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 °ts the data. Unfortunately, di/erent individuals may draw di/erent lines and, 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 must be able to both express what is meant by a ±close °t²and to express any specialized information about the accuracy of speci°c observations. 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 , respectively. For any value of the regressor, the predicted (°tted) 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- ual). (In providing a close °t of the line to the data, note that there are many ways to measure distance from a °tted line. Any point on the °tted line corresponds to ° 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 .) A close °t corresponds to small residuals. But how should we measure whether 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 the value of the estimators that force the sum to minus in°nity, so it makes no sense to discuss minimizing the sum.
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with opposite signs. It would be possible to have large residuals with opposite signs, resulting in a sum of zero without any indication of a close °t.
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