Kafu Wong University of Hong Kong

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Ka-fu WongUniversity of Hong Kong
Volatility Measurement, Modeling, and Forecasting
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Importance of volatility

• Good volatility forecasts are crucial for the
  implementation and evaluation of asset and
  derivative pricing theories as well as trading
  and hedging strategies.
• Two assets
• an risky and a riskless (i.e., volatility 0)
• Risky asset generally has a higher expected
  return than the riskless assets.
• We would like to invest in a portfolio consisting
  of the two assets.
• When the risky asset has a very high volatility,
  the portfolio will consist of the riskless asset
  only.
• When the risky asset has a very low volatility,
  the portfolio will consist of more risky assets.

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Importance of volatility

• The variance of inflation may have impact on
  various macro and investment decisions.
• High variance in inflation may also imply welfare
  loss.
• Previous studies have tried to measure the
  time-varying variance of inflation.

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Clustering of volatility

• It is a well-established fact, dating back to
  Mandelbrot (1963) and Fama (1965), that financial
  returns display pronounced volatility clustering.
• Therefore, models of volatility should allow such
  clustering.

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Example AR(1)
yt f yt-1 et
et WN(0, s2)
AR(1)
yt f (f yt-2 et-1) et
Repeatd substitution
f2 yt-2 f et-1 et
f2 (f yt-3 et-2) f et-1 et
f3 yt-3 f2 et-2 f et-1 et

et fet-1 f2et-2 f3et-3 f4et-4 f5et-5

E(et) 0, E(yt) 0
Var(et) E(et E(et))2 s2 Var(yt) E(yt
E(yt))2 s2(1 f f2 f3 f4 )
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Homoskedasticity vs. Heteroskedasticity
So far, innovation are assumed to be i.i.d.
It is possible to allow variance to change across
observations, i.e., Heteroskedasticity.
Information available at time t-1
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A general linear process
Consider a general linear process
Need not be i.i.d.
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Two examples
Consider a general linear process
AR(1)
yt f yt-1 et
yt et fet-1 f2et-2 f3et-3 f4et-4
bi fi
MA(2)
Need not be i.i.d.
yt et ?1et-1 ? 2et-2
b01, b1 ?1, b2 ?2, b3b40
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Unconditional means and variances
Consider a general linear process
AR(1)
yt f yt-1 et
yt et fet-1 f2et-2 f3et-3 f4et-4
bi fi
E(yt) E(et) fE(et-1) f2E(et-2) 0
V(yt) V(et) f2V(et-1) f4V(et-2)
MA(2)
yt et ?1et-1 ? 2et-2
b01, b1 ?1, b2 ?2, b3b40
E(yt) E(et) ?1E(et-1) ?2E(et-2) 0
V(yt) V(et) ?12V(et-1) ?22V(et-2)
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Conditional variances change with horizon of
forecast but are not time-varying given a horizon.
Consider a general linear process
Conditional information
MA(2)
Conditional mean is time-varying
yt et ?1et-1 ? 2et-2
b01, b1 ?1, b2 ?2, b3b40
E(yt?t-1) ?1et-1 ?2et-2
h-step ahead forecast is time-varying
E(yt1?t) ?1et ?2et-1
E(yt2?t1) ?1et1 ?2et
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Conditional variances change with horizon of
forecast but are not time-varying given a horizon.
Consider a general linear process
Conditional information
MA(2)
Conditional variance is not time-varying
yt et ?1et-1 ? 2et-2
b01, b1 ?1, b2 ?2, b3b40
Conditional prediction error variance
E(yt-E(yt?t-1) )2?t-1 E(et2 ?t-1) s2
Non-time-varying!
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ARCH(p) process
Examples (1) ARCH(1) st2 w g1 et-12 (2)
ARCH(2) st2 w g1 et-12 g2 et-22
ARCH(p)
AutoRegressive Conditional Heteroskedasticy of
order p
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ARCH(p) process
Examples (1) ARCH(1) st2 w g1 et-12 (2)
ARCH(2) st2 w g1 et-12 g2 et-22
ARCH implies volatility clustering. That is,
large changes tend to be followed by large
changes and small by small, of either sign.
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ARCH(p) process
Some properties
(1) Unconditional mean
(2) Unconditional variance
(3) Conditional variance
Examples (1) ARCH(1) st2 w g1 et-12 (2)
ARCH(2) st2 w g1 et-12 g2 et-22
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ARCH(1)

