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CHAPTER – III
STOCK MARKET VOLATILITY: THE MEASUREMENT
3.1. Introduction
The development of the theory of portfolio selection by Markowitz (1952) is the
foundation to the various practices in today’s financial markets. He translated the
basic idea of economics [“in order to obtain something we have to forego something
else”] into finance theory, which implies that there exists a risk-return tradeoff. In
simple words, if an investor wants higher return on a project, he has to accept a higher
degree of risk as well. As risk is measurable with respect to some benchmark, the
investor defines his utility function according to his attitude towards risk.
A measure that is immediately brought in while doing so is the dispersion of various
outcomes. Thus, the notion of risk is related to variance. It is well known that
variance is a statistic that can be determined from the past observations (returns). In
almost all financial models, variance is used as proxy for variability. We may
measure the variability based on past prices that conform to the present variability as
closely as possible. Basically, this has been conceptualized as ‘VOLATILITY’ in
financial market. Thus volatility, as the concept, may be treated as synonymous with
variability in general or variance in particular.
There have been a lot of empirical studies to test volatility in the stock markets
globally. Research has proved that stock markets have become more volatile in the
recent times due to the emergence of “New Economy” stocks, which are valued
highly as compared to their “Old Economy” counterparts on the expectations of
giving very high returns in future. Thus this high expectation has brought about wide
fluctuations in the prices making the markets turbulent.
Volatility is fluctuation, sometimes a significant fluctuation of something related to
risk. There is risk involved in investing and fluctuation is a part of it. Significant
volatility, something many investors have experienced recently, may have both
negative as well as positive results on the investment portfolios. As investors’
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fundamental assumptions about stocks change, stock prices can move quickly,
especially in today’s wired marketplace. Much of today’s volatility is simply a result
of the marketplace reacting to fundamental overvaluations.
In short, volatility in and of itself is relatively benign. It is the consequence of the
volatility that matters. If an investor has time to ride out the market’s ups and downs,
volatility may be of little consequence. The shorter the time frame, however, the
more harmful volatility can be.
The factors that affect volatility may be categorized as those affecting the long-term
and those affecting the short-term volatility. Several economic factors may cause
slow changes in stock market volatility, where the changes become noticeable over
many months or years. These are known as the long-term factors, such as corporate
leverage, personal leverage and the condition of the economy.
Several bursts in stock volatility in some markets around the world have spurred the
interest in stock volatility during the past few years. The boom and subsequent crash
in the Indian stock market in 1992 and 2001, the US stock market crashes of October
1987 and October 1989 the Mexican currency crisis in 1994, Asian currency crisis in
1997, the Russian crisis of 1998, the 1999 Brazilian crisis, the 2001 Argentinean
crisis, the 2002 Turkish crisis, the subprime financial crisis in 2008 and the European
Debt crisis in 2009 are prominent examples. These bursts of volatility are hard to
relate to longer-term phenomena such as recessions or leverage. Instead, most people
have tried to relate them to the structure of securities trading. Hence, these are
categorized as the factors that may cause short-term volatility, a few of which are
trading volume, trading halts (circuit breakers and circuit filters), computerized
trading, noise trading, international linkages, market makers, takeovers, supply of
equities, the press and other factors.
Volatility per se is not unnatural or unwanted. However, excessive volatility caused
by irrational or speculative behavior of the traders and investors, trading mechanism
imperfections and lack of information transparency is not desirable. If stock market
volatility increases, it may have important consequences for investors and policy-
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makers. Investors may equate higher volatility with greater risk and may alter their
investment decisions due to increased volatility. Policy-makers may be concerned
that stock market volatility would spill over into the real economy and harm economic
performance. Alternatively, policy-makers may feel that increased stock volatility
threatens the viability of financial institutions and the smooth functioning of financial
markets.
Stock return volatility hinders economic performance through consumer spending
(Garner 1988), and may also affect business investment spending (Gretler and
Hubbard 1989), where the investors may perceive a rise in stock market volatility as
an increase in the risk in equity investments. Hence, investors may shift their funds to
less risky assets. This move could result in rise in cost of capital of firms (Arestis et
al 2001). According to Bekaert (1995), in segmented capital markets, a country’s
volatility is a critical input in the cost of capital. Volatility may also be used as a
decision making criterion, where one would invest in those assets that yield the
highest return per unit of risk (Wessels 2006).
Further, extreme stock return volatility could disrupt the smooth functioning of the
financial system and lead to structural or regulatory changes. Systems that work well
with normal return volatility may be unable to cope with extreme price changes.
Changes in market rules or regulations may be necessary to increase the resiliency of
the market in the face of greater volatility.
However, increase in volatility per se cannot be criticized. Increased volatility may
simply reflect fundamental economic factors or information and expectations about
them. In fact, the more quickly and accurately prices reflect new information; the
more efficient would be pricing of securities and thereby the allocation of resources.
A market in which prices always “fully reflect” available information is called
“efficient” where share prices fluctuate randomly about their “intrinsic” values.
The stock market in India has had its fair share of crises engendered by excessive
speculation resulting in excessive volatility. Undoubtedly, the enthusiasm of
investors in the early 1990’s to some extent has been replaced by a growing concern
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about the excessive volatility of the Indian stock market in recent years. The
widespread concern of the exchange management, brokers and investors alike has
underlined the importance of being able to measure and predict stock market
volatility. Only then can effective monitoring mechanisms be put in place, which
would help in avoidance of such episodes in future.
The industrial development of a nation largely depends on the allocative efficiencies
of the stock market, which acts as a barometer of a country’s health. Indian stock
market has a long history but as an agent of development it has only a few glorious
occasions to its credit to reckon. A number of committees were appointed to review
functioning of stock market that submitted innumerable suggestions to minimize
speculative activities thereby fluctuation of share prices. Ironically all these efforts by
and large ultimately failed to control a rational share price movement that is crippling
functioning of the market for long.
3.2. Review of Literature
Volatility is the degree to which asset prices tend to fluctuate. It is the variability or
randomness of asset prices, i.e. the dispersion of returns of an asset from its mean
return. Stock market volatility measures the size and frequency of fluctuations in a
broad stock market price index (Madhusudan Karmakar 2006) (Mishra et al 2010).
It also has a significant forecasting power for real GDP growth (Campbell et al 2001).
The stock market returns follow a deterministic path implying that stock returns
oscillate between excess and under return, passing through the mean stock return
(Seth and Saloni 2005).
According to Poon and Granger (2003) volatility has a very wide sphere of influence
including investment, security valuation, risk management and policy making,
emphasizing on the importance of volatility forecasting. Research has also shown
that capital market liberalization policies too, are likely to affect volatility.
Rao and Tripathy (2008) found that the market would react very sharply to economic,
political and policy issues.
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Batra (2004) examined the economic significance of changes in the pattern of stock
market volatility in India during the period of financial reforms.
According to Schwert (1989 a, b) time variation in market volatility can often be
explained by macroeconomic and micro structural factors.
Volatility in national markets is determined by world factors and part determined by
local market effects, assuming that the national markets are globally linked (Pratip
Kar et al 2000).
Using time varying market integration parameter, Bekaert and Harvey (1995) showed
that world factors have an increased influence on volatility with increased market
integration.
Mandelbrot (1963) observed volatility clustering and leptokurtosis as common
observations in financial time series. Moreover, a highly significant large JB statistic
confirms that the return series is not normally distributed.
Harvey (1995) points out that in many emerging markets, time series return data do
not follow normal distribution.
But according to Obaidullah (1991), time series return data in Indian Stock Markets
are normally distributed.
Bollerslev (1986) introduced model Generalized Autoregressive Conditional
Heteroskedasticity (GARCH). The GARCH model allows the conditional variance to
be dependent upon its own lags. He estimated GARCH(1,1) model on the quarterly
data set of U.S. inflation for the period 1948-II to 1983-IV. The results suggested the
presence of GARCH effect in the inflation data. It indicated that the volatility of
inflation was persistent.
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To compute the conditional variance of sample return series, GARCH (1,1) model has
been applied by Susan Thomas (1995), Madhusudan Karmakar (2003) and Puja Padhi
(2005).
Brailsford and Faff (1996) find that the GARCH models are superior to other models
to forecast Australian monthly stock index volatility.
Brooks (1998) found that the FARCH models outperform other techniques while
modeling volatility.
According to Jean and Peters (2001), GJR and APARCH give better forecasts than
symmetric GARCH. But increased performance of the forecasts could not be clearly
observed when using non-normal distributions.
Jayanth R Varma (1999) tested the relevance of GARCH-GED (Generalized Auto-
Regressive Conditional Heteroskedasticity with Generalized Error Distribution
residuals) model and the EWMA (Exponentially Weighted Moving Average) model
and evaluated their performance in the VaR framework in Indian stock market.
David X Li (1999) presented a new approach to calculating Value at Risk (VaR) using
skewness, kurtosis and the standard deviation explicitly. The new approach was
found to capture the extreme tail much better than the standard VaR calculation
method used in Riskmetrics.
When a time-varying risk premium is incorporated into the analysis, the view that
“historical prices have been consistently too volatile and their returns too high” cannot
be supported (Angela Black and Patricia Fraser 2003).
Harvinder Kaur (2004) studied the extent and pattern of stock return volatility of the
Indian Stock Market and concluded that the most volatility found during the months
of April followed by March and February could be due to the presentation of the
Union Budget.
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The GARCH (1, 1) model has been found to be the overall superior model based on
most of the symmetric loss functions, though ARCH has been found to be better than
the other models for investors who are more concerned about under predictions than
over predictions (Deb et al 2003).
According to Ravi Madapati (2005), the conditional heteroskedastic models fit the
Indian data quite satisfactorily, providing good forecasts of volatility.
Madhusudan Karmakar (2005) reported the presence of leverage effect in the Indian
stock market, but which model can best capture the leverage effect has been left for
further research.
Bhaskkar Sinha (2006) found the EGARCH for BSE Sensex and GJR-GARCH for
NSE Nifty best for modeling volatility clustering and persistence of shock.
Mahajan and Singh (2008) examined the empirical relationship between return,
volume and volatility in Indian stock market using GARCH (1,1) and EGARCH (1,1)
estimated for Nifty index.
Rao, Kanagaraj and Tripathy (2008) found that stock future derivatives were not
responsible for increase or decrease in spot market volatility and conclude that there
could be other market factors that have helped the increase in Nifty volatility.
Atanu Das et al (2009) compared the predictive power of Stochastic Volatility Model
(SVM) and Kalman Filter (KF) based approach vis-à-vis EWMA and GARCH based
approaches with data from Indian security indices.
Somsanker Sen (2010) has explored the movements of Volatility on S&P CNX
NIFTY.
Vipul Singh and Ahmad (2011) compared several GARCH family models in order to
model and forecast the conditional variance of S&P CNX Nifty Index with special
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focus on the fitting of first order GARCH models to Nifty financial daily return series
and explaining financial market risk.
Rajan (2011) discussed different mathematical models to model the volatility of the
stock market in general and apply it to Indian context to pin down one that captures
the irregular behavior of the Indian stock market.
Asad Ahmad and Rana (2012) attempted to determine the forecasting performance of
symmetric and asymmetric volatility forecasting models in terms of error estimators
using the intra-day of highly liquid stocks in the Indian stock market. Superiority of
forecasting performance of asymmetric GARCH model over symmetric model has
been established.
The issue of changes in volatility of stock returns in emerging markets, in particular,
has received considerable attention in recent years due to various reasons. The market
participants need this measure for reasons like portfolio management, pricing of
options, predicting asset return series, forecasting confidence intervals, financial risk
management, etc. The issue of volatility and risk has also become increasingly
important in recent times to financial practitioners, regulators and researchers. In this
context, the present study is carried out to understand the volatility behavior of the
Indian stock markets.
