//  File MIT-Lab-20-01-04-05.txt

// Title
Instrumental_Variables_-_Omitted_Variables

//  Omitted Variables or Measurement Error
      OmitIV

//  List for Y on X Constant, Coefficient, and Error Term Variance
      10 30 10 10
      -2.0 2.0 2.0 2.0
      -4.0 4.0 4.0 4.0
      100  500  200  300

// List Sample Size
      50 150 25 50
      
// Correlation Coefficients: X1Z, X2Z, and X1X2
      .50  .75  .25  .50
      .00  .10  .10  .00
     -.30  .60  .30  .30

// Correlation Coefficient Betas
//   The order is X1Z X2Z X1X2
    .493 .000 -.296    .721 .000 -.289    .549 .091 -.363    .827 .091 -.398
    .366 .000  .000    .532 .000  .000    .383 .091 -.038    .568 .091 -.057
    .311 .000  .186    .457 .000  .183    .305 .091  .162    .461 .091  .145
    .290 .000  .348    .459 .000  .367    .267 .091  .333    .429 .091  .326


// Measurement Error X1Z Correlation Betas
    .500 .662
// Measurement Error Variance
     1.0 3.0 1.0 2.0
     
// Data Check: Needed to account for violatile IV behavior
// Simulation ignores repetition in which the estimate differs
// from the actual value by more than the Data Check value.
    25

//  Problem Specs: abcd Corr[X1,Z] Corr[X2,Z] Corr[X1,X2] Coef1Value Coef2Value SampleSize
//   a: Pause checkbox
//   b: Both 0-Both Xs    1-Only X1
//   c: Parameter to estimate. 0, 1, or 2: 0-Constant  1-X1  2-X2
//   d: Estimation procedure. 0-OLS 1-IV




// OLS - Biased and Not Consistent
` 0110 .50 .00 .60 2 5 50
The Const, Coef and Err Var lists on the extreme left describe the relationship
 between explanatory variables (X1 and X2) and dependent variable (Y):
_  Y = 10 + 2X1 + 4X2 + Err
The list Corr X1&X2 list indicates that the two explanatory variables are 
 positively correlated; the correlation coefficient equals .60. Also, the sample
 size equals 50 and that the Both Xs check box is selected.


` 0110 .50 .00 .60 2 5 50
1. Only one explanatory variable is included. Only the explanatory variable
 X1 is included; X2 is omitted. We know that an omitted variable causes the ordinary least
 squares (OLS) estimation procedure to be biased whenever
_     the omitted variable affects the dependent variable.
and
_     the omitted variable is correlated with an included variable.
Both of these conditions are met. 
`
Let us confirm that bias results. Click Start
 and then after many, many repetitions click Stop. What does the mean of
 the coefficient estimates equal? What is the actual value of the coefficient?
 
` 0110 .50 .00 .60 2 5 100
2. Even though the ordinary least squares (OLS) estimation procedure is biased,
 it might still be consistent. To investigate this possibility the sample size has
 been increased from 50 to 100. Click Start
 and then after many, many repetitions click Stop. What does the mean of
 the coefficient estimates equal?
 
` 0110 .50 .00 .60 2 5 150
3. The sample size has been increased from 100 to 150. Click Start
 and then after many, many repetitions click Stop. What does the mean of
 the coefficient estimates equal?
_
4. When an omitted variable casues the ordinary least squares (OLS) estimation
 procedure to be biased, Is it consistent?