Chapter 8 Forecasting with Multiple Regression Business Forecasting Chapter 8 Forecasting with Multiple Regression

Chapter Topics The Multiple Regression Model Estimating the Multiple Regression Model—The Least Squares Method The Standard Error of Estimate Multiple Correlation Analysis Partial Correlation Partial Coefficient of Determination

Chapter Topics (continued) Inferences Regarding Regression and Correlation Coefficients The F-Test The t-test Confidence Interval Validation of the Regression Model for Forecasting Serial or Autocorrelation

Chapter Topics Equal Variances or Homoscedasticity Multicollinearity (continued) Equal Variances or Homoscedasticity Multicollinearity Curvilinear Regression Analysis The Polynomial Curve Application to Management Chapter Summary

The Multiple Regression Model Relationship between one dependent and two or more independent variables is a linear function. Population Y-intercept Population slopes Random Error Dependent (Response) Variable Independent (Explanatory) Variables

Interpretation of Estimated Coefficients Slope (bi) Estimated that the average value of Y changes by bi for each 1 unit increase in Xi, holding all other variables constant (ceterus paribus). Example: If b1 = −2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X1), given the inches of insulation (X2). Y-Intercept (b0) The estimated average value of Y when all Xi = 0.

Multiple Regression Model: Example (°F) Develop a model for estimating heating oil used for a single family home in the month of January, based on average temperature and amount of insulation in inches.

Multiple Regression Equation: Example Excel Output For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 4.86 gallons, holding insulation constant. For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 15.07 gallons, holding temperature constant.

Multiple Regression Using Excel Stat | Regression … EXCEL spreadsheet for the heating oil example.

Simple and Multiple Regression Compared Coefficients in a simple regression pick up the impact of that variable (plus the impacts of other variables that are correlated with it) and the dependent variable. Coefficients in a multiple regression account for the impacts of the other variables in the equation.

Simple and Multiple Regression Compared: Example Two simple regressions: Multiple Regression:

Standard Error of Estimate Measures the standard deviation of the residuals about the regression plane, and thus specifies the amount of error incurred when the least squares regression equation is used to predict values of the dependent variable. The standard error of estimate is computed by using the following equation:

Coefficient of Multiple Determination Proportion of total variation in Y explained by all X Variables taken together. Never decreases when a new X variable is added to model. Disadvantage when comparing models.

Adjusted Coefficient of Multiple Determination Proportion of variation in Y explained by all X variables adjusted for the number of X variables used and sample size: Penalizes excessive use of independent variables. Smaller than . Useful in comparing among models.

Coefficient of Multiple Determination Adjusted R2 Reflects the number of explanatory variables and sample size Is smaller than R2

Interpretation of Coefficient of Multiple Determination 96.32% of the total variation in heating oil can be explained by temperature and amount of insulation. 95.71% of the total fluctuation in heating oil can be explained by temperature and amount of insulation after adjusting for the number of explanatory variables and sample size.

Using The Regression Equation to Make Predictions Predict the amount of heating oil used for a home if the average temperature is 30° and the insulation is 6 inches. The predicted heating oil used is 304.39 gallons.

Predictions Using Excel Stat | Regression … Check the “Confidence and Prediction Interval Estimate” box EXCEL spreadsheet for the heating oil example.

Residual Plots Residuals vs. Residuals vs. Time May need to transform Y variable. May need to transform variable. May need to transform variable. Residuals vs. Time May have autocorrelation.

Residual Plots: Example May be some non-linear relationship. No Discernible Pattern

Testing for Overall Significance Shows if there is a linear relationship between all of the X variables together and Y. Use F test statistic. Hypotheses: H0: 1 = 2 = … = k = 0 (No linear relationship) H1: At least one i  0 (At least one independent variable affects Y.) The Null Hypothesis is a very strong statement. The Null Hypothesis is almost always rejected.

Testing for Overall Significance (continued) Test Statistic: where F has k numerator and (n-k-1) denominator degrees of freedom.

