Lecture Managerial economics (Ninth edition): Chapter 4 – Thomas, Maurice

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