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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