Financial Econometricsyear

puk

1942

1943

1944

1945

1946

1947

1948

1949

1950

1951

1952

1953

1954

1955

1956

1957

1958

1959

1960

1961

1962

1963

1964

1965

1966

1967

1968

1969

1970

1971

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

pus

2.751

2.851

2.911

2.971

3.091

3.277

3.526

3.629

3.742

4.082

4.456

4.592

4.683

4.887

5.137

5.318

5.488

5.511

5.568

5.76

6.01

6.123

6.327

6.622

6.883

7.064

7.393

7.79

8.289

9.071

9.718

10.602

12.303

15.285

17.814

20.637

22.349

25.343

29.901

33.451

36.331

37.998

39.891

42.318

43.758

45.583

s

16.3

17.3

17.6

18

19.54

22.34

24.08

23.85

24.08

25.98

26.55

26.75

26.88

26.78

27.18

28.15

28.92

29.16

29.62

29.92

30.26

30.62

31.03

31.56

32.46

33.4

34.8

36.67

38.84

40.51

41.85

44.45

49.33

53.84

56.94

60.61

65.22

72.57

82.38

90.93

96.5

99.6

103.9

107.6

109.6

113.6

4.04

4.04

4.04

4.03

4.03

4.03

4.03

3.69

2.8

2.8

2.79

2.81

2.81

2.79

2.8

2.79

2.81

2.81

2.81

2.8

2.81

2.8

2.79

2.8

2.79

2.75

2.39

2.39

2.4

2.44

2.5

2.45

2.34

2.22

1.8

1.75

1.92

2.12

2.33

2.02

1.75

1.52

1.34

1.3

1.47

1.64

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

47.817

51.536

56.412

59.723

61.957

62.944

64.463

66.697

68.307

70.45

72.865

73.988

76.176

77.526

78.818

81.098

83.513

85.871

88.615

92.414

96.082

95.589

100

105.207

108.56

111.87

114.51

118.3

124

130.7

136.2

140.3

144.5

148.2

152.4

156.9

160.5

163

166.6

172.2

177.1

179.9

184

188.9

195.3

201.6

207.34

215.3

214.54

218.06

224.94

229.59

232.96

236.74

1.78

1.64

1.78

1.77

1.77

1.5

1.53

1.58

1.56

1.64

1.66

1.62

1.52

1.44

1.5

1.64

1.83

1.82

1.84

2

1.85

1.57

1.55

1.6

1.59

1.56

1.65

Financial Econometrics

Coursework assignment

According to the Purchasing Power Parity theory of nominal exchange rate determination, at

time t a particular bundle of goods should cost exactly the same either:

(i)

(ii)

if it is purchased in the UK for a given price in £, say PtUK = £100; or

if it is purchased in the US from the proceeds of converting £100 into $ at the

current nominal exchange rate.

If the purchase price in the US is PtUS = $150, PPP implies that the nominal exchange rate

should be St = 1.5, i.e. £1 = $1.50.

An implication of PPP is that if price inflation is running at different rates in the two

countries, the nominal exchange rate should adjust so that the PPP condition is maintained.

For example, suppose UK price inflation is 10\% between year t and year t+1, and US price

inflation is 5\%.

In the UK, the bundle of goods will cost PtUK

+1 = £100×1.1 = £110

In the US, the bundle of goods will cost PtUS

+1 = $150×1.05 = $157.5

Therefore the nominal exchange rate required for the PPP condition to be maintained is St+1 =

1.4318 (=1.50×1.05/1.1). i.e. £1 = $1.4318. The £ has depreciated in value against the $.

An alternative way of expressing the PPP condition is to say that the real exchange rate,

PtUK S t

defined as Q t = US , should always be constant.

Pt

+ s t − p US

Applying a log transformation, q t = p UK

t

t

= ln( PtUK ) , p US

= ln( PtUS ) , st = ln(St)

where Qt = ln(Qt), p UK

t

t

Many empirical tests for the validity of the PPP theory of exchange rate determination have

focused on:

either the stationarity/non-stationarity of qt (PPP qt should be stationary),

US

or the existence/non-existence of a cointegrating relationship between p UK

and st (PPP

t , pt

US

p UK

and st should be cointegrated).

t , pt

The Excel file exchrate.xlsx contains yearly time-series data for the period 1942-2014 for the

following series:

1

puk = UK consumer/retail price index series, PtUK

pus = US consumer/retail price index series, PtUS

s0 = nominal $/£ exchange rate (US dollars per GBP), St

After loading the data into Stata, generate the natural logarithms of the three series (denoted

below using lower-case symbols). Generate the real exchange rate series, and its natural

logarithm. Suggested Stata variable names for the log series are lpuk, lpus, ls and lq.

