Measuring asymmetric volatility and stock returns in the Philippine Stock Market
Daniel S. Hofileña and Maria Francesca D. Tomaliwan
Economics Department, De La Salle University- Manila
2401 Taft Avenue, Manila 1004, Philippines
Abstract
This paper aims to study the relationship between expected stock market returns and conditional
volatility in the Philippines while also analyzing the asymmetric effects of good and bad news on
conditional volatility. Contrary to the significant risk-return trade-off prescribed by traditional
finance theory, there are an increasing number of empirical evidence on developed markets
claiming a negative relationship between stock market returns and volatility. However, only a
few studies were made on emerging markets such as the Philippines. By applying an asymmetric
volatility model such as the Exponential GARCH-In-Mean (EGARCH-M)and GJR GARCH
(Threshold-GARCH) models to the weekly Wednesday returns of the Philippine Stock Exchange
Composite Index over the period of January 5, 1994 to December 26, 2012,we found the
existence of a negative relationship, although insignificant, between stock market returns and
conditional volatility. Our results also showed that conditional volatility react to good and bad
news asymmetrically. That is, the arrival of bad news was found to have a greater impact on
conditional volatility than the arrival of good news. Since the study was conducted on a marketwide level, the researchers surmised that asymmetric volatility may be the result of a downmarket effect. We recommend that given asymmetric volatility, policy makers should continue to
regulate the financial market to ensure a smooth integration of the Philippine stock market to the
world economy.
Key Words: Asymmetric volatility, Risk-returns trade-off, EGARCH-M model, ThresholdGARCH model,the Philippines
_________________
Authors’ Email:
[email protected]
[email protected]
I. Introduction
It is well-known in financial research that stock return volatility is highly persistent. At
the same time, existing literature cannot find a definite relationship between asset returns and its
variance (used as a proxy for risk). As Li et al. (2005) pointed out, the relationship whether it is
positive or negative has been controversial. Theoretically, asset pricing models (Sharpe, 1964;
Linter, 1965; Mossin, 1966; Merton, 1973, 1980) link returns of an asset to its own variance or to
the covariance between the returns of stock and market portfolio.
As summarized by Baillie and DeGennarro (1990), most asset-pricing models (e.g.,
Sharpe, 1964; Linter, 1965; Mossin, 1966; Merton, 1973) show a positive relationship of
expected returns and volatility. Yet, there are also many empirical studies that implicates a
negative relationship between returns and volatility such as Black (1976), Cox and Ross (1976),
Bekaert and Wu (2000), Whitelaw (2000), Li et al. (2005) and Dimitrios and Theodore (2011).
For example, Li et al. (2005) found that there seemed to be a significant negative relationship
between expected returns and volatility in 6 out of the 12 largest international stock markets.
Bekaert and Wu (2000) explain that it appears that returns and conditional volatility are
negatively correlated in the equity markets which meant volatility is asymmetric.
Due to an increasing number of empirical evidences saying that negative (positive)
returns are generally associated with upward (downward) revisions of conditional volatility, this
phenomenon is often referred to as asymmetric volatility in the literature. (Goudarzi, 2011)As
Wu (2001) was quoted saying, “the presence of asymmetric volatility is most apparent during
stock market crashes when a large decline in stock price is associated with a significance
increase in market volatility.” The theory that considers the relationship between volatility and
equity price is called the “down market effect” which states variation in market volatility is
driven by variation in market conditions.
As mentioned earlier, numerous studies have been made in the literature with a focus on
asymmetric stock volatility among developed countries with mature markets. Only in the last
decade or so had studies been made on developing countries such as Aggarwal et al. (1999),
Bekaert and Wu (2000), Kassimatis (2002), Goudarzi, H. (2011) and N’dri (2007). Furthermore,
if asymmetric volatility is present during a crisis then it should also be noted that this event does
not only impact developed markets but emerging markets as well. Guinigundo (2010) reported
that by end-December 2008, the benchmark Philippine Stock Exchange Index (PSEi) had
declined by 48.3%, year-on-year. Thus, this study aims to investigate the asymmetric relation
between stock returns and its volatility in the Philippines. To model the asymmetry in stock
market volatility and allow the possibility to measure the different impact on the conditional
variance of bad and good news, the Exponential GARCH-In-Mean (EGARCH-M) model
proposed by Nelson (1991) and Engle et. al (1987) was used.
The rest of this paper is organized as follows. Section 2 is the review of literature. The
theoretical framework about the Philippine stock market and a brief background on the model
used are presented in Section 3. Section 4 shows the economic modelling while Section 5
describes data used in the study. Results which include the descriptive statistics and some
diagnostic inference could be seen in Section 6 while the empirical findings and analysis could
be found on Section 7.
II. Review of Related Literature
A. Conditional Volatility Measurements and Empirical Evidences
Models
of
conditional
heteroskedasticity
(i.e.
Autoregressive
Conditionally
Heteroskedastic (ARCH) and Generalized ARCH (GARCH models) have been
comprehensively employed to explain the behavior of the world’s developed stock
markets. For example, the U.S. stock return’s conditional volatility has broadly been
examined, most notably by French et al. (1987). Most of these studies find a significant
risk premium associated with the conditional volatility of excess returns.
The GARCH type models have been employed to explain the behavior of smaller
European and Emerging Stock Markets (ESM). Koutmos et al., (1993) examine the
stochastic behavior of the Athens Stock Exchange composite index. Using models of
conditional heteroskedasticity such as the EGARCH-M, they find that the volatility of
weekly returns is an asymmetric function of past shocks. They also report that there is no
evidence that domestic investors require higher returns for this increased risk.
Additionally, they find that in contrast to the existing evidence for the U. S. market,
positive return innovations have a greater effect on conditional return volatility than
negative innovations–a "reverse leverage effect.”
On the other hand, Aggarwal et al. (1999) examine shifts in volatility of the emerging
stock market returns and the events that are associated with the increased volatility. They
examine ten of the largest emerging markets in Asia and Latin America. Their results
show that large changes in volatility are related to important country-specific, political,
social and economic events.
