The data set is obtained from Datastream and consists of MSCI (Morgan Stanley Capital International) daily returns on the BRICS countries. BRICS is the acronym for an association of five major emerging national economies: Brazil, Russia, India, China and South Africa. The grouping was originally known as “BRIC” before the inclusion of South Africa in 2010.
These countries are all developing or newly industrialized countries and they are all characterized by large, fast-growing economies and a strong influence on regional and global business. The data covers the period from 19 July 1999 to 16 July 2015.
2.1. In-Sample Analysis
Let , denote the time series of the daily returns and assume, for simplicity, that the unconditional and conditional mean are zero.
In order to model the dynamic of
, a GARCH(1,1) model has been used; it has been established as an adequate model to obtain good performances in terms of fitting and forecasting. The canonical GARCH(1,1) model is:
where
is a sequence of
i.i.d. random variables with mean zero and unit variance. Conditions on
,
and
need to be imposed for the previous equation to be well defined. In particular,
and
are imposed to ensure that the conditional variance
is positive. Moreover,
ensures that the process is stationary. For a GARCH(1,1) process, the unconditional variance is defined as
.
The parameters of model (
1) are estimated by using the Quasi Maximum Likelihood Estimation in which the likelihood corresponding to the assumed distribution of
is maximized under the previous assumptions.
We are interested in testing whether the unconditional variance is constant over the available sample since a constant unconditional variance implies a not stable GARCH process governing conditional volatility. In order to identify possible structural breaks, a test for a single break in the presence of conditional heteroskedasticity has been employed. It is based on a statistic proposed in
Sansó et al. (
2004) which takes into account both the fourth order moment of the process and persistence in the variance. The test is based on the following statistic:
where
for
is the cumulative sum of squares of
and
and
is a consistent estimator of
, the long-run fourth order moment of
.
Under quite general conditions, in
Sansó et al. (
2004), it has shown that:
where
is a Brownian bridge and
is a standard Brownian motion. Finite-sample critical values for the test can be determined by simulation.
The
statistic is a generalization of the IT statistic proposed in
Inclan and Tiao (
1994), generally used to test the constancy of the unconditional variance of a time series. In particular,
makes adjustments to the IT statistic to allow
to obey a wide class of dependent processes, including GARCH processes, under the null.
In order to obtain a consistent estimator of the long-run fourth order moment of
, which is also the long-run variance of the zero mean random variable
, a non-parametric approach based on the Bartlett kernel (see also
Rapach and Strauss 2008) has been used used. In particular, it is:
where
and
. This estimator depends on the bandwidth
m which can be selected using the procedure in
Newey and West (
1994).
In order to extend the single break point method to multiple ones, a binary segmentation algorithm has been implemented. It is based on successive application of the test to sub-series obtained consecutively after a change-point is found. The procedure starts by applying the detection method to the whole series. If no change-point is found, the procedure is stopped; otherwise, the data are split into two segments and the detection method is applying to each of them. The procedure is repeated until no further change-points are detected. The choice of a binary segmentation algorithm is justified by its simplicity and efficiency; it is very fast and it could be implemented with a low computational cost. However, the procedure could produce spurious break points because of the presence of extreme observations which can be erroneously interpreted as being change points (see
Ross 2013). To partially solve this problem and to better identify the break points location, a pruning procedure, in the spirit of the ICCS algorithm (
Inclan and Tiao 1994), has been implemented. In the case that
m breaks have been detected at times
, …,
with
and
, the pruning procedure can be implemented as follows:
The detection method is applied to the segment for .
If no change-point is found in the segment the break at is not considered a change point. If a new change point is detected, it replaces the old one at .
The procedure is repeated until the number of change points does not change and the points found in each new step are ”close” to those on the previous step.
The main problem, when a detection method is applied with a searching algorithm, is that the use of the same critical value for any segments may distort the performance of the iterative procedure. To overcome this problem, the response surfaces’ methodology has been used (see
MacKinnon 1994 for details).
Table 2 reports the identified breaks dates.The proposed procedure identified five breaks for Brazil, Russia and India, ten breaks for China and only two breaks for South Africa. It is evident that the volatility may change in each country according not only to global, but also to specific financial, economic, social and political events.
