08.leastsquares - Lecture 8: Ordinary Least Squares...

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    Lecture 8:  Ordinary Least Squares Estimation BUEC 333  Summer 2009 Simon Woodcock
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    From Last Day Recall our population regression function: Because the coefficients ( β ) and the errors ( ε i ) are population quantities, we don’t observe them. Sometimes our primary interest is the coefficients themselves β k measures the marginal effect of variable X ki on the dependent variable Y i . Sometimes we’re more interested in predicting Y i. if we have sample estimates of the coefficients, we can calculate predicted values: In either case, we need a way to estimate the unknown β’ s. That is, we need a way to compute from a sample of data It turns out there are lots of ways to estimate the β’ s (compute ). By far the most common method is called ordinary least squares (OLS) . i ki k i i i i X X X X Y ε β + + + + + + = 3 3 2 2 1 1 0 ki k i i i X X X Y ˆ ˆ ˆ ˆ ˆ 2 2 1 1 0 + + + + = s ' ˆ s ' ˆ
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    What OLS does Recall that we can write: where e i are the residuals. these are the sample counterpart to the population errors ε i they measure how far our predicted values ( ) are from the true Y i think of them as prediction mistakes We want to estimate the β’s in a way that makes the residuals as small as possible. we want the predicted values as close to the truth as possible OLS minimizes the sum of squared residuals : i i i ki k i i i ki k i i i e Y e X X X X X X Y + = + + + + + = + + + + + = ˆ ˆ ˆ ˆ ˆ 2 2 1 1 0 2 2 1 1 0 β ε i Y ˆ ( 29 = = - = n i n i i i i Y Y e 1 1 2 2 ˆ minimizes OLS
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    Why OLS? OLS is “easy”
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08.leastsquares - Lecture 8: Ordinary Least Squares...

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