04 Population Regression Models

04 Population Regression Models - Economics 140A Population...

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Economics 140A 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, termed regressors, the evolution of which we generally do not choose to model. With a dependent variable in hand, we next select which feature of the conditional distribution of Y t given the regressors that we wish to model. Much of econometrics is concerned with modeling the location of the conditional In this class we focus exclusively on models of conditional location. We begin with the case in which we model (the mean of) Y t as a linear function of only one other variable X t . The variable X t is termed the regressor, to distinguish it from the dependent variable. To express (the mean of) Y t as a linear function of the regressor X t we write Y t = & 0 + 1 X t + U t ; (0.1) which is termed the population regression model. (As we will see below, whether (0.1 ) captures the conditional mean or conditional median, depends on an as- sumption about U t .) The quantities & 0 and 1 numbers) and the random variable U t is the error. The word regression may seem model) modeled children±s height ( Y t ) as a function of parents±height ( X t ) . He discovered a positive relation, but the estimate of 1 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 capture the fact that the chil- dren of the wealthy (wealth and height were highly correlated) were ²regressing³ Y t and X t are jointly normal with approximately equal variances. In fact, we could reverse taller than the children±s average have parents who are less than one inch taller than the average for parents. While mathematically true, the example points to a logical paradox often termed Galton±s regression fallacy. Note that if
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This note was uploaded on 09/04/2011 for the course ECON 140a taught by Professor Staff during the Fall '08 term at UCSB.

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04 Population Regression Models - Economics 140A Population...

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