Volatility persistence in the Russian stock market

Abstract This paper applies a fractional integration framework to analyse the stochastic behaviour of two Russian stock market volatility indices (namely the originally created RTSVX and the new RVI that has replaced it) using daily data over the period 2010–2018. The empirical findings are consistent and imply in all cases that the two series are mean-reverting, i.e. they are not highly persistent and the effects of shocks disappear over time. This is true regardless of whether the errors are assumed to follow a white noise or autocorrelated process; this is confirmed by the rolling window estimation, and it holds for both subsamples, before and after the detected break. On the whole, it seems shocks do not have permanent effects on volatility in the Russian stock market.


Introduction
Financial market instabilities have become more frequent and acute in the era of globalisation (Bordo et al., 2001), and have raised concerns about the benefits of traditional portfolio diversification strategies. Those involving instruments based on the VIX volatility index (which is negatively correlated to equity returns) are thought to be particularly effective during periods of market turmoil for tail risk hedging (Whaley, 1993). The VIX is especially attractive to investors with high skewness preferences (Barberis and Huang, 2008). Unlike credit derivative instruments, the liquidity of VIX derivatives improves during periods of markets turmoil, when investors are in search of hedging instruments (Bahaji and Aberkane, 2016). The existing literature also shows the diversification benefits of VIX exposures in institutional investment portfolios (Szado, 2009). In particular, a VIX short future exposure in a benchmark portfolio triggers a positive expansion of the efficient frontier (Chen et al., 2011); moreover, the addition of VIX futures to pension fund equity portfolios can significantly improve their in-sample performance, whilst incorporating VIX instruments into long-only equity portfolios significantly enhances Value-at-Risk optimisation (Briere et al., 2010).
Most of the studies mentioned above focus on the developed economies. By contrast, the present paper provides new evidence for the VIX in an emerging economy such as Russia. Moreover, it considers both the old and the new VIX constructed for the Russian stock market and analyses in depth the statistical properties of both (long-range dependence, non-linearities and breaks) in a fractional integration framework.
Understanding the behaviour of the VIX is important because this index can be used as a predictor of stock returns and volatility, economic activity and financial instability. Further, it can be the basis of portfolio diversification strategies designed by domestic and international institutional investors. Specifically, the choice of the hedging effectiveness measure aimed at capturing the tail risk in the portfolio depends on the stochastic properties of the VIX. This is the motivation for the present study, which examines two different VIX measures (RTSVX and RVI) in a comparative framework in the case of Russia, a country for which very little evidence is available at present. The newly constructed RVI has replaced the originally created RTSVX in order to comply with the latest international financial industry standards and take into account feedback from market participants (see Section 2 for more details).
The layout of the paper is as follows. Section 2 provides background information on the Russian VIX, Section 3 outlines the empirical methodology, Section 4 describes the data and the empirical findings. Section 5 offers some concluding remarks.

The VIX in the Russian Stock Market
The idea of constructing a volatility index using option prices was first formulated at the time of the introduction of exchange trade index options in 1973. In subsequent years, the original methodology of Gastineau (1977), Cox and Rubinstein (1985)  (1) where Т 1 and T 2 are the time to expiration expressed as a fraction of a year consisting of 365 days for the nearby and far option series respectively; Т 30 andТ 365 stand for 30 and 365 days respectively, expressed as a fraction of a year; σ 1 2 and σ 2 2 are the variance of the nearby and next option series respectively.
There is only a limited number of studies on the Russian stock market, possibly because of the lack of long series of reliable data. As Mirkin and Lebedeva (2006) point out, Russian companies are more dependent on debt financing than equity financing since only about 6 percent of listed companies are traded in the largest Russian exchange; ownership in the equity market is highly concentrated; the Russian bond and equity markets are easily accessible to international investors and the corporate bond market has proven to be highly profitable without any defaults. Russian financial markets are rather stable and integrated in terms of international capital flows (Peresetsky and Ivanter, 2000); the degree of financial liberalisation in Russia determines the strength of its international integration (Hayo and Kutan, 2005); since the Russian stock market is not cointegrated with the US one investors should focus on the Russian VIX for predicting Russian stock market returns (Mariničevaitė & Ražauskaitė, 2015); in general, they have become more knowledgeable about the effects of the VIX on stock price indices for developed and emerging economies (Natarajan et al., 2014). 1

