01 Population Regression Models Notes

# 01 Population Regression Models Notes - Economics 241B...

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Unformatted text preview: Economics 241B Population Regression Models A starting point for work in econometrics is to select a variable of interest, often termed the dependent variable, most frequently denoted Y t . The phrase dependent variable arises, because we attempt to model how Y t depends on other variables, the evolution of which we generally do not choose to model. A regression model is a set of restrictions on the joint distribution of the dependent variable and regressors. We begin with a collection of four baseline assumptions that form the classic regression model. For the classic regression model, we can determine an optimal estimator of the parameters of interest (most often the coe¢ cients). With a dependent variable in hand, we next select which feature of the distri- bution of Y t that we wish to model. Most frequently (exclusively in this class) we choose to model the mean of Y t rather than the median or variance. The Linearity Assumption Assumption 1 (linearity): The regression model is linear in the coe¢ cients with an additive error term. Remarks: For observation t , the population regression model is written as Y t = & 1 X t 1 + & 2 X t 2 + & & & + & K X tK + U t ( t = 1 ; 2 ;:::;n ) where X t 1 = 1 allows for an intercept. The quantities & 1 through & K are un- known coe¢ cients (&xed numbers) and the random variable U t is the error. The coe¢ cients have the interpretation of partial derivatives & 2 = @Y t @X t 2 ; that is ¡The coe¢ cient & 2 measures the e/ect on Y t of a one unit change in X t 2 holding all other regressors constant.£ The linearity assumption is not as restrictive as it may &rst seem, because the dependent variable and the regressors can be transformations of the variables in question log( WAGE t ) = & 1 + & 2 S t + & 3 TENURE t + & 4 EXPR t + U t ; where WAGE is the wage rate for the individual, S education in years, TENURE years on the current job and EXPR is experience in the labor force. The equation is in semi-log form because only the dependent variable is in logs. The coe¢ cients now have the form of percentage changes, rather than changes in levels. For example, if & 3 = : 02 then an additional year of tenure has the e/ect of raising the wage rate by 2 percent. Certain other forms of nonlinearities may be accommodated as well. If the marginal e/ect of job tenure tapers o/ as the level of job tenure get higher, then adding TENURE 2 as an additional regressor captures the e/ect. If the coe¢ cient on TENURE 2 is & 5 , then the marginal e/ect of tenure is & 3 + 2 & 5 & TENURE: If & 5 is negative, the marginal e/ect of tenure declines with the level of tenure. The word regression may seem odd. About a century ago, Sir Francis Galton (in one of the &rst uses of such a model) modeled children¡s height ( Y t ) as a function of parents¡height ( X t ) . He discovered a positive relation, but the estimate of & 2 was less than one. He chose to focus on the fact that the tallest parents had children whose height was closer to the average, and coined the term regression to...
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01 Population Regression Models Notes - Economics 241B...

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