Sudhanshu K. Mishra

A Brief History of Production Functions

This paper gives an outline of evolution of the concept and econometrics of production function, which was one of the central apparatus of neoclassical economics. It shows how the famous Cobb-Douglas production function was indeed invented by von Thunen and Wicksell, how the Constant Elasticity of Substitution (CES) production function was formulated, how the elasticity of substitution was made a variable and finally how Sato’s function incorporated biased technical changes. It covers almost all specifications proposed during 1950 to 1975, as well as the linear-exponential (LINEX) production functions and incorporation of energy as an input. The paper is divided into single product functions, joint product functions, and aggregate production functions. It also discusses the ‘capital controversy’ and its impacts.

Least absolute deviation estimation of linear econometric models: A literature review

Econometricians generally take for granted that the error terms in the econometric models are generated by distributions having a finite variance. However, since the time of Pareto the existence of error distributions with infinite variance is known. Works of many econometricians, namely, Meyer & Glauber (1964), Fama (1965) and Mandlebroth (1967), on economic data series like prices in financial and commodity markets confirm that infinite variance distributions exist abundantly. The distribution of firms by size, behaviour of speculative prices and various other recent economic phenomena also display similar trends. Further, econometricians generally assume that the disturbance term, which is an influence of innumerably many factors not accounted for in the model, approaches normality according to the Central Limit Theorem. But Bartels (1977) is of the opinion that there are limit theorems, which are just likely to be relevant when considering the sum of number of components in a regression disturbance that leads to non-normal stable distribution characterized by infinite variance. Thus, the possibility of the error term following a non-normal distribution exists. The Least Squares method of estimation of parameters of linear (regression) models performs well provided that the residuals (disturbances or errors) are well behaved (preferably normally or near-normally distributed and not infested with large size outliers) and follow Gauss-Markov assumptions. However, models with the disturbances that are prominently non-normally distributed and contain sizeable outliers fail estimation by the Least Squares method. An intensive research has established that in such cases estimation by the Least Absolute Deviation (LAD) method performs well. This paper is an attempt to survey the literature on LAD estimation of single as well as multi-equation linear econometric models.

A Comparative Study of Various Inclusive Indices and the Index Constructed By the Principal Components Analysis

Construction of (composite) indices by the PCA is very common, but this method has a preference for highly correlated variables to the poorly correlated variables in the data set. However, poor correlation does not entail the marginal importance, since correlation coefficients among the variables depend, apart from their linearity, also on their scatter, presence or absence of outliers, level of evolution of a system and intra-systemic integration among the different constituents of the system. Under-evolved systems often throw up the data with poorly correlated variables. If an index gives only marginal representation to the poorly correlated variables, it is elitist. The PCA index is often elitist, particularly for an under-evolved system. In this paper we consider three alternative indices that determine weights given to different constituent variables on the principles different from the PCA. Two of the proposed indices, the one that maximizes the sum of absolute correlation coefficient of the index with the constituent variables and the other that maximizes the entropy-like function of the correlation coefficients between the index and the constituent variables are found to be very close to each other. These indices alleviate the representation of poorly correlated variables for some small reduction in the overall explanatory power (vis-a-vis the PCA index). These indices are inclusive in nature, caring for the representation of the poorly correlated variables. They strike a balance between individual representation and overall representation (explanatory power) and may perform better. The third index obtained by maximization of the minimal correlation between the index and the constituent variables cares most for the least correlated variable and in so doing becomes egalitarian in nature.

Performance of Differential Evolution and Particle Swarm Methods on Some Relatively Harder Multi-Modal Benchmark Functions

This paper aims at comparing the performance of the Differential Evolution (DE) and the Repulsive Particle Swarm (RPS) methods of global optimization. To this end, some relatively difficult test functions have been chosen. Among these test functions, some are new while others are well known in the literature. We use DE method with the exponential crossover scheme as well as with no crossover (only probabilistic replacement). Our findings suggest that DE (with the exponential crossover scheme) mostly fails to find the optimum in case of the functions under study. Of course, it succeeds in case of some functions (perm#2, zero-sum) for very small dimension, but begins to falter as soon as the dimension is increased. In case of DCS function, it works well up to dimension = 5. When we use no crossover (only probabilistic replacement) we obtain better results in case of several of the functions under study. In case of Perm#1, Perm#2, Zero-sum, Kowalik, Hougen and Power-sum functions, a remarkable advantage is there. Whether crossover or no crossover, DE falters when the optimand function has some element of randomness. This is indicated by the functions: Yao-Liu#7, Fletcher-Powell, and “New function#2”. DE has no problems in optimizing the “New function #1”. But the “New function #2” proves to be a hard nut. However, RPS performs much better for such stochastic functions. When the Fletcher-Powell function is optimized with non-stochastic c vector, DE works fine. But as soon as c is stochastic, it becomes unstable. Thus, it may be observed that an introduction of stochasticity into the decision variables (or simply added to the function as in Yao-Liu#7) interferes with the fundamentals of DE, which works through attainment of better and better (in the sense of Pareto improvement) population at each successive iteration. The paper concludes: (1) for different types of problems, different schemes of crossover (including none) may be suitable or unsuitable, (2) Stochasticity entering into the optimand function may make DE unstable, but RPS may function well.

