代写Instrumental Variables EstimationStage Least Squares
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代写Instrumental Variables EstimationStage Least Squares
From Chapter 15 you will learn
Motivation: Omitted variables in a simple regression model
IV estimation of multiple regression model
Two stage least square
Testing for endogeneity and testing for overidentifying restrictions
The endogeneity problem is endemic in social sciences/economics
In many cases important personal variables cannot be observed
These are often correlated with observed explanatory information
The endogeneity problem is endemic in social sciences/economics
In many cases important personal variables cannot be observed
These are often correlated with observed explanatory information
Measurement error may lead to endogeneity
The endogeneity problem is endemic in social sciences/economics
In many cases important personal variables cannot be observed
These are often correlated with observed explanatory information
Measurement error may lead to endogeneity
Jointly detremined dependent variables are endogenous
The endogeneity problem is endemic in social sciences/economics
In many cases important personal variables cannot be observed
These are often correlated with observed explanatory information
Measurement error may lead to endogeneity
Jointly detremined dependent variables are endogenous
Solutions to endogeneity problems considered so far:
•Proxy variables method for omitted regressors
•Fixed effects methods if 1) panel data is available, 2) endogeneity is time-constant, and 3) regressors are time-constant
The endogeneity problem is endemic in social sciences/economics
In many cases important personal variables cannot be observed
These are often correlated with observed explanatory information
Measurement error may lead to endogeneity
Jointly detremined dependent variables are endogenous
Solutions to endogeneity problems considered so far:
•Proxy variables method for omitted regressors
•Fixed effects methods if 1) panel data is available, 2) endogeneity is time-constant, and 3) regressors are time-constant
代写Instrumental Variables EstimationStage Least Squares
Instrumental variables method (IV)
IV is the most well-known method to address endogeneity problems
Example: Education in a wage equation
Definition of a instrumental variable:
1) It does not appear in the regression
2) It is highly correlated with the endogenous variable
3) It is uncorrelated with the error term
Reconsideration of OLS in a simple regression model
A simple consistency proof for OLS under exogeneity:
Assume existence of an instrumental variable :
Inference with IV estimation
Assume homoscedasticity holds: E(u 2|z) = Var(u) = s 2.
Example: Father‘s education as an IV for education
Other IVs for education that have been used in the literature:
The number of siblings
1) Correlated with education because of resource constraints; 2) Uncorrelated with innate ability
College proximity when 18 years old
Correlated with education because more education if lived near college; 2) Uncorrelated with error
Month of birth
1) Correlated with education because of compulsory school attendance laws, 2) Uncorrelated with error
Properties of IV with a poor instrumental variable
IV may be much more inconsistent than OLS if the instrumental variable is not completely exogenous and only weakly related to
Computing R-squared after IV estimation
where SSR is the sum of squared IV residuals, and SST is the total sum of squares of y.
If SSR > SST, R-squred after IV estimation will be negative.
IV estimation in the multiple regression model
Conditions for instrumental variable
1) Does not appear in regression equation
2) Is uncorrelated with error term
3) Is partially correlated with endogenous explanatory variable
Computing IV estimates in the multiple regression case:
Two Stage Least Squares (2SLS) estimation
It turns out that the IV estimator is equivalent to the following procedure, which has a much more intuitive interpretation:
Why does Two Stage Least Squares work?
All variables in the second stage regression are exogenous because y2 was replaced by a prediction based on only exogenous information
By using the prediction based on exogenous information, y2 is purged of its endogenous part (the part that is related to the error term)
Properties of Two Stage Least Squares
The standard errors from the OLS second stage regression are wrong. However, it is not difficult to compute correct standard errors.
If there is one endogenous variable and one instrument then 2SLS = IV
The 2SLS estimation should be used if there is more than one endo-genous variable and at least as many instruments
Example: 2SLS in a wage equation using two instruments
Testing for endogeneity of explanatory variables
Testing overidentification restrictions (testing exogeneity of IVs)
§Suppose s variables are available to be potential IVs for p endogenous variables, but are all the s variables really exogenous?
§If s > p, we can test whether (s - p) of the s instruments are exogenous (i.e., uncorrelated with the structural error u), that is, s - p overidentification restrictions. (If s = p, the model is just identified: we cannot test whether the instruments are exogenous).
§Test procedure:
(a) Estimate the structural model using 2SLS and obtain the residuals.
(b) Regress the residuals on all the exogenous variables and obtain the R 2 to form LM = nR 2 ~ where q = s – p (q ³ 1).
H 0: all IVs are exogenous vs. H 1: at least 1 IV is endogenous
(c) If LM > , reject H 0 at the a significance level.
Example: Test for overidentification restrictions
Results of OLS regression of the redisuals against exper, exper 2, motheduc and fatheduc:
Compute the LM statistic: LM = nR 2 = 428´0.0009 = 0.3853 (p-value = 0.535 for )
Statistic inference: Can‘t reject H 0 that motheduc and fatheduc are all exogenous.
Example: Test for overidentification restricitons
Regression of the redisuals against exper, exper 2, motheduc , fatheduc and huseduc
Compute the LM statistic: LM = nR 2 = 428´0.0026 = 1.113 (p-value = 0.574 for )
Statistic inference: Can‘t reject H 0 that motheduc , fatheduc and huseduc are all exogenous.
代写Instrumental Variables EstimationStage Least Squares