Conditional heteroskedasticity

7/25/2019 Conditional heteroskedasticity

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Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I-GARCH, EGARC

Lecture 5: Conditional Heteroskedasticity Models

Drew Creal

February 2, 2016

Drew Creal (Chicago Booth) February 2, 2016 1 / 104
http://goforward/http://find/http://goback/


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Outline

1 Time Series Analysis

2 Stylized facts on Volatility Clustering

3 Realized Variance

4 ARCH models

5 GARCH models

6 I-GARCH, EGARCH, GARCH-M, GJR models7 Value-at-Risk

8 References

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Outline



3 Realized Variance

4 ARCH models

5 GARCH models

6 I-GARCH, EGARCH, GARCH-M, GJR models7 Value-at-Risk

8 References

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ARMA(p,q)

so far, we have focused on modeling the conditional mean:

rt = E [rt|rt1, rt2, . . . ; ]+t the benchmark models for the conditional mean are ARMA models

rt=0+1rt1+. . .+prtp+t 1t1+. . . qtq given estimates, 0, 1, . . . , q

1, q, we can calculate the residuals

t=vt(0) recursively from the initial condition.

the key modeling goal is to make sure that the residuals{t} arewhite noise


Ti S i A l i S li d f V l ili Cl i R li d V i ARCH d l GARCH d l I GARCH EGARC
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Why White Noise?

consider an AR(1):

rt=0+1rt1+t suppose the AR(1) is misspecified

then the forecast errors are correlated over time, i.e.:

vt(0) = t

vt+1(0) = t+1

...

that means your forecast is not optimal (youre not using all theinformation optimally)

forecast errors should be white noise


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Volatility Clustering

suppose we have a good model for the conditional mean t (e.g.ARMA(p, q))

now we look at the squared residuals{

2

t} to test for conditionalheteroskedasticity for returns on most financial assets, there is a lot more

autocorrelationin conditional second moments than in the

conditional first moments. volatility is predictable!


Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I GARCH EGARC
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Volatility?

In most asset markets, volatility varies dramatically over time.

episodes of high volatility seem to be clustered.

we dont simply observe volatility

how do we measure vol?

1. implied volatility (back out volatility from option prices)

2. (non-parametric) realized volatility (e.g., realized volatility of stockreturns over one-month using daily data)

3. (parametric) model volatility


Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I GARCH EGARC
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VIX-Implied Volatility

1987 1990 1993 1995 1998 2001 2004 2006 2009 201210

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70VIX CBOE SPX Volatility

Annualized measure of 30-day vol. VIX, designed to measure the markets expectationof 30-day volatility implied by at-the-money S&P 100 Index (OEX) option prices.

Monthly data. 1990-2009. VIX white paper


Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I-GARCH EGARC
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Why do we care about Volatility?

why do we care about modeling vol?

1. portfolio allocation

2. risk management

3. option pricing


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Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I GARCH, EGARC

ARCH model ofEngle (1982)

Definition

The ARCH(m) model is given by:

t=tt, 2t =0+1

2t

1+. . .+m

2t

m

where t is i.i.d., mean zero, has variance of one , 0 > 0 and i 0 fori> 0

the standard normal is a common choice for t

the (standardized) Students t is another option for t

large shocks are followed by more large shocks


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y y y g ,

Building a Volatility Model

1. estimate a model for the conditional mean (e.g. ARMA model)

2. use the residuals tfrom the mean equation to test for ARCH effects

3. specify a volatility model for ARCH effects

4. check the fitted model


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y y y g ,

Outline



3 Realized Variance

4 ARCH models

5 GARCH models

6

I-GARCH, EGARCH, GARCH-M, GJR models7 Value-at-Risk

8 References


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y y y g

Monthly Returns on Market Squared

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log stock returns on market -squared. Monthly data. 1926-2012.


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Stock Returns

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0.4PACF for squared returns

Month

log stock returns on CRSP-VW. Monthly data. 1926-2012.


