# Answers for Wooldridge

MULTIPLE REGRESSION After completing this chapter, you should be able to: understand model building using multiple regression analysis apply multiple regression analysis to business decision-making situations analyze and interpret the computer output for a multiple regression model test the significance of the independent variables in a multiple regression model use variable transformations to model nonlinear relationships recognize potential problems in multiple regression analysis and take the steps to correct the problems. ncorporate qualitative variables into the regression model by using dummy variables. Multiple Regression Assumptions The errors are normally distributed The mean of the errors is zero Errors have a constant variance The model errors are independent Model Specification Decide what you want to do and select the dependent variable Determine the potential independent variables for your model Gather sample data (observations) for all variables The Correlation Matrix Correlation between the dependent variable and selected independent variables can be found using Excel:

Tools / Data Analysis… / Correlation Can check for statistical significance of correlation with a t test Example A distributor of frozen desert pies wants to evaluate factors thought to influence demand Dependent variable: Pie sales (units per week) Independent variables: Price (in $) Advertising ($100’s) Data is collected for 15 weeks Pie Sales Model Sales = b0 + b1 (Price) + b2 (Advertising) Interpretation of Estimated Coefficients Slope (bi) Estimates that the average value of y changes by bi units for each 1 unit increase in Xi holding all other variables constant

Example: if b1 = -20, then sales (y) is expected to decrease by an estimated 20 pies per week for each $1 increase in selling price (x1), net of the effects of changes due to advertising (x2) y-intercept (b0) The estimated average value of y when all xi = 0 (assuming all xi = 0 is within the range of observed values) Pie Sales Correlation Matrix Price vs. Sales : r = -0. 44327 There is a negative association between price and sales Advertising vs. Sales : r = 0. 55632 There is a positive association between advertising and sales Scatter Diagrams

Computer software is generally used to generate the coefficients and measures of goodness of fit for multiple regression Excel: Tools / Data Analysis… / Regression Multiple Regression Output The Multiple Regression Equation Using The Model to Make Predictions Input values Multiple Coefficient of Determination Reports the proportion of total variation in y explained by all x variables taken together Multiple Coefficient of Determination Adjusted R2 R2 never decreases when a new x variable is added to the model This can be a disadvantage when comparing models

What is the net effect of adding a new variable? We lose a degree of freedom when a new x variable is added Did the new x variable add enough explanatory power to offset the loss of one degree of freedom? Shows the proportion of variation in y explained by all x variables adjusted for the number of x variables used (where n = sample size, k = number of independent variables) Penalize excessive use of unimportant independent variables Smaller than R2 Useful in comparing among models Multiple Coefficient of Determination Is the Model Significant? F-Test for Overall Significance of the Model

Shows if there is a linear relationship between all of the x variables considered together and y Use F test statistic Hypotheses: H0: ? 1 = ? 2 = … = ? k = 0 (no linear relationship) HA: at least one ? i ? 0 (at least one independent variable affects y) F-Test for Overall Significance Test statistic: where F has (numerator) D1 = k and (denominator) D2 = (n – k – 1) degrees of freedom H0: ? 1 = ? 2 = 0 HA: ? 1 and ? 2 not both zero ( = . 05 df1= 2 df2 = 12 Are Individual Variables Significant? Use t-tests of individual variable slopes Shows if there is a linear relationship between the variable xi and y

Hypotheses: H0: ? i = 0 (no linear relationship) HA: ? i ? 0 (linear relationship does exist between xi and y) H0: ? i = 0 (no linear relationship) HA: ? i ? 0 (linear relationship does exist between xi and y) t Test Statistic: (df = n – k – 1) Inferences about the Slope: t Test Example H0: ? i = 0 HA: ? i ? 0 Confidence Interval Estimate for the Slope Standard Deviation of the Regression Model The estimate of the standard deviation of the regression model is: Standard Deviation of the Regression Model The standard deviation of the regression model is 47. 46 A rough prediction range for pie sales in a given week is

