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**Ch. 14: The Multiple Regression Model building**

Idea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi) Multiple Regression Model with k Independent Variables: Y-intercept Population slopes Random Error

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The coefficients of the multiple regression model are estimated using sample data with k independent variables Interpretation of the Slopes: (referred to as a Net Regression Coefficient) b1=The change in the mean of Y per unit change in X1, taking into account the effect of X2 (or net of X2) b0 Y intercept. It is the same as simple regression. Estimated (or predicted) value of Y Estimated intercept Estimated slope coefficients

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**Graph of a Two-Variable Model**

Three dimension Y Slope for variable X1 X2 Slope for variable X2 X1

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**Example: Simple Regression Results Multiple Regression Results **

Check the size and significance level of the coefficients, the F-value, the R-Square, etc. You will see what the βnet of β effects are.

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**Using The Equation to Make Predictions**

Predict the appraised value at average lot size (7.24) and average number of rooms (7.12). What is the total effect from 2000 sf increase in lot size and 2 additional rooms?

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**Coefficient of Multiple Determination, r2 and Adjusted r2**

Reports the proportion of total variation in Y explained by all X variables taken together (the model) Adjusted r2 r2 never decreases when a new X variable is added to the model This can be a disadvantage when comparing models

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**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

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**Multiple Regression Assumptions**

The errors are normally distributed Errors have a constant variance The model errors are independent Errors (residuals) from the regression model: ei = (Yi β Yi) These residual plots are used in multiple regression: Residuals vs. Yi Residuals vs. X1i Residuals vs. X2i Residuals vs. time (if time series data)

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**Two variable model Y Yi Residual = ei = (Yi β Yi) Yi x2i X2 x1i**

Sample observation Yi Residual = ei = (Yi β Yi) < Yi < x2i X2 x1i < The best fit equation, Y , is found by minimizing the sum of squared errors, οe2 X1

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**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: Ξ²i = 0 (no linear relationship) H1: Ξ²i β 0 (linear relationship does exist between Xi and Y) Test Statistic: Confidence interval for the population slope Ξ²i

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**Is the Overall 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: Ξ²1 = Ξ²2 = β¦ = Ξ²k = 0 (no linear relationship) H1: at least one Ξ²i β 0 (at least one independent variable affects Y) Test statistic:

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**Testing Portions of the Multiple Regression Model**

To find out if inclusion of an individual Xj or a set of Xs, significantly improves the model, given that other independent variables are included in the model Two Measures: Partial F-test criterion The Coefficient of Partial Determination

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**Contribution of a Single Independent Variable Xj**

SSR(Xj | all variables except Xj) = SSR (all variables) β SSR(all variables except Xj) Measures the contribution of Xj in explaining the total variation in Y (SST) consider here a 3-variable model: SSR(X1 | X2 and X3) = SSR (all variablesX1-x3) β SSR(X2 and X3) SSRR Model SSRUR Model

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**The Partial F-Test Statistic**

Consider the hypothesis test: H0: variable Xj does not significantly improve the model after all other variables are included H1: variable Xj significantly improves the model after all other variables are included Note that the numerator is the contribution of Xj to the regression. If Actual F Statistic is > than the Critical F, then Conclusion is: Reject H0; adding X1 does improve model

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**Coefficient of Partial Determination for one or a set of variables**

Measures the proportion of total variation in the dependent variable (SST) that is explained by Xj while controlling for (holding constant) the other explanatory variables

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**Regression intercepts are different if the variable is significant **

Using Dummy Variables A dummy variable is a categorical explanatory variable with two 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 If more than two levels, the number of dummy variables needed is (number of levels - 1)

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**Different Intercepts, same slope**

Fire Place No Fire Place Fire Place (X2 = 1) Y (sales) If H0: Ξ²2 = 0 is rejected, then βFire Placeβ has a significant effect on Values b0 + b2 No Fire place (X2 = 0) b0

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**Interaction Between Explanatory Variables**

Hypothesizes interaction between pairs of X variables Response to one X variable may vary 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 Ξ²1 With interaction term, effect of X1 on Y is measured by Ξ²1 + Ξ²3 X2 Effect changes as X2 changes

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**Slopes are different if the effect of X1 on Y depends on X2 value**

Example: Suppose X2 is a dummy variable and the estimated regression equation is = 1 + 2X1 + 3X2 + 4X1X2 Y Y = 1 + 2X1 + 3(1) + 4X1(1) = 4 + 6X1 X2 = 1: Y = 1 + 2X1 + 3(0) + 4X1(0) = 1 + 2X1 X2 = 0: X1 0.5 1 1.5 Slopes are different if the effect of X1 on Y depends on X2 value

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