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Dynamics of Variance-Monthly Returns

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Dynamics of Variance-Monthly Returns

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Daily Returns on Market Squared

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log stock returns on market -squared. Daily data. 1988-2012.


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Dynamics of Variance -Daily Returns

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log stock returns on CRSP-VW. Daily data. 1926-2012.


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Dynamics of Variance-Daily Returns

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log stock returns on CRSP-VW. Daily data. 1970-2012.


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IBM

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log stock returns on IBM -squared. Monthly data. 1988-2009.


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IBM

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ACF for |log returns|

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ACF for squared log returns

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PACF for squared log returns

Month

log stock returns on IBM. Monthly data. 1988-2009.


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MSFT

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log stock returns on MSFT -squared. Monthly data. 1988-2012.


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MSFT

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ACF for |log returns|

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ACF for squared log returns

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PACF for squared log returns

Month

log stock returns on MSFT. Monthly data. 1988-2009.


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Changes in log U.S. $

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0.1Changes in log USD (TradeWeighted Index)

log changes in Dollar -Trade-Weighted Index. Monthly data. 1971-2009.


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Changes in log U.S. $-squared

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U.S. $

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Yen

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log changes in Yen/USD squared. Daily data. 1971-2009.


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Yen

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log changes in Yen/USD. Daily data. 1971-2009.


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Testing for ARCH

estimate a model for the conditional mean: e.g. ARMA(p, q)

use the squared residuals to test for ARCH

specify a volatility model if ARCH effects are documented


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Testing for ARCH(1)

we can test for autoregressive conditional heteroscedasticity using aLB Q-test:

suppose the model for the conditional mean is ARMA(p, q)

estimate the model

run a Q-test on the estimated squared residuals{2t} test the null that:

H0 :1 =2 = . . .=m =0

where idenotes the autocorrelation of the squared residuals if you cannot reject the null, no need for ARCH machinery


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Testing for ARCH(2)

we can test for autoregressive conditional heteroscedasticity using aLM test:

suppose the model is ARMA(p, q)

estimate the model

run a LM-test on the estimated squared residuals{2t} test the null that i =0, i=1, 2, . . . ,m:

2t =0+12t1+. . .+m

2tm+et, t=m+1, . . . ,T

test the null that:

H0 :1=2 = . . .=m =0

use the usual F-test, reject the null ifF exceeds 2() the (1 )-thupper percentile


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Volatility

volatility clusters

volatility is stationary

leverage effect (asymmetry) negative returns seem to be followed by larger increases in volatility

than positive returns

first emphasized byBlack (1976)

Black (1976) suggested that negative returns (decreases in price)change a companys debt/equity ratio, increasing their leverage.


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Leverage Effects: Monthly Returns on Market Squared

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0negative log returns on market

log stock returns on market -squared. Monthly data. 1926-2012.


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Models of Volatility

Definition

The model is given by

rt=t+t

where t is governed by an ARMA(p, q) and where the conditionalvariance oftvaries according to:

V[t|

Ft1

] =

2

t


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Outline



3 Realized Variance

4 ARCH models

5 GARCH models

6 I-GARCH, EGARCH, GARCH-M, GJR models

7 Value-at-Risk

8 References


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High-Frequency Data

let rmt

denote the log of the returns over a period of time

may denote 1 day, 1 week, 1 month.

suppose over the time , we observe equally spaced log-returns{rt,i}ni=1 at ahigher frequency

rmt =n

i=1

rt,i

for log returns, the variance of the return over time is given by:

V (rmt | Ft1) =n

i=1

V (rt,i| Ft1) +2i


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Realized Variance (Volatility)

the estimate for the variance is given by:

2t,m = ni=1(rt,i rt)2n 1 in practice, is often 1 day or 1 month.

assumption: estimator assumes errors are i.i.d. meaning that volatilityis roughly constant over .

note: the term realized varianceis not used consistently in theliterature.

many finance papers will call this realized volatility. In econometrics,realized volatility is the square root of realized variance.