3.3. Objectives of the Chapter
The objective of the chapter is to understand the volatility behavior and to measure
the volatility levels of the Indian stock market, through the CNX Nifty. However, the
following are set as the sub objectives for the chapter:
1. To compute the historical volatility levels of CNX Nifty using the classical,
range based and drift independent volatility estimators
2. To subject the CNX Nifty prices to autocorrelation tests
3. To estimate the conditional variance of sample return series through GARCH
model
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3.4. Database and Methodology
The National Stock Exchange captures 83 per cent transactions of the cash segment
and 79 per cent of the derivatives segment, thus the prominent index of NSE, the
CNX Nifty is treated as the principal stock index of the country. Hence for analysis,
CNX Nifty was selected.
The classical estimator is calculated using
a. The daily close prices of Nifty for a period of 20 years from January 4, 1993
to December 31, 2012 totaling 4922 trading days.
b. The daily open prices of Nifty for a period of 17 years from November 3, 1995
to December 31, 2012 totaling 4283 trading days.
The range-based and drift-independent volatility estimators are calculated using the
daily open, high, low and close prices of Nifty for a period of 17 years from
November 3, 1995 to December 31, 2012 totaling 4283 trading days.
The daily open, high, low and close prices of Nifty are obtained from the official
website of NSE ignoring the days when there was no trading.
The price changes are calculated from the last day the market was open.
The autocorrelation tests, AR(1) and GARCH(1,1) models were all calculated by
using the statistical package ‘E Views 6’.
3.5. Volatility and its Measure
A stock’s price moves in accordance with investors’ continuously changing and
contradictory expectations about the company’s future financial performance. There
is a perpetual uncertainty associated with every stock. But it is considered normal
behavior, and is reflected in the share price movement. However, the extent of this
movement is not the same for all stocks. Some stocks tend to move more, and this
difference has implications on their investment potential. In order to gauge how such
behavior impacts share prices, its movement – specifically, the volatility intrinsic to it
– needs to be measured.
There are two basic ways to measure volatility: historical and implied. Historical
volatility is calculated by using the standard deviation of underlying asset price
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changes from close to close (or open to open) of trading for the past few months.
Implied volatility is a computed value that measures an option’s volatility, rather than
the underlying asset. A fair value of an option may be calculated by entering the
historical volatility of the underlying asset into an option-pricing model. The
computed fair value may differ from the actual market price of the option. In brief,
historical volatility gauges price movement in terms of past performance and implied
volatility approximates how much the market place thinks prices will move. An
important advantage of estimating the implied volatility is that it requires no historical
data collection. The volatility is readily available with merely one day’s price
information.
3.5.1. Historical Volatility
The simplest way to estimate volatility is to use the basic definition that volatility is
the standard deviation of logarithmic asset returns. Given daily or weekly asset
prices, it is a simple matter to compute the corresponding daily or weekly return and
compute their standard deviation. The mean and the standard deviation of a set of
data are usually reported together. In a certain sense, the standard deviation is a
"natural" measure of statistical dispersion if the center of the data is measured about
the mean. This is because the standard deviation from the mean is smaller than from
any other point. Volatility is traditionally estimated using closing price data, hence
this method is also called the classical estimator or the optimal (maximum likelihood)
estimator which is obtained from random walk model.
The current value of the Standard Deviation may be used to estimate the importance
of a move or set expectations. This assumes that price changes are normally
distributed with a classic bell curve. Even though price changes for securities are not
always normally distributed, chartists may still use normal distribution guidelines to
gauge the significance of a price movement. In a normal distribution, 68% of the
observations fall within one standard deviation. 95% of the observations fall within
two standard deviations. 99.7% of the observations fall within three standard
deviations. Using these guidelines, traders may estimate the significance of a price
movement. A move greater than one standard deviation would show above average
strength or weakness, depending on the direction of the move.
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Figure 3.1: Normal Distribution guidelines
3.5.1a. Return
Daily stock returns were calculated by the logarithmic difference in the price, using
the following formula where 𝒓𝒕 and 𝑰𝒕 indicate return and index values respectively at
time t:
𝒓𝒕 = 𝒍𝒏 (𝑰𝒕
𝑰𝒕−𝟏) 𝑿 𝟏𝟎𝟎 (𝒆𝒒𝒏 𝟑. 𝟏)
Arithmetic and logarithmic returns are not equal, but are approximately equal for
small returns. The difference between them is large only when percent changes are
high. For example, an arithmetic return of +50% is equivalent to a logarithmic return
of 40.55%, while an arithmetic return of -50% is equivalent to a logarithmic return of
-69.31%.
Logarithmic returns are often used by academics in their research. The main
advantage is that the continuously compounded return is symmetric, while the
arithmetic return is not; positive and negative percent arithmetic returns are not equal.
This means that an investment of $100 that yields an arithmetic return of 50%
followed by an arithmetic return of -50% will result in $75, while an investment of
$100 that yields a logarithmic return of 50% followed by a logarithmic return of -50%
it will remain $100. Foreseeing the reasons of consistency of log returns in utility and
in uniformity the study also used log returns for the analysis.
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3.5.1b. Average Annual Returns
The average annual returns are calculated and presented in Table 3.1, where it may be
observed that the returns have been negative almost all the years in the decade of
1990s but during the next decade the average returns have recorded positive in almost
all the years.
The important points of consideration are:
a. The market showed negative returns during the years 1995 to 2000 which may
be attributed to the boom and subsequent crash of the Indian stock market in
1999-2000
b. Though the market could recover during the year 1997, it could not maintain it
for the next year due to the Asian Financial Crisis which started in 1997 and
impacted global stock markets to crash
c. During the year 1999 the market plunged to a very high level of return due to
the effect of boom but subsequently dropped when the Indian stock markets
crashed due to the Dot-com bubble in the year 2000 and was impacted by the
economic effects of September 11 attacks in the year 2001
d. Since the year 2002 the markets have shown considerably positive returns,
though fluctuating
e. The sudden drop of the returns to the maximum level during the year 2008
could be attributed to the subprime financial crisis that affected countries
across the globe
f. The market could withstand this crisis and revert back at the earliest giving the
maximum returns during the year 2009
g. Stock markets around the world plummeted during late July and early August
2011, and were volatile for the rest of the year resulting in a negative return
h. The market is found to revert back at the earliest from the various affects it
faced and resulted in positive average returns
i. It could be concluded that the market is improving its efficiency to withstand
and recover from any sort of crises
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Table 3.1: Annual Average Returns from Nifty close values
Year Return (%) No. of Obs.
1993 0.147616 213
1994 0.054668 230
1995 -0.111598 236
1996 -0.004190 249
1997 0.074904 244
1998 -0.079768 250
1999 0.202898 254
2000 -0.063368 250
2001 -0.071191 248
2002 0.012754 251
2003 0.213290 254
2004 0.039949 254
2005 0.123498 251
2006 0.134108 250
2007 0.175397 249
2008 -0.296624 246
2009 0.232082 243
2010 0.065503 252
2011 -0.114414 247
2012 0.097407 251
1993-2012 0.041620 4922
Source: Compiled Data
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3.5.1c. Standard Deviation
It is the simplest and most common type of calculation that benefits from only using
reliable prices from closing auctions. It may also be calculated using only the
opening auction values for each transacting day. This classical estimator is calculated
using the following formula, where 𝒓𝒕 denotes the logarithmic return (either open or
close) on day t and E denotes the average return for the period.
= √𝟏
𝑻 − 𝟏∑(𝒓𝒕 − 𝑬)𝟐
𝑻
𝒕=𝟏
(𝒆𝒒𝒏 𝟑. 𝟐)
Following are the advantages of the Classical Estimator:
a. It has well understood sampling properties
b. It is very simple to use
c. It is free from obvious sources of error and bias on the part of market activity
d. It is easy to convert to a form involving typical daily moves
Following are the disadvantages of the Classical Estimator:
a. Inadequate use of readily available information in its estimation
b. It converges information very slowly
3.5.1d. Volatility Measure: Classical Estimators
The classical estimators, open-open standard deviation and close-close standard
deviation, are calculated using
- The daily close prices of Nifty for a period of 20 years from January 4, 1993
to December 31, 2012 totaling 4922 trading days.
- The daily open prices of Nifty for a period of 17 years from November 3, 1995
to December 31, 2012 totaling 4283 trading days.
The classical estimators, using the standard deviation method as stated above, are
obtained and the changes in standard deviation over the study period are tabulated in
Table 3.2. A closer examination of the standard deviation in closing prices of Nifty
and the standard deviation in opening prices of Nifty reveals that two phases i.e.
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1999-2000 and 2008-2009 get highlighted with standard deviation being highest,
representing a very high volatility in the index prices, both open as well as close.
Table 3.2: Classical Estimators
Year C-C St. dev O-O St. dev No. of Obs.
1995 1.242143 1.481455 236
1996 1.527411 1.566099 249
1997 1.798488 1.843867 244
1998 1.777206 1.762365 250
1999 1.837389 2.035663 254
2000 2.001889 2.032186 250
2001 1.630300 1.643363 248
2002 1.060891 1.067565 251
2003 1.232220 1.251100 254
2004 1.762618 1.778010 254
2005 1.113604 1.116737 251
2006 1.650139 1.673507 250
2007 1.601395 1.613973 249
2008 2.808280 2.789615 246
2009 2.142748 2.125210 243
2010 1.024103 1.008786 252
2011 1.321311 1.445137 247
2012 0.954654 1.000615 251
1993-2012 1.640411 1.692721 4922
Source: Compiled Data
These two phases may be attributed to
(a) the boom and subsequent crash in the Indian stock market during 1999-2000, and
(b) to the sub-prime financial crisis in 2008
which affected the Indian market during the period selected for the study.
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3.5.2. Advanced Volatility Measures
Close-to-close volatility is usually used as it has the benefit of using the closing
auction prices only. Should other prices be used, then they could be vulnerable to
manipulation or a “fat fingered” trade. However, a large number of samples need to
be used to get a good estimate of historical volatility, and using a large number of
closing values may obscure short-term changes in volatility. There are however,
different methods of calculating volatility using some or all of the open, high, low and
close values.
The advanced volatility models may further be categorized as
i. Range based volatility models
ii. Drift Independent volatility models
3.5.3. Range Based Volatility Models
Instead of depending on only one representative value (snapshot price) of a day’s
trading, the range based volatility models consider embodying more information in
the calculation methodology; hence include few or all of the intraday information
available. Thus these models use the Open price, High price, Low price and the Close
price for estimating the volatility of the returns from the select scrip or index.
3.5.3a. Parkinson’s High-Low Volatility Measure
Building on the work of Feller (1951), Parkinson (1980) suggested a range estimator,
which was a counterpart to the traditional one. The Parkinson number, or High Low
Range Volatility, developed by the physicist, Michael Parkinson, in 1980 aims to
estimate the Volatility of returns for a random walk using the high and low prices
over the entire day instead of just a ‘snapshot’ price at the end of the day, embodying
more information. It was claimed to attain the same accuracy with about 80% less
data and with a relative efficiency of 5.2 when the traditional estimator was taken as
the benchmark, thus it was said to be “far superior to” the traditional estimator. It
attempts to estimate the volatility of returns for an asset following a diffusion process
(geometric random walk) by using only the high and low of the period. Essentially,
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the simple formula, gives the distribution of the maxima and the minima of the asset
returns. Parkinson's number P for an asset is given by:
𝑝 = √1
4 𝑇 𝑙𝑛(2)∑(𝑙𝑛(
𝑇
𝑡=1
𝐻𝑡 𝐿𝑡⁄ ))2 (𝑒𝑞𝑛 3.3)
An important use of the Parkinson number is the assessment of the distribution prices
during the day as well as a better understanding of the market dynamics. Comparing
the Parkinson number and periodically sampled volatility helps understand the
tendency towards mean reversion in the market as well as the distribution of stop-
losses. This estimator has an efficiency of 5.2 times the classic close-to-close
estimator (standard deviation).
