3) Relative Frequency Distribution* for bˆ when b = Relative frequency of bˆ 1 ˆ Least-squares estimate of b (b) 4 6... High R2 indicates Y and X are highly correlated 4 Managerial Economics F-Test • Used to test for significance of overall regression equation • Compare F-statistic to critical Fvalue from F-table • Two degrees of freedom, n – k & k – 1