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Unformatted text preview: Economics 140A Regression Model Estimators As discussed last time, we begin with the population regression Y t = & + & 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 &tand to express any specialized information about the accuracy of speci&c observations. We then recognize that the error is likely always present and we must deter- mine how to estimate the coe cients in light of its presence. Let B and B 1 be estimators of & 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 + 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 6 = & and B 1 6 = & 1 . Second, even if B = & and B 1 = & 1 our prediction Y P t di/ers from Y t by U t . Thus Y t & Y P t = ( & & B ) + ( & 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 &...
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This note was uploaded on 09/04/2011 for the course ECON 140a taught by Professor Staff during the Fall '08 term at UCSB.
- Fall '08