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Some Terminology, cont.
β0 :intercept parameter β1 :slope parameter means that a one-unit increase in x changes the expected value of y by the amount β1,holding the other factors in u fixed.
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For each observation in this sample, it will be the case that: yi = b0 + b1xi + ui
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E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x)
y
f(y)
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A possible model for E [Wages|Education] is a linear one E [Wages|Education] = β0 + β1Education, where β0 and β1 are unknown parameters that we would like to estimate. β1 is the change of the mean (expected value) of Wages for one additional year of Education.
Two methods to estimate
Part 1 Regression Analysis with Cross-Sectional Data
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Ch2 The Simple Regression Model
y = b0 + b1x + u
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The model can be written in the more familiar econometric terms Wages = β0 + β1Education + u, This model is known as The Simple Regression Model. It is linear in the parameters β0 and β1 (and in the explanatory variables).
This is not a restrictive assumption, since we can always use b0 to normalize E(u) to 0.
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Zero Conditional Mean
In the simple linear regression of y on x, we typically refer to x as the



Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Covariate, or Control Variables
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Basic idea of regression is to estimate the population parameters from a sample Let {(xi,yi): i=1, …,n} denote a random sample of size n from the population Example : A particular realization of a sample is:
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1 Motivation for the linear regression model
Economic Theory suggests interesting relations between variables. Example 1: Returns to education A model of human capital investment predicts that getting more education should lead to higher wages: Wages = f (Education). However, let us look at a data set: US national survey of people in the labour force that already completed their education, 526 2016/9/22 Department of Statistics-Zhaoliqin 3 people.
Population regression line, sample data points and the associated error terms
y y4 E(y|x) = b0 + b1x . u4 {
y3 y2
u2 {.
.} u3
y1
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x4
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Hence, we are interested in studying is the mean of wages given the years of Education that will be denoted as E [Wages|Education] Following economic theory, we assume a specific model for E[Wages|Education]
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A possibility is to look at means of wages conditional on the years of Education.
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Graph above show:People with the same years of education have different hourly wages. How can we study if the evidence of the data supports Economic Theory?
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2 A Simple Assumption
The average value of u, the error term, in the population is 0. That is, E(u) = 0(assume that things such as average ability and land quality to have same effect on y)

Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Regressand
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Some Terminology, cont.
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4 The Simple Regression Model: estimation
The parameters β0 and β1 are unknown and are the population parameters. Our objective is to find estimators for β0 and β1. Recall that before we had
We need to make a crucial assumption about how u and x are related We want it to be the case that knowing something about x does not give us any information about u, so that they are completely unrelated. That is, that E(u|x) = E(u) = 0, which implies E(y|x) = b0 + b1x(Population Regression Function [PRF]