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Problem_Set_3_Econ_103

May 3, 2022

1

ECON 103 – Problem Set 3

This problem set is devoted to the study the properties of the simple regression model discussed in

Chapter~4.

You will need the following datasets from the course website or the below link (which have uploaded

for you):

1. food

2. wa_wheat

You can review how to load a dataset in R by checking the Section 0 of the Jupiter Notebook called

Week_5_Econ_103_Lab_Class 2021.ipynb

1.1

Due: May 3, 2022

Theory questions

1.2

Question 1 (4.2, Page 158)

Consider the following estimated regression equation (standard errors in parentheses):

y? = 10.00 + 1.00x

(se) (1.23) (0.117)

R2 = 0.756

According to the regression, we hqave that b?1 = 10.00 and b?2 = 1.00 Rewrite the estimates that

would result if:

(a) All values of x were divided by 20 before estimation.

(b) All values of y were multiplied by 50 before estimation.

(c) All values of y and x were multiplied by 20 before estimation.

(a)

(b)

(c)

1

[1]: # Use this R cell if you want to make calculations

hi

[ ]:

1.3

Data analysis questions

1.4

Question 2

Consider the data set of food expenditure and income used in class. Let y denotes Food expenditure

and x denotes the family weekly income. Do the following tasks:

(a) Evaluate the sample correlation between x and y, that is, rxy and compute the

2 . (You can obtain the correlation besquared value of this correlation, that is, rxy

tween variables by using the command cor(food$food_exp, food$income). Also

check Section 3 of the the Notebook Week_5_Econ_103_Lab_Class 2021.ipynb)

(b) Evaluate the linear regression Y = ?1 + ?2 · X + ? and the linear regression X =

?1 + ?2 · Y + ?. Are the ?2 estimates the same? Are the R2 the same? How does

2 ? (check Section 3 of

the R2 compare with the squared value of the correlation rxy

the the Notebook Week_5_Econ_103_Lab_Class 2021.ipynb)

(c) Evaluate the sample standard deviation of x and y, that is, ?x , ?y and

generate the transformed variables y? = y/?y and x? = x/?x . (You

can obtain the standard errors by using the following commands sy Insert Cell Below

Note also that you can change the description of the cell using the drop-down?

,?menu at the top-right corder of the cell.

# Note that your data is already loaded due to the previous exercise.

# If you attemt thie question before question 3, then use the following?

,?commands to load the data:

# The following commands are useful to load the data

#library(PoEdata) # Load Library that contains the data set

#data(food)

# Data set called wa_wheat is loaded

# Note that these commands do not generate any output

# The following commands are useful to show some basic description of the data

#names(food)

# displays the names of the variables in the data set

#head(food)

# Displays the first 10 rows of data

#summary(food)

# Basic Statistical description of the data

1.6

Question 4 (q. 4.4, page 158)

The general manager of an engineering firm wants to know whether a technical artists experience

influences the quality of his or her work. A random sample of 50 artists is selected and their years of

work experience and quality rating (as assessed by their supervisors) are recorded. Work experience

(EXP ER) is measured in years and quality rating (RAT IN G) takes a value in the interval one

to four, with 4 = very good and 1 = very poor. Two models are estimated by least squares. The

estimates and standard errors are

4

Model 1: RAT

IN G = 3.446 ? 0.001459 (EXP ER ? 35)2 ; N = 50

(se)

(0.0375)

(0.0000786)

Model 2: RAT

IN G = 1.4276 + 0.5343 log(EXP ER); N = 50

(se)

(0.1333)

(0.04333)

In this question you do not need to evaluate a regression, but simply use the information on the

regressions presented above.

(a) For each model, predict the rating of a worker with 10 years of experience.

(To answer this question, you may want to see Section 8 of the notebook

Week_5_Econ_103_Lab_Class 2021.ipynb)

(b) Using each model, estimate the expected marginal effect of another year of

experience on the expected worker rating for a worker with 10 years experience. (To answer this question, you may want to see Section 8 of the notebook

Week_5_Econ_103_Lab_Class 2021.ipynb)

(c) Using each model, construct a 95% interval estimate for the average marginal

effect found in (c). To answer this question, you will need to compute the estimated standard erro of your marginal effect, which turns out to be a simple

linear transformation of the estimator b2 . You will also need to compute the critical value tc . You may find it usefull to check section 1 of the the notebook

Week_5_Econ_103_Lab_Class 2021.ipynb. The section computes the confidence

interval for ? = c1 b1 + c2 b2 where c1 = 1 and c2 = 20. This question is simpler as

the marginal effect is linear transformation of b2 only.

