代写Instrumental Variables EstimationStage Least Squares

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    • 代写Instrumental Variables EstimationStage Least Squares
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    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
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    *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
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    *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
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    *Example: Education in a wage equation
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    *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
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    *Reconsideration of OLS in a simple regression model
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    *A simple consistency proof for OLS under exogeneity:
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    *Assume existence of an instrumental variable   :
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    *Inference with IV estimation
      Assume homoscedasticity holds: E(u 2|z) = Var(u) = s 2.  
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    *Example: Father‘s education as an IV for education
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    *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
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    *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
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    *Computing R-squared after IV estimation
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      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.
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    *IV estimation in the multiple regression model
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    *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
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    *Computing IV estimates in the multiple regression case:
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    *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:
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    *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
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    *Example: 2SLS in a wage equation using two instruments
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    *Testing for endogeneity of explanatory variables
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    *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.
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    *Example: Test for overidentification restrictions
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      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. 
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    *Example: Test for overidentification restricitons
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          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
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