Financial Econometrics – Slides-12: Further Issues for GARCH & Realized Volatility
Asymmetric GARCH Exponential GARCH GARCH in mean
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Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Financial Econometrics
Slides-12: Further Issues for GARCH & Realized Volatility
School of Economics1
1©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Lecture Plan
©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material
• Asymmetric GARCH: Leverage effect
• Quantify the effect of standardised shock and avoid positivity restrictions:
• Measure the risk premium effect: GARCH-M model
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
GARCH Extensions
Asymmetric GARCH models
I Motivation: a negative shock to financial time series is likely to cause volatility to
rise by more than a positive shock of the same magnitude
I This is due to leverage effects, i.e. a fall in the value of a firm’s stock causes the
firm’s debt to equity ratio to rise, which makes the future stream of dividends
more volatile
I Standard GARCH models assume a symmetric response of volatility to positive
and negative shocks since by squaring the lagged error term the sign is lost:
In GARCH(1,1): σ2t = α0 + α1µ
t−1, the impact µt−1 on σ
symmetric.
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Asymmetric GARCH: Motivation
• In equity markets, however, bad news (-ve shock) tends to cause more
volatility than good news (+ve shock), aka “asymmetric effect” or
“leverage effect”.
• Desirable to allow for asymmetric effect in GARCH
Topic 6. GARCH Extensions
• Asymmetric GARCH
– Introduction
• In GARCH(1,1):
𝜎𝜎𝑡𝑡2 = 𝛼𝛼0 + 𝛼𝛼1𝜀𝜀𝑡𝑡−1
2 + 𝛽𝛽1𝜎𝜎𝑡𝑡−1
the impact of 𝜀𝜀𝑡𝑡−1 on 𝜎𝜎𝑡𝑡2
is symmetric.
• In equity markets, however, bad news (-ve shock) tends
to cause more volatility than good news (+ve shock),
aka “asymmetric effect” or “leverage effect”.
• Desirable to allow for asymmetric effect in GARCH.
School of Economics, UNSW Slides-09, Financial Econometrics 3
-2 -1 0 1 2
News Impact Curve
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Asymmetric GARCH
I The Threshold GARCH (TGARCH) model. Glosten, Jagannathan and Runkle
[JF, 1993, 48(5), p1779-1801] propose a so-called TGARCH model (GJR) in
which the conditional variance equation is given by
σ2t = α0 + α1µ
t−1It−1 + β1σ
where It−1 is a dummy variable: It−1 = 1 if µt−1 < 0 and It−1 = 0 otherwise. If leverage effects are present γ > 0
– If µt−1 < 0, its effect on σ t is α1 + γ If µt−1 ≥ 0, its effect on σ2t is α1 - The asymmetric effect exists if and only if γ > 0. Reduced back to GARCH if
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Example: GJR/TGARCH
Example: estimates TGARCH/GJR model for returns for S&P500 index with
robust standard errors
Extra topics MBF: Modelling volatility
Extensions of GARCH models
Asymmetric GARCH models
Figure 17: Estimates AR(1)-TGARCH(1,1) model with robust standard
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
News impact curve
I Graphical representation of the degree of asymmetry of volatility to positive and
negative shocks: the curves are drawn by using the estimated conditional
variance equation of the model under consideration.
I Calculate the values of the conditional variance σt over a range of past error
terms. Set the lagged conditional variance at the unconditional variance
I Example: News impact curve from estimates TGARCH model for returns for
S&P500 index
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Example: GJR/TGARCH
News impact curve from estimates TGARCH model
Extra topics MBF: Modelling volatility
Extensions of GARCH models
Asymmetric GARCH models
Figure 18: News impact curve from estimates TGARCH model
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Properties of the TGARCH/GJR model
Properties of the TGARCH/GJR model
I Unconditional variance:
µt|Ωt−1 ∼ N(0, σ2t ), σ
t = α0 + α1µ
t−1It−1 + β1σ
• E(µ2t−1) = E
µ2t−1|Ωt−2
• E(σ2t ) = α0 +
• Stationarity: E
1− (α1 + β1 + 12γ)
• The above is valid when the conditional distribution of µt|Ωt−1 is
symmetric.
