Monte Carlo simulation was carried out taking into account both the variation in the parameter estimates and in the residuals accounting for cross-equation correlation assuming that both parameter estimates and the residuals are jointly distributed normally.
Figures 14.3 and 14.4 depict the value functions obtained using the direct and the meta-modelling approaches. Of the meta-modelling specifications used, only the second-order polynomial non-log specification is presented. As we note above, higher order polynomials can be unstable in the DP algorithm as they give highly variable predictions at the boundaries of the state space
and this is what occurred in the model specified here; the log and third-order polynomial specifications did not converge so those results are not presented. As seen in Figure 14.4, even the second-order specification gives rather unintuitive results with the value function falling as higher stocks are reached. The failure to achieve convergence in some of the meta-modelling specifications may surprise some readers who have been assured of the convergence of the successive approximation approach to solving DP problems. However, the contraction mapping properties of the algorithm do not necessarily hold when the value function is predicted over a continuous state space. An alternative solution approach that might have been more successful would be to implement the collocation method advocated by Judd (1998). However, this method requires the specification of a functional form for the value function. The functional form preferred by Judd is a Chebyshev polynomial. However, since Chebyshev polynomials must be defined over a rectangular grid, they are unsuitable for the non-uniform grid that we favor in this application to avoid extrapolation to unrealistic state-space combination.
Figure 14.5 presents the DPSim results, the simulated optimal policy paths for both the direct and meta-modelling approaches. At each point along these
paths an optimal policy is found by solving the static optimization problem, equation (2.1), with ut and xt+i found by GBFSM and the value function, v(xt+\), obtained from the solution of the optimization problem as presented in Figures 14.3 and 14.4. Once an optimal policy is identified, that policy is introduced into GBFSM to determine the state of the fishery in the next year. Both simulations start with the same stock at t=0, but because the policies differ the predicted stocks underlying the simulated policy paths differ. As seen in Figure 14.6, the red snapper stock recovers significantly when policies from the direct-method are followed, while the stock stays essentially stable if policies from the meta-modelling approach are used. The reason for the differences in the policies chosen must be attributable to the value functions presented in Figures 14.3 and 14.4. Because of the upward slope in the value function identified using the direct method, there is an identified benefit to increasing the stock. In contrast, the value function found using meta-modelling approach was not monotonically increasing so that stock enhancements were not treated favorably when equation (2.1) was evaluated to identify the optimal policies at each point in time.
The above figures give reason to believe that in this application the direct method is more reliable than the meta-modelling method. What is the reason for this? Recall that either method introduces potential errors. The direct method introduces errors through the incorrect prediction of the disaggregated stocks in period t, which then leads to errors in the prediction of ut and xt+1. The meta method introduces errors more directly through the inaccurate prediction of ut and xt+1. One cannot say a priori which errors will be more important. To compare the relative accuracy of the two methods, we compared
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