------Dr. Junsoo Lee
Why Volatility clustering? What are the issues?
· Time varying risk premia
· Heteroskedastic variance
o not constant variance
· News arrivals are serially (auto) correlated.
o News tends to cluster in time.
· Asymmetric reactions (leverage effects):
o “People react more when prices fall.”
· Non-linearity in the model
o Time deformation (economic activity does not match calendar time)
· Leptokurtic distribution
o Fat-tails and excess peakness at the mean
Volatility Models
1. Moving Average Models
m-day historic volatility estimate
2 = rt-i2
where rt-i is the m most recent returns
rt = Dlog(Pt) = log(Pt) - log(Pt-1) : first difference of the price
data = growth rates of price
Questions:
o How to determine m?
o Equal weights for each term, rt-i2?
o Could we lose valuable information by smoothing out the series?
2. Exponentially Weighted Moving Averages (EWMA)
2 = a rt-12 + (1 - a)
2
= a (1 - a)i-1 rt-i2
Note: This is the same as exponential smoothing method
(except that yt is replaced with rt-12).
Recall from Lecture 3:
t = a yt-1 + (1 - a) t-1
t-1= a yt-2 + (1 - a)
t-2
t-2= a yt-3 + (1 - a) t-3
= a yt-1 + (1-a)ayt-2
+ (1-a)2ayt-3+
….. + (1-a)kayt-k+1
Similarly,
2 = (1-l) li-1rt-i2 (with l = 1-a)
Remarks:
o If a = 1, it’s a naive model for volatility with rt-12.
o If a is close to 1, recent values of rt-i2 are heavily weighted.
o The predicted volatility remains constant as the estimate at T.
3. Auto-Regressive Conditional Heteroskedasticity (ARCH) Models
“Express ut2 in terms of past values of ut2 (lagged squared residuals).”
ut2 = w + a1ut-12 + … + aqut-q2 (no error term!)
(1) Mean Equation: Usual ARMA models or others
The Model Explaining "Y" (the "mean" equation):
yt = c + f yt-1 + ut .. AR(1) model, for instance
Var(ut | Wt-1 ) = ht
.. conditional variance of ut
whereWt-1 is the information set available at t-1.
(2) Variance Equation: ARCH(Q) equation
The Model Explaining the Conditional Variance:
ht = w + a1ut-12 + … + aqut-q2
® These two equations consist of an AR(1)-ARCH(Q) model!
[AR(1) Mean & ARCH(Q) Variance equations]
More examples:
ARCH(2): ht = w + a1ut-12 + a2ut-22
ARCH(3): ht = w + a1ut-12 + a2ut-22
+ a3ut-32
4. GARCH (Generalized ARCH) Models
Additionally include lagged variance terms, ht-j, j=1,..,P.
GARCH(P, Q): ht = w + a1ut-12 + … + aqut-q2
+ b1ht-1 + …. + bpht-p
Note: People often use GARCH (1,1) models, without having to search for optimal models.
GARCH(1, 1): ht = w + a1ut-12 + b1ht-1
A complete specification of the AR(1)-GARCH(1,1) model, for example:
yt = c + f yt-1 + ut .. AR(1) model, for instance
Var(ut | Wt-1 ) = ht
.. conditional variance of ut
ht = w + a1ut-12 + b1 ht-1 ... GARCH(1,1), for instance
More examples:
GARCH(1, 2): ht = w + a1ut-12 + a2ut-22 + b1ht-1
GARCH(2, 1): ht = w + a1ut-12 + b1ht-1
+ b2ht-2
Summary Questions:
1. Why conditional?
2. Time-varying volatility? Non-Constant variance?
3. Time-varying risk premia?
4. Is Volatility AUTO-CORRELATED?
5. Can we estimate/forecast the time-varying volatility?
Estimating ARCH-GARCH Models
Main questions (as you may expect from ARMA models)
· How to choose P and Q?
· Testing for ARCH effects?
