w8 volatilityii l

8/17/2019 W8 VolatilityII L

1/34

ECON3350/7350Volatility Models-II

Alicia N. Rambaldi

Week 8

1/34


2/34

In this lecture

Readings

Introduction

ARCH(p) and GARCH(p,q)Linear Models with ARCH Errors

Extensions to the Basic GARCH ModelEGARCH

TGARCHTesting for Leverage Eff ectsExample

(G)ARCH-in Mean

IGARCH ModelsStochastic Volatility Models

Realised Volatility

Coming Up

2/34


3/34

Reference Materials

Author Title Chapter Call No

Enders, W AppliedEconometricTime Series,

3e

3 HB139 .E552015

Brooks,C Introductory

Econometricsfor Finance,

3e

9 HG173 .B76

2014

Verbeek, M A Guide to

ModernEconomet-rics,4e

8.10 HB139

.V465 2012

3/34


4/34

ARCH(q) and GARCH(p,q)

I ARCH (q ) process

E {2t |=t −1} = α0 + α12t −1 + α22t −2 + ... + αq 2t −q = α0 + α(L)

2t −1

where α(L) is a lag polynomial of order q − 1, α(1) < 1,α0 > 0.

I GARCH (p , q ) process

ht = α0

+

q

X j =1

α j 2

t −

j +

p

Xi =1

β i ht −i

= α0 + α(L)2t −1 + β (L)ht −1

For a GARH(1,1), a non-negative ht requires α0,α1 and β 1 to

be non-negative.4/34


5/34

Regressions and Autoregressions

I Consider a regression for the mean of y t with ARCH (q ) errors:

y t = β 1 + β 2x 2,t + ... + β K x K ,t + t

t = ν t

q α0 + α12t −1 + ... + αq

2t −q

where ν t ∼ N (0, 1), α(1) < 1, α0 > 0 andE [t ν t −s ] = 0 ∀ t , s I Consider an autoregressive model, AR (1), of the mean of y t

with GARCH (1, 1) errors:

y t = a0 + a1y t −1 + t |a1| < 1

t = ν t

q α0 + α12t −1 + β 1ht −1

α0,α

1 and β

1 to be non-negative, α

1 + β

1


6/34

Extensions to the Basic GARCH Model

I

Since the GARCH model was developed, a huge number of extensions and variants have been proposed.

I Problems with GARCH(p,q) Models:- Non-negativity constraints may still be violated- GARCH models cannot account for leverage eff ects

I Some of the most important extensions are:

I Asymmetric Models TGARCH and EGARCH (account forleverage eff ects),

I (G)ARCH-M models (particularly suited to study assetmarkets)

I The IGARCH model (a constrained model that accounts forthe persistence of volatility)

6/34


7/34

The EGARCH Model

I EGARCH was suggested by Nelson (1991). The variance equation isgiven by

log (ht ) = α0 + β 1loght −1 + λt −1

h0.5t −1+ α1

t −1h0.5t −1

I Advantage - Since we model the log (ht ), then even if theparameters are negative, ht will be positive.

I s t −1 = t −1

h0.5t −1and |s t −1| =

t −1h0.5t −1 are iid standard normal. s t −1 and

|s t −1|− E (|s t −1|) are zero mean iid. E (|s t −1|) =p

2/π undernormality.

The eff ect of a shock in t on log (ht )If t −1

h0.5t −1is positive, eff ect is (α1 + λ)

If t −1

h0.5t −1is negative, eff ect is (α1 − λ)

Leverage eff ect as long as λ

6= 0 and λ < 0.

7/34


8/34

EGARCH notation in Enders

I Note that Enders uses diff erent notation. However, thenotation above is consistent with that used in EViews.

I Enders’ notation of the EGARCH is as follows

log (ht ) = α0 + β 1loght −1 + α1t −1

h0.5t −1

+ λ1 t −1h

0.5t −1

I In this case the assumption is that λ1 > 0 and α1 )I If the random shock is t > 0 then the eff ect is λ1 + α1 (


9/34

Representing The Leverage Eff ect

-When λ < 0 positive shocksgenerate less volatility than negative

shocks (“bad news”).

