Eric Zivot Econ 424 Fall 2014

Eric Zivot
Econ 424
Fall 2014
Problem Set #3
Bivariate distributions, Time Series Concepts, and Matrix Algebra
Due: Friday 10/17/14 at 8 PM (PST) via Canvas
Readings
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My lecture notes on review of probability, time series concepts, and matrix
algebra
Ruppert, chapter 7 (sections 1 – 5), chapter 9 (sections 1-2, 4 and 7)
ZLM, chapters 5 and 7
R Cookbook, chapter 5 (data structures) and 8 (probability)
Introduction to R (pdf document on webpage), chapter 5 (Arrays and Matrices)
Beginners Guide to R, chapter 3, section 5 (Manipulating objects)
Programs and Data
The following files are located on the class homework page:
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424lab3.r
probReview.r
matrixReview.r
timeSeriesConcepts.r
(R commands/hints for lab3)
(R script used for in class examples)
(R script used for in class examples)
(R script used for in class examples)
Instructions
In this lab you will
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Use R to evaluate bivariate distributions, simulate some simple time series models
and do simple matrix algebra computations
Exercises
The following questions require R. On the class web page are the R script files
424lab3.r, probReview.r, timeSeriesConcepts.r, and
matrixReview.r. The file 424lab3.r contains hints for completing the assignment
and the latter files contains R code for replicating the in-class examples illustrated in the
powerpoint files timeSeriesConceptsPowerPoint.pdf,
matrixReviewPowerPoint.pdf. Copy and paste all statistical results and graphs
into a MS Word document (or your favorite word processor) while you work, and add
any comments and answer all questions in this document.
Start MS Word and open a blank document. You will save all of your work in this
document.
I. Bivariate distributions
Let X and Y be distributed bivariate normal with
 X  0.05, Y  0.025,  X  0.10,  Y  0.05
(a) Using R package mvtnorm function rmvnorm(), simulate 100 observations from
the bivariate distribution with  XY  0.9 . Using the plot() function create a scatterplot
of the observations and comment on the direction and strength of the linear association.
Using the function pmvnorm(), compute the joint probability Pr  X  0, Y  0  .
(b) Using R package mvtnorm function rmvnorm(), simulate 100 observations from
the bivariate distribution with  XY  0.9 . Using the plot() function create a
scatterplot of the observations and comment on the direction and strength of the linear
association. Using the function pmvnorm(), compute the joint probability
Pr  X  0, Y  0  .
(c ) Using R package mvtnorm function rmvnorm(), simulate 100 observations from
the bivariate distribution with  XY  0 . Using the plot() function create a scatterplot
of the observations and comment on the direction and strength of the linear association.
Using the function pmvnorm(), compute the joint probability Pr  X  0, Y  0  .
II. Simulating Time Series Data
Consider the MA(1) model
Yt  0.05   t   t 1 , | |<1

 t ~ iid N 0,  0.10 
2

a) Using the R function arima.sim(), simulate and plot 250 observations of the
MA(1) with   0.5, 0.9 . Briefly comment on the behavior of the simulated data
series.
Now consider the AR(1) model
Yt  0.05   (Yt 1  0.05)   t , |  | 1

 t ~ iid N 0,  0.10
2

a) Using the R function arima.sim(), simulate and plot 250 observations of the
AR(1) with   0.5, 0.9 . Briefly comment on the behavior of the simulated data
series.
III. Matrix Algebra
(a) Create the matrices and vectors
1 4 7 
 4 4 0
1 
5 
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A  2 4 8, B  5 9 1, x  2, y  2
6 1 3
2 2 5
3
7 
(b) Compute the transposes of the matrices and vectors
(c) Compute A+B, A-B, 2*A, Ax, y 'Ax
(d) Consider the system of equations
x  y 1
2x  4 y  2
Plot the two lines and note the solution to the system of equations (hint: use the R
function abline()). Write the system using matrix notation as Az = b and solve
for z.
(e) Consider creating a portfolio of three assets denoted A, B and C. Assume the
following information
0.10 0.30
0.10
 0.01
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  0.04,   0.30 0.15  0.20
0.02
0.10  0.20 0.08
Compute the expected return and variance for an equally weighted portfolio
portfolio (i.e., xA = xB = xC = 1/3).
IV. Ruppert Exercises:
Chapter 7, (section 15), exercises 1 and 2.
Chapter 9 (section 17), exercise 3.