Choudhry (1996), also models conditional heteroskedasticity to study volatility, risk
premia and persistence of volatility in six emerging stock markets before and after the
1987 stock market crash. He documents changes in the ARCH parameters before and
after the crash, but finds no evidence of risk premium and volatility persistence.
Brooks et al. (1997) and Appiah-Kusi and Pescetto (1998) focus on some of the African
Stock markets. They studied the effects of political regime change on stock market
volatility of South Africa. Using the EGARCH model, Appiah-Kusi and Pescetto (1998)
find that most of the African Stock markets are characterized by changes in volatility
levels, with some periods having extremely high volatility. The evidence they present
suggests there are considerable asymmetries in the response of volatility to news. These
asymmetries, however, are not always consistent with traditional leverage effect
explanation, since they find that sometimes good news causes a more accentuated
reaction than comparable bad news.
B. Philippine Stock Market
There is much need to study the literature in terms of the emerging markets. There are at
least four distinguishing features of emerging market returns: higher sample average
returns, low correlations with developed market returns, more predictable returns, and
higher volatility (Bekaert and Wu, 2000).
As a background, the Philippine Stock Exchange Composite Index (PSEi) is a weighted
average of 30 companies whose selection is based on a set criteria. The index measures
the relative change of the free float adjusted market capitalization of the 30 of the most
active and largest common stocks listed at the Stock Exchange (PSE,2011).
Studies made on the volatility of the PSEi had been limited. Bautista (2003) reported that
the Philippine high stock return volatility preceded a bust cycle. His study showed the
sensitivity of the Philippine stock market to drastic changes in the political environment
as well. Asai and Unite (2008) also studied the Philippine stock market volatility but with
its correlation to trading volume. The result showed that first, there is a negative
correlation between stock return volatility and variance of trading volume. Second,
there is a lack of effect of information arrivals onthe level of trading volume.
III. Theoretical Framework
A. Stock Returns and Volatility
Authors such as Black (1976), French, Schwert, &Stambaugh (1987), Poterba &
Summers (1986), Christie (1982), Merton (1980), suspected that stock returns are related
to changes in conditional volatility.
However, it is yet to be determined whether the causality runs from conditional volatility
to asset returns, or vice-versa. Bekaert (2000) highlighted the two different perspectives.
1.
The time varying risk premium.
This theory puts forth that conditional volatility causes the shocks in asset returns.
This means that an increase in conditional volatility will make the stock riskier.
The investor will then revise his expectations about the return of the asset which
in turn will cause a drop in the stock price. A volatility increase is a result of the
inflow of bad news while a dampening of volatility is caused by an influx of good
news.
2.
Changes in stock returns cause changes in conditional volatility.
The views included in this type of causality are the leverage effect, the down
market effect, and the constant costs view. The paper would focus on the down
market effect as discussed later.
In this paper, our primary concern is the time-varying risk premium theory which can be
measured through a GARCH-M type model. Yet we will also tackle the down market
effect since this study will be done on a market level rather than on a firm level. This will
be done through the incorporation of asymmetric volatility in the conditional variance
equation.
B. Stock Returns and Volatility Relationship
There are many views that reflect the relationship of the two variables, most of them are
conflicting. Whether these contradicting findings in empirical literature are a result of
misspecification of GARCH type models, since stock returns are found to be negatively
skewed and fat-tailed, remains an issue of great debate.
It may be confusing, but it is not uncommon to find different results about the peculiar
relationship. Taking note of Glosten et. al. (1993), it is yet to be determined whether
investors ask for a larger risk premium on average during the times when the security is
more volatile (risky). Hence a negative and positive relationship of conditional volatility
and asset returns may be theoretically correct. The proceeding parts tackle that
underlying theories of the nature of this relationship.
B.1. Positive Relationship
Linter (1965) states that the minimum expected return for an asset is a function of
risk-free rate, the market price of risk, and the variance of its own returns, among
other factors. Wherein the minimum expected rate of return is linearly related to
the risk of returns. This claim is also supported by Mossin (1986) who also
tackled the price of risk- if the investor is to take on a higher expected yield, he
has to carry a larger risk burden.
Glosten et. al. (1993) offered a more practical perspective of this relationship. For
example, during times of high volatility in the market, there are no risk-free
opportunities that are available to the investors. Therefore instead of the price of
the asset falling, it may go up considerably.
In empirical literature, French, Schwert, and Stambaugh (1987) found evidence of
this relationship as long as the level of volatility is predictable, as measured by
forecasts done through an ARMA model. Bae et. al. and Campbell and Hentschel
(1992) also found a positive relationship of risk and return after modelling for
regime changes in volatility.
B.2. Negative Relationship
Most empirical literature find that there is an inverse relationship between
conditional volatility and asset returns. The study of Glosten et. al. (1993) who
accounted for seasonality, asymmetric volatility, and included nominal interest
rates as an explanatory variable.
French et al. (1987) found a negative relationship of unpredictable volatility and
asset returns. Nelson (1991) found a negative yet insignificant relationship by
using an exponential ARCH model. He explains this relationship through the
volatility feedback effect. An increase in anticipated volatility of future income
will increase the required rate of return of the investor, since now he perceives the
stock as being more risky. This in turn, will decrease the present value of the
asset. Li et. al. (2005) also found a negative and significant relationship by using a
semi-parametric GARCH-M model.
It is important to note that Poterba & Summers (1986) finds that volatility has
little or no effect on stock prices especially in periods of low volatility.
C. Asymmetric Volatility
Many authors including Nelson (1991), Li et. al. (1993), Bekaert (2000), Figlewskie &
Wang (2000), Guidi & Gupta (2012) and Glosten et. al. (1993) found evidence of
asymmetric volatility in different markets. The underlying concept is that negative shocks
increase conditional volatility more than positive shocks, hence there is asymmetry on the
impact of good and bad news on the riskiness of the stock market. There are three
different theories that tend to explain this phenomenon namely leverage effect, down
market effect and constant costs effect. This paper as mentioned would be focusing on
the down market effect in light of the Global Financial Crisis.
C.1.
