1、Managerial EconomicsManagerial Economics4-1Simple Linear Regression Simple linear regression model relates dependent variable Y to one independent(or explanatory)variable XYabXaYYYXIntercept parameter()gives value of where regression line crosses-axis(valueof when is zero)Slope parameter(b)gives the
2、 change in Y associated with a one-unit change in X,bY/X Managerial EconomicsManagerial Economics4-2Method of Least Squares The sample regression line is an estimate of the true regression line abParameter estimates are obtained bychoosing values of&that minimizethe sum of squared residualsiiY YYThe
3、 residual is the difference between theactual&fitted values of,YabXManagerial EconomicsManagerial Economics4-3iS,.ASam pl e r egr essi on l i ne11 5734 9719Sample Regression Line (Figure 4.2)A08,0002,00010,0004,0006,00010,00020,00030,00040,00050,00060,00070,000Advertising expenditures(dollars)Sales(
4、dollars)SiS46,376eiiS60,000Managerial EconomicsManagerial Economics4-4 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 parameterUnbiased Estimators
5、 ababThe estimates of&do not generally equal the true values of&ab&are random variables computed usingdata from a random sampleManagerial EconomicsManagerial Economics4-5Relative Frequency Distribution*(Figure 4.3)0821046113579Rel at i ve f requency of bLeast-squares est i m at e of b(b)*Also called
6、 a probability density function(pdf)Relative Frequency Distribution*for when 5bbManagerial EconomicsManagerial Economics4-6 Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i.e.,b 0)Statistical SignificancebbEven if =0 it is possible that thesample
7、 will produce an estimate that is different from zero Test for statistical significance using t-tests or p-valuesManagerial EconomicsManagerial Economics4-7 First determine the level of significance Probability of finding a parameter estimate to be statistically different from zero when,in fact,it i
8、s zero Probability of a Type I Error 1 level of significance=level of confidencePerforming a t-TestManagerial EconomicsManagerial Economics4-8Performing a t-Test 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
9、of parameters estimatedbbttS-ratio is computed as bSbwhere is the standard error of the estimate Managerial EconomicsManagerial Economics4-9Performing a t-Test If absolute value of t-ratio is greater than the critical t,the parameter estimate is statistically significantManagerial EconomicsManageria
10、l Economics4-10Using 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 Also the probability of finding significance when none existsManagerial EconomicsManagerial E
11、conomics4-11Coefficient 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 correlatedManagerial EconomicsManagerial Economics4-12F-Test Used to test for signific
12、ance of overall regression equation Compare F-statistic to critical F-value 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 significantManagerial EconomicsManagerial Economics4-13Multiple Regress
13、ion 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 variableManagerial EconomicsManagerial Economics4-14 Use when curve fitting scatter plotQuadratic Regression Model
14、s2YabXcX2ZXFor linear transformation computenew variable YabXcZEstimate is U-shaped or U-shaped Managerial EconomicsManagerial Economics4-15Log-Linear Regression ModelsbcYaX ZUse when relation takes the form:YbXPercentage change in Percentage change in YcZPercentage change in Percentage change in lnYlnablnXclnZTransform by taking natural logarithms:bc and are elasticities
