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日期：2022-12-16 03:58

Econometrics for Macro and Finance

Empirical Project 2022

Dennis Kristensen, UCL

1 Instructions

The mark for this empirical project is worth 20% of your total mark for the module.

You will be awarded a mark of 0% or Grade F if you (1) do not attempt the sum-

mative assessment component or (2) attempt so little of the summative assessment

component that it cannot be assessed. Please check the UCL Academic Manual (Sec-

tion 3.11) for information on the consequences of not submitting or engaging with

any of your assessment components.

If you are a re-sitting student or taking deferred assessment the academic regula-

tions for 2017/18 apply to you. In this case if you do not complete or take an

assessment component that is worth more than 20% of the total assessment you will

be considered incomplete. This means that you cannot pass the module. If this is

your rst attempt, you may be entitled to LSA in the component. Please discuss with

the Departmental Tutor (f.witte@ucl.ac.uk) if you are unsure of the consequences

for you.

If you have extenuating circumstances that a¤ect your ability to engage with any

of the module assessment components, please apply for alternative arrangements to

the Economics Department as soon as possible. See details in Section 6 of the Aca-

demic Manual and send your request to economics.ug@ucl.ac.uk.

If you have a disability or long-term medical condition, you may be entitled to ad-

justments for assessments. This may include an extension for this essay. Please see

Section 5 of the Academic Manual for information on how to apply for adjustments.

Contact the Departmental Tutor, Dr Frank Witte (f.witte@ucl.ac.uk) and the UG

Admin team (economics.ug@ucl.ac.uk). Do not contact the course lecturer about

this.

Please follow these instructions below so that we can ensure anonymity in marking

and ensure compliance with UCL assessment policies. We will only be able to give

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you credit for your project if you follow these instructions. IF THE INSTRUC-

TIONS ARE NOT FOLLOWED, YOU MAY RECEIVE A MARK OF

ZERO

1. You are allowed to work in groups of up to three persons. Please elect one

group member to submit the project for the group.

2. All answers must be uploaded via Turnitin by Noon on 16th December

2022.

3. All marking on Turnitin is anonymised. Do not put your name or student

number anywhere on your submitted answer - either in the document or in the

le name

4. Submit the project with the submission cover sheet on the rst page. Put the

candidate numbers for all group members on the submission coversheet. Your

candidate number is not you student number!

5. You should include your Stata commands in the appendix. If you use a di¤er-

ent software for the project, you should state which programme was used and

present your code and results in the appendix.

6. Your submission should be no more than 2000 words in length, including foot-

notes but excluding bibliographies, tables, ?gures, and the appendix which

should include your Stata commands. You must state the number of words at

the top of the ?rst page of your submission.

7. Upload your submission on Turnitin as a word document or pdf. Please make

sure to allow su￠ cient time should problems arise with Turnitin.

Here is some additional information about submissions and marking:

1. If your essay is longer than 2000 words the following Faculty guidelines on

penalties for over-long work will be applied:

For work that exceeds a speci?ed maximum length by less than 10% the

mark will be reduced by ?ve percentage marks, but the penalised mark

will not be reduced below the pass mark, assuming the work merited a

Pass.

For work that exceeds a speci?ed maximum length by 10% or more the

mark will be reduced by ten percentage marks, but the penalised mark

will not be reduced below the pass mark, assuming the work merited a

Pass.

2. Late work will be marked but will be subject to UCL rules as set out in Section

3.12 of the Academic Manual (click here more information). For the avoidance

of doubt, a working day means a 24-hour period from the 1pm deadline.

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3. It is your responsibility to ensure that your work is your own. Action will be

taken if there is any plagiarism concern, including failure to provide a complete

reference list with your work. See Section 3.14 of the Academic Manual for

more information on the consequences of the work not being your own (click

here more information).

4. If you are normally entitled to Reasonable Adjustments, such as extra time in

exams, you may be entitled to extra time for this Assessed Essay. You will need

to have a SORA in place for this to be taken into account. Please see Section

5 of the Academic Manual for the process to follow if you have not already

done so (click here more information). Make sure to follow the process as

early as possible in Term 1. The responsible marker will know which candidate

numbers have a SORA and this will be taken into account when reviewing

the timing of submissions. Contact the Departmental Tutor, Dr Frank Witte

(f.witte@ucl.ac.uk) and the UG Admin team (economics.ug@ucl.ac.uk).

Do not contact the course lecturer about this.

5. Check the submission inbox for con?rmation that your essay has been sub-

mitted. Once your submission has been accepted you will return to the ?My

Submissions?tab where you will be able to see the details of your submission.

If your submission is not con?rmed for some reason, or you are having issues

uploading the document, get in touch with ISD (servicedesk@ucl.ac.uk) as

soon as possible to ?gure out what the problem might be.

Any matters a¤ecting your ability to submit on time should be directed to the Depart-

mental Tutor (f.witte@ucl.ac.uk) rather than the module lecturer to ensure anonymity

is retained.

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2 Questions

The goal of this empirical project is to analyse the relationship between the GBP/USD

spot exchange rate and the one month forward exchange rate in order to better un-

derstand investors?risk preferences and at the same time build a forecasting model

for the spot exchange rate. First, an introduction to spot rates and forward rates in

the GBP/USD currency market:

The spot exchange rate is the amount of GBP you have to pay in order to receive

1 USD instantaneously. For example, the spot exchange rate on 23rd November 2022

was 0.8411 meaning that you had to pay 0.8411 GBP to receive 1 USD the same day.

