QBA25622 ASSIGNMENT PROJECT

Discipline of Finance, UTS.

Semester 1, 2015.

Subject Coordinator: Dr Otto Konstandatos This assignment has several parts of unequal value. Some parts are technical, and are designed to help

you understand the connection between Mathematics and Econometric Regression Analysis.

You must type or neatly hand-write such answers in the spaces provided.

Other parts are applied, and are designed to help you develop practical econometric skills. You will

apply regression analysis as done by finance professionals using real world data. These parts will test

your econometric modelling skills using multiple regression and Eviews.

For these parts you may be required to report your Eviews output in the spaces below.

This assignment can be done in groups of up to five students. Your group members do not need to be

part of the same tutorial group. You must include this cover sheet. No names may be added to the

group lists apart from the names that appear below at hand-in. Submit only one assignment for each

group. DO NOT hand in a separate assignment for each group member. Late assignments will

require a Special Consideration application and will otherwise not be accepted.

Due Time/Date/Venue: The Student Centre will be collecting assignments this semester. The assignment drop box is located in Building 5. There are no individual collection boxes for each

subject – just ONE chute for all subjects. It is important to identify your assignment CLEARLY.

Submit your assignment by 12:00pm Thursday 28 May at Level 1 (street entry) Block C of

Building 5, located on the SIDE of the Student Centre (along the corridor) and NOT over the

counter. The assignment box is accessible 24 hours.

Name Student Number

Tutor’s Name:

Tutorial Day and Time:

Date stamp or tutor’s signature and date

Group Work Protocol

Group work and group assignments are an important part of business education. The

purpose of group work goes beyond the requirements of the group project itself. Group

work is also intended to develop awareness of the dynamics of teamwork and your role in

a team environment. In undertaking group work it is expected that each team member

will communicate with other team members in a professional and courteous manner.

Resolving any conflict or difficulties that arise within the team is also part of the group

dynamics that may emerge during the project. It is expected that each team member will

take responsibility for managing and reducing any conflict that may arise. If the team

members are unable to resolve tensions or conflicts then this MUST be discussed with the

lecturer/tutor as soon as it becomes obvious that a resolution within the team cannot be

agreed.

In the learning environment of group work we are seeking to ensure that you develop and

demonstrate both academic skills and important group work skills such as:

Commitment to working with others (eg, undertaking a fair share of the work,

sharing ideas, doing the tasks allocated, attending meetings)

Collaboration and inclusiveness (eg encouraging and supporting others, respecting

others; recognising the skills and valuing the contribution of others, helping

resolve conflicts)

Contribution to establishing and working towards a common outcome (eg

establishing and supporting team goals, plans, rules, roles, decisions)

Group Work Declaration Form

Individuals must sign a declaration form, otherwise the project report will not be

marked. Complete only Part 1 or Part 2. Do not complete both. This sheet should be

completed and handed in along with your group assignment.

Part 1 I believe that all members of the group have contributed fairly to this assignment, and each member should

receive the same mark for the assignment.

Your name:.............................................................................Signature............................. ............................

Group member names:

Part 2

I believe that not all members of the group have contributed fairly to this assignment. I believe the

proportion of the total workload that each has contributed is indicated below.

Your name..........................................................................Signature...........................................................

Group member names: Proportion of Workload:

……………………… ……….….%

………………………. …………..%

……………………… ……….….%

………………………. …………..%

………………………. …………..%

Total = 100%

Please note that if you have signed Part 2 you must attach a statement explaining:

Why you believe a group member/group members have not contributed their fair share to

the project

A breakdown of tasks you and others have completed in the project

An estimate of the time you think each of those tasks has taken.

Any other positive or negative contributions made by you and others

This information will be shown to other group members so that they have an opportunity

to respond.

While each group member’s comments will be taken into consideration, the final

decision on how the marks are awarded will remain the right of the course coordinator.

Background.

The Cobb-Douglas production function is widely used by economists to represent the functional

relationship between a country’s real output (GDP) with the inputs Capital (K) and Labour (L). It is

named after American economists Charles Cobb (1875 – 1949) and Paul Douglas (1892 -1976), who

first tested this functional from against statistical evidence in the early twentieth century.

As we saw in lectures, the Cobb-Douglas function takes the form:

𝑌 = 𝐶 𝐾𝛼𝐿𝛽 (1)

You are supplied with empirically gathered data, summarising the normalised GDP output for

Australia from the years 1965 to 2008 with the corresponding indexes for Labor and Capital inputs for

each year. Your task is to undertake a study to estimate the Cobb-Douglas production function from

the given set of data in the spreadsheet ausdata.xlsx. The variables are as follows:

Y = Index of total production, namely the value of all goods produced in the given year.

K = Index of total value of all Capital input in the given year.

L = Index of total value of all Labour input in the given year.

The constant C is the total factor productivity. It represents all the factors influencing total production

not accounted for by the inputs Capital and Labour.

The constant 𝛼 is called the Output Elasticity for Capital. The constant 𝛽 is the Output Elasticity for

Labour. The output elasticities determine the ceteris-paribus responsiveness of Total Production

(output Y) to a change in levels of Capital and Labour respectively. When the output elasticities

satisfy the extra restriction:

𝛼 + 𝛽 = 1 (2)

the model will display constant returns to scale, so that a doubling of the inputs of Labor and Capital

will lead to a doubling of production.

Your task is to use econometric modelling to estimate the constants C, 𝛼 and 𝛽 for Australia’s Cobb-

Douglas production function (1), using the linearised model:

ln 𝑌 = 𝐵0 + 𝐵1 ln 𝐾 + 𝐵2 ln 𝐿 + 𝑢𝑡 (3)

Please enter all your answers into the provided spaces below.

