程序代写代做代考 Economics 430

程序代写 CS代考

Economics 430
Lecture 9
Autoregressive Conditional Heteroscedasticity Models
(ARCH/GARCH)
1

Today’s Class
• The ARCH Family • ARCH Models
– The ARCH(1) Process – The ARCH(p) Process
• GARCH Models
– The GARCH(1,1) Process
– The GARCH(p,q) Process
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The ARCH Family
• Def: ARCH(1) = Autoregressive Conditional Heteroscedasticity Process of order 1:
Yt = μt|t-1+εt, where μt|t-1 is the conditional mean and εt is a white noise process s.t:
εt= σt|t-1zt and zt ~WN(0,1), where σ2t|t-1= E[ε2t| It-1] = ω+α1ε2t-1
= Conditional variance of εt subject to ω>0, α1 ≥0
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ARCH Models
For every ARCH process, we need to address the following 3 questions:
1. WhatdoesatimeseriesofanARCHprocess look like?
2. WhatdothecorrespondingACFsandPACFs look like?
3. Whatistheoptimalforecast?
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ARCH Models
Example: ARCH(1) Process
• (1) What does a time series of an ARCH(1) process look like?
• Consider the following ARCH(1) process: Yt = μt|t-1+εt =μt|t-1+ σt|t-1zt, σ2t|t-1= ω+αε2t-1
• We can plot this process for μt|t-1 = 2, zt ~N(0,1), ω = 2, and different values of α, e.g., for α=0.3,
• We can show that as α increases, the series becomes more volatile.
α=0.6, α=0.9.
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Simulated ARCH(1) Process
Yt =2+εt
σ2t|t-1= 2 + 0.3ε2t-1
Yt =2+εt
σ2t|t-1= 2 + 0.6ε2t-1
Yt =2+εt
σ2t|t-1= 2 + 0.9ε2t-1
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Descriptive Statistics of an ARCH(1) Process and Standardized Process
z = Yt – μ t σt|t-1
μ (unconditional) remains constant ~ 2
σ (unconditional) increases with increasing α
If this is a true ARCH(1)
process, then we expect
the distribution of the standardized process to ~N(0,1)
Fail to reject the normality hypothesis
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Autocorrelation Functions of Simulated ARCH(1) Processes
Yt
Y2t
Yt =2+εt
σ2t|t-1= 2 + 0.3ε2t-1
• The series Yt is uncorrelated
• No dynamics in the conditional
meanACF and PACF0
• The series Y2t is correlated with ACF and PACF AR process
• Observe dynamics implied by the ARCH(1) Process

Autocorrelation Functions of Simulated ARCH(1) Processes
Yt =2+εt
σ2t|t-1= 2 + 0.3ε2t-1
Yt /σt|t-1
Y2t /σ2t|t-1
No more correlations left. Both the ACF and PACF are ‘clean’. Therefore, the ARCH(1) is accurate in capturing the dynamics of the Yt process.

ARCH Models
Forecasting in an ARCH(1) Process
• Consider first the 1-step-ahead variance forecast, h = 1,ARCH(1):
σ2t+1|t= ω + αε2t
h=2: σ2t+2|t= ω + ασ2t+1|t
  
σ2t+h|t= ω(1+α+α2++αh-2) + αh-1σ2t+1|t as h∞, σ2t+h|t = ω/(1-α)
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ARCH Models
• In general, for an ARCH(p) process: Yt = μt|t-1+εt =μt|t-1+ σt|t-1zt , where
σ2t|t-1= ω+α1ε2t-1 +α2ε2t-2 ++αpε2t-p
for ω>0, αi ≥0,i=1, 2, …, p.
Note: α1 + α2++αp = Persistence in Variance
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Example: ARCH(p) Process
Daily SP500 Returns and Autocorrelations of Squared Returns
Q: What type of a process is this?
High persistence
Try an ARCH(8) or ARCH(9)
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GARCH Models
Example: GARCH(1,1) Process
• (1) What does a time series of a GARCH(1,1) process look like?
• Consider the following GARCH(1,1) process:
Yt = μt|t-1+εt =μt|t-1+ σt|t-1zt, σ2t|t-1= ω+αε2t-1+
where ω>0, α≥0, and β≥0.
Also depends on the most recent level of volatility
βσ2t-1|t-2
For example, if β=0.7, we interpret this as saying that
70% of yesterday’s variance carries over to today’s variance.
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GARCH Models
Example: GARCH(1,1) Process
• The key advantage of introducing the term βσ2t-1|t-2 is parsimony!Fewer parameters to estimate than ARCH.
– For example, the previous ARCH(9) model for S&P500 returns suggested an ARCH(9) –need 10 estimates- yet we can do the same with a GARCH(1,1) -only need 3 estimates.
• A GARCH(1,1) process is equivalent to an ARCH(∞) process with exponentially decreasing weights {α, αβ, αβ2,…}.
• The Persistence of the GARCH(1,1) Process is equal to α + αβ + αβ2+…= α/(1-β).
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GARCH Models
Example: GARCH(1,1) Process
Yt = 2 + εt and σ2t|t-1= 2+αε2t-1+ βσ2t-1|t-2 • Case1:LowPersistenceprocess:α=0.4β=0.4,
persistence =α/(1-β) = 0.67
• Case2:HighPersistenceprocess:α=0.1,β=0.88, persistence =α/(1-β) = 0.83
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Simulated GARCH(1,1) Process
Low
Persistence
High
Persistence
For a high (low) persistence process, once the volatility is high, it tends to remain high (low).
For the low
persistence process, only 40% of the past volatility is transferred to the current volatility.

Simulated GARCH(1,1) Process
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Simulated GARCH(1,1) Process
Yt = 2 + εt and σ2t|t-1= 2+αε2t-1+ βσ2t-1|t-2
Low persistencefaster decay High persistenceslower decay
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ARCH(3)
ARCH(9)

Example: GARCH(1,1) Process
Daily SP500 Returns and Autocorrelations of Squared Returns
Q: Was GARCH(1,1) a good model fit?

Example: GARCH(1,1) Process Autocorrelation Function of the Standardized Squared Residuals
from GARCH(1,1) for S&P500 Daily Returns
A: Yes!

GARCH Models
Forecasting in a GARCH(1,1) Process
• Consider first the 1-step-ahead variance forecast, h = 1,
GARCH(1,1): σ2t+1|t= ω + αε2t + βσ2t|t-1 h=2: σ2t+2|t= ω + (α+β)σ2t+1|t
  
σ2t+h|t= ω(1+ (α+β)+ (α+β)2+++ (α+β)h-2) + (α+β)h-1σ2t+1|t
as h∞, σ2t+h|t = ω/(1-(α+β)) = σ2(unconditional variance).
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GARCH Models GARCH(p,q)
Yt = εt
εt|Ωt-1 ~N(0, σ2t )
σ2t = ω+α(L)ε2t+β(L)σ2t
ω>0, Σ αi +Σβi <1 GARCH(p,q) ≈ ARCH(∞) 22

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