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Properties of the TGARCH/GJR model
Properties of TGARCH/GJR: persistence
• Let ωt = µ2t − σ
t , then µ
t has a representation:
µ2t = α0 + (α1 + β1 + γIt−1)µ
t−1 + ωt − β1ωt−1
• When the shocks are zero, ie, ωt = 0 for all t, by substitution,
τ=0(α1 + β1 + γIτ )µ
• E (Iτ |Ωτ−1) = 12 by symmetry.
• On average, the impact of µ20 on µ
Πt−1τ=0(α1 + β1 + γIτ )
(α1 + β1 + γE [It−1|Ωt−2]) Πt−2τ=0(α1 + β1 + γIτ )
= (α1 + β1 + γ/2)E
Πt−2τ=0(α1 + β1 + γIτ )
= · · · = (α1 + β1 + γ/2)t .
• Half-life time, tH , is defined as tH =
ln(α1+β1+γ/2)
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Example: TGARCH/GJR
eg. NYSE composite return: γ̂ = 0.1977, significant α̂1 negative, insignificant
Topic 6. GARCH Extensions
• Asymmetric GARCH
eg. NYSE composite return:
𝛾𝛾� = 0.1977, significant
𝛼𝛼�1 negative, insignificant
School of Economics, UNSW Slides-09, Financial Econometrics 7
-5.0 -2.5 0.0 2.5
Series: Standardized Residuals
Sample 3 1931
Observations 1929
Mean -0.015283
Median -0.007326
Maximum 3.437926
Minimum -6.279817
Std. Dev. 0.999823
Skewness -0.465783
Kurtosis 4.617748
Jarque-Bera 280.1008
Probability 0.000000
Type in Eviews upper panel:
arch(1,1,h,thrsh=1) rc c ar(1)
Topic 6. GARCH Extensions
• Asymmetric GARCH
eg. NYSE composite return:
𝛾𝛾� = 0.1977, significant
𝛼𝛼�1 negative, insignificant
School of Economics, UNSW Slides-09, Financial Econometrics 7
-5.0 -2.5 0.0 2.5
Series: Standardized Residuals
Sample 3 1931
Observations 1929
Mean -0.015283
Median -0.007326
Maximum 3.437926
Minimum -6.279817
Std. Dev. 0.999823
Skewness -0.465783
Kurtosis 4.617748
Jarque-Bera 280.1008
Probability 0.000000
Type in Eviews upper panel:
arch(1,1,h,thrsh=1) rc c ar(1)
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Example: TGARCH/GJR
Example: Test for asymmetry
eg. NYSE composite return: Asymmetric news impact. GJR is preferred by AIC/SIC.
Test for asymmetry,
LR = 2 (logLU − logLR) = 2 [(−2472.7)− (−2523.6)] = 97.8
Topic 6. GARCH Extensions
• Asymmetric GARCH
– GJR model
eg. NYSE composite return:
Asymmetric news impact.
GJR is preferred by AIC/SIC.
Test for asymmetry,
LR = 2(logLU − logLR) = 2[(−2474.7)−(−2523.6)] = 97.8
“H0: symmetry” is rejected.
School of Economics, UNSW Slides-09, Financial Econometrics 8
-2 -1 0 1 2
News Impact Curve
GARCH(1,1)
log Likelihood AIC SIC
AR(1)-GARCH(1,1) -2523.6 2.622 2.636
AR(1)-GJR -2474.7 2.572 2.589 Topic 6. GARCH Extensions
• Asymmetric GARCH
– GJR model
eg. NYSE composite return:
Asymmetric news impact.
GJR is preferred by AIC/SIC.
Test for asymmetry,
LR = 2(logLU − logLR) = 2[(−2474.7)−(−2523.6)] = 97.8
“H0: symmetry” is rejected.
