Moments of truncated <Emphasis Type="Italic">t</Emphasis> and <Emphasis Type="Italic">F</Emphasis> distributions

Truncated distributions commonly arise in economics and related areas, see, for example, Lee (Econ Lett 3:165–169, 1979), Lien (Econ Lett 19:243–247, 1985; Econ Lett 20:45–47, 1986), Burdett (Econ Lett 52:263–267, 1996), Sercu (Insur: Math and Econ 20:79–95, 1997), Abadir and Magdalinos (Econom Theory 18:1276–1287, 2002), and Horrace (J Econom 126:335–354, 2005). In this note, we consider the most commonly encountered truncated distributions with heavy tails: the truncated t distribution and the truncated F distribution. For each of these distributions, we derive explicit expressions for the moments and estimation procedures by the method of moments and the method of maximum likelihood. An application is illustrated to a popular data set in the econometric literature.      

Economic models based on Pareto and Gamma random variables

The distribution of products of random variables arises explicitly in economics and related areas. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the exact distribution of the product XY is derived when X and Y are independent random variables and arise from the two most applied models for economic data. The associated estimation procedures and percentage points are provided.     

Estimating aggregate autoregressive processes when only macro data are available

The aggregation of individual random AR(1) models generally leads to an AR(∞$\infty$) process. We provide two consistent estimators of aggregate dynamics based on either a parametric regression or a minimum distance approach for use when only macro data are available. Notably, both estimators allow us to recover some moments of the cross-sectional distribution of the autoregressive parameter. Both estimators perform very well in our Monte-Carlo experiment, even with finite samples.     

Optimal hedge ratio estimation and hedge effectiveness with multivariate skew distributions

This article proposes to use the three multivariate skew distributions (generalized hyperbolic distribution, multivariate skew normal distribution, and multivariate skew Student-t distribution) for estimating the minimum variance hedge ratio in a dynamic setting. Three criteria for measuring hedge effectiveness are employed: hedging instrument effectiveness, overall hedge effectiveness, and relative-to-optimal hedge ratio effectiveness (RHRE). Three portfolios of spot and futures series are formed for empirical analysis. The outcomes confirm that the three multivariate skew distributions are more helpful in deciding the minimum variance hedge ratio, especially the generalized hyperbolic distribution, than the symmetrical normal and Student-t distributions. This outperformance is significant especially at critical market moments and it is indicated by three hedge effectiveness measures. This advantage is held without the cost of lowering portfolio return. In addition, there is speculation possibility existing in the portfolio hedged by the traditional optimal hedge ratio and this potential can be detected especially by RHRE.     

Modelling income distribution using the log Student’s t distribution: New evidence for European Union countries

Income distribution remains a crucial topic in economic analysis, among other reasons, due to the increase in inequality in recent years, as one of the effects of the Great Recession. In this context, proposing parametric models that represent the full distribution through a small number of parameters arouses great interest as an instrument for economic analysis. This paper studies the ability of log Student’s t distribution to model the size distribution of income due to its potential to reproduce the effect of a mode around low-incomes as well as its precision in capturing the degree of kurtosis of empirical distributions. These characteristics make the log-t an ideal analysis tool, for instance, for exploring the effects of anti-poverty policies. The model has been fitted to income data for the EU25 and for several years. The conclusion is that the log Student’s t distribution offers the best fit in the vast majority of cases.     

A new model of income distribution: the κ-generalized distribution

This paper proposes a three-parameter statistical model of income distribution by exploiting recent developments on the use of deformed exponential and logarithm functions as suggested by Kaniadakis (Phys A 296:405–425, 2001; Phys Rev E 66:056125, 2002; Phys Rev E 72:036108, 2005). Formulas for the shape, moments and standard tools for inequality measurement are given. The model is shown to fit remarkably well the personal income data for Great Britain, Germany and the United States in different years, and its empirical performance appears to be competitive with that of other existing distributions.     

