Econ613 - Introduction to R - Solved

1.1      Introduction Using this sample code,

install.packages("BB") library(BB) source("A1.R") ?for ??rpareto dir() 1+1 2/2 save.image("misc.RDATA") 1:10 30%%4 setwd("/Users/ms486/Dropbox/Papers/Progress") getwd() ls() 2/0 log(-1) sum(1:10)

                                                                  Exercise 1          Introduction
1.    Create a directory for this class and store your script “a0.R”

2.    Install the packages, Hmisc, gdata,boot,xtable,MASS,moments,snow,mvtnorm

3.    Set your working directory

4.    List the content of your directory and the content of your environment

5.    Check whether 678 is a multiple of 9

6.    Save your environment

7.    Find help on the function mean, cut2

8.    Find an operation that returns NaN (Not A Number)

1.2     Objects
Vectors, Matrix, Arrays
vec0 = NULL vec1 = c(1,2,3,4) vec2 = 1:4 vec3 = seq(1,4,1) vec4 = rep(0,4)

sum(vec1) str(vec1) prod(vec1) mat1 = mat.or.vec(2,2) mat2 = matrix(0,ncol=2,nrow=2,byrow=T) mat3 = cbind(c(0,0),c(0,0)) mat4 = rbind(c(1,1),c(0,0)) mat5 = matrix(1:20,nrow=5,ncol=4) mat5[1:2,3:4] mat5[1,] arr1 = array(0,c(2,2)) dim(mat4) dim(vec2) length(vec2)

length(mat1) class(mat4)

                                                       Exercise 2           Object Manipulation
1.    Print Titanic, and write the code to answer these questions (one function (sum) , one

operation)

(a)    Total population

(b)    Total adults

(c)    Total crew

(d)    3rd class children

(e)    2nd class adult female

(f)     1st class children male

(g)    Female Crew survivor

(h)    1st class adult male survivor

2.    Using the function prop.table, find

(a)    The proportion of survivors among first class, male, adult

(b)    The proportion of survivors among first class, female, adult

(c)    The proportion of survivors among first class, male, children

(d)    The proportion of survivors among third class, female, adult

                                                     Exercise 3           Vectors - Introduction
1.    Use three different ways, to create the vectors

(a)     a = 1,2,...,50

(b)    b = 50,49,...,1

Hint : rev

2.    Create the vectors

(a)     a = 10,19,7,10,19,7,...,10,19,7 with 15 occurrences of 10,19,7

(b)    b = 1,2,5,6,...,1,2,5,6 with 8 occurrences of 1,2,5,6

Hint : rep

3.    Create a vector of the values of log(x)sin(x) at x = 3.1,3.2,...,6

4.    Using the function sample, draw 90 values between (0,100) and calculate the mean. Re-do the same operation allowing for replacement.

5.    Calculate

 

6.    Create a vector of the values of exp(x)cos(x) at x = 3, 3.1, ...6.

                                                         Exercise 4           Vectors - Advanced
1.    Create two vectors xV ec and yV ec by sampling 1000 values between 0 and 999.

2.    Suppose xV ec = (x1,...,xn) and yV ec = (y1,...,yn)

(a)     Create the vector (y2 − x1,...,yn − xn−1) denoted by zV ec.

(b)    Create the vector ( ) denoted by wV ec.

(c)     Create a vector subX which consists of the values of xV ec which are ≥ 200.

(d)    What are the index positions in yV ec of the values which are ≥ 600.

                                                                         Exercise 5          Matrix
                                                                     1        1        3

1.    Create the matrix

(a)    Check that A3=0 (matrix 0).

(b)    Bind a fourth column as the sum of the first and third column

(c)    Replace the third row by the sum of the first and second row (d) Calculate the average by row and column.

2.    Consider this system of linear equations:

2x + y + 3z = 10
(1)
x + y + z = 6
(2)
x + 3y + 2z = 13
(3)
3.    Solve this equation.

                                                                      Exercise 6          Functions
1.    Write a function fun1 which takes two arguments (a,n) where (a) is a scalar and n is a positive integer, and returns

 

2.    Consider the function



2

 x + 2x + |x|

f(x) =                    x2 + 3 + log(1 + x)

 x2 + 4x − 14
if x < 0; if 0 ≤ x < 2; if x ≥ 2.
(4)
Evaluate the function at -3, 0 and 3.

                                                                         Exercise 7          Indexes
1.    Sample 36 values between 1 and 20 and name it v1

2.    Use two different ways to create the subvector of elements that are not in the first position of the vector. Hint: which and subset can not be used. Check x[a] and x[-a].

3.    Create a logical element (TRUE or FALSE), v2, which is true if v1 > 5. Can you convert this logical element into a dummy 1 (TRUE) and 0 (FALSE)?

