Theorems on the decomposition of a large nonlinear convex separable economic system in the dual direction

Large mathematical programming problems often arise as the result of the economic planning process. When such a problem is not only large, but nonlinear as well, there is a need to make it more manageable by breaking it down into several smaller and more easily handled subproblems. The subproblems are solved separately with the coordination activity carried out by a master problem. A decomposition method can be seen as a dialogue between the master problem and the subproblems, where the flow of information back and forth between the former and the latters results in a series of approximations converging to the solution of the overall problem. Such a decomposition method was elaborated by Benders [1] for linear programmes and generalized to nonlinear convex separable programmes by Kronsjö [4] and by Geoffrion [3]. After considering our basic nonlinear programming problem from a two-stage minimization point of view, we review the Kronsjö nonlinear decomposition algorithm. Then we establish some properties of a function related to this algorithm.I am grateful to Professor T.O.M. Kronsjö for inspiring this work and giving valuable help.     

A nonlinear approach to US GNP

A univariate nonlinear model is estimated for US GNP that on many criteria outperforms standard linear models. The estimated model is of the threshold autoregressive type and contains evidence of asymmetric effects of shocks over the business cycle. In particular the nonlinear model suggests that the post-1945 US economy is significantly more stable than the pre-1945 US economy.     

Statistical Proofs of Some Matrix Theorems *

Books on linear models and multivariate analysis generally include a chapter on matrix algebra, quite rightly so, as matrix results are used in the discussion of statistical methods in these areas. During recent years a number of papers have appeared where statistical results derived without the use of matrix theorems have been used to prove some matrix results which are used to generate other statistical results. This may have some pedagogical value. It is not, however, suggested that prior knowledge of matrix theory is not necessary for studying statistics. It is intended to show that a judicious use of statistical and matrix results might be of help in providing elegant proofs of problems both in statistics and matrix algebra and make the study of both the subjects somewhat interesting. Some basic notions of vector spaces and matrices are, however, necessary and these are outlined in the introduction to this paper.     

关于逆函数的性质与定理

本文给出了逆函数的定义,介绍了它的性质,并证明了关于逆函数的几个定理。     

The theta model: a decomposition approach to forecasting

This paper presents a new univariate forecasting method. The method is based on the concept of modifying the local curvature of the time-series through a coefficient ‘Theta’ (the Greek letter θ), that is applied directly to the second differences of the data. The resulting series that are created maintain the mean and the slope of the original data but not their curvatures. These new time series are named Theta-lines. Their primary qualitative characteristic is the improvement of the approximation of the long-term behavior of the data or the augmentation of the short-term features, depending on the value of the Theta coefficient. The proposed method decomposes the original time series into two or more different Theta-lines. These are extrapolated separately and the subsequent forecasts are combined. The simple combination of two Theta-lines, the Theta=0 (straight line) and Theta=2 (double local curves) was adopted in order to produce forecasts for the 3003 series of the M3 competition. The method performed well, particularly for monthly series and for microeconomic data.     

A new approach to risk-return trade-off dynamics via decomposition

This paper revisits the puzzling time series relation between risk premium and conditional volatility by proposing a flexible risk-return trade-off that allows for a variety of possible shapes and incorporates potential nonlinearities inherent in excess return dynamics. We derive this flexible risk-return relation using the decomposition approach of Anatolyev and Gospodinov (2010), which splits excess returns into the product of absolute returns and signs. Using this decomposition strategy, we study four major international financial markets. The empirical results support a significant and positive risk-return trade-off that is driven by conditional volatility, market timing and the interdependence between the two components, which is generically related to return skewness.     

Kernel method to estimate nonlinear structural equation models

Quality & Quantity - The purpose of this paper is to provide a nonparametric kernel method to estimate nonlinear structural equation models involving the functional effects between the latent...     

