RDP 9601: Why Does the Australian Dollar Move so Closely with the Terms of Trade? Appendix C: Monte Carlo Simulations

This appendix outlines the Monte Carlo simulations used to generate results reported in Tables 5a, 5b and 6 in the text. Simulations are conducted assuming four different data generating processes (DGPs), which we examine in turn.

C.1 The Real Exchange Rate

To begin, we test the null hypotheses that our models of the real exchange rate have no explanatory power, that is, that = 0 in each of the equations:

To test these null hypotheses, we assume a data generating process (DGP₁) with the log terms of trade, tot_t, following the preferred specification identified in Section 2.2, and embodying the hypothesis that the real exchange rate is unforecastable:

Let and the estimates be . Both equations are estimated by OLS, with equation (C2) estimated over the full terms of trade sample, 1969:Q3 to 1994:Q2, and (C3) estimated over the post-float period, 1984:Q1 to 1994:Q2. The off-diagonal elements of V are derived from the correlation coefficient, _1,2, of the estimated errors, _1,t and _2,t over the post-float period 1984:Q1 to 1994:Q2. The estimates are:

and

The Monte Carlo distributions are generated by running 5,000 trials with each trial (i = 1, …, 5,000) proceeding as follows.

1. Draw a vector sequence of observations from a bivariate normal distribution with mean 0, covariance matrix , n = 58 and T = 44. (n is the length of the pre-float terms of trade sample; T is the post-float period, 1984:Q1 to 1994:Q4.)
2. Generate sequences of observations , according to ,
   using (tot₅,tot₄,tot₃,tot₂,tot₁) to start the autoregression.
3. For time t, (t = 71 corresponds to 1987:Q1; t = 100 to 1994:Q2) use the sequence of synthetic terms of trade data, to estimate the terms of trade model (C2) and thereby generate a vector of parameter estimates , and hence derive , k = 0,…, 8, t = 71,…, 100.
4. For time t, , use the sequences of synthetic data, and to estimate a preferred real exchange rate model. For model A, this is equation (6) in the text, . For model B, it is one of the three specifications in equation (8), depending on the time, t. Use this preferred real exchange rate model, together with estimates of , derived above, to generate , x = A,B, k = 1,…, 8, t = 71,…, 100.
5. Estimate the regressions , t = 71,…,100, by OLS for all values k = 1,…, 8, and for the two models and generate the OLS t-statistics, , x = A, β.^[10]

These 5,000 observations of t_OLS and t_OLS form the Monte Carlo distributions under the null hypothesis that the models of the real exchange rate have no explanatory power. Based on these Monte Carlo distributions, the results of the hypothesis tests, H₀₁:β = 0, for k = 1,…, 8, are shown in column (x) in Table 5a for model A and in Table 5b for model B.

We turn now to the second data generating process (DGP₂) which assumes that the real exchange rate model, estimated over the post-float period 1984:Q1 to 1994:Q2, is the true model. For this data-generating process, we again assume that the terms of trade follow model (C2) with parameter values (C4). The real exchange rate model (C3) is, however, replaced by:

with parameter values, estimated by OLS over the post-float period 1984:Q1 to 1994:Q2, given by:

The Monte Carlo distributions under this data generating process are again generated by running 5,000 trials with each trial (i = 1, …, 5,000) proceeding as before, but with the following modifications. In step 2, the sequences of observations , are now derived from and .

In step 5, after running the regressions , x = A,B, k = 1,…,8 we now collect the coefficients, .

These 5,000 observations of for the two models, x = A,B and for each value of k form the Monte Carlo distributions under the assumption that the real exchange rate is described by model (C6) with parameter values (C7). They are used to test the null hypotheses, H₀₂:β = 1, for k = 1,…, 8, with the results shown in column (xi) of Tables 5a and 5b.

C.2 Excess Returns

The third data generating process, DGP₃, combines the terms of trade and real exchange rate models defined by equations (C2) and (C6) with parameter values (C4) and (C7) – used for DGP₂ – with models for the Australian-foreign quarterly inflation differential, , and the one-year (four-quarter) Australian-foreign nominal interest differential, . is defined by = − , where = − ) and is the trade-weighted foreign quarterly inflation rate, =, and is defined analogously.

Simple bi-variate auto-regressive time-series models are fitted for and by OLS using data from 1987:Q1 to 1994:Q4 and allowing up to six lags for each variable. Eliminating insignificant lags leads to these models:

where ε3_t and ε4_t are uncorrelated i.i.d. N(0,1) random variables.

Assuming this data generating process, DGP₃, Monte Carlo distributions are again generated by running 5,000 trials with each trial i proceeding as for DGP₂, but with the following modifications. As well as generating sequences of observations , , we now also generate sequences of observations and using (C8) and (C9).

As for DGP₂, for time t, , we use the sequences of synthetic data, and to estimate real exchange rate model B (which involves estimating one of the three specifications in equation (8), depending on the time, t). With this real exchange rate model, together with estimates of E_t(), we generate , t = 71,…, 100.

At time t, the expected and actual excess returns to holding a one-year Australian dollar bond rather than a trade-weighted basket of foreign bonds, are and which are equations (12) and (10) in the text. By construction, the expected real interest differential using backward-looking inflationary expectations is , while the actual real interest differential is . Thus, for trial i, the expected and actual excess returns are (approximately) and .

For trial i, we use the values of E_t() and , to run the regression = αⁱ + βⁱ E_t() + ε_t. The 5,000 values of βⁱ form the Monte Carlo distribution with which the hypotheses, H₀₁:β = 0 and H₀₂:β = 1 are tested. The results are shown in the first row of Table 6.

The fourth data generating process, DGP₄, combines the terms of trade and real exchange rate models defined by equations (C2) and (C6) with parameter values (C4) and (C7), with models for the Australian-foreign quarterly inflation differential, , and the two-year (eight-quarter) Australian-foreign nominal interest differential, .

As before, simple bi-variate auto-regressive time-series models are fitted for and by OLS using data from 1987:Q1 to 1994:Q4 and allowing up to six lags for each variable. Eliminating insignificant lags leads to these models:

where, again, ε_3t and ε_4t are uncorrelated i.i.d. N(0,1) random variables.

The Monte Carlo distributions are derived as for DGP₃, with the following modifications. The expected and actual excess returns to holding a two-year Australian dollar bond are and which are equations (14) and (13) in the text. The expected real interest differential using backward-looking inflationary expectations is , while the actual real interest differential is . Thus, for trial i, the expected and actual excess returns are (approximately) and .

For trial i, we use the values of E_t () and , to run the regression . Again, the 5,000 values of βⁱ form the Monte Carlo distribution with which the hypotheses, H₀₁:β = 0 and H₀₂:β = 1 are tested. The results are shown in the second row of Table 6.

Footnote

As explained earlier, the actual exchange rate data ends in 1994:Q4, and hence the sample size is 30 for k = 1,2, but progressively less for forecasts further ahead (for k = 8, it is only 24). This pattern is replicated for the synthetic data. [10]