Aggregational Gaussianity and Barely Infinite Variance in Crop Prices

2008

Consider a class of power transformed and threshold GARCH(p,q) (PTTGRACH(p,q)) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa (2004) and includes the standard GARCH model and many other models as special cases. We first establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has finite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH(p,q) model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH(p,q) models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy tailed errors. Furthermore the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the financial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH(p,q), we give in the appendix a necessary and sufficient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.

International Research Journal …, 2012

Many methods in finance rest upon the assumption that asset returns follow a normal distribution. However, finance data often depart from the normal distribution. Since stable distributions can accommodate both fat tails and asymmetry, they often provide a very good fit to empirical data. This paper examines the statistical distributions of high frequency (intra-daily) TRY/USD foreign exchange changes and the daily-hourly volatility of the return series by employing normal and stable GARCH models before and after the global financial crisis. Empirical evidence supports that a GARCH model with stable Paretian innovations fits returns better than the normal distribution; empirical evidence also supports that while the global financial crisis has a negative impact on the distribution of returns, it does not affect volatility of an emerging market, namely the Turkish Interbank Foreign Exchange Market. 57 crucially on distributional specifications. Modern finance relies heavily on the assumption that the random variable under investigation follows a normal distribution . Yet, finance data often depart from the normal model in that their marginal distributions are fat-tailed. In order to overcome the problem of fat tails, and introduced stable distributions to finance. The peaked and heavy-tailed nature of the return distribution led the authors to reject the standard hypothesis of normally distributed returns in favor of the stable distribution. Since then, the stable distributions have been used to model both the unconditional and conditional return distributions. It is now commonly accepted that financial asset returns are indeed heavy-tailed .

Studies in Nonlinear Dynamics & Econometrics, 1998

Journal of Statistical Planning and Inference, 2008

Conditionally heteroskedastic time series given by y k = k k are frequently used in econometrics. The conditional variance 2 k is defined by a parametric function of past observations and volatilities. Since several conditionally heteroskedastic time series models have been suggested in the literature, we want to test if a given model fits well the data. The method we propose in this paper is based on comparing the distributions of the observed and implied volatilities. Our results can be used to assess the validity of the GARCH(p, q) model.

1998

2021

Here we have analysed a GARCH(1,1) model with the aim to fit higher order moments for different companies’ stock prices. When we assume a gaussian conditional distribution, we fail to capture any empirical data. We show instead that a double gaussian conditional probability better captures the higher order moments of the data. To demonstrate this point, we construct regions (phase diagrams) in higher order moment space, where a GARCH(1,1) model can be used to fit the higher order moments and compare this with empirical data from different sectors of the economy. We found that, the ability of the GARCH model to fit higher order moments is dictated by the time window our data spans. Primarily, if we have a time series, using a GARCH(1,1) model with a double gaussian conditional probability (a GARCH-double-normal model) we cannot necessarily fit the statistical moments of the time series. Highlighting, that the GARCH-double-normal model only allows fitting of specific lengths of time s...

2004

Abstract. In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavy-tailed innovations. This means that the innovations are regularly varying with index α ∈ (2,4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and √ n-rates breaks down. This was recently observed by Hall and Yao (2003). It is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavytailed case nevertheless very much parallels the normal asymptotic theory. In the light-tailed case, the limit theory is based on the CLT for stationary ergodic finite variance martingale difference sequences. In the heavy-tailed case such a general result does not exist, but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for GARCH processes...

2004

Testing for normality is of paramount importance in many areas of science since the Gaussian distribution is a key hypothesis in many models. As the use of semi–moments is increasing in physics, economics or finance, often to judge the distributional properties of a given sample, we propose a test of normality relying on such statistics. This test is proposed in three different versions and an extensive study of their power against various alternatives is conducted in comparison with a number of powerful classical tests of normality. We find that semi–moments based tests have high power against leptokurtic and asymmetric alternatives. This new test is then applied to stock returns, to study the evolution of their normality over different horizons. They are found to converge at a “log-log” speed, as are moments and most semi–moments. Moreover, the distribution does not appear to converge to a real Gaussian.

This paper,develops,a new,econometric,framework,for investigating,how,the sensitivity of the financial market volatility to shocks varies with the volatility level. For this purpose, the paper first introduces,the square-root (SQ) GARCH model,for financial time series. It is an ARCH analogue,of the continuous-time,square-root stochastic volatility model,popularly,used in derivatives pricing and hedging.,The variance,of variance,is a linear function of the conditional,variance,in the SQGARCH and of the square of it in the GARCH. After showing some implications of this difference, the paper introduces the constant-elasticity-of-variance (CEV) GARCH model, which allows more flexible fitting of variance-of-variance dynamics. The paper develops conditions for stationarity, the existence of finite moments, β-mixing, and other properties of the conditional variance process via the general state-space Markov chains approach. In particular, the paper generalizes the strict stationarity condi...

Journal of Time Series …, 2004