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Lecture 3 lecture notes

# Lecture 3 lecture notes - Lecture 3 A Simple Linear...

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Lecture 3. A Simple Linear Regression Model A simple (one-variable) linear regression (SLR) model is given by Y j = β 0 + β 0 X j + ² j , j = 1 , . . . , n, (3.1) where Y j is a dependent variable and X j is an independent (explanatory) variable; β 0 and β 1 are called an intercept and a slop respectively. The Gauss-Markov Theorem . If ² j are uncorrelated random variables with common variance, then of all possible estimators β * 0 and β * 1 that are linear functions of Y t , the least squares (LS) estimators have the smallest variance. Using historical data Y 1 , . . . , Y N we want to obtain optimal estimates β * 0 and β * 1 and further use this information to obtain the forecasts of Y . Let us denote the estimated (historical) values of Y j by Y * j for j = 1 , . . . , n , i.e. Y * = β * 0 + β * 1 X j . Then using the LS approach, we minimize the sum of squares of estimated residuals (SSE) SSE = n X j =1 Y j - Y * j · 2 = n X j =1 ( Y j - β * 0 - β * 1 X j ) 2 (3.2) Hence, we need to obtain the first partial derivatives of SSE with respect to each of β * 0 and β * 1 and set them both equal to 0, solving simultaneously. The solution is given by β * 0 = Y n - β * 1 ¯ x n (3.3) and β * 1 = S XY SS X = n j =1 X j - X n · ‡ Y j - Y n · n j =1 X j - X n · 2 (3.4) It is clear from 3.3 that the two estimates β * 0 and β * 1 are related. (In fact, we should not

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