Simple Regression

Kunio Takezawa
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Kunio Takezawa: National Agricultural and Food Research Organization

Chapter Chapter 3 in Learning Regression Analysis by Simulation, 2014, pp 109-162 from Springer

Abstract: Abstract When the data $$\{(x_{i},y_{i})\}$$ (1 ≤ i ≤ n) are given, a 0 and a 1 are derived by minimizing the residual sum of squares (RSS) in a procedure called a simple regression: $$\displaystyle{ RSS =\sum _{ i=1}^{n}{(y_{ i} - a_{0} - a_{1}x_{i})}^{2} =\sum _{ i=1}^{n}e_{ i}^{2}, }$$ where $$(y_{i} - a_{0} - a_{1}x_{i})$$ ( = e i ) is a residual. This process yields the regression equation: $$\displaystyle{ y =\hat{ a}_{0} +\hat{ a}_{1}x, }$$ where a 0 is the intercept and a 1 is the gradient (slope). Each data point is represented as $$\displaystyle{ y_{i} =\hat{ a}_{0} +\hat{ a}_{1}x_{i} + e_{i}. }$$ Values such as a 0 and a 1 are called regression coefficients. The “ $$\widehat{}$$ ” (hat) of $$\hat{a}_{0}$$ and $$\hat{a}_{1}$$ indicates that these values are estimates.

Keywords: Null Hypothesis; Regression Coefficient; Simulation Data; Probability Density Function; Prediction Error (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-4-431-54321-3_3

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