CS考试辅导 Financial Econometrics - Slides-10: Modeling Return Volatility: Testing/E

Financial Econometrics – Slides-10: Modeling Return Volatility: Testing/Estimating/Forecasting ARCH and Introduction to GARCH

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

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Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Financial Econometrics
Slides-10: Modeling Return Volatility: Testing/Estimating/Forecasting

ARCH and Introduction to GARCH

School of Economics1

1©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Lecture Plan

• ARCH LM-test
• Forecasting with ARCH
• Generalised ARCH: why and how

Formulation of GARCH: parameter restrictions
• Properties of GARCH(1,1)

• Mean, variance, ARMA(1,1) representation
• ML estimation of GARCH

• Forecasting with GARCH

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ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

ARCH-LM TEST

LM test for ARCH effect

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

ARCH-LM TEST

LM test for ARCH effect: Example

eg. NYSE composite return:

1 Estimate the model for mean (eg. AR(1)) and save the residual series µ̂t.

2 OLS auxilary regression: µ̂2t = γ0 + γ1µ̂
t−1 + · · ·+ γqµ̂2t−q + errort

Save the R2. (q depends on T and data frequency)

3 T ′ = T − q, with q = 5 reject when T ′R2 exceeds χ2(5)

Topic 5. Modelling Return Volatility: ARCH

• Conditional Volatility
– LM test for ARCH effect

• Estimate the model for mean (eg. AR(1)) and save the
residual series 𝑒𝑒𝑡𝑡.

• OLS auxiliary regression
𝑒𝑒𝑡𝑡2 = 𝑐𝑐0 + 𝑐𝑐1𝑒𝑒𝑡𝑡−1

2 + ⋯+ +𝑐𝑐𝑞𝑞𝑒𝑒𝑡𝑡−𝑞𝑞2 + error𝑡𝑡
and save 𝑅𝑅𝑎𝑎2. (𝑞𝑞 depends on 𝑇𝑇 and data frequency)
• Reject “H0: no ARCH” if 𝑇𝑇 − 𝑞𝑞 𝑅𝑅𝑎𝑎

2 exceeds 𝜒𝜒(𝑞𝑞)

eg. NYSE composite return: LM test with q = 5

Performed on “V” to check the adequacy of variance equation

School of Economics, UNSW Slides-7, Financial Econometrics 21

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Forecasting with ARCH Models

• Using repeated substitutions, we can make multi-step forecasts for the
return and its volatility

• Example. AR(1)-ARCH(2)

yt = c+ φ1yt−1 + µt, µt|Ωt−1 ∼ N(0, σ2t )
t = α0 + α1µ

yt+1|t = c+ φ1yt,

yt+2|t = c+ φ1yt+1|t, · · ·
t+1|t = α0 + α1µ

t+2|t = α0 + α1σ

t+1|t + α2µ

t+3|t = α0 + α1σ

t+2|t + α2σ

t+1|t, · · ·

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Forecasting with ARCH Models

Forecasting with ARCH models: Example

Topic 5. Modelling Return Volatility: ARCH

• Conditional Volatility
– Forecasting with ARCH models

eg. NYSE composite return:
AR(1)-ARCH(5) forecasts
revert to unconditionals
(mean reverting)

School of Economics, UNSW Slides-7, Financial Econometrics 23

250 500 750 1000 1250 1500 1750

1870 1880 1890 1900 1910 1920 1930

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ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Forecasting with ARCH Models

Remember the limitations of ARCH!

Advantages of ARCH
• It is able to capture ’clustering’ in return series or the autocorrelation in squared

• It facilitates volatility forecasting.
• It explains, partially, non-normality in return series.

Limitations of ARCH
I In ARCH(q), the q may be selected by AIC, SIC or LR test. The correct value of

q might be very large. The model might not be parsimonious. (eg. ARCH(1)
would not work for the composite return)

I The conditional variance σ2t cannot be negative: Requires non-negativity constraints on
the coefficients. Sufficient (but not necessary) condition is: αi ≥ 0 for all i = 0, 1, 2, · · · q. Especially for
large values of q this might be violated

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH Models: Introduction

Generalised ARCH (GARCH) models allow the conditional variance to
depend upon previous own lags.

