VI' Autoregressive conditional heteroscedasticity ARCH

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VI. Autoregressive conditional heteroscedasticity
(ARCH)

• Modeling volatility

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ARCH (Engle, 1982) and three stylized facts

• Conditional variance change over time, sometimes
  quite substantially
• There is volatility clustering large (small)
  changes in unpredictable returns tend to be
  followed by large (small) changes
• The unconditional distribution of returns has
  fat tails giving a relatively large probability
  of outliers relative to the normal distribution.

Engle, Robert F. (1982) Autoregressive
Conditional Heteroskedasticity with Estimates of
the Variance of U.K. Inflation, Econometrica,
50, 987?008.
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Returns are unpredictable but its variances show
volatility clustering
UK stock price index
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Modelling Volatility - SP500
Returns are unpredictable but its variances show
volatility clustering
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Empirical applications

• The primary area of application is in financial
  econometrics
• Originally modeling the conditional variance
  inflation
• Shocks affecting variances of stock market
  returns ? market risk premium
• Increasing variance of excess returns on bonds ?
  risk premium?
• Market volatility drives individual stock return
  volatility?individual stock returns

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Empirical applications

• US stock market volatility has a spillover
  effect to the UK and Japanese stock markets
• Deviations of log difference of various nominal
  exchange rates, lnEt-lnEt-1, from a RW may be
  uncorrelated, but there is a tendency for
  large(small) deviations to cluster together

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Daily returns of the yen/dollar exchange rate
high-volatility periods alternate with periods of
relative calm
20-day moving average of the squared changes,
1986-1995
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Basic concepts

• Conditional variance is not constant over time
• Conditional variance may affect the conditional
  mean
• A regression model for the mean should include
  some functions of the conditional variance.

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Why is conditional variance not constant over
time?

• A multitude small shocks to particular industries
• Large events
• oil price shock,
• Oct. 1987 stock market crash,
• changes of governments and organizations
• All potentially affect the volatility of returns.

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How could it be modeled so that it responds to
time-varying shocks?

• Engles answer an ARCH process
• AR(1)

If the conditional variance of et is constant,
then
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How could it be modeled so that it responds to
time-varying shocks?

• The (one step) ahead forecast variance is
  invariant to past values of et or e2t
• Heteroscedasticity depends on past value of yt.

Relaxing it!
A large (positive or negative) value of yt-1
leads to a large variance of yt but here is no
memory apart from this one period.
Conditional variance of yt
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Generalization 1 ARCH(1) ? ARCH(q)

• Problems
• many coefficients to estimate
• non-negativity constraints
• Variance cannot be negative so estimated ais all
  need to be positive to ensure definitely positive
  variance for all errors

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Generalization 2 regression model
The mean regression function
Conditional variance of yt
The ARCH specification to regression disturbance
ARCH(1)
ARCH(q)
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• A large shock in t-1, et-1, leads to a large
  conditional variance in t, st2.
• As shock in t will be large, yt will rise so
  will yt1. The effect on yt1 depends on .
• yts will be affected, but the effect dies out
  as s?8.

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ARCH(q)
Restrictions
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GARCH(1,1)

• The GARCH(1,1) model extends the ARCH(1) model to
  let st2 depend on its own lag as well as the lag
  of the squared error.
• By repeated substitutionGARCH(1,1)
• GARCH(1,1) restricted infinite order ARCH model

Bollerslev, Tim (1986). Generalized
Autoregressive Conditional Heteroskedasticity,
Journal of Econometrics 31, 307-27.
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GARCH(p,q)
Unconditional variance is nonnegative and finite
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Estimation ARCH(1)
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MLE

• Step 1 form
• estimate b by OLS, ignoring ARCH
• Step 2 the sequence of st can be calculated from
• Step 3 st2 used to evaluate the log likelihood.
  It can be maximized numerically using the method
  of hill-climbing or BHHH algorithm.

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LM test for ARCH effects

• Estimate
• obtain the OLS residuals and form
• Auxiliary regression regressing
• TR2 is asymptotically distributed as ?2 (q)
• T is the number of obs.

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Forecasting From GARCH Models

• Best linear predictor using It

• Using the chain-rule of forecasting and the fact
  that

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Forecasting From GARCH Models
Iterating the above gives for k gt 2
Remark Standard errors may be computed using
simulation methods.
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Extension GARCH-in-Mean (GARCH-M)

• Idea Modern finance theory suggests that
  volatility may be related to risk premia on
  assets
• The GARCH-M model allows time-varying volatility
  to be related to expected returns

Engle, Robert F., David M. Lilien, and Russell P.
Robins (1987) Estimating Time Varying Risk
Premia in the Term Structure The ARCH-M Model,
Econometrica 55, 391?07.
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Extension GARCH-M

• An increase in risk, given by the conditional
  standard deviation leads to a rise in the mean
  return.
• The value of a gives the increase in returns
  needed to compensate for a give increase in risk.
• So it is a measure of risk aversion.

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Extension Asymmetric GARCH Leverage Effects

• Suppose there is a negative shock to the equity
  return of a company
• This increases the leverage of the firm (equity
  value down, debt unchanged)
• So the risk of the equity has risen
• A positive shock to the equity reduces leverage
  and has a negative impact on risk
• A negative error has a larger effect than a
  positive error

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TARCH or Threshold ARCH Model

• Introduced independently by Glosten, Jaganathan,
  and Runkle (1993)
• Leverage effect would suggest g gt 0
• Non-negativity constraint is a0gt0, a1gt0, b1gt0 and
  a1 g gt0

Glosten, L.R., R. Jagannathan, and D. Runkle
(1993) On the Relation between the Expected
Value and the Volatility of the Normal Excess
Return on Stocks, Journal of Finance, 48,
1779?801.
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SP500 Example

• Fit a GJR model to monthly SP returns
• suppose variance last period was 0.8
• ut-1 0.5 implies st2 1.645
• ut-1-0.5 implies st2 1.796 leverage effect

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News Impact Curves
Engle, Robert F. and Victor K. Ng (1993)
Measuring and Testing the Impact of News on
Volatility, Journal of Finance, 48, 1022?082.

• NICs plot this impact of a shock (news) on
  conditional variance

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Suggesting readings

• Engle, R.F. (2000). "What Good is a Volatility
  Model," unpublished manuscript, Stern School of
  Business, NYU. http//faculty.washington.edu/ezivo
  t/econ512/Engle20and20Patton.pdf
• Engle, R.F. (2001). "GARCH 101 The Use of
  ARCH/GARCH Model in Applied Economics," Journal
  of Economic Perspectives, 15(4), 157-168.
• Granger, C. and S.-H. Poon (2001). "Forecasting
  Financial Market Volatility," unpublished
  manuscript, Strathclyde University.
  http//faculty.washington.edu/ezivot/econ512/Forec
  astingFMVolatilityPoonGranger.pdf