RDP 2000-06: Inflation Targeting and Exchange Rate Fluctuations in Australia Appendix C: Design of the Stochastic Simulations

The stochastic simulations involve solving the model (described in Appendix A) over 100 periods allowing for the effect of a new draw of random shocks in each period. Here we describe the properties of the shocks and the procedure which we used to generate the simulations of the model for each specification of the policy rule.

For an estimated model, the distribution of the shocks is usually based on the estimated distribution of the residuals from each of the equations. The advantage of this approach is that the shocks used to simulate the model embody the actual historical correlations of the shocks – making the results of such simulations more quantitatively significant. The approach which we used to generate the shocks is based on Bryant, Hooper and Mann (1993, pp 240–241) and was also used by de Brouwer and O'Regan (1997).

The approach involves calibrating each vector of shocks to the estimated variance-covariance matrix of the equation residuals. The matrix was calculated over the common sample 1985:Q1–1998:Q4. Residuals from the import price equation were excluded because, for the purposes of the simulations, we assume that all foreign nominal shocks are captured in shocks to the world price Equation (B5). In each period, shocks to the remaining 10 variables in the model were obtained by passing a 10×1 vector of realisations from the standard normal distribution through a Cholesky decomposition of the 10×10 residual variance-covariance matrix.^[16] The seed of the random number generator was kept constant in order to generate the same set of shocks for each simulation. In this way, differences in the stochastic behaviour of the economy under each simulation can only be attributed to differences in the specification of the policy rule.

Armed with a set of shocks, the process for generating the stochastic simulations starts with the assumption that the policy-maker sets the policy instrument at the beginning of each period, before the current period's shocks have arrived. Then, at each point in time, we simultaneously solve for each of the endogenous variables in the model, including the policy instrument, over a 25-period horizon, assuming there are no future shocks and the exogenous variables follow the data-generating processes reported in Appendix B. Beyond 25 periods, we close the model by assuming that aggregate and non-tradeable inflation have returned to target and actual output is at potential. Our experiments indicated that a 25-period horizon was sufficiently far in the future that imposing these end conditions did not affect current outcomes.

The forward solution at each point in time involves a path for each of the endogenous variables over the next 25 periods, conditional on the specification of the policy rule and the assumption of no further shocks. The first realisation from the path for the policy instrument is then taken to be the actual setting for the policy instrument in the current period. The actual outcomes for the remaining endogenous variables are then determined by applying a set of shocks to the exogenous variables and the solutions for the endogenous variables in the current period. We then move forward one period in time, and repeat the whole process, using the actual outcomes just derived as part of the historical starting values for the next 25-period solution to the model. We repeat this process for 100 periods, revealing a different set of shocks each period.

By repeating the entire process, we are able to generate stochastic simulations of the model for each specification of the policy rule. From the simulated outcomes for each policy rule, we can then summarise various aspects of the stochastic behaviour of the model by calculating unconditional moments of the endogenous variables. In order to construct the efficient policy frontier in inflation and output variability space, we calculate the standard deviation of annual inflation and the output gap (both expressed in percentage points). These standard deviations are calculated around the population means (2½ per cent for inflation, and zero for the output gap) in order to account for small-sample biases that can cause the sample means to deviate from population means. Each specification of the policy rule therefore provides us with a single point in inflation and output variability space. The efficient policy frontier is mapped out by joining the set of outcomes that minimise the variability in either inflation or the output gap for given variability of the other.

The simulations were performed entirely in Mathematica (version 4.0) based on code used to generate the stochastic simulations in de Brouwer and Ellis (1998). The computational burden involved in generating these simulations was heavy – each 100 period simulation (for a given specification of the policy rule) took just over 2 minutes to generate. In order to obtain our results within a reasonable period of time, we relied on the earlier results of de Brouwer and Ellis (1998) to reduce the dimensions of the grid search over possible feedback coefficients in the benchmark policy rule. Nevertheless, we still searched at least 300 sets of weights in order to produce a reliable estimate of each efficient frontier.

Footnote

While we did not perform the same experiment ourselves, de Brouwer and O'Regan (1997, footnote 17) – using a similar model to ours – reported that their efficient policy frontiers did not change significantly when the covariances between the shocks were set to zero, so that only the variances of the shocks mattered. [16]