Uwe Cantner, Bernd Ebersberger, Horst Hanusch, Jens J. Krüger and Andreas Pyka (2001)
Empirically Based Simulation: The Case of Twin Peaks in National Income
Journal of Artificial Societies and Social Simulation
vol. 4, no. 3,
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Received: 30-Jun-00 Accepted: 11-May-01 Published: 30-Jun-01
Figure 1. Income distribution for selected years ((1999)Pyka/Krüger/Cantner 1999) |
Figure 2. Relative technological scale |
Figure 3. Basic mechanism of a master-equation |
(1) |
(2) |
(3) |
(4) |
Eq. (1) | α = 0.775 |
d = 3.5 | |
Eq. (2) | g = 0.5 |
b = 15 | |
γ = 0.225 | |
Eq. (3) | o = 0.27 |
Figure 4. Simulated productivity distribution over time |
Figure 5. Productivity distributions for selected iterations (phase portrait) |
.
where the operator #(.) represents the number of members of a particular class and years for which the argument is true. In the following these probabilities are abbreviated by and for each class. The centers of the 20 classes on the scale [0,1] are given by x, i.e.
.
Table 1: Theoretical Specification: | |||
Coefficient | t-Statistic | p-Value | |
β_{1} (= α) | -0.0783 | -0.2647 | 0.7949 |
β_{2} (= -d) | 2.3486 | 0.8586 | 0.4041 |
β_{3} (= γ) | 0.7895 | 0.8317 | 0.4186 |
β_{4} (= b.g) | -1.7279 | -0.5203 | 0.6104 |
β_{5} (= -b) | -4.4012 | -0.1079 | 0.9155 |
0.9000 | |||
ln L | 45.7433 | ||
Note: t-statistics and p-values are based on heteroskedasticity consistent standard errors | |||
Table 2: Theoretical Specification: | |||
Coefficient | t-Statistic | p-Value | |
φ_{1} (= 0) | 0.2296 | 6.7165 | 0.0000 |
0.3004 | |||
ln L | 27.4072 | ||
Note: t-statistics and p-values are based on heteroskedasticity consistent standard errors | |||
Figure 6. Simulation results of the theoretical specification |
Table 3: Specification of the Functional Search Algorithm | |
Characteristic | Default value |
objective | find functional form and parameters simultaneously for o(x) and pr(x) |
number of individuals in population | 200 |
set of primitive functions | +, -, /, *, ^, exp |
set of terminals | Constants c drawn from [0,1], x |
fitness function (to be maximized) | R^{2} |
termination criterion | Number of generations > 200 |
genetic operators | Subtree-mutation, subtree-crossover, copying of the best individual, reproduction |
Table 4: Empirical Specification: | |||
Coefficient | t-Statistic | p-Value | |
θ_{1} | 1.5453 | 32.5218 | 0.0000 |
0.9026 | |||
ln L | 43.6456 | ||
Note: t-statistics and p-values are based on heteroskedasticity consistent standard errors | |||
Table 5: Empirical Specification: | |||
Coefficient | t-Statistic | p-Value | |
δ_{1} | 0.1936 | 4.5565 | 0.0002 |
δ_{2} | 0.5235 | 2.4320 | 0.0257 |
0.4066 | |||
ln L | 29.5940 | ||
Note: t-statistics and p-values are based on heteroskedasticity consistent standard errors | |||
Figure 7. Simulation results of the empirical specification |
^{2} For a detailed formal description of the model see Cantner/Pyka (1998).
^{3} The term functional search dates back to Schmertmann (1996) whereas Koza (1992) speaks of symbolic regression. Kargupta and Sarkar (1999) use the term function induction for a related procedure. We do not want to insinuate that functional search is used as a purely inductive device of analysis nor do we want to get confused with traditional regression analysis, hence we refer to the methodology as functional search as to stress the search-like character of the procedure.
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