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Realized Variance

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AnnualizedVol

Realized Vol

Annualized (Monthly) Realized Vol. Daily data. 1926-2012.


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Realized Variance and Implied Vol

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Realized Vol and Implied Vol (VIX)

Realized

VIX

Annualized (Monthly) Realized Vol and the VIX. Daily data. 1983-2012.


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Outline



3 Realized Variance

4 ARCH models

5 GARCH models


7 Value-at-Risk

8 References


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ARCH(1)

Definition

The ARCH(1) process is given by:

t = tt, 2t =0+1

2t1

where t is i.i.d., mean zero, has variance of one , > 0 and i 0 fori> 0


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Variance

the unconditional mean is zero:E[t] = E(E[t|Ft1]) = E(tE[t|Ft1])

the unconditional variance is:

V[t] = E 2t= E E 2t|Ft1 = E 0+12t1 because of stationarity, we get that:

V[t] =0+1V[t]

as a result, the variance is:

V[t] = 01 1

we require that 0

1 < 1


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Fourth Moment

the unconditional fourth moment is :

E[4t] = E(E[4t|Ft1]) = E(34t) =3E

0+1

2t1

2 using stationarity, this implies

m4= 320(1+1)

(1 1)(1 321)which means 0 21 < 1/3

the unconditional kurtosis is given by:

E[4t]

V[t]2 =3

1 211 321

> 3

positive kurtosiseven though innovations are conditionally Gaussian


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Normality

ARCH(1) process is conditionally normal ift

N (0, 1)

ARCH(1) process is not unconditionally normal

time variation in the variance generates fat tails


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AR format

the ARCH(m) model in AR format is given by:

2t+1=0+12t+. . .+m

2tm+1+ t+1

where t=2t 2t is an uncorrelated series with mean zero. this is like an AR(m), except that t is not i.i.d.

the PACF is a useful too for determing the right lag length



1 h d f
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1-step ahead forecast

the ARCH(m) model is given by:

2t+1 =0+1

2t+. . .+m

2tm+1

hence the forecast is simply:

2t(1) =0+12t+. . .+m

2tm+1



2 h d f
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2-step ahead forecast


2t+2=0+12t+1+. . .+m

2tm+2

hence the forecast is simply:

2t(2) =0+12t(1) +. . .+m

2tm+2

note that we have used the following:

Et2t+1|Ft= Et 2t+12t+1|Ft=2t+1Et 2t+1|Ft



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h-step ahead forecast


2t+h =0+12t+h+. . .+m

2t

m+h

General expression:

2t(h) =0+12t(h 1) +. . .+m2t(hm)

where 2t(hj) =

2t+hj ifhj< 0



B ildi ARCH d l
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Building an ARCH model

we can pick the order of an ARCH model by looking at PACFat{2t}; you might need a large number of lags

for a well-specified ARCH model, the standardized residual:

t= tt

should be white noise

this can be tested by examiningt1. check the volatility spec by doing a Q-test on2t2. check the mean spec by doing a Q-test ont



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Weakness of ARCH Models

1. symmetry in effects of positive and negative shocks on vol

2. ARCH model is restrictive (21 < 1/3 for ARCH(1))

3. ARCH model: mechanical description of volatility (no economics)

4. sometimes many lags are needed to describe vol dynamics

Matlab function



ARCH(5)
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ARCH(5)

Table: ARCH(5)

Parameter Value s.e. t-stat

Constant 3.09E-05 6.65E-07 46.472

ARCH1 0.099307 0.005071 19.5838ARCH2 0.144208 0.008707 16.5623ARCH3 0.141901 0.009809 14.4668

ARCH4 0.168116 0.01012 16.6119ARCH5 0.150002 0.010459 14.3424

ARCH(5). Daily data. CRSP-VW.1971-2012.