(a)

(b)

(c)

[4]: #

#

#

#

#

Use this R cell for your regressions

You can add additional code cells for sake of organization of your answer

To add a code cell, simply click on:

Insert-> Insert Cell Below

Note also that you can change the description of the cell using the drop-down?

,?menu at the top-right corder of the cell.

1.7

Question 5 (q. 4.8, page 159)

The first three columns in the file wa_wheat contain observations on wheat yield in the Western

Australian shires Northampton, Chapman Valley, and Mullewa, respectively. There are 48 annual

observations for the years 1950 – 1997. The name of the varibale that contains data on the crop

yields of the Chapman Valley Shire is called chapman. For the Chapman Valley Shire, consider

the four itens below:

The tasks performed in this exercise requires the commands discussed in the Notebook

Week_5_Econ_103_Lab_Class 2021.ipynb Please study the Notebook before attempting the question.

5

(a) This item consistis of three tasks where Y syands for crop yield of the Chapman Valley Shire: (a.1) Estimate the Linear model Y = ?1 + ?2 time + ? (a.2)

Plot the residuals of the regression (for commands, you can check Section 5

of Week_5_Econ_103_Lab_Class 2021.ipynb) (a.3) Test the normality of the

residuals using the Jarque-Bera test (for commands, you can check Section 5 of

Week_5_Econ_103_Lab_Class 2021.ipynb)

(b) This item consistis of three tasks in the same fashion as in the item (a): (b.1)

Estimate the Linear-log model Y = ?1 + ?2 log(time) + ? (b.2) Plot the residuals

of the regression (b.3) Test the normality of the residuals using the Jarque-Bera

test (For commands related to the estimation of the Linear-log model, see Section 4

of Week_5_Econ_103_Lab_Class 2021.ipynb. See Section 5 to use the command

jarque.bera.test(ehat) #(in package ‘tseries’) that performs the Jarquebera Test.)

(c) Taking into consideration (i) plots of the fitted equations, (ii) plots of the residuals,

(iii) error normality tests, and (iv) values for R2 , which equation do you think is

preferable? (This is a theoretical question, no code needed)

(a) You can simply show the code in the cell below

(b) You can simply show the code in the cell below

[5]: #

#

#

#

#

Use this code cell for your answer

You can add additional code cells for sake of organization of your answer

To add a code cell, simply click on:

Insert-> Insert Cell Below

Note also that you can change the description of the cell using the drop-down?

,?menu at the top-right corder of the cell.

# The following commands are useful to load the data

library(PoEdata) # Load Library that contains the data set

data(wa_wheat)

# Data set called wa_wheat is loaded

# Note that these commands do not generate any output

# The following commands are useful to show some basic description of the data

names(wa_wheat)

# displays the names of the variables in the data set

#head(wa_wheat)

# Displays the first 10 rows of data

summary(wa_wheat)

# Basic Statistical description of the data

library(tseries) # This command loads the lybrary tseries that has the?

,?Jarque-Bera Test

# See Section 5 of the Notebook Week_4_Econ_103_Lab_Class.ipynb for an example?

,?of Jarque-Bera test

# The Jarque-Bera test is given by:

#jarque.bera.test(e) # Note that the variable “e” in the test denotes the?

,?residuals of a regression

1. northampton 2. chapman 3. mullewa 4. greenough 5. time

6

northampton

Min.

:0.3024

1st Qu.:0.9124

Median :1.0419

Mean

:1.1687

3rd Qu.:1.3023

Max.

:2.3161

time

Min.

: 1.00

1st Qu.:12.75

Median :24.50

Mean

:24.50

3rd Qu.:36.25

Max.

:48.00

chapman

Min.

:0.4167

1st Qu.:0.8586

Median :1.0133

Mean

:1.0724

3rd Qu.:1.2203

Max.

:2.0244

mullewa

Min.

:0.3965

1st Qu.:0.7871

Median :0.9706

Mean

:0.9841

3rd Qu.:1.1928

Max.

:1.7992

Registered S3 method overwritten by ‘quantmod’:

method

from

as.zoo.data.frame zoo

[ ]:

[ ]:

[ ]:

[ ]:

7

greenough

Min.

:0.4369

1st Qu.:0.9141

Median :1.0955

Mean

:1.1531

3rd Qu.:1.3285

Max.

:2.2353

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Tags:

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