Following are the advantages of the Parkinson’s Estimator:
a. Using daily range seems sensible and provides completely separate
information from using time based sampling such as closing prices
Following are the disadvantages of the Parkinson’s Estimator:
a. It is appropriate only for measuring volatility of a Geometric Brownian
Motion process
b. It particularly cannot handle trends and jumps
c. It systematically underestimates volatility
3.5.3b. Garman-Klass Open Close Volatility Measure
The relative efficiency of an estimator is defined as the ratio of variance of the
benchmark estimator to the variance of the estimator under consideration. Garman
and Klass (1980) suggested several estimators and tested their relative efficiencies in
their paper. The preferred estimator was constructed by normalized high, low and
closing prices relative to the open prices. This estimator combined the traditional
estimator and Parkinson’s estimator, thus incorporating more intraday information.
𝑔𝑘 = √1
𝑇∑ [
1
2(𝑙𝑛
𝐻𝑡
𝐿𝑡)
2
− (2 𝑙𝑛2 − 1) (𝑙𝑛𝐶𝑡
𝑂𝑡)
2
]
𝑇
𝑡=1
(𝑒𝑞𝑛 3.4)
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This estimator is up to 7.4 times as efficient as the close-to-close estimator (the exact
efficiency improvement is dependent on the sample size) but is also biased due to the
discrete sampling leading to a low estimate of the range. However, this estimator is
more biased than the Parkinson estimator.
Following are the advantages of the Garman-Klass Estimator:
a. It is upto 7.4 times more efficient than the close to close estimator
b. It makes the best use of commonly available price information
Following are the disadvantages of the Garman-Klass Estimator:
a. It is even more biased than Parkinson’s estimator
3.5.3c. Yang Zhang extension to Garman-Klass Volatility Measure
Yang and Zhang offered an extension to the Garman and Klass historical
volatility estimator. The equation was modified to include the logarithm of the open
price divided by the preceding close price. As a result, this modification allows the
volatility estimator to account for the opening jumps, but as the original function, it
assumes that the underlying follows a Brownian motion with zero drift (the historical
mean return should be equal to zero). The estimator tends to overestimate the
volatility when the drift is different from zero, however, for a zero drift motion, this
estimator has an efficiency of 8 times the classic close-to-close estimator (standard
deviation).
𝑦𝑧 = √1
𝑇∑ [(𝑙𝑛
𝑂𝑡
𝐶𝑡−1)
2
+ 1
2(𝑙𝑛
𝐻𝑡
𝐿𝑡)
2
− (2 𝑙𝑛2 − 1) (𝑙𝑛𝐶𝑡
𝑂𝑡)
2
]
𝑇
𝑡=1
(𝑒𝑞𝑛 3.5)
Where
= volatility
T = total number of trading days
𝐶𝑡−1 = the closing price of previous day
𝐶𝑡 = the closing price
𝑂𝑡 = the opening price
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𝐻𝑡 = the high price
𝐿𝑡 = the low price
Ln = the natural log
Parkinson (1980), Garman and Klass (1980) made an assumption that the underlying
asset followed a continuous Brownian motion process, which was the shortcoming of
their estimator. The fact that prices of financial instruments are only observable at
discrete time intervals contradicts their assumption and creates possible sources of
bias; therefore the fidelity of the observed high and low prices becomes questionable.
In an empirical study of stock price variability, Beckers (1983) reinforced this idea
that Parkinson’s estimator would be biased downward by non-continuous prices,
because when sampling discretely, it is common for the observed low price to be
higher than the “true” lowest price and for a similar situation to apply to the observed
high price. Similar findings were made by Edwards (1988), studying the S&P 500
and Value Line cash indices, and Wiggins (1991) investigating individual stocks.
Wiggins (1992) stated that this bias was not particularly serious for an actively traded
instrument with small price increments, however. He took S&P 500 futures prices as
an example in his empirical analysis.
3.5.3d. Volatility Measure: Range Based Estimators
The range-based estimators, Parkinson’s volatility measure, Garman-Klass volatility
measure and Yang-Zhang’s correction to Garman-Klass volatility measure are
obtained and the results are tabulated below. An examination of the results obtained
in Table 3.3 indicates that the three range based volatility measures also depict the
same trend as that of the close to close volatility. Even these measures show a higher
volatility during the two said phases of
(a) the boom and subsequent crash in the Indian stock market during 1999-2000, and
(b) to the sub-prime financial crisis in 2008
which affected the Indian market during the period selected for the study.
115
Table 3.3: Range Based Estimators
Year Parkinson GK GK-YZ No. of Obs.
1995 0.977948 0.860752 0.934904 236
1996 1.147219 1.075719 1.219436 249
1997 1.308221 1.219637 1.447892 244
1998 1.515057 1.420693 1.438012 250
1999 1.580481 1.516513 1.685942 254
2000 2.016158 2.024442 2.036289 250
2001 1.569462 1.546917 1.552033 248
2002 1.01514 1.002154 1.006063 251
2003 1.199727 1.181123 1.191931 254
2004 1.63223 1.581951 1.599249 254
2005 1.084088 1.068069 1.074678 251
2006 1.610891 1.591949 1.596541 250
2007 1.514174 1.477489 1.482379 249
2008 2.599011 2.517717 2.524942 246
2009 1.84509 1.722613 1.72747 243
2010 0.933496 0.903525 0.934773 252
2011 1.103295 1.085559 1.330034 247
2012 1.032948 1.102186 1.219126 251
1993-2012 1.505712 1.46364 1.516674 4922
Source: Compiled Data
It may be observed that the Garman Klass model shows an overall lesser volatility
than Parkinson’s volatility measure. This is because Parkinson assumed continuous
trading and considers only the high and low values of the daily prices of the index
where as the Garman Klass model considers the open and close values also in addition
to the others. Hence the overall volatility for a year according to Garman Klass model
is lesser when compared to the volatility shown by the Parkinson’s model.
116
The Yang Zhang’s correction to the Garman Klass model of volatility estimation,
considers the opening jumps as well, along with the open, high, low and close values
of the index prices. Despite that, as the model is not drift independent, the volatility
resulted using Yang Zhang’s correction is almost similar to the volatility resulted by
the Parkinson’s and the Garman-Klass’ models.
3.5.4. Drift Independent Volatility Models
These models are also range based models, but are different in terms of the treatment
of the drift in developing the model. All of the range based volatility models assume
the drift (or the average return) to be zero for the series under consideration for
volatility measurement. Volatility of securities possessing or depicting drift, or non-
zero mean, was not found appropriately measured if the measurement tool was any of
the classical models or any of the range based models. Such securities require a more
sophisticated measure of volatility.
3.5.4a. Rogers-Satchell Volatility Measure
The Rogers-Satchell function [Rogers and Satchell (1991) and Rogers, Satchell and
Yoon (1994)] is a volatility estimator that properly measures the volatility for
securities with non-zero mean. As a result, it provides better volatility estimation
when the underlying is trending.
𝑟𝑠 = √ 1
𝑇∑(𝑙𝑛 (𝐻𝑡 𝐶𝑡) 𝑙𝑛(𝐻𝑡 𝑂𝑡) + ⁄⁄
𝑇
𝑡=1
𝑙𝑛(𝐿𝑡 𝐶𝑡) 𝑙𝑛(𝐿𝑡 𝑂𝑡))⁄⁄ (𝑒𝑞𝑛 3.6)
However, this estimator does not account for opening jumps in price (Gaps), hence
underestimates the volatility. The function uses the open, close, high, and low price
series in its calculation and it has only one parameter, which is ‘the period to use’ to
estimate the volatility. This estimator has an efficiency of 8 times the classic close-to-
close estimator (standard deviation).
Following are the advantages of the Roger-Satchell Estimator:
a. It allows for the presence of trends
117
Following are the disadvantages of the Roger-Satchell Estimator:
a. It still cannot deal with opening jumps
3.5.4b. Yang Zhang Drift Independent Volatility Measure
In 2000 Yang and Zhang created a volatility measure that handles both opening jumps
and drift. It is the sum of the overnight volatility (close-to-open volatility) and a
weighted average of the Rogers-Satchell volatility and the open-to-close volatility.
The assumption of continuous prices does mean the measure tends to slightly under
estimate the volatility.
𝑦𝑧 = √𝑜2 + 𝑘𝑐
2 + (1 − 𝑘)𝑟𝑠2 (𝑒𝑞𝑛 3.7)
Where
𝑜2 =
1
𝑇 − 1 ∑ [ln (
𝑂𝑡
𝐶𝑡−1) − ln (
𝑂𝑡
𝐶𝑡−1)
]
2𝑇
𝑡=1
𝑐2 =
1
𝑇 − 1 ∑ [ln (
𝐶𝑡
𝑂𝑡) − ln (
𝐶𝑡
𝑂𝑡)
]
2𝑇
𝑡=1
𝑘 = 0.34
1.34 + 𝑁 + 1𝑁 − 1
𝑟𝑠2 = Rogers-Satchell’s Variance
In some simulations it may have efficiency 14 times greater than the close-to-close
volatility. But this is highly dependent on the proportion of volatility caused by
opening jumps. If these jumps dominate, the estimator performs no better than the
close-to-close estimator.
Following are the advantages of the Yang-Zhang Estimator:
a. It is specifically designed to have minimum estimation error
b. It can handle both drift and jumps
c. It is most efficient in its use of available data
Following are the disadvantages of the Yang-Zhang Estimator:
a. The performance degrades to that of close to close estimator when process is
dominated by jumps
118
3.5.4c. Volatility Measure: Drift Independent Estimators
The Drift Independent Estimators, Rogers-Satchell’s volatility measure and Yang
Zhang’s volatility measure are obtained and the results are tabulated below.
Table 3.4: Drift Independent Estimators
Year R-S Y-Z No. of Obs.
1995 0.860370 0.796883 236
1996 1.063661 0.795669 249
1997 1.180066 0.795671 244
1998 1.435104 0.795673 250
1999 1.497977 0.795656 254
2000 2.098462 0.795662 250
2001 1.571860 0.795660 248
2002 1.003531 0.795650 251
2003 1.161203 0.795653 254
2004 1.558296 0.795645 254
2005 1.047189 0.795654 251
2006 1.596040 0.795656 250
2007 1.505478 0.795671 249
2008 2.603048 0.795747 246
2009 1.659962 0.795664 243
2010 0.906830 0.795663 252
2011 1.073396 0.795707 247
2012 1.309246 0.795652 251
1993-2012 1.481189 0.795459 4922
Source: Compiled Data
The results in Table 3.4 reveal that Roger Satchell’s measure is not different from the
other measures, and shows the higher volatility during the two said phases. Though
the Roger Satchell’s measure handles drift, it does not handle opening jumps. Hence
the results from this model are quite similar to those of the other models.
119
It becomes imperative to take a look at the results from the Yang Zhang model, which
handles both drift as well as the opening jumps. This was clearly observed from the
volatility values depicted by this method through all the years of the study period.
The overall volatility has been similar and consistent through each of the years of the
study period. Even for the whole duration taken at a time, there was not much
difference in the volatility displayed by this method.
From the above observations it is concluded that Yang Zhang’s measure of volatility
is better and consistent in calculating the overall risk of the index under consideration
as it considers overnight jumps as well as handles drift in the return series. This
measure also shows a maximum efficiency of 14 times the classical close to close
volatility measure.
Table 3.5: Summary of Volatility Estimates
Estimate Prices
Taken
Handle
Drift?
Handle overnight
jumps?
Efficiency
(max)
Close to close C No No 1
Parkinson HL No No 5.2
Garman-Klass OHLC No No 7.4
Rogers-Satchell OHLC Yes No 8
Garman-Klass
Yang-Zhang ext. OHLC No Yes 8
Yang-Zhang OHLC Yes Yes 14
Source: http://www.todaysgroep.nl/media/236846/measuring_historic_volatility.pdf
All these estimators are built on a strict assumption that an asset price follows a
Geometric Brownian Motion which is certainly not the case in real markets.
