On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data

This paper summarizes some results on random fuzzy variables with existing expectation and variance, called random fuzzy variables of second order. Using the Frechét-principle and – via support functions – the embedding of convex fuzzy sets into a Banach space of functions it especially presents a unified view on expectation and variance of random fuzzy variables. These notions are applied in developing linear statistical inference with fuzzy data. Detailed investigations are presented concerning best linear unbiased estimation in linear regression models with fuzzy observations. Received: November 1999     

G@RCH 2.2: An Ox Package for Estimating and Forecasting Various ARCH Models

This paper discusses and documents G@RCH 2.2, an Ox package dedicated to the estimation and forecast of various univariate ARCH–type models including GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH, HYGARCH, FIEGARCH and FIAPARCH specifications of the conditional variance and an AR(FI)MA specification of the conditional mean.
These models can be estimated by Approximate (Quasi) Maximum Likelihood under four assumptions: normal, Student– t , GED or skewed Student errors. Explanatory variables can enter both the conditional mean and the conditional variance equations. h –step–ahead forecasts of both the conditional mean and the conditional variance are available as well as many mispecification tests.
We first propose an overview of the package's features, with the presentation of the different specifications of the conditional mean and conditional variance. Then further explanations are given about the estimation methods. Measures of the accuracy of the procedures are also given and the GARCH features provided by G@RCH are compared with those of nine other econometric softwares. Finally, a concrete application of G@RCH 2.2 is provided.     

Stock returns and volatility: Another look

Most empirical work examining the intertemporal mean-variance relationship in stock returns has tended to use relatively simple specifications of the mean and especially of the conditional variance. We augment the information set to include economic variables that other researchers have found to be important and use GARCH-M models to explore the relation between volatility and expected stock returns. We find that the additional variables have little impact on the conditional variance and that any intertemporal relationship between volatility and stock returns is weak or unstable. Our results signal the need for theoretical models of the intertemporal volatility-return relationship, and call for further studies of the determinants of the conditional variance of stock returns.     

Distributions characterized through conditional expectations

Using conditional expectations, we present results that lead to the characterization of several distributions. Both absolutely continuous random variables and discrete random variables are considered. In the case of absolutely continuous random variables, the results lead to the characterization of a family of distributions while in the case of discrete random variables, the distribution is almost uniquely determined under the stated conditions.     

Designs for smoothing spline ANOVA models

Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique. This paper considers the design problem for specific types of SS-ANOVA models. As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior. Three examples for additive and interaction spline models are provided for illustration. A comparison of the asymptotic designs, the 2^d factorial designs, and the glp designs is given by numerical computation. Received May 2000     

A note on exchangeability of random variables

A sufficient condition for the induced exchangeability or partial exchangeability of linear functions of exchangeable random variables is presented. The use of this result is illustrated through the establishment of conditional exchangeability for two sets of dependent random variables that are important in constructing conditionally distribution–free test procedures for two distinctly different problems.     

Compound gamma,beta and F distributions

Summary In this paper a compound gamma distribution has been derived by compounding a gamma distribution with another gamma distribution. The resulting compound gamma distribution has been reduced to the Beta distributions of the first kind and the second kind and to theF distribution by suitable transformations. This includes theLomax distribution as a special case which enjoys a useful property. Moment estimators for two of its parameters are explicitly obtained, which tend to a bivariate normal distribution. The paper contains expressions for a bivariate probability density function, its conditional expectation, conditional variance and the product moment correlation coefficient. Finally, all the parameters of the compound gamma distribution are explicitly expressed in terms of the functions of the moments of the functions of random variables in two different ways. This note is based on a technical report prepared by the author while he was with the Procter and Gamble Company.     

Modelling volatility by variance decomposition

In this paper, we propose two parametric alternatives to the standard GJR-GARCH model of Glosten et al. (1993), based on additive and multiplicative decompositions of the variance. They allow the variance of the model to have a smooth time-varying structure. The suggested parameterizations describe structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. The main focus is on the multiplicative decomposition of the variance into an unconditional and conditional components. Estimation of the multiplicative model is discussed in detail. An empirical application to daily stock returns illustrates the functioning of the model. The results show that the ‘long memory type behaviour’ of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance.     

