Download ps2 - uc-davis economics

Prof. Oscar Jorda
Due Date Theory: Wednesday, February 2
Due Date Applied: Monday, February 7
ECN 140 - ECONOMETRICS
Problem Set 2
THEORY
Instructions: The goal of the problem set is to understand what you are doing rather than just getting the
correct result. Please show your work clearly and neatly. No credit will be given to late homework,
regardless of the excuse.
[1]
The joint probability density function of two discrete random variables, X and Y, is given by the
following table:
X
Y
1
1/8
1/4
1/8
2
4
6
3
1/24
1/4
1/24
9
1/12
0
1/12
[a] Find the marginal probability density function of Y.
[b] Find the conditional probability density function of Y given X = 2
[c] Find the covariance of X and Y.
[d] Are X and Y independent?
[2]
Let X1, X2,… Xn, be independent random variables that all have the same probability distribution,
with mean b and variance 2. Let
X 
1
n
n
X
i
i 1
[a] Prove that E[ X ]  b
[b] Prove that V ( X )   2 / n
[3]
Consider the joint probability density function
f ( x, y)  (0.6) x (0.4)1 x (0.3) y (0.52)1 y 2 xy
where the possible values for X and Y are x = 0, 1 and y = 0, 1. Find:
[a] f(0,0), f(0,1), f(1,0), f(1,1)
[b] The marginal density functions f(x) and f(y)
[c] The conditional density function f(y|x = 0)
[d] E(X) and E(Y)
[e] V(X) and V(Y)
[f] COV(X,Y) and the correlation between X and Y
[g] E(X + Y) and V(X + Y)
[4]
A Stock market analyst wanted to test whether the rate of return of purchasing stock in a certain
company exceeded the average return for the market as a whole. In other words, is the average excess
return (company's rate of return minus the market's average rate of return) positive or negative? We
therefore want to test whether the average returns are zero or not. The excess rate of return was computed
for 13 periods and the average was 3.1 percent with a standard deviation of 1 percent. Test the hypothesis at
the 5% level of significance, assuming that excess returns are normally distributed. Derive a 95%
confidence interval for the average excess returns.
[5]
Let X and Y represent the rates of return (in percent) on two stocks. You are told that X ~ N(15, 25)
and Y ~ N(8, 4), and that the correlation coefficient between the two rates of return is -0.4. Suppose you
want to hold the two stocks in your portfolio in equal proportion.
[a] What is the probability distribution of the return on the portfolio?
[b] Is it better to hold this portfolio or to invest in only one of the two stocks? Why?
EViews Exercise (50 points)
For this exercise you will need to download the excel file ``ps2.xls.'' This file contains data from the
Current Population Survey of the U.S.
Bureau of the Census. The following variables are contained:
LNWAGE = log of wage rate of employed workers in dollars per hour
SEX = 1 if the person is female, 0 otherwise
ED = years of education
AGE = age of employee
NONWH = 1 if the person is non-Hispanic, nonwhite; 0 otherwise
HISP = 1 if the person is Hispanic; 0 otherwise.
There are 206 observations in the data-set..
1.Give sample statistics for the variables in this study. Note, you should transform the variable LNWAGE
into dollars per hour (call the
variable WAGE) to obtain a more intuitive description.
2.Give the sample statistics of the variables WAGE, AGE and ED sub-sampled according to: SEX,
NONWH, HISP respectively.
Comment on the differences.
3.Calculate the correlation matrix for all the variables (leave LNWAGE as is).
4.Calculate the correlation between LNWAGE and ED for sub-samples based on SEX, NONWH, HISP.
5.Use pie charts to describe the racial composition of your data
6.In a 1 paragraph summary, comment on the wage differentials between: Males and Females; Whites and
Non-Whites (note, you will
need to include Hispanics into the Non-White category) and Non-Whites and Hispanics.
7.Find data for Real Fixed Private Investment in the Internet (you can use the sources listed in my web
page). Load the data into Eviews
and produce a graph and summary statistics.