//  File MIT-Lab-20-03-04-05.txt

// Title
Instrumental_Variables_-_Omitted_Variables:_Bad_Instrument

//  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






// IV: Bad Instrument - Biased and Not Consistent
` 0111 .50 .10 .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 Corr X1&Z
 specifies the correlation coefficient for the included explanatory variable, X1,
 and the instrument, Z. 
The Corr X2&Z
 specifies the correlation coefficient for the omitted explanatory variable, X2,
 and the instrument, Z. 
The Corr X1&X2
 specifies the correlation coefficient for the two explanatory variables, X1 and X2. 



` 0111 .50 .10 .60 2 5 50
Only the first explantory variable will be included in the regression.
_
The list Corr X1&X2 list indicates that the two explanatory variables are 
 positively correlated; the correlation coefficient equals .60.
_
Consequently,
 the first explanatory variable will be a "problem" explanatory variable because
 it will be correlated with the error term.


` 0111 .50 .10 .60 2 5 50
The instrument variable estimation procedure has been selected.
_
Focus on the Corr X1&Z list. The correlation coefficient for the "problem" variable, X1,
 and the instrument, Z, equals .50. The instrument, Z, is correlated 
 with the "problem" explantory variable, X1.
_
1. Is the first "good" instrument condition satisfied?


` 0111 .50 .10 .60 2 5 50
Now, focus on the Corr X2&Z list. The correlation coefficient of the 
 omitted variable, X2, and the instrument, Z, 
 equals .10. The instrument and the omittted variable are correlated.
_
2. Is the second "good" instrument condition satisfied?

_
3. The sample size equals 50. Click Start and then after many, many 
 repetitions click Stop. What do the mean and variance of the 
 coefficient estimates equal?

` 0111 .50 .10 .60 2 5 100
4. The sample size has been increased from 50 to 100. 
 Click Start and then after many, many repetitions click Stop. 
 What do the mean and variance of the coefficient estimates equal?

` 0111 .50 .10 .60 2 5 150
5. The sample size has been increased from 100 to 150. 
 Click Start and then after many, many repetitions click Stop. 
 What do the mean and variance of the coefficient estimates equal?
_
6. Is the instumental variable (IV) estimation procedure for the coefficient
 estimate unbiased?
_
7. When the second "good" instrument condition is violated, is the 
 instumental variable (IV) estimation procedure for the coefficient
 estimate consistent?