120
3.6. Skewness
The first thing usually noticed about a distribution’s shape is whether it has one mode
(peak) or more than one. If it is uni-modal (has just one peak), like most data sets, the
next thing noticed is whether it is symmetric or skewed to one side. If the bulk of the
data is at the left and the right tail is longer, we say that the distribution is skewed
right or positively skewed; if the peak is toward the right and the left tail is longer, we
say that the distribution is skewed left or negatively skewed.
Figure 3.2: Different forms of Skewness
It may be noted that the mean and standard deviation have the same units as the
original data, and the variance has the square of those units. However, the skewness
has no units; it is a pure number, like a z-score.
𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = 𝑛
(𝑛 − 1)(𝑛 − 2) ∑ (
𝑥𝑡 − ��
𝑠)
3
(𝑒𝑞𝑛 3.8)
3.7. Kurtosis
The height and sharpness of the peak relative to the rest of the data are measured by a
number called kurtosis. Higher values indicate a higher, sharper peak; lower values
indicate a lower, less distinct peak. This occurs because, as Wikipedia’s article on
kurtosis explains, higher kurtosis means more of the variability is due to a few
extreme differences from the mean, rather than a lot of modest differences from the
mean.
121
𝐾𝑢𝑟𝑡𝑜𝑠𝑖𝑠 = {𝑛(𝑛 + 1)
(𝑛 − 1)(𝑛 − 2)(𝑛 − 3)∑ (
𝑥𝑡 − ��
𝑠)
4
} − 3 (𝑛 − 1)2
(𝑛 − 2)(𝑛 − 3) (𝑒𝑞𝑛 3.9)
In the words of Kevin P. Balanda and H.L. MacGillivray “Increasing kurtosis is
associated with the movement of probability mass from the shoulders of a distribution
into its center and tails.” The mean and standard deviation have the same units as the
original data, and the variance has the square of those units. However, the kurtosis
has no units, but is a pure number, like a z-score.
Figure 3.3: Different forms of Kurtosis
The reference standard is a normal distribution, which has a kurtosis of 3. In token of
this, often the excess kurtosis is presented as simply kurtosis-3.
A normal distribution has kurtosis exactly 3 (excess kurtosis exactly 0). Any
distribution with kurtosis ≈ 3 (excess ≈ 0) is called mesokurtic.
A distribution with kurtosis < 3 (excess kurtosis < 0) is called platykurtic.
Compared to a normal distribution, its central peak is lower and broader, and
its tails are shorter and thinner.
A distribution with kurtosis > 3 (excess kurtosis > 0) is called leptokurtic.
Compared to a normal distribution, its central peak is higher and sharper, and
its tails are longer and fatter.
122
From Table 3.6 it may be observed that skewness (or distribution shape) of the return
series of Nifty has been shifting between positive and negative for individual years
taken into consideration. But the overall period of 20 years showed a negative
skewness in the distribution of Nifty returns.
Table 3.6: Skewness and Kurtosis
Year Skewness Kurtosis No. of Obs.
1993 -0.33569 0.33082 213
1994 0.63399 1.96098 230
1995 -0.10597 0.75154 236
1996 0.70025 1.09674 249
1997 0.05845 7.56417 244
1998 -0.09430 1.61662 250
1999 0.04450 2.24681 254
2000 -0.10528 1.49105 250
2001 -0.46151 2.26191 248
2002 0.07770 1.45728 251
2003 -0.33654 0.47011 254
2004 -1.80181 14.39703 254
2005 -0.51667 0.59184 251
2006 -0.61989 2.73131 250
2007 -0.25819 1.55815 249
2008 -0.28344 1.68816 246
2009 1.50835 12.62100 243
2010 -0.27696 0.67026 252
2011 0.27044 0.05748 247
2012 0.07563 0.66164 251
1993-2012 -0.13085 5.99630 4922
Source: Compiled Data
It may also be observed from this table that Kurtosis was beyond 3 in the years 1997,
2004 and 2009 when individual years are taken into consideration. For the overall
123
period of 20 years, Kurtosis ended up at 5.9963 which is almost double the required
standard value of 3. This clearly proves that the Nifty return distribution is
Leptokurtic in nature. As this is one of the important characteristics of financial time
series data, it enables the applicability of ARCH / GARCH.
3.8. Jarque Bera Test
In statistics, the Jarque Bera test is a goodness-of-fit test of whether sample data have
the skewness and kurtosis matching a normal distribution. The test is named
after Carlos Jarque and Anil K. Bera. It is used to check hypothesis about the fact that
a given sample xS is a sample of normal random variable with unknown mean and
dispersion. As a rule, this test is applied before using methods of parametric statistics
which require distribution normality.
This test is based on the fact that skewness and kurtosis of normal distribution equal
zero. Therefore, the absolute value of these parameters could be a measure of
deviation of the distribution from normal. The test statistic JB is defined as
𝐽𝐵 = 𝑛
6 (𝑆2 +
1
4 (𝐾 − 3)2) (𝑒𝑞𝑛 3.10)
where n is the number of observations (or degrees of freedom in general); S is the
sample skewness, and K is the sample kurtosis.
The hypotheses are set as:
H0: The return series are normally distributed
H1: The return series are not normally distributed
If the data come from a normal distribution, the JB statistic asymptotically has a chi-
squared distribution with two degrees of freedom, so the statistic may be used
to test the hypothesis that the data are from a normal distribution. The null
hypothesis is a joint hypothesis of the skewness being zero and the excess
kurtosis being zero. Samples from a normal distribution have an expected skewness of
0 and an expected excess kurtosis of 0 (which is the same as a kurtosis of 3). As the
definition of JB shows, any deviation from this increases the JB statistic.
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3.8a. Empirical Investigation: Jarque Bera
The descriptive statistics of Log Returns of Nifty close values for the 20 year period
are given below.
Figure 3.4: Histogram of Log Returns of Daily Nifty Close Values
Table 3.7: Descriptive Statistics of Log Returns of
Nifty Close Values from 4 Jan 1993 to 31 Dec 2012
Mean 0.041620
Median 0.076938
Standard Deviation 1.640411
Kurtosis 5.996300
Skewness -0.130850
Minimum -13.05386
Maximum 16.33432
Count 4922
Jarque-Bera 7369.966
Probability 0.000000
Source: Compiled Data
According to Table 3.7, the basic statistics indicate that the mean return (0.041620) is
closer to zero, when relatively compared to the standard deviation (1.6404111). The
return series is negatively skewed for the 20 year period. The Kurtosis, which
0
200
400
600
800
1,000
1,200
1,400
1,600
-15 -10 -5 0 5 10 15 20
Freq
uenc
y
125
measures the magnitude of the extremes, is greater than three, which means that the
return series are leptokurtic in shape, with higher and sharper central peak, and longer
and fatter tails than the normal distribution. The daily stock returns are thus not
normally distributed. The null hypothesis “the return series are normally distributed”
of Jarque-Bera test was rejected proving the returns to be not normally distributed.
Hence, ARCH / GARCH modeling is suggested.
3.9. ARCH Models
The econometric challenge is to specify how the information is used to forecast the
mean and variance of the return, conditional on the past information. The main
constraint of the ARIMA model is the assumption that the disturbance term t in a
model has a constant conditional variance through time. The studies of Mandelbrot
(1963) and Fama (1965) have recognized that such an assumption will not be valid
when studying stock returns. Hence, a more flexible model is required to describe the
volatility of the data.
Conventional econometric analysis assumes the variance of the disturbance terms as
constant over time. The assumption is very limiting for analyzing financial series
because of volatility clustering. The models capable of dealing with variance of the
series are required. The researchers engaged in forecasting time series such as stock
prices and foreign exchange rates observed autocorrelation in the variance at t with its
values lagged one or more periods. If the error variance is related to the squared error
term in the previous term, such autocorrelation is known as autoregressive conditional
heteroskedasticity (ARCH).
While many specifications have been considered for the mean return and have been
used in efforts to forecast future returns, virtually no methods were available for the
variance before introduction of ARCH models. The primary description tool was the
rolling standard deviation. This is a standard deviation calculated using a fixed
number of the most recent observations. It assumes that the variance of tomorrow’s
return is an equally weighted average of the squared residuals from the previous days.
The assumption of equal weights seems unattractive; as one would think that the more
126
recent events would be more relevant and therefore should have higher weights.
Furthermore, the assumption of zero weights for observations more than one month
old is also unattractive (Engle, 2001).
The ARCH model proposed by Engle (1982) allows the data to determine the best
weights to use in forecasting the variance. A useful generalization of this model is the
GARCH introduced by Bollerslev (1986). This model is also a weighted average of
past squared residuals, but it has declining weights that never go completely to zero.
It gives models that are easy to estimate and has proven successful in predicting
conditional variances (Engle and Patton, 2001).
Stock returns are characterized by statistical distributions. Most high frequency
financial time series are found deviating from normality. One of the key assumptions
of the ordinary regression is that variance of the errors is constant throughout the
sample which is known as homoskedasticity. Violation of this assumption indicates
the problem of heteroskedasticity. Findings of heteroskedasticity in stock returns are
well documented by Fama (1965), Engle (1982) and Bollerslev (1986). These studies
have found that the stock return data is typically characterized by following empirical
regularities:
a. Serial Correlation in the Returns
It is a measure of relationship between successive errors. Serial correlation in
the returns indicates that successive returns are not independent.
b. Thick Tails
Skewness and Kurtosis measure the shape of a probability distribution. It
measures the degree of asymmetry, with symmetry implying zero skewness.
Positive skewness indicates a relatively long right tail compared to the left tail
and negative skewness indicates the opposite. Kurtosis indicates the extent to
which probability is concentrated in the center and especially at the tail of the
distribution rather than in the shoulders which are the regions between center
and the tails. Every normal distribution has skewness equal to 0 and kurtosis
of 3. Kurtosis in excess of 3 indicates heavy tails, an indicator of leptokurtosis.
127
Asset returns tend to be leptokurtotic, i.e. too many values near the mean and
in the tails of the distribution when compared with the normal distribution.
The documentation of this empirical regularity is presented in Mandelbrot
(1965). Such regularity has also been observed in the present study (Table
3.6, pg 103).
c. Volatility Clustering
Large changes tend to be followed by large changes, of either sign and small
changes tend to be followed by small changes [(Engle, 1982) and (Bollerslev,
1986)]. This is known as volatility clustering. Statistically, volatility
clustering implies a strong autocorrelation in squared returns.
d. Leverage Effect
Volatility seems to react differently to a big price increase or a big price drop.
This property is known as ‘leverage effect’; and plays an important role in the
development of volatility models. The negative asymmetry in the distribution
of return questions the assumption of an underlying normal distribution. The
so-called ‘leverage effect’ first observed by Black (1976) refers to the
tendency for stock prices to be negatively correlated with changes in stock
volatility. A firm with outstanding debt and equity typically becomes more
highly leveraged when value of the firm falls. This raises the equity return
volatility.
e. Forecastable Events
Forecastable releases of important information are associated with high
volatility. There are also important predictable changes in volatility across the
trading day.
The main weaknesses of ARCH model are
a. The model assumes that positive and negative shocks have the same effects on
volatility because it depends on the square of the previous shock
b. It over predicts the volatility because it responds slowly to large isolated
shocks in time series data (Tsay, 2005).
128
c. ARCH model hardly provides any new insight for understanding the source of
variations of a financial time series. It only provides the mechanical way of
describing the behavior of the conditional variance.
d. ARCH model is rather restrictive. For instance, a 12 of an ARCH(1) model
must be in the interval [0, 1/3] if series is to have finite fourth moment. The
constraint becomes complicated for high order ARCH models.
3.10. Stationarity in Series
A stationary time series is one whose statistical properties such as mean, variance,
autocorrelation, etc. are all constant over time. Most statistical forecasting methods
are based on the assumption that the time series may be rendered approximately
stationary through the use of mathematical transformations. A stationarized series is
relatively easy to predict. A simple prediction that its statistical properties will be the
same in the future as they have been in the past may be made. The predictions for the
stationarized series may then be "untransformed," by reversing whatever
mathematical transformations were previously used, to obtain predictions for the
original series. Thus, finding the sequence of transformations needed to stationarize a
time series often provides important clues in the search for an appropriate forecasting
model.