Down Market Effect
Figlewskie & Wang (2000) found inconsistencies while testing for the leverage
effect. They argued that if there is really a change in the financial leverage of the
firm because of changes in stock returns, there should be a permanent shift in
volatility. However it was shown that these volatility changes die down rather
quickly. Second, the authors analyzed whether changes in leverage that resulted
from the issuance or retirements of bonds and shares stock volatility. Their results
show that there is little or no impact on stock volatility. It appears that conditional
volatility is more related with falling stock prices. Thus it lead them to conclude
that the leverage effect is a misnomer, but rather it is more apt to call it the “down
market effect ".
This finding has also been supported by Aydemir et. al. (2006) wherein there is
evidence that variation in market volatility is driven by variation in market
conditions and financial leverage does not affect the skewness of returns.
Research by Baekert and Wu (2000) also proved that financial leverage enhances
asymmetry but it is only second to the feedback affect.
IV. Econometric Methodology
A. Measuring asymmetric volatility
For the following models, the innovations are assumed to be Gaussian distributed.
Moreover, we assume that the conditional variance follows an Exponential Generalized
Autoregressive Conditional Heteroskedasticity (EGARCH) process (Nelson, 1991). The
EGARCH model, a refinement of the GARCH model does not impose a non-negativity
constraint on market variance, and allows for conditional variance to respond
asymmetrically to return innovations of different signs.
The GARCH (Bollerslev, 1986) family of models assumes that the market conditions its
expectation of market variance on both past conditional market variance and past market
innovations. Bollerslev (1986) proposed an extension of the ARCH type models in order
to allow longer memory and a more flexible lag structure. Thus, Generalized Auto
Regressive Conditional Heteroskedasticity GARCH type models were born. A GARCH
(p,q) process is given by:
𝑘
ℎ
𝑅𝑡 = 𝛼0 + ∑ 𝛽𝑖 𝑋𝑖 + ∑ 𝜓𝑗 𝑅𝑡−𝑗 + 𝜖𝑡 (𝟏)
𝑖=1
𝑗=1
𝑤ℎ𝑒𝑟𝑒 𝜖𝑡 |Ωt−1 ~N(0, ht )
𝑝
𝑞
ht = α + ∑ 𝛽𝑖 ht−i + ∑ 𝜃𝑖 𝜖𝑡−𝑗 2 (𝟐)
𝑖=1
𝑗=1
𝑤ℎ𝑒𝑟𝑒 ht = σ2t |Ωt−1 (conditional variance dependent on the information set ′Ωt−1 ′)
With the following conditions
𝑝 ≥ 0, 𝑞 > 0
𝛼 ≥ 0, 𝜃 ≥ 0, 𝛽 ≥ 0
and:
𝜖𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑅𝑡 𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡, 𝑎𝑛𝑑
𝑋 ′ 𝑠 𝑎𝑟𝑒 𝑒𝑥𝑝𝑙𝑎𝑛𝑎𝑡𝑜𝑟𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠.
Equation (1) is called the mean equation while Equation (2) is called the equation for the
conditional variance. It can be clearly seen that the GARCH (p,q) models the conditional
variance as the function of both the squared innovations and its own past values. Notice
that when p=0, Equation (2) reduces to an ARCH(q) model.
However, it is important to note, that this equation restricts the parameters to be strictly
non-negative in order to satisfy the condition of a positive variance. This means that the
regular GARCH type models only capture the magnitude of the shocks, and tend to
neglect its sign.
A.1 EGARCH Model
To answer such a problem of not capturing signs, Nelson (1991) modified the GARCH
which led to the EGARCH model. By modifying 𝜖𝑡 or the residuals of the mean equation
such that
𝜖𝑡
√ht
= 𝑧𝑡 (𝟑)
𝑤ℎ𝑒𝑟𝑒 𝑧𝑡 ~𝑖𝑖𝑑 (0,1)and is called the standardized residuals
Furthermore, the EGARCH model is given by:
∞
ln(ht ) = αi + ∑ 𝛽𝑘 𝑔(𝑧𝑡−𝑘 ), 𝛽 ≤ 1 (𝟒)
𝑘=1
𝑤ℎ𝑒𝑟𝑒 𝑔(𝑧𝑡 ) = 𝜃𝑧𝑡 + 𝛾[|𝑧𝑡 | − 𝐸|𝑧𝑡 |]
Thus upon simplification, the EGARCH variance equation becomes
q
ln(σ2t |Ωt−1 )
q
p
= αi + ∑ 𝛾𝑗 [|𝑧𝑡−𝑗 | − 𝐸|𝑧𝑡−𝑗 |] + ∑ 𝜃𝑗 𝑧𝑡−𝑗 + ∑ ∆i ln(σ2t−i |Ωt−i−1 )
j=1
j=1
(𝟓)
i=1
Equation (4) employs the natural logarithm of the conditional variance in order to ensure
that the conditional variance remains non-negative. This is in contrast to the previous
approach of GARCH type models which impose conditions that the variables must be
strictly positive so that a linear combination of such will also be positive. Given this
freedom, 𝑔(𝑧𝑡 )will now be able to accommodate asymmetric volatility.
The parameter denoted by ' 𝜃′ capture the effect of the sign of the innovation. While
' 𝛾′measures the impact of the current innovation with its long run average, we can say
that it captures the magnitude (size) of the innovation. By incorporating an AR process
for the conditional variance, to allow for a longer memory, we arrive at Equation (5).
However it is important to note that Stata takes the 𝐸|𝑧𝑡−𝑗 | 𝑎𝑠 √2/𝜋 (Stata,2009)
Overall, the EGARCH model, unlike the linear GARCH models, uses the natural
logarithm of the conditional variance to relax the non-negativity constraint of the model's
coefficients and to allow for the persistence of shocks to the conditional variance.
A.2Threshold-GARCH Model
In order to verify the existence of asymmetric volatility in stock returns, we employ
another model first proposed by Glosten, Jagannathan, & Runkle (1993). By assigning a
dummy variable to negative returns, they were able to allow asymmetric effects of good
and bad news on conditional volatility. It is also known as Threshold (G)ARCH since we
consider 𝜖𝑡−1 = 0 as a point of separation of the impacts of negative and positive shocks.
