Lecture12-13

# Lecture12-13 - 39 Lecture 12 Ordinary Least Squares and...

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39 Lecture 12 – Ordinary Least Squares and Multiple Regression 1) Multiple regression - a multiple regression equation is an equation with more than one independent variable. The general form of the multiple regression equation can be written as follows: YX XX k t t =+ + ++ + ββ β β ε .... tt t k 01 1 2 2 . There is really only one change that we must be clear about when we move from a single independent variable to many independent variables. This concerns the interpretation of the 's, i.e., the coefficients on the independent variables. We now interpret these coefficients in the following way: β tells us the amount that the dependent variable, Y , changes when there is a one unit change in independent variable , β i t X it holding all other independent variables constant . Our estimated regression l ine now becomes: \$ \$\$ \$ \$ X k k t X t + β β 2 . Our estimate of the error term is still eY = Y . Y =− \$ t t \$ \$ β β 2 −−− ttk X k t Our criteria for selecting our estimators , i.e., the s, is also the same, i.e., we want to minimize . \$ β e t 2 There is little sense in providing formulas for these estimators. The reason is that the formulas will change depending upon the number of independent variables we have in the equation. Later we will look at the equations in the case of two independent variables. In all cases, however, we will obtain the estimates for these estimators by using the computer, which we will learn to do in the next class. (Those of you who have taken linear algebra may be interested to know that there are formulas that can be given which make use of a matrix.) 2) R-SQUARED is also defined in exactly the same way. That is, R-SQUARED = ESS/TSS. ESS and TSS are also defined in exactly the same way as before. Recall that ESS = \$ YY t d 2 i . The equation above tells us that Y is now found by using an equation with k independent variables. So what R- SQUARED tells us now is the proportion of the deviation of the dependent variable from its typical value which is explained by all of the independent variables taken together. Intuitively, \$ t R-SQUARED is telling us the correlation between the dependent variable and all of the independent variables taken together .

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## This note was uploaded on 03/03/2011 for the course ECO 230 taught by Professor Yongjinpark during the Spring '11 term at Conn College.

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Lecture12-13 - 39 Lecture 12 Ordinary Least Squares and...

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