计算机代写 Slides_13_T2_2021.pdf

Slides_13_T2_2021.pdf

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

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Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Financial Econometrics

Slides-13: Remainaing Issues for GARCH and Alternative

School of Economics1

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

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Lecture Plan

• Measure the risk premium e↵ect: GARCH-M model
• Deal with structural break in volatility
• Seasonality and distributional assumptions
• Inclusion of other volatility measures
• SV models

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

GARCH in mean

I Risk premium e↵ect: investing in a riskier asset should be rewarded by a
higher expected return.

I In the context of a market index: investing in a riskier (more volatile)
period should be rewarded by a higher expected return.

⌅ In AR(1)-GARCH, the mean equation yt = c+ �yt�1 + µt: implies
the expected return = yt = c+ �yt�1, which is unrelated to the
volatility or risk measure �t.

⌅ Motivation: investors should be rewarded for taking additional risk
by obtaining a higher return

I GARCH-M is used to account for the risk premium

yt = c+ ��t�1 + µt µt|⌦t�1 ⇠ N(0,�2t )
t = ↵0 + ↵1µ

where � measures the risk premium e↵ect.
(See Lundblad (2007, JFE, p123-150) among others.)

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

eg. NYSE composite return No evidence for the “risk premium” e↵ect in any
of GARCH(1,1), TGARCH/GJR and EGARCH.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Structural break in Volatility

• The composite return series appears to have a change in its volatility

• The change is permanent.
• If ignored, it can result in

– over-estimating the persistence measure (↵1 + �1);
– making the unconditional variance estimate inconsistent;
– reducing the quality of forecasts, and VaR.

• Important to detect and account for the structural break.

Topic 6. GARCH Extensions

• Structural break in volatility
– Break in volatility

• The composite return series
appears to have a change
in its volatility level.

• The change is permanent.
If ignored, it can result in

– over-estimating the persistence measure (𝛼𝛼1 + 𝛽𝛽1);
– making the unconditional variance estimate inconsistent;
– reducing the quality of forecasts, and VaR.

• Important to detect and account for the structural

School of Economics, UNSW Slides-09, Financial Econometrics 19

1996 1998 2000 2002

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GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Test for structural break

I As the variance is closely related to squared returns, we may check the
break in an AR model for the squared returns, using the CUSUM test.

I Model stability: Its structure changes over time?:
• Recursive parameter estimates. Monitor changes in parameter

estimates over time.

{y1, · · · , y⌧}, {y1, · · · , y⌧+1}, · · · , {y1, · · · , yT }
�̂(⌧), �̂(⌧ + 1), �̂(T )

• Recursive residuals: e⌧+1|⌧ = y⌧+1 �X⌧+1�̂(⌧)

{y1, · · · , y⌧}, {y1, · · · , y⌧+1}, · · · , {y1, · · · , y⌧}
e⌧+1|⌧ , e⌧+2|⌧+1, eT |T�1

• If the model is stable/correct: w⌧+1|⌧ =

se(e⌧+1|⌧ )

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Model stability test: CUSUM

CUSUM test (cumulative sum of standardised recursive residuals)

w⌧+1|⌧ , t = K + 1,K + 2, · · · , T � 1

Reject stability if it goes outside the 95% bands.
Eviews: View/Stability Tests/Recursive Estimates after a linear regression is
Test in volatility break: eg. AR(5) for the composite return squared:

t = a0 + a1r

t�1 + · · ·+ a5r

t�5 + errort

CUSUM test rejects the null hypothesis of no break.

Topic 6. GARCH Extensions

• Structural break in volatility
– Test for a break in volatility

• As the variance is closely related to squared returns, we
may check the break in an AR model for the squared
returns, using the CUSUM test.

eg. AR(5) for the composite return squared:
𝑟𝑟𝑡𝑡2 = 𝑎𝑎0 + 𝑎𝑎1𝑟𝑟𝑡𝑡−1

2 +⋯+ 𝑎𝑎5𝑟𝑟𝑡𝑡−5
2 + error𝑡𝑡

CUSUM test rejects the null
hypothesis of no break.

School of Economics, UNSW Slides-09, Financial Econometrics 20

250 500 750 1000 1250 1500 1750

CUSUM 5% Significance

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Structural break in volatility

Find the break point

Topic 6. GARCH Extensions

• Structural break in volatility
– Find the break point

1) Run the restricted regression (no break) and save the log
likelihood as ℓ0.

