Aggregational Gaussianity and Barely Infinite Variance in Financial Returns

RePEc: Research Papers in Economics, 2010

This paper aims at reconciling two apparently contradictory empirical regularities of financial returns, namely the fact that the empirical distribution of returns tends to normality as the frequency of observation decreases (aggregational Gaussianity) combined with the fact that the conditional variance of high frequency returns seems to have a unit root, in which case the unconditional variance is infinite. We show that aggregational Gaussianity and infinite variance can coexist, provided that all the moments of the unconditional distribution whose order is less than two exist. The latter characterises the case of Integrated GARCH (IGARCH) processes. Finally, we discuss testing for aggregational Gaussianity under barely infinite variance.

It is widely accepted that some of the most accurate predictions of aggregated asset returns are based on an appropriately specified GARCH process. As the forecast horizon is greater than the frequency of the GARCH model, such predictions either require time-consuming simulations or they can be approximated using a recent development in the GARCH literature, viz. analytic conditional moment formulae for GARCH aggregated returns. We demonstrate that this methodology yields robust and rapid calculations of the Value-at-Risk (VaR) generated by a GARCH process. Our extensive empirical study applies Edgeworth and Cornish-Fisher expansions and Johnson SU distributions, combined with normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets; it validates the accuracy of the analytic approximations to GARCH aggregated returns and derives GARCH VaR estimates that are shown to be highly accurate over multiple horizons and significance levels. JEL Code: C53, G17 1 1 INTRODUCTION

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 when fitting the first three even moments of financial time series. We show instead that a double gaussian conditional probability distribution better captures the higher order moments of the data. To demonstrate this point, we construct regions (phase diagrams) in the fourth and sixth order standardised moment space, where a GARCH(1,1) model can be used to fit these moments and compare them with the corresponding moments from empirical data for different sectors of the economy. We found that, the ability of the GARCH model with a double gaussian conditional distribution to fit higher order moments is dictated by the time window our data spans. We can only fit data collected within specific time window lengths and only with certain parameters of the conditional double ga...

Studies in Nonlinear Dynamics & Econometrics, 1998

Conditional returns distributions generated by a GARCH process, which are important for many problems in market risk assessment and portfolio optimization, are typically generated via simulation. This paper extends previous research on analytic moments of GARCH returns distributions in several ways: we consider a general GARCH model -the GJR specification with a generic innovation distribution; we derive analytic expressions for the first four conditional moments of the forward return, of the forward variance, of the aggregated return and of the aggregated variance -corresponding moments for some specific GARCH models largely used in practice are recovered as special cases; we derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments; and we demonstrate empirically that some excellent approximate predictive distributions can be obtained from these analytic moments, thus precluding the need for time-consuming simulations. JEL Code: C53 1 1 INTRODUCTION Forward-looking physical return distributions have attracted a vast academic research literature because they have a great variety of financial applications to market risk assessment and portfolio optimization techniques. Since it has been recognized that time series of asset returns are not well described by normal, independent processes. Typically, their conditional distributions are non-normal and they exhibit volatility clustering, so are not independent.

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...

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.

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 .

1998

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.