Methodology
The concept of long memory was originally introduced by Granger (1980Granger ( , 1981, Granger and Joyeux (1980) and Hosking (1981), and allows the differencing parameter required to make a series stationary I(0) to take a fractional value. Assuming that u t is a covariance-stationary I(0) process (denoted as u t ≈ I(0)) with a spectral density function that is positive and bounded at all frequencies, x t is said to be integrated of order d (and denoted as x t ≈ I(d)), if it can be represented as (2) with x t = 0 for t ≤ 0, and where is the lag operator ( ) and d can be any real value. Then, u t is I(0) and x t is I(d), and d measures of the persistence of the series. In such a case, one can use the following Binomial expansion for the polynomial on the left-hand side of (2) for all real d: , and thus, noting that L j x t = x t-j , .
The main advantage of this model, which became popular in the late 1990s and early 2000s (see Baillie, 1996;Gil-Alana and Robinson, 1997;Michelacci and Zaffaroni, 2000;Gil-Alana and Moreno, 2004;Abbritti et al., 2016;etc.), is that it is more general than standard models based on integer differentiation: it includes the stationary I(0) and 1 Other papers studying the Russian stock market and its volatility include Goriaev and Zabotkin (2006), Luukka et al. (2016) and Korhonen and Peresetsky (2016).
nonstationary I(1) series as particular cases of interest when d = 0 and 1 respectively, but also nonstationary though mean-reverting processes if the differencing parameter is in the range [0.5, 1).
We estimate the fractional differencing parameter d along with the rest of the parameters in the model by using the Whittle function in the frequency domain (Dahlhaus, 1989;Robinson, 1994) under the assumption that the estimated errors are uncorrelated and autocorrelated in turn. In particular, we adopt a parametric method that involves imposing a structure on the error term. Robinson's (1994) test is most suitable in this case very convenient in this context since it is valid for any range of values of d and therefore it does not require preliminary differencing; moreover, it allows the inclusion of deterministic terms such as an intercept and a time trend, and its limit distribution is standard normal. 2

Data and Empirical Results
We analyse daily transaction level data for both the old (RTSVX) and new (RVI) volatility indices obtained from the Moscow exchange web database; the sample period goes from 7 December 2010 to 12 December 2014 and 6 January 2014 to 9 February 2018 respectively. Appendix 1 provides some descriptive statistics. RTSVX has a slightly higher mean but is less volatile than RVI; further, it has a lower kurtosis coefficient, but a higher skewness one.

4a. The RTSVX index
As a first step we estimate the following model: 2 See Gil-Alana and Robinson (1997) for a description of the functional form of the version of the tests of Robinson (1994) used in this paper.
where y t is the series of interest, in this case the original volatility index and the logtransformed data. Three specifications are considered, namely i) without deterministic terms (i.e. α = β = 0 a priori in (3)); (ii) with an intercept (α is estimated and β = 0 a priori), and iii) with an intercept and a linear time trend (as in equation (3)), and assuming that the errors are uncorrelated (white noise) and autocorrelated (Bloomfield, 1973) in turn.
[Insert Table 1 about here] Table 1 shows the estimated values of d with their 95% confidence intervals.
The t-stats imply that the time trend is not a significant regressor, therefore the selected [Insert Figure 1 about here] , ... , 2 , 1 , Under the assumption of autocorrelation, the estimates of d are initially around 0.8, and then decrease from the subsample  till the end of the sample; all of them are below 1, implying mean-reverting behaviour.
Next, we test for breaks using the approach suggested by Bai and Perron (2003)  We then split the sample in two subsamples accordingly. The results for the two cases of uncorrelated and autocorrelated errors are presented, respectively, in Tables 2 and 3. The estimates of d are significantly below 1 in both subsamples, with both white noise and autocorrelated errors, and for both the original and the logged data. Tables 2 and 3 about here] 4b.

[Insert
The RVI index [Insert Table 4

Conclusions
This paper has applied a fractional integration framework to analyse the stochastic behaviour of two Russian stock market volatility indices, namely the originally created RTSVX and the new RVI that has replaced it (for both of which very limited evidence was previously available), using daily data over the period 2010-2018. The chosen approach is more general than those based on the I(0) v. I(1) dichotomy and provides useful information on the long-memory properties and degree of persistence of the series being analysed.
The empirical findings are consistent and imply in all cases that the two series are mean-reverting, i.e. their degree of persistence is limited and the effects of shocks disappear over time. This is consistent with the results reported in Cont and Fonseca (2002) and others on volatility in stock markets, it is true regardless of whether the errors are assumed to follow a white noise or autocorrelated process, and it holds for both subsamples, before and after the detected break. The rolling window estimation reveals the presence of some degree of time variation, but does not affect the general conclusion about the behaviour of the two series under examination.
This type of volatility index can also be seen as a measure of market fear, which therefore does not seem to be permanently affected by shocks in the case of the Russian stock market. Moreover, given the fact that the effects of shocks are not long-lived there does not seem to be any need of strong policy measures to push the series back to their original trends. Finally, our findings represent useful information for investors aiming to design appropriate portfolio diversification strategies.    ([931-1430]. In the case of autocorrelation, the only noticeable change takes place at the 40 th subsample corresponding to .
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