4.    Create a matrix m1 [6 × 6] which is filled by row using the vector v1.

5.    Create the following object

x = c(rnorm(10),NA,paste("d",1:16),NA,log(rnorm(10)))

6.    Test for the position of missing values, and non-finite values. Return a subvector free of missing and non-finite values.

                                                          Exercise 8           Data Manipulation
1.    Load the library AER, and the dataset (data(”GSOEP9402”)) to be named dat.

2.    What type of object is it? Find the number of rows and column? Can you provide the names of the variables?

3.    Evaluate and plot the average annual income by year.

4.    Create an array that illustrates simultaneously the income differences (mean) by gender, school and memployment.

                                                              Exercise 9          First regression
1.    Load the dataset (data(”CASchools”)) to be named dat1.

2.    Using the function lm, run a regression of read on the following variables: district, school, county, grades, students, teachers, calworks, lunch, computer, expenditure, income and english. Store this regression as reg1.

3.    Can you run a similar regression by specifying,

formula = y ~ x. lm(formula)

Create reg2, that uses only the 200 first observations.

                                                        Exercise 10           Advanced indexing
1.    Create a vector lu of 200 draws from a pareto distribution (1,1). How many values are higher than 10. Replace these values by draws from a logistic distribution (6.5,0.5).

2.    Create a vector de of 200 draws from a normal distribution (1,2). Set de = log(de), and count the number of missing values or negative values. Replace these values by draws from a normal distribution (0,1) truncated at 0. hint:truncnorm

3.    Create two vectors, orig and dest as 200 draws from a uniform distribution [0,1].

4.    Create two matrices, hist and dist as 200*200 draws from a uniform distribution [0,1].

5.    Consider this function

 

where

suj,l = log(origj + destl + distj,l)/(1 + log(origj + destl + distj,l))
(6)
sej,l = exp(origj + destl + histj,l)/(1 + exp(origj + destl + histj,l))
(7)
6.    Create the matrices su and se.

7.    Set r = 0.05. Create a function to evaluate qjl(.). Evaluate qjl(9245) for all pairs (j,l).

8.    Create gridw, which consists of a sequence from 9100 to 55240 of length 50.

9.    Using the function sapply, evaluate qjl. Store the ouput into an array of dimension (50 × 200 × 200). How long does it take to evaluate qjl() for each value of w?

List
li                      = list()

li[[1]] = mat1

li[[2]] = Titanic li1              = list(x=mat1,y=Titanic)

li1$x li2$y

Dataframe
data=data.frame(x=rnorm(100),y=runif(100)) data browse(data) edit(data) data[,1] data[1,] data$x names(data) attach(data) x detach(data) y

Tests and Conversion
is.na()

is.list() as.list() is.factor() as.factor() is.matrix() is.vector() is.array() is.finite() a==b a=>b a<=b

                                                        Exercise 11           Tests and indexing
1.    Test if c(1,2,3) is an array? a vector? a matrix?

2.    x0 = rnorm(1000); Using the function table() count the number of occurrences of x0 > 0,

x0 > 1, x0 > 2, x0 > 0.5, x0 < 1 and x0 > −1

3.    x1 = cut2(runif(100,0,1),g=10) levels(x1)=paste("q",1:10,sep="")

4.    Test whether or not x1 is a factor?

5.    Verify that ”q1” has 10 occurences.

6.    Convert x1 into a numeric variables. What happens to the levels?

7.    rand = rnorm(1000)

8.    Using the function which() find the indexes of positive values.

9.    Create the object w of positive values of x using:

(a)    Which

(b)    Subset

(c)    By indexing directly the values that respect a condition

1.3      Basic functions
Table 1: Basic Functions

Function
Description
abs(x)
absolute value
sqrt(x)
square root
ceiling(x)
ceiling(3.475) is 4
floor(x)
floor(3.475) is 3
trunc(x)
trunc(5.99) is 5
round(x, digits=n)
round(3.475, digits=2) is 3.48
signif(x, digits=n)
signif(3.475, digits=2) is 3.5
log(x)
logarithm
exp(x)
ex
substr(x, start=n1, stop=n2)
Extract or replace substrings in a character vector.

x = ”abcdef” , substr(x, 2, 4) is ”bcd”
grep(pattern, x )
Search for pattern in x.
sub(pattern, replacement, x)
Find pattern in x and replace with replacement text.
strsplit(x, split)
Split the elements of character vector x at split.
strsplit(”abc”, ””)
returns 3 element vector ”a”,”b”,”c”
paste(..., sep=””)
Concatenate strings
toupper(x)
Uppercase
tolower(x)
Lowercase
1.4     Language
if (condition) statement for (i in range) statement while (condition) statement fun = function(input) {calculation return(output)} fun = function(input) {calculation output}

                                                               Exercise 12           Programming

Write a program that asks the user to type an integer N and compute u(N) defined with :

u(0)=1 u(1)=1 u(n+1)=u(n)+u(n-1)

