Write on purchasing power parity theory of nominal exchange

Question

Assignment Problem: Financial Econometrics

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.

Problem 1: 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).

Problem 2: 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) p_tⁱ (ii) p_t^US (iii) s_t (iv) q_tⁱ

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 auto regressions for (i), (ii), but not for (iii), (iv). Remember to use the same number of observations in all DF/ADF auto regressions.

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 p_tⁱ, p_t^US 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).

Problem 3:  Estimate a VAR model for (Δp_tⁱ, Δp_t^US, Δs_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 Δp_t^US and Δs_t do not Granger cause current values of

(ii) Lagged values of Δp_tⁱ and Δs_t do not Granger cause current values of Δp_t^US.

(iii) Lagged values of Δp_tⁱ and Δp_t^US do not Granger cause current values of Δs_t.

Obtain graphs of the impulse response functions for the effect on Δs_t over the next two years unit shocks to Δp_tⁱ and Δp_t^US. Do the impulse response functions appear consistent with the PPP theory?

Problem 4: Test for the existence of a co-integrating relationship between p_t, p_t^US and s_t, using the Engle-Granger two-step residuals-based procedure:

Obtain the estimated co-integrating regression: p_tⁱ = Π₁^ +  Π₂^p_t^US + Π₃^s_t + v_t^ using all observations available.

Save the residuals v^t , 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.

Problem 5: Test for the existence of a cointegrating relationship between p_tⁱ, p_t^US 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 {p_tⁱ, p_t^US and 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?

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