Another reason for trying to stationarize a time series is to be able to obtain
meaningful sample statistics such as means, variances, and correlations with other
variables. Such statistics are useful as descriptors of future behavior only if the series
is stationary. For example, if the series is consistently increasing over time, the
sample mean and variance will grow with the size of the sample, and they will always
underestimate the mean and variance in future periods. And if the mean and variance
of a series are not well-defined, then neither shows correlation with other variables.
For this reason a caution is required while trying to extrapolate regression models
fitted to non stationary data.
Most business and economic time series are far from stationary when expressed in
their original units of measurement, and even after deflation or seasonal adjustment
they will typically still exhibit trends, cycles, random-walking, and other non-
129
stationary behavior. If the series has a stable long-run trend and tends to revert to the
trend line following a disturbance, it may be possible to stationarize it by de-trending
(e.g., by fitting a trend line and subtracting it out prior to fitting a model, or else by
including the time index as an independent variable in a regression or ARIMA
model), perhaps in conjunction with logging or deflating. Such a series is said to
be trend-stationary.
However, sometimes even de-trending is not sufficient to make the series stationary,
in which case it may be necessary to transform it into a series of period-to-period
and/or season-to-season differences. If the mean, variance, and autocorrelations of the
original series are not constant in time, even after detrending, perhaps the statistics of
the changes in the series between periods or between seasons will be constant. Such a
series is said to be difference-stationary. Sometimes it may be hard to tell the
difference between a series that is trend-stationary and one that is difference-
stationary, and a so-called unit root test may be used to get a more definitive answer.
Thus, before estimating ARCH models for a financial time series, taking two steps is
necessary. First check for unit roots in the residuals and second test for ARCH
effects. The input series for ARMA needs to be stationary before we may apply Box-
Jenkins methodology. The series first needs to be differenced until it is stationary.
This needs log transforming the data to stabilize the variance. Since the raw data are
likely to be non-stationary, an application of ARCH test is not valid. For this reason, it
is usual practice to work with the logs of the changes of the series rather than the
series itself.
3.10.1. Unit root test process
The presence of unit root in a time series is tested using Augmented Dickey- Fuller
test. It tests for a unit root in the univariate representation of time series. For a return
series Rt, the ADF test consists of a regression of the first difference of the series
against the series lagged k times as follows:
∆𝑟𝑡 = α + 𝛿 𝑟𝑡−1 + ∑ 𝛽𝑖
𝑝
𝑖=1
∆𝑟𝑡−𝑖 + ε𝑡 (𝑒𝑞𝑛 3.11)
130
Where
∆𝑟𝑡 = 𝑟𝑡 − 𝑟𝑡−1 Or ∆𝑟𝑡 = ln(𝑅𝑡)
The null and alternative hypotheses are as follows:
H0 = The series contains unit root
H1 = The series is stationary
The acceptance of null hypothesis implies non-stationary. If the ADF test rejects the
null hypothesis of a unit root in the return series, that is if the absolute value of ADF
statistics exceeds the McKinnon critical value the series is stationary and we may
continue to analyze the series. Before estimating a full ARCH model for a financial
time series, it is necessary to check for the presence of ARCH effects in the residuals.
3.10.2. Empirical Investigation: Unit Root Test
Augmented Dickey-Fuller Test is conducted on the Nifty close values at “Level” and
“First Difference” which resulted in the attainment of stationarity in the series at first
difference level. The results are presented in Table 3.8.
Table 3.8: Result of ADF Test
At Level t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -0.267402 0.9272
Test critical values: 1% level -3.431497
5% level -2.861932
10% level -2.567021
At First Difference t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -24.38754 0.000
Test critical values: 1% level -3.431503
5% level -2.861935
10% level -2.567023
*MacKinnon (1996) one-sided p-values.
Source: Compiled Data
131
Figure 3.5: Graph of NSE Nifty Close Prices
Figure 3.6: Graph of NSE Nifty Log Returns
From Figure 3.6, the series is found to have a constant mean showing the stationarity
of the data. It is seen that the returns fluctuated around the mean value, which is close
to zero. The series has a non constant variance, i.e. heteroskedasticity, which is the
typical feature of a financial time series data. Volatility clustering in the returns was
observed, where periods of low volatility are followed by periods of low volatility and
periods of high volatility are followed by periods of high volatility. Statistically,
volatility clustering implies a strong autocorrelation in squared returns.
As the series follows the characteristics of financial time series data, i.e.
heteroskedasticity, leptokurtosis and serial correlation, a linear model would not be
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
1000 2000 3000 4000
CLOSE
-15
-10
-5
0
5
10
15
20
1000 2000 3000 4000
LNRET
132
able to capture the volatility of the series. Hence non linear models such as ARCH /
GARCH have been used for modeling the volatility on the Indian Stock Market.
3.11. Durbin-Watson Statistic
The Durbin–Watson statistic is a test statistic used to detect the presence of
autocorrelation (a relationship between values separated from each other by a given
time lag) in the residuals (prediction errors) from a regression analysis. It is named
after James Durbin and Geoffrey Watson. The small sample distribution of this ratio
was derived by John von Neumann (von Neumann, 1941). Durbin and Watson (1950,
1951) applied this statistic to the residuals from least squares regressions, and
developed bounds tests for the null hypothesis that the errors are serially uncorrelated
against the alternative that they follow a first order autoregressive process. Later, John
Denis Sargan and Alok Bhargava developed several non Neumann–Durbin–Watson
type test statistics for the null hypothesis that the errors on a regression model follow
a process with a unit root against the alternative hypothesis that the errors follow a
stationary first order auto regression (Sargan and Bhargava, 1983).
If 𝑒𝑡 is the residual associated with the observation at time t, then the test statistic is
𝑑 = ∑ (𝑒𝑡 − 𝑒𝑡−1)2𝑇
𝑡=2
∑ 𝑒𝑡2𝑇
𝑡−1
(𝑒𝑞𝑛 3.12)
Where, T is the number of observations. Since d is approximately equal to 2(1 − r),
where r is the sample autocorrelation of the residuals, d = 2 indicates no
autocorrelation. The value of d always lies between 0 and 4. If the Durbin–Watson
statistic is substantially less than 2, there is evidence of positive serial correlation. As
a rough rule of thumb, if Durbin–Watson is less than 1.0, there may be cause for
alarm. Small values of d indicate successive error terms are, on average, close in
value to one another, or positively correlated. If d > 2, successive error terms are, on
average, much different in value from one another, i.e., negatively correlated. In
regressions, this may imply an underestimation of the level of statistical significance.
To test for positive autocorrelation at significance 𝛼, the test statistic d is compared to
lower and upper critical values (𝑑𝐿 and 𝑑𝑈):
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If d < 𝑑𝐿 , there is statistical evidence that the error terms are positively
autocorrelated.
If d > 𝑑𝑈, there is no statistical evidence that the error terms are positively
autocorrelated.
If 𝑑𝐿 < d < 𝑑𝑈, the test is inconclusive.
Positive serial correlation is serial correlation in which a positive error for one
observation increases the chances of a positive error for another observation.
To test for negative autocorrelation at significance 𝛼 , the test statistic (4−d) is
compared to lower and upper critical values (𝑑𝐿and 𝑑𝑈):
If (4−d) < 𝑑𝐿, there is statistical evidence that the error terms are negatively
autocorrelated.
If (4−d) > 𝑑𝑈 , there is no statistical evidence that the error terms are
negatively autocorrelated.
If 𝑑𝐿 < (4−d) < 𝑑𝑈, the test is inconclusive.
Negative serial correlation implies that a positive error for one observation increases
the chance of a negative error for another observation and a negative error for one
observation increases the chances of a positive error for another.
The critical values, 𝑑𝐿 and 𝑑𝑈 , vary by level of significance (α), the number of
observations, and the number of predictors in the regression equation. Their
derivation is complex—statisticians typically obtain them from the appendices of
statistical texts.
3.12. Ljung – Box Q Statistic
The Ljung–Box test is commonly used in autoregressive integrated moving
average (ARIMA) modeling. It is important to note that this test is applied to
the residuals of a fitted ARIMA model, and not the original series. In such
applications the hypothesis actually being tested is that the residuals from the ARIMA
model have no autocorrelation. When testing the residuals of an estimated ARIMA
model, the degrees of freedom need to be adjusted to reflect the parameter estimation.
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For example, for an ARIMA(p,0,q) model, the degrees of freedom should be set to h –
p – q.
The Ljung–Box test, named after Greta M. Ljung and George E. P. Box, is a type
of statistical test of whether any of a group of autocorrelations of a time series is
different from zero. Instead of testing randomness at each distinct lag, it tests the
"overall" randomness based on a number of lags, and is therefore a portmanteau test.
This test is sometimes known as the Ljung–Box Q test, and it is closely connected to
the Box–Pierce test, which is named after George E. P. Box and David A. Pierce. In
fact, the Ljung–Box test statistic was described explicitly in the paper that led to the
use of the Box-Pierce statistic, and from which that statistic takes its name. The Box-
Pierce test statistic is a simplified version of the Ljung–Box statistic for which
subsequent simulation studies have shown poor performance.
Simulation studies have shown that the Ljung–Box statistic is better for all sample
sizes including small ones. The Ljung–Box test is widely applied in econometrics and
other applications of time series analysis.
The null and alternative hypotheses of the Ljung–Box test may be defined as follows:
H0: The data are independently distributed (i.e. the correlations in the population
from which the sample is taken are 0, so that any observed correlations in the
data result from randomness of the sampling process).
Ha: The data are not independently distributed.
The test statistic is:
𝑄 = 𝑛 (𝑛 + 2) ∑
𝑘2
𝑛 − 𝑘
ℎ
𝑘=1
(𝑒𝑞𝑛 3.13)
where n is the sample size, is the sample autocorrelation at lag k, and h is the
number of lags being tested. For significance level α, the critical region for rejection
of the hypothesis of randomness is
𝑄 > 1−𝛼,ℎ2
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where is the α-quantile of the chi-squared distribution with h degrees of
freedom.
3.12.1. Arch effect test process
Consider the k-variable linear regression model.
𝑦𝑡 = 𝛽1 + 𝛽2 𝑥𝑡 + … . + 𝛽𝑘 𝑥𝑘 𝑡 + 𝑢𝑡 (𝑒𝑞𝑛 3.14)
In addition, assume that conditional on the information available at time (t-1), the
disturbance term distributed as
𝑢𝑡 ~ [0, (α0 + α1 𝑢𝑡−12 )] (𝑒𝑞𝑛 3.15)
That is, ut is normally distributed with zero mean and
𝑉𝑎𝑟 (𝑢𝑡) = (α0 + α1 𝑢𝑡−12 ) (𝑒𝑞𝑛 3.16)
That is the variance of ut follows an ARCH (1) process. The variance of u at time t is
dependent on the squared disturbance at time (t-1), thus giving the appearance of
serial correlation. The error variance may depend not only on one lagged term of the
squared error term but also on several lagged squared terms as follows:
𝑉𝑎𝑟 (𝑢𝑡) = σ𝑡2 = α0 + α1 𝑢𝑡−1
2 + α2 𝑢𝑡−22 + … . + α𝑝 𝑢𝑡−𝑝
2 (𝑒𝑞𝑛 3.17)
If there is no autocorrelation in the error variance, we have
𝐻0 ∶ α1 = α2 = ⋯ = α𝑝 = 0 (𝑒𝑞𝑛 3.18)
In such a case, 𝑉𝑎𝑟 (𝑢𝑡) = α0, and we do not have the ARCH effect.