Parametric Volatility
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Parametric Volatility t

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140Conditional Annualized Volatility

arch(5)

ARCH(5). Daily data. CRSP-VW.1971-2012. use the [V, logL] = infer(EstMdl, ycommand.



Parametric Volatility against Realized Vol
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Parametric Volatility against Realized Vol.

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realized

arch(5)




Standardized Residuals ( / )2
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Standardized Residuals(t/t) .

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Standardized Residuals t/t
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Standardized Residuals t/t.

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AIC/BIC
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AIC/BIC.

check the AIC/BIC:

[V, logL] = infer(EstMdl, y)

[aic, bic] =aicbic(logL, 6, size(y, 1))

aic=7.0162e+04

bic=7.0126e+04



ARCH(10)
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ARCH(10)

Table: ARCH(10)

Parameter Value standard error t-stat

Constant 2.04E-05 7.56E-07 26.927

ARCH1 0.083715 0.004496 18.6207ARCH2 0.113967 0.008698 13.1022ARCH3 0.080502 0.009147 8.80109ARCH4 0.095266 0.00937 10.1667ARCH5 0.107659 0.011581 9.29659ARCH6 0.075962 0.009292 8.1747ARCH7 0.052205 0.008305 6.28631

ARCH8 0.083885 0.00877 9.56551ARCH9 0.073778 0.008694 8.4864

ARCH10 0.049606 0.008884 5.58371




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arch(10)

ARCH(10). Daily data. CRSP-VW.1971-2012. use the [V, logL] =infer(EstMdl, ycommand.



Parametric Volatility against Realized Variance.
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Parametric Volatility against Realized Variance.

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realized

arch(10)




Standardized Residuals(t/t)2.
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S (t/ t)

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Standardized Residuals t/t .
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t/ t

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AIC/BIC.
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/

check the AIC/BIC:


[aic, bic] =aicbic(logL,11, size(y, 1))

aic=7.0538e+04

bic=

7.0465e+04

ARCH(10) has smaller AIC and BIC than ARCH(5)



AIC/BIC.
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/

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6.75x 10

4 AIC

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4 BIC

lags

ARCH(lags). Daily data. CRSP-VW.1971-2012.



Outline
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3 Realized Variance

4 ARCH models

5 GARCH models


7 Value-at-Risk

8 References



GARCH model ofBollerslev (1986)
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Definition

Consider t=rt t. The t follows a GARCH(m, s) model if

t=

tt, 2

t =

0+

m

i=1i2ti

+s

j=1j2tj

where t is i.i.d., mean zero, has variance of one , 0 > 0 and i 0 fori> 0 and j 0 for j> 0, and

max(m

,s

)i=1

(i+ i) < 1



GARCH is ARMA for vol
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define:t=2t 2t

hence2t =

2t t

this delivers an ARMA-like representation:

2t=0+max(m,s)

i=1

(i+ i)2ti

s

j=1

jtj+ t

where E(t) =0 and E(ttj) =0 but is not white noise

the unconditional variance is simply:

E2t

= 0

1max(m,s)i=1 (i+ i)Drew Creal (Chicago Booth) February 2, 2016 66 / 104


GARCH(1,1) ofBollerslev (1986)
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consider GARCH(1,1) model:

t = tt t N (0, 1)2t+1 = 0+1

2t+1

2t

large 2t, 2t leads to large

2t+1

the unconditional kurtosis is given by:

E[4t]

V[t]2 =3 1

(1+1)2

1 (1+1)2 221 > 3 positive kurtosis even though innovations tare Gaussian



Forecasting with GARCH(1,1)
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2t+1=0+12t+1

2t

the one-step ahead forecast of volatility:

2t(1) =0+12t+12t

the two-step ahead forecast of volatility:

2t(2) =0+ (1+1)[2t(1)]

where we used the following expression

2t+1=0+ (1+1)2t +1

2t

2t 1



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2t+1=0+12t+1

2t

the one-step ahead forecast of volatility:

2t(1) =0+12t+12t

the two-step ahead forecast of volatility:

