代写代考 ECON7350 Modelling Volatility - II

The University of Queensland
(School of Economics) Applied Econometrics for Macro and Finance Week 9 1 / 25
ECON7350 Modelling Volatility – II

Recap of ARCH(q) and GARCH(p, q) Models
ARCH(q) process:
ht ≡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 >0forj=0,…,q.
GARCH(p, q) process:
ht ≡E(ε2t |ε2t−1,…,ε2t−q)=α0 +􏰈αjε2t−j +􏰈βjht−j,
= α0 + α(L)ε2t−1 + β(L)ht−1.
In practice, focus is on GARCH(1, 1), where non-negative ht is obtained by restricting α0 ≥ 0,
α1 ≥0,andβ1 ≥0.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 2 / 25
qp j=1 j=1

Regressions and Autoregressions
Consider a regression for the mean of yt with ARCH(q) errors:
yt =β1 +β2×2,t +···+βkxk,t +εt,
ε =ν􏰑α +αε2 +···+αε2 , t t 0 1 t−1 q t−q
where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s.
Consider an auto-regression for the mean of yt with GARCH(1, 1) errors:
yt = a0 + a1yt−1 + εt, εt = νt􏰔ht,
ht = α0 + α1ε2t−1 + β1ht−1,
where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 3 / 25

Extensions to the Basic GARCH Model
Huge number of extensions and variants of the GARCH model have been proposed in the past three decades to address issues with the basic model:
practical problems: non-negativity constraints difficult to impose; conceptual limitations: 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.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 4 / 25

The EGARCH Model
EGARCH (suggested by Nelson, 1991) specifies the conditional variance equation as:
lnht =α0 +β1lnht−1 +λ􏰔h +α1􏰀􏰔h 􏰀. t−1 􏰀 t−1􏰀
ε 􏰀􏰀ε􏰀􏰀 t−1 􏰀 t−1 􏰀
Advantage of EGARCH is ht will be positive for all values of parameters (due to the specification in logs).
is iid standard normal; |νt| = 􏰀􏰀√ 􏰀􏰀 is iid half-normal with E(|νt|) = 2/π.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 5 / 25

Leverage Effect in EGARCH
The effect of a shock in εt on ht:
is positive, the effect on lnht is α1 +λ; is negative, the effect on lnht is α1 −λ.
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 prices fall than when they rise; if λ = 0 then no leverage effect exists and if λ > 0 then a reverse effect exists.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 6 / 25

The Leverage Effect
The standardised ν ≡ εt is unit free t √ht
and permits a more natural interpreta- tion of the size and persistence of the shocks.
Source: Enders (2015)
(School of Economics) Applied Econometrics for Macro and Finance Week 9 7 / 25

Forecasting Volatility with the EGARCH
Assuming parameters are known, the natural predictor is 􏰐ht+j ≡ exp (E(ln ht+j | It)). Since E(νt+j |It) = 0 and E(|νt+j||It) = 􏰔2/π for j > 0,
E(ln ht+j | It) = α0 + α1􏰔2/π + β1E(ln ht+j−1 | It), =(α0 +α1􏰔2/π)(1+β1 +···+βj−2)
The long-run forecast (using the natural predictor with |β1| < 1) is: + βj−1(ln ht + λνt + α1|νt|), 1 􏰒α0 +α1􏰔2/π􏰓 lim 􏰐ht+j = exp . j→∞ 1−β1 (School of Economics) Applied Econometrics for Macro and Finance The TGARCH or GJR Model The threshold GARCH (TGARCH) model (developed by Glosten, et al., 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 εt−1 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; interpreted as leverage effect when λ > 0.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 9 / 25

Testing for Leverage Effects with GARCH
Estimate a GARCH and store the standardized residuals 􏰑
ν􏰐 =ε􏰐/ h, fort=1,…,T. t t 􏰐t
Use an F-test from the regression: 2
ν􏰐t =a0 +a1ν􏰐t−1 +a2ν􏰐t−2 +···+ut.
Under H0 : a1 = a2 = · · · = 0, the standardized squared 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: ν􏰐t = a0 + a1dt−1 + ut.
H0 : a1 = 0 indicates no leverage effect; usual t-statistic / critical values apply.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 10 / 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:
H1 :λ<0fortheEGARCHorλ>0fortheTGARCH.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 11 / 25

TGARCH Example
Data: monthly S&P 500 returns, 1979M12–1998M6. Source: Brooks (2002). Estimated model is (test statistics in parentheses)
y􏰐 = 0.172, t
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.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 12 / 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 .
−1 −0.5 0 0.5 1
GARCH TGARCH
(School of Economics) Applied Econometrics for Macro and Finance Week 9 13 / 25

(G)ARCH-in-Mean
We expect risk to be compensated by a higher expected return; so why not let the expected return be partly determined by risk?
Engle, et al. (1987) suggested the ARCH-in-Mean, or ARCH-M, specification: yt = μt + εt,
where α0, α1, β and δ > 0 are constants; w1, . . . , wq are weights assigned to past q squared
errors(e.g.,setwj =(5−j)/10forj=1,…,4).
(School of Economics) Applied Econometrics for Macro and Finance Week 9 14 / 25
μt =β+δ􏰔ht =β+δ􏰖􏰕α0 +α1􏰈wjε2t−j,

ARCH-M Components
The ARCH-M is composed of the following.
yt: observable 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.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 15 / 25

ARCH-M Intuition
The expected excess return from holding the long-term asset must be just equal to the risk premium:
E(yt+1 | It) = μt+1.
The assumption is that the risk premium is an increasing function of the conditional variance
ht+1 = E(ε2t+1 | It).
The greater the conditional variance of returns, the greater the compensation necessary
to induce the agent to hold the long-term asset.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 16 / 25

Engle, et al. (1987) ARCH-M Estimates The ARCH-M estimates are:
μ􏰐t = −0.0241 + 0.687 􏰐ht,
(−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)
Estimate of 1.64 implies the unconditional variance is infinite (although conditional variance is finite).
Risk premium is time-varying.
During volatile periods, the risk premium rises as risk-averse agents seek assets that are
conditionally less risky.
(School of Economics) Applied Econometrics for Macro and Finance Week 9 17 / 25

Stochastic Volatility Models
An alternative to conditional heteroscedacity models are the stochastic volatility models (SV). The foundation of SV models differs from conditional heteroscedasticity 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.
(School of Economics) Applied Econometrics for Macro and Finance 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), where ht now denotes log volatility.
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
(School of Economics) Applied Econometrics for Macro and Finance 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),
ht =φht−1 +ηt, ηt ∼N(0,ση2),
and 0 < φ < 1. It is convenient to treat ht as the signal or state (in analogy to signal processing), which we wish to estimate. Volatility is mapped to observed returns, and it is assumed to be a stationary AR(1) process in this case. (School of Economics) Applied Econometrics for Macro and Finance Week 9 23 / 25 Realized Volatility In (G)ARCH, E-GARCH, T-GARCH, (G)ARCH-M and SV models, the conditional variance is modelled as latent (i.e. not observed). (G)ARCH 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. (School of Economics) Applied Econometrics for Macro and Finance 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 trade-off between accuracy and micro-structure noise. Ex-post volatility measured this can be regarded as observable; it can be taken as data and modelled directly. Further reading: McAleer, M. and Medeiros, M. C. (2008), “Realized Volatility: A Review”, Econometric Reviews, 27(1), pp. 10–45. (School of Economics) Applied Econometrics for Macro and Finance Week 9 25 / 25 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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