A Sufficient Condition for Generalized Lorenz Order

In this paper, a sufficient condition for non-negative random variables to be ordered in the Generalized Lorenz sense is presented. This condition does not involve inverse distribution functions. Applications of this result to several income distribution models are given. Journal of Economic Literature Classification Numbers: D36, D69.     

Simulating the spatial distribution of population and emissions to 2100

Urbanization and economic development have important implications for many environmental processes including global climate change. Although there is evidence that urbanization depends endogenously on economic variables, long-term forecasts of the spatial distribution of population are often made exogenously and independent of economic conditions. It is common for research concerning long-run projections of global environmental change to use population density as the primary means to spatially distribute emissions projections. However, researchers typically utilize year 1990 cross-sectional population data to distribute their emissions projections for both the short- and long-term, without projecting any changes in population density. Thus, a beta distribution for individual countries/regions is estimated to describe the geographical distribution of population using a one-degree-by-one-degree latitude–longitude global population data set. Cross-sectional country/regional data are then used to estimate an empirical relationship between parameters of the beta distribution and macroeconomic variables as they vary among countries/regions. This conditional beta distribution allows the simulation of a changing distribution of population, including the growth of urban areas, driven by economic forecasts until the year 2100.     

Nonparametric Estimation of Sample Selection Models

Sample selection models provide an important way of accounting for economic decisions that combine discrete and continuous choices and of correcting for nonrandom sampling. Nonparametric estimators for these models are developed in this paper. These can be used for estimating shapes and important economic quantities, as in standard nonparametric regression. Endogeneity of regressors of interest is allowed for. Series estimators for these models are developed, which are useful for imposing additivity restrictions that arise from selection corrections. Convergence rates and asymptotic normality results are derived. An application to returns to schooling among Australian young females is given.     

RIDGE ESTIMATORS FOR PROBIT REGRESSION: WITH AN APPLICATION TO LABOUR MARKET DATA

In this paper we propose ridge regression estimators for probit models since the commonly applied maximum likelihood (ML) method is sensitive to multicollinearity. An extensive Monte Carlo study is conducted where the performance of the ML method and the probit ridge regression (PRR) is investigated when the data are collinear. In the simulation study we evaluate a number of methods of estimating the ridge parameter k that have recently been developed for use in linear regression analysis. The results from the simulation study show that there is at least one group of the estimators of k that regularly has a lower mean squared error than the ML method for all different situations that have been evaluated. Finally, we show the benefit of the new method using the classical Dehejia and Wahba dataset which is based on a labour market experiment.     

Income inequality analysis in the period of economic transformation in poland

This paper presents some statistical methods of income inequality analysis based on the theoretical income distribution models that are well-fitted to the empirical models. As theoretical curves, the lognormal, gamma, Burr Type XII, and the Dagum models were used. They were applied to the earnings distributions in Poland in the period of economic transformation from a centrally-planned to a market economy. On the basis of the Dagum model, showing the perfect consistency with the considered earnings distributions, the maximun likelihood estimators of inequality parameters and economic distance ratios between men and women were calculated.     

On the consequences of trend for simultaneous equation estimation

It is shown that the presence of trend in the exogenous variables violates some standard assumptions that govern large sample asymptotics in simultaneous equations, and that contrary to the standard analysis Ordinary Least Squares may be consistent with trended data. Necessary and sufficient conditions are derived for this to happen, and in case of consistency the asymptotic distributions for a wide class of k-class estimators are shown to be identical.     

