Examples on Simple and Multiple Linear Regression

© 2002 The Wadsworth Group
Direct vs Inverse Relationships
• Direct relationship:
– As x increases, y increases. – The graph of the model rises from left to right. – The slope of the linear model is positive.
– Coefficient of correlation. – Coefficient of determination.
• Construct confidence intervals and carry out hypothesis tests involving the slope of the regressios Regression Line
• Least Squares Regression Line:
ˆ = b + bx y 0 1 1
– Slope
( x y ) – n× x× y i i b = 1 ( x 2 ) – n× x2 i – y-intercept
CHAPTER 15 Simple Linear Regression and Correlation
to accompany
Introduction to Business Statistics
fourth edition, by Ronald M. Weiers
Presentation by Priscilla Chaffe-Stengel Donald N. Stengel
© 2002 The Wadsworth Group

Introductory Econometrics
4
Motivation : Advantage
It can explain more of the variation in the dependent variable.
It can incorporate more general functional form.
If other factors that affecting y are not correlated with x, changing x can ensure that u is not changed, and the effect of x on y can be identified.
Multiple regression analysis is more amenable to ceteris paribus analysis because it allows us to explicitly control for many other factors that simultaneously affect the dependent variable.
6
Motivation: An Example
Consider a model that says family
consumption is a quadratic function of family income:
Cons = b0 + b1 inc+b2 inc2 +u
Now the marginal propensity to consume is approximated by
pcolGPA: predicted values of college grade point average

Regression Analysis
Chapter 11
Regression and Correlation
Techniques that are used to establish whether there is a mathematical relationship between two or more variables, so that the behavior of one variable can be used to predict the behavior of others. Applicable to “Variables” data only.
run
axis.
b
0
X
A simple linear relationship can be described mathematically by
Y = mX + b
Simple Linear Regression
slope =
rise run
=
(6 - 3)
1
=
(10 - 4)
2
Y
rise
5
run intercept = 1
Rent
Step 1: Scatter plot
2500 2300 2100 1900 1700 1500 1300 1100 900 700 500
500 700 900 1100 1300 1500 1700 1900 2100
Size
Scatter plot suggests that there is a ‘linear’ relationship between Rent and Size
High

20
（四）自变量的筛选
基本思路：尽可能将回归效果显著的自变量选入方程 中，作用不显著的自变量排除在外。 （1）全局择优法（all possible subsets selection）： （2）逐步选择法
前进法（Forward selection） 后退法（Backward elimination） 逐步法（Stepwise）
Sum of Square s
1
Re g re ssi o n
13 3.71 1
df Me an Square
4
33 .4 28
F
S i g.
8.278 .000a
Re si dua l
88 .8 41
22
4.03 8
To tal
22 2.55 2
26
a. P redict ors: (Const ant ), 总 胆 固醇 x1, 胰 岛 素x3, 糖 化 血红 蛋 白 x4, 甘 油 三脂 x2
β0为回归方程的常数项（constant)，表示各自变量均为0时y的平 均值；
m为自变量的个数； β1、β2、βm为偏回归系数（Partial regression coefficient）
意义：如β1 表示在X2、X3 …… Xm固定条件下，X1 每增减 一个单位对Y 的效应（Y 增减β个单位）。 e为去除m个自变量对Y影响后的随机误差，称残差（residual)。
Sig. .047 .701 .099 .036 .016
将总胆固醇(X1) 剔除。 注意：通常每次只剔除关系最弱的一个因素。
对于同一资料，不同自变量的t值可以相互比较，t的绝对
值越大，或P越小，说明该自变量对Y所起的作用越大。