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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 distribution, although recent work in °nance has focused on the conditional scale. 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 are unknown coe¢ cients (°xed numbers) and the random variable U t is the error. 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 ° 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 to be like the children of the poor. Such a °nding is to be expected if Y t and X t are jointly normal with approximately equal variances. In fact, we could reverse the regression and also °nd a slope of less than 1, implying that children an inch 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 U t = 0 for each observation t , then (0.1) is the equation for a line with intercept ° 0 and slope ° 1 . Although the coe¢ cients are unknown, any two observations on Y t and X t would reveal the population value of the coe¢ cients.

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

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