(Enders, 2004)
Considering the TGARCH process:
2
2
ℎ𝑡 = 𝛼0 + 𝛼1 𝜖𝑡−1
+ 𝜗𝐷𝑡−1 𝜖𝑡−1
+ 𝛽ℎ𝑡−1 (6)
Where: ℎ𝑡 − 𝑡ℎ𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡
𝛼1 − 𝑡ℎ𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑓𝑜𝑟 𝑡ℎ𝑒 𝐴𝑅𝐶𝐻(1)𝑝𝑟𝑜𝑐𝑒𝑠𝑠
𝐷𝑡−1 = 1 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 𝜖𝑡−1 < 0
𝛽 − 𝑡ℎ𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑒𝑛𝑡 𝑓𝑜𝑟 𝑡ℎ𝑒 𝐺𝐴𝑅𝐶𝐻(1) 𝑝𝑟𝑜𝑐𝑒𝑠𝑠
2
When 𝜖𝑡−1 < 0, the effect of 𝜖𝑡−1on ℎ𝑡 is (𝛼1 + 𝜗)𝜖𝑡−1
and when 𝜖𝑡−1 ≥ 0 the effect of
2
𝜖𝑡−1 is 𝛼1 𝜖𝑡−1
.It can be clearly seen here, if that the coefficient 𝜗 is positive and
statistically significant, past lagged negative innovations (bad news) have a greater
impact on volatility than lagged positive innovations (good news).
However, Stata presents another variation of the TGARCH process wherein it modifies
the dummy variable 𝐷𝑡−1 on Equation (6) so that it will represent positive shocks rather
than negative shocks. (Stata,2009)
𝐷𝑡−1 = 1 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 𝜖𝑡−1 > 0
In our paper, in order to determine asymmetric volatility, we will look at the coefficient
' 𝜗′ . If ′𝜗′ is positive and statistically significant, we can conclude that positive
innovations create more volatility than negative innovations. However, if ′𝜗′ is negative
and statistically significant then we can conclude that negative innovations indeed
produce higher volatility than good news.
B. Measuring asymmetric volatility and expected stock returns
The EGARCH-M model developed by Nelson (1991) incorporates volatility in the mean
equation. That is, modifying equation (2) we have:
𝑘
ℎ
𝑅𝑡 = 𝛼0 + ∑ 𝛽𝑖 𝑋𝑖 + 𝛿ℎ𝑡 + ∑ 𝜓𝑗 𝑅𝑡−𝑗 + 𝜖𝑡 (𝟕)
𝑖=1
𝑗=1
Equation (7) postulates that aside from the explanatory variables and the auto-regressive
terms, the return of the asset at time t is also affected by its conditional variance. The
study will estimate and interpret the parameter ′𝛿′ for this will determine both the nature
and significance of the impact of conditional volatility on stock returns.
C. Determination of the ARMA component of the Mean Equation
In determining the right model that fits the mean equation, employing the methodology
employed by Box and Jenkins (1976) we can determine the correct ARMA
(Autoregressive- Moving Average) process using three steps:
C.1. Identification.
In order to find out the order of the ARMA process, we use the autocorrelation function
(ACF) and the partial autocorrelation function (PACF). The ACF measures the
correlation of lag(t) and lag (t+k), while the PACF measures the correlation of lag(t) and
lag(t+k) while controlling for the effects of intermediate lags. (Gujarati,2009).
Enders(2004) created a table in order to determine the best-fitting ARMA model for the
time-series process
Process
ARMA(p,q)
AR(p)
MA(q)
ACF
Decay (direct or oscillatory)
beginning at lag q.
Decays toward zero. Coefficients
may oscillate
Significant spikes through lag q.
All ACF for lags greater than q =0
PACF
Decay (direct or oscillatory)
after lag p.
Spikes through lag p. All
PACF=0 for lags greater than
p.
Declines exponentially.
C.2. Estimation
After the identification of the appropriate lags of the ARMA process, estimation can be
done through either simple least squares method or nonlinear estimation methods.
(Gujarati,2009)
C.3. Diagnostic Checking
This basically involves testing the residuals of the estimated ARMA process if it follows
a white noise process. (Gujarati,2009). However in the case of this paper, we will also
utilize the different Information Criteria and techniques of determining the correct lag
length in ad-hoc estimation. These processes will also be used in determining the lag
length of the EGARCH model.
D. Determination of the 'Best Fitting' volatility model
In order to determine the most appropriate EGARCH model, we employ the following
techniques:
D.1. Akaike Information (AIC) and Schwartz Bayesian (SBC/ BIC/ SBIC) Criterion
We look at the two information criterion in our search for a more parsimonious model.
The AIC is defined as
𝐴𝐼𝐶 = 𝑇𝑙𝑛(𝑅𝑆𝑆) + 2𝑛(8)
and the SBC as
𝑆𝐵𝐶 = 𝑇𝑙𝑛(𝑅𝑆𝑆) + 𝑛𝑙𝑛(𝑇)(9)
where:
𝑇 − 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
𝑅𝑆𝑆 − 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑑𝑒𝑙
𝑛 − 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑏𝑒𝑖𝑛𝑔 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑
As it can be seen in the equations, the criterions test whether adding more lags is worth
its weight in reducing the RSS. The intuition behind this is that by adding another
explanatory variable, n will increase and if it does not add significantly to the explanatory
power of the model, both AIC and SBC will increase. It is important to note however,
that SBC will point out to the more parsimonious model than AIC and it has superior
large sample size properties than AIC. While AIC tends to select an overparameterized
model. In an ideal situation, both Information Criterion will be very small (it can also be
negative). As the model fit improves, both criterion will approach−∞. A model is said to
be superior than the other if it yields a lower AIC or SBC. (Enders, 2004)
D2. Ad-Hoc estimation method
According to the method used by Alt(1942) and Tinbergen (2006)in trying to search for
the appropriate lag length, one can add lags sequentially. This procedure halts once the
regression coefficients of the lagged variables start becoming statistically insignificant or
when the coefficients start to switch signs.