Modeling Nonlinear Autoregressive Distributed Lag Models: A New Approach

It is a common practice in econometrics that estimation is carried out in terms of the reduced form parameters and the structural form parameters are retrieved using the functional relationship between structural form parameters and the reduced form parameters. The reduced form of many useful economic models is a nonlinear distributed lag model (NLADL) with error structure which may have autocorrelation. In addition, the relationship between the reduced form and the structural parameters is often nonlinear and in a ratio form. In such situations existing sampling theory estimation procedures result in estimators for the structural parameters which do not have finite moments and do not possess optimal sampling properties. As an alternative, we propose a two step Bayesian estimation method. The Bayesian method has great potential and allows us to obtain the posterior probability density functions of all parameters of interest. In particular, its application for the analysis of adaptive expectation partial adjustment models for which the reduced form is a NLADL model, has been found extremely useful. An application to Nerlove’s supply response function supports the proposed methodology.

     

Testing for Granger non-causality in heterogeneous panels

This paper proposes a very simple test of Granger (1969) non-causality for heterogeneous panel data models. Our test statistic is based on the individual Wald statistics of Granger non causality averaged across the cross-section units. First, this statistic is shown to converge sequentially to a standard normal distribution. Second, the semi-asymptotic distribution of the average statistic is characterized for a fixed T sample. A standardized statistic based on an approximation of the moments of Wald statistics is hence proposed. Third, Monte Carlo experiments show that our standardized panel statistics have very good small sample properties, even in the presence of cross-sectional dependence.     

Score-driven Markov-switching EGARCH models: an application to systematic risk analysis

We introduce new Markov-switching (MS) dynamic conditional score (DCS) exponential generalized autoregressive conditional heteroscedasticity (EGARCH) models, to be used by practitioners for forecasting value-at-risk (VaR) and expected shortfall (ES) in systematic risk analysis. We use daily log-return data from the Standard & Poor’s 500 (S&P 500) index for the period 1950–2016. The analysis of the S&P 500 is useful, for example, for investors of (i) well-diversified US equity portfolios; (ii) S&P 500 futures and options traded at Chicago Mercantile Exchange Globex; (iii) exchange traded funds (ETFs) related to the S&P 500. The new MS DCS-EGARCH models are alternatives to of the recent MS Beta-t-EGARCH model that uses the symmetric Student’s t distribution for the error term. For the new models, we use more flexible asymmetric probability distributions for the error term: Skew-Gen-t (skewed generalized t), EGB2 (exponential generalized beta of the second kind) and NIG (normal-inverse Gaussian) distributions. For all MS DCS-EGARCH models, we identify high- and low-volatility periods for the S&P 500. We find that the statistical performance of the new MS DCS-EGARCH models is superior to that of the MS Beta-t-EGARCH model. As a practical application, we perform systematic risk analysis by forecasting VaR and ES.

Abbreviation Single regime (SR); Markov-switching (MS); dynamic conditional score (DCS); exponential generalized autoregressive conditional heteroscedasticity (EGARCH); value-at-risk (VaR); expected shortfall (ES); Standard & Poor's 500 (S&P 500); exchange traded funds (ETFs); Skew-Gen-t (skewed generalized t); EGB2 (exponential generalized beta of the second kind); NIG (normal-inverse Gaussian); log-likelihood (LL); standard deviation (SD); partial autocorrelation function (PACF); likelihood-ratio (LR); ordinary least squares (OLS); heteroscedasticity and autocorrelation consistent (HAC); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan-Quinn criterion (HQC).     

Forecasting volatility in the Indian equity market using return and range-based models

In this article, we assess the time-varying volatility of the National Stock Exchange in the Indian equity market using unconditional estimators and asymmetric conditional econometric models. The volatility estimate and forecast is computed from the interday return and intraday range-based data of the exchange’s flagship index, CNX NIFTY, for the time period spanning 1 January 2009 through 31 December 2013. These are our findings: First, we determine that the time-varying volatility of the index is asymmetric with qualities of stationarity and leptokurtic distribution. Second, the one-step-ahead volatility forecast derived from the univariate time series parameters through the GJR-GARCH ?????process indicates that the model evaluation criteria of the autoregressive process tends towards range-based models vis-à-vis a return-based model. The validity of this methodology is further analysed with the superior predictive ability test, the outcome of which supports the use of range-based conditional models. Finally, among the evaluated range-based model variants, the model confidence set procedure favours the Yang–Zhang estimator as being better suited to forecast the exchange’s volatility than the ones by Parkinson, Garman–Klass and Rogers–Satchell.     

Statistical Equilibrium Wealth Distributions in an Exchange Economy with Stochastic Preferences

We describe an exchange market consisting of many agents with stochastic preferences for two goods. When individuals are indifferent between goods, statistical mechanics predicts that goods and wealth will have steady-state gamma distributions. Simulation studies show that gamma distributions arise for a broader class of preference distributions. We demonstrate this mathematically in the limit of large numbers of individual agents. These studies illustrate the potential power of a statistical mechanical approach to stochastic models in economics and suggest that gamma distributions will describe steady-state wealths for a class of stochastic models with periodic redistribution of conserved quantities. Journal of Economic Literature Classification Numbers: C15, C62, C73, D3, D5.     

Modeling of Income and Indicators of Poverty and Social Exclusion Using the Generalized Beta Distribution of the Second Kind

There are three reasons why estimation of parametric income distributions may be useful when empirical data and estimators are available: to stabilize estimation; to gain insight into the relationships between the characteristics of the theoretical distribution and a set of indicators, e.g. by sensitivity plots; and to deduce the whole distribution from known empirical indicators, when the raw data are not available. The European Union Statistics on Income and Living Conditions (EU‐SILC) survey is used to address these issues. In order to model the income distribution, we consider the generalized beta distribution of the second kind (GB2). A pseudo‐likelihood approach for fitting the distribution is considered, which takes into account the design features of the EU‐SILC survey. An ad‐hoc procedure for robustification of the sampling weights, which improves estimation, is presented. This method is compared to a non‐linear fit from the indicators. Variance estimation within a complex survey setting of the maximum pseudo‐likelihood estimates is done by linearization (a sandwich variance estimator), and a simplified formula for the sandwich variance, which accounts for clustering, is given. Performance of the fit and estimated indicators is evaluated graphically and numerically.     

PAIRWISE DIFFERENCE ESTIMATION WITH NONPARAMETRIC CONTROL VARIABLES*

This article extends the pairwise difference estimators for various semilinear limited dependent variable models proposed by Honoré and Powell (Identification and Inference in Econometric Models. Essays in Honor of Thomas Rothenberg Cambridge: Cambridge University Press, 2005) to permit the regressor appearing in the nonparametric component to itself depend upon a conditional expectation that is nonparametrically estimated. This permits the estimation approach to be applied to nonlinear models with sample selectivity and/or endogeneity, in which a “control variable” for selectivity or endogeneity is nonparametrically estimated. We develop the relevant asymptotic theory for the proposed estimators and we illustrate the theory to derive the asymptotic distribution of the estimator for the partially linear logit model.     

Limit Theorems for Estimating the Parameters of Differentiated Product Demand Systems

We provide an asymptotic distribution theory for a class of generalized method of moments estimators that arise in the study of differentiated product markets when the number of observations is associated with the number of products within a given market. We allow for three sources of error: sampling error in estimating market shares, simulation error in approximating the shares predicted by the model, and the underlying model error. It is shown that the estimators are CAN provided the size of the consumer sample and the number of simulation draws grow at a large enough rate relative to the number of products. We consider the implications of the results for the Berry, Levinsohn and Pakes (1995) random coefficient logit model and the pure characteristics model analysed in Berry and Pakes (2002) . The required rates differ for these two frequently used demand models. A small Monte Carlo study shows that the differences in asymptotic properties of the two models are reflected, in quite a striking way, in the models' small sample properties. Moreover the limit distributions provide a good approximation to the actual Monte Carlo distribution of the parameter estimates. The results have important implications for the computational burden of the two models.