Multiple Regression

Kunio Takezawa
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Kunio Takezawa: National Agricultural and Food Research Organization

Chapter Chapter 4 in Learning Regression Analysis by Simulation, 2014, pp 163-224 from Springer

Abstract: Abstract When data $$\{(x_{i1},x_{i2},\ldots,x_{iq},y_{i})\}$$ (1 ≤ i ≤ n) are given, multiple regression derives the values of {a j }(1 ≤ j ≤ q) by minimizing $$\displaystyle{ \mathit{RSS} =\sum _{ i=1}^{n}{(y_{ i} - a_{0} -\sum _{j=1}^{q}a_{ j}x_{ij})}^{2} =\sum _{ i=1}^{n}e_{ i}^{2}. }$$ Here $$e_{i} = y_{i} - a_{0} -\sum _{j=1}^{q}a_{j}x_{ij}$$ are called residuals. The acronym RSS stands for the residual sum of squares. Equation (4.1) yields the regression equation: $$\displaystyle{ \hat{y} = \hat{a}_{0} +\sum _{ j=1}^{q}\hat{a}_{ j}x_{j}, }$$ where $$\{\hat{a}_{j}\}$$ are estimates of the regression coefficients, {x j } the predictor variables, and $$\hat{y}$$ the estimates of target variables. Using Eq. (4.2), Eq. (4.1) is transformed into $$\displaystyle{ \mathit{RSS} =\sum _{ i=1}^{n}{(y_{ i} -\hat{a}_{0} -\sum _{j=1}^{q}\hat{a}_{ j}x_{ij})}^{2}. }$$

Keywords: Null Hypothesis; Regression Coefficient; Simulation Data; Probability Density Function; Alternative Hypothesis (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-4-431-54321-3_4

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