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) if it is purchased in the UK for a given price in £, say = £100; or
(ii) 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 = $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 = £100×1.1 = £110
In the US, the bundle of goods will cost = $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, defined as , should always be constant.

Applying a log transformation,

where Qt = ln(Qt), , , st = ln(St)

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),

or the existence/non-existence of a cointegrating relationship between , and st (PPP ? , and st should be cointegrated).
The Excel file realexch.xlsx contains yearly time-series data for the period 1942-2014 for the following series:

puk = UK consumer/retail price index series,
pus = US consumer/retail price index series,
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) (ii) (iii) st (iv) qt

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

The tests completed in Q1 may produce evidence to suggest that one or more of , 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 ( , , ?st ) using the 68 observations for 1947-2014 only.

Use the multivariate Akaike Information Criterion to select the appropriate order (lag-length) 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 and ?st do not Granger cause current values of .
(ii) Lagged values of and ?st do not Granger cause current values of .
(iii) Lagged values of and do not Granger cause current values of ?st.
3. Test for the existence of a cointegrating relationship between , and st using the Engle-Granger two-step residuals-based procedure:

Obtain the estimated cointegrating regression: using observations 1942-2014.

Save the residuals , 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. Test for the existence of a cointegrating relationship between , and st using the Johansen Vector Error Correction Model (VECM) procedure:

Use the multivariate Akaike Information Criterion to select the appropriate order (lag-length) for the three-variable VAR(p) model for { , , 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?
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. Your Stata output should be presented in an Appendix to the written report.

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.
year puk pus s
1942 2.751 16.3 4.04
1943 2.851 17.3 4.04
1944 2.911 17.6 4.04
1945 2.971 18 4.03
1946 3.091 19.54 4.03
1947 3.277 22.34 4.03
1948 3.526 24.08 4.03
1949 3.629 23.85 3.69
1950 3.742 24.08 2.8
1951 4.082 25.98 2.8
1952 4.456 26.55 2.79
1953 4.592 26.75 2.81
1954 4.683 26.88 2.81
1955 4.887 26.78 2.79
1956 5.137 27.18 2.8
1957 5.318 28.15 2.79
1958 5.488 28.92 2.81
1959 5.511 29.16 2.81
1960 5.568 29.62 2.81
1961 5.76 29.92 2.8
1962 6.01 30.26 2.81
1963 6.123 30.62 2.8
1964 6.327 31.03 2.79
1965 6.622 31.56 2.8
1966 6.883 32.46 2.79
1967 7.064 33.4 2.75
1968 7.393 34.8 2.39
1969 7.79 36.67 2.39
1970 8.289 38.84 2.4
1971 9.071 40.51 2.44
1972 9.718 41.85 2.5
1973 10.602 44.45 2.45
1974 12.303 49.33 2.34
1975 15.285 53.84 2.22
1976 17.814 56.94 1.8
1977 20.637 60.61 1.75
1978 22.349 65.22 1.92
1979 25.343 72.57 2.12
1980 29.901 82.38 2.33
1981 33.451 90.93 2.02
1982 36.331 96.5 1.75
1983 37.998 99.6 1.52
1984 39.891 103.9 1.34
1985 42.318 107.6 1.3
1986 43.758 109.6 1.47
1987 45.583 113.6 1.64
1988 47.817 118.3 1.78
1989 51.536 124 1.64
1990 56.412 130.7 1.78
1991 59.723 136.2 1.77
1992 61.957 140.3 1.77
1993 62.944 144.5 1.5
1994 64.463 148.2 1.53
1995 66.697 152.4 1.58
1996 68.307 156.9 1.56
1997 70.45 160.5 1.64
1998 72.865 163 1.66
1999 73.988 166.6 1.62
2000 76.176 172.2 1.52
2001 77.526 177.1 1.44
2002 78.818 179.9 1.5
2003 81.098 184 1.64
2004 83.513 188.9 1.83
2005 85.871 195.3 1.82
2006 88.615 201.6 1.84
2007 92.414 207.34 2
2008 96.082 215.3 1.85
2009 95.589 214.54 1.57
2010 100 218.06 1.55
2011 105.207 224.94 1.6
2012 108.56 229.59 1.59
2013 111.87 232.96 1.56
2014 114.51 236.74 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) if it is purchased in the UK for a given price in £, say = £100; or
(ii) 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 = $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 = £100×1.1 = £110
In the US, the bundle of goods will cost = $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, defined as , should always be constant.