In our post estimation, we use the various tests proposed by Engle and Ng (1993). The
tests indicate whether the squared normalized residuals can be predicted by variables in
the past that are not included in the volatility model. If these variables can predict our 𝑧𝑡2 ,
then our model is not correctly specified. The sign bias test considers a dummy
−
variable, 𝑆𝑡−1
that takes a value of one whenever 𝜖𝑡−1 < 0 and zero otherwise. The
−
negative size bias test on the other hand, uses the variable 𝑆𝑡−1
𝜖𝑡−1 which captures the
effects of large and small negative returns on the volatility that are not being predicted in
+
the model. While the positive size bias test uses the variable 𝑆𝑡−1
𝜖𝑡−1 in order to capture
the impacts of small and large positive returns on volatility that the underlying volatility
+
−
model does not capture, where 𝑆𝑡−1
= 1 − 𝑆𝑡−1
. Hence we can form the following
equation:
+
−
−
𝑧𝑡2 = 𝛼 + 𝜓1 𝑆𝑡−1
+ 𝜓2 𝑆𝑡−1
𝜖𝑡−1 + 𝜓3 𝑆𝑡−1
𝜖𝑡−1 + 𝛽 ′ 𝑋𝑖 + 𝜀𝑡 (10)
where:
𝑧𝑡2 − 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠
𝛽 ′ − 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝑜𝑓 𝑜𝑡ℎ𝑒𝑟 𝑒𝑥𝑝𝑙𝑎𝑛𝑎𝑡𝑜𝑟𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑚𝑜𝑑𝑒𝑙
(a) The sign bias test is the t-ratio for the coefficient '𝜓1 ′, when 𝜓2 & 𝜓3 = 0.
(b) The negative size bias test is the t-ratio for the coefficient '𝜓2 ′, when 𝜓1 &𝜓3 = 0
(c) The positive size bias test is the t-ratio ′𝜓3 ′, when 𝜓1 &𝜓2 = 0
We can also test for the joint significance of the three variables, the test statistic is T
multiplied by the R-squared of the regression. The LM statistic follows a chi-square
distribution with three degrees of freedom (when 𝛽 is set to zero) .If the underlying
volatility model that was used is correct, then 𝜓1 = 𝜓2 = 𝜓3 = 0,𝛽 ′ = 0, and 𝜀 is i.i.d.
V. Data
The raw data comprised of the Philippine Stock Exchange Composite Index (PSEi)’s
weekly closing prices from January 5, 1994 and December 28, 2012. The weekly closing prices
were taken on Wednesdays or at the previous business day if the Wednesday is a holiday. Being
a snapshot of the market’s overall condition, the PSEi is composed of the 30 largest and most
active common stocks listed at the exchange based on their free float-adjusted market
capitalization. (PSE, 2011) All data in the study were obtained from the Philippine Stock
Exchange (PSE) and had a total of 989 observations.
Despite daily return data being preferred to weekly or monthly return data, however
“daily data are deemed to contain ‘too much noise’ and is affected by the day-of-the-week effect
while monthly data are also affected by the month-of-the-year effect”. (Roca, 1999)Ramchand
and Susmel(1998), Aggarwal et al. (1999), and Tay and Zhu (2000) were among the large
number of studies that have employed weekly data instead of daily or monthly data in order to
provide a sufficient number of observations required to estimate the GARCH models without the
noise of daily data.
As such, the weekly return series is generated from the following equation:
Rt = (100)*(ln(Pt)-ln(Pt-1))
(11)
where ln is the natural logarithm operator; t represents time in weeks; Rt is the return for period
t,; Pt is the index closing price for period t. Each return series is therefore expressed as a
percentage. Modeling an index in this manner is typical in the literature (Nelson, 1991). As seen
in Figure 1, it shows you the graph of the residuals of weekly Wednesday return of the PSEi.
VI. Results and Methodology
Pre-testing
Table 1 depicts the results of the normality test and the descriptive statistics for the
weekly returns. Under assumptions of normality, skewness and kurtosis have asymptotic
distributions of N(0) and N(3) respectively (Xu, 1999). Empirical distributions of weekly returns
differ significantly from a normal distribution. There was an indication of negative skewness
(Skw= -.158015) which indicates that the index declines occur more often than it increases. The
kurtosis coefficient was positive, having a relatively high value for the return series (Kurt =
4.975861) this points out, that the distribution of returns is leptokurtic. The weekly return series
being negatively skewed implies that the distribution is not symmetric. Furthermore, by testing
for the joint significance of skewness and kurtosis using the procedure proposed by D'Agostino
et al. (1990), the null-hypothesis of normality was rejected.
The Augmented Dicky-Fuller (ADF)statistic indicates that the stock return series is
stationary. The ADF test statistics rejected the hypothesis of the existence of a unit root in the
returns series at 1% level of significance. The absolute value of the ADF statistic 30.786
exceeds the MacKinnon critical value of 3.430. These properties of the PSEi returns series were
consistent with other financial times series in the literature.
The Breusch-Godfrey LM Test would test for autocorrelation of the error terms of the
series. The LM test included 24 lags of the return series as seen in Table 2. It did not reject the
null hypothesis of no serial correlation, the test statistics confirmed the absence of
autocorrelation in the first and the higher orders but showed autocorrelation in the 2nd, 3rd and 8th
lag. Overall, the PSEi returns exhibit little correlation. The existence of autocorrelation on some
lags is possibly due to non-synchronous trading of the stocks that make up the PSEi. (Lo and
MacKinlay, 1990) It means that closing weekly prices of each stock that comprised the PSEi are
not set simultaneously at the end of the business day which is against the usual assumption that
these stocks are sampled simultaneously.
Lastly, before the estimation of ARCH models, the ARCH-LM test was done to indicate
the presence of ARCH effect in the squared residuals of the mean equation. The ARCH type
models were estimated for PSEi returns series using the robust method of BollerslevWooldridge’s quasi-maximum likelihood estimator (QMLE). Based on Table 3, there are ARCH
effects on a .001 level of significance on the residuals of the 1stlag,.01 level of significance on
the residuals of the 2nd lag and .05 level of significance on the residuals of the 3rd, 7th and 19th lag
Determination of the Mean Equation and the 'Best Fitting' Model
By selecting the best ARMA order in the mean equation, we began by taking a look at the
autocorrelation function (ACF) of the returns. Based on Figure 2, we can say that it starts to
decline geometrically after lag (3) with significant spikes on lag 2 & 3, however it is important to
note that the first lag does not exhibit a significant spike. Furthermore, taking a look at the
partial autocorrelation function (PACF) of the returns on Figure 3, we can come up to the same
conclusion. According to Enders (2004), such behavior of both the PACF and ACF indicate an
ARMA(3,3) process. However, one might argue that we should have used an ARMA(8,8) model
yet we chose the more parsimonious models in order to lessen the toll on the degrees of freedom.
