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EC228.F2009.nn03

# EC228.F2009.nn03 - Wooldridge Introductory Econometrics 4th...

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Unformatted text preview: Wooldridge, Introductory Econometrics, 4th ed. Chapter 2: The simple regression model Most of this course will be concerned with use of a regression model: a structure in which one or more explanatory variables are consid- ered to generate an outcome variable, or de- pendent variable.We begin by considering the simple regression model, in which a single ex- planatory, or independent, variable is involved. We often speak of this as ‘two-variable’ regres- sion, or ‘Y on X regression’. Algebraically, y i = β + β 1 x i + u i (1) is the relationship presumed to hold in the pop- ulation for each observation i. The values of y are expected to lie on a straight line, depending on the corresponding values of x. Their values will differ from those predicted by that line by the amount of the error term, or disturbance, u, which expresses the net effect of all factors other than x on the outcome y- that is, it re- flects the assumption of ceteris paribus. We often speak of x as the ‘regressor’ in this rela- tionship; less commonly we speak of y as the ‘regressand.’ The coefficients of the relation- ship, β and β 1 , are the regression parameters, to be estimated from a sample. They are pre- sumed constant in the population, so that the effect of a one-unit change in x on y is assumed constant for all values of x. As long as we include an intercept in the rela- tionship, we can always assume that E ( u ) = 0 , since a nonzero mean for u could be absorbed by the intercept term. The crucial assumption in this regression model involves the relationship between x and u. We consider x a random variable, as is u, and con- cern ourselves with the conditional distribution of u given x. If that distribution is equivalent to the unconditional distribution of u, then we can conclude that there is no relationship between x and u- which, as we will see, makes the es- timation problem much more straightforward. To state this formally, we assume that E ( u | x ) = E ( u ) = 0 (2) or that the u process has a zero conditional mean . This assumption states that the unob- served factors involved in the regression func- tion are not related in any systematic manner to the observed factors. For instance, con- sider a regression of individuals’ hourly wage on the number of years of education they have completed. There are, of course, many factors influencing the hourly wage earned beyond the number of years of formal schooling. In work- ing with this regression function, we are as- suming that the unobserved factors–excluded from the regression we estimate, and thus rel- egated to the u term–are not systematically related to years of formal schooling. This may not be a tenable assumption; we might con- sider “innate ability” as such a factor, and it is probably related to success in both the edu- cational process and the workplace. Thus, in- nate ability–which we cannot measure without some proxies–may be positively correlated to the education variable, which would invalidate...
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EC228.F2009.nn03 - Wooldridge Introductory Econometrics 4th...

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