L7.pdf - APPLIED STATISTICS Model Diagnostics for Linear Regression II Dr Tao Zou Research School of Finance Actuarial Studies Statistics The Australian

L7.pdf - APPLIED STATISTICS Model Diagnostics for Linear...

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APPLIED STATISTICS Model Diagnostics for Linear Regression II Dr Tao Zou Research School of Finance, Actuarial Studies & Statistics The Australian National University Last Updated: Fri Aug 25 09:13:00 2017 1 / 37
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Overview R-Squared and Adjusted R-Squared Graphical Tools for Model Diagnostics 4. Leverage plot. 5. Standardized (Studentized) residuals versus fitted values plot. 6. Cook’s distance plot. Weighted Regression 2 / 37
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References 1. F.L. Ramsey and D.W. Schafer (2012) Chapter 8, Chapter 10 and Chapter 11 of The Statistical Sleuth 2. ANU STAT2008 Lecture Notes 3. W.H. Green (2012) Econometric Analysis 4. The slides are made by R Markdown . 3 / 37
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Review: Sum of Squared Errors (SSE) The sum of squared errors (SSE) for a MLR μ { Y | X } = β 0 + β 1 X 1 + · · · + β k X k , where X = ( X 1 , · · · , X k ) , is defined by SSE = n i = 1 res 2 i = n i = 1 ( Y i - ˆ Y i ) 2 . In Tutorial 2, we have shown for SLR, the sample variance of the residuals is s 2 res = 1 n - 1 n i = 1 ( res i - res ) 2 = 1 n - 1 n i = 1 Y i - ˆ Y i 2 = 1 n - 1 SSE , which measures the variation in the residuals , where res = 1 n n i = 1 res i = 0 . This result is also true for MLR. Thus, SSE also measures the variation in the residuals. 4 / 37
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Total Sum of Squares (SST) In any dataset, there will be variation in the values of the response variable . This variation can be measured by the sample variance of the response values s 2 Y = 1 n - 1 n i = 1 ( Y i - ¯ Y ) 2 , where ¯ Y = 1 n n i = 1 Y i . We call the total sum of squares (SST) of the response SST = n i = 1 ( Y i - ¯ Y ) 2 , which the variation in the values of the response variable too. 5 / 37
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Sum of Squares due to Regression (SSR) In Tutorial 2, we have shown for SLR, the sample variance of the fitted values is s 2 ˆ Y = 1 n - 1 n i = 1 ˆ Y i - ¯ Y 2 . which measures the variation in the fitted values . This result is also true for MLR. We call the sum of squares due to regression (SSR) SSR = n i = 1 ˆ Y i - ¯ Y 2 , which the variation in the fitted values too. 6 / 37
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Partitioning Variability Intuitively, n i = 1 ( Y i - ¯ Y ) 2 = n i = 1 { ( Y i - ˆ Y i ) + ( ˆ Y i - ¯ Y ) } 2 = n i = 1 ( Y i - ˆ Y i ) 2 + n i = 1 ˆ Y i - ¯ Y , namely SST = SSE + SSR . One important interpretation of a regression model is that it explains variation in the values of the response variable (SST). The variation in the values of the response variable (SST) can be split into the following two parts: the variation in the residuals (SSE) + the variation in the fitted values (SSR). The variation in the fitted values (SSR) is explained by the regression model. The variation in the residuals (SSE) is the unexplained variation. 7 / 37
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R-Squared R-squared for MLR μ { Y | X } = β 0 + β 1 X 1 + · · · + β k X k , where X = ( X 1 , · · · , X k ) , is the % of the total response variation explained by the regression model R 2 = SSR SST = 1 - SSE SST . 0 % R 2 100 % . Hence, if R 2 is close to 100%, the regression model can explain the variation in response a lot. The regression model is “good”.
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