The null hypothesis is tested by the usual F test but the ARCH-LM test of Engle 1982
is a common test in this regard. Under ARCH-LM test the null and alternative
hypotheses for Nifty stock index are as follows:
𝐻0 ∶ α1 = 0 𝑎𝑛𝑑 α2 = 0 𝑎𝑛𝑑 α3 = 0 𝑎𝑛𝑑 … . α𝑞 = 0 (𝑒𝑞𝑛 3.19)
𝐻1 ∶ α1 ≠ 0 𝑎𝑛𝑑 α2 ≠ 0 𝑎𝑛𝑑 α3 ≠ 0 𝑎𝑛𝑑 … . α𝑞 ≠ 0 (𝑒𝑞𝑛 3.20)
Null hypothesis in this case is homoskedasticity or equality in the variance.
Acceptance of this hypothesis implies that, there is no ARCH effects in the under
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process series. In other words, the data do not show volatility clustering i.e. there is
no heteroskedasticity or time varying variance in the data.
Since an ARCH model may be written as an AR model in terms of squared residuals,
a simple Lagrange Multiplier (LM) test for ARCH effects may be constructed based
on the auxiliary regression as in equation 3. 16. Under the null hypothesis that there
is no ARCH effects: 𝐻0 = α1 = α2 = ⋯ = α𝑃 = 0
The test statistic is
𝐿𝑀 = 𝑇 . 𝑅2 ~ 2(𝑃) (𝑒𝑞𝑛 3.21)
where T is the sample size and R2 is computed from the regression equation 3.16
using estimated residuals. That is in a large sample, TR2 follows the Chi-square
distribution with degrees of freedom equal to the number of autoregressive terms in
the auxiliary regression. The test statistic is defined as TR2 (the number of
observations multiplied by the coefficient of multiple correlation) from the last
regression, and it is distributed as a 𝑞2 (Gujarati, 2007).
Thus, the test is one of a joint null hypothesis that all q lags of the squared residuals
have coefficient values that are not significantly different from zero. If the value of
the test statistic is greater than the critical value from the χ2 distribution, then one can
reject the null hypothesis. The test may also be thought of as a test for autocorrelation
in the squared residuals.
If P-value is smaller than the conventional 5% level, the null hypothesis that there are
no ARCH effects will be rejected. In other words, the series under investigation
shows volatility clustering or persistence (Brooks, 2002). If the LM test for ARCH
effects is significant for a time series, one could proceed to estimate an ARCH model
and obtain estimates of the time varying volatility σ2 based on past history.
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3.12.2. Serial Correlation effect test process
In Ordinary Least Squares (OLS) regression, time series residuals are often found to
be serially correlated with their own lagged values. Serial correlation means
(a) OLS is no longer an efficient linear estimator,
(b) Standard errors are incorrect and generally overstated, and
(c) OLS estimates are biased and inconsistent if a lagged dependent variable is used as
a regressor.
This test is an alternative to the Q-Statistic for testing for serial correlation. It is
available for residuals from OLS, and the original regression may include
autoregressive (AR) terms. Unlike the Durbin-Watson Test, the Breusch-Godfrey
Test may be used to test for serial correlation beyond the first order, and is valid in the
presence of lagged dependent variables.
The null hypothesis of the Breusch-Godfrey Test is that there is no serial correlation
up to the specified number of lags. The Breusch-Godfrey Test regresses the residuals
on the original regressors and lagged residuals up to the specified lag order. The
number of observations multiplied by R2 is the Breusch-Godfrey Test statistic.
3.12.3. Empirical Investigation: Autocorrelation tests
NSE Nifty is selected as proxy of the Indian stock market and data is collected for
Nifty for a period of 20 years from 4 January 1993 to 31 December 2012. According
to the Jarque-Bera test for normality, the Nifty series showed a p-value of 0 rejecting
the null hypothesis “Return series are normally distributed”.
Autocorrelation tests have been performed on the return series and presented in Table
3.9. As the DW statistic value (1.821049) of log returns is greater than the upper limit
of the DW table value (1.779), the null hypothesis of “the error terms are positively
autocorrelated” cannot be accepted. It may thus be concluded that there is no
statistical evidence that the error terms are positively autocorrelated. Hence the DW
statistic could not provide a strong evidence of the presence of autocorrelation in the
Nifty return series.
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Table 3.9: Results of Auto Correlation Tests
DW Statistic 1.821049
Ljung Box Q Statistic 39.265
Probability of Q Statistic 0.000
Serial correlation LM test
Prob. of Chi square (2 df) 0.000
Heteroskedasticity test for ARCH effect
Prob. Of Chi square (1 df) 0.000
Source: Compiled Data
Q(k), the Ljung Box statistic, identified the presence of first order autocorrelation in
the returns with a lag of 15 days. The Q statistic showed a p-value of 0 rejecting the
null hypothesis “There is no autocorrelation in the log return series”. Thus it could be
concluded that there exists an autocorrelation in the log return series.
3.13. ARIMA Estimation for Stock Returns
The general ARMA model was described in the 1951 thesis of Peter Whittle,
Hypothesis testing in time series analysis, who used mathematical analysis (Laurent
series and Fourier analysis) and statistical inference. ARMA models were
popularized by a 1971 book by George E. P. Box and Jenkins, who expounded an
iterative (Box–Jenkins) method for choosing and estimating them. This method was
useful for low-order polynomials of degree three or less.
There are two distinct steps in the estimation of the parameters of the model. The first
is to estimate mean equation with the help of Box-Jenkins methodology, and the
second step is to estimate the parameters of variance equation of ARCH and GARCH
class of models.
The general model introduced by Box and Jenkins (1976) includes autoregressive as
well as moving average parameters and includes differencing in the formulation of the
model. The important three types of parameters in the model are:
139
i. The autoregressive parameters (p)
ii. The number of differencing passes (d)
iii. Moving average parameters (q)
In the notation introduced by Box and Jenkins, models are summarized as ARIMA (p,
d, q); so, if a model is described as (0, 1, 2) means that it contains 0 (zero)
autoregressive (p) parameters and 2 moving average (q) parameters which were
computed for the series after it was differenced once.
To estimate mean equation, the Autoregressive Integrated Moving Average (ARIMA)
model, as developed by Box and Jenkins, has been widely applied in a variety of
economic and financial time series. Box-Jenkins method consists of the following
steps to estimate ARIMA mean equation.
a. Identification
Here the appropriate values of p, d and q are found out using autocorrelation
(ACF) and partial autocorrelation (PACF) functions1
b. Estimation
Having identified the appropriate values of p and q, the next stage is to estimate
the parameters of the autoregressive and moving average terms included in the
model
c. Diagnostic Checking
After choosing particular ARIMA model and having estimated its parameters, the
next step is to see whether chosen ARIMA model fits the data reasonably well.
Residuals from this model are examined to see if they are white noise, and if they
are, then accept the particular fit of the model, otherwise refine the model.
1 ACF is correlation between observations of a stationary process as function of the time
interval between them. PACF is also a measure of correlation used to identify the extent of
relationship between current values of a variable with earlier values of that same variable
while holding the effects of all other time lags constant.
140
If the Box–Jenkins model is a good model for the data, the residuals should satisfy
these assumptions.
a. Model diagnostics for Box–Jenkins models is similar to model validation for
non-linear least squares fitting.
b. That is, the error term At is assumed to follow the assumptions for a stationary
univariate process.
c. The residuals should be white noise (or independent when their distributions
are normal) drawings from a fixed distribution with a constant mean and
variance.
One way to assess if the residuals from the Box–Jenkins model follow the
assumptions is to generate statistical graphics (including an autocorrelation plot) of
the residuals. Plotting the mean and variance of residuals over time and performing a
Ljung-Box test or plotting autocorrelation and partial autocorrelation of the residuals
are also helpful to identify misspecification.
However, if these assumptions are not satisfied, one needs to fit a more appropriate
model by going back to the model identification step and try to develop a better
model. Hopefully the analysis of the residuals may provide some clues as to a more
appropriate model.
3.13.1. Fitting ARIMA Model
ARMA models in general, after choosing p and q, can be fitted by least
squares regression to find the values of the parameters which minimize the error term.
It is generally considered good practice to find the smallest values of p and q which
provide an acceptable fit to the data. For a pure AR model the Yule-Walker
equations may be used to provide a fit.
Finding appropriate values of p and q in the ARMA(p,q) model may be facilitated by
plotting the partial autocorrelation functions for an estimate of p, and likewise using
the autocorrelation functions for an estimate of q. Further information may be gleaned
141
by considering the same functions for the residuals of a model fitted with an initial
selection of p and q. Brockwell and Davis (p. 273) recommend using AIC for
finding p and q.
3.13.2. Application of the Model
ARMA is appropriate when a system is a function of a series of unobserved shocks
(the MA part) as well as its own behavior. For example, stock prices may be shocked
by fundamental information as well as exhibiting technical trending and mean-
reversion effects due to market participants.
3.14. ARCH Model Specification for Nifty
ARCH models are capable of modeling and capturing many of the stylized facts of the
volatility behavior usually observed in financial time series including time varying
volatility or volatility clustering (Zivot and Wang, 2006).
The serial correlation in squared returns, or conditional heteroskedasticity (volatility
clustering), may be modeled using a simple autoregressive (AR) process for squared
residuals. For example, let yt denote a stationary time series such as financial returns,
then yt is expressed as its mean plus a white noise if there is no significant
autocorrelation in yt itself:
𝑦𝑡 = 𝑐 + ε𝑡 (𝑒𝑞𝑛 3.22)
where c is the mean of yt, and ε t is the standardized residuals which are independent
and identically distributed with mean zero.
To allow for volatility clustering or conditional heteroskedasticity, assume that
𝑉𝑎𝑟𝑡−1 (ε𝑡)2 = σ𝑡2 (𝑒𝑞𝑛 3.23)
where 𝑉𝑎𝑟𝑡−1 (ε𝑡)2 denotes the variance conditional on information at time t-1, and
σ𝑡2 = α0 + α1 ε𝑡−1
2 + … . + α𝑝 ε𝑡−𝑝2
Since ε t has a zero mean, the above equation may be rewritten as:
ε𝑡2 = α0 + α1 ε𝑡−1
2 + … . + α𝑝 ε𝑡−𝑝2 + 𝑢𝑡 (𝑒𝑞𝑛 3.24)
Where 𝑢𝑡 = ε𝑡2 − 𝐸𝑡−1 (ε𝑡
2) is a zero mean white noise process.
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The above equation represents an AR (p) process for (ε𝑡2), and the model in equation
3.14 is known as the autoregressive conditional heteroskedasticity (ARCH) model of
Engle (1982), which is usually referred to as the ARCH(p ) model. Before estimating
a full ARCH model for a financial time series, it is necessary to test for the presence
of ARCH effects in the residuals. If there are no ARCH effects in the residuals, then
the ARCH model is unnecessary and mis-specified.
In order to identify the ARCH characteristics in Nifty, the conditional return should
be modeled first; the general form of the return may be expressed as a process of
autoregressive AR (p), up to (p) lags, as follows:
𝑅𝑡 = α0 + ∑ α1
𝑝
𝑖=1
𝑅𝑡−1 + ε𝑡 (𝑒𝑞𝑛 3.25)
This general form implies that the current return depends not only on (𝑅𝑡−1) but also
on the previous (p) return value (𝑅𝑡−𝑝).
The next step is to construct a series of squared residuals (ε𝑡2) based on conditional
return to drive the conditional variance. Unlike the OLS assumption of a constant
variance of (ε𝑡, 𝑠), ARCH models assume that (ε𝑡, 𝑠) have a non constant variance or
heteroscedasticity, denoted by (ℎ𝑡2) . After constructing time series residuals, we
modeled the conditional variance in a way that incorporates the ARCH process of
(ε 2) in the conditional variance with q lags. The general forms of the conditional
variance, including (q) lag of the residuals is as follows:
𝜎𝑡2 = 𝛽0 + ∑ 𝛽1
𝑞
𝑡=1
𝑡−12 (𝑒𝑞𝑛 3.26)
The above equation is what Engle (1982) referred to as the linear ARCH (q) model
because of the inclusion of the (q) lags of the (ε𝑡2) in the variance equation. This
model suggests that volatility in the current period is related to volatility in the past
periods.