· It is a non-linear model, so how to estimate it?
Choosing P and Q
· Look at the ACF and PACF of squared residuals, ut2 of the ARMA model (see Case Study)
· Ljung-Box Q2(m) statistics using ut2.
· AIC / SBC (can be practical!)
· Your call: say, GARCH(1,1)
Note: If your GARCH model is optimal, the resulting squared residuals from GARCH models should look like a white noise process.
(ACF/PACF of ut2lie within the band; Q2(m) statistics; other tests)
Testing for ARCH effects
LM test for ARCH effects.
From ht = w + a1ut-12 + … + aqut-q2
, we can have
H0: a1 = a2 = … = aq = 0.
Then, ht = w (constant!)
Thus, the testing hypothesis is
H0: a1 = a2 = … = aq = 0 (no ARCH effects)
1st. Run the usual ARMA models, and obtain the residual
t (t2).
2nd. Regress t2 on t-12,
t-22, …, t-q2
t2 = c + c1t-12
+ c2t-22 + + cqt-q2 + error
LM = T R2 ~ cq (chi-square with d.f. q.)
“Reject the null of no ARCH effects if LM > critical values.”
(i) It is important to test on the residuals from the mean equation (ARMA models), not from GARCH models.
(ii) We do not test directly for GARCH effects. If ARCH effects exist, GARCH models can be also considered
Estimating GARCH models
Estimating GARCH models
We use the Maximum likelihood Estimation (MLE), since the GARCH model is non-linear. Using iterative algorithm such as BHHH or others, we search for optimal parameter values while maximizing the log likelihood function:
Log Lik = log f(yi|Wt-1) where f(.) is the density function.
All details can be skipped..
EXAMPLE (SAS code):
FILENAME INDATA 'INFLATION.XLS';
PROC IMPORT OUT= WORK.INFLATION
DATAFILE= INDATA DBMS=EXCEL2000 REPLACE;
GETNAMES=YES;
RUN;
PROC AUTOREG;
MODEL Y = YLAG / ARCHTEST GARCH=(q=1, p=1);
RUN;
SAS OUTPUT:
The AUTOREG Procedure
Dependent Variable Y
Ordinary Least Squares Estimates
SSE 596.02691 DFE 162
MSE 3.67918 Root MSE 1.91812
SBC 687.240352 AIC 681.040619
Regress R-Square 0.0601 Total R-Square 0.0601
Durbin-Watson 2.2057
Q and LM Tests for ARCH Disturbances
Order Q Pr > Q LM Pr > LM
1 0.0839 0.7721 0.0701 0.7912
2 5.3519 0.0688 5.1615 0.0757
3 6.0898 0.1073 5.7805 0.1228
4 8.6146 0.0715 7.1844 0.1265
5 19.2066 0.0018 15.9118 0.0071
6 20.9009 0.0019 16.6870 0.0105
7 23.2149 0.0016 16.8749 0.0182
8 23.2781 0.0030 17.8524 0.0224
9 41.5336 <.0001 28.6017 0.0008
10 41.6789 <.0001 28.6881 0.0014
11 49.4676 <.0001 30.1777 0.0015
12 49.8010 <.0001 31.4906 0.0017
At lag 6, for instance,
Q(6)test (20.91, p-value = 0.0019)
Ho: ut2 is a white noise up to lags 6.
p-value is less than 5% ® Ho is .
Thus, ut2does / does not look like a white noise.
LM test for ARCH effects (16.69, p-value = 0.0105)
Ho: no ARCH effect up to lags 6.
p-value is less than 5% ® Ho is .
Thus, there is / is no ARCH effect.
SAS OUTPUT (continued):
Standard Approx
Standard Approx Variable
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 -0.001241 0.1498 -0.01 0.9934
YLAG 1 -0.2455 0.0763 -3.22 0.0016 YLAG
... This is a mean equation estimate.
