//  File MIT-Lab-18-07-04-05.txt. Edition 1/21/2013.

// 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 "good"
 instrument conditions are satisfied.
_
As reported in the Corr X&E list, the explanatory variable/error term correlation
 coefficient equals .30. The explanatory variable and the error term are
 positively correlated; the explanatory variable/error term premise is violated.

`
By default the two "good" instrument conditions are satisfied.
_
Instrument/"Problem" Variable Correlation: As the Corr X&Z reports, the "problem"
 explanatory variable/instrument correlation coefficient equals .50. The "problem"
 explanatory variable and the instrument are correlated.
_
Instrument/Error Term Independence: As the Corr X&Z list reports, the
 instrument/error term correlation coefficient equals .00. The instrument
 and the error term are independent.
 
 
` 0011 .50 .00 .30 2 5 50
1. Initially, the sample size equals 50.
 Click Start and then after many, many repetitions click Stop.
_
1a. What are the mean and variance of the numerical values of the coefficient estimates
 when the sample size equals 50?
_
1b. Is the instrumental variable (IV) estimation procedure unbiased?


` 0011 .50 .00 .30 2 5 100
2. The sample size has been increased from 50 to 100.
_
2a. What are the mean and variance of the numerical values of the coefficient estimates
 when the sample size equals 100?
_
2b. Is the instrumental variable (IV) estimation procedure unbiased?



` 0011 .50 .00 .30 2 5 150
3. The sample size has again been increased from 100 to 150.
_
3a. What are the mean and variance of the numerical values of the coefficient estimates
 when the sample size equals 150?
_
3b. Is the instrumental variable (IV) estimation procedure unbiased?


` 0011 .50 .00 .30 2 5 150
4. When the explanatory variable/error term independence premise is violated and
 the "good" instrument conditions are satisfied is the Instrumental variable (IV)
 estimation procedure for the coefficient value 
_______a. unbiased? Explain.
_______b. consistent? Explain.
_
Now, we will illustrate the importance of the "good" instrument conditions.
 
` 0011 .75 .00 .30 2 5 150
5. Instrument/"Problem" Variable Correlation: The instrument is now more correlated
 with the "problem" explanatory variable (Corr X&Z has been increased from .50 to .75).
 By default the sample size is 150. Click Start and than after many, many repetitions
 click Stop. Compare the mean and variance of the numerical values with your answers
 to question 3a. Has the increase in correlation between the instrument and "problem"
 explanatory variable made the instrument more or less effective? Explain.


` 0111 .75 .10 .30 2 5 50
6. Instrument/Error Term Independence: The instrument is now correlated with
 the error term (Corr Z&E now equals .10).
_
6a. 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?
`
6b. Repeat this process for a sample size of 100 and then a sample size of 150. For
 each sample size, what do the mean and variances equal?
_
6c. Is the instrumental variable (IV) estimation procedure consistent when the instrument
 and error term are correlated? Explain.