ECON3350/7350 Volatility Models – II
Eric Eisenstat
The University of Queensland
Lecture 8
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 9
1 / 25
Volatility Modeling Extensions
Recommended readings
Author
Title
Chapter
Call No
Enders Brooks
Verbeek
Applied Econometric Time Series, 4e Introductory Econo- metrics for Finance, 3e
A Guide to Modern Econometrics
3 9
8.10
HB139 .E55 2015 HG173 .B76 2014
HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 2 / 25
ARCH(q) and GARCH(p, q)
ARCH(q) process:
E(ε2t |ε2t−1,…,ε2t−q)=α0 +α1ε2t−1 +α2ε2t−2 +···+αqε2t−q,
= α 0 + α ( L ) ε 2t − 1 , whereα(L)isalagpolynomialoforderq−1,α(1)<1,αj >0for
j = 0, . . . , q. GARCH(p, q) process:
qp
h t = α 0 + α j ε 2t − j + β j h t − j ,
j=1 j=1
= α0 + α(L)ε2t−1 + β(L)ht−1.
In practice, focus is on GARCH(1, 1), where non-negative ht is obtained by
restrictingα0 ≥0,α1 ≥0,andβ1 ≥0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 3 / 25
Regressions and Autoregressions
Consider a regression for the mean of yt with ARCH(q) errors: yt =β1 +β2×2,t +···+βkxk,t +εt,
ε =να +αε2 +···+αε2 ,
t t 0 1 t−1
where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s.
q t−q
Consider an auto-regression for the mean of yt with GARCH(1, 1) errors:
yt = a0 + a1yt−1 + εt, εt = νtht,
ht = α0 + α1ε2t−1 + β1ht−1, where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 4 / 25
Extensions to the Basic GARCH Model
Huge number of extensions and variants of the GARCH model proposed in the past three decades:
practical problems with basic GARCH: non-negativity constraints difficult to impose;
conceptual limitations in GARCH: cannot account for leverage effects. Some of the most important extensions are:
Asymmetric models: TGARCH and EGARCH (account for leverage effects);
(G)ARCH-M models: particularly suited to study asset markets;
IGARCH models: impose constraints that account for the persistence of volatility.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 5 / 25
The EGARCH Model
EGARCH (suggested by Nelson, 1991) specifies the variance equation as:
lnht =α0 +β1lnht−1 +λh +α1h . t−1 t−1
ε ε t−1 t−1
Advantage of EGARCH is ht will positive for all values of parameters (due to the specification in logs).
εt−1
is iid standard normal; |νt−1| = √ is iid half-normal
ht−1
εt−1
νt−1 = √
with E(|νt−1|) = 2/π.
ht−1
Eric Eisenstat (School of Economics) ECON3350/7350
Week 9
6 / 25
Leverage Effect in EGARCH
The effect of a shock in εt on ht:
εt−1
if √ εt−1
is positive, the effect on lnht is α1 +λ; is negative, the effect on lnht is α1 −λ.
ht−1 ht−1
if √
Effect of a negative shocks is greater than the effect of a positive shock iff
positive shock negative shock
α1+λ < α1−λ , λ < 0.
This is the leverage effect: volatility increases more when stock prices fall than when they rise; if λ = 0 then no leverage effect exists and if λ > 0 then a reverse effect exists.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 7 / 25
The Leverage Effect
The standardised √
is
εt−1
Source: Enders (2015)
ht−1 unit free and permits a more
natural interpretation of the size and persistence of the shocks.
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 9
8 / 25
Volatility Prediction–EGARCH
Assuming the parameters are known, the natural predictor is
ht =exp(lnht)=exp(α0 +β1lnht−1 +λνt−1 +α1|νt−1|), Et(ht+j) = Et (exp(lnht+j)), j > 0.
When in addition shocks are Gaussian,
lnht =(1−β1)α0 +β1lnht−1 +g(νt−1), g(ν )=λν +α |ν |−2/π.
t−1 t−1 1 t−1
The h-step ahead forecast is given by
h =hβ1 exp((1−β )α )exp(g(ν)).
t+ht 10 t
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 9 / 25
The TGARCH or GJR Model
The threshold GARCH (TGARCH) model, developed by Glosten, Jaganathan and Runkle (1994), is
ht = α0 + α1ε2t−1 + λdt−1ε2t−1 + β1ht−1, 1 ifεt−1<0,
dt−1 = 0 otherwise.
We require α0 ≥ 0 and α1 ≥ 0 for non-negativity.
The effect of an εt−1 shock on ht:
if εt−1 ≥ 0, dt−1 = 0 and the effect is α1ε2t−1;
if εt−1 < 0, dt−1 = 1 and the effect is (α1 + λ)ε2t−1; interpret as leverage effect when λ > 0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 10 / 25
Testing for Leverage Effects with GARCH
Estimate a GARCH and store the standardized residuals ε
t
νt = , for t = 1,…,T.
ht Use an F-test from the regression:
2
νt =a0 +a1νt−1 +a2νt−2 +···+ut.
Under the null H0 : α1 = α2 = · · · , the standardized square errors are uncorrelated with history of levels, and therefore, no leverage effects exist. Under H1, there are some leverage effects present. The usual F-statistic and critical values can be used.
Alternatively, use a t-test in the regression: 2
νt =a0+a1dt−1+ut,
where H0 : a1 = 0 indicates no leverage effect and the usual t-statistic and
critical values are applicable.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 11 / 25
Testing for Leverage Effects with EGARCH or TGARCH
Recall that in both the EGARCH and TGARCH, leverage effects are controlled by a single parameter λ.
