代写Instrumental Variables EstimationStage Least Squares

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

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:

1) Correlated with education because of resource constraints; 2) Uncorrelated with innate ability

Correlated with education because more education if lived near college; 2) Uncorrelated with error

1) Correlated with education because of compulsory school attendance laws, 2) Uncorrelated with error

Properties of IV with a poor instrumental variable

Computing R-squared after IV estimation

If SSR > SST, R-squred after IV estimation will be negative.

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:

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

§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

Example: Test for overidentification restricitons

代写Instrumental Variables EstimationStage Least Squares