oh4 - LECTURE 4 THE SIMPLE REGRESSION MODEL Regression...

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LECTURE 4: THE SIMPLE REGRESSION MODEL Regression Analysis : The study of the relationship be- tween one variable ( dependent variable ) and one or more other variables ( independent, or explanatory, vari- ables ) using a (typically linear) regression model. What do we use regression analysis for? ° To estimate the mean or average value of the dependent variable, given the values of the independent variables. ° What is the average income for people with a high school diploma? What is the average income for peo- ple with a college degree? ° To test a hypothesis implied by economic theory ° If we increase the price will the quantity demanded fall? ° To predict, or forecast, the mean value of the dependent variable given the independent variables. ° What will happen to GDP if we change the interest rate? 1
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THE POPULATION REGRESSION FUNCTION DETERMINISTIC POPULATION REGRESSION FUNC- TION ° Population Regression Function, E ( Y j X ) ; is the conditional mean of the dependent variable ( Y ) given any value of the independent variable ( X ) ° In general, E [ Y j X ] can have any shape as a function of X: ° We typically choose to think that the population regres- sion function is a linear function of the conditioning vari- able(s), that is we specify a linear regression model. ° The bivariate linear regression model takes the form E ( Y j X ) = ° 1 + ° 2 X ° ° 1 and ° 2 are the unknown population regression parameters or regression coe¢ cients ° ° 1 is the intercept. It measures E ( Y j X = 0) : 2
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° ° 2 is the slope. It measures the average marginal change in Y given a small change in X: ° 1 = E ° dY dX ± ° ° 1 and ° 2 are unknown and they are the primary objects of interest. 3
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STOCHASTIC POPULATION REGRESSION FUNCTION ° Equivalently the linear regression model can be written as Y = ° 1 + ° 2 X + " where " is an unobservable stochastic, or random, error term (or disturbance). ° " is an random variable, i.e. it has some distribution ° " is nothing but the deviation of any realization of Y from its conditional mean, E ( Y j X ) ; i.e.
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