The limiting distribution of extremal exchange rate returns

Several nonnested fat-tailed distributions have been advocated for modelling exchange rate returns. Instead of directly estimating these nonnested distributions we investigate the extremal distribution of the returns. The advantage is that the parameter which characterizes the amount of tail fatness can be estimated without maintaining a specific distribution, and hence enables one to test hypotheses. The parameter of the limit law is estimated by employing nonparametric procedures based on order statistics. The appropriateness of these procedures is assessed. Given this estimate one can derive bounds on the returns for very low probabilities on an excess. Such information is useful in evaluating the volatility of exchange rates.     

Model‐free evaluation of directional predictability in foreign exchange markets

We examine directional predictability in foreign exchange markets using a model‐free statistical evaluation procedure. Based on a sample of foreign exchange spot rates and futures prices in six major currencies, we document strong evidence that the directions of foreign exchange returns are predictable not only by the past history of foreign exchange returns, but also the past history of interest rate differentials, suggesting that the latter can be a useful predictor of the directions of future foreign exchange rates. This evidence becomes stronger when the direction of larger changes is considered. We further document that despite the weak conditional mean dynamics of foreign exchange returns, directional predictability can be explained by strong dependence derived from higher‐order conditional moments such as the volatility, skewness and kurtosis of past foreign exchange returns. Moreover, the conditional mean dynamics of interest rate differentials contributes significantly to directional predictability. We also examine the co‐movements between two foreign exchange rates, particularly the co‐movements of joint large changes. There exists strong evidence that the directions of joint changes are predictable using past foreign exchange returns and interest rate differentials. Furthermore, both individual currency returns and interest rate differentials are also useful in predicting the directions of joint changes. Several sources can explain this directional predictability of joint changes, including the level and volatility of underlying currency returns. Copyright © 2007 John Wiley & Sons, Ltd.     

Record values from generalized Pareto distribution and associated inference

In this paper, we derive exact explicit expressions for the single, double, triple and quadruple moments of the upper record values from a generalized Pareto distribution. We then use these expressions to compute the mean, variance, and the coefficients of skewness and kurtosis of certain linear functions of record values. Finally, we develop approximate confidence intervals for the location and scale parameters of the generalized Pareto distribution using the Edgeworth approximation and compare them with the intervals constructed through Monte Carlo simulations. Received: June 1999     

Higher moment exchange rate exposure of S&P500 firms

We examine the impact of higher order moments of changes in the exchange rate on stock returns of U.S. large-cap companies in the S&P500. We find a robust negative effect of exchange rate volatility on S&P500 company returns. The consumer discretionary and the consumer staples sectors have significant negative exposure to exchange rate volatility suggesting that exchange rate volatility affects stock returns through the channel of international operations. In terms of industries, the household products and personal products industries have significant negative exposure as well. The impact in the financial sector suggests that derivatives and hedging activity can mitigate exposure to exchange rate volatility. We find weak evidence that exchange rate skewness has an effect on S&P500 stock returns, but, find evidence that exchange rate kurtosis affects returns of companies that are more exposed to exchange rate volatility.     

Assessing misspecified asset pricing models with empirical likelihood estimators

Hansen and Jagannathan (1997) compare misspecified asset pricing models based on least-square projections on a family of admissible stochastic discount factors. We extend their fundamental contribution by considering Minimum Discrepancy projections where misspecification is measured by a family of convex functions that take into account higher moments of asset returns. The Minimum Discrepancy problems are solved on dual spaces producing a family of estimators that captures the least-square problem as a particular case. We derive the asymptotic distributions of the estimators for the Cressie–Read family of discrepancies, and illustrate their use with an assessment of the Consumption Asset Pricing Model.     

预测VaR:高阶矩可行域未必越广越好

高阶矩可行域反映了分布函数对样本高阶矩特征(非对称和尖峰、厚尾等)的适应能力,是影响VaR预测绩效的重要因素之一。本文以中、美、英、日四个股市的收益为样本,实证比较了五种高阶矩可行域各不相同的分布函数的VaR预测绩效。结果发现,可行域过于宽广(狭窄)的分布易于高(低)估样本的偏度和峰度,并高(低)估VaR,而可行域适中的广义偏斜-t和偏斜-t分布的预测绩效相对较好。此结果可为后续研究改进VaR的预测绩效提供有益借鉴。     

Sources of volatility persistence: A case study of the U.K. pound/U.S. dollar exchange rate returns

This paper utilizes a new approach to examine the inherent nonlinear dynamics of the exchange rate returns volatility. Specifically, we utilize a regime switching threshold (i) generalized autoregressive conditional heteroskedasticity (RS-TGARCH) and (ii) a fractional generalized autoregressive conditional heteroskedasticity (RS-TFIGARCH) model. The RS-TGARCH model is found to be adequate in analyzing the first two moments of the U.K. pound/U.S. dollar monthly exchange rate returns series. The RS-TFIGARCH is found to be adequate for the daily returns series. The volatility persistence and leverage effects associated with exchange rate returns series are jointly tested by means of a Wald Chi-square test.     

