ECMT2130-无代写
时间:2023-04-30
Page 1The University of Sydney
ECMT2130
Financial Econometrics:
Autoregressive Conditional
Heteroscedasticity
Sinan Deng
The University of Sydney
Page 2The University of Sydney
Recall: Stationary AR(1)
Consider
! = " + #!$# + !
where ! ∼ i.i.d. 0, % .
- The unconditional mean of ! is
! = "1 − # , ∀.
- The conditional mean of ! is!$# ! = " + #!$#, ∀.
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Point forecast
One-step-ahead optimal forecast at time :2!&#∣!∗ = ! !&# = " + #!.
The optimal forecast for the next period will be updated over time, as new
observations on are realized.
The vast improvement in forecasts due to time-series models stems from the
use of conditional mean.
- Unlike the unconditional mean, the conditional mean is time-varying.
- Point forecast incorporates the updated information.
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Quantify the "risk"
What is the risk/uncertainty of our point forecast 2!&#∣!∗ ?
When we forecast tomorrow, we want to know, ideally, the conditional
distribution of !&# given today’s information ℱ! :!&# ∣ ℱ! ∼ ? distribution
Point forecast provides forecast of the mean (1st moment) of this conditional
distribution:
!&# ∣ ℱ! : = ! !&#
A simple "risk" measure: variance of this conditional distribution
Var !&# ∣ ℱ! : = Var! !&# = ! !&# − ! !&# %= ! !&#% − ! !&# %.
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Interval forecast
Given ! !&# and Var! !&# , the one-step-ahead two standard deviation
interval forecast for !&# is
! !&#⏟
level
± 2 Var! !&#⏟
width
.
If we assume !&# ∣ ℱ! follows a normal distribution which is fully
characterized by the first two moments, we have!&# ∣ ℱ! ∼ ! !&# , Var! !&# .
Then the 95% one-step-ahead interval forecast of !&# is! !&# ± 1.96 Var! !&# .
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Conditional homoskedasticity
Homoskedasticity: econometrics jargon for identical/constant variances
(across individuals or across time).
In all ARMA models that you have seen, we have
! ∣ ℱ!$# ∼ 0, % .
That is, !$# ! = 0 and Var!$# ! = % : both conditional mean and
conditional variance are time-invariant.
Under this assumption, for all ,Var! !&# = ! !&# − ! !&# %= ! !&#% = Var! !&# = ! !&# % = %
which is also time-invariant. Therefore, we say ! and ! are conditionally
homoscedastic.
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Interval forecast of AR(1)
In the above AR 1 model,
!&# ∣ ℱ! ∼ " + #!⏟
cond. mean
, %⏟
cond. var
.
This allows us to construct the one-step-ahead two standard deviation
interval forecast: " + #!⏟
level
± 2⏟
width
.
The level of this interval forecast change over time, while the width of the
interval remains time-invariant!
How do these interval forecasts perform in practice?
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Point and interval forecasts (Oct-Dec 2009)
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Point and interval forecasts (2008-2009)
Observations outside the interval: 20.8% for S&P 500 returns,
11.2% for 3-month T-bill returns
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Better models?
The ARMA models are a class of conditional mean models: conditional
mean of the time series is explicitly modeled as
- a linear combination of past values (AR models), or
- a linear combination of realised past shocks (MA models), or
- both (ARMA models).
However, without further specification, ARMA models yield a constant
conditional variance: time-invariant risk.
One might expect a better forecast of the risk
- if additional information from the past were allowed to affect the
forecast variance (conditional variance) as well.
- Robert Engle’s 2003 Nobel prize: a testament to the importance of
conditional variance models.
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Fit a conditional homoscedastic model
Let ! be the above daily return series. Fit the AR 1 model! = ! + !
with
- ! = ! ∣ ℱ!$# = " + #!$#, and
- ! ∣ ℱ!$# ∼ 0, % .
where ", # and % are parameters to be estimated. Estimation results are:
We can see !ϕ! insignificant and !ϕ" very small.
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AR(1) fitted residual of NASDAQ daily returns
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Implication of conditional homoskedasticity
If ! is conditionally homoskedastic, i.e., !$# !% = %, as assumed in the
model, what will happen?
Define another process ! by! = !% − !$# !% .
This implies !% = !$# !% + ! = % + !.
is a white noise plus a constant!
- !% should be serially uncorrelated.
- What do the data tell?
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Summary
A model assuming conditional homoskedasticity is invalidated by the
empirical facts.
- ! should be conditionally heteroskedastic
- We should allow serial correlation in !%
Heuristically, we may consider AR 1 model for !% :!% = + !$#% + !
where ! is the m.d.s. constructed above. This way we have!$# !% = + !$#% .
Together with !$# ! = 0, we have! ∣ ℱ!$# ∼ 0, + !$#%
ARCH solution
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Framework
Conditional mean and variance model is the work-horse in financial time
series analysis:
! = ! + !
where ! ∼ 0 and ! ∣ ℱ!$# ∼ 0, !% .
In this model,
- !: = !$# ! is the conditional mean, and
- !%: = Var!$# ! = Var!$# ! is the conditional variance of !.