• Let µt be the error term or shock in a model.
ARCH(q): Var(µt|Ωt−1) = σ2t ,

t = α0 + α1µ

t−2 + · · ·+ αqµ

is not parsimonious as a large q is often required.

• If σ2t−1 is a summary of volatility info in Ωt−2, then
Ωt−1 = {µt−1, µt−2, µt−3, · · · } = {µt−1,Ωt−2} ≈ {µt−1, σ2t−1} (volatility

• This leads to the GARCH(1,1) model:

t = α0 + α1µ

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH: Introduction

• More generally, GARCH(p, q) model
Var(µt|Ωt−1) = σ2t ,

t = α0 + α1µ

t−1 + · · ·+ αqµ

t−1 + · · ·+ βpσ

where the parameters should satisfy:

(1) Positivity constraint: α0 > 0, αi ≥ 0, βj ≥ 0 for all i = 1, · · · , q and
j = 1, · · · , p

(2) Finite Variance

• In practice, the models for asset returns rarely go beyond GARCH(1,1).

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Properties of GARCH

The generalisation implied by GARCH can be seen from backward iterating
the GARCH(1,1) model:

This shows that the GARCH model is an ARCH(∞) with geometrically
declining coefficients (for |β1| < 1). Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH(1,1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility GARCH models Alternatively, if we define the surprise in the squared innovations as t − σ2t , the GARCH(1,1) model can be rewritten as µ2t − ωt = α0 + α1µ2t−1 + β1 µ2t−1 − ωt−1 µ2t = α0 + (α1 + β1)µ t−1 + ωt − β1ωt−1 which shows that the squared errors follow an ARMA(1,1) model. As the root of the autoregressive part is α1 + β1, the squared residuals are stationary provided |α1 + β1| < 1. Under stationarity, E unconditional variance of µt is given by σ2 = α0 + α1σ 1− (α1 + β1) Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH(1,1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility GARCH models Two general cases can be distinguished I α1 + β1 < 1 → the unconditional variance is defined, i.e. I α1 + β1 ≥ 1 → the unconditional variance is not defined, i.e. The latter case is denoted non-stationarity in variance I Variance does not converge to an unconditional mean I The special case where α1 + β1 = 1 is known as a unit root in variance or integrated GARCH (IGARCH) Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH(1,1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • GARCH(1,1): µt|Ωt−1 ∼ N(0, σ2t ), t = α0 + α1µ α0 > 0, α1 ≥ 0, β1 ≥ 0, α1 + β1 < 1 • Its conditional variance is time varying: E(µt|Ωt−1) = 0, Var(µt|Ωt−1) = σ2t , CI(95%) = E(yt+1|Ωt−1) + 2σt • µt is a White Noise: E(µt) = 0, Var(µt) = α01−(α1+β1) , Cov(µt, µt−j) = 0 • But it is NOT an independent WN or iid WN. It is NOT unconditionally Normally distributed: kurt(µt) > 3

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Properties of GARCH(1,1)

• GARCH(1,1) can be expressed in terms of standardised shocks νt:
µt = σtνt and νt ∼ iid N(0, 1)
• When model is correct, ν2t should have no autocorrelation.

Advantages of the GARCH model (compared to ARCH)

I Avoids overfitting, i.e. a higher order ARCH model may have a more
parsimonious GARCH representation

I Due to less estimated parameters, violations of the non-negativity
constraint are less likely

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

For instance, estimate the following AR(1)-GARCH(1,1) model

yt = µ+ φyt−1 + µt
µt = νtσt νt ∼ N (0, 1)
σ2t = α0 + α1µ

OLS is inappropriate

I OLS minimises the RSS,

yt − µ̂− φ̂yt−1

which is a function of the parameters in the conditional mean
equation only and not in the conditional variance equation

I In fact, OLS assumes that the residuals are homoscedastic,
i.e. all slope coefficients in the conditional variance equation
are set to zero

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

Maximum Likelihood

I Make assumptions about conditional distribution of µt , e.g.

νt ∼ N (0, 1) such that µt ∼ N

This means that conditional on information available at t − 1,
µt is normally distributed with mean zero and variance σt
with the latter being known at time t − 1. Note that this does
not imply that the unconditional distribution of µt is
normal, as σt becomes a random variable if we do not
condition on all information available on t − 1.

