INTRODUCING ASYMMETRY INTO THE SKEWNESS AND KURTOSIS DYNAMICS OF THE NAGARCHSK MODEL OF LEON ET AL. 2004

 
29.12.2015
 
Department of Economics
 
Hakan Eratalay

On December 14th, MDM Professor of Financial Econometrics Hakan Eratalay gave a presentation as part of the EUSP Department of Economics’ research seminar. Professor Eratalay presented his report based on an article co-authored with Department of Economics graduate Vitaly Merso.

Most financial data present themselves as a time series—the value of any parameter (stock markets, oil prices, exchange rates) at successive moments in time. One of the most important financial indicators is the return on any financial asset for any period of time (usually a day or a month). Return depends on an asset’s market value at the beginning or end of a selected period of time, and is thereby determined by the behavior of a large number of players on the market. Thus, returns are random variables and are characterized by their distribution, the knowing of which can accurately predict their future values. But the true distribution of return is unknown, and in practice only certain of its parameters (moments) can be assessed—for example, the average value (first moment) or the variance (second moment). Hence most models of financial econometrics are dedicated to finding the best estimates that most accurately describe the characteristics of unknown yield distributions and allow us to make predictions.

Although it’s believed that the distribution of returns on any asset looks like a normal distribution (Gaussian bell), it differs in important and essential ways. In particular, in contrast to a normal distribution in which these values are zero, any real distribution of returns has a negative skewness (normalized third moment) and a positive kurtosis (parameter related to the fourth moment). This negative skewness means that the left “tail” of the distribution is longer than the right—that is, the market attaches a greater likelihood of asset prices decreasing rather than increasing. Positive kurtosis indicates that such a distribution has thick “tails”—the market will attach greater certainty to extreme return values than it would in a normal distribution. Thus, in models of financial econometrics we must pay special attention not only to the first two moments, but to the distribution’s higher moments as well.

A class of models for assessing time series popular in financial econometrics is GARCH (Generalized AutoRegressive Conditional Heteroscedasticity). GARCH is based on the assumption that the conditional variance of a time series changes over times and depends on past values of the series and on past values of the variance. Models that additionally include the effect of asymmetry well known in financial markets (negative returns have a stronger impact on dispersion than equal modulo positive returns—in a falling market dispersion is higher than in a growing one) are called NAGARCH (Nonlinear Asymmetric GARCH). The paper (Leon et al., 2004) described the NAGARCHSK model (NAGARCH with Skewness and Kurtosis), which accounts not only for variance but also considers that higher moments (skewness and kurtosis) can change over time and can also depend on past values of the time series and on their own past values.

In his report professor Eratalay presented the next step in the development of methods for assessing time series. The proposed NAGARCHNASK model is a modification of the NAGARCHSK model, in which nonlinear asymmetric effects affecting kurtosis and asymmetry are introduced. The model was evaluated according to daily quotations of currency pairs—ruble/dollar, ruble/euro, euro/dollar—prices for Brent crude, and the MMVB index. The results show that in many series nonlinear asymmetric effects are indeed present in the dynamics of kurtosis and asymmetry, which may serve as an indicator of investor and trader attitudes toward risk. Furthermore, a comparison of various specifications shows that the quality and predictive power of the model that accounts for these effects is greater than that of models considering the third and fourth moments of distribution with constants (NAGARCH) or symmetries (NAGARCHSK).

Mikhail Pakhnin