Using this premise, we also refrained from modeling the mean equation with an MA(3)
component since we are incorporating the past innovations in the variance equation. We also test
for this conclusion using the different methods proposed by Box Jenkins (1976) and Gujarati
(2009). Thus, we compared 4 models namely:ARMA(3,3), AR(3), AR(2), & AR(1).
Based on Table 4, it is clear that the AR(3) model yields the lowest values for the AIC
and the BIC. Furthermore, the ARMA(3,3) model does not exhibit and significant coefficients
and there is erratic behavior in the part of the coefficient of its AR(3) component.
After selecting the best ARMA order which is the AR(3) process, it is now time for fit the
best EGARCH model. To do that, we will test for the best fitting EGARCH lag order for our
variance equation. We tested for the following specifications: EGARCH (1,1), EGARCH (1,2),
EGARCH(1,3), EGARCH (1,4) and EGARCH (2,1).
Basing on the information criterion, the EGARCH(1,3) model dominates over the 4
others. However, the sign of the second lag of the GARCH component or GARCH (2) switched
from positive to negative. This means that higher realized conditional volatility dampens current
conditional volatility, this is against our apriori expectation of the GARCH effect, wherein there
is volatility clustering since high past conditional volatility contributes positively to current
volatility. This leads us to select the second best model, EGARCH(1,4) unfortunately it also
experiences the same problem as the former.
Next, we compare the EGARCH(1,1) and the EGARCH(1,2). It is clearly seen that
EGARCH(1,1) model dominates over the EGGARCH (1,2) in terms of the information criterion.
However, we select the EGARCH(1,2) model for the following reasons. The first is that the
difference between the information criterion of both models is very small, for the AIC it is .1 and
the BIC it is 5, therefore adding a second lag for the EGARCH component did not penalize as
much. The second Gujarati (2009) suggested the ad-hoc estimation procedure that states that we
can continue on adding lags until the coefficients become statistically insignificant and/or
switches signs.
Finally in previous tests for the lag order of the ARMA model and the ARCH effect tests,
we discovered that the first lag does not have a statistically significant effect while the second
and third lags do. This may imply that there is a slow incorporation of information in the returns
of the PSEi. In order to acknowledge this, we chose a higher ordered EGARCH (1,2).
Diagnostic Tests
We tested whether the squared standardized are independent andidentically distributed.
This is what we call our post-estimation results to verify if EGARCH (1,2) is the best fitted
model. We used four tests proposed by Engle and Ng (1993): the Size-Bias test, the NegativeSize-Bias test, the Positive-Size-Bias test and joint test of the three.
As seen in Table 6, for the size bias test that the 't' test for the coefficient of sign bias test
−
𝜓1 𝑆𝑡−1
is statistically insignificant with a p-value of .514. Therefore it implies that our model was
able to capture leverage effects. This conclusion can be supported by running the test described
by Enders (2004), the methodology here is to run the squared standardized residuals taken from
the underlying volatility model with lagged standardized residuals. If the joint test for
significance of the coefficients of the lagged standardized residuals, then the underlying
volatility does not capture remaining leverage effects in the residuals. Upon doing this, the pvalue of the F-test is 0.1301.
+
In the same table, the coefficients of both the positive size bias test (𝜓3 𝑆𝑡−1
𝜖𝑡−1) and the
−
negative size bias test (𝜓2 𝑆𝑡−1
𝜖𝑡−1 ) are statistically insignificant at a 5% level. Therefore it can
be said that the underlying volatility model used, EGARCH(1,2), captures the effects of both
large and small negative(positive) returns. Testing for the joint significance of '𝜓1 ′, ′𝜓2 ′,&′𝜓3 ′
concludes that jointly, all three coefficients are equal to zero (p-value of .5910).
VII.
Empirical Findings and Analysis
Findings indicate that the PSEi returns can be reasonably modelled with an AR(3)
component for the mean equation and an EGARCH(1,2) for the variance equation . The results
are presented in Table 7. From our post-estimation testing, we conclude that EGARCH-M (1, 2)
is adequate and is indicative of asymmetric volatility in the PSEi.
The modified model based on the results of the EGARCH-M (1, 2) are presented:
𝑅𝑡 = .181 + .0230𝑅𝑡−1 + .0592𝑅𝑡−2 + .0450𝑅𝑡−3 − .0104ht + 𝜖𝑡 (𝟏𝟐)
ln(ht ) = .0907 + .129[|𝑧𝑡−1 | − 𝐸|𝑧𝑡−1 |] − .115(𝑧𝑡−1 ) + .260 ln(ht−1 ) + .703ln(ht−2 )(𝟏𝟑)
In order to verify the traditional notion of the risk-return trade-off, it is imperative to
examine the coefficient(δ) of the conditional variance (ℎ𝑡 ) in the mean equation (eq. 12). In Table 7,
we report the point estimate of the price of market risk (δ).
As our results show, the parameter of the conditional variance in equation 12 is -.0104. This
implies that there is a negative, yet statistically insignificant relationship between conditional
volatility and returns on the PSEi. This result is consistent with the findings of Fama and Schwert
(1977), Campbell (1987), Nelson(1991), Glosten Jagannathan and Runkle (1993) whose studies
showed a negative relationship between stock returns and its variance.
The coefficient 𝑜𝑓 [|𝑧𝑡−1 | − 𝐸|𝑧𝑡−1 |] with a value of 0.129 is significant at a 0.001 level.
The positive coefficient implies that large unpredictable innovations (whether positive or negative)
have destabilizing effects on the conditional variance of stock returns.
