For each observation in this sample, it will be the case that
yi = b0 + b1xi + ui
10
Population regression line, sample data points
and the associated error terms
y = b0 + b1x + u
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
of the OLS estimators
2
What is the simple regression model?
y = b0 + b1 x + u
3
Some Terminology
In the simple linear regression model,
where y = b0 + b1x + u, we typically
第一章 简单回归模型
y = b0Hale Waihona Puke + b1x + u
要求： 1、普通最小二乘估计方法(OLS) 2、OLS的统计特性
1
Contents
What is the simple regression model? How to derive the ordinary least
squares (OLS) estimates? Properties of OLS statistics and R2 Unbiasedness of OLS and Variances
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
5
A Simple Assumption
y = b0 + b1x + u
The average value of u, the error term, in the population is 0. That is,
E(u) = 0
This is not a restrictive assumption,
Cov(x,u) = E(xu) = 0
Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y)
12
Deriving OLS continued
We can write our 2 restrictions just in
9
Ordinary Least Squares
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
terms of x, y, b0 and b1 , since y = b0 + b1x + u,
u = y – b0 – b1x
E(y – b0 – b1x) = 0 E[x(y – b0 – b1x)] = 0
These are called moment restrictions
y y4
E(y|x. ) = b0 + b1x
u4 {
y3 y2
u2{.
.} u3
y1
.} u1
x1
x2
x3
x4
x
11
Deriving OLS Estimates
To derive the OLS estimates we need to realize that our main assumption of E(u|x) = E(u) = 0 also implies that
since we can always use b0 to
normalize E(u) to 0
wage = b0 + b1educ + u
6
Zero Conditional Mean
y = b0 + b1x + u
We need to make a crucial assumption about how u and x are related
refer to y as the
Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Response Variable, or Regressand
4
Some Terminology, cont.
y
f(y)
.
.
E(y|x) = b0 + b1x
Population
Regression
Function
x1
x2
How to estimate the
parameters b0 and b1?
8
How to derive the ordinary least squares (OLS) estimates?
E(u|x) = E(u) = 0, which implies
E(y|x) = b0 + b1x, which is often called
Population Regression Function (PRF)
7
E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x)