Consider the case of time fixed effects only, i.e., Yit = 0 + ...

[codyclark]

Consider the case of time fixed effects only, i.e., Yit = 0 + 1Xit + 3St + uit,
 
  First replace 0 + 3St with t. Next show the relationship between the t and t in the following equation
 
   Yit = 0 + 1Xit + 2B2t + ... + TBTt + uit,
 
  where each of the binary variables B2, , BT indicates a different time period. Explain in words why the two equations are the same. Finally show why there is perfect multicollinearity if you add another binary variable B1. What is the intuition behind the fact that the OLS estimator does not exist in this case? Would that also be the case if you dropped the intercept?
  What will be an ideal response?

Question 2

To standardize a variable you
 
  A) subtract its mean and divide by its standard deviation.
  B) integrate the area below two points under the normal distribution.
  C) add and subtract 1.96 times the standard deviation to the variable.
  D) divide it by its standard deviation, as long as its mean is 1.

[dajones82]

Answer to Question 1

Answer: Yit = 1Xit + t + uit. The relationship is 1 = 0, and t = 0 + t for t  2. Consider time period t, then the population regression line for that period is t + 1Xit, with 1 being the same for all time periods, but the intercept varying from time period to time period. The variation of the intercept comes from factors which are common to all entities in a given time period, i.e., the St. The same role is played by 2,  T, since B2t BTt are only different from zero during one period. There is perfect multicollinearity if one of the regressors can be expressed as a linear combination of the other regressors. Define B0t as a variable that equals one for all period. In that case, the previous regression can be rewritten as

Yit = 0 B0t + 1Xit + 2B2t + ... + TBTt + uit.

Adding B1 with a coefficient here results in

Yit = 0 B0t + 1Xit + 1B1t + 2B2t ... + TBTt + uit.

But B0t = B1t + B2t + ... + BTt, and hence there is perfect multicollinearity. Intuitively, whenever any one of the binary variable equals one in a given period, so does the constant. Hence the coefficient of that variable cannot pick up a separate effect from the data. Dropping the intercept from the regression eliminates the problem.

Answer to Question 2

Answer: A