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Annex III.1 A Formal Description of the Empirical Methodology
This annex provides a formal description of the empirical methodology used in the paper.
Let Xt denote a vector containing the first differences of the relative terms of trade, relative output, the real effective exchange rate, and the relative price level.1 Then, we can write the reduced-form VAR as
Where B(L) is a 4x4 matrix of lag polynomials. This VAR can be inverted to obtain the following moving average representation:
In order to be able to give an economic meaning to the estimation results, one has to derive an alternative moving average representation where the shocks are mutually uncorrelated and can be interpreted as fundamental macroeconomic shocks, that is:
The relationship between the reduced-form and the structural parameters are evident from the comparison of equations (2) and (3), namely: ηt = A0−1ε1 and Aj = CjA0, for j = 1, 2,… Since the variance covariance matrix, Ώ, is symmetric, the identity A0A0 = Ώ entails ten restrictions on the sixteen elements of A0. Consequently, the identification of the A0 matrix requires six additional restrictions which are imposed by constraining particular long-run multipliers in the system to be zero.
One can write the set of long-run multipliers as the matrix A(1) = [A0 + A1 + A2+…]. or alternatively, A(1) = [I + C1 + C2 + …]* A0. Hence, given the estimates of Cj for j = 1, 2, by constraining a particular long-run multiplier, one imposes a linear restriction on the elements of the A0 matrix. As described above, in this paper it is assumed that terms of trade shocks alone have a permanent effect on the level of the terms of trade, that nominal and demand shocks have no long run-effect on the level of output, and finally, that nominal shocks do not have a permanent effect on the level of the real exchange rate. These assumptions restrict the elements (1,2), (1,3), (1,4), (2,3), (2,4), and (3,4) of A(1) to be zero, and make the A0 matrix uniquely identified.
Finally, because of the lower triangular structure of the A(1) matrix, one can interpret η11, η21, η31, and η41 as the underlying terms of trade, supply, demand, and nominal shocks, respectively.
The actual data used in this paper are described in Footnote 9.
Prepared by Giovanni Dell’ Ariccia (x38135) who is available to answer questions.
An intuitive way to distinguish between the last two shocks is to think of demand shocks as impacting the IS curve and of nominal shocks as affecting the LM curve.
In November 2000 the real effective exchange rate was about three standard deviations below the level predicted by the tracking equation used by most analysts, based on terms of trade and interest rate differentials (see Beechey et al., 2000). Although similar episodes of divergence have occurred in the past, they were before the floating of $A.
As in line 61 of the IFS.
However, econometric problems associated with omitted variables and endogeneity of the regressors may bias standard tests of coefficient stability in this case (see Rigobon, 2000).
Macfarlane, I., “Recent Influences on the Exchange Rate” November 9, 2000. Available at www.rba.gov.au.
Some analysts have compared the expected growth gap between the United States and Australia to movements in the exchange rate and find that they track movements in 2000 quite well. However, over a longer period (i.e., the 1990s), movements in the expected growth differential does not perform well in explaining the behavior of the $A. See Asia Economic Viewpoints, Chase Manhattan Bank, September 2000.
Lack of a complete set of data for 2000 precluded its inclusion in the sample. All data in this paper are from the IFS, the OECD, and the Reserve Bank of Australia. Relative output for Australia, Canada, and New Zealand is constructed using each country’s domestic real GDP and the trade—weighted average of the real GDP of its trading partners (although not complete, the data in this paper covers well above 90 percent of total trade for each country). Similarly, the relative price level is constructed using domestic CPI and a trade—weighted average of the CPI of partner countries. Finally, the real effective exchange rate was computed using the nominal effective exchange rate and the relative price level.
Supply shocks are the main innovations affecting relative output in the medium run. To that extent, they can be interpreted as shocks to expectations.