Both ! and !% are known at time − 1. The specifications of ! and !%
vary across models.
Note: ARMA models effectively specify ! to depend on the available data,
but keep silent about the potential time-varying conditional variance !%.
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The model
The ARCH 1 model specifies ! as! = ! ⋅ !
with
(conditional) volatility: ! = + !$#% ,
innovation/shock: ! ∣ ℱ!$# ∼ 0,1 .
where ℱ!$# = !$#, !$%, … , " = !$#, !$%, … , ", " , 0,1 is some
distribution with mean zero and unit variance, and > 0, ≥ 0 are two
unknown parameters.
Important: Conditional on ℱ!$#, ! is known,_ !_ is random.
Conditional distribution of ! :! ∣ ℱ!$# ∼ 0, !% .
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! is a m.d.s. and hence a WN
We see that
!$# ! = 0, ∀.
So, ! is a martingale difference sequence.
Recall that as a m.d.s., ! has unconditional mean 0 , and is uncorrelated with all
the variables in ℱ!$#. This implies that
- ! = 0, and
- ! is serially uncorrelated: !!$+ = 0, ∀ ≥ 1, ∀.
Therefore, ! ∼ 0 . Unconditional variance of !
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Unconditional variance of !
We see that
!$# !% = !%.
By the law of iterated expectation,
!% = !$# !% = !%
if !% exists.
With ! = 0, it implies Var ! = !% : = ‾%.
So, using our usual notation we have ! ∼ 0, ‾% .
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{!} follows AR(1)
Decompose !% as !% = !$# !% + !
where ! = ! − !$# !% is a m.d.s., and hence ! ∼ 0 .
!% = !% + != + !$#% + !
which means !% is an AR 1 process whose persistency is governed by .
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Stationary variance
If < 1, !% follows a weakly stationary AR 1 process with
!% = 1 − .
Given ! = 0, we have Var ! = 1 − .
Another way: Under the stationarity of !% , we take expectation on both sides.
!% = + !$#%= + !% ⇒ Var ! = !% = 1 − .
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Unconditional homoskedasticity
The ARCH process ! is unconditionally homoskedastic:
Var ! = 1 − , ∀.
Conditional heteroskedasticity
The ARCH process ! is also conditionally heteroskedastic:Var!$# ! = !% = + !$#%
which is varying across time. This is why the model got its name -
Autoregregressive Conditional Heteroskedastic (ARCH) model.
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Example: AR(1) - ARCH(1)
If ! is modeled as ! = " + #!$# + !
where ! is a martingale diff sequence with parameters satisfying stationarity
restrictions.
- conditional mean ! = " + #!$#,
- conditional variance !% = + !$#% .
Note: we have shown above that ! ∼ 0, ,#$- .
Hence we deduce from the stationary AR 1 results that
! = "1 − # , Var ! = / 1 − 1 − #% = 1 − 1 − #% .
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Example: AR(1) - ARCH(1)
The conditional mean process ! has mean ! = " + # !$# = " + # "1 − # = "1 − # = ! ,
and varianceVar ! = Var " + #!$# = #%Var !$# = #%1 − 1 − #%
As for the conditional variance process !% , it has mean !% = Var ! = 1 − .
Hence we verifyVar ! = Var ! + Var ! = #%1 − 1 − #% + 1 − = 1 − 1 − #%
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Variance Decomposition
Clearly, the mean of ! is fully determined by the conditional mean component !. However, this is not true for the variance of !.
We have Var ! = Var ! + Var ! withVar !Var ! = #% and Var !Var ! = !%Var ! = 1 − #%.
in the above example.
The magnitude of # determines the relative proportion of Var ! explained
by the variance of conditional mean and average conditional variance
respectively.
What did we see in practice?
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AR(1) for Returns
Estimation results:
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Dominance by the Residual
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Generalization: ARCH
The simple ARCH(1) model can be easily generalized to an ARCH model:
! ∣ ℱ!$# ∼ 0, !%
where
!% = + #!$#% +⋯+ .!$.%
with > 0, / ≥ 0 and ∑/0#. / < 1.
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Properties of ARCH
For an ARCH process ! , we have
- ! is a white noise.
- !% ∼ AR : !% = + #!$#% +⋯+ .!$.% + !
where ! is a white noise process.
- Unconditional variance
Var ! = !% = 1 − # −⋯− . .
- Conditional variance
Var!$# ! = !% = + #!$#% +⋯+ .!$.% .
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ARCH Test
OLS is appropriate and efficient if the errors in the regression are
conditionally homoskedastic (no ARCH effect).
MLE is preferred if there is an ARCH effect, but it requires nonlinear
optimization, or iterative procedures. So we don’t want to do it unless there is
indeed an ARCH effect in the regression error term.
Engle (1982) proposed a Lagrange Multiplier test for the ARCH effect in the
regression errors. The null hypothesis is
": # = ⋯ = . = 0,
which is no ARCH effect.
An example of detecting ARCH effect for asset returns:
mathworks.com/help/econ/detect-arch-effects. html
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