Applying a log transformation,

where Qt = ln(Qt), , , st = ln(St)

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),

or the existence/non-existence of a cointegrating relationship between , and st (PPP ? , and st should be cointegrated).
The Excel file realexch.xlsx contains yearly time-series data for the period 1942-2014 for the following series:

puk = UK consumer/retail price index series,
pus = US consumer/retail price index series,
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) (ii) (iii) st (iv) qt

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

The tests completed in Q1 may produce evidence to suggest that one or more of , 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 ( , , ?st ) using the 68 observations for 1947-2014 only.

Use the multivariate Akaike Information Criterion to select the appropriate order (lag-length) 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 and ?st do not Granger cause current values of .
(ii) Lagged values of and ?st do not Granger cause current values of .
(iii) Lagged values of and do not Granger cause current values of ?st.
3. Test for the existence of a cointegrating relationship between , and st using the Engle-Granger two-step residuals-based procedure:

Obtain the estimated cointegrating regression: using observations 1942-2014.

Save the residuals , 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. Test for the existence of a cointegrating relationship between , and st using the Johansen Vector Error Correction Model (VECM) procedure:

Use the multivariate Akaike Information Criterion to select the appropriate order (lag-length) for the three-variable VAR(p) model for { , , 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?
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. Your Stata output should be presented in an Appendix to the written report.

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.
year puk pus s
1942 2.751 16.3 4.04
1943 2.851 17.3 4.04
1944 2.911 17.6 4.04
1945 2.971 18 4.03
1946 3.091 19.54 4.03
1947 3.277 22.34 4.03
1948 3.526 24.08 4.03
1949 3.629 23.85 3.69
1950 3.742 24.08 2.8
1951 4.082 25.98 2.8
1952 4.456 26.55 2.79
1953 4.592 26.75 2.81
1954 4.683 26.88 2.81
1955 4.887 26.78 2.79
1956 5.137 27.18 2.8
1957 5.318 28.15 2.79
1958 5.488 28.92 2.81
1959 5.511 29.16 2.81
1960 5.568 29.62 2.81
1961 5.76 29.92 2.8
1962 6.01 30.26 2.81
1963 6.123 30.62 2.8
1964 6.327 31.03 2.79
1965 6.622 31.56 2.8
1966 6.883 32.46 2.79
1967 7.064 33.4 2.75
1968 7.393 34.8 2.39
1969 7.79 36.67 2.39
1970 8.289 38.84 2.4
1971 9.071 40.51 2.44
1972 9.718 41.85 2.5
1973 10.602 44.45 2.45
1974 12.303 49.33 2.34
1975 15.285 53.84 2.22
1976 17.814 56.94 1.8
1977 20.637 60.61 1.75
1978 22.349 65.22 1.92
1979 25.343 72.57 2.12
1980 29.901 82.38 2.33
1981 33.451 90.93 2.02
1982 36.331 96.5 1.75
1983 37.998 99.6 1.52
1984 39.891 103.9 1.34
1985 42.318 107.6 1.3
1986 43.758 109.6 1.47
1987 45.583 113.6 1.64
1988 47.817 118.3 1.78
1989 51.536 124 1.64
1990 56.412 130.7 1.78
1991 59.723 136.2 1.77
1992 61.957 140.3 1.77
1993 62.944 144.5 1.5
1994 64.463 148.2 1.53
1995 66.697 152.4 1.58
1996 68.307 156.9 1.56
1997 70.45 160.5 1.64
1998 72.865 163 1.66
1999 73.988 166.6 1.62
2000 76.176 172.2 1.52
2001 77.526 177.1 1.44
2002 78.818 179.9 1.5
2003 81.098 184 1.64
2004 83.513 188.9 1.83
2005 85.871 195.3 1.82
2006 88.615 201.6 1.84
2007 92.414 207.34 2
2008 96.082 215.3 1.85
2009 95.589 214.54 1.57
2010 100 218.06 1.55
2011 105.207 224.94 1.6
2012 108.56 229.59 1.59
2013 111.87 232.96 1.56
2014 114.51 236.74 1.65