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Financial Econometrics

 

Coursework assignment

 

General instructions

 

The word limit for this assignment is 2500 words excluding tables, figures, references, and the appendix. There will be a penalty for exceeding the word limit (20 marks). The assignment submission deadline will be announced later. Assignments submitted after the submission deadline will not be accepted and a zero mark will be awarded.

 

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. 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 (log file) and the do-file should be presented in an Appendix to the written report. [10 marks]

 

Please ask if you are experiencing difficulties. Note that plagiarism constitutes a serious offence and will result in a zero mark. All assignments are electronically checked for plagiarism.

 

Recommendations for choosing a group/doing group work:

 

1. Get a good balance of working skills (e.g., writing, numerical, presentation skills)

2. Agree on meeting times and stick to them.

3. Make sure that everybody has read the material prior to the meeting.

4. Assign tasks and duties to each group member to avoid ‘free riding’.

5. The group is jointly responsible for being unable to deliver the assignment on time.

 

Good luck!

 

 

 

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 country i for a given price in €, say Pti

= €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 S_t = 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 country i’s price inflation is 10% between year t and year t+1, and US price inflation is 5%.[1]

 

In country i, the bundle of goods will cost Pt+1i

= €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 S_t+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 Qti=PtiStPtUS

, should always be constant.

 

Applying a log transformation, qti=pti+
st–ptUS

 

where qti=ln⁡(Qti)

, pti=ln⁡(Pti)

, ptUS=ln⁡(PtUS)

, s_t = ln(S_t)

 

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

 

either the stationarity/non-stationarity of qti

 (PPP Þ qti

 should be stationary),

 

or the existence/non-existence of a cointegrating relationship between Pti

,
PtUS

,  and s_t (PPP Þ Pti

,
PtUS,

 and s_t should be cointegrated).

 

The Excel file “Euro-US ppp data.xlsx” contains monthly time-series data for the period January 2002 – April 2017 for the following series:

 

            p_i = Country i consumer price index series[2]

, Pti

p_us = US consumer price index series, PtUS

s = nominal $/€ exchange rate (US dollars per EUR), S_t

For the following exercises, your group will be allocated one of the countries that were part of the European Monetary Union prior to 2002. This means that you should only use the series corresponding to the price levels for that given country, the price level for the US, and the nominal $/€ exchange rate.

 

1.         [10 marks] Write a short literature review (around 500 words) about the Purchasing Power Parity theory of nominal exchange rate determination, and its implications for a monetary union (as the European Monetary Union).

 

2.         [25 marks] Read the data into Stata and create a Stata data file. Using the supplied do file template, write your own code to do the following exercises (remember that you will have to present this do file as part of the assignment). Create a log file (to be presented as part of the assignment as well).

 

Generate the natural logarithms of the three series (denoted below using lower-case symbols). Generate the real exchange rate series, and its natural logarithm.

 

Test each of the following series for stationarity or non-stationarity using the same number of observations in each estimation, using a Dickey-Fuller or Augmented Dickey-Fuller unit root test:

 

            (i)        pti

                    (ii)       ptUS

                  (iii)      s_t          (iv)      qti

 

            In each case, use the Akaike Information Criterion to select the appropriate order (lag-length) for the DF/ADF(p) test, starting from p=18 and reducing p in steps of one as far as possible. Include a time-trend in the Dickey-Fuller autoregressions for (i), (ii), but not for (iii), (iv). Remember to use the same number of observations in all DF/ADF autoregressions.

 

            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 pti

, ptUS

, and s_t is I(2), or even I(0). 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). 

 

3.         [20 marks] Estimate a VAR model for (∆pti

,
∆ptUS

, Ds_t ).

 

            Use the multivariate Akaike Information Criterion to select the appropriate order (lag-length) for the VAR(p) model, starting from p=15 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 Ds_t do not Granger cause current values of      ∆pti

.

            (ii)       Lagged values of ∆pti

 and Ds_t do not Granger cause current values of

.

            (iii)      Lagged values of ∆pti

 and

 do not Granger cause current values of Ds_t.

Obtain graphs of the impulse response functions for the effect on Ds_t over the next two years unit shocks to ∆pti

 and

. Do the impulse response functions appear consistent with the PPP theory?

 

4.         [15 marks] Test for the existence of a cointegrating relationship between pti

,
ptUS

 and s_t, using the Engle-Granger two-step residuals-based procedure:

 

            Obtain the estimated cointegrating regression: pti=π1+π2ptUS+π3st+vt

        using all observations available.

 

            Save the residuals

, and test for stationarity using the Engle-Granger adaptation of the ADF test. Determine the optimal lag-length for the test as in exercise (2), by using the Akaike Information, starting from p=18 and reducing p in steps of one as far as possible. Remember to use the same number of observations in the DF/ADF autoregressions (to make the AIC comparable across regressions). Do not include a constant or a trend in neither the DF/ADF autoregressions, nor the Engle-Granger test (the dependent variable is the first difference of the residuals in the former, and the residuals in the latter, both of which are untrended and centered around zero).

 

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

 

5.         [20 marks] Test for the existence of a cointegrating relationship between pti

,
ptUS

 and s_t, 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 {pti

,
ptUS

, s_t }, starting from p=15 and reducing p in steps of one as far as possible. Use the same number of observations in each model to make the models comparable. 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.

 

Regardless of your findings with respect to the evidence of cointegration, 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?