Let X denote a random variable that takes the value of 1 if a person hascompleted college education and the value 0 if the person has not. Let Y denote therandom variable that indicates this person's hourly wage in dollars. Consider the jointprobability distribution between X and Y in the population, given by Table 1. Probabilities for Possible Values of X and Y
Possible values of Y
Possible values of X 10 15 20
0 0.25 0.15 0.10
1 0.15 0.15 0.20
That is, if we draw one observation from this population, we have 25 percent chance of secing a person who has not completed college (X=0) and has an hourly wage of 10 dollars. In other words, P(X = 0, Y = 10) = 0.25. Similarly. P(X = 0.Y = 15) = 0.15, P(X = 0, y = 20) = 0.10, P(x = 1, Y = 10) = 0.15, and so on.
Recall that a conditional expectation is the expected value of a random variable given an event. For example, the conditional expectation of Y given X = 0 is
E[Y|X = 0] = P (Y = yi| X = 0)y;
where yi are the possible values of Y. Recall also that the conditional probability of Y given X = r is defined as
P(Y = y|X = ) = P(X = 1, Y = y) P(X = 1) P(X = r, Y = y) E, P(X = 2, Y = y;)
We say that two random variables X and Y are independent if the conditional probabilities of Y given X = I are the same regardless of the value of x.
(a) Compute the conditional probabilities P Y = 10)X = 0), PſY = 15 X = 0), and P(Y = 20 X = 0). Compute the conditional probabilities P(Y = 10 X = 1), P(Y = 15|X = 1), and PſY = 20 X = 1). Are X and Y independent?
(b) Compute the conditional expectation of Y given X = 0. Compute the conditional expectation of Y given X = 1.
(c) The relationship between X and Y in the population can be written as the linear model
Y = βo + β1X +u
where βo and β1, are parameters, and the error term u has a conditional mean of zero given any realized value of x. What are the values of βo and β1, that satisfy the joint probabilities of Table 1 and the conditional mean zero property of the error term?
Hint: take conditional expectations of both sides of the linear equation given possible values of X. By doing this, you should get a system of two equations and two unknowns.]
(d) Recall that the conditional variance of Y given X is
Var(Y|X) = E[Y – E(YX)[x]2
Using the values of βo and β1, found in (c), compute the conditional variance of u given X = 0. Conditional variance of u given X = 1. Is u homoskedastic?