On construction of robust composite indices by linear aggregation

In this paper we construct thirteen different types of composite indices by linear combination of indicator variables (with and without outliers/data corruption). Weights of different indicator variables are obtained by maximization of the sum of squared (and, alternatively, absolute) correlation coefficients of the composite indices with the constituent indicator variables. Seven different types of correlation are used: Karl Pearson, Spearman, Signum, Bradley, Shevlyakov, Campbell and modified Campbell. Composite indices have also been constructed by maximization of the minimal correlation. We find that performance of indices based on robust measures of correlation such as modified Campbell and Spearman, as well as that of the maxi-min based method, is excellent. Using these methods we obtain composite indices that are autochthonously sensitive and allochthonously robust. This paper also justifies a use of simple mean-based composite indices, often used in construction of human development index.

Performance of Repulsive Particle Swarm Method in Global Optimization of Some Important Test Functions: A Fortran Program

The Repulsive Particle Swarm (RPS) method of global optimization is perhaps the simplest to understand and implement. Due to its simplicity, it can be easily modified to suit the purpose and therefore, it has better prospects as well. The method has been frequently used in the field of artificial intelligence. It is well founded on philosophical and methodological grounds (bounded rationality and efficacy of decentralized decision-making to reach the global best) also. The method of RPS has been programmed (in FORTRAN) and run to optimize 32 test functions (such as Ackley, Beale, Booth, Dixon & Price, Easom, Griewank, Himmelblau, Hump, Levy, Michalewics, Rastrigin, Rosenbrock, Schwefel, Shubert, Trid, etc). The program has successfully optimized these functions. The paper also provides graphical presentations of most of these functions and the FORTRAN codes of RPS method.

Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization

In this paper an attempt has been made to fit the Gielis curves (modified by various functions) to simulated data. The estimation has been done by two methods - the Classical Simulated Annealing (CSA) and the Particle Swarm (PS) methods - of global optimization. The Repulsive Particle Swarm (RPS) optimization algorithm has been used. It has been found that both methods are quite successful in fitting the modified Gielis curves to the data. However, the lack of uniqueness of Gielis parameters to data (from which they are estimated) is corroborated. From a technical viewpoint, this exercise may be considered as an application of CSA and RPS to extremely nonlinear least-squares curve-fitting to data that may exhibit a large number of local optima.

Global Optimization By Particle Swarm Method: A Fortran Program

Programs that work very well in optimizing convex functions very often perform poorly when the problem has multiple local minima or maxima. They are often caught or trapped in the local minima/maxima. Several methods have been developed to escape from being caught in such local optima. The Particle Swarm Method of global optimization is one of such methods. A swarm of birds or insects or a school of fish searches for food, protection, etc. in a very typical manner. If one of the members of the swarm sees a desirable path to go, the rest of the swarm will follow quickly. Every member of the swarm searches for the best in its locality - learns from its own experience. Additionally, each member learns from the others, typically from the best performer among them. Even human beings show a tendency to learn from their own experience, their immediate neighbours and the ideal performers. The Particle Swarm method of optimization mimics this behaviour. Every individual of the swarm is considered as a particle in a multidimensional space that has a position and a velocity. These particles fly through hyperspace and remember the best position that they have seen. Members of a swarm communicate good positions to each other and adjust their own position and velocity based on these good positions. The Particle Swarm method of optimization testifies the success of bounded rationality and decentralized decisionmaking in reaching at the global optima. It has been used successfully to optimize extremely difficult multimodal functions. Here we give a FORTRAN program to find the global optimum by the Repulsive Particle Swarm method. The program has been tested on over 90 benchmark functions of varied dimensions, complexities and difficulty levels.

Multicollinearity and Maximum Entropy Leuven Estimator

Multicollinearity is a serious problem in applied regression analysis. Q. Paris (2001) introduced the MEL estimator to resolve the multicollinearity problem. This paper improves the MEL estimator to the Modular MEL (MMEL) estimator and shows by Monte Carlo experiments that MMEL estimator performs significantly better than OLS as well as MEL estimators.

Estimation of Zellner-Revankar Production Function Revisited

Zellner and Revankar in their paper “Generalized Production Functions” introduced a production function, which was illustrated by fitting the generalized Cobb-Douglas function to the U.S. data for Transportation Equipment Industry. For estimating the production function, they used a method in which one of the parameters (theta) is repeatedly chosen at the trial basis and other parameters are estimated so as to obtain the global optimum of the likelihood function. We show that this method of Zellner and Revankar (ZR) is caught into a local optimum trap and the estimated parameters reported by ZR are somewhat sub-optimal. Using the Differential Evolution (DE) and the Repulsive Particle Swarm (RPS) methods, we re-estimate the parameters of the ZR production function with data used by ZR and show that our estimates of parameters are better than those of ZR. We also find that the returns to scale do not vary with the size of output in the manner reported by ZR.