If you ?rst plan to use USD sometime in the future, say, in one month time, you

can instead enter a so?called forward contract where you receive 1 USD in one month

time at a price agreed upon today. This way you avoid any risk of the exchange rate

increasing over the next month time. Such contracts are, for example, used by British

?rms conducting business in the US to hedge against exchange rate ?uctuations.

Today?s 1 month forward exchange rate in the GBP/USD market is therefore the

price in GBP at which investors are willing to buy 1 USD with delivery 1 month

later. For example, the one month forward rate on 23rd November 2022 was 0.8296

meaning that you had to pay 0.8296 GBP to receive 1 USD in one month time.

In the following let:

St = the GBP/USD spot exchange rate (e.g., 0.8411 GBP per USD).

Ft = the GBP/USD one month forward exchange rate (e.g., 0.8296 GBP per

USD).

Time series data of St and Ft can be found on the Bank of England?s website.

We will here work with data of the spot exchange rate and the 1 month forward

exchange rate observed at a monthly frequency from 31 January 2000?30 June 2021.

Speci?cally, follow these steps:

Go to https://www.bankofengland.co.uk/boeapps/database/ and click on

"Interest & exchange rates data"

Spot exchange rate:

1. Choose "Spot exchange rates" and then "US Dollar"

2. Tick the box of: End month - XUMLUSS - Monthly

3. Choose as data range 31 January 2000?30 June 2021.

4. Download data in your desired format

Forward exchange rate:

1. Choose "￡ sterling against US dollar forward rates"

2. Under "1 month" tick the box of: End month - XUMLDS1 - Monthly

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3. Choose as data range 31 January 2000?30 June 2021.

4. Download data in your desired format

Combine the two data sets and import them to Stata

Generate a time variable, here denoted t, that keep tracks of the di¤erent

months in our sample. The ?rst value of t should therefore be 2000m1 and

the last value should be 2021m6.

1. We wish to build a model for St and Ft and examine its empirical content. As

a benchmark, we here establish the relationship between St and Ft under the

scenario that investors are risk?neutral.

(a) Consider the following two investment strategies:

Strategy I: Buy 1 USD this month (at time t); convert your 1 USD back

to GBP one month later (at time t+ 1)

Strategy II: Buy 1 USD this month and at the same time buy a one month

forward contract which promises you 1=Ft GBP for 1 USD. You then use

the forward contract to convert your dollar amount back to GBP at time

t+ 1.

The two strategies leave you with the following amount of GBP, respec-

tively, at time t+ 1:

Pay-o¤ of Strategy I = St=St+1;

Pay-o¤ of Strategy II = St=Ft;

At time t, is the pay-o¤ of each of the two strategies certain or uncertain?

Explain.

(b) Let It be the investor?s information set at time t (which contains St and

Ft). Argue that if all investors are risk neutral then the following identity

should hold:

E [St=St+1jIt] = St=Ft: (1)

(c) Let st = log (St) and ft = log (Ft). Use the log-approximation x

log (x) + 1 to show that eq. (1) is (approximately) equivalent to

E [st+1jIt] = ft: (2)

Interpret this identity.

2. Plot each of the following three time series

st = log (St) ; ft = log (Ft) ; et = st ft1;

as function of t and comment on the features that you observe. In particular,

do you see any evidence of stochastic trends in the three series? Should we

worry about them being non?stationary? Explain.

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3. Report the ?rst 24 sample autocorrelations of each of the same three time series.

Comment on your ?ndings.

4. Suppose that et is stationary and mixing. Show that (2) implies that

Cov (et; etk) = 0 for all k 1: (3)

Test the following null hypothesis:

Cov (et; et1) = Cov (et; et2) = = Cov (et; et24) = 0:

Conclude.

5. Clearly, investors are not risk?neutral. A more general relationship is

E [st+1jIt] = t + ft; (4)

where the coe￠ cients t captures the investors?risk preferences. If investors

are risk?averse, which sign do you expect t to have?

6. To examine the empirical content of the pricing relationship (4), suppose that

t = is constant and consider the following two regression models:

st = + ft1 + "t;

st = + (ft1 st1) + "t: (5)

where we wish to test = 1. Explain why testing = 1 using a t-statistic

will most likely requires non?standard inference tools if you use the ?rst regres-

sion model, while the same test based on the second model can be done using

standard inference tools.

7. Estimate (5) and report the ?tted model together with heteroskedasticity?

robust standard errors together with other relevant statistics. Discuss the re-

ported output. Test the null of = 1 at a 5% level. Conclude.

8. Compute the residuals from the regression in Q.7 and report the 24 ?rst sample

autocorrelations. Based on these, do you trust the test that you carried out in

Q.7?

9. Report two sets of HAC standard errors for the regression in Q.7 using the

Newey?West estimator with the cut?o¤m chosen in two di¤erent ways:

(a) Choose m based on the sample autocorrelations that you reported in Q.8.

You should here justify your choice.

(b) Choose m according to the recommendation found in Stock and Watson?s

"Introduction to Econometrics".

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Compare the two sets of HAC standard errors to each other and the ones that

you reported in Q.7. Test the null that = 1 using the HAC standard errors

found in (a). Conclude.

10. Examine whether the Random Walk hypothesis for st is supported by data

where you use as alternative an ADL(p; q) model for st with p lags of st and q

lags of ft included. Justify your choices of p and q. Conclude.