-The standardised t −1/h0.5t −1 is unitfree and permits a more natural

interpretation of the size and

persitence of the shocks.

Source: Enders (2015)

9/34


10/34

The EGARCH model (cont.)

I It is possible to extend the EGARCH model by includingadditional lags.

I Note that we can rewrite as:

log (ht ) = α0 + β 1loght −1 + (α1 + λ)s t −1 if t −1 > 0

log (ht ) = α0 + β 1loght −1 + (α1 − λ)s t −1 if t −1 0 while λ


11/34

Volatility Prediction - EGARCHI Assuming the parameters are known and the innovations are

Gaussian, we proceed with a natural predictor

ht = exp [log (ht )] = exp

α0 + β 1loght −1 + λ

s t −1 + α1

s t −1

E t ht + j = E t {exp [log (ht + j )]} j > 0

I The Gaussian case can be re-written to find a more explicitexpression (see for example Tsay (2002), pp 105)

log (ht ) = (1− β 1)α0 + β 1loght −1 + g (s t −1)

g (s t −1) = λs t −1 + α1(

s t −1−p 2/π)I The one-step ahead forecast is given by

ht +h = h

2β1

t

exp [(1−β 1)α0] exp [g (

s

t

)]

11/34


12/34

The TGARCH or GJR Model

I Due to Glosten, Jaganathan and Runkle (1994), the

threshold-GARCH:

ht = α0 + α12t −1 + λd t −1

2t −1 + β 1ht −1

where,

d t −1 = 1 if t −1 0.

12/34


13/34

Testing for Leverage Eff ects

I Estimate an ARCH or GARCH (Use a "diagnostic" approach

to the test)

I Form the standardised residuals (from ARCH or GARCH)

s t = e t /ĥ0.5

t for t = 1,...,T

I An F - statistic, H 0 : a1 = a2 = .. = 0 using the regression

s 2t = a0 + a1s t −1 + a2s t −2 + ...

If there are no leverage eff ects the squared errors should beuncorrelated with the level of the error term and the

computed F statistic will fail to reject the null hypothesis.I A t −test H 0 : a1 = 0 using the regression

s 2t = a0 + a1d t −1 + εt

13/34


14/34

Testing for Leverage Eff ects (cont.)

I Estimate the TARCH or EGARCH model,

ht = α0 + α12t −1 + λd t −1

2t −1 + β 1ht −1

or

log (ht ) = α0 + β 1loght −1 + λt −1

h0.5t −1+ α1

t −1h0.5t −1 I Test:

H 0 : λ = 0

I using a t − test I PLEASE NOTE TYPO IN ENDERS (It should read

H 0 : α1 = 0, page 157)

14/34


15/34

Example

An Example of the use of a GJR ModelData: monthly S&P 500 returns, December 1979- June 1998.Source: Brooks (2002)

y t = 0.172

(3.198)

ht = 1.243 + 0.0152t −1 + 0.498ht −1 + 0.6042t −1d t −1

(t − stat ) (16.37) (0.44) (14.99) (5.77)

Testing the leverage eff ect

H 0 : λ = 0

Computed t = 5.77. The computed value is larger than 1.645 (fora one-tail test) and thus we conclude that there are significant

leverage eff

ects.15/34

E l


16/34

Example

GJR Model (cont.)

I Data: monthly S&P 500 returns, December 1979- June 1998.

I

The news impact curve plots the next period volatility (ĥt )that would arise from various positive and negative values of 2t

16/34


17/34

(G)ARCH-in Mean

I We expect a risk to be compensated by a higher return. So

why not let the return of a security be partly determined by itsrisk?

I Engle, Lilien and Robins (1987) suggested the ARCH-Mspecification.

y t = µt + t

µt = β + δ ht

µt = β + δ (α

0 +

p

Xi =1

αi 2

t −

i )

where,β , δ , α0 and αi are constants, and δ > 0.

