103lect4hand

103lect4hand - Estimation - I As before, we are going to...

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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 0 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. (this is ensured by random sampling) – A3) (technical) X i and Y i have finite fourth moments – i.e. E [ X i 4 ] < and E [ Y i 4 ] < . 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. Estimation - III • The most common way of estimating the parameters β 0 and β 1 is called Ordinary Least Squares (OLS) • The OLS estimators of β 0 and β 1 , which we will call β 0 and β 1 , minimize the following quantity: • In words, β 0 and β 1 minimize the sum of squared distances between the values Y i and the OLS Regression Line β 0 + β 1 X i . Intuitively, this is just saying that β 0 and β 1 are chosen to make the line “as close as possible” to the points on the scatterplot. Graphic illustration.
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103lect4hand - Estimation - I As before, we are going to...

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