//  File MIT-Lab-18-02-04-05.txt. Edition 1/21/2013.

// Title
Explantory_Variable/Error_Term_Correlation

//  Omitted Variables or Measurement Error
      ExplErrCorr

//  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


` 0010 .00 .00 .00 2 5 60
Objective: Show that when the explanatory variable/error term independence
 premise is violated, the ordinary least squares (OLS) estimation procedure
 for the coefficient value is biased.
_
That is, when the explanatory variable is correlated with the error term, the
 ordinary least squares (OLS) estimation procedure for the coefficient value
 is biased.

`
The Const, Coef and Err Var lists on the extreme left describe the relationship
 between explanatory variable (X) and the dependent variable (Y):
_  Y = 10 + 2X + Err
The list Corr X&E list specifies the correlation coefficient of the
 the explanatory variable and the error term. Also, the sample
 size equals 50.


` 0010 .00 .00 .00 2 5 50
1. Initially, .00 is selected from the Corr X&E list. The explanatory variable and
 the error term are independent. Click Start and then after many repetitions click Stop.
_
1a. Does the  simulation suggest that the ordinary least squares (OLS) estimation
 procedure for the coefficient value is unbiased or biased when the explanatory
 variable and error term are independent? 
_
1b. If the estimation procedure is biased, is it biased up or down? Explain.


` 0010 .00 .00 .30 2 5 50
2. Next, note that .30 is selected from the Corr X&E list. The explanatory variable and
 the error term are positively correlated. Click Start and then after many repetitions click Stop.
_
2a. Does the  simulation suggest that the ordinary least squares (OLS) estimation
 procedure for the coefficient value is unbiased or biased when the explanatory
 variable and error term are independent? 
_
2b. If the estimation procedure is biased, is it biased up or down? Explain.



` 0010 -.00 .00 -.30 2 5 50
3. Next, note that -.30 is selected from the Corr X&E list. The explanatory variable and
 the error term are negatively correlated. Click Start and then after many repetitions click Stop.
_
3a. Does the  simulation suggest that the ordinary least squares (OLS) estimation
 procedure for the coefficient value is unbiased or biased when the explanatory
 variable and error term are independent? 
_
3b. If the estimation procedure is biased, is it biased up or down? Explain.