To analyze asymmetric volatility, we take a look at the coefficient of 𝑧𝑡−1 . Based on our
results, the coefficient takes on a value of -.115 and is statistically significant at the 0.001 level.
This means that whenever the innovation 𝜖𝑡−1 is negative, which implies that 𝑧𝑡−1 is negative,
conditional volatility will tend to rise. Conversely, when the innovation 𝜖𝑡−1 is positive, which
means there was an inflow of good news, conditional volatility will tend to fall. This confirms
our hypothesis that there is asymmetric volatility in the returns of the PSEi. In other words, there
is an existence of the 'down market effect' or the 'leverage effect'. Extending on this outcome, it
might indicate that bad news creates speculative bubbles, particularly when the socioeconomic and
political circumstances are very unpredictable. (Ogum et. al., 2008).
We also look at whether shocks in the conditional volatility persist over time. Our findings
show that the GARCH (∆) parameters of .260 and .703 are statistically significant at .05 and .001
level of significance. This confirms our notion that shocks in the conditional volatility persist,
although it dies down quickly. This finding is not surprising and has already been documented
(Campbell, 1998). This suggests that once volatility increases, it is likely to remain high for over two
weeks based on the models used and their results. Again, this supports the spirit of the ARCH
models, which acknowledges the presence of volatility clustering.
Furthermore, in order to verify the asymmetry of the impacts of good and bad news on
conditional volatility, we also present the results of the Threshold GARCH (TGARCH)model.
Based on previous literature, it can be concluded that TGARCH (1, 1) is adequate in detecting
asymmetric volatility in the returns of the PSEi.
The following coefficients for the mean and variance equation were estimated using the
TGARCH (1, 1):
𝑅𝑡 = −.304531 + .0541989𝑅𝑡−1 + .0715812𝑅𝑡−2 + .064076𝑅𝑡−3 + .0333968ℎ𝑡 + 𝜖𝑡 (𝟏𝟒)
2
2
ℎ𝑡 = .3352319 + .0728316𝜖𝑡−1
− .0982595𝐷𝑡−1 𝜖𝑡−1
+ .9444847ℎ𝑡−1 (𝟏𝟓)
From Table 8, we notice that the coefficient ′𝜗′ which represents asymmetry is negative with a
value of -.0982595. First, it is important to highlight that Stata presents another variation of the
TGARCH process wherein it modifies the dummy variable 𝐷𝑡−1 on Equation 6 so that it will
represent positive shocks rather than negative shocks. (Stata, 2009). Therefore, the sign of ′𝜗′
supports our finding that the arrival of bad news (negative innovations) induce greater volatility
than the arrival of good news (positive innovations).
Looking at the intercept of the mean equation (equation 14) and the coefficients of the AR(3)
component, it implies that the distribution of the returns of the PSEi are skewed to the left. This
means that there is a greater probability of incurring negative returns on the index. This finding
is also in line with our descriptive statistics.
VIII. Conclusion
The volatility of PSEi stock returns from January 1994 to December 2012 have been
investigated and modeled using two asymmetric conditional volatility models, the EGARCH-M
(1, 2) and the TGARCH-M (1, 1). We found that PSEi returns series exhibit leverage effects and
showed persistence of chocks in the conditional volatility are persistent for at least 2 weeks.
These are in line with findings done on developed stock markets. Our results also showed a
negative yet insignificant relationship between risk and return like many recent studies.
The presence of asymmetric volatility in the returns of the PSEi, gives light to the
presence of the down market effect as modeling the index does not necessarily capture the
changes in the leverage of each firm. Furthermore, Figlewskie & Wang (2000) argued when
firms change their leverage, there should be a permanent shift in volatility. However, as we have
shown, persistence in volatility changes die down after at least 2 weeks.
The down market effect states that conditional volatility is more related to falling stock
prices rather than changes in the financial leverage of firms. It may explain why the market was
highly volatile during the periods covering the two global financial crises (Please see figure 4).
Given the sensitivity of the PSEi to the inflow of information, we recommend that policy makers
maintain a well-regulated financial market in order to facilitate a smooth integration of the
Philippine market with the global economy. By taking immediate action to neutralize the effects
brought about by economic reverses and social or political upheavals, we can promote the
Philippines as a haven for safe investments.
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X.
Appendix
Figure 1: Weekly PSEi Returns Residuals
RESID
16
12
8
4
0
-4
-8
-12
-16
94
96
98
00
02
04
06
08
10
Figure 2: Autocorrelation Function (ACF) of the PSEi return series
12
Figure 3: Partial Autocorrelation Function (PACF)of the PSEi return series
-20
-10
0
returns
10
20
Figure 4: Time series line representing the weekly returns of the Index
1994w11996w11998w12000w12002w12004w12006w12008w12010w12012w1
date
Table 1. Descriptive Statistics and Normality Test of Return Series Data for PSEi
Statistics
Mean
Median
Maximum
Minimum
Standard Deviation
Skewness
Kurtosis
Adj. chi2D’Agostino Test of normality
Variable: Return
.058373
.004023
13.79998
-16.16304
3.476322
-.158015
4.975861
43.28***
(0.0000)
Note: *** indicates significance at the 0.001 level.
Table 2.Breusch- Godfrey Serial Correlation LM Test
Return Series: Constant and Lags
Constant
p(1)
p(2)
p(3)
p(4)
p(5)
p(6)
p(7)
p(8)
p(9)
p(10)
p(11)
p(12)
p(13)
p(14)
p(15)
p(16)
Coefficients
-0.000565
0.023803
0.083642***
0.075107***
-0.039108
-0.004407
-0.038426
0.003851
0.108251***
-0.028854
-0.020882
0.011479
-0.023639
0.033747
-0.027314
0.001858
-0.047418
p(17)
p(18)
p(19)
p(20)
p(21)
p(22)
p(23)
p(24)
0.041753
0.002127
-0.047129
-0.018487
-0.02436
0.01211
0.007985
-0.015475
Null hypothesis= No Serial Correlation
Note: *** indicates significance at the 0.001 level.