Combining wavelet decomposition with machine learning to forecast gold returns

This paper combines the discrete wavelet transform with support vector regression for forecasting gold-price dynamics. The advantages of this approach are investigated using a relatively small set of economic and financial predictors. I measure model performance by differentiating between a statistically-motivated out-of-sample forecasting exercise and an economically-motivated trading strategy. Disentangling the predictors with respect to their time and frequency domains leads to improved forecasting performance. The results are robust compared to alternative forecasting approaches. My findings on the relative importances of such wavelet decompositions suggest that the influences of short-term and long-term trends are not stable over the full evaluation period.     

Small‐sample testing inference in symmetric and log‐symmetric linear regression models

This paper deals with the issue of testing hypotheses in symmetric and log‐symmetric linear regression models in small and moderate‐sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi‐squared asymptotic distribution. Bartlett and Bartlett‐type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett‐type correction for the gradient test. We show that the corrections are also valid for the log‐symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett‐corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.     

Local identification of ARMAX structures subject to nonlinear constraints

Summary We obtain necessary and sufficient conditions for the local identification of a simultaneous ARMAX structure whose parameters are subject to nonlinear constraints. A global identification condition is also obtained. The results generalize theWegge [1965] andRothenberg [1971] identification criteria for a classical simultaneous equations model.This paper is preliminary and confidential. Please do not quote its contents without the written permission of the author.     

Asymmetric outcome in a symmetric dynamic duopoly

In a dynamic duopoly, will an initial asymmetry between firms increase or decrease over time? We examine this issue within a stochastic dynamic alternate-move duopoly model that explicitly accounts for action and reaction between firms. We consider two firms that are symmetric with regard to all primitives such as demand, cost and production functions, and which are subject to the same stochastic environment. The only asymmetry is with regard to their initial capital stocks which in turn asymmetrically influences their current and future profit and investment possibilities. We offer a characterization of the stochastic steady state and its supporting ergodic set for each firm. We are then able to identify the precise restrictions on the initial conditions under which the two firms either converge or diverge in the long run.     

On the asymmetry of the symmetric MAPE

Several authors have suggested that the use of the mean absolute percentage error (MAPE) as a measure of forecast accuracy should be avoided because they argue it treats forecast errors above the actual observation differently from those below this value. To counter this, the use of a symmetric (or modified) MAPE has been proposed. This paper shows that, in its treatment of negative and positive errors, the proposed modification is far from symmetric, particularly where these errors have large absolute values. It also shows that, under some circumstances, a non-monotonic relationship can occur between the symmetric MAPE and the absolute forecast errors.     

A market game with symmetric limit orders

We extend the market game with symmetric limit orders studied in Weyers (2003, 2004) to a many-good setup. Our limit orders are symmetric in terms of payment and determine a unique consistent price system for every strategy profile. The limit orders studied in the previous literature—see Dubey (1982), Simon (1984) and Mertens (2003)—share none of these properties. It is shown that three mild market-thickness conditions imply that the set of symmetric Nash equilibrium outcomes coincides with the set of price-taking equilibrium outcomes. First, the Dubey and Shubik (1978) refinement is used to eliminate no-trade as an equilibrium. Second, any price-taking equilibrium has trade in each market. Third, there are at least two agents of each type, where a type is determined by preferences and endowments. The last two conditions enable applying the Bertrand argument. This paper thus provides new insights to Bertrand’s (1883) classic critique of Cournot and the associated problem of capacity constraints raised by Edgeworth (1897).     

Are “Anti-Folk Theorems” in repeated games nongeneric?

Folk Theorems in repeated games hold fixed the game payoffs, while the discount factor is varied freely. We show that these results may be sensitive to the order of limits in situations where players move asynchronously. Specifically, we show that when moves are asynchronous, then for a fixed discount factor close to one there is an open neighborhood of games which contains a pure coordination game such that every Perfect equilibrium of every game in the neighborhood approximates to an arbitrary degree the unique Pareto dominant payoff of the pure coordination game.