//  File MIT-Lab-18-07-04-05.txt. Edition 5/15/2012.

// Title
Explantory_Variable/Error_Term_Correlation

//  Omitted Variables or Measurement Error
      ExplErrCorrIV

//  List for Y on X Constant, Coefficient, and Error Term Variance
      10 30 10 10
      -2.0 2.0 2.0 2.0
      -5.0 5.0 5.0 5.0
      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
//   abcd
//   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


` 0011 .50 .00 .30 2 5 50
Objective: Show that  while the instrumental variable (IV) estimation procedure
 for the coefficient value is biased when the explanatory variable/error term
 independence premise is violated, it is consistent whenever the conditions for
 a "good" instrument are met.
_
Note that the explanatory variable/error term correlation coefficient is specified as .30.
 The explanatory variable and the error term are positively correlated; the explanatory
 variable/error term premise is violated.


` 0011 .50 .00 .30 2 5 50
1. Initially, the sample size equals 50.
 Click Start and then after many, many repetitions click Stop.
_
What are the mean and variance of the numerical values of the coefficient estimates
 when the sample size equals 50?


` 0011 .50 .00 .30 2 5 100
2. Next, note that the sample size has been increased from 50 to 100.
_
What are the mean and variance of the numerical values of the coefficient estimates
 when the sample size equals 100?



` 0011 .50 .00 .30 2 5 150
3. Now, note that the sample size has been increased from 100 to 150.
_
What are the mean and variance of the numerical values of the coefficient estimates
 when the sample size equals 150?
_
4. When the explanatory variable/error term independence premise is violated,
 is the Instrumental variable (IV) estimation procedure for the coefficient
 value
_______a. unbiased? Explain.
_______b. consistent? Explain.

 
` 0011 .75 .00 .30 2 5 150
Now, we shall illustrate the importance of the "good" instrument conditions.
_
5. The instrument is now more correlated with the problem variable (Corr X1&Z has been
 from .50 to .75). Increase the sample to 80 and then to 100. In each case, click
 Start and than after many, many repetitions calick Stop. Has the
 increase in correlation between the instrument and problem variable made the
 instrument more or less effective? Explain.


` 0111 .75 .10 .30 2 5 50
6. The instrument is now correlated with the omitted variable (Corr X2&Z no 
 longer equals 0, it now equals .10). A sample size of 50 is specified. Click Start and
 then after many, many repetitions click Stop. What are the mean and variance of
 the numerical values of the coefficient estimates when the sample size equals 50?
_
Repeat this process for a sample size of 100 and then a sample size of 150.
_
Is the instrumental variable (IV) estimation procedure consistent when the instrument
 and error term are correlated? Explain.