17/34

(G) ( )


18/34

(G)ARCH-in Mean (cont.)

I In Engle, Lilien and Robins (ELR)

I y t =excess return from holding a long-term asset relative to aon-period treasury bill

I µt =risk premium necessary to induce the risk-averse agent tohold the long-term asset rather than the one-period bond.

I t = unforecastable shock to the excess return on thelong-term asset.

18/34

Th ELR ARCH M M d l ( )


19/34

The ELR ARCH-M Model (cont.)

I The expected excess return from holding the long-term assetmust be just equal to the risk premium:

E t −1y t = µt

I The assumption is that the risk premium is an increasingfunction of the conditional variance of t

I The greater the conditional variance of returns, the greater thecompensation necessary to induce the agent to hold thelong-term asset .

19/34

Th (G)ARCH M M d l ( )


20/34

The (G)ARCH-M Model (cont.)

I A general (G)ARCH-M model is given by:

y t = x 0

t β + γ g (ht ) + t +r

X j =1

θ j t − j

ht = α0 +

q X j =1

α j 2t − j +

p Xi =1

β i ht −i

I Usual g (ht ) are:

I log (ht ),√

ht , (ht )m m = 1, 2, ..

20/34

E l


21/34

Example

Example. ELR - Excess yields on six-month treasury bills

I r t quarterly yield on a three-month treasury bill held from t to

t + 1.I Rolling over all proceeds, at the end of two quarters and

individual investing $1 at the beginning of the t will have(1 + r t )(1 + r t +1) dollars.

I R t quarterly yield on a six-month treasury bill, buying andholding the six-month bill for the full two quarters will result in(1 + R t )

2 dollars.

I The excess yield, y t , due to holding the six-month bill is

y t = (1 + R t )2

− (1 + r t +1)(1 + r t )which is approximately equal to:

y t = 2R t − r t +1 − r t 21/34


22/34

Example

(cont.)

y t = 0.142 + t

(4.04)

The excess yield of 0.142% per quarter is more than four standard

deviations from zero.

LM ARCH = 10.1 which is larger than the critical value from the χ2(1)

at 1% (6.635).

However, the post-1979 period showed higher volatility than theearlier period.

22/34


23/34

Example

The ARCH-M estimates are:

ŷ t = −0.0241 + 0.687ĥt (−1.29) (5.15)

ĥt = 0.0023 + 1.64(0.42t −1 + 0.3

2t −2 + 0.2

2t −3 + 0.1

2t −4)

(1.08) (6.30)

Results:

I Risk premium are time-varying

I Estimate of 1.64 implies the unconditional variance is infinite

(the conditional variance if finite)I During volatily periods, the risk premium rises as risk-averse

agents seek assets that are conditionally less risky.

23/34

IGARCH Models


24/34

IGARCH Models

I Sometimes when we estimate GARCH(1,1) models we findthat the sum of the estimated α1 and β 1 coefficients is closeto 1.

I Due to Nelson (1990), constraining α1 + β 1 = 1 yields the

Integrated-GARCH or IGARCH (1, 1) model

ht = α0 + (1 − β 1)2t −1 + β 1Lht I It yields a more parsimonious representation of the error

process (there is one less parameter to estimate)

I forces ht to act like a process with a unit root.

24/34

IGARCH Models (cont )


25/34

IGARCH Models (cont.)

I However, solving for ht

ht = α0/(1−β1) + (1−β

1)∞

Xi =0

β i

12

t −

1−

i

I Unlike a true nonstationary process, ht is a geometricallydecaying function of current and past realisations of the 2t sequence.

25/34

IGARCH (cont )


26/34

IGARCH (cont.)

I

As it is a decaying function of current and past values of theshocks it can be estimated like any other GARCH (p , q )

I The one −step-ahead forecast of the conditional variance is

E (ht +1|

=t ) = α0 + ht

I The j −step-ahead forecast of the conditional variance is

E (ht + j |=t ) = j α0 + ht I Thus, appart from the intercept the forecast of the conditional

variance for the next period is the current value of theconditional variance. The unconditional variance is infinite.

