Topic 1.2 Estimation.pptx - Topic 1.2 Simple linear...

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Topic 1.2: Simple linear regression (Estimation) 1
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1.2 Estimation Our goal is to estimate the PRF based on the SRF, as accurately as possible But how is the SRF determined? The method of estimation used to achieve that goal: Ordinary Least Squares (OLS) method To understand the OLS method, must first understand the least-squares principle 2
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Estimation Figure 3.1: given n pairs of (X, Y) observations, we need to determine the SRF in such a manner that the SRF is as close as possible to the actual Y To do this, can use either of the two criterion to choose the SRF (1) the minimum residual sum criterion (2) the least-squares sum criterion 3
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(1) the minimum residual sum criterion Choose the SRF in such a way that the sum of the residuals is as small as possible See Fig 3.1 But, this is not a very good criterion Cos the residuals receive the same weight in the sum 4
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(1) the minimum residual sum criterion The residuals uhat_2 & uhat_3, as well as uhat_1 & uhat_4 receive the same weight in the sum: uhat_1 + uhat_2 + uhat_3 + uhat_4 But from Fig 3.1, we notice that uhat_2 & uhat_3 are much closer to the SRF than uhat_1 & uhat_4 5
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(1) the minimum residual sum criterion That is, all the residuals receive equal importance no matter how close or how far apart the observations are from the SRF An unwanted consequence: it’s possible to get a sum of zero, i.e., residuals of the same magnitude but with opposite signs will cancel out each other This will give a wrong impression of no residuals 6
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