2t(2) =0+ (1+1)[2t(1)]

in general:

2t(h) =0+ (1+1)[2t(h 1)]



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Result

The multi-step volatility forecast2t(h)

2t(h) =0+ (1+1)[2t(h 1)]

converges to the unconditional variance ash increases



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for a well-specified GARCH model, the standardized residual:

t=

tt

should be white noise

this can be tested by examining

t

1. check the volatility specification by doing a Q-test on

2t

2. check the mean spec. by doing a Q-test ont



GARCH(1,1)
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Table: GARCH(1,1)


Constant 1.16E-06 2.23E-07 5.18121GARCH1 0.909294 0.002906 312.933

ARCH1 0.07998 0.001725 46.3601

GARCH(1,1). Daily data. CRSP-VW.1971-2012.



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0 5 10 15 20 250.02

0

0.02

0.04

0.06

0.08

0.1


Days0 5 10 15 20 25

0.04

0.03

0.02

0.01

0

0.01

0.02


Days

0 5 10 15 20 250.02

0.01

0

0.01


Days0 5 10 15 20 25

0.02

0.01

0

0.01


Days




AIC/BIC.
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check the AIC/BIC:


[aic, bic] =aicbic(logL,10, size(y, 1))

aic=7.0738e+04

bic=7.0665e+04 GARCH(1,1) has smaller AIC and BIC than ARCH(10) you could use the BIC/AIC criterion to find the best GARCH(p,q)



Forecasting
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forecasting using the GARCH(1,1) model

suppose we want to forecast the daily variance over the next 500

periods use the observed returns y to initialize the forecast

Vf =forecast(EstMdl, 500,Y0, y)



Forecast.
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1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 110000

0.5

1

1.5

2

2.5

3

3.5x 10

3 Conditional Variance Forecasts for VWCRSP Returns

Inferred

Forecast




Forecast.
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0 50 100 150 200 250 300 350 400 450 5006.5

7

7.5

8

8.5

9

9.5

10

10.5

11x 10

5 Conditional Variance Forecasts for VWCRSP Returns




Outline
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3 Realized Variance

4 ARCH models

5 GARCH models


7 Value-at-Risk

8 References

Drew Creal (Chicago Booth) February 2, 2016 81 / 104 Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I-GARCH, EGARC

I-GARCH
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Definition

Consider t=rt t. The t follows a I-GARCH(1, 1) model if

t=tt, 2t =0+ (11)

2t1+1

2t1

where t is i.i.d., mean zero, has variance of one. 0 < < 1

the unconditional variance is not defined!

hard to justify for returns

Drew Creal (Chicago Booth) February 2, 2016 82 / 104 Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I-GARCH, EGARC

Forecasting with I-GARCH(1,1)
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in general for GARCH(1,1):

2t(h) =0+ (1+1)[2t(h 1)]

now set 1+1 =1 repeated substitution yields:

2t(h) = (h

1)0+ [

2t(1)]

effects on future vols are persistent!

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GARCH(1,1)-M
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DefinitionConsider

t=rt t 2t.The t follows a GARCH(1, 1)-M model if

t=tt, 2t =0+1

2t1+1

2t1

where t is i.i.d., mean zero, has variance of one. 0 < < 1.

Mstands for GARCH in the mean.

the parameter measures a variance risk premium

the risk premium increases when volatility increases


E-GARCH model ofNelson (1991)
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DefinitionConsider t=rt t. The t follows a E-GARCH(1, 1) model if

t=tt, (1 1B) ln 2t = (1 )0+g(t1)

where t is i.i.d., mean zero, has variance of one.

g(t1) = (+)t E(|t|) ift 0g(t1) = ( )t E(|t|) ift< 0

built-in asymmetry to capture leverage effect

we expect < 0


E-GARCH(1,1)
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Table: E-GARCH(1,1)


Constant -0.17256 0.011062 -15.599GARCH1 0.980926 0.001135 864.546

ARCH1 0.13937 0.005217 26.7121Leverage1 -0.07601 0.003161 -24.0473

EGARCH(1,1). Daily data. CRSP-VW.1971-2012.