26/34

Stochastic Volatility Models


27/34

Stochastic Volatility Models

I An alternative class of models to those in the GARCH (p , q )family are the stochastic volatility models (SV).

I They diff er principally because the conditional variance is not adeterministic function of past information.

I They are particularly designed to deal with volatility clustering.

I SV models contain a second error term which enters theconditional variance specification and are written asstate-space models

27/34

Stochastic Volatility Models (cont )


28/34

Stochastic Volatility Models (cont.)

I The simple forms can be estimated using maximum likelihoodor Monte Carlo methods via Kalman filtering

I A SV model is defined as

at = ht t

(1 − β 1L − . . .β mLm) ln ht = α0 + ν t where at are returns, ν t

∼N (0,σ2ν ), t

∼N (0, 1),

α0,β 1, . . . , β m are parameters

28/34

Stochastic Volatility Models (cont )


29/34

Stochastic Volatility Models (cont.)

Example

A SV model for the daily returns of an asset price, y t is:

y t = a + σexp (1

2

θt )t t ∼

N (0, 1)

θt +1 = φθt + ηt ηt ∼ N (0, σ2η) 0


30/34

Realised Volatility

I The conditional variance is latent (ie not observed)

I Models from the GARCH (p , q ) family (and SV type models)are estimates of the latent conditional variance

I Criticism are that standard latent volatility models fail todescribe in an adequate manner the low, but slowly decreasing,

autocorrelations in the squared returns that are associatedwith high excess kurtosis of returns.

I Search for an adequate framework has led to the analysis of high frequency intraday data.

I Merton (1980) noted that the variance over a fixed intervalcan be estimated arbitrarily, although accurately, as the sum of squared realisations, provided the data are available at asufficiently high sampling frequency

30/34

Realised Volatility (cont )


31/34

Realised Volatility (cont.)

I Andersen and Bollerslev (1998) showed that ex-post daily

foreign exchange volatility is best measured by aggregating 288squared five minute returns.

I The five-minute frequency is a trade-off between accuracy,which is theoretically optimized using the highest possiblefrequency, and microstructure noise that can arise through thebid-ask bounce, asynchronous trading, infrequent trading, andprice discreteness, among other factors

I Ignoring the remaining measurement error, which can beproblematic, the ex-post volatility essentially becomes

“observable.”I As volatility becomes “observable,” it can be modeled directly,

rather than being treated as a latent variable

31/34

Realised Volatility (cont.)


32/34

Realised Volatility (cont.)I Consider a simple discrete time model in which the daily returns of

a given asset are typically characterized as

r t = h0.

5t ηt

where ηt is a sequence of independently and normally distributedrandom variables with zero mean and unit variance, ηt NID (0, 1)and h0.5t is the (time-varying) standard deviation of daily returns.

I Suppose that, in a given trading day t , the logarithmic prices are

observed tick-by-tick.I Consider a grid Λt = {τ 0,..., τ nt } containing all observation

points, andI p t ,i , i = 1,...,nt , to be the i

th (logarithmic) price observationduring day t , where nt is the total number of observations at

day t .I Furthermore, suppose that r t ,i = p t ,i − p t ,i −1 is the i th

intraperiod return of day t , such that

r t =

nt

Xi =0

r i ,t

32/34

Two unbiased estimators of RV


33/34

uI The realised variance is defined as the sum of all available

intraday high frequency squared returns given by

RV (all )t =

nXi =0

r 2t ,i

I The squared daily return can be written as

r 2t =

nXi =0

r t ,i

!2

I They are unbiased as it can be shown that

E (r 2t |=t ,0) = E (RV (all )t |=t ,0) = ht (=t ,0 is the information setavailable prior to the start of day t ).

I Further reading, see for example:I McAleer, Michael and Medeiros, Marcelo C.(2008) ’Realized

Volatility: A Review’, Econometric Reviews , 27: 1, 10 — 45

33/34

Coming Up


34/34

g p

I Vector Autoregressive Models

34/34

w8 volatilityii l

Documents