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Unformatted text preview: Lecture Notes 3 Econ 410 Introduction to Econometrics 1 Linear Regression with One Regressor The linear regression model with one regressor postulates a linear relationship between the random variables X and Y . This linear relationship can be expressed as: X Y 1 + = Ex.: Y is the random variable earnings and X is the random variable years of schooling. The linear relationship between these two variables can be written as: schooling of Years Earnings + = 1 Is the behavior of Y exactly explained by the relationship: X Y 1 + = ? Probably not. Indeed, the random variable X is probably only one of the elements affecting Y . Ex.: Assume X and Y are defined as in the previous example. Are earnings dependent only on the number of years of education? No, other variables influence the earnings of an individual, for instance how smart he/she is, his/her ability at work, etc. Earnings can also be affected by random reasons having to do, for instance, with the performance on the day of the interview, etc. So, we need to account for the other factors that influence the random variable Y . We will see later how to include them explicitly, for now we add them altogether: i i i u X Y + + = 1 Ex.: Assume X and Y are defined as above. Then i Y , i X and i u would be the earnings, the years of education and the other factors influencing the earnings for the i th person respectively. This last equation is the linear regression model with a single regressor; Y is the dependent variable, while X is called independent variable or regressor. The first part of the equation, i X 1 + , is called population regression line and represents a relationship that holds in average in the population. According to this equation, if we knew the value of the independent variable X we would predict that the value of the dependent variable Y is: X 1 + . The intercept and the slope 1 are called coefficients or parameters of the population regression line. The slope 1 is the change in Y associated with a one-unit change in X , while the intercept is the value of Y when = X (that is the value at which the line crosses the Y axis) . The term i u is the error term and incorporates all the factors for which the value of Y for the i th person is different from what predicted by the population regression line. Lecture Notes 3 Econ 410 Introduction to Econometrics 2 What is the expected change in Y following a change in X ? What is the expected level of Y for a given level of X ? We can use the population regression line to answer these questions. Indeed: 1 , the slope of the population regression line, is the expected effect of a one-unit change in X on Y . In our example, 1 is the expected change in earnings if we undertake one additional year of schooling....
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- Fall '08