1.

Test each of the following series for stationarity or non-stationarity using the 68

observations for 1947-2014 only, using a Dickey-Fuller or Augmented Dickey-Fuller

unit root test:

(i)

p UK

t

(ii)

p US

t

(iii)

st

(iv)

qt

In each case, use the Akaike Information Criterion to select the appropriate order (laglength) for the DF/ADF(p) test, starting from p=4 and reducing p in steps of one as far

as possible. Include a time-trend in the Dickey-Fuller autoregressions for (i), (ii) and

(iii), but not for (iv). Observations from years before 1947 can be used to create any

lagged variables used in the DF/ADF autoregressions, but all DF/ADF

autoregressions should be estimated over the 68 observations for 1947-2014 only.

For any series of the series that you find to be non-stationary, determine the order of

integration by repeating the unit root test on the first-differences of the same series,

and (if necessary) the second-differences.

Comment on the implications of (iv) for the validity of the PPP theory.

US

The tests completed in Q1 may produce evidence to suggest that one or more of p UK

t , pt

and st is I(2). In the following questions, however, for simplicity and for consistency with the

PPP theory, we will assume that all three of these series have the same order of integration

I(1).

2.

Estimate a VAR model for ( p UK

, p US

t

t , st ) using the 68 observations for 19472014 only.

Use the multivariate Akaike Information Criterion to select the appropriate order (laglength) for the VAR(p) model, starting from p=4 and reducing p in steps of one as far

as possible.

Using your chosen model specification, carry out Granger causality tests of the

following null hypotheses:

(i)

Lagged values of p US

and st do not Granger cause current values of

t

p UK

.

t

(ii)

Lagged values of p UK

and st do not Granger cause current values of p US

t

t .

(iii)

Lagged values of p UK

and p US

do not Granger cause current values of st.

t

t

2

3.

US

Test for the existence of a cointegrating relationship between p UK

and st

t ,pt

using the Engle-Granger two-step residuals-based procedure:

= ˆ 1 + ˆ 2 p US

+ ˆ 3s t + v̂ t

Obtain the estimated cointegrating regression: p UK

t

t

using observations 1942-2014.

Save the residuals v̂ t , and test for stationarity using the Engle-Granger adaptation of

the ADF test, using the observations 1947-2014 only. Observations from years before

1947 can be used to create any lagged residuals used in the DF/ADF autoregressions,

but the DF/ADF autoregressions should be estimated over the 68 observations for

1947-2014 only.

Comment on the implications of this cointegration test for the validity of the PPP

theory.

4.

US

Test for the existence of a cointegrating relationship between p UK

and st

t ,pt

using the Johansen Vector Error Correction Model (VECM) procedure:

Use the multivariate Akaike Information Criterion to select the appropriate order (lagUS

length) for the three-variable VAR(p) model for { p UK

t , p t , st }, starting from p=4

and reducing p in steps of one as far as possible. Use the observations 1947-2014

only. Select the lag-length, p*, that produces the smallest value of MAIC.

Compute the Johansen rank and maximal eigenvalue cointegration tests based on the

VECM derived from the VAR(p*) representation. Comment on the implications of

this cointegration test for the validity of the PPP theory.

Estimate the VECM based on the VAR(p*) model with one cointegrating vector. Are

the signs and statistical significance of the coefficients in the cointegrating vector

consistent with the PPP theory?

Submission date: Monday 9 January 2017

3

General instructions

Your submission should comprise a written report, containing all relevant estimation and

hypothesis test results presented in report form (as if you were writing for a journal article),

and a brief commentary outlining the estimations and tests, and commenting on the results.

There is no specific word limit for the commentary; a few sentences for each question will be

sufficient. Marks will be awarded for clarity and presentation. You are encouraged to devise

informative means for presenting the results, including well-designed tables. Look at journal

articles which report empirical research for ideas on how to present the results of estimations

and statistical tests. Your Stata output should be presented in an Appendix to the written

report. Please submit one hard-copy to the Administrative Centre, and one electronic copy on

Blackboard.

You may collaborate in working out how to run the estimations on Stata. However, you are

required to write up your report independently, and I will require a separate and independent

submission from each person. Please ask if you are experiencing difficulties.

John Goddard

November 2016

4

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