(not with ARCH)
GARCH Estimates
SSE 597.943653 Observations 164
MSE 3.64600 Uncond Var 3.399403
Log Likelihood -328.63655 Total R-Square 0.0571
SBC 682.772425 AIC 667.273092
Normality Test 6.2109 Pr > ChiSq 0.0448
The AUTOREG Procedure
Standard Approx Variable
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 -0.0132 0.1292 -0.10 0.9187
YLAG 1 -0.3002 0.0988 -3.04 0.0024 YLAG
ARCH0 1 0.1088 0.1763 0.62 0.5372
ARCH1 1 0.0972 0.0799 1.22 0.2235
GARCH1 1 0.8708 0.1211 7.19 <.0001
... This is the GARCH equation estimate.
We write the result as:
yt = -0.0132 - 0.3002 yt-1 + t
Var(ut | Wt-1 ) = ht .. conditional variance of ut
ht = 0.1088 + 0.0972 t-12 + 0.8708 ht-1 .
Technical Details
Conditional versus Unconditional Expectation
(1) Mean
Example: AR(1) model
yt = c + f yt-1 + ut
Unconditional Mean:
m = E(yt) = E(c + f yt-1 + ut ) = c + f m + 0 since E(ut) = 0
Thus, m =
Conditional Mean:
E(yt|Wt-1) = c + f yt-1 + 0 since E(yt|Wt-1) = 0
… c and yt-1 is included in |Wt-1.
Question: Find
(i) the unconditional mean of yt.
(ii) conditional mean of yt.
(2) Variance
Example: GARCH(1,1) model
ht = w + a1ut-12 + b1 ht-1
Unconditional Variance:
s2 = Var(ut) = E(ut2) = E(w + a1ut-12 + b1 ht-1)
= w + a1 E(ut-12) + b E(ht-1) = w +
a1 s2+ b1s2
Thus, s2 =
Conditional Variance:
Var(ut | Wt-1 ) = E(ut2| Wt-1 ) = ht
= w + a1ut-12 + b1 ht-1
Question: Find
(i) the unconditional variance of yt.
(ii) conditional variance of yt.
Extensions of GARCH Models
1. GARCH-M model (GARCH in Mean model)
Add ht in the mean equation:
yt = c + f yt-1 +
d ht + ut
If d > 0 and it is significant (t-test), there is a trade-off between the mean (return) and the conditional variance (time varying risk).
EXAMPLE:
PROC AUTOREG;
MODEL Y = YLAG / ARCHTEST GARCH = ( q=1, p=1, MEAN=SQRT);
RUN;
Standard Approx Variable
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 0.3501 0.5520 0.63 0.5259
YLAG 1 -0.3032 0.0990 -3.06 0.0022 YLAG
ARCH0 1 0.1045 0.1719 0.61 0.5431
ARCH1 1 0.0981 0.0774 1.27 0.2051
GARCH1 1 0.8709 0.1170 7.44 <.0001
DELTA 1 -0.2273 0.3321 -0.68 0.4937
We write the result as:
yt = 0.3501 - 0.3032 yt-1- 0.2273
+ t
Var(ut | Wt-1 ) = ht .. conditional variance of ut
ht = 0.1045 + 0.0981 t-12 + 0.8709 ht-1 .
2. E-GARCH Model (Exponential GARCH)
Two additions:
(i) Take a log form and
(ii) Add an additional term for a leverage effect (asymmetric effect).
log ht = = w + b1log ht-1 + qVt-1 + g{|Vt-1| - E|Vt-1|}
where Vt-1 = ut/
More questions:
(i) Why exponential? To guarantee Positive variance.
ht = exp(R.H.S.) > 0 always
(ii) Asymmetric effect
qVt-1 ® q < 0 in usual cases
if Vt-1 < 0, log ht increases ® ht increases.
if Vt-1 > 0, log ht decreases ® ht decreases.