Characterizations of logarithmic series distrimtrans

The distributions of X, Y and (X. Y ), where X and Y are random variables with probability functions of a logarithmic series law, are characterized by the regression function of X on Y and the conditional distribution of Y given X. Moreover, characterizations are given for binomial or Pascal conditional distributions in terms of the regression function of X on Y and the marginal distribution of X.     

One-sample tests for a generalized Fréchet variance of a fuzzy random variable

A procedure to test hypotheses about the population variance of a fuzzy random variable is analyzed. The procedure is based on the theory of UH-statistics. The variance is defined in terms of a general metric to quantify the variability of the fuzzy values about its (fuzzy) mean. An asymptotic one-sample test in a wide setting is developed and a bootstrap test, which is more suitable for small and moderate sample sizes, is also studied. Moreover, the power function of the asymptotic procedure through local alternatives is analyzed. Some simulations showing the empirical behavior and consistency of both tests are carried out. Finally, some illustrative examples of the practical application of the proposed tests are presented.     

Asymptotic and Bootstrap techniques for testing the expected value of a fuzzy random variable

In this paper we will consider hypothesis-tests for the (fuzzy-valued) mean value of a fuzzy random variable in a population. For this purpose, we will make use of a generalized metric for fuzzy numbers, and we will develop an approach for normal fuzzy random variables, and two different approaches for the case of fuzzy random variables taking on a finite number of different values. A real-life example illustrates the use of the last two approaches. Finally, a comparison between the introduced techniques is developed by means of simulation studies leading to close inferential conclusions.Acknowledgements.The research in this paper has been partially supported by MCYT Grants BFM2002-01057 and BFM2001-3494. Their financial support is gratefully acknowledged. The authors are sincerely grateful to their colleague Gil González-Rodríguez for all his comments and suggestions in connection with this paper; his scientific support has been very valuable. The authors want also thank the referees of the first version of the paper because of their useful hints to improve it.     

Dynamic quasi concave performance measures

We define Conditional quasi concave Performance Measures (CPMs), on random variables bounded from below, to accommodate for additional information. Our notion encompasses a wide variety of cases, from conditional expected utility and certainty equivalent to conditional acceptability indexes. We provide the characterization of a CPM in terms of an induced family of conditional convex risk measures. In the case of indexes these risk measures are coherent. Then, Dynamic Performance Measures (DPMs) are introduced and the problem of time consistency is addressed. The definition of time consistency chosen here ensures that the positions which are considered good tomorrow are already considered good today. Finally, we investigate the relation between time consistency for a DPM and weak acceptance consistency for the induced families of risk measures.     

A note on consistency and unbiasedness of point estimation with fuzzy data

Based on the SLLN for fuzzy random variables in uniform metric d, some asymptotical properties of point estimation with fuzzy random samples are investigated. The results of this paper establish a corresponding version on the consistency and unbiasedness of point estimation with n-dimensional fuzzy samples under considering a kind of fuzzy statistic.     

A note on the distribution of the product of zero‐mean correlated normal random variables

The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero‐mean correlated normal random variables as a variance‐gamma random variable, from which an explicit formula for the probability density function is immediate.     

The incompatibility of achieving a fully specified linear model with assuming that residual dependent-variable variance is random

The residual dependent-variable variance in experiments is not “random error”, as it is often assumed to be, but merely “unaccounted for variance”, because what is random is inexplicable in terms of any possible set of independent-variables and this is something that ultimately is only empirically determinable. So, if there is any unaccounted for dependent-variable variance, an experiment’s set of independent-variables is certainly under-specified and perhaps mis-specified because of the confounding of variables included in this set by causally relevant variables not included in the set. Thus, the proper first empirical test of any linear model is whether it leaves any residual dependent-variable variance, and if it does then none of its independent variables can yet logically justifiably be claimed to predict or causally explain any of the dependent-variable variance whatsoever.     