For example in the case of AR(1) model, If β1 is positive, it suggests that if volatility
was high in the previous period, it will continue to be high in the current period,
indicating volatility clustering. If β1 is zero, then there is no volatility clustering.
143
To determine the value of q or the ARCH model order, we use the model selection
criterion such as AIC (Akaike Information Criterion). The decision rule is to select the
model with the minimum value of information criterion. This condition is necessary
but not enough because the estimate meets the general requirements of an ARCH
model. The model to be adequate should have coefficients that are significant. If this
requirement is met, then the specified model is adequate and said to fit the data well.
In an ARCH(q) model, old news arrived at the market more than q periods ago has no
effect on current volatility at all. In the ARCH model, the variance of the current
error is an increasing function of the magnitude of lagged errors, irrespective of their
sign. ARCH model implies that a large (small) variance tends to be followed by a
large (small) variance.
3.14.1. Empirical Investigation: Estimation of ARIMA
In this section, estimation of mean equation for the Indian stock market, Nifty, is
discussed followed by volatility model specifications. The mean equation is
estimated for Nifty using Box-Jenkins methodology.
3.14.2. Step 1: Identification
The input series for ARIMA needs to be stationary, which has been attained for the
select market, Nifty, by using the first difference level (Table 3.8). The steps
involved in finding out the appropriate values of p, d and q. The Autocorrelation
Function (ACF) and Partial Autocorrelation Function (PACF) are computed and
presented in Table 3.10 below. Ljung-Box-Pierce Q Statistic is highly significant,
and it indicates the first order serial correlation in the return series.
The Partial Autocorrelation Function of the correlogram of log return series of Nifty
in Table 3.10 shows significance at lag 1, which means that AR(1) would be
applicable for modeling the mean.
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Table 3.10: Correlogram of Log Returns
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
|* | |* | 1 0.089 0.089 39.265 0.0000
| | | | 2 -0.033 -0.041 44.613 0.0000
| | | | 3 0.000 0.007 44.613 0.0000
| | | | 4 0.015 0.013 45.75 0.0000
| | | | 5 -0.004 -0.007 45.829 0.0000
| | | | 6 -0.044 -0.043 55.494 0.0000
| | | | 7 0.009 0.017 55.935 0.0000
| | | | 8 0.024 0.018 58.794 0.0000
| | | | 9 0.024 0.022 61.691 0.0000
| | | | 10 0.038 0.037 68.673 0.0000
| | | | 11 -0.028 -0.035 72.576 0.0000
| | | | 12 -0.019 -0.013 74.318 0.0000
| | | | 13 0.015 0.017 75.497 0.0000
| | | | 14 0.036 0.033 81.971 0.0000
| | | | 15 0.005 0.002 82.087 0.0000
Source: Compiled Data
145
Figure 3.7: Actual and Theoretical Autocorrelation
For modeling the mean, two ways may be followed, which includes categories named
“with intercept” (where a constant is used) and “without intercept” (where constant is
not used). Out of these two categories, the Akaike information criterion (AIC) is
observed and the category showing least value of AIC will be used for running the
AR(1) model. It is observed that the AIC is least in case of “with intercept” according
to Table 3.11. Hence, the AR(1) model was run with intercept and the model is
presented in Table 3.12.
Table 3.11: Basis for taking intercept in the estimation
AR(1) Coeff. prob. Constant prob. Akaike Info.
With intercept 0.08929 0 0.042113 0.0997 3.820204
Without intercept 0.089883 0 - - 3.820349
Source: Compiled Data
-.05
.00
.05
.10
2 4 6 8 10 12 14 16 18 20 22 24
Actual Theoretical
Auto
corre
latio
n
-.05
.00
.05
.10
2 4 6 8 10 12 14 16 18 20 22 24
Actual Theoretical
Parti
al a
utoc
orre
latio
n
146
3.14.3. Step 2: Estimation
AR(1) model has been used with the Box Jenkins methodology to model the
conditional mean equation. Let Yt denote the first differenced Nifty return series. The
regression when run with AR(1) resulted in the following tentatively identified AR
model:
𝑌𝑡 = 𝑐 + 𝛼1 𝑌𝑡−1 + 𝑡 (𝑒𝑞𝑛. 3.27)
Using E Views 6, the following estimates are obtained:
𝑌𝑡 = 0.042113 + 0.08929 𝑌𝑡−1 + 𝑡 (𝑒𝑞𝑛. 3.28)
From this estimated equation in Table 3.12, it may be observed that the AR(1) value
is significant rejecting the null hypothesis “The residuals are not stationary”. AR(1)
value, i.e. 𝛼1 = 0.08929, is positive which suggests that if volatility was high in the
previous period, it will continue to be high in the current period and vice-versa. Thus
the estimated AR(1) equation indicates volatility clustering in the series.
Table 3.12: AR(1) model
Variable Coefficient Std. Error t-Statistic Prob.
C 0.042113 0.025575 1.646647 0.0997
AR(1) 0.08929 0.014198 6.288664 0.0000*
R-squared 0.007976 Mean dependent var 0.042071
Adjusted R-squared 0.007774 S.D. dependent var 1.640273
S.E. of regression 1.633885 Akaike info criterion 3.820204
Sum squared resid 13131.66 Schwarz criterion 3.822847
Log likelihood -9397.613 Hannan-Quinn criter. 3.821131
F-statistic 39.5473 Durbin-Watson stat 1.992952
Prob(F-statistic) 0.0000
Source: Compiled Data
147
3.14.4. Step 3: Diagnostic Checking
Having chosen the ARIMA model as specified in equation 3.27, and having estimated
its parameters in equation 3.28, the next step involves checking whether the chosen
model fits the data reasonably well. One simple diagnostic measure is to obtain
residuals from the equation 3.28 and obtain the ACF and PACF of these residuals up
to lag of 15 days.
The estimated ACF and PACF from the residuals as well as the squared residuals of
the equation showed no significance for the existence of serial correlation upto the
15th
lag. Presented in Tables 3.13 and 3.14, the correlogram of autocorrelation and
partial autocorrelation for both residuals and squared residuals of the equation give
the impression that the residuals estimated from equation 3.28 are purely white noise.
These residuals were further tested for ARCH effects using ARCH in
Heteroskedasticity test and for Serial Correlation effect using the Serial Correlation
LM Test. The F statistic is reported significant in both cases at 5% level of
significance, rejecting the null hypotheses of no heteroskedasticity and no serial
correlation. Thus, this test further suggests for the use of non linear model for
capturing volatility.
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Table 3.13: Correlogram of Residuals
Q-statistic probabilities adjusted for 1 ARMA term(s)
Autocorrelation Partial Correlation
AC PAC Q-Stat Prob
| | | | 1 0.003 0.003 0.0586
| | | | 2 -0.042 -0.042 8.6853 0.003
| | | | 3 0.001 0.001 8.6906 0.013
| | | | 4 0.016 0.014 9.9453 0.019
| | | | 5 -0.001 -0.001 9.9522 0.041
| | | | 6 -0.045 -0.044 19.972 0.001
| | | | 7 0.012 0.012 20.656 0.002
| | | | 8 0.021 0.017 22.878 0.002
| | | | 9 0.019 0.02 24.64 0.002
| | | | 10 0.039 0.041 32.013 0.000
| | | | 11 -0.03 -0.03 36.529 0.000
| | | | 12 -0.018 -0.017 38.104 0.000
| | | | 13 0.014 0.013 39.128 0.000
| | | | 14 0.035 0.035 45.264 0.000
| | | | 15 0.002 0.005 45.277 0.000
Source: Compiled Data
149
Table 3.14: Correlogram of Squared Residuals
Q-statistic probabilities adjusted for 1 ARMA term(s)
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
| | | | 1 0.017 0.017 1.3498
| | | | 2 0.005 0.005 1.462 0.227
| | | | 3 0.008 0.007 1.7484 0.417
| | | | 4 -0.008 -0.008 2.0384 0.564
| | | | 5 0.007 0.007 2.2467 0.69
| | | | 6 -0.005 -0.005 2.3752 0.795
| | | | 7 -0.003 -0.003 2.4252 0.877
| | | | 8 -0.016 -0.016 3.7079 0.813
| | | | 9 -0.018 -0.017 5.2907 0.726
| | | | 10 0.001 0.001 5.292 0.808
| | | | 11 -0.007 -0.007 5.5624 0.851
| | | | 12 -0.016 -0.016 6.8205 0.813
| | | | 13 0.01 0.01 7.2719 0.839
| | | | 14 -0.009 -0.009 7.7038 0.862
| | | | 15 -0.014 -0.014 8.6274 0.854
Source: Compiled Data
150
3.15. GARCH Model
The problem with applying the original ARCH model is the non-negativity constraint
on the coefficient parameters of βi's to ensure the positivity of the conditional
variance. However, when a model requires many lags to model the process correctly,
the non-negativity may be violated. To avoid the long lag structure of the ARCH (q)
developed by Engle (1982), Bollerslev (1986) generalized the ARCH model, the so-
called GARCH, by including the lagged values of the conditional variance. Thus,
GARCH(p,q) model specifies the conditional variance to be a linear combination of q
lags of the squared residuals (ε𝑡−12 ) from the conditional return equation and p lags
from the conditional variance σ𝑡−12 . Then, the GARCH(p,q) specification may be
written as follows:
σ𝑡2 = β0 + ∑ β1
𝑞
𝑖=1
ε𝑡−𝑖2 + ∑ 𝛽2
𝑝
𝑗=1
σ𝑡−𝑗2 (𝑒𝑞𝑛 3.29)
where the coefficients 𝛽1, 𝛽2 > 0 and (𝛽1 + 𝛽2) < 1
The coefficients are all assumed to be positive to ensure that the conditional variance
σ𝑡2 is always positive. This model is known as the generalized ARCH or GARCH(p,q)
model. When q = 0, the GARCH model reduces to the ARCH model.
To show the significance of the explanation of conditional variance of one lag of both
ε𝑡2 and σ𝑡
2 i.e. ε𝑡−12 and σ𝑡−1
2 , the GARCH process should be employed by estimating
the conditional return to drive ε𝑡2, and then the estimation of the conditional variance
by using the following equation:
σ𝑡2 = β0 + β1 ε𝑡−1
2 + α1 σ𝑡−12 (𝑒𝑞𝑛 3.30)
The adequacy of the GARCH model may be examined by standardized residuals,
(/𝜎), where (σ) is the conditional standard deviation as calculated by the GARCH
model, and(ε ) is the residuals of the conditional return equation.
If the GARCH model is well specified, then the standardized residuals will be
Independent and Identically Distributed (IID). To show this, two-step test is needed.
The first step is to calculate the Ljung-Box Q-Statistics (LB) on the squared
observation of the raw data. This test may be used to test for remaining serial
151
correlation in the mean equation and to check the specification of the mean equation.
If the mean equation is correctly specified, all Q-statistics should not be significant.
The next step is to calculate the Q-statistics of the squared standardized residuals.
This test may also be used to test for remaining ARCH in the variance equation and to
check the specification of the variance equation. If the variance equation is correctly
specified, all Q-statistics should not be significant. Put another way, if the GARCH is
well specified, then the LB statistic of the standardized residuals will be less than the
critical value of the Chi-square statistic.
The test for mean equation specification is thought of as a test for autocorrelation in
the standardized residuals. The test has a null hypothesis that there is no
autocorrelation up to order k of the residuals. If the value of the test statistic is greater
than the critical value from the Q-statistics, then the null hypothesis can be rejected.
Alternatively, if p-value is smaller than the conventional significance level, the null
hypothesis that there are no autocorrelation will be rejected. In other words, the series
under investigation shows volatility clustering or volatility persistence. The same is
true for variance equation .The only difference is that in this case the test will be done
on squared standardized residuals.
