1
1.1
BASIC FORM OF ENDOGENEITY PROBLEM
Omitted Variables Bias
𝛽"!"#$ = 𝛽! + 𝛽%
𝐶𝑜𝑣(𝑥!& , 𝑥%& )
𝑉𝑎𝑟 (𝑥!& )
1.
在多元回归中加⼊变量
2.
利⽤同⼀观测个体在不同时间点上的观测数据
3.
固定效应可控制不随时变，或者不随个体变化的遗漏变量
4.
如果遗漏变量不可观测，利⽤⼯具变量或代理变量
5.
采⽤随机对照实验
1.2
Reversed Causality
1.3
Selection Bias
1.4
Measurement Error
1.5
Misspecification
Name
Functional
Interpretation
Linear
Y=β1+β2X
1 unit change in X will induce a β2 unit change in Y
Linear-log Y=β1+β2lnX 1 percent change in X will induce a β2/100 unit change in Y
Log-linear lnY=β1 +β2X 1 unit change in X will induce a 100β2 percent change in Y
Double-log lnY=β1+β2lnX 1 percent change in X will induce a β2 percent change in Y
1
2
2.1
SOLUTION OF MODERATING EFFECT
Definition of Moderating Effect
Effect of a predictor variable (X) on a criterion(Y) depends on a third variable
(M), the moderator.
Y = 𝑏! + 𝑏" 𝑋 + 𝑏# 𝑀 + 𝑏$ (𝑋 ∗ 𝑀) + 𝜀
The slope and intercept of regression of Y on X depends upon the specific value of
M. Hence, there is a different line for every individual value of M (simple regression
line). If b3 is significant, the relationship between X and Y depends on the level of M.
The b3 coefficient reflects the interaction between X and M only if the lower order
terms b1X and b2M are included in the equation. Leaving out these terms confounds
the additive and multiplicative effects, producing misleading results.
2.2
Types of Moderating Effect
Distinguishment between moderating effect and heterogeneity analysis is the
moderator, if moderator isn’t dummy variable, that is moderating effect; if moderator
is dummy variable, that is heterogeneity analysis. There are two equivalent ways to
evaluate whether an interaction is present: Test whether the increment in the squared
multiple correlation (R2) given by the interaction is significantly greater than zero. Test
whether the coefficient b3 differs significantly from zero.
主效应符号(main terms)和交叉项符号(interaction term)的关系，同号促进，
异号抑制。主效应符号(main terms)和交叉项符号(interaction term)的关系为异
号时，如果主效应和调节变量同号，⼆者可能存在理论上的替代效应。
2
主效应符号
交叉项符号
调节变量符号
含义
M 显著为正
强化作⽤
M 显著为负
强化作⽤
M 显著为正
削弱（替代）
M 显著为负
削弱作⽤
M 显著为正
削弱作⽤
M 显著为负
削弱（替代）
M 显著为正
强化作⽤
M 显著为负
强化作⽤
交叉项 X*M 显著为正
X 对 Y 显著正向影响
交叉项 X*M 显著为负
交叉项 X*M 显著为正
X 对 Y 显著负向影响
交叉项 X*M 显著为负
2.3
Example of Moderating Effect by Stata
𝑉(𝛼" + 𝛼# 𝑥# ) = 𝑉(𝛼" ) + 𝑥## 𝑉(𝛼# ) + 2𝑥# 𝐶𝑜𝑣(𝛼" , 𝛼# )
𝑠. 𝑒 = 6𝑉(𝛼" ) + 𝑥## 𝑉(𝛼# ) + 2𝑥# 𝐶𝑜𝑣(𝛼" , 𝛼# )
Aiken and West (1991)
3
example
To investigate the role of FDI on the relationship between GDP and financial
development.
𝑌8 = 𝑏! + 𝑏" 𝑙𝑔𝑑𝑝 + 𝑏# 𝑙𝑓𝑑𝑖 + 𝑏$ 𝑙𝑐𝑝𝑖 + 𝑏% (lfcaital)
+𝑏& (avsch1) + 𝑏' (𝑙𝑔𝑑𝑝 ∗ 𝑙𝑓𝑑𝑖) + 𝜀
𝑙𝑑𝑐𝑝𝑠 = 1.473113 + 0.3628707𝑙𝑔𝑑𝑝 − 0.004733𝑙𝑓𝑑𝑖 − 0.315757𝑙𝑐𝑝𝑖
+ 0.2101182𝑙𝑓𝑐𝑎𝑝𝑖𝑡𝑎𝑙 + 0.0796701𝑎𝑣𝑠𝑐ℎ𝑙
+ 0.0464659(𝑙𝑔𝑑𝑝 ∗ 𝑙𝑓𝑑𝑖)
W
𝛿𝑙𝑑𝑐𝑝𝑠
= 0.3628707 + 0.0464659𝑙𝑓𝑑𝑖
W
𝛿𝑙𝑔𝑑𝑝
4
𝛽Y =
W
𝛿𝑙𝑑𝑐𝑝𝑠
= 0.3628707 + 0.0464659 ∗ 1.02617 = 0.4105
W
𝛿𝑙𝑔𝑑𝑝
σ
[# ()*+,- = var]β8" _ + lfdi# ∗ var]β8' _ + 2lfdi ∗ cov]β8" β8' _
().*,
𝜎c # /01234 = 0.00130778 + 1.02617# ∗ 0.00046553 + 2 ∗ 1.02617 ∗ 0.00019862
/0513
𝑠𝑒 = 6𝜎c # /01234 = √0.002205 = 0.047
/0513
t=
2.4
𝛽Y
0.4105
=
= 8.73
𝑠𝑒
0.047
Explanation of Centering
5
Centering Variables
Centering means subtracting the mean from a variable, leaving deviation scores.
There are advantages to be gained from centering independent variables: Centering can
make otherwise uninterpretable regression coefficients meaningful, and Centering
reduces multicollinearity among predictor variables.
Centering to reduce multicollinearity is particularly useful when the regression
involves squares or cubes of IVs. Centering has no effect at all on linear regression
coefficients (except for the intercept) unless at least one interaction term is included.
The more the IVs are correlated, the smaller their regression weights and the larger their
standard errors tend to be.
6
3
SOLUTION OF GMM
7