کنترل بهینه الگوهای اقتصاد سنجی دینامیکی غیرخطی: یک الگوریتم و برنامه

OPTCON is an algorithm for the optimal control of nonlinear stochastic systems which is particularly applicable to econometric models. It delivers approximate numerical solutions to optimum control problems with a quadratic objective function for nonlinear econometric models with additive and multiplicative (parameter) uncertainties. The algorithm was programmed in C#C# and allows for deterministic and stochastic control, the latter with open-loop and passive learning (open-loop feedback) information patterns. The applicability of the algorithm is demonstrated by experiments with a small quarterly macroeconometric model for Slovenia. This illustrates the convergence and the practical usefulness of the algorithm and (in most cases) the superiority of open-loop feedback over open-loop controls.

Optimum control theory has a great number of applications in many areas of science from engineering to economics. In particular, there are many studies on determining optimal economic policies for econometric models. Most of these optimum control applications use algorithms for linear dynamic systems or those that do not take the full stochastic nature of the econometric model into account. Examples of the former are Kendrick (1981) and Coomes (1987), and the references in Amman (1996) and Chow, 1975 and Chow, 1981 for the latter. An algorithm that is explicitly aimed at providing (approximate) solutions to optimum control problems for nonlinear econometric models and other dynamic systems with different kinds of stochastics is OPTCON, as introduced by Matulka and Neck (1992). However, so far OPTCON has been severely limited by being based on very restrictive assumptions about the information available to the decision-maker. In particular, learning about the econometric model while in the process of controlling the economy was ruled out by assumption. In reality, however, new information arrives in each period, and econometric models are regularly re-estimated using this information. Therefore, extensions of the OPTCON algorithm to include various possibilities of obtaining and using new information about the system to be controlled are highly desirable. The present extension of the OPTCON algorithm from open-loop control only (OPTCON1) to the inclusion of passive learning or open-loop feedback control where the estimates of the parameters are updated in each period results in the algorithm OPTCON2. It can deliver approximately optimal solutions to dynamic optimization (optimum control) problems for a rather large class of nonlinear dynamic systems under a quadratic objective function with stochastic uncertainty in the parameters and in the system equations under both kinds of control schemes. In the open-loop feedback part, it is assumed that new realizations of both random processes occur in each period, which can be used to update the parameter estimates of the dynamic system, i.e. of the econometric model. Following Kendrick’s (1981) approach, the parameter estimates are updated using the Kalman filter in order to arrive at more reliable approximations to the solution of stochastic optimum control problems. Whether this hope will materialize depends upon the comparative performance of open-loop feedback vs. open-loop control schemes in actual applications. Some indication of this will be provided by comparing the two control schemes within a control problem for a small econometric model. This also serves to show that the OPTCON2 algorithm and its implementation in C# actually deliver plausible numerical solutions, at least for a small problem, with real economic data. The paper has the following structure. In Section 2, the class of problems to be tackled by the algorithm is defined. Section 3 briefly reviews the OPTCON1 algorithm. Section 4 explains the OPTCON2 algorithm. In Section 5, the small econometric model for Slovenia SLOVNL is introduced, the applicability and convergence of OPTCON2 as implemented in C# is shown, and the quality of open-loop and open-loop feedback (passive learning) controls in Monte Carlo simulations for this model are compared. Section 6 concludes. More details about the mathematics of the algorithm are given in Blueschke-Nikolaeva et al. (2010).

The present extension of the OPTCON algorithm, OPTCON2, calculates open-loop feedback in addition to open-loop control policies. It was programmed in the computer language C# and shown to converge for a small econometric model. The main improvement lies in learning about stochastically disturbed parameters during the control process. A comparison of open-loop control (without learning) and open-loop feedback control (with passive learning) shows that weighted open-loop feedback control outperforms open-loop control in a majority of the cases investigated for the small econometric model of Slovenia. The next task is to apply OPTCON2 to larger and better macroeconometric models (in terms of their theoretical and statistical quality). Additional comparisons of the policy performance with respect to the postulated objective function are desirable; for example, it may be interesting to calculate controls by straightforward heuristic optimization procedures (see Gilli and Winker, 2009, Winker and Gilli, 2004 and Lyra et al., 2010, among others) and assess their performance compared to the more sophisticated ones calculated by OPTCON2. Moreover, major extensions will have to include various schemes of active learning to deal with the dual nature of the control under uncertainty along the lines of Kendrick (1981) for the linear case. Another challenge consists in incorporating rational (forward-looking) expectations and hence a non-causal structure in the dynamic system; see Amman and Kendrick (2000) for the linear case.