In general, all of the BRICS countries, as the rest of the emerging markets, largely stood at the fringes of the global financial crisis that started in 2007 with the US Subprime market collapse and developed into a full-blown international banking crisis with the collapse of the investment bank Lehman Brothers on September 2008. This crisis affected global stock markets, where securities suffered large losses during 2008 and early 2009 with a reduction of volatility after this period. Therefore, in the period from 2007 to 2009, in all of the BRICS countries, there is evidence of the presence of structural break in volatility with different peaks.
Regarding individual BRICS countries, the identified breaks can be explained by looking at specific domestic events. According to
Morales and Gassie-Falzone (
2011), the first break identified for Brazil is in 2002 and it could be due to the Brazilian stock market crash and the pressures in the run up to the presidential election. The Indian and Russian markets share a common trend with a break point in April and March 2001, respectively, which is connecting with the dot-com bubble effects and to the energy crisis (
Morales and Gassie-Falzone 2011). Moreover, for Russia, a high magnitude of unsystematic risk was observed in 2001–2002 when the Russian stock market was hardly on the radar of international portfolio managers (
Nivorozhkin and Castagneto-Gissey 2016). Regarding India, the breaks in August 2011 is related to the ”August 2011 stock markets fall”, which is the sharp drop in stock prices due to fears of contagion of the European sovereign debt crisis. The effect of this crisis continued for the rest of the year; the break in March 2012 corresponds to the end of the effect of the crisis and a consequential reduction of volatility. The results for the Chinese market seem to be quite different; the large number of identified breaks and their locations could be also explained by regional volatility. As pointed out in
Zhou et al. (
2012), from 1996 to 2009, the Chinese stock market was not much influenced by other markets because it was not completely open to foreign investors. More precisely (see
Li (
2015) for a complete review of the market), in 1999, the government formally put forward the pilot plan of transferring state-owned stocks, but it did not work well, and the scheme was a shock to the stock market. However, due to the discrepancy between the market’s expectations and the implementation plan, the pilot project was led to some variations on the initial proposal from 2001 to 2003. In 2004, to solve the non-tradable shares issue some institutional reforms were made and, during 2006 and 2007, the Chinese market experienced the emergence of the stock market ‘bubble’. From 2007 to 2009, the market was affected by the global financial crisis which caused a peak in the volatility. In general, volatility spillovers among the Chinese and the other Asiatic markets, in particular Japanese and Indian markets, are more distinctive than those among the Chinese and Western markets are. This consideration could explain the presence of breaks in August 2011 and at the beginning of 2012 as previously pointed out for the Indian market. The last break in March 2015 could be explained by the Chinese stock market turbulence which began with the popping of the stock market bubble. For South Africa, the only identified structural breaks are those linked to the global financial crisis in the period 2007–2009.
Figure 1 shows the two-standard-deviation bands for each of the regimes defined by the structural breaks.
Table 3 presents the full-sample GARCH(1,1) unconditional variance, as well as its values for the sub-samples defined by the structural breaks identified by the binary segmentation algorithm with the
test, for all the considered series. As expected, a substantial change in the unconditional variance is quite evident. Note that, for Brazil, the unconditional variance of subsample 5 collapses to zero indicating that, in this period, the model is an IGARCH(1,1) without trend (
), a model with the so-called ”persistent variance” property in which the current information remains important for the forecasts of the conditional variances for all horizons. For the other periods, the unconditional variance varies from 1.57, in the subsample 6 to 34.28 in subsample 4. For Russia, the unconditional variance of the GARCH(1,1) full sample model is equal to 5.10, whereas, in the subsamples, it varies from 1.99 to 58.93, a value more than 10 times larger than that observed in the full sample. In the case of India, as for Brazil, in the subsample 5, the estimated model is an IGARH(1,1) without a trend as suggested by the zero value of the unconditional variance. A significant variability among the identified subsamples is still evident. For China, the unconditional variance of the GARCH(1,1) full sample model is equal to 3.44 and, again, significant differences are observable in the eleven identified subsamples. The same feature is also noticeable for South Africa in which the unconditional variance varies from 0.99 to 3.75.
These features confirm the relevance of variance breaks in the analysis of MSCI daily returns for all five of the BRICS countries.