The direct approachrequires estimation of functions mapping a state-vector, x, into a vector of states for the full set of state variables in the simulation model. Three aggregate variables are used: the recreational catch per day of effort (CPUE) in the previous year, xc, the stocks at the beginning of the year of young red snapper (2-years old or less), xy, and adult red snapper (3-years old and above), xa. The CPUE variable, xc, coincides exactly with a variable in the simulation model so that a one-to-one mapping is possible for this variable. For the population variables, however, the size of the array X is much larger than could be incorporated into a DP model. The red snapper population in GBFSM is composed of 720 depth-cohort groups. Using the simulated data, a conditional expectation function was estimated for each depth-cohort combination, xdc:

E(ln Xdc) = aodc + aide ln xc + a2dc ln xy + a3dc ln xa (4.1)

The double-log specification was used to avoid the prediction of negative stocks.

Overall, these estimated equations were able to predict the cohort populations quite accurately. The goodness of fit is the most important diagnostic variable. As can be seen in Figure 14.2, for cohorts less than 13 years old, most of the R2s exceeded 80%. Although the R2s drop off for older cohorts, the failure of the model to predict these older stocks is less critical since the population is dominated by younger fish due to natural and fishing mortality. On average across the simulations, only about 5% of all fish were 14 years or older. These results indicate that the distribution of Xt conditional on xt is relatively "tight" so that it is possible to predict Xt quite accurately, giving a high degree of confidence in the direct approachfor solving the DP model. Higherorder polynomial specifications of the approximating function were evaluated

5There are many details that are suppressed here. A detailed description of GBFSM and its calibration are available at http://gbfsm.tamu.edu. The discount rate used in the analysis is the federally mandated rate of 7% per annum (Executive Office of the President).

and these specifications improved the overall goodness of fit. However, such specifications resulted in some unrealistic predicted values of X, particularly at points on the edge of the state space. Monte Carlo methods were used to esti-

mate the expectations Ev(xt+\) and Eut. The distribution f (xt) is made up of 720 jointly distributed variables. This distribution was estimated by drawing 'time =50 observations of vectors of residuals from the econometric model so that the ith observation of each depth-cohort group was ln X%dc = aodc + aide ln x%c + a2dc ln xt + a3dc ln x%a + 4C

in this case since it avoids the need to make an ad hoc assumption about the distribution (e.g., joint normality), which almost certainly does not hold true.

The vectors predicted using equation (4.2) do not typically sum exactly to their associated aggregate variable, e.g., xly = EydJ2e<2 Xde. Hence, after obtaining this initial prediction Xde using equation (4.2), all cohorts were proportionally scaled up or down so that for the cohort vectors used in the GBFSM it holds that A(X) = x.

A three-dimensional 20 x 20 x 20 uniform grid was generated to approximate the state space. However, in the simulated data there was a high degree of correlation among the three aggregate state variables and many potential combinations of state variables were never observed. To take advantage of this and to avoid making prediction in portions of the state space that are never realistically observed, only vertices of the grid that were adjacent to points in the data were included in the state space. This reduced the number of points in the state space from 8,000 vertices to only 519.

4.4 Solving the DP problem: The meta-modelling approach

The meta modelling specification was parameterized with the same data used for the direct method that is discussed above. Four functions were estimated, three to predict each of the state variables, xt+1, and a fourth to predict the fishery's annual surplus. The independent variables in each of these functions are the state of the fishery in year t as described by the aggregate state variables, xt, and the control variables, Zt. We experimented with a variety of specifications, all of which can be represented by the function:

Vk = at + ^ ^ bijk(vi)j + ^ ^ cijk(viVj), i=1 j=1 i=1 j=i+1

where yk is a predicted variable, an element of xt+1 or surplus in period t, np is the order of the polynomial, and the v's are the independent variables with v1 through v3 equal to the elements of the state space, and v4 and v5 equal to values of the control variables. Double-log specifications were also used, in which case, for example, yk = ln(Xi!-+1) and v1 = ln(xj). Using second and third-order polynomials, a total of four specifications were estimated.

Table 14.1. R-squared Values for the Specifications Used in the Meta-Modelling Approach

Equation

Table 14.1. R-squared Values for the Specifications Used in the Meta-Modelling Approach

Equation

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