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# lect4_103_2010_Compatibility_Mode_ - 1 Estimation I • As...

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Unformatted text preview: 4/7/2010 1 Estimation - I • As before, we are going to try to use our random sample to learn about the population model. • More specifically, we are going to use our sample to try to estimate the parameters β and β 1 . Why? Well, β 1 , for example, measures the effect of changing X on Y , exactly what we are trying to learn. Estimation - II • We are going to perform our estimation under 3 key assumptions: – A1) Corr( X i , u i ) = Cov( X i , u i ) = 0. This is saying that the errors u i are uncorrelated with the regressors X i . – A2) ( X i , Y i ) are i.i.d. (independent and identically distributed). This is the case if we have a random sample. – A3) (technical) X i and Y i have finite fourth moments – i.e. E [ X i 4 ] < u221e and E [ Y i 4 ] < u221e . In practice, this means that large outliers , i.e. values of X i and Y i that are far outside the range of the data, are very unlikely. • Later, we are going to pay particular attention to Assumption A1, and the question of whether it holds. But for now, lets just assume that it does. 4/7/2010 2 Estimation - III • The most common way of estimating the parameters β and β 1 is called Ordinary Least Squares (OLS) • The OLS estimators of β and β 1 , which we will call β and β 1 , minimize the following quantity: • In words, β and β 1 minimize the sum of squared vertical distances between the values Y i and the OLS Regression Line β + β 1 X i . Intuitively, this is making the Regression line “as close as possible” to the points on the scatterplot. Graphic illustration....
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lect4_103_2010_Compatibility_Mode_ - 1 Estimation I • As...

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