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0 5 10 15 20 250.02

0

0.02

0.04

0.06

0.08

0.1


Days

0 5 10 15 20 250.06

0.04

0.02

0

0.02


Days

0 5 10 15 20 250.02

0.01

0

0.01


Days0 5 10 15 20 25

0.02

0.01

0

0.01


Days



Glosten, Jagannathan, and Runkle (1993)
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DefinitionConsider t=rt t. The tfollows a GJR-GARCH(1, 1) model if

t=tt, 2t =0+1

2t1+1

2t1+It1

where t is i.i.d., mean zero, has variance of one.

It =

0 ift 01 ift< 0

simple way to capture leverage effect

we expect > 0


Outline
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1 Time Series Analysis2 Stylized facts on Volatility Clustering

3 Realized Variance

4 ARCH models

5 GARCH models


7 Value-at-Risk

8 References


VaR

D fi iti
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Definition

Let the r.v. Lt,t+h denote the loss of a financial position over a timehorizont t+h. The Value at Risk over the time horizon h is theamount of money you could lose with probability p. This is given by:

p=Pr[Lt,t+h

VaR] = 1

Pr[Lt,t+h < VaR] .

For a given probability p, how much could a position lose? This is the amount ofmoney (value) that is at risk.

in applications, we need

1. the probability (say p=.01)2. the time horizon h (say h=10 days)3. the frequency of the data4. the loss cdfFt,t+h5. the size of the position


Remarks
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By convention, VaR is reported as a positive number.

VaR is the 1 pquantile function of the loss distribution. For a long position, a loss is a negative return.

Option #1: we can look at the distribution ofrtand calculate the1 p-th quantile.

Option #2: we can look at the distribution ofrt, calculate the p-thquantile, and take the absolute value.

Warning: be careful when calculating this. The distribution of returnsis typically not symmetric.


Simple example

Suppose you bought $10000 of MSFT
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Suppose you bought $10000 of MSFT.

Let rt+1 denote the log-return over the next month. Our model is:

rt+1 N (0.05, 0.01) The one-period return CDF is: Ft,t+1(rt+1) = (rt+1

|0.05, 0.01)

Let p=0.01. Calculate the quantile F1t,t+1(0.01) = 0.1145. Thevalue at risk is:

VaR0.01 = |10000 0.1145| = 1145 Let p=0.05. Calculate the quantile F1t,t+1(0.05) = 0.1826. The

value at risk is:

VaR0.05 = |10000 0.1826| = 1826


Comments
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The quantile function is known (or easily calculated) when returns areconditionally normal or conditionally Students t.

In most models used for risk management, the conditional distribution

of returns is not known. We need to calculate VaR by simulationmethods.

RiskMetrics uses the IGARCH(1,1) model with normally distributederrorstand no drift t=0. This is a special case where returns are

conditionally normal at all horizons. (Next example.)


RiskMetrics

D fi
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Definition

The RiskMetrics Value at Risk assumes the daily log return

rt|Ft1 N(t, 2t)where

t=0, 2t =2t1+ (1 )r2t1, 1 > > 0Hence,

rt=t=tt t N(0, 1)

is an IGARCH(1,1) process without drift.

The combination of no drift, normal errors, and IGARCH implies thatthe conditional distribution is known at all horizons.

