A theorem on regression coefficients obtained from successively extended sets of variables

Abstract

1. Introductory. - The theorem of the present note is intended as a comment on multiple regression analysis as employed in the study of causal relations. Trying to formulate in general terms, our theorem will refer to the following situation: We are concerned with a variable phenomenon, say ξ₀, which is regarded as causally determined by a number of other variables, say ξ₁, ξ₂,…; it is desired to find out a numerical formula for the dependence, so that ξ₀ can be calculated (more or less approximately) when knowing ξ₁, ξ₂ …; the dependence is studied by the use of multiple regression analysis; the influencing factors ξ₁, ξ₂,... are numerous and intercorrelated.     

A note on the forward measure

For a Markov process , the forward measure over the time interval is defined by the Radon-Nikodym derivative , where is a given non-negative function and is the normalizing constant. In this paper, the law of under the forward measure is identified when is a diffusion process or, more generally, a continuous-path Markov process.     

A note on the government budget restraint

In a recent article in this Journal, Christ presented a dynamic macroeconomic model which has the striking implication of instability under bond financing of budgetary deficits. In this paper, we show that the stability (and other fundamental) properties of Christ's model are closely related to the specification of the exogenous fiscal variables. This point is demonstrated through an analysis of Christ's model in which several definitions of government spending are taken as exogenous. As an alternative to this approach, we explore the implications of models in which alternative policy goals are specified as exogenous — goals such as a balanced or a zero gap between actual and potential output.     

A note on takeover success prediction

A takeover success prediction model attempts to use information that is publicly available at the time of the announcement in order to predict the probability that a takeover attempt will succeed. This paper develops a takeover success prediction model by comparing two techniques: the traditional logistic regression model and the artificial neural network technology. To alleviate the problem of bias from the sampling variation, we validate our results through re-sampling. Our empirical results indicate that 1). Arbitrage spread, target resistance, deal structure and transaction size are the dominating factors that have impacts on the outcome of a takeover attempt. 2). Neural network model outperforms logistic regression in predicting failed takeover attempts and performs as well as logistic regression in predicting successful takeover attempts.     

A note on essential smoothness in the Heston model

This note identifies a gap in the proof of Corollary 2.4 in Forde and Jacquier (Finance Stoch., 2011) which arises because the essential smoothness of the family (X _t /t) _t≥1 can fail for the log-spot process X in the Heston model, and it describes how to circumvent the issue by applying a standard argument from large deviation theory.     

A note on the valuation of asset management firms

Market capitalization relative to assets under management is often used to value asset management firms. Huberman’s (2004) dividend discount model implies that cross-sectional variations in this metric are explained by cross-sectional differences in operating margins, and yet we find no evidence of this in our data set. We show that a superior model—inspired by the work of Berk and Green (2004)—includes also the level of fees as an explanatory variable. This approach dramatically increases the fit of our valuation model and casts doubt on the relevance of the so-called Huberman puzzle.     

A note on the returns from minimum variance investing

Disappointed with the performance of market weighted benchmark portfolios yet skeptical about the merits of active portfolio management, investors in recent years turned to alternative index definitions. Minimum variance investing is one of these popular concepts. I show in this paper that the portfolio construction process behind minimum variance investing implicitly picks up risk-based pricing anomalies. In other words the minimum variance tends to hold low beta and low residual risk stocks. Long/short portfolios based on these characteristics have been associated in the empirical literature with risk adjusted outperformance. This paper shows that 83% of the variation of the minimum variance portfolio excess returns (relative to a capitalization weighted alternative) can be attributed to the FAMA/FRENCH factors as well as to the returns on two characteristic anomaly portfolios. All regression coefficients (factor exposures) are highly significant, stable over the estimation period and correspond remarkably well with our economic intuition. The paper also shows that a direct combination of market weighted benchmark portfolio and risk based characteristic portfolios will provide a statistically significant improvement over the indirect pickup via the minimum variance portfolio.     

A note on the consistency of maximum likelihood estimators

Abstract

A sequence of maximum likelihood estimators based on a sequence of independent but not necessarily identically distributed random variables is shown to be consistent under certain assumptions. Some examples are given to show that these assumptions are easy to verify and not very restrictive.     

