Residual Sum of Squares

3 Key excerpts on "Residual Sum of Squares"

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  eBook - PDF

  Applied Regression Analysis

  • Norman R. Draper, Harry Smith(Authors)
  • 2014(Publication Date)
  • Wiley-Interscience

    (Publisher)

  equal (sum of squares)/(degrees of freedom), can be divided by S2 to provide an F-ratio for testing the hypothesis that the true values of the coefficients whose estimates gave rise to the extra sum of squares are zero. The expected value of an extra sum of squares is evaluated in Appendix 68. The extra sum of squares principle is actually a special case of testing a general linear hypothesis. In the more general treatment the extra sum of squares is calculated from the residual sums of squares and not the regression sum of squares. Since the total sum of squares Y'Y is the same for both regression calculations, we would obtain the same result numerically whether we used the difference of regression or residual sums of squares. See Eq. (6.1.8). 6.1. THE "EXTRA SUM OF SQUARES" PRINCIPLE 151 Two Alternative Forms of the Extra SS We can decide to remove the correction factor n y2 or not remove it, before we take the difference between two sums of squares to get an extra sum of squares. For example, suppose our initial model (Model 1) is (6.1.1) and we want the extra SS for b 3 , b 4 , and b s given b o, b 1 , and b2 • The reduced model (Model 2) is (6.1.2) For Model 1, the regression SS is (6.1.3) the correction factor is n y2, and the total SS is Y'Y, Thus the residual SS is Y'Y - S1' For model 2, the regression SS is (6.1.4) where we no longer show that the Model 2 b's could be different from the Model 1 b's of same subscript, though in general they are.

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  Principles of Econometrics

  • R. Carter Hill, William E. Griffiths, Guay C. Lim(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  208 CHAPTER 5 The Multiple Regression Model EXAMPLE 5.3 Error Variance Estimate for Hamburger Chain Data In the hamburger chain example, we have K = 3. The esti-mate for our sample of data in Table 5.1 is ̂ σ 2 = ∑ 75 i = 1 ̂ e 2 i N − K = 1718.943 75 − 3 = 23.874 Go back and have a look at Table 5.2. There are two quantities in this table that relate to the above calculation. The first is the sum of squared errors SSE = N ∑ i = 1 ̂ e 2 i = 1718.943 The second is the square root of ̂ σ 2 , given by ̂ σ = √ 23.874 = 4.8861 Both these quantities typically appear in the output from your computer software. Different software refer to it in different ways. Sometimes ̂ σ is referred to as the standard error of the regression . Sometimes it is called the root mse (short for the square root of mean squared error). 5.2.3 Measuring Goodness-of-Fit For the simple regression model studied in Chapter 4, we introduced R 2 as a measure of the proportion of variation in the dependent variable that is explained by variation in the explanatory variable. In the multiple regression model the same measure is relevant, and the same formulas are valid, but now we talk of the proportion of variation in the dependent variable explained by all the explanatory variables included in the model. The coefficient of determination is R 2 = SSR SST = ∑ N i = 1 ( ̂ y i − y ) 2 ∑ N i = 1 ( y i − y ) 2 = 1 − SSE SST = 1 − ∑ N i = 1 ̂ e 2 i ∑ N i = 1 ( y i − y ) 2 (5.12) where SSR is the variation in y “explained” by the model ( sum of squares due to regression ), SST is the total variation in y about its mean (sum of squares, total), and SSE is the sum of squared least squares residuals (errors) and is that part of the variation in y that is not explained by the model.

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  Statistics

  An Introduction Using R

  • Michael J. Crawley(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Before we can work out the standard errors, however, we need to know the value of the error variance s 2, and for this we need to carry out an analysis of variance. Partitioning Sums of Squares in Regression: SSY = SSR + SSE The idea is simple: we take the total variation in y, SSY, and partition it into components that tell us about the explanatory power of our model. The variation that is explained by the model is called the regression sum of squares (denoted by SSR), and the unexplained variation is called the error sum of squares (denoted by SSE ; this is the sum of the squares of the lengths of the red lines that we drew on the scatterplot on p. 119). Then SSY = SSR + SSE (the proof is given in Box 7.5). Box 7.5. Proof that SSY = SSR + SSE Let us start with what we know. The difference between and is the sum of the differences and, as you can see from the figure: Inspection of the figure shows clearly that. It is not obvious, however, that the sums of the squares of these three quantities should be equal. We need to prove that. First, square the terms on both sides of the equation to get From both sides, subtract to leave Apply summation then set to zero: Now group the terms with and without : Because is a constant, we can take it outside the summation along with the 2: We already know that from Box 7.2, so all that remains is to prove is that. This requires more algebra, because we have to replace with. First multiply through the bracket to get, then replace the : This is the gory bit

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