Does conditional market skewness resolve the puzzling market risk-return relationship?

In this paper, we employ instrumental variables methods that allow time-varying risk and reward-to-risk to test various conditional asset pricing models. We find a negative partial relation between the market excess return and conditional market variance. In contrast with recent findings, we show that this negative relationship is not due to the omission of the hedge term associated with the ICAPM. However, conditional market skewness seems to partly account for this negative risk-return relationship.     

Modelling response error in school effectiveness research

Statistical modelling of school effectiveness in educational research is considered. Variance component models are generally accepted for the analysis of such studies. A shortcoming is that outcome variables are still treated as measured without an error. Unreliable variables produce biases in the estimates of the other model parameters. The variability of the relationships across schools and the effects of schools on students' outcomes differ substantially when taking the measurement error in the dependent variables of the variance component models into account. The random effects model can be extended to handle measurement error using a response model, leading to a random effects item response theory model. This extended random effects model is in particular suitable when subjects are measured repeatedly on the same outcome at several points in time.     

The impacts of investor sentiment on returns and conditional volatility of international stock markets

One of the main arguments of behavioral finance is that some properties of asset prices are most probably regarded as deviations from fundamental value and they are generated by the participation of traders who are not fully rational, thus called noise traders. Noise trader theory postulates that sentiment traders have greater impact during high-sentiment periods than during low-sentiment periods, and sentiment traders miscalculate the variance of returns undermining the mean-variance relation. The main objective of this research is to construct a model to evaluate the returns and conditional volatility of various stock market indexes considering the changes in the investor sentiment by measuring the effects of noise trader demand shocks on returns and volatility. EGARCH model is used to determine whether earning shocks have more influence on the conditional volatility in high sentiment periods weakening the mean–variance relation. This paper takes an international approach using weekly market index returns of U.S., Japan, Hong Kong, U.K., France, Germany, and Turkey. Weekly trading volumes of these indexes are regressed against a group of macroeconomic variables and the residuals are used as proxies for investor sentiment and significant evidence is found that there is asymmetric volatility in these market indexes and earning shocks have more influence on conditional volatility when the sentiment is high.     

An integrated fuzzy mathematical model and principal component analysis algorithm for forecasting uncertain trends of electricity consumption

This paper introduces an integrated algorithm for forecasting electricity consumption (EL) based on fuzzy regression, time series and principal component analysis (PCA) in uncertain markets such as Iran. The algorithm is examined by mean absolute percentage error, analysis of variance (ANOVA) and Duncan Multiple Range Test. PCA is used to identify the input variables for the fuzzy regression and time series models. Monthly EL in Iran is used to show the superiority of the algorithm. Moreover, it is shown that the selected fuzzy regression model has better estimated values for total EL than time series. The algorithm provides as good results as intelligent methods. However, it is shown that the algorithm does not require utilization of preprocessing methods but genetic algorithm, artificial neural network and fuzzy inference system require preprocessing which could be a cumbersome task to deal with ambiguous data. The unique features of the proposed algorithm are three fold. First, two type of fuzzy regressions with and without preprocessed data are prescribed by the algorithm in order to minimize the bias. Second, it uses PCA approach instead of trial and error method for selecting the most important input variables. Third, ANOVA is used to statistically compare fuzzy regression and time series with actual data.     

What׳s news in News? A cautionary note on using a variance decomposition to assess the quantitative importance of news shocks

This paper points out a conceptual difficulty in using a variance decomposition to assess the quantitative importance of news shocks. A variance decomposition will attribute to news shocks movements in endogenous variables driven both by news about future exogenous fundamentals that has yet to materialize (what I call “pure news”) as well as movements driven by realized changes in fundamentals that were anticipated in the past (what I call “realized news”). I present a stylized model in which news about yet unrealized changes in fundamentals is irrelevant for output dynamics, but in which a variance decomposition may nevertheless attribute a large share of the variance of output to news shocks. I then revisit the quantitative importance of news in the model of Schmitt-Grohe and Uribe (2012). In their model news shocks account for 40 percent of the variance of output growth, but this is mostly driven by realized news.