Pie sales in the sample were in the 300 to 500 per week range, so this range is probably too large to be acceptable. The analyst may want to look for additional variables that can explain more of the variation in weekly sales OUTLIERS If an observation exceeds UP=Q3+1. 5*IQR or if an observation is smaller than LO=Q1-1. 5*IQR where Q1 and Q3 are quartiles and IQR=Q3-Q1 What to do if there are outliers? Sometimes it is appropriate to delete the entire observation containing the oulier. This will generally increase the R2 and F test statistic values Multicollinearity Multicollinearity: High correlation exists between two independent variables

This means the two variables contribute redundant information to the multiple regression model Including two highly correlated independent variables can adversely affect the regression results No new information provided Can lead to unstable coefficients (large standard error and low t-values) Coefficient signs may not match prior expectations Some Indications of Severe Multicollinearity Incorrect signs on the coefficients Large change in the value of a previous coefficient when a new variable is added to the model A previously significant variable becomes insignificant when a new independent variable is added

The estimate of the standard deviation of the model increases when a variable is added to the model Output for the pie sales example: Since there are only two explanatory variables, only one VIF is reported VIF is < 5 There is no evidence of collinearity between Price and Advertising Qualitative (Dummy) Variables Categorical explanatory variable (dummy variable) with two or more levels: yes or no, on or off, male or female coded as 0 or 1 Regression intercepts are different if the variable is significant Assumes equal slopes for other variables The number of dummy variables needed is (number of levels – 1)

Dummy-Variable Model Example (with 2 Levels) Interpretation of the Dummy Variable Coefficient Dummy-Variable Models (more than 2 Levels) The number of dummy variables is one less than the number of levels Example: y = house price ; x1 = square feet The style of the house is also thought to matter: Style = ranch, split level, condo Dummy-Variable Models (more than 2 Levels) Interpreting the Dummy Variable Coefficients (with 3 Levels) Nonlinear Relationships The relationship between the dependent variable and an independent variable may not be linear Useful when scatter diagram indicates non-linear relationship

Example: Quadratic model The second independent variable is the square of the first variable Polynomial Regression Model where: ?0 = Population regression constant ?i = Population regression coefficient for variable xj : j = 1, 2, …k p = Order of the polynomial (i = Model error Linear vs. Nonlinear Fit Quadratic Regression Model Testing for Significance: Quadratic Model Test for Overall Relationship F test statistic = Testing the Quadratic Effect Compare quadratic model with the linear model Hypotheses (No 2nd order polynomial term) (2nd order polynomial term is needed) Higher Order Models Interaction Effects

Hypothesizes interaction between pairs of x variables Response to one x variable varies at different levels of another x variable Contains two-way cross product terms Effect of Interaction Without interaction term, effect of x1 on y is measured by ? 1 With interaction term, effect of x1 on y is measured by ? 1 + ? 3 x2 Effect changes as x2 increases Interaction Example Hypothesize interaction between pairs of independent variables Hypotheses: H0: ? 3 = 0 (no interaction between x1 and x2) HA: ? 3 ? 0 (x1 interacts with x2) Model Building Goal is to develop a model with the best set of independent variables

Easier to interpret if unimportant variables are removed Lower probability of collinearity Stepwise regression procedure Provide evaluation of alternative models as variables are added Best-subset approach Try all combinations and select the best using the highest adjusted R2 and lowest s? Idea: develop the least squares regression equation in steps, either through forward selection, backward elimination, or through standard stepwise regression The coefficient of partial determination is the measure of the marginal contribution of each independent variable, given that other independent variables are in the model

Best Subsets Regression Idea: estimate all possible regression equations using all possible combinations of independent variables Choose the best fit by looking for the highest adjusted R2 and lowest standard error s? Aptness of the Model Diagnostic checks on the model include verifying the assumptions of multiple regression: Each xi is linearly related to y Errors have constant variance Errors are independent Error are normally distributed Residual Analysis The Normality Assumption Errors are assumed to be normally distributed Standardized residuals can be calculated by computer

Examine a histogram or a normal probability plot of the standardized residuals to check for normality Chapter Summary Developed the multiple regression model Tested the significance of the multiple regression model Developed adjusted R2 Tested individual regression coefficients Used dummy variables Examined interaction in a multiple regression model Described nonlinear regression models Described multicollinearity Discussed model building Stepwise regression Best subsets regression Examined residual plots to check model assumptions

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