D C l (Chi B th) F b 2 2016 97 / 104 Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I-GARCH, EGARC

VaR in RiskMetrics

Using the independence assumption for t :
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Using the independence assumption for t:

V (rt[h]|Ft)=h

i=1

V (t+i|Ft)

where V (t+i|Ft)= E2t+i|Ft

can be computed recursively.

from the I-GARCH(1,1) specification:

2t =2t1+ (1 )2t1(2t1 1) t

this implies that the conditional variance can be stated as:

2t+i=2t+i

1+ (1

)2t+i

1(

2t+i

1

1), i=2, . . . , h

and hence E 2t+i|Ft=2t+1, i=2, . . . , h hence, the conditional variance of the long-horizon return is given by:

V (rt[h]| Ft) =hV (t+1| Ft) =h2t+1D C l (Chi B th) F b 2 2016 98 / 104


VaR in RiskMetrics

For the IGARCH(1,1) model under these assumptions, we have
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For the IGARCH(1,1) model under these assumptions, we have

rt[h] N 0, h2t+1 Important assumptions:

normal errors: sum of normal r.v.s is normal. no drift volatility does not mean-revert (it has no mean)

suppose the tail probability is 5% the daily VaR under RiskMetrics is :

VaR=Amount of Position 1.65t+1

the VaR at the h-day horizon under RiskMetrics is :VaR(h) = Amount of Position 1.65ht+1

the square root of time rule...


Discussion of RiskMetrics
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simple, makes risk transparent and tangible

normality assumption is a weakness

square root of time rule is an artefact of the assumption that t=0

in general:

VaR(h) =Amount of Position

1.65ht+1 h

VaR(h)

=kVar(1)


Calculation of VaR from a general GARCH model

F d l d t k th CDF f th l t h i h
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For many models, we do not know the CDF of the losses at horizon h.We must use simulation.

Start at time twith known estimate 2t and mean t.1. For i=1, . . . ,M, simulate future returns

r(i)

t+1, r

(i)

t+2, . . . , r

(i)

t+h

which requires simulating (i)t+1,

(i)t+2, . . . ,

(i)t+h and

2,(i)t+1,

2,(i)t+2, . . . ,

2,(i)t+h

2. For each return sequence i, calculate the loss L(i)t,t+h at horizon h on the position.

3. Sort the lossesL

(i)t,t+h

Mi=1

from smallest to largest. This is an approximation to

the CDF of the loss distribution.

4. As your estimator of VaR, take the nearest integer (1 p) M in the sorted valuesas the quantile.

The larger the value ofM the more accurate the estimate.

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VaR.
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MillionsofDollars

EGARCH(1,1). Daily data. CRSP-VW.1971-2012. VaR for $1bn position in stocks.

D C l (Chi B h) F b 2 2016 102 / 104 Time Series Analysis Stylized facts on Volatility Clustering Realized Variance ARCH models GARCH models I-GARCH, EGARC

Outline

S A l
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3 Realized Variance

4 ARCH models

5 GARCH models


7 Value-at-Risk

8 References



Black, F. (1976).Studies of stock market volatility changes.In Proceedings of the American Statistical Association, Business and Economic Statistics Section, Volume 177, pp. 181.
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Bollerslev, T. (1986).Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics 31(3), 307327.

Engle, R. F. (1982).

Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation.Econometrica 50(4), 9871007.

Glosten, L. R., R. Jagannathan, and D. E. Runkle (1993).On the relation between the expected value and the volatility of the nominal excess return on stocks.

The Journal of Finance 48(5), 17791801.

Nelson, D. B. (1991).Conditional heteroskedasticity in asset returns: a new approach.Econometrica 59(2), 347370.



Matlab
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to specify Engles original ARCH(Q) model, use the equivalentGARCH(0,Q) specification.

specify the ARCH(Q) modelas follows:

model=garch(0,Q)

math works link

estimate the ARCH(Q) modelas follows:

[EstMdl,EstParamCov, logL, info] =estimate(Mdl,Y)

math works link

Back

http://www.mathworks.com/help/econ/specify-garch-models-using-garch.htmlhttp://www.mathworks.com/help/econ/garch.estimate.htmlhttp://www.mathworks.com/help/econ/garch.estimate.htmlhttp://www.mathworks.com/help/econ/specify-garch-models-using-garch.htmlhttp://find/

Conditional heteroskedasticity

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