School of Economics, UNSW Slides-09, Financial Econometrics 8
-2 -1 0 1 2
News Impact Curve
GARCH(1,1)
log Likelihood AIC SIC
AR(1)-GARCH(1,1) -2523.6 2.622 2.636
AR(1)-GJR -2474.7 2.572 2.589
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Example: TGARCH/GJR
Example: Forecasts
eg. NYSE composite return: forecasts
σ2t is still persistent, but less than GARCH(1,1).
γ = 0.985, tH = 45.9 (days)
Topic 6. GARCH Extensions
• Asymmetric GARCH
– GJR model
eg. NYSE composite return: forecasts
𝜎𝜎𝑡𝑡2 is still persistent, but less than GARCH(1,1).
𝛼𝛼1 + 𝛽𝛽1 +
𝛾𝛾 = 0.985, 𝑡𝑡𝐻𝐻 = 45.9 (days)
School of Economics, UNSW Slides-09, Financial Econometrics 9
1870 1880 1890 1900 1910 1920 1930
1870 1880 1890 1900 1910 1920 1930
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Example: TGARCH/GJR
Example: VaR
eg. NYSE composite return: VaR
Portfolio valued at $1m at T = 2002− 08− 29.
AR(1)-GJR : σT+1 = 1.577, yT+1|T = 0.0185.
The 1% quantile of νt: Q0.01 = −2.678
yT+1|T − 2.678σT+1
Topic 6. GARCH Extensions
• Asymmetric GARCH
– GJR model
eg. NYSE composite return: VaR
Portfolio valued at $1m at T = 2002-08-29.
AR(1)-GJR: 𝜎𝜎𝑇𝑇+1 =1.577, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.0185.
The 1% quantile of 𝑣𝑣𝑡𝑡: 𝑞𝑞0.01 = −2.678
𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.678𝜎𝜎𝑇𝑇+1 ×$1m = −$42,048
School of Economics, UNSW Slides-09, Financial Econometrics 10
𝜎𝜎𝑇𝑇+1 𝑦𝑦𝑇𝑇+1|𝑇𝑇 𝑞𝑞0.01 VaR
AR(1)-ARCH(5) 1.253 0.050 −2.774 −34260
AR(1)-GARCH(1,1) 1.642 0.051 −2.873 −46660
AR(1)-GJR 1.577 0.019 −2.678 −42048
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Exponential GARCH
I In GARCH, positivity restrictions on parameters make the ML estimation
difficult. Why not exponential?
I In GARCH, new info is incorporated via the term
Why not separate the news ν2t−1 from non-news σ
I EGARCH (Nelson, 1991, Econometrica, 59(2), p347-370)
• Exponential functional form: no need to worry about positivity;
• Separation of the effect of pure news;
• Incorporation of asymmetric effect.
Slides-12 UNSW
Asymmetric GARCH Exponential GARCH GARCH in mean
Exponential GARCH
• Model: µt|Ωt−1 ∼ N(0, σ2t ),
ln(σ2t ) = α0 + α1|νt−1|+ γνt−1 + β1ln(σ
−1 < β1 < 1, νt−1 = µt−1/σt−1 - if νt−1 < 0, its effect on ln(σ t ) is (α1 − γ)|νt−1|. if νt−1 ≥ 0, its effect on ln(σ2t ) is (α1 + γ)|νt−1|. - Negative shocks cause more volatility if and only if γ < 0. Reduced to symmetry if γ = 0. - σ2t = (σ β1exp {α0 + α1|νt−1|+ γνt−1} Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Exponential GARCH: persistence • µt|Ωt−1 ∼ N(0, σ2t ), ln(σ2t ) = α0 + α1|νt−1|+ γνt−1 + β1ln(σ −1 < β1 < 1, νt−1 = µt−1/σt−1 - By substitution, ln(σ2t ) ≈ β 1 (α1|ν0|+ γν0) . Initial impact of the shock ν0 on ln(σ 1) : (α1|ν0|+ γν0) . - The time for the initial impact to halve: 1 (α1|ν0|+ γν0) = (α1|ν0|+ γν0) - Half-life time: tH = Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example: EGARCH eg. NYSE composite return:AR(1)-EGARCH γ̂ = −0.1573, significant Topic 6. GARCH Extensions • Asymmetric GARCH eg. NYSE composite return: AR(1)-EGARCH 𝛾𝛾� = −0.1573, significant School of Economics, UNSW Slides-09, Financial