02 Regression Model Estimators

02 Regression Model Estimators - Economics 241B Regression...

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Economics 241B Regression Model Estimators As discussed last time, we begin with the population regression Y t = & 0 + 1 X t + U t : How then should we infer (estimate) the value of the coe¢ cients? We could plot the observations on graph paper and draw a line that we feel most closely even for one individual the same line may not be drawn for two identical sets of observations. We wish to produce a method of inferring the value of the coe¢ cients that is: 1) reproducible, and 2) easily communicated. To be easily communicated We recognize that the error is likely always present and we must determine how to estimate the coe¢ cients in light of its presence. Let B 0 and B 1 be estimators of & 0 and 1 value of the dependent variable given by the regression model is Y P t = B 0 + B 1 X t : Because the error is not zero, there are two reasons why Y P t will not equal Y t . First, if U t is not zero, then in general B 0 6 = & 0 and B 1 6 = 1 . Second, even if B 0 = & 0 and B 1 = 1 our prediction Y P t di/ers from Y t by U t . Thus Y t Y P t = ( & 0 B 0 ) + ( 1 B 1 ) X t + U t ± U P t ; where U P t is the predicted value of the unobserved error (often termed the resid- Y P t ;X t ± while the actual data are ( Y t ;X t ) . We implicitly use vertical distance. Geometrically one could use horizontal distance, if predicting X t , or orthogonal distance, if predicting a linear combination of X t and Y t .) or not the residuals are small? One idea is simply to sum the residuals and choose values of the estimators that make P n t =1 ( Y t B 0 B 1 X t ) as close to zero as possible. 1 Yet predictions can be both too large and too small, yielding residuals 1 Some texts discuss minimizing the sum. Of course, we would minimize the sum by choosing minimizing the sum.
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with opposite signs. It would be possible to have large residuals with opposite
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This note was uploaded on 12/26/2011 for the course ECON 241b taught by Professor Staff during the Fall '08 term at UCSB.

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02 Regression Model Estimators - Economics 241B Regression...

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