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1 on the preceding page. 6) is true, it is useful to break y into two components. The piece ␤0 ϩ ␤1x is sometimes called the systematic part of y — that is, the part of y explained by x— and u is called the unsystematic part, or the part of y not explained by x. 6) in the next section for motivating estimates of ␤0 and ␤1. 5. 1). To do this, we need a sample from the population. Let {(xi,yi): iϭ1,…,n} denote a random sample of size n from the population. 9) for each i. Here, ui is the error term for observation i because it contains all factors affecting yi other than xi .

In any case, we will see examples of this model in later chapters. 3 gives the interpretation of ␤1. In the log-level model, 100 и␤1 is sometimes called the semi-elasticity of y with respect to x. 11, in the log-log model, ␤1 is the elasticity of y with respect to x. 3 warrants careful study, as we will refer to it often in the remainder of the text. The Meaning of “Linear” Regression The simple regression model that we have studied in this chapter is also called the simple linear regression model.

3. Later on, we will explore the relationship between the errors and the residuals. 48) is also needed for some of the statistical derivations. 7. 2, the OLS slope and intercept estimates are not defined unless we have some sample variation in the explanatory variable. We now add variation in the xi to our list of assumptions. 3 (Sample Variation in the Explanatory Variable) The sample outcomes on x, namely, {xi , i ϭ 1,…,n}, are not all the same value. This is a very weak assumption—certainly not worth emphasizing, but needed nevertheless.

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An Intro. to Modern Japanese [Book 1 - Grammar] by R. Bowring, et. al.,

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