• st2 w g1 et-12
• Note that
• Eet2 E E(et2?t-1) E(st2) s2
• E(et-E(et))2 ?
• Est2 w g1 Eet-12
• s2 w g1 s2
• s2 w / (1- g1)

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How to simulation ARCH(1)?

• Suppose we are interested in generating T
  observations of et that has the property of
  ARCH(1).
• et N(0,st2), where st2 w g1 et-12
• (1) Fixed the parameters. Compute the
  unconditional variance of et.
• s2 w / (1- g1)
• (2) Generate T1 observations of standard normal
  random variables, v0, v1, ., vT
• (3) Generate et recursively
• For t0, st2 s2, et vt st
• For t1, st2 w g1 et-12, and et vt st
• For t2, st2 w g1 et-12, and et vt st

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The inflation example of Engle (1982)
log of the quarterly manual wage rates
First difference of the log of the quarterly
consumer price index
Lagged 4 periods
Engle, Robert F. (1982) Autoregressive
Conditional Heteroscedasticity with Estimates of
the Variance of United Kingdom Inflation,
Econometrica, 50(4) 987-1007.
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The inflation example of Engle (1982)OLS
regression
Restriction imposed.
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The inflation example of Engle (1982)ML
estimation with ARCH(1)
The ARCH model comes closer to truly random
residuals after standardizing for their
conditional distributions.
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GARCH(p,q)
Backward substitution on st2 yields
A infinite-order ARCH process with some
restriction in the coefficients. (Analogy An
ARMA(p,q) process can be written as MA(8)
process.)
GARCH can be viewed as a parsimonious way to
approximate a high order ARCH process
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Important properties of GARCH(p,q)(1)
Unconditional variance is fixed but conditional
variance is time-varying
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Important properties of GARCH(p,q)(2)
Unconditional distribution of conditionally
Gaussian GARCH is symmetric and leptokurtic.
Real-world financial asset returns, are often
found to symmetrically distributed and have a
fatter tail than Gaussian distribution.
Ordinary Gaussian distribution does not provide
a good approximation of the asset returns, but
the Gaussian distribution with GARCH does.
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Important properties of GARCH(p,q)(3)
Conditional prediction error variance varies with
conditional information set.
unbiased forecast
Conditional variance of the prediction error
Conditional variance approaches unconditional
variance
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Important properties of GARCH(p,q)(3) et follows
GARCH implies et2 follows an ARMA .
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Extension of ARCH and GARCH ModelsThreshold GARCH

• When the lagged return is positive (good news
  yesterday), D0, so the effect of the lagged
  squared return on the current conditional
  variance is simply a.
• When the lagged return is negative (negative news
  yesterday), D1, so the effect of the lagged
  squared return on the current conditional
  variance is simply a g.
• Allowance for asymmetric response has proved
  useful for modeling leverage effects in stock
  returns, which occur when g lt 0.

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Extension of ARCH and GARCH Modelsexponential
GARCH

• Volatility is drive by both the size and sign of
  shocks (both positive and negative). Hence, the
  model allows for asymmetric response depending on
  the sign of news.
• When the shock is positive, the impact of
  (et-1/st-1) on ln(st2) is
• a g
• When the shock is negative, the impact of
  (et-1/st-1) on ln(st2) is
• -a g

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Extension of ARCH and GARCH ModelsGARCH with
exogenous variables

• Financial market volume, for example, often helps
  to explain market volatility.