A note on diversification and the reduction of dispersion

The purpose of this paper is to note that the question of optimal diversification cannot be answered simply by determining the average variability of equally allocated investment. Empirical results are presented which show that it is possible to obtain the same level of average variation with far greater average portfolio returns and fewer securities in the portfolio by using an alternative allocation scheme.     

A note on the neutrality of temporary monetary disturbances

In the rational expectations analysis of Lucas and Barro, the quantity of money is subject to random shocks that are specified to be unanticipated and permanent in character. The present note provides some simple examples of alternative money supply specifications that lead to non-neutrality of perceived temporary monetary growth through the channel of expected inflation. Subsequently, the discussion demonstrates that this non-neutrality is not robust to an alternative specification of private monetary behavior, the permanent balance model. The key difference is that the initial model involves commodity supply and demand which depend on current real balances while in the subsequent model these depend on permanent balances. Some final remarks are directed to the idea that the distinction between current and permanent balances in this simple model could be linked to alternative roles of money in more detailed, optimizing analyses.     

A note on videos and accounting education

Doctoral students are often uncomfortable making oral presentations before their peers and the instructor. This paper describes a recent innovation employed in the doctoral programme of a major university which we feel has been useful not only in increasing the communication skills of our doctoral students, but has also enhancing the quality of doctoral students' dissertations as well as the teaching and presentation skill of staff involved in the programme.     

A note on generalized distortion risk measures

A generalized distortion risk measure is introduced as power of the mean absolute deviation power of a distorted random variable with respect to a location parameter. This class of risk measures extends both the distortion risk measure by Wang and Denneberg and the class of financial risk measures by Pedersen and Satchell, which itself contains the class of Stone. Integral representations and a stop–loss order preserving property of a special up-side risk measure are derived.     

A note on inventory reorder point determination

Managerial decisions regarding timing of inventory replenishment are as important as determining the economic order size in inventory management. Yet most of the introductory texts on cost/managerial accounting and production/operations management do not adequately discuss reorder point determination, especially the condition on lead time demand that is necessary for valid application of the commonly reported formula for the reorder point. A clear understanding of the reorder point determination is important to students, teachers, and practicing managers alike. This note attempts to explain the underlying condition and modification needed in the reorder point formula.     

A note on Wick products and the fractional Black-Scholes model

In some recent papers (Elliott and van der Hoek 2003; Hu and Øksendal 2003) a fractional Black-Scholes model has been proposed as an improvement of the classical Black-Scholes model (see also Benth 2003; Biagini et al. 2002; Biagini and Øksendal 2004). Common to these fractional Black-Scholes models is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Itô integral is replaced by the Wick integral, and proofs have been presented that these fractional Black-Scholes models are free of arbitrage. These results on absence of arbitrage complelety contradict a number of earlier results in the literature which prove that the fractional Black-Scholes model (and related models) will in fact admit arbitrage. The objective of the present paper is to resolve this contradiction by pointing out that the definition of the self-financing trading strategies and/or the definition of the value of a portfolio used in the above papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful. In particular we show that in the framework of Elliott and van der Hoek 2003, a naive buy-and-hold strategy does not in general qualify as self-financing. We also show that in Hu and Øksendal 2003, a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative value.Received: August 2004, Mathematics Subject Classification (2000): 91B28, 60H05JEL Classification: G10Support of the first author from the Jan Wallander and Tom Hedelius foundation is gratefully acknowledged. The research of the second author is supported by the Swedish Research Council.     

A note on the efficiency of the binomial option pricing model

We discuss the efficiency of the binomial option pricing model for single and multivariate American style options. We demonstrate how the efficiency of lattice techniques such as the binomial model can be analysed in terms of their computational cost. For the case of a single underlying asset the most efficient implementation is the extrapolated jump-back method: that is, to value a series of options with nested discrete sets of early exercise opportunities by jumping across the lattice between the early exercise times and then extrapolating from these values to the limit of a continuous exercise opportunity set. For the multivariate case, the most efficient method depends on the computational cost of the early exercise test. However, for typical problems, the most efficient method is the standard step-back method: that is, performing the early exercise test at each time step.