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OHCh2 - Chapter 2 Ordinary Least Squares Revised on 1...

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Chapter 2: Ordinary Least Squares Revised on September 25, 2007
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1 Estimating Single-Independent-Variable Mod- els with OLS We now discuss how to estimate the unknown parameters ° 0 and ° 1 . Estimating these two unknown parameters by choosing ° 0 and ° 1 in Y i = ^ ° 0 + ^ ° 1 X i + e i (1) is equivalent to °tting a straight line to the data. Visually inspecting the data and °tting a line is not a good idea as it is too arbitrary and imprecise. We choose ° 0 and ° 1 in such a way that the residual
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e i = Y i ° ^ Y i (2) Y i ° ( ^ ° 0 + ^ ° 1 X i ) (3) is as "small" as possible under a certain criterion. Note that the residual, e i , is the vertical deviation of the corresponding data point from the °tted line. There are at least three di±erent criteria that one could use: (1) Minimizing the sum of the residuals : The problem of minimizing the sum of residuals is not well de°ned, we can make the sum of residuals as small as we please if we are allowed to make the sum negative. (2) Minimizing the sum of absolute values of the residuals : This criterion is technically di²cult to use because the absolute value function is not
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di±erentiable at zero. Some people, however, still use it. We will not use
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