Econometrics 14 Type in Eviews upper panel: arch(1,1,h,egarch) rc c ar(1) -5.0 -2.5 0.0 2.5 Series: Standardized Residuals Sample 3 1931 Observations 1929 Mean -0.005748 Median 0.000146 Maximum 3.521650 Minimum -6.035894 Std. Dev. 1.002586 Skewness -0.377514 Kurtosis 4.385068 Jarque-Bera 200.0118 Probability 0.000000 Topic 6. GARCH Extensions • Asymmetric GARCH eg. NYSE composite return: AR(1)-EGARCH 𝛾𝛾� = −0.1573, significant School of Economics, UNSW Slides-09, Financial Econometrics 14 Type in Eviews upper panel: arch(1,1,h,egarch) rc c ar(1) -5.0 -2.5 0.0 2.5 Series: Standardized Residuals Sample 3 1931 Observations 1929 Mean -0.005748 Median 0.000146 Maximum 3.521650 Minimum -6.035894 Std. Dev. 1.002586 Skewness -0.377514 Kurtosis 4.385068 Jarque-Bera 200.0118 Probability 0.000000 Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example: EGARCH Topic 6. GARCH Extensions eg. NYSE composite return: Asymmetric news impact. �̂�𝛽1=0.9645, 𝑡𝑡𝐻𝐻 = 20.2 (days). Revert to mean quickly. School of Economics, UNSW Slides-09, Financial Econometrics 15 -2 -1 0 1 2 News Impact Curve: EGA 1870 1880 1890 1900 1910 1920 1930 1870 1880 1890 1900 1910 1920 1930 Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example: EGARCH eg. NYSE composite return:VaR. Portfolio valued at $1m at T = 2002− 08− 29. AR(1)-EGARCH:σT+1 = 1.482, yT+1|T = 0.0124 The 1% quantile of νt : Q0.01 = −2.678 yT+1|T − 2.678σT+1 × $1m = −39, 565 Topic 6. GARCH Extensions • Asymmetric GARCH eg. NYSE composite return: VaR Portfolio valued at $1m at T = 2002-08-29. AR(1)-EGARCH: 𝜎𝜎𝑇𝑇+1 =1.482, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.0124. The 1% quantile of 𝑣𝑣𝑡𝑡: 𝑞𝑞0.01 = −2.678 𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.678𝜎𝜎𝑇𝑇+1 ×$1m = −$39,565 School of Economics, UNSW Slides-09, Financial Econometrics 16 𝜎𝜎𝑇𝑇+1 𝑦𝑦𝑇𝑇+1|𝑇𝑇 𝑞𝑞0.01 VaR AR(1)-ARCH(5) 1.253 0.050 −2.774 −34260 AR(1)-GARCH(1,1) 1.642 0.051 −2.873 −46660 AR(1)-GJR 1.577 0.019 −2.678 −42048 AR(1)-EGARCH 1.482 0.012 −2.678 −39565 Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean GARCH in mean I Risk premium effect: 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+ φyt−1 + µt: implies the expected return = yt = c+ φyt−1, which is unrelated to the volatility or risk measure σt. � 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+ δσt−1 + µt µt|Ωt−1 ∼ N(0, σ2t ) σ2t = α0 + α1µ where δ measures the risk premium effect. (See Lundblad (2007, JFE, p123-150) among others.) Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean eg. NYSE composite return No evidence for the “risk premium” effect in any of GARCH(1,1), TGARCH/GJR and EGARCH. Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean • We completed the ARCH/GARCH extensions that capture: • Leverage effect/Asymmetry in the returns volatility • Positivity of the volatility and the impossibility constraints • Next... how about structural change in volatility? Slides-12 UNSW Asymmetric GARCH Properties of the TGARCH/GJR model Example: TGARCH/GJR Exponential GARCH GARCH in mean 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com