If either model is estimated, only a simple one-tailed t-test on λ is needed to test for leverage effects:
H0 :λ=0,
H1 :λ<0fortheEGARCHorλ>0fortheTGARCH.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 12 / 25
TGARCH Example
Data: monthly S&P 500 returns, 1979M12–1998M6. Source: Brooks (2002).
Estimated model is
y = 0.172, t
(3.198)
ht = 1.243 + 0.015ε2t−1 + 0.604dt−1ε2t−1 + 0.498ht−1.
(16.37) (0.44) (5.77) (14.99)
Test for leverage effects with H0 : λ = 0 against H1 : λ > 0; t-stat is
larger than 1.645, so conclude significant leverage effects.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 13 / 25
TGARCH Example
The news impact curve plots the next period volatility (ht+1) that would arise from various positive and negative values of ε2t .
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
−1 −0.5 0 0.5 1
GARCH TGARCH
Eric Eisenstat
(School of Economics) ECON3350/7350 Week 9 14 / 25
(G)ARCH-in Mean
We expect a risk to be compensated by a higher return; so why not let the return be partly determined by risk?
Engle, Lilien and Robins (1987) suggested the ARCH-M specification:
yt = μt + εt,
μt =β+δht =β+δα0 +α1wjε2t−j,
q j=1
where α0, α1, β and δ > 0 are constants; w1, . . . , wq are weights assigned topastqsquarederrors(ELRsetwj =(5−j)/10forj=1,…,4).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 15 / 25
(G)ARCH-in Mean
In ELR,
yt: excess return from holding a long-term asset relative to a one-period treasury bill;
μt: risk premium necessary to induce the risk-averse agent to hold the long-term asset rather than the one-period bond;
εt: unforecastable shock to the excess return on the long-term asset.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 16 / 25
The ELR ARCH-M Model
The expected excess return from holding the long-term asset must be just equal to the risk premium:
Et−1yt = μt.
The assumption is that the risk premium is an increasing function of the
conditional variance of ε2t .
The greater the conditional variance of returns, the greater the compensation necessary to induce the agent to hold the long-term asset.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 17 / 25
ELR Data
ELR use excess yields on six-month treasury bills:
rt: quarterly yield on a three-month treasury bill held from t to t + 1;
rolling over all proceeds; at the end of two quarters, an individual investing $1 at the beginning of t will have (1 + r)(1 + rt+1) dollars
Rt: quarterly yield on a six-month treasury bill, buying and holding the six-month bill for the full two quarters will result in (1 + Rt)2 dollars.
The excess yield yt due to holding the six-month bill is yt = (1+Rt)2 −(1+rt+1)(1+rt),
which is approximately equal to
yt = 2Rt − rt+1 − rt.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 18 / 25
ELR ARCH Test
Estimates of a constant mean and constant variance model: y = 0.142 + ε .
(4.04)
The excess yield of 0.142% per quarter is more than four standard
deviations from zero.
LMARCH = 10.1, which is larger than the critical value of 6.635 obtained
from χ21 at the 1% significance level.
The post-1979 period showed higher volatility than the earlier period.
tt
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 19 / 25
ELR ARCH-M Estimates
The ARCH-M estimates are:
y = −0.0241 + 0.687h , t t
(−1.29) (5.15)
ht = 0.0023 + 1.64 0.4ε2t−1 + 0.3ε2t−2 + 0.2ε2t−3 + 0.1ε2t−4 .
(1.08) (6.30)
Results:
Risk premium is time-varying.
Estimate of 1.64 implies the unconditional variance is infinite (although conditional variance is finite).
During volatile periods, the risk premium rises as risk-averse agents seek assets that are conditionally less risky.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 20 / 25
Stochastic Volatility Models
An alternative to GARCH are the stochastic volatility models (SV). The foundation of SV models differs from GARCH in that conditional
variance is not a deterministic function of past shocks.
SV models are particularly designed to deal with volatility clustering.
SV models contain a second error term that enters the conditional variance specification; they are commonly written as state-space models.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 21 / 25
Stochastic Volatility Models
A typical SV model is defined as
y =a +ehtε, ε ∼N(0,1), t02t t
β(L)ht =α0 +ηt, ηt ∼N(0,ση2).
β(L) is a lag polynomial of order m; the parameters in the model are
a0,α0,β1,…,βm.
Parameters of the SV model can be estimated using maximum likelihood
or simulation methods.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 22 / 25
SV Example
A SV model for the daily returns of an asset price yt can be specified as y =a +ehtε, ε ∼N(0,1),
t02t t
ht =φht−1 +ηt, ηt ∼N(0,ση2),
and 0 < φ < 1.
The signal or state, ht, is the logarithmic volatility, which we wish to
estimate.
Volatility is mapped to observed daily returns, and it is assumed to be a
stationary AR(1) process in this case.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 23 / 25
Realized Volatility
The conditional variance is latent (i.e. not observed).
GARCH and SV type models produce estimates of the latent conditional
variance.
Latent conditional variance models do not capture high excess kurtosis of
returns—i.e., low but persistent autocorrelations in squared returns. Merton (1980): we can accurately estimate the variance over a fixed
interval with sufficiently high frequency data.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 24 / 25
Realized Volatility
Andersen and Bollerslev (1998): estimate ex-post daily foreign exchange volatility by sum of squared returns, sampled at 5-minute intervals.
The 5-minute frequency is a tradeoff between accuracy and microstructure noise.
Ex-post volatility measured this can be regarded as observable; it can be taken as data and modeled directly.
Further reading:
McAleer, M. and Medeiros, M. C. (2008), “Realized Volatility: A Review”, Econometric Reviews, 27(1), pp. 10–45.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 9 25 / 25