Table 3. ARCH LM Test
Coefficients
Return Series: Constant and Lags
4.3862***
Constant
0.135993***
p(1)
0.088274**
p(2)
0.074849*
p(3)
-0.045195
p(4)
-0.022454
p(5)
0.038201
p(6)
0.077281*
p(7)
0.055292
p(8)
-0.004427
p(9)
0.021361
p(10)
0.034271
p(11)
-0.023008
p(12)
-0.031024
p(13)
0.057364
p(14)
0.0642
p(15)
-0.038412
p(16)
0.01088
p(17)
0.015231
p(18)
0.071549*
p(19)
0.022604
p(20)
0.037079
p(21)
-0.020784
p(22)
-0.014858
p(23)
0.014092
p(24)
Null hypothesis= No ARCH Effect
Note: *** indicates significance at the 0.001 level, ** indicates significance at the 0.01 level, *indicates
significance at the 0.05 level
Table 4. Model Selection
ARMA(3,3)
AR(3)
AR (1)
-0.156
0.0144
(0.551)
(0.573)
AR (2)
-0.0865
0.0793**
(0.722)
(0.004)
AR (3)
-0.457
0.0746**
(0.051)
(0.008)
MA (1)
0.172
(0.501)
MA(2)
0.164
(0.474)
MA (3)
0.554*
(0.017)
Constant
3.446***
3.453***
(0.000)
(0.000)
Number of Observations
987
987
Akaike Information Criterion
5259.2
5257.1
Schwartz Bayesian Criterion
5298.3
5281.6
AR(2)
0.0203
(0.411)
0.0810**
(0.003)
AR(1)
0.0223
(0.363)
3.462***
(0.000)
987
5260.6
5280.2
3.474***
(0.000)
987
5265.1
5279.7
Note: *** indicates significance at the 0.001 level, ** indicates significance at the 0.01 level, *indicates
significance at the 0.05 level
Constant
ARCH-M
AR (1)
AR (2)
AR (3)
ARCH (1)
EGARCH
(1,1)
returns
0.0578
(0.841)
-0.000278
(0.992)
0.0360
(0.289)
0.0624*
(0.050)
0.0486
(0.139)
-0.0727***
(0.000)
Table 5. EGARCH Comparison
EGARCH
EGARCH
(1,2)
(1,3)
returns
returns
0.181
0.245
(0.511)
(0.302)
-0.0104
-0.0161
(0.684)
(0.467)
0.0230
0.0278
(0.500)
(0.401)
0.0592*
0.0688*
(0.049)
(0.043)
0.0450
0.0273
(0.174)
(0.402)
-0.115***
-0.0564***
(0.000)
(0.000)
EGARCH
(1,4)
returns
0.392
(0.086)
-0.0313
(0.144)
0.0119
(0.723)
0.0763*
(0.017)
0.0286
(0.378)
-0.0991***
(0.000)
ARCH (2)
Gamma (1)
0.0859***
(0.000)
0.129***
(0.000)
0.119***
(0.000)
0.197***
(0.000)
0.972***
(0.000)
0.260*
(0.044)
0.703***
(0.000)
1.860***
(0.000)
-1.740***
(0.000)
0.850***
(0.000)
0.0673**
(0.002)
0.0907**
(0.005)
0.0718**
(0.001)
0.977***
(0.000)
0.0451
(0.258)
-0.901***
(0.000)
0.842***
(0.000)
0.0898**
(0.004)
987
987
987
987
987
5147.9
5148.0
5123.1
5118.6
5149.9
5191.9
5196.9
5176.9
5177.3
5203.8
Gamma (2)
GARCH (1)
GARCH (2)
GARCH (3)
GARCH (4)
Constant
Number of
Observations
Akaike
Information
Criterion
Schwartz
Bayesian
Criterion
EGARCH
(2,1)
returns
0.117
(0.685)
-0.00512
(0.846)
0.0318
(0.366)
0.0578
(0.068)
0.0483
(0.132)
-0.124***
(0.001)
0.0590
(0.116)
0.126
(0.061)
-0.0486
(0.463)
0.978***
(0.000)
0.0543**
(0.010)
Note: It is important to note however that the following models did not converge: EGARCH (3,1) (3,2), (2,2)
Note: *** indicates significance at the 0.001 level, ** indicates significance at the 0.01 level, *indicates
significance at the 0.05 level
Table 6. Post- Estimation Result
EGARCH (1,2)
t-statistics
−
Size Bias Test (𝜓1 𝑆𝑡−1 )
0.65
(0.514)
+
Positive Size Bias Test(𝜓3 𝑆𝑡−1 𝜖𝑡−1 )
-.16
(0.869)
−
Negative Size Bias Test (𝜓2 𝑆𝑡−1
𝜖𝑡−1 )
.15
(0.885)
Joint Significance('𝜓1 ′, ′𝜓2 ′, &′𝜓3 ′)
(.5910)
Table 7. Estimated EGARCH-M (1,2) Result
Equation Coefficient
Value
AIC
BIC/SIC
Mean
Intercept
0.181
Equation
(.511)
In-Mean
-0.0104
(.684)
AR (1)
0.0230
(0.500)
AR (2)
0.0592*
(0.049)
AR (3)
0.0450
5148.0
5196.9
(0.174)
Variance
Intercept
0.0907**
Equation
(.005)
ARCH (1)
-0.115***
(0.000)
GARCH (1)
0.260*
(0.044)
GARCH (2) 0.703***
(0.000)
Gamma
0.129***
(0.000)
Note: *** indicates significance at the 0.001 level, ** indicates significance at the 0.01 level, *indicates
significance at the 0.05 level
Table 8. Estimated TGARCH-M Result
Equation Coefficient
Value
Mean
Intercept
-.3045311
Equation
(.297)
In-Mean
.0333968
(.231)
AR (1)
0.0541989
(0.090)
AR (2)
0.0715812*
(0.023)
AR (3)
0.0640765
(0.052)
Variance
Intercept
0.3352319***
Equation
(0.000)
ARCH (1) 0.0728316***
(0.000)
GARCH (1) 0.9444847***
(0.000)
TARCH (1) -0.0982595***
(0.000)
Note: *** indicates significance at the 0.001 level, ** indicates significance at the 0.01 level, *indicates
significance at the 0.05 level