Part 1: Deriving the Least Squares Estimators

Recall, in Linear Least Squares, we have to estimate the ‘line of best fit’ which

minimizes the ‘sum of squared deviations’ of the data. This is equivalent to choosing

parameters ),( 10 bb which minimises the function of two variables (namely SSR):

n

i

ii xbbybbfSSR1

2

1010 )(),( .

You will use calculus to show that ),( 10 bbf is minimized for the choices:

xbyb

xx

yyxx

bn

i

i

i

n

i

i

100

1

2

1

11ˆ,

)(

)()(ˆ

where

n

i

ixn

x1

1 and

n

i

iyn

y1

1.

You may assume the following summation properties in your solution (see Lecture 5):

If k is a constant, nkkn

i

1

.

Given iu and iv for ni 1 , then

n

i

i

n

i

i

n

i

ii vbuabvau111

)( .

Differentiating a summation (the ‘derivative of the sums’ equals ‘the sum of the

derivatives’):

),(

),( and ),(

),(1111

n

i

in

i

i

n

i

in

i

iy

yxfyxf

yx

yxfyxf

x

Q1 (1 mark): In lectures we saw that the ‘total deviation about the mean’ is always zero,

i.e.

n

i

i xx1

0)( . Expand the summation across the brackets and apply this result to the

iy to prove that: )()()(11

yyxyyxx i

n

i

ii

n

i

i

.

Ans:

Q2 (1 mark) Prove that (xi − x )2𝑛

𝑖=1= xi (xi − x )

𝑛

𝑖=1

Ans:

Q3 (2 marks): By differentiating the summation, show that 0),( 10

0

bbf

b when

xbyb 100ˆ .

Ans:

Q4 (1 mark): Differentiate the summation with respect to 1b to get a summation

expression for ),( 10

1

bbfb

.

Ans:

Q5 (2 marks): Prove that by choosing b0 and b1 to minimize

n

i

ii xbby1

2

10 )( you

obtain the least squares estimators, namely:

xbyb

xx

yyxx

bn

i

i

i

n

i

i

100

1

2

1

11ˆ

)(

)()(ˆ

Ans:

Q6. (1 mark): Given the data in the supplied spreadsheet, can you estimate the Cobb-Douglas

production function in the form of Equation (1) (𝑌 = 𝐶 𝐾𝛼𝐿𝛽 ) directly using Eviews and Linear

Least Squares? If not, why not?

Ans:

Q7. (1 mark): A production function has constant returns to scale if doubling the inputs of

Capital (K) and Labour (L) doubles the total output (Y). If the output elasticities satisfy restriction

(2), prove that the Cobb-Douglas production function (1) exhibits constant returns to scale.

Is this still true if the restriction given by Equation (2) is relaxed?

Ans:

Q8. (1 mark): Take the natural logarithm of both sides of the equation to obtain a linear model of

the form:

0 1 2ln( ) ln( ) ln( )Y B B K B L for some coefficients 𝐵0,𝐵1 and 𝐵2. What is the relationship between these coefficients and the

parameters of the original non-linear model given by Equation (1)?

Ans:

Q9. (1 mark): Read the data into Eviews and transform the supplied data to estimate the coefficients

𝐵0, 𝐵1 and 𝐵2 in the linearised regression given by Equation (3). Cut and paste your Eviews output

below. (Note: Eviews uses the function log(x) for Natural logarithm, i.e. log base e).

Ans:

Q10. (1 mark) Comment on the sign of the estimated coefficients 𝐵1 and 𝐵2, and state whether this is

what you expect. Ignore significance at this stage.

Ans:

Q11 (1 mark) Formulate and carry out an appropriate hypothesis test, to test whether Capital (K)

affects Australia’s output, Y. Use the t-statistic approach, at the =0.05 level.

Ans:

Q12 (1 mark) Formulate and carry out a hypothesis test for testing whether Labour (L) affects

Australia’s economic output, Y. Use the p-values approach, at the =0.05 level.

Include any Eviews output in your answer.

Ans:

Q13 (1 mark) Test the following joint hypothesis about the coefficients B1 and B2:

H0: 01 B and 02 B , H1: 01 B or 02 B , with =0.05.

Along with the previous results, what do you conclude about Capital and Labour’s significance for

explaining Australia’s GDP output?

Ans:

Q14 (2 marks) Test the following hypotheses for the coefficients B1 and B2. Clearly specify the

rejection region if you are using critical values, and clearly state your conclusions.

When using p-values, calculate and compare your p-values to the test size then state your conclusion.

Assume the Central Limit Theorem holds.

(a) H0: 01 B , H1: 01 B , with =0.05 using the critical-value approach.

(b) H0: 01 B , H1: 01 B , with =0.05 using the critical-value approach.

(c) H0: 02 B , H1: 02 B , with =0.05 using the p-value approach.

(d) H0: 02 B , H1: 02 B , with =0.05 using the p-value approach.

Ans:

Q15 (1 mark) An economist thinks that the Australian economy displays increasing returns to scale,

and consults you for help using your econometric expertise. Formulate a hypothesis test on the

regression coefficients in Q8 to test for increasing returns to scale using the data.

Do not perform the hypothesis test.

Ans:

Q16 (2 marks) Using re-parametrisation, convert the regression in Q8 to perform the hypothesis test

you formulated in Q15. Set up and run the re-parametrised regression model in Eviews. Copy and

paste the results of your re-parametrised Eviews regression and state your conclusion, at the 1% level.

Ans: