3. 3
UNIT 4: OUTLINE
• Introduction
• Characteristics of Volatility
• Structure of an Arch Model
• Model Building
• Arch Models
• Garch Model—Generalized Arch
• Allowing for an Asymmetric Effect
• Garch-in-Mean and Time-Varying Risk Premium
4. 4
Introduction
An important measure in finance is the risk associated with an asset
and asset volatility is perhaps the most commonly used risk measure.
Most financial studies involve returns, instead of prices, of assets.
Direct statistical analysis of financial prices is difficult, because
consecutive prices are highly correlated, and the variances of
prices often increase with time. This makes it usually more
convenient to analyze changes in prices (return). Results for
changes can easily be used to give appropriate results for prices.
5. 5
Introduction
Definition:
Volatility is a rate at which the price of a security increases or
decreases for a given set of returns. It shows the range to which the
price of a security may increase or decrease.
Computing volatility from historical series of actual prices
• Prices of an asset at (T + 1) time points {Pt ,t = 0, 1, 2, . . . , T}
• Returns of the asset for T time periods
• Rt = log(Pt/Pt−1), t = 1, 2, . . . ,T
Note: In volatility modeling, the financial series must be converted to
returns of the assets using the above formula.
6. 6
Description:
Volatility measures the risk of a security. It is used in option pricing formula to
gauge the fluctuations in the returns of the underlying assets. Volatility indicates
the pricing behavior of the security and helps estimate the fluctuations that may
happen in a short period of time. If the prices of a security fluctuate rapidly in a
short time span, it is termed to have high volatility. If the prices of a security
fluctuate slowly in a longer time span, it is termed to have low volatility.
In this unit, our interest is to study how to model the volatility of an asset return. we
are
concerned with stationary series, but with conditional variances that change over
time.
The model we focus on is called the autoregressive conditional heteroskedastic
(ARCH)
model. The name — ARCH — conveys the fact that we are working with time-
varying variances (heteroskedasticity) that depend on (or are conditional on)
lagged effects (autocorrelation).
7. 7
CHARACTERISTICS OF VOLATILITY
From Figure 1, the following features can be observed:
• The values of these series change rapidly from period to period in an
apparently unpredictable manner; we say the series are volatile.
• There are periods when large changes (volatility) are followed by
further large changes and periods when small changes are followed
by further small changes. In this case the series are said to display
time-varying volatility as well as ‘‘clustering’’ of changes.
• Volatility evolves over time in a continuous manner—that is, volatility
jumps are rare.
• Volatility does not diverge to infinity—that is, volatility varies within
some
fixed range. Statistically speaking, this means that volatility is often
stationary.
1 1
8. 8
• Figure 1: Returns to shares in Brighten Your Day Lighting
• Volatility seems to react differently to a big price increase or a big price drop, referred to as
the leverage effect.
• Figure 2 shows the histograms of the returns. All returns display non-normal properties. Note
that there are more
observations around the mean and in the tails. Distributions with these properties, more
peaked around the mean and relatively fat tails, are said to be leptokurtic.
9. 9
Note: The ARCH model can be used to capture changing volatility and the
leptokurtic nature of the distribution for a financial series.
10. 10
STRUCTURE OF AN ARCH MODEL
Let yt be the log return of an asset at time index t. The basic idea
behind volatility study is that the series {yt} is either serially uncorrelated
or with minor lower order serial correlations, but it is a dependent
series.
To put the volatility models in proper perspective, it is informative to
consider the conditional mean and variance of yt given It−1; that is
• (1)
• (2)
• (3)
11. 11
Equations (2 and 3) describe the autoregressive conditional
heteroskedastic (ARCH) class of models. The second equation
says that the error term is conditionally normal where represents
the information available at time with mean 0 and time-varying
variance, denoted as , following popular terminology. The third
equation models as a function of a constant term and the
lagged error squared
12. 12
MODEL BUILDING
• Building a volatility model for an asset return series consists of four steps:
• Specify a mean equation by testing for serial dependence in the data and,
if necessary, building an econometric model (e.g., an ARMA model) for the
return series to remove any linear dependence.
• Use the residuals of the mean equation to test for ARCH effects.
• Specify a volatility model if ARCH effects are statistically significant, and
perform a joint estimation of the mean and volatility equations.
• Check the fitted model carefully and refine it if necessary.
In what follows, we describe each step of the modeling procedure in detail and introduce
various volatility models.
Testing for ARCH effects
Before we can model volatility, we need to test for its presence in the returns series. A
Lagrange multiplier (LM) test is often used to test for the presence of ARCH effects.
To perform this test, first estimate the mean equation, which can be a regression of
the variable on a constant or may include other variables. Then save the estimated residuals
and obtain their squares .
ˆt
e
13. 13
To test for first-order ARCH, regress on the squared residuals
lagged ,
where vt is a random term. The null and alternative hypotheses are
• If there are no ARCH effects, then and the fit of will be poor, and the equation will
be low. If there are ARCH effects, we expect the magnitude of to depend on its
lagged values, and the will be relatively high. The LM test statistic is where T is the
sample size, 9 is the number of terms on the right-hand side of the equation, and is
the coefficient of determination.
14. 14
If the null hypothesis is true, then the test statistic is distributed (in
large samples) as , where q is the order of lag, and is the number of
complete observations; in this case, . If , then we reject the null
hypothesis that and conclude that ARXH effects are present.
Hypothesis
Ho: There is no ARCH effect in the data
H1: There is an ARCH effect
15. 15
ARCH LM-test; Null hypothesis: no ARCH effects
data: vv
Chi-squared = 22.203, df = 12, p-value = 0.03531
Conclusion
Since the p-value (0.03531) is less than the significance level, 0.05, we reject
the null hypothesis. Thus, there is an ARCH effect in the data.
16. 16
ARCH MODELS
The first model that provides a systematic framework for volatility modeling is the
ARCH model of Engle (1982). The basic idea of ARCH models is that (i) the shock
et of an asset return is serially uncorrelated, but dependent, and (ii) the dependence of
et can be described by a simple quadratic function of its lagged values. Specifically,
an ARCH(q) model assumes that
NB: The coefficients, and, have to be positive to ensure a positive variance. The
coefficient must be less than 1, or ht will continue to increase over time, eventually
exploding.
17. 17
Illustration
ARCH(2) Model
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 2.135e+00 2.048e-01 10.425 <2e-16 ***
a1 1.455e-01 6.312e-02 2.305 0.0212 *
a2 1.181e-14 4.381e-02 0.000 1.0000
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
How many squared residuals lagged should be included in the ARCH model?? That is
the order determination of ARCH model
18. 18
Order Determination: Two approaches
If an ARCH effect is found to be significant:
• PACF approach: One can use the PACF of to determine the
ARCH order. The number of significant lags/spikes of PACF of
suggests the order of the ARCH model. NB: This approach may
not be effective when the sample size is small.
• Information criteria approach: Many competing models can be
fitted to the data, the model or order with the minimum
information criterion (whether AIC, SIC etc) is selected as the
appropriate order of the ARCH model.
19. 19
ARCH(9) ARCH(1) ARCH(2) ARCH(3)
AIC -2.212817 -2.241224 -2.234569 -2.225419
BIC -2.043678 -2.195095 -2.173064 -2.148538
Shibata -2.217466 -2.241586 -2.235209 -2.226413
HQ -2.144522 -2.222598 -2.209734 -2.194376
Interpretation
ARCH 1 is the best model because it has the minimum AIC, BIC HQ.
20. 20
Forecasting Volatility using ARCH model
Once we have estimated the model, we can use it to forecast next period’s
return rt+1 and the conditional volatility ht+1. When one invests in shares (or
stocks), it is important to choose them not just based on their mean returns,
but also on the basis of their risk. Volatility gives us a measure of their risk.
NB: This forecast return equation gives the estimated return that—because it
does not change over time—is both the conditional and unconditional mean
return.
The estimated error in period t, given by can then be used to obtain the
estimated conditional variance:
^
𝑟𝑡+1=^
𝛽𝑜
^
h𝑡=^
𝛼𝑜+^
𝛼1
^
𝑒𝑡−1
2
21. 21
Illustration
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,0)
Mean Model : ARFIMA(0,0,0)
Distribution : norm
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.014954 0.005469 2.7344 0.00625
omega 0.004382 0.000666 6.5824 0.00000
alpha1 0.343705 0.110568 3.1086 0.00188
LogLikelihood : 250.6553
22. 22
Forecast
Rt = 0.015
Ht = 0.004382 + 0.343705et^2
Therefore at a returns of 100, we should expect a volatility of
Ht=0.004382 + 0.343705(100 – 0.015)^2
23. 23
GARCH MODEL—GENERALIZED ARCH
One of the shortcomings of an ARCH(q) model is that there are q
+ 1 parameters to estimate. If q is a large number, we may lose
accuracy in the estimation. The generalized ARCH model, or
GARCH, is an alternative way to capture long lagged effects
with fewer parameters. It is a special generalization of the ARCH
model, GARCH(p,q).
This is GARCH (p,q) model, where p is the number of lagged h
terms and q is the number of lagged terms. The order of
GARCH(p,q) can be determined using the information criteria
approach.
^
h𝑡= 𝛼𝑜 +𝛼1 𝑒𝑡 − 1
2
+ ...+𝛼𝑞 𝑒𝑡 − 𝑞
2
+ 𝛽1 h𝑡 − 1+ ...+ 𝛽𝑝 h𝑡 −𝑝
24. 24
Note:
We also note that we need for stationarity; if , we have a so-called ‘‘integrated
GARCH’’ process, or IGARCH.
The GARCH(1,1) model is a very popular specification because it fits many data series
well. It tells us that the volatility changes with lagged shocks but there is also
momentum in the system working via ht-1.
One reason why this model is so popular is that it can capture long lags in the shocks
with only a few parameters. A GARCH(1,1) model with three parameters () can
capture similar effects to an ARCH(q) model requiring the estimation of (q+1)
parameters, where q is large, say .
25. 25
GARCH(1,1) model
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : std
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 1.04927 0.038843 27.0130 0.000000
omega 0.39670 0.075256 5.2714 0.000000
alpha1 0.49511 0.069273 7.1472 0.000000
beta1 0.24075 0.062236 3.8683 0.000110
shape 99.99304 60.137624 1.6627 0.096365
LogLikelihood : -736.1875
26. 26
INTERPRETATION
Both are statistically significant at any level of significance.The persistent of
movement in the volatility is determined by the sum of these two
coefficients , which is 0.73586. This indicates that movement in the
conditional variance are persistent, implying relatively long lasting periods of
high volatility.
27. 27
Since the GARCH (1,1) model is stationarity, therefore there no need to fit integrated models (e.g., i-
GARCH).
Obtaining the fitted volatility series
v1=sigma(garchfit)
ts.plot(as.numeric(v1))
28. 28
ALLOWING FOR AN ASYMMETRIC EFFECT
A standard ARCH model treats bad ‘‘news’’ (negative et-1 < 0) and good ‘‘news’’ (positive et-
1 > 0) symmetrically: that is, the effect on the volatility ht is the same . However, the effects of
good and bad news may have asymmetric effects on volatility. In general, when negative
news hits a financial market, asset prices tend to enter a turbulent phase and volatility
increases, but with positive news volatility tends to be small and the market enters a period of
tranquility. There is the need to test for effect of news, we call that leverage effects.
Testing for Leverage Effects
Engle and Ng (1993) have proposed a set of tests for asymmetry in volatility, known as sign
and size bias tests. In practice, the Engle-Ng tests are usually applied to the residuals of a
GARCH fit to the data.
29. 29
Define S−1
t as an indicator dummy that takes the value 1 if ˆut−1 < 0 and zero
otherwise. The test for sign bias is based on the significance or otherwise of φ1 in
It could also be the case that the magnitude or size of the shock will affect
whether the response of volatility to shocks is symmetric or not. In this case, a
negative size bias test would be conducted, negative size bias is argued to be
present if φ1 is statistically significant in the regression:
30. 30
Example
The Engle-Ng sign bias tests is performed to test for leverage
effect. The output is given below:
t-value prob sig
Sign Bias 0.5642978 0.0180707
Negative Sign Bias 1.7406252 0.0237068
Positive Sign Bias 1.3926256 0.16435825
Joint Effect 7.9497604 0.04706150
Conclusion
Since the p-value associated with the sign bias and negative sign
bias are less than 0.05 significance level, we conclude that there is
leverage effect or effect of news for the returns series. Therefore,
we can fit the specialized GARCH models for leverage effects.
31. 31
The threshold ARCH model, or T-ARCH or at time gjrGARCH, is one example where positive
and negative news are treated asymmetrically. In the T-GARCH version of the model, the
specification of the conditional variance is
where is known as the asymmetry or leverage term. When , the model collapses to the
standard GARCH form. Otherwise, when the shock is positive (i.e., good news) the effect on
volatility is , but when the news is negative (i.e., bad news) the effect on volatility is . Hence,
if is significant and positive, negative shocks have a larger effect on ht than positive shocks.
Note: Other asymmetric models are E-GARCH and APARCH.
h𝑡=𝛿+𝛼1𝑒𝑡−1
2
+𝛾𝑑𝑡−1𝑒𝑡−1
2
+𝛽1 h𝑡 −1
32. 32
Examples
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : gjrGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : std
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.99531 0.041694 23.8720 0.000000
omega 0.35198 0.065095 5.4072 0.000000
alpha1 0.26451 0.063072 4.1937 0.000027
beta1 0.28971 0.059288 4.8865 0.000001
gamma1 0.49621 0.136514 3.6348 0.000278
shape 99.99980 49.023870 2.0398 0.041368
LogLikelihood : -730.7837
33. 33
Intepretation:
The gamm1 is the coefficient of effect of news. Gamma1 (0.49621) is
greater than zero, it gives an indication of leverage effect. Again, if the
p-value associated to the gamma1 is significant, then there is an
evidence of leverage effect.
The T-ARCH or gjrGARCH is written as:
36. 36
E-GARCH MODEL
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : std
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 1.001488 0.045939 21.80036 0.000000
omega 0.037565 0.039386 0.95375 0.340211
alpha1 -0.190152 0.056989 -3.33664 0.000848
beta1 0.662627 0.047880 13.83944 0.000000
gamma1 0.690388 0.084726 8.14847 0.000000
shape 99.993615 57.220644 1.74751 0.080549
LogLikelihood : -733.1157
37. 37
Comparison of models
The information criteria of these models can be compared in order
to select the best model to the study the leverage effect. Thus, the
model with the minimum information criteria is selected as the best
model.
gjrGARCH(1,1)
Information Criteria
------------------------------------
Akaike 2.9471
Bayes 2.9977
Shibata 2.9469
Hannan-Quinn 2.9670
39. 39
GARCH-IN-MEAN AND TIME-VARYING RISK PREMIUM
Another popular extension of the GARCH model is the ‘‘GARCH-in-mean’’
model. The positive relationship between risk (often measured by volatility)
and return is one of the basic tenets of financial economics. As risk increases,
so does the mean return. Intuitively, the return to risky assets tends to be
higher than the return to safe assets (low variation in returns) to compensate
an investor for taking on the risk of buying the volatile share. However, while
we have estimated the mean equation to model returns, and have
estimated a GARCH model to capture time-varying volatility, we have not
used the risk to explain returns. This is the aim of the GARCH-in-mean models.
40. 40
The equations of a GARCH-in-mean model are shown below:
The first equation is the mean equation; it now shows the effect of the
conditional variance on the dependent variable. In particular, note that
the model postulates that the conditional variance ht affects yt by a
factor . The other two equations are as before.
41. 41
ILLUSTRATION
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : norm
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.46516 0.163114 2.8518 0.004348
archm 0.61078 0.160521 3.8050 0.000142
omega 0.37594 0.061397 6.1231 0.000000
alpha1 0.46997 0.068447 6.8662 0.000000
![34
Diagnostic Tests
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1] 0.8372 0.3602
Lag[2*(p+q)+(p+q)-1][2] 2.4815 0.1941
Lag[4*(p+q)+(p+q)-1][5] 4.2571 0.2237
d.o.f=0
H0 : No serial correlation
Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3] 1.455 0.500 2.000 0.2277
ARCH Lag[5] 3.204 1.440 1.667 0.2615
ARCH Lag[7] 3.645 2.315 1.543 0.4006](https://image.slidesharecdn.com/unit4slides-1-250715160400-47889862/85/UNIT-4-SLIDES-Asigbetse-Maximum-Fredrick-34-320.jpg)