Skewed Normal Variance-Mean Models for Asset Pricing and the Method of Moments

Financial returns (log-increments) data, Y _t , t = 1,2,…, are treated as a stationary process, with the common distribution at each time point being not necessarily symmetric.
We consider as possible models for the common distribution four instances of the General Normal Variance-Mean Model (GNVM), which is described by Y | V ∼ N ( a ( b + V ), c ²V + d² ) where V is a nonnegative random variable and a, b, c and d are constants. When V is Gamma distributed and d = 0, Y has the skewed Variance-Gamma distribution (VG). When V follows a Half Normal distribution and c = 0, Y has the well-known Skew Normal (SN) distribution. We also consider two cases where V is Exponentially distributed. Bounds for skewness and kurtosis in each case are found in terms of the moments of the V . These are useful in determining whether the Method of Moments for a given model is feasible. The problem of overdetermination of parameters via estimating equations is examined. 5 data sets of actual returns data, chosen because of their earlier occurrence in the literature, are analysed using each of the 4 models.     

An objective function for simulation based inference on exchange rate data

The assessment of models of financial market behaviour requires evaluation tools. When complexity hinders a direct estimation approach, e.g., for agent based microsimulation models, simulation based estimators might provide an alternative. In order to apply such techniques, an objective function is required, which should be based on robust statistics of the time series under consideration. Based on the identification of robust statistics of foreign exchange rate time series in previous research, an objective function is derived. This function takes into account stylized facts about the unconditional distribution of exchange rate returns and properties of the conditional distribution, in particular, autoregressive conditional heteroscedasticity and long memory. A bootstrap procedure is used to obtain an estimate of the variance-covariance matrix of the different moments included in the objective function, which is used as a base for the weighting matrix. Finally, the properties of the objective function are analyzed for two different agent based models of the foreign exchange market, a simple GARCH-model and a stochastic volatility model using the DM/US-$ exchange rate as a benchmark. It is also discussed how the results might be used for inference purposes. Research has been supported by the DFG grant WI 20024/2-1/2. We are indebted to two anonymous referees of this journal, Leigh Tesfatsion, Patrick Burns and other participants of the CEF’06 conference in Limassol for helpful comments on preliminary versions of this paper.     

Parametric pricing of higher order moments in S&P500 options

A general parametric framework based on the generalized Student t‐distribution is developed for pricing S&P500 options. Higher order moments in stock returns as well as time‐varying volatility are priced. An important computational advantage of the proposed framework over Monte Carlo‐based pricing methods is that options can be priced using one‐dimensional quadrature integration. The empirical application is based on S&P500 options traded on select days in April 1995, a total sample of over 100,000 observations. A range of performance criteria are used to evaluate the proposed model, as well as a number of alternative models. The empirical results show that pricing higher order moments and time‐varying volatility yields improvements in the pricing of options, as well as correcting the volatility skew associated with the Black–Scholes model. Copyright © 2004 John Wiley & Sons, Ltd.     

Weekly pattern in higher moments: An empirical test in Hong Kong stock market

This paper provides a direct test on the day-of-the-week effect on higher moments of stock returns and compares across different industrial sectors of the Hong Kong market. Empirical results show that daily returns of six different industrial sectors on all weekdays are non-normally distributed. The hypothesis of equal higher moments is rejected by most pairs of weekdays, particularly the Monday-Tuesday pair, for all indices, supporting the existence of the day-of-the-week effect on higher moments. The results also show that the weekly pattern on volatility and higher moments cannot help explain the weekly pattern on mean returns through the concept of risk premium. Further analysis shows that Rogalski’s effect exists on the higher moments because the day-of-the-week effect exists only in non-January months.     

Moments structure of ? 1-stochastic volatility models

We consider Taylor??s stochastic volatility model (SVM) when the innovations of the hidden log-volatility process have a Laplace distribution (? ₁ exponential density), rather than the standard Gaussian distribution (? ₂) usually employed. Recently many investigations have employed ? ₁ metric to allow better modeling of the abrupt changes of regime observed in financial time series. However, the estimation of SVM is known to be difficult because it is a non-linear with an hidden markov process. Moreover, an additional difficulty yielded by the use of ? ₁ metric is the not differentiability of the likelihood function. An alternative consists in using a generalized or efficient method-of-moments (GMM/EMM) estimation. For this purpose, we derive here the moments and autocovariance function of such ? ₁-based stochastic volatility models.     

On the distribution of city sizes

The city size distribution in many countries is remarkably well described by a Pareto distribution. We derive conditions that standard urban models must satisfy in order to explain this regularity. We show that under general conditions urban models must have (i) a balanced growth path and (ii) a Pareto distribution for the underlying source of randomness. In particular, one of the following combinations can induce a Pareto distribution of city sizes: (i) preferences for different goods follow reflected random walks, and the elasticity of substitution between goods is 1; or (ii) total factor productivities of different goods follow reflected random walks, and increasing returns are equal across goods.     

Multilateral aggregation-theoretic monetary aggregation over heterogeneous countries

We derive fundamental theory for measuring monetary service flows aggregated over countries within a multicountry area. We develop three increasingly restrictive approaches: (1) the heterogeneous agents approach, (2) the multilateral representative agent approach, and (3) the unilateral representative agent approach. These results are being used by the European Central Bank in construction of its Divisia monetary aggregates database, with convergence from the most general to the more restrictive approaches expected as economic convergence within the area proceeds. Our theory permits monitoring the effects of policy over a multicountry area, while also monitoring the distribution effects of policy among the countries.     

Mean–variance econometric analysis of household portfolios

We investigate households' portfolio choice using a microeconometric approach derived from mean–variance optimization. We assume that households have heterogeneous expectations on the distribution of excess returns and that they cannot take short positions in risky assets. Assuming two such assets, we derive an explicit solution of the model characterized by four possible portfolio regimes, which are analyzed using two structural probit and tobit specifications with three latent state variables. Both specifications are estimated by weighted maximum likelihood on a cross‐section of US households drawn from the 2004 SCF. The tobit specification is simulated in order to evaluate the regressors' effects on regime probabilities and asset demands. We also assess to what extent the predicted state variables are consistent with the self‐reported expected returns and risk aversion elicited from the SCF questionnaire. Copyright © 2009 John Wiley & Sons, Ltd.     

Extreme returns and intensity of trading

Asymmetric information models of market microstructure claim that variables such as trading intensity are proxies for latent information on the value of financial assets. We consider the interval‐valued time series (ITS) of low/high returns and explore the relationship between these extreme returns and the intensity of trading. We assume that the returns (or prices) are generated by a latent process with some unknown conditional density. At each period of time, from this density, we have some random draws (trades) and the lowest and highest returns are the realized extreme observations of the latent process over the sample of draws. In this context, we propose a semiparametric model of extreme returns that exploits the results provided by extreme value theory. If properly centered and standardized extremes have well‐defined limiting distributions, the conditional mean of extreme returns is a nonlinear function of the conditional moments of the latent process and of the conditional intensity of the process that governs the number of draws. We implement a two‐step estimation procedure. First, we estimate parametrically the regressors that will enter into the nonlinear function, and in a second step we estimate nonparametrically the conditional mean of extreme returns as a function of the generated regressors. Unlike current models for ITS, the proposed semiparametric model is robust to misspecification of the conditional density of the latent process. We fit several nonlinear and linear models to the 5‐minute and 1‐minute low/high returns to seven major banks and technology stocks, and find that the nonlinear specification is superior to the current linear models and that the conditional volatility of the latent process and the conditional intensity of the trading process are major drivers of the dynamics of extreme returns.     

Variance swaps,non-normality and macroeconomic and financial risks

This paper studies the determinants of the variance risk premium and discusses the hedging possibilities offered by variance swaps. We start by showing that the variance risk premium responds to changes in higher order moments of the distribution of market returns. But the uncertainty that determines the variance risk premium – the fear by investors to deviations from normality in returns – is also strongly related to a variety of macroeconomic and financial risks associated with default, employment growth, consumption growth, stock market and market illiquidity risks. We conclude that the variance risk premium reflects the market willingness to pay for hedging against these financial and macroeconomic sources of risk. An out-of-sample asset allocation exercise shows that the inclusion of the variance swap reduces the modified value-at-risk with respect to a portfolio holding exclusively the equity market portfolio.     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

On the convergence of moments of geometric and harmonic means

The moments of the geometric mean of n independent and identically distributed random variables are shown to converge as n→∞. Rates of convergence are determined for the first moment and the variance. The results relate to recent work on long term investment returns when yearly rates of return are randomly varying. Application is made to moments of the harmonic mean.     

Volatility forecast comparison using imperfect volatility proxies

The use of a conditionally unbiased, but imperfect, volatility proxy can lead to undesirable outcomes in standard methods for comparing conditional variance forecasts. We motivate our study with analytical results on the distortions caused by some widely used loss functions, when used with standard volatility proxies such as squared returns, the intra-daily range or realised volatility. We then derive necessary and sufficient conditions on the functional form of the loss function for the ranking of competing volatility forecasts to be robust to the presence of noise in the volatility proxy, and derive some useful special cases of this class of “robust” loss functions. The methods are illustrated with an application to the volatility of returns on IBM over the period 1993 to 2003.