I The conditional distribution of yt is then also normal, given by

f (yt |yt−1, . . . , µt−1, . . .) =

with µt = yt − µ− φyt−1 and σ2t = α0 + α1µ2t−1 + β1σ2t−1.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

I The loglikelihood function is given by the sum over all t of
the log of the conditional distribution of yt

log (2π)− T

I The ML estimator is obtained by maximising the loglikelihood
with respect to the unknown parameters (µ, φ, α0, α1, β1).

I Analytical solution not possible: use numerical procedures
I These algorithms ‘search’ over the parameter space, from an

initial guess, until a maximum for the loglikelihood function is

I Potential problem: the loglikelihood function may have several
local maxima such that alternative initial guesses may yield
different results.

I In practice: use linear regression to get initial estimates of the
parameters in the conditional mean equation and choose some
(alternative) parameter value for the parameters in the
conditional variance equation 6= 0.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

Figure 6: The problem of local maxima

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

I Fortunately, first order conditions are, under some weak
assumptions, valid even when νt is not normally distributed.

I The parameter estimates are still consistent
I Adjustments have to be made to the standard errors, i.e. use

Bollerslev-Wooldridge variance-covariance matrix, also known
as Quasi Maximum Likelihood Estimation, which is robust
for non-normality.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Example 1: GARCH(1,1) Estimation

Topic 5. Modelling Return Volatility: GARCH

– ML Estimation of GARCH(1,1)

eg. NYSE composite return

School of Economics, UNSW Slides-08, Financial Econometrics 8

-5.0 -2.5 0.0 2.5

Series: Standardized Residuals
Sample 3 1931
Observations 1929

Mean -0.048341
Median -0.039867
Maximum 2.850528
Minimum -6.601836
Std. Dev. 0.996820
Skewness -0.547486
Kurtosis 4.973199

Jarque-Bera 409.3080
Probability 0.000000

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Example 1: GARCH(1,1) Estimation

Topic 5. Modelling Return Volatility: GARCH

– ML Estimation of GARCH(1,1)

eg. NYSE composite return (continued)
Large 𝛽𝛽1 estimate: about 0.9
Small 𝛼𝛼1 estimate: about 0.1
𝛼𝛼1 + 𝛽𝛽1 estimate: very close to 1

GARCH(1,1) is preferred by AIC and SIC.

School of Economics, UNSW Slides-08, Financial Econometrics 9

AR(1)-ARCH(5) 2.664 2.687

AR(1)-GARCH(1,1) 2.622 2.636

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Example 1: GARCH(1,1) Estimation

Topic 5. Modelling Return Volatility: GARCH

– ML Estimation of GARCH(1,1)

eg. NYSE composite return (continued)
GARCH(1,1) 𝜎𝜎𝑡𝑡 plot is smoother than ARCH(5).
Large 𝛽𝛽1 estimate implies persistence:
𝜎𝜎𝑡𝑡 tends to continue at the current level.
𝜎𝜎𝑡𝑡2 = 𝛼𝛼0 + 𝛼𝛼1𝜀𝜀𝑡𝑡−1

2 + 𝛽𝛽1𝜎𝜎𝑡𝑡−1

School of Economics, UNSW Slides-08, Financial Econometrics 10

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GARCH(1,1)

250 500 750 1000 1250 1500 1750

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ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Summary facts about GARCH models

• GARCH(1,1) is usually preferred to ARCH or higher order GARCH,
because of its parsimony.

• Usually, GARCH β1 estimate is about 0.9 or more and α1 + β1 estimate is
very close to 1, for daily returns.

• Standardised residuals are usually non-normal, with negative skewness and
excessive kurtosis.

• GARCH(1,1) is able to capture clustering in returns but unable to account

Asymmetry: negative returns tend to cause more volatility;

Non-normality; Structural change

• Coefficient restrictions are hard to impose in MLE

Slides-10 UNSW

ARCH: Test and Forecasting
ARCH-LM TEST
Forecasting with ARCH Models

GARCH Models
Properties of GARCH
GARCH Estimation

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