//  File MIT-Lab-14-04-05-05.txt. Edition 1/20/2013.

// Title
Irrelevant_Explanatory_Variable

//  Irrelevant
      Omit

//  List for Y on X Constant, Coefficient, and Error Term Variance
      10 30 10 10
      -2 2 2 2
      -5 5 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  .90  .30  .30

// Correlation Coefficient Betas
//   The order is X1Z X2Z X1X2
    .000 .000 -.458    .721 .000 -.289    .549 .091 -.363    .827 .091 -.398
    .000 .000  .000    .532 .000  .000    .383 .091 -.038    .568 .091 -.057
    .000 .000  .239    .532 .000  .000    .383 .091 -.038    .568 .091 -.057
    .000 .000  .428    .457 .000  .183    .305 .091  .162    .461 .091  .145
    .000 .000  .674    .673 .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



` 0110 .00 .00 .00 2 0 50

Objective: Illustrate the effect of irrelevant explanatory variables. 
_
Good news: The ordinary least squares (OLS) estimation procedure
 for the value of the relevant variable's coefficient is unbiased
 regardless of whether or not the irrelevant variable is included.
_
Bad news: The variance of the probability distribution for the
 relevant variable increases whenever the irrelevant variable is
 included.

`
By default, the actual coefficient of the first explanatory variable
 equals 2.0. The first explanatory variable, X1, is the relevant explanatory
 variable.
_
The actual coefficient of the second explanatory variable equals .0
 The second explanatory variable, X2, is the irrelevant variable.
_ 

The Coef1 radio button is selected indicating that the estimates
 for the first, the relevant, explanatory variable's coefficient
 will be reported.

` 0110 .00 .00 .00 2 0 50
The Only X1 checkbox is selected indicating that only the relevant
 explanatory variable is included in the regression; the irrelevant
 variable, X2, is omitted.
_
1a. Note that the two explanatory variables are not correlated:
 .00 is selected from the CorrX1&X2 list. Click Start and then,
 after many, many repetitions click Stop. Do the results suggest that
 the estimation procedure for the coefficient value for the relevant
 explanatory variable, X1, is biased or unbiased? What is the variance
 of the estimates.

` 0110 .00 .00 .30 2 0 50
1b. Note that the two explanatory variables are now correlated;
 .30 is selected from the CorrX1&X2 list. Click Start and then,
 after many, many repetitions click Stop. Do the results suggest that
 the estimation procedure for the coefficient value of the
 relevant explanatory variable, X1, is biased or unbiased?
 What is the variance of the estimates.

` 0110 .00 .00 .30 2 0 50

1c. Next, select .60, and then .90, and then -.30 from the CorrX1&X2 list. 
 In each case, Click Start and then, after many, many repetitions 
 click Stop. Do the results suggest that the estimation procedure
 for the coefficient value of the relevant explanatory variable, X1,
 is biased or unbiased? 
_
1d. How does the correlation of the relevant and irrelevant explanatory
 variables affect the variance of the estimates of the relevant variable, X1?
 How does the correlation affect the reliabilty of the estimate for the
 relevant explanatory variable, X1?




` 0010 .00 .00 .00 2 0 50
The Both X's checkbox is selected indicating that both the first and second
 explanatory variables are included in the regression; that is, the irrelevant
 variable is included.
_
2a. Note that the two explanatory variables are not correlated:
 .00 is selected from the CorrX1&X2 list. Click Start and then,
 after many, many repetitions click Stop. Do the results suggest that
 the estimation procedure for the coefficient value is biased or unbiased?
 What is the variance of the estimates.
 

` 0010 .00 .00 .30 2 0 50
2b. Note that the two explanatory variables are now correlated;
 .30 is selected from the CorrX1&X2 list. Click Start and then,
 after many, many repetitions click Stop. Do the results suggest that
 the estimation procedure for the coefficient value of the
 relevant explanatory variable, X1, is biased or unbiased?
 What is the variance of the estimates.

` 0010 .00 .00 .30 2 0 50
2c. Next, select .60, and then .90, and then -.30 from the CorrX1&X2 list. 
 In each case, Click Start and then, after many, many repetitions 
 click Stop. Do the results suggest that the estimation procedure
 for the coefficient value of the relevant explanatory variable, X1,
 is biased or unbiased? 
_
2d. How does the correlation of the relevant and irrelevant explanatory
 variables affect the variance of the estimates of the relevant variable, X1?
 How does the correlation affect the reliabilty of the estimate for the
 relevant explanatory variable, X1?