Test for Overall Significance Excel Output: Example p value k = 2, the number of explanatory variables. n - 1

Test for Overall Significance Example Solution H0: 1 = 2 = … = k = 0 H1: At least one i  0  = 0.05 df = 2 and 12 Critical Value: Test Statistic: Decision: Conclusion:  F 157.24 (Excel Output) Reject at  = 0.05  = 0.05 There is evidence that at least one independent variable affects Y. F 3.89

Test for Significance: Individual Variables Shows if there is a linear relationship between the variable Xi and Y. Use t Test Statistic. Hypotheses: H0: i = 0 (No linear relationship.) H1: i  0 (Linear relationship between Xi and Y.)

t Test Statistic Excel Output: Example t Test Statistic for X1 (Temperature) t Test Statistic for X2 (Insulation)

t Test : Example Solution Does temperature have a significant effect on monthly consumption of heating oil? Test at  = 0.05. H0: 1 = 0 H1: 1  0 df = 12 Critical Values: Test Statistic: Decision: Conclusion: t Test Statistic = -15.084 Reject H0 at  = 0.05 Reject H Reject H 0.025 0.025 There is evidence of a significant effect of temperature on oil consumption. t −2.1788 2.1788

Confidence Interval Estimate for the Slope Provide the 95% confidence interval for the population slope 1 (the effect of temperature on oil consumption). -5.56  1  -4.15 The estimated average consumption of oil is reduced by between 4.15 gallons and 5.56 gallons for each increase of 1° F.

Contribution of a Single Independent Variable Let Xk be the independent variable of interest Measures the contribution of Xk in explaining the total variation in Y.

Contribution of a Single Independent Variable From ANOVA section of regression for: From ANOVA section of regression for: Measures the contribution of in explaining Y.

Coefficient of Partial Determination of Measures the proportion of variation in the dependent variable that is explained by Xk , while controlling for (Holding Constant) the other independent variables.

Coefficient of Partial Determination for (continued) Example: Model with two independent variables

Coefficient of Partial Determination in Excel Stat | Regression… Check the “Coefficient of partial determination” box. EXCEL spreadsheet for the heating oil example.

Contribution of a Subset of Independent Variables Let Xs be the subset of independent variables of interest Measures the contribution of the subset Xs in explaining SST.

Contribution of a Subset of Independent Variables: Example Let Xs be X1 and X3 From ANOVA section of regression for: From ANOVA section of regression for:

Testing Portions of Model Examines the contribution of a subset Xs of explanatory variables to the relationship with Y. Null Hypothesis: Variables in the subset do not improve significantly the model when all other variables are included. Alternative Hypothesis: At least one variable is significant.

Testing Portions of Model (continued) One-tailed Rejection Region Requires comparison of two regressions: One regression includes everything. Another regression includes everything except the portion to be tested.

Partial F Test for the Contribution of a Subset of X variables Hypotheses: H0 : Variables Xs do not significantly improve the model, given all other variables included. H1 : Variables Xs significantly improve the model, given all others included. Test Statistic: with df = m and (n-k-1) m = # of variables in the subset Xs .

Partial F Test for the Contribution of a Single Hypotheses: H0 : Variable Xj does not significantly improve the model, given all others included. H1 : Variable Xj significantly improves the model, given all others included. Test Statistic: With df = 1 and (n−k−1) m = 1 here

Testing Portions of Model: Example Test at the  = 0.05 level to determine if the variable of average temperature significantly improves the model, given that insulation is included.

Testing Portions of Model: Example H0: X1 (temperature) does not improve model with X2 (insulation) included. H1: X1 does improve model  = 0.05, df = 1 and 12 Critical Value = 4.75 (For X2) (For X1 and X2) Conclusion: Reject H0; X1 does improve model.

Testing Portions of Model in Excel Stat | Regression… Calculations for this example are given in the spreadsheet. When using Minitab, simply check the box for “partial coefficient of determination. EXCEL spreadsheet for the heating oil example.

Do We Need to Do This for One Variable? The F Test for the inclusion of a single variable after all other variables are included in the model is IDENTICAL to the t Test of the slope for that variable. The only reason to do an F Test is to test several variables together.

The Quadratic Regression Model Relationship between the response variable and the explanatory variable is a quadratic polynomial function. Useful when scatter diagram indicates non-linear relationship. Quadratic Model: The second explanatory variable is the square of the first variable.

Quadratic Regression Model (continued) Quadratic model may be considered when a scatter diagram takes on the following shapes: Y Y Y Y X1 X1 X1 X1 2 > 0 2 > 0 2 < 0 2 < 0 2 = the coefficient of the quadratic term.

Testing for Significance: Quadratic Model Testing for Overall Relationship Similar to test for linear model F test statistic = Testing the Quadratic Effect Compare quadratic model: with the linear model: Hypotheses: (No quadratic term.) (Quadratic term is needed.)

Heating Oil Example (°F) Determine if a quadratic model is needed for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.