2) Set 𝜏𝜏 = .15𝑇𝑇 (15% trim). Define the break dummy as 𝐵𝐵𝑡𝑡,𝜏𝜏, which
is 0 for 𝑡𝑡 < 𝜏𝜏 and 1 for 𝑡𝑡 ≥ 𝜏𝜏. 3) Run the unrestricted regression 𝑟𝑟𝑡𝑡2 = 𝑎𝑎0 + 𝑎𝑎1𝑟𝑟𝑡𝑡−1 2 +⋯+ 𝑎𝑎5𝑟𝑟𝑡𝑡−5 2 + 𝜓𝜓𝐵𝐵𝑡𝑡,𝜏𝜏 + error𝑡𝑡 and save the log likelihood ℓ𝜏𝜏 and 𝐿𝐿𝑅𝑅𝜏𝜏 = 2(ℓ𝜏𝜏 − ℓ0). 4) Set 𝜏𝜏 = 𝜏𝜏 + 1. If 𝜏𝜏 ≤ .85𝑇𝑇 (15% trim), go to 3). Otherwise go to 5). 5) The break point is estimated as the 𝜏𝜏 associated with the greatest 𝐿𝐿𝑅𝑅𝜏𝜏. It could be used as a test: the null of no break is rejected if max LR > cv.
The cv for 15% trim is 8.85, see Andrews (1993, Etrca, p821-856).

School of Economics, UNSW Slides-09, Financial Econometrics 21

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GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Structural break in volatility

Find the break point
eg. AR(5) for the composite return squared:

r2t = a0 + a1r
t�1 + · · ·+ a5r

t�5 + Bt,⌧ + errort

Topic 6. GARCH Extensions

• Structural break in volatility
– Find the break point

eg. AR(5) for the composite return squared:
𝑟𝑟𝑡𝑡2 = 𝑎𝑎0 + 𝑎𝑎1𝑟𝑟𝑡𝑡−1

2 +⋯+ 𝑎𝑎5𝑟𝑟𝑡𝑡−5
2 + 𝜓𝜓𝐵𝐵𝑡𝑡,𝜏𝜏 + error𝑡𝑡

The break point = 566.
AR(5) with the break passes the CUSUM test

School of Economics, UNSW Slides-09, Financial Econometrics 22

250 500 750 1000 1250 1500 1750

750 1000 1250 1500 1750

CUSUM 5% Significance

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Break in volatility models

Incorporating breaks in volatility models

Incorporate a break in GARCH

I Once the break point is known and the break dummy Bt,⌧ is defined, the
break should be included in the conditional variance.

I GARCH(1,1) :
�2t = ↵0 + ↵1µ

t�1 + Bt,⌧

I TGARCH/GJR :
�2t = ↵0 + ↵1µ

t�1It�1 + �1�

t�1 + Bt,⌧

I TGARCH/GJR :
ln(�2t ) = ↵0 + ↵1|⌫t�1|+ �⌫t�1 + �1ln(�2t�1) + Bt,⌧

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Seasonality: January E↵ect

⌅ Including a dummy in the variance equation,
• GARCH(1,1):
�2t = ↵0 + ↵1µ

�2t = ↵0 + ↵1µ

t�1It�1 + �1�

• EGARCH: ln(�2t ) = ↵0 + ↵1|⌫t�1|+ �⌫t�1 + �1ln(�2t�1) + �Jt

where Jt is 1 if t is in January and 0 otherwise.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Non-normality

⌅ Normality alternatives
• In our examples, normality is usually rejected owing to

– heavy tails (Kurtosis> 3) and
– negative skewness

in the distribution of the standardised shock ⌫t.
• Alternative distributions may be assumed

– Student’s t: t(n)
with heavy tails but symmetry.
t(n) ⇡ N(0, 1) when the df n ! 1

– Mixture distributions: heavy tails and asymmetry.

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GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Mixture of two normals

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GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Mixture of gaussians

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GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Incorporate other volatility measures

⌅ Range and implied volatility
• In addition to µt�1 or ⌫t�1, other volatility measures may have predictive

power for conditional variance.

Typically, the range (100ln(high/low)) and implied volatility (IV) are
informative measures of volatility.

• For EGARCH, we may specify
ln(�2t ) = ↵0 + ↵1|⌫t�1|+ �⌫t�1 + �1ln(�2t�1) + a1rngt�1 + a2ivt�1,
where the range (rng) and IV (iv) are included.

! It is good for 1-step ahead forecast. However, we need models for the
range and IV to do multi-step ahead forecasts.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Example: Range and Implied Volatility in EGARCH

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Stochastic volatility (SV) model: Latent Volatility

• In GARCH type models, the shock µt�1 or ⌫t�1 can be recovered
from the mean equation. The conditional variance, as a function of
µt�1 is ”observable”.

yt = µ+ �t⌫t,

ln(�2t ) = ↵0 + �1ln(�
t�1) + ⌘t, ⌘t ⇠ iid N(0,!

the conditional variance �2t is latent (unobservable):
• there are two shocks: ⌫t and ⌘t. Often used in theoretical options

pricing literature;
• it is di�cult to estimate (likelihood evaluation is challenging)
• it is awkward for forecasting, as �2t is conditional on an unobservable

information set.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

• We have seen a variety of models for conditional volatility for
niveriate returns models

• Next… Multivariate Volatility models: Portfolio management,
hedging strategies…

Slides-13 UNSW

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