Econometrics_计量经济学课件_第十四讲 模型变量设定.ppt

True Model
Y 1 2X2 u Y 1 2 X2 3 X3 u
Yˆ b1 b2 X 2
Correct specification, no problems
Fitted Model
Yˆ b1 b2 X 2 b3 X 3
Correct specification, no problems
If Y depends only on X2, and we fit a simple regression model, we will not encounter any problems, assuming of course that the Gauss-Markov conditions are satisfied.
Autumn 2006
SPECIFICATION OF
4
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Consequences of Variable Misspecification
Consequences of Variable Misspecification True Model
Y 1 2X2 u Y 1 2 X2 3 X3 u
Yˆ b1 b2 X 2
Fitted Model
Yˆ b1 b2 X 2 b3 X 3
To keep the analysis simple, we will assume that there are only two possibilities. Either Y
Yˆ b1 b2 X 2
Correct specificationபைடு நூலகம் no problems
Coefficients are biased (in general). Standard errors are invalid.
Yˆ b1 b2 X 2 b3 X 3
Correct specification, no problems
true model is multiple.
Autumn 2006
SPECIFICATION OF
6
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Consequences of Variable Misspecification
True Model
Y 1 2X2 u Y 1 2 X2 3 X3 u
Yˆ b1 b2 X 2
Correct specification, no problems
Fitted Model
Yˆ b1 b2 X 2 b3 X 3
Correct specification, no problems
regressed on X2, which of course is Cov(X2, X3)/Var(X2).
Autumn 2006
SPECIFICATION OF
12
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
X 2)
3
)
Y
effect of X3
direct effect of X2, holding X3 2 constant
3
apparent effect of X2, acting as a mimic for X3
X2
X3
The intuitive reason is that, in addition to its direct effect 2, X2 has an apparent indirect
Consequences of Variable Misspecification
True Model
Y 1 2X2 u Y 1 2 X2 3 X3 u
Yˆ b1 b2 X 2
Correct specification, no problems
Fitted Model
Yˆ b1 b2 X 2 b3 X 3
Fitted Model
The omission of a relevant explanatory variable causes the regression coefficients to be
biased and the standard errors to be invalid.
Autumn 2006
Autumn 2006
SPECIFICATION OF
9
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Y 1 2X2 3X3 u
Yˆ b1 b2 X 2
E (b2
)
2
3
Cov( X 2 , Var( X
X 2)
3
)
In the present case, the omission of X3 causes b2 to be biased by an amount
3 Cov(X2, X3)/Var(X2). We will demonstrate this first intuitively and then mathematically.
Autumn 2006
SPECIFICATION OF
11
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Y 1 2X2 3X3 u
Yˆ b1 b2 X 2
E (b2
)
2
3
Cov( X 2 , Var( X
Y 1 2X2 3X3 u
Yˆ b1 b2 X 2
E (b2
)
2
3
Cov( X 2 , Var( X
X 2)
3
)
Y
effect of X3
direct effect of X2, holding X3 2 constant
3
apparent effect of X2, acting as a mimic for X3
SPECIFICATION OF
8
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Y 1 2X2 3X3 u
Yˆ b1 b2 X 2
E (b2
)
2
3
Cov( X 2 , Var( X
X 2)
3
)
Y
effect of X3
direct effect of X2, holding X3 2 constant
3
apparent effect of X2, acting as a mimic for X3
X2
X3
The ability of X2 to mimic X3 is determined by the slope coefficient obtained when X3 is
Likewise we will not encounter any problems if Y depends on both X2 and X3 and we fit the multiple regression.
Autumn 2006
SPECIFICATION OF
5
REGRESSION VARIABLES
Y 1 2X2 3X3 u
Yˆ b1 b2 X 2
b2
Cov( X 2 ,Y Var( X 2)
)
Cov( X 2 ,[1 2 X 2
Var( X 2)
3
X3
u])
Cov( X 2 , 1 ) Cov( X 2 , 2 X 2 ) Cov( X 2 , 3 X 3 ) Cov( X 2 , u)
In the next one we will do the opposite and examine the consequences of fitting a multiple
regression when the true model is simple.
Autumn 2006
SPECIFICATION OF
regression model in terms of explanatory variables.
Autumn 2006
SPECIFICATION OF
2
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Consequences of Variable Misspecification
True Model
Y 1 2X2 u Y 1 2 X2 3 X3 u
Yˆ b1 b2 X 2
Autumn 2006
SPECIFICATION OF
1
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Consequences of Variable Misspecification True Model
effect as a consequence of acting as a proxy for the missing X3.
Autumn 2006
SPECIFICATION OF
10
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
depends only on X2, or it depends on both X2 and X3.
Autumn 2006
SPECIFICATION OF
3
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
7
REGRESSION VARIABLES
VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
Consequences of Variable Misspecification
True Model
Y 1 2X2 u Y 1 2 X2 3 X3 u
Correct specification, no problems
Fitted Model
Yˆ b1 b2 X 2 b3 X 3
Correct specification, no problems
In this sequence we will examine the consequences of fitting a simple regression when the
第十四讲 模型变量设定
Specification of Regression Variables
• VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE
• VARIABLE MISSPECIFICATION II: INCLUSION OF AN IRRELEVANT VARIABLE
Y 1 2X2 u Y 1 2 X2 3 X3 u
Yˆ b1 b2 X 2
Fitted Model
Yˆ b1 b2 X 2 b3 X 3
In this sequence and the next we will investigate the consequences of misspecifying the
X2
X3
The strength of the proxy effect depends on two factors: the strength of the effect of X3 on
Y, which is given by 3, and the ability of X2 to mimic X3.