ECON206_1011_01_Handout_11

ECON206_1011_01_Handout_11 - ECON 206 METU- Department of...

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ECON 206 January 03, 2011 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 1 LECTURE 11 SIMPLE REGRESSION MODEL - III Outline of today’s lecture: I. Variance of the random variable u ..................................................................................... 1 II. A Measure of Goodness of Fit: Coefficient of Determination ( 2 R ) . .............................. 2 A. Relationship between 2 R and Correlation Coefficient . .............................................. 7 III. Sampling Distribution of the Least Squares Estimates . ................................................... 9 IV. Multiple Regression Analysis . ....................................................................................... 10 V. Two Explanatory Variable Regression . .......................................................................... 11 VI. Testing Hypotheses . ....................................................................................................... 12 A. Testing Individual Coefficients . .................................................................................. 12 One-Tailed Test .......................................................................................................... 12 Two-Tailed Test . .......................................................................................................... 13 B. Testing Several Coefficients Jointly . .......................................................................... 14 VII. Confidence Intervals for β k ........................................................................................... 16 I. Variance of the random variable u The formulae of the variance of 0 β ± and 1 ± involve the variance of the random term u , which we have denoted by 2 σ . However, the true variance of t u can not be computed since the values of t u are not observable. But we may obtain an unbiased estimate of 2 from the expression 2 2 2 1 2 T t t u S T = == ± ± where t u ± is the OLS residual, and hence it is given by tt t uY Y =− ± ± .
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ECON 206 January 03, 2011 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 2 II. A Measure of Goodness of Fit: Coefficient of Determination ( 2 R ) We now consider the goodness of fit of the fitted regression line to a set of data; that is, we shall find out how “well” the sample regression line fits the data. The coefficient of determination 2 r (two-variable case) or 2 R (multiple regression) is a summary measure that tells how well the sample regression line fits the data. We need to know how “good” is the fit of the regression line to the sample observations of Y and X, that is to say we need to measure the dispersion around the regression line. This knowledge is essential, because the closer the observations to the line, the better the goodness of fit, that is better is the explanation of the variations of Y by the changes in the explanatory variables. Below, we will show prove that a measure of the goodness of fit is the square of the correlation coefficient, 2 r , which shows the percentage of the total variation of the dependent variable that can be explained by the independent variable X.
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ECON 206 January 03, 2011 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 3 Figure 2. Breakdown of the variation of t Y into two components. By fitting the line 01 ˆˆ tt YX β =+ we try to obtain the explanation of the variations of the dependent variable Y produced by the changes of the explanatory variable X . However, the fact that the observations deviate from the estimated line shows that the regression line explain only a part of the total variation of the dependent variable. A part of the variation, defined as t uY Y =− ± ± , remains unexplained.
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This note was uploaded on 03/22/2012 for the course ECON 106 taught by Professor Kücüksenel during the Spring '12 term at Middle East Technical University.

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ECON206_1011_01_Handout_11 - ECON 206 METU- Department of...

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