For every additional car placed in service estimate how much annual reve nue

# For every additional car placed in service estimate

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For every additional car placed in service, estimate how much annual reve- nue will change. Fox Rent A Car has 11,000 cars in service. Use the estimated regression equation to predict annual revenue for Fox Rent A Car. Answers: ˆ y = - 17 . 005 + 12 . 966 x ; increase by \$12,966; \$126 million Tymon Sloczyński Linear Regression
Introduction Least squares Inference Residuals Definition ( Residual ) The difference between the observed value and the predicted value of the dependent variable, y i - ˆ y i . The error in using ˆ y i to estimate y i . Definition ( Sum of squares due to error, SSE ) SSE = ( y i - ˆ y i ) 2 This object is minimized by OLS. Tymon Sloczyński Linear Regression
Introduction Least squares Inference Further definitions Suppose you do not have any good explanatory variables for y . How would you make predictions in this situation? You would use ¯ y . Definition ( Total sum of squares, SST ) SST = ( y i - ¯ y ) 2 Definition ( Sum of squares due to regression, SSR ) SSR = ( ˆ y i - ¯ y ) 2 Theorem ( Relationship between SSE, SST, and SSR ) SST = SSR + SSE Tymon Sloczyński Linear Regression
Introduction Least squares Inference Caution These objects — SST, SSR, and SSE — are sometimes called dif- ferently, and this might be quite confusing. For example, the eco- nometrics textbook that is typically used at Brandeis, Introduction to Econometrics by James H. Stock and Mark W. Watson, uses the following terms: Total sum of squares (TSS); so the term is the same but the abbreviation is different (TSS instead of SST) Explained sum of squares (ESS) instead of sum of squares due to regression (SSR) Sum of squared residuals (SSR) instead of sum of squares due to error (SSE) You must be very careful. “SSR” refers to different objects in these two textbooks. Tymon Sloczyński Linear Regression
Introduction Least squares Inference Deviations about ˆ y and ¯ y Tymon Sloczyński Linear Regression
Introduction Least squares Inference Measuring goodness of fit In many applications, we might be interested in evaluating whether our estimated regression equation fits the data well — we wish to measure goodness of fit . How can we approach this? Think about two extremes : perfect fit and worst fit. Perfect fit: every value of y i lies on the estimated regression line. Then, for each i , y i - ˆ y i = 0. Consequently, SSE = 0; hence, SST = SSR ; hence SSR SST = 1. Worst fit: larger values of SSE correspond to worse fit. Re- call that SSE = SST - SSR . So when does SSE attain its maximum value? When SSR = 0; hence, SSR SST = 0. Thus, these two extremes correspond to SSR SST = 1 and SSR SST = 0. Tymon Sloczyński Linear Regression
Introduction Least squares Inference Measuring goodness of fit — coefficient of determination Definition ( Coefficient of determination ) r 2 = SSR SST Interpretation: percentage of the total sum of squares that can be explained by a given regression.

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