Question

y = β0 + β1x + u

If yi is earnings of individual i and xi indicates whether individual i is a recent immigrant, what do you think is in u? Do you think E[u|x] = 0? What sign of bias would you expect in an OLS regression of y on x?

Answer 1

Here, Y = B₀ + B₁x + u

Where, Y: income of an individual

            x: Whether an individual is an immigrant or not

Clearly x is a qualitative variable where,

x = 1 if Immigrant

   = 0, otherwise

a) For x = 1, u = Y_(hat) - B_0(hat) - B_1(hat)

      For x = 0, u = Y_(hat) - B_0(hat)

b) Now, for x = 1, E[u|x] = Y_(hat) - B_0(hat) - B_1(hat)

      For x = 0, E[u|x] = Y_(hat) - B_0(hat)

Clearly, u is binomial in nature and no longer follows N(0, σ²)

C) The variance is no longer homoscedastic because of the different u’s. Thus, estimated variance of OLS estimators will be biased.