Intro to Econometric Theory Heinz School, Carnegie Mellon University 90-906, Spring 2003-4

Homework #1

1. Consider a sample of people who were asked to report their sex (Everyone answers, and everyone answers either male or female). From their answers, we construct the following matrix:

X = [ 1 M F ]  (1)

In the matrix, each row corresponds to one person’s response; 1 is a column of 1s, M is a column with a 1 in the ith row if the ith person answered male and a zero otherwise, and F is a column with a 1 in the ith row if person i answered female and a zero otherwise.

Is X′X invertible. Why or why not?

2. We discussed in class that variance matrixes are always positive semidefinite but not always positive definite. In the case of scalar random variables, a singular variance matrix means a zero variance — that is, that the random variable is not really random at all, but fixed.

Now, let’s think about what a positive semi-definite but not positive definite (hereafter PSDNPD) variance matrix means for the random vector it corresponds to. Let x be a random vector with PSDNPD variance matrix VX.

(a) Must there be a fixed non-zero vector α, such that α′V Xα=0? Why or why not?

(b) Suppose there is such an α, what would the variance of α′x be?

(c) What does this tell us about α′x?

(d) Does this mean that α′x = 0?

(e) So, what does it mean for a random vector to have a PSDNPD variance matrix?

3. We want to estimate the proportion of US households with broadband Internet access. We randomly and independently call up 1,000 households and ask them whether or not they do. We make a 1, 000 × 1 vector x which has a 1 in the ith row if the ith household answered yes and a 0 if no. Assume that all households in the US have a phone, all answer our call, and all truthfully answer the question.

Consider the following two estimators for p, the proportion of US households who, in fact, have broadband:

(a) \overline{x}, also known as the sample proportion.

(b) 0.73 for surveys with sample size ¡ ten million, and \overline{x} for surveys with sample size ¿ ten million.

Which is unbiased or are both? Which is consistent or are both? What is my point?

4. Suppose we have an estimator of per-capita income in the US, Î, and an estimator of the share of income saved, ŝ. The true values of these parameters are μ_{I} and μ_{s}. These two estimators are derived somehow from a survey of size n, and we have proved the following result for these estimators:

\sqrt{n} (( Î ; ŝ ) − ( \begin{matrix}μ_{I} \\ μ_{s} \end{matrix} )) N (( \begin{matrix}0 \\ 0 \end{matrix} ) , [ \begin{matrix}400 & 20 \\ 20 & 9 \end{matrix} ])  (2)

Please construct an approximate (why approximate?) 95% confidence interval for the per-capita savings, μ_{I}μ_{s}.