L11 Ordinary Least Squares Regression

L11 Ordinary Least Squares Regression - Ordinary Least...

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Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions. In this part of the course we are going to study a technique for analysing the linear relationship between two variables Y and X. We have n pairs of observations (Y i X i ), i = 1, 2, . .,n on the relationship which, because it is not exact, we shall write as: 1,. ..., ii i y xin α β = + +∈ = In this relationship α , β and ε i i = 1,. ., n are fundamentally unobservable and we would like to estimate α and β . Approach: The idea is to select estimates of α and β lets call them α * and β * which yield a straight line (called the regression line) Y = α * + β *X which minimises a measure of the aggregate distance of the points (Y i X i ), i = 1, 2, . .,n to that line in X Y space, where Y is measured on the vertical axis. The measure we use is the sum of squared vertical distances which we shall call the Error Sum of Squares (ERSS) so that α * and β * are solutions to the problem: () 2 , 1 min n ti i ERSS Y X αβ ∗∗ = ⎛⎞ =− + ⎜⎟ ⎝⎠ The multivariate calculus is employed to solve this problem by setting the partial derivatives of ERSS with respect to α * and β * to zero (called the first order conditions) and solving thus: 1 1 20 n i n i ERSS YX ERSS X = = + = + = i The solutions to which are:
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() 1 1 n ii i n i YX YYX X XX αβ β ∗∗ = = =− = Note the formula for β * has many alternative equivalent versions which may be seen by observing that for the numerator: 11 1 1 nn n n i i i i i i YYX X X X Y X Yn Y X == = = −= = ∑∑ and for the denominator: 2 2 1 n i i i i X X X XX X n X = so that various combinations of numerator and denominator formulae yield 12 alternative representations of β * . Assumptions in the Ordinary Least Squares model. Note that while α , β and ε i , i = 1,. ., n are fundamentally unobservable we only concern ourselves with estimating α and β which define the relationship between Y and X. The ε i i = 1,. ., n are considered “errors” which accommodate all the other influences on Y not accounted for in α + β X as such we assume them to be random and to obey the following four assumptions: 1) E( ε i ) = 0 for all i. This assumption really says that the average or net effect of all the other influences on Y not accounted for in α + β X is constant and zero for each observation (i=1,. .,n). 2) V( ε i ) = σ 2 > 0 for all i.
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This note was uploaded on 03/23/2011 for the course ECONOMICS 209 taught by Professor Kambourov during the Spring '11 term at University of Toronto- Toronto.

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L11 Ordinary Least Squares Regression - Ordinary Least...

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