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Lecture 4

# Lecture 4 - LECTURE 4 THE SIMPLE REGRESSION MODEL...

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LECTURE 4: THE SIMPLE REGRESSION MODEL Regression Analysis : The study of the relationship between one variable ( dependent variable ) and one or more other variables ( independent, or explanatory, variables ) 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. o What is the average income for people with a high school diploma? o What is the average income for people with a college degree? To test a hypothesis implied by economic theory o 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. o What will happen to GDP if we change the interest rate? THE POPULATION REGRESSION FUNCTION DETERMINISTIC POPULATION REGRESSION FUNCTION Population Regression Function , E(Y|X), is the conditional mean of the dependent variable (Y) given any value of the independent variable (X). In general, E(Y|X) can have any shape as a function of X. We typically choose to think that the population regression function is a linear function of the conditioning variable(s), that is, we specify a linear regression model . The bivariate linear regression model takes the form E(Y|X) = β 0 + β 1 X o β 0 and β 1 are the unknown population regression parameters or regression coefficients

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o β 0 is the intercept . It measures E(Y|X = 0). o β 1 is the slope . It measures the average marginal change in Y given a small change in X. β 1 = E ( dX dY ) o β 0 and β 1 are unknown and they are the primary objects of interest. STOCHASTIC POPULATION REGRESSION FUNCTION Equivalently the linear regression model can be written as Y = β 0 + β 1 X + ε where ε is an unobservable stochastic, or random, error term ( or disturbance ). o ε is a random variable, i.e. it has some distribution o ε is nothing but the deviation of any realization of Y from its conditional mean, E(Y|X), i.e.
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