Heckman Correction Technique: Obtaining Unbiased Estimates with Selective Samples, Summaries of Introduction to Econometrics

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Heckman correction

technique - a short

introduction

Bogdan Oancea

INS Romania

Problem definition

 We have a population of N individuals and we want to fit a linear model to this population: but we have only a sample of only n individuals not randomly selected  Running a regression using this sample would give biased estimates.  The link with our project – these n individuals are mobile phone owners connected to one MNO;  One possible solution : Heckman correction.

1 1 1

y x

Heckman correction

 Two things are important here:

1. When OLS estimates based on the selected sample will suffer selectivity bias?
2. How to obtain non-biased estimates based on the selected sample?

Heckman correction

 Preliminary assumptions:  ( x,y 2 ) are always observed;  y 1 is observed only when y 2 =1 ( sample selection effect );  ( ε, v ) is independent of x and it has zero mean (E ( ε, v ) =0) ;  v ~ N(0,1);  , correlation between residuals ( it is a key assumption ) : it says that the errors are linearly related and specifies a parameter that control the degree to which the sample selection biases estimation of β 1 (i.e., will introduce the selectivity bias). E (  |  )        0

Heckman correction

 This means that follows a truncated normal distribution and we can use a well know result: where z follows a standard normal distribution, a is a constant, is the standard normal pdf and the standard normal cumulative distribution function  λ is called the inverse Mills ratio. 1 ( ) ( ) ( | ) a a E z z a 

   ( ) ( )

 x

x x x x E v v x    

 ^ 

Heckman correction

 We obtained the parametric expression of the expected value of y 1 conditional on observable x and selection selectivity ( y 2

 The last term in the above equation gives the selectivity correction.  Plugging the term into the initial equation we could get an unbiased estimation of β 1 (as well as ).

1 2 1 1

E y x y   x      x 

 ( x  )

 This regression will give an unbiased estimate of β 1 (as well as ).  In this last regression equation the dependent variables are x 1 and.  Support for practical implementation:  R package sampleSelection ;  Stata heckman function;  Eviews implementation;  ( x  )

A practical procedure ( heckit )

Steps forward

 This technique seems suitable to produce unbiased estimates in presence of selectivity which is the case of mobile phone subscribers (a mobile phone subscriber would have y 2 =1 in our presentation);  Investigate if this technique could be applied (and where) to our project;  At a first glance eq. (1) from David’s internal document could be a place where we can apply this correction technique since our estimations are only for the persons that have a phone registered to the MNO under consideration.