1.    Evaluate 12 + 22 + 32 + ...4002.

Table 2: Apply functions

Functions
Usage
apply
Apply Functions Over Array Margins
by
Apply a Function to a Data Frame Split by Factors
eapply
Apply a Function Over Values in an Environment
lapply
Apply a Function over a List or Vector
mapply
Apply a Function to Multiple List or Vector Arguments
rapply
Recursively Apply a Function to a List
tapply
Apply a Function Over a Ragged Array
2.    Evaluate 1 × 2 + 2 × 3 + 3 × 4 + ... + 249 × 250

3.    Create a function ”crra” with two arguments (c,θ) that returns  . Add an if condition such that the utility is given by the log when θ ∈ [0.97,1,03] ≈ 1

4.    Create a function ”fact” that returns the factorial of a number

                                                           Exercise 13           Apply Functions
1.    Using this object,

m = matrix(c(rnorm(20,0,10), rnorm(20,-1,10)), nrow = 20, ncol = 2)

Calculate the mean, median, min, max and standard deviation by row and column.

2.    Using the dataset iris in the package ”datasets”, calculate the average Sepal.Length by Species. Evaluate the sum log of Sepal.Width by Species.

3.    y1 = NULL; for (i in 1:100) y1[i]=exp(i) y2 = exp(1:100) y3 = sapply(1:100,exp)

(a)     Check the outcome of these three operations.

(b)    Using proc.time() or system.time(), compare the execution time of these three equivalents commands.

Table 3: Statistical distributions

                                                        name             description

 

dname( )     density or probability function pname( )    cumulative density function qname( )     quantile function rname( )        random deviates

 

Table 4: Statistical Functions



 
 
 

Function                                                          Description

mean(x, trim=0,na.rm=FALSE)             mean of object x sd(x), var(x)   standard deviation, variance of object(x)

median(x)                                                       median

quantile(x, probs)                                           x is the numeric vector and probs is a numeric vector with probabilities

range(x)                                                           range

sum(x)                                                              sum

diff(x, lag=1)                                                    lagged differences, with lag indicating which lag to use

min(x)               minimum max(x)           maximum

Table 5: Statistical distributions

Distribution

 
R name
Beta
beta
Lognormal
lnorm
Binomial
binom
Negative Binomial
nbinom
Cauchy
cauchy
Normal
norm
Chisquare
chisq
Poisson
pois
Exponential
exp
Student t
t
F
f
Uniform
unif
Gamma
gamma
Tukey
tukey
Geometric
geom
Weibull
weib
Hypergeometric
hyper
Wilcoxon
wilcox
Logistic
logis
 

1.5      Statistics

                                             Exercise 14            Simulating and Computing
1.    Simulate a vector x of 10,000 draws from a normal distribution. Use the function summary to provide basic characteristics of x.

2.    Create a function dsummary that returns, the minimum, the 1st decile, the 1st quartile, the median, the mean, the standard deviation, the 3rd quartile, the 9th decile, and the maximum.

3.    Suppose X ∼ N(2,0.25). Evaluate f(0.5),F(2.5),F−1(0.95)

4.    Repeat if X has t-distribution with 5 degrees of freedom.

5.    Suppose X ∼ P(3,1), where P is the pareto distribution. Evaluate f(0.5),F(2.5),F−1(0.95)

                                                                     Exercise 15          Moments
Consider a vector V = rnorm(100,−2,5).

1.    Evaluate n as the length of V.

2.    Compute the mean 

3.    Compute the variance 

4.    Compute the skewness 

5.    Compute the kurtosis 

                                                                           Exercise 16           OLS
1.        Create a matrix X of dimension (1000,10). Fill it with draws from a beta distribution with shape1 parameter 2, and shape 2 parameter 1. Make sure that there is no negative.

2.        Create a scalar denoted by σ2 and set it to 0.5. Generate a vector β of size 10. Fill it with draws from a Gamma distribution with parameters 2 and 1.

Table 6: Matrix operation

                                 Function (Operator)      Description

 

A ∗ B     Element wise multiplication A% ∗ %B matrix multiplication

                                 t(A)                                        Transpose

diag(a) Create a diagonal matrix with a elements diag(A)          Return the diagonal of A Solve(A)         inverse of A

 

3.        Create a vector  of 1000 draws from a normal distribution.

4.        Create 

5.        Recover βˆ = (X0X)−1(X0Y )

6.        Evaluate . Plot the histogram (filled in grey) and the kernel density of the

distribution of the error term.

7.        Estimate , and 

 q

8.        Create param that binds (β,         V (βb)). Using the command lm, check these estimates.

9.        Construct a confidence interval for β.

10.    Redo the exercise by setting σ2 = 0.01. How are your confidence intervals for β.