Under the GARCH (p, q) model, the conditional variance of ε t, σt2, depends on the
squared residuals in the previous p periods, and the conditional variance in the
previous q periods. Usually a GARCH (1, 1) model with only three parameters in the
conditional variance equation is adequate to obtain a good model fit for financial time
series (Zivot and Wang, 2006)
Many variants of the GARCH model have been proposed in the literature thereafter,
including the following:
a. GARCH (the simplest GARCH)
b. M-GARCH (GARCH in Mean)
c. E-GARCH (Exponential GARCH)
d. T-GARCH (Threshold GARCH)
e. C-GARCH (Component GARCH)
152
f. CT-GARCH (Asymmetric Component Threshold GARCH)
g. P-GARCH (Power GARCH)
h. I-GARCH (Integrated GARCH)
i. V-GARCH (Variance GARCH)
j. GJR-GARCH (Glosten, Jagannathan and Runkle)
k. GARCH-GED (Generalized Error Distribution residuals)
ARCH and GARCH models assume conditional heteroskedasticity with
homoskedastic unconditional error variance, i.e. the changes represent temporary and
random departure from a constant unconditional variance.
The advantage of GARCH model is that it captures the tendency in financial data for
volatility clustering. It therefore, enables us to make the connection between
information and volatility explicit since any change in the rate of information arrival
to the market will change the volatility in the market. Thus, unless information
remains constant, which is hardly the case, volatility must be time varying even on a
daily basis.
In GARCH process, unexpected returns of the same magnitude (irrespective of their
sign) produce same amount of volatility. But Engle and Ng (1993) argue that if a
negative return shock causes more volatility than a positive return shock of the same
magnitude, the GARCH model under predicts the amount of volatility following bad
news and over predicts the same amount of volatility following good news.
3.15.1. Stationarity and Persistence
The GARCH (p,q) is defined as stationary when {( 𝛼1 + 𝛼2 + … … + 𝛼𝑞) +
(𝛽1 + 𝛽2 + … … + 𝛽𝑝)} < 1. The large GARCH lag coefficients 𝛽𝑝 indicate that
shocks to conditional variance takes a long time to die out, so volatility is ‘persistent’.
Large GARCH error coefficient 𝛼𝑞 means that volatility reacts quite intensely to
market movements and so if 𝛼𝑞 is relatively high and / or 𝛼𝑞 is relatively low, then
volatilities tend to be ‘spiky’.
153
If (α + β) is close to unity, then a shock at time t will persist for many future periods.
A high value of it implies a ‘long memory’. Non-negativity constraints on 𝛼𝑞, 𝛽𝑝 and
𝛼0 can create difficulties in estimating GARCH models. Furthermore those non-
negativity constraints imply that increasing 𝑡2 in any period increases ℎ𝑡+𝑚 for all
ruling out the random oscillatory behavior in the process.
3.15.2. Modeling the Conditional Variance
After confirming the presence of clustering volatility and ARCH effect on the log
return series, the GARCH model was run. The model was run with an error
distribution function of “student’s t with fixed df” and “student’s t” separately and
results were compared with each other to find which of these best fitted the model and
presented in Table 3.15. The best fit is identified based on the Akaike information
criterion (AIC), i.e. the function showing smaller absolute value of AIC will be
chosen as the best fitted model.
Table 3.15: Comparison of both functions for the best fitted model
Student's t with df Student's t
@SQRT GARCH significant significant
RESID(-1)^2 significant significant
GARCH(-1) significant significant
Akaike info criterion -0.71204 -0.71298
Schwarz criterion -0.70412 -0.70374
Residual Test
ARCH effect accept H0* accept H0*
Serial Correlation test accept H0** accept H0**
* H0 is framed as “There is no ARCH effect”
** H0 is framed as “There is no Serial Correlation effect”
Source: Compiled Data
154
From an examination of the data presented in Table 3.15, we find that all
requirements for the model run are fulfilled by both the functions. It is observed that
the absolute AIC value is smaller for the “Student’s t with df” function which is
identified as the best fitted model for Nifty for the select period.
Thus, as per the specified equation 3.30, the best fitted model for Nifty for the select
period according to the best fit AIC function i.e. ‘Student’s t with df’ is presented in
Table 3.16 below. The GARCH model estimated according to equation 3.30 is also
given in this table.
Table 3.16: Estimated GARCH Equation at Student’s t with df
Dependent Variable: LNRET
Method: ML - ARCH (Marquardt) - Student's t distribution
Sample: 1 4922
Included observations: 4922
Convergence achieved after 12 iterations
Presample variance: backcast (parameter = 0.7)
t-distribution degree of freedom parameter fixed at 10
GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)
Variable Coefficient Std. Error z-Statistic Prob.
@SQRT(GARCH) 0.369844 0.051541 7.175735 0.0000
C 0.035825 0.008072 4.438016 0.0000
U 0.999837 0.001408 710.2163 0.0000
Variance Equation
C 0.000891 0.00015 5.954406 0.0000
RESID(-1)^2 0.113218 0.00994 11.39011 0.0000
GARCH(-1) 0.862069 0.010934 78.84313 0.0000
R-squared 0.986073 Mean dependent var 0.04162
Adjusted R-squared 0.986059 S.D. dependent var 1.640411
S.E. of regression 0.19369 Akaike info criterion -0.71204
Sum squared resid 184.4275 Schwarz criterion -0.70412
Log likelihood 1758.34 Hannan-Quinn criter. -0.70926
F-statistic 69611.96 Durbin-Watson stat 1.825068
Prob(F-statistic) 0.0000
Source: Compiled Data
155
Figure 3.8: Descriptive statistics of Residuals of fitted GARCH(1,1) Model
Figure 3.9: Volatility Clustering of fitted GARCH(1,1) Model
0
200
400
600
800
1,000
1,200
-6 -4 -2 0 2 4 6
Series: Standardized Residuals
Sample 2 4922
Observations 4921
Mean -0.037177
Median -0.028802
Maximum 7.475823
Minimum -6.281666
Std. Dev. 1.019179
Skewness -0.095965
Kurtosis 5.524925
Jarque-Bera 1314.744
Probability 0.000000
-2
-1
0
1
2
3
-20
-10
0
10
20
1000 2000 3000 4000
Residual Actual Fitted
156
According to the GARCH model (eqn 3.30)
σ𝑡2 = β0 + β1 ε𝑡−1
2 + α1 σ𝑡−12
The value of RESID(-1)^2 represents the β1 value and the value of GARCH(-1)
represents the α1 value of the GARCH model. According to the estimated GARCH
equation, the values of β1 and α1 are identified as 0.113218 and 0.862069
respectively.
It becomes imperative to observe that the α1 value is large enough in magnitude, to
imply that volatility reacts intensely to the market movements. The sum of these
parameter estimates (0.975287) is very close to but smaller than unity. A sum smaller
than unity, indicates that stationarity condition is not violated. The sum closer to
unity, indicates a long persistence of shocks in volatility.
3.15.3. Half Life of Volatility Persistence
These parameter estimates of GARCH(1,1) model are further used to calculate the
half life of volatility persistence by using the following formula:
ln(0.5)
ln(𝛼 + 𝛽) (𝑒𝑞𝑛 3.31)
The half life of volatility persistence (shock) is calculated and found to be 27.699
days. This implies that the shocks in volatility would die out within an approximate
time of 28 days.
It may be observed that the lag coefficient of conditional variance β1(0.113218) is
identified to be greater than the error coefficient C (0.000891), which implies that
volatility is not spiky. It also indicates that the volatility does not decay speedily and
tends to die out slowly.
Thus, it may be concluded from this estimation that volatility in the Nifty movements
reacts intensely to the market movements and the long persistence in volatility
indicates that the Indian market is inefficient; hence information is not reflected in the
stock process quickly.
157
3.16. Factors Affecting Volatility
Several economic factors may cause slow changes in stock market volatility. These
include corporate leverage, personal leverage, business conditions, trading volume
and trading halts, computerized trading, noise trading, international linkages and
others.
3.16.1. Corporate leverage
Financial and operating leverage effect the volatility of stock returns. Financial
leverage indicates the use of debt financing to increase the expected return and risk of
equity capital. The use of fixed assets to increase the expected profitability and risk
of production and marketing activities is known as operating leverage. The standard
deviation of stock returns of all equity firms simply equals the standard deviation of
the returns to its assets. If the firm issue debt to buy back its share, the volatility of its
stock returns will increase because the stock holders still have to bear most of the risk.
Similarly, large amounts of operating leverage will make the value of the firm more
sensitive to economic conditions. When a demand for the company’s products falls
off unexpectedly, the profits of a firm with large fixed costs will fall more than the
profits of the firm that avoids large capital investments causing higher stock return
volatility.
3.16.2. Personal Leverage
It refers to the use of personal debt to increase the expected return and risk of an
individual’s investment portfolio. Much recent debate has focused on the effects of
margin requirements on the volatility of aggregate stock prices.
3.16.3. Business Conditions
There is strong evidence that stock volatility increases during economic recessions.
This relationship may in part reflect operating leverage, as recessions are typically
associated with excess capacity and unemployment. Fixed costs for the economy
would have the effect of increase in the volatility of stock returns during periods of
low demand.
158
3.16.4. Trading Volume & Trading Halts
It has been argued that increased trading activity and stock return volatility occur
together. With high trading volume, prices fluctuate rapidly causing volatility. Stock
exchanges apply market-wide or scrip-specific trading halts for providing a ‘cooling-
off’ period that allows the investors to re-evaluate the market information and to re-
formulate his investment strategy.
A trading halt may be applied in the form of a circuit breaker or a circuit filter.
Circuit filters are also referred to as price limits since these are applied whenever the
stock price change exceeds certain preset percentage limit in either direction. There is
no theoretical basis for determining whether the imposition of circuit breakers will
have the desired effect of reducing stock market volatility. But it is believed that they
will pacify volatility.
3.16.5. Computerized Trading
The sophistication in information technology has made it much easier for larger
number of people to learn about and react to information very quickly thereby
increasing liquidity. The liquidity of organized securities market plays an important
part in supporting the value of traded securities allowing changing price quickly. This
will induce variation in price causing the volatility.
3.16.6. Noise Trading
Noise trading is trading on noise as if it were information. It is essential to the
existence of liquid markets providing the essential missing ingredient. The more the
noise trading, the more liquid the markets will be in the sense of having frequent
trades that allows us to observe prices. But noise trading actually puts noise into the
prices. The price of a stock reflects both the information that the information traders
trade on and the noise that noise traders trade on. The farther away the stocks get
from their fundamental values, the more aggressive the noise traders become which
creates more volatility in the market.
159
3.16.7. International Linkages
The movements of the Indian stock prices are influenced by the movements of stock
prices on other overseas markets and vice versa. Due to greater integration of stock
markets of the world, international developments like recent Financial crisis of USA
and Iraq, and the crisis in the European countries, would have varying degree of
impacts on the Indian stock markets and other stock markets of the world.
3.16.8. Other factors
The extensive coverage of stock market news by media indicates that they are an
important source of information and comment upon the stock market. This brings
about changes in the buying and selling decisions of investors thereby affecting the
price of the stock market, creating volatility.
160
3.17. Conclusions
This study is carried out to understand the volatility behavior of the Indian stock
market by computing historical volatility levels of Nifty using classical, range based
and drift independent volatility estimators. The study also aims at performing
autocorrelation tests and estimate conditional variance of the sample return series
through GARCH(1,1) model.
On the whole, the analysis of 20 year data starting from January 1993 to December
2012 established two phases in volatility in Nifty, namely, the boom and subsequent
crash of the Indian stock market during 1999-2000 and the subprime financial crisis
that cropped up across the globe during 2008-2009. Excepting these two phases, the
20 year period exhibited a trend closer to stationarity.
The 28 day persistence in volatility of Nifty return series indicate that volatility in the
Indian stock market reacts intensely to the market movements and takes long time to
die out the shocks it faces from the market movements. It may mean that the Indian
market is inefficient; hence information is not reflected in the stock process quickly.
161
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