Heating Oil Example: Residual Analysis (continued) Possible non-linear relationship No Discernible Pattern

Heating Oil Example: t Test for Quadratic Model (continued) Testing the Quadratic Effect Model with quadratic insulation term: Model without quadratic insulation term: Hypotheses (No quadratic term in insulation.) (Quadratic term is needed in insulation.)

Example Solution Is quadratic term in insulation needed on monthly consumption of heating oil? Test at  = 0.05. H0: 3 = 0 H1: 3  0 df = 11 Critical Values: Do not reject H0 at  = 0.05. Reject H Reject H 0.025 0.025 There is not sufficient evidence for the need to include quadratic effect of insulation on oil consumption. Z −2.2010 2.2010 0.2786

Validation of the Regression Model Are there violations of the multiple regression assumption? Linearity Autocorrelation Normality Homoscedasticity

Validation of the Regression Model (Continued…) The independent variables are nonrandom variables whose values are fixed. The error term has an expected value of zero. The independent variables are independent of each other.

Linearity How do we know if the assumption is violated? Perform regression analysis on the various forms of the model and observe which model fits best. Examine the residuals when plotted against the fitted values. Use the Lagrange Multiplier Test.

Linearity (continued) Linearity assumption is met by transforming the data using any one of several transformation techniques. Logarithmic Transformation Square-root Transformation Arc-Sine Transformation

Serial or Autocorrelation Assumption of the independence of Y values is not met. A major cause of autocorrelated error terms is the misspecification of the model. Two approaches to determine if autocorrelation exists: Examine the plot of the error terms as well as the signs of the error term over time.

Serial or Autocorrelation (continued) Durbin–Watson statistic could be used as a measure of autocorrelation:

Serial or Autocorrelation (continued) Serial correlation may be caused by misspecification error such as an omitted variable, or it can be caused by correlated error terms. Serial correlation problems can be remedied by a variety of techniques: Cochrane–Orcutt and Hildreth–Lu iterative procedures

Serial or Autocorrelation (continued) Generalized least square Improved specification Various autoregressive methodologies First-order differences

Homoscedasticity One of the assumptions of the regression model is that the error terms all have equal variances. This condition of equal variance is known as homoscedasticity. Violation of the assumption of equal variances gives rise to the problem of heteroscedasticity. How do we know if we have heteroscedastic condition?

Homoscedasticity Plot the residuals against the values of X. When there is a constant variance appearing as a band around the predicted values, then we do not have to be concerned about heteroscedasticity.

Homoscedasticity Constant Variance Fluctuating Variance

Homoscedasticity Several approaches have been developed to test for the presence of heteroscedasticity. Goldfeld–Quandt test Breusch–Pagan test White’s test Engle’s ARCH test

Homoscedasticity Goldfeld–Quandt Test This test compares the variance of one part of the sample with another using the F-test. To perform the test, we follow these steps: Sort the data from low to high of the independent variable that is suspect for heteroscedasticity. Omit the observations in the middle fifth or one-sixth. This results in two groups with . Run two separate regression one for the low values and the other with high values. Observe the error sum of squares for each group and label them as SSEL and SSEH.

Homoscedasticity Goldfeld-Quandt Test (Continued…) Compute the ratio of If there is no heteroscedasticity, this ratio will be distributed as an F-Statistic with degrees of freedom in the numerator and denominator, where k is the number of coefficients. Reject the null hypothesis of homoscedasticity if the ratio exceeds the F table value.

Multicollinearity High correlation between explanatory variables. Coefficient of multiple determination measures combined effect of the correlated explanatory variables. Leads to unstable coefficients (large standard error).

Multicollinearity How do we know whether we have a problem of multicollinearity? When a researcher observes a large coefficient of determination ( ) accompanied by statistically insignificant estimates of the regression coefficients. When one (or more) independent variable(s) is an exact linear combination of the others, we have perfect multicollinearity.

Detect Collinearity (Variance Inflationary Factor) Used to Measure Collinearity If is Highly Correlated with the Other Explanatory Variables.

Detect Collinearity in Excel Stat | Regression… Check the “Variance Inflationary Factor (VIF)” box. EXCEL spreadsheet for the heating oil example Since there are only two explanatory variables, only one VIF is reported in the Excel spreadsheet. No VIF is >5 There is no evidence of collinearity.

Chapter Summary Developed the Multiple Regression Model. Discussed Residual Plots. Addressed Testing the Significance of the Multiple Regression Model. Discussed Inferences on Population Regression Coefficients. Addressed Testing Portions of the Multiple Regression Model.

Chapter Summary Described the Quadratic Regression Model. (continued) Described the Quadratic Regression Model. Addressed the violations of the regression assumptions.