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Extension of ARCH and GARCH ModelsGARCH-in-Mean
(i.e., GARCH-M)
Conditional mean regression

• High risk, high return.

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Estimating, Forecasting, and Diagnosing GARCH
Models

• Diagnostic
• Estimate the model without GARCH in the usual
  way.
• Look at the time series properties of the squared
  residuals.
• Correlogram, AIC, SIC, etc.
• ARMA(1,1) in the squared residuals implies
  GARCH(1,1).

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Estimating, Forecasting, and Diagnosing GARCH
Models

• Estimation Usually use maximum likelihood with
  the assumption of normal distribution.
• Maximum likelihood estimation finds the parameter
  values that maximize the likelihood function

• Forecast
• In financial applications, volatility forecasts
  are often of direct interest.

• Better forecast confidence interval

1-step-ahead conditional variance
vs.
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Application Stock Market Volatility

• Objective Model and forecast the volatility of
  daily returns on the New York Stock Exchange
• Data
• Daily returns on the New York Stock Exchange
  (NYSE) form January 1, 1988, through December 31,
  2001.
• Excluding holidays, there are 3531 observations.
• Estimation 1-3461
• Forecast 3462-3531.

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Time Series Plot, NYSE Returns
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Histogram and Related Diagnostic Statistics, NYSE
Returns
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Correlogram, NYSE Returns
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Time Series Plot, Squared NYSE Returns
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Correlogram, Squared NYSE Returns
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AR(5) Model, Squared NYSE Returns
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ARCH(5) Model, NYSE Returns
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Correlogram, Squared Standardized ARCH(5)
residuals, NYSE Returns
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GARCH(1,1) Model, NYSE Returns
et-12
st-12
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Correlogram, Squared Standardized GARCH(1,1)
residuals, NYSE Returns
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Estimated Conditional Standard Deviation,
GARCH(1,1) Model, NYSE Returns
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Estimated Conditional Standard Deviation,
Exponential Smoothing, NYSE Returns
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Conditional Standard Deviation, History and
Forecast, GARCH(1,1) Model
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Conditional Standard Deviation, Extended History
and Extended Forecast, GARCH(1,1) Model
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Is GARCH(1,1) enough most of the time?

• 330 GARCH-type models are compared in terms of
  their ability to forecast the one-day-ahead
  conditional variance.
• The models are evaluated out-of-sample using six
  different loss functions, where the realized
  variance is substituted for the latent
  conditional variance.

Hansen, Peter R. and Asger Lunde (2005) A
Forecast Comparison Of Volatility Models Does
Anything Beat A GARCH(1,1)? Journal of Applied
Econometrics, 20 873-889.
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Is GARCH(1,1) enough most of the time?

• Data
• DM spot exchange rate data,
• the estimation sample spans the period from
  October 1, 1987 through September 30, 1992 (1254
  observations) and
• the out-of-sample evaluation sample spans the
  period from October 1, 1992 through September 30,
  1993 (n 260).
• IBM stock returns,
• the estimation period spans the period from
  January 2, 1990 through May 28, 1999 (2378 days)
  and
• the evaluation period spans the period from June
  1, 1999 through May 31, 2000 (n 254).

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Specifications of the conditional variance
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Loss functions for forecast evaluation
MSE2 and R2Log are similar to R2 of the MZ
regressions.
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The test
Loss of alternatiave GARCH models.
Loss of GARCH(1,1)
Giving benefits of the doubt to the benchmark,
i.e., GARCH(1,1).
The maintained hypothesis is that GARCH(1,1) is
better unless there is strong evidence against it.
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Superior Predictive Ability and Reality Check
for data snooping
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The test resultssuperior predictive ability (SPA)
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IBM datasuperior predictive ability and reality
check for data snooping
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Does Anything Beat A GARCH(1,1)?

• No. So, use GARCH(1,1) if no other information
  is available.

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End