EXAMPLE:
PROC AUTOREG;
MODEL Y = YLAG / ARCHTEST GARCH=(q=1, p=1, TYPE=EXP);
RUN;
The AUTOREG Procedure
Standard Approx Variable
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 0.0907 0.1096 0.83 0.4080
YLAG 1 -0.2852 0.0854 -3.34 0.0008 YLAG
EARCH0 1 0.0168 0.0317 0.53 0.5967
EARCH1 1 0.1390 0.0723 1.92 0.0546
EGARCH1 1 0.9857 0.0231 42.60 <.0001
THETA 1 1.1074 0.8273 1.34 0.1807
We write the result as:
yt = 0.0907 - 0.2852 yt-1 + t
Var(ut | Wt-1 ) = ht .. conditional variance of ut
log ht = 0.0168 + 0.9857 log ht-1 + 0.1390Vt-1
+ 1.1074{|Vt-1| - E|Vt-1|}
3. I-GARCH model (Integrated Garch)
Rearranging ht = w + a1ut-12 + b1 ht-1, we can have
ut2 = w + (a1 + b1) ut-12 + xt - b1xt-1
with xt = ut2 - ht such that E(xt) = E(ut2 - ht ) = E(xt-1) = 0
Measures of Persistence = a1 + b1
If a1 + b1 = 1, we have
ut2 = w + ut-12 (like a random walk model in variance!)
Then, the predicted variance is
ht+s = w + ht+s-1 = … = w·s+ ht
ht+s = ht if w = 0
This implies a persistence effect, as in the non-stationary (unit root) model.
This is an empirical phenomenon that we frequently observe in estimating GARCH models, especially in using high frequency data.
People who believe the I-GARCH model impose the restriction a1 + b1 = 1, and estimate one of these parameters (a1) and obtain the other (b1= 1 - a1).
4. Leverage-GARCH (or also called, T-GARCH or Threshold-GARCH)
GJR model (Glosten, Jaganathan and Runkle, 1993)
We add an additional term which appears only when ut-1 < 0 (negative shock):
Asymmetric effect
ht = w + a1ut-12 + b1 ht-1 + q It-1ut-12
where It-1 = 1 if ut-1 < 0 and It-1
= 0 otherwise
If q > 0, we say that there is a leverage effect.
Bad news (ut-1 < 0) has an effect of (a1+q)ut-12 on the variance.
Good news (ut-1 ³ 0) has an effect of a1 ut-12 on the variance.
If q < 0, vice versa.
One can test If q = 0 or q > 0 (t-test).
The AUTOREG procedure supports the following variations of the GARCH models:
- generalized ARCH (GARCH) , integrated GARCH (IGARCH)
- exponential GARCH (EGARCH) , GARCH-in-mean (GARCH-M)
5. Other Extensions
Component GARCH models
Multivariate GARCH models
Smooth Transition GARCH models (Lee..)
Many other Brothers and Sisters GARCH models..
Forecasting with GARCH Models
Example with GARCH(1,1)
ht = w + a1ut-12 + b1 ht-1
with unconditional Variance s2 =
Write:
ht = s2 + a1 (ut-12
- s2) + b1 (ht-1 - s2)
ht+s = s2 + a1 (ut+s-12
- s2) + b1 (ht+s-1 - s2)
Then, the predicted value for ht+s is:
ht+s= w + (a1 + b1) (ht+s-1 - s2)
Review Questions on ARCH-GARCH Models
1. What are major limitations of using MA or EWMA models for volatility?
2. What are economic reasoning for (i) conditional and (ii) heteroskedasticity (not constant variance) implied in the ARCH-GARCH models.
3. Briefly explain about the GARCH-M model and underlying reasons for this model.
4. Briefly explain about the T-GARCH model and underlying reasons for this model.
5. Briefly explain about the E-GARCH model and underlying reasons for this model.
6. Discuss how you can test for (existence of) ARCH effects.
7. Using the excess rates of returns, which was defined as the difference between the rate of returns of interest rates and the rate of return of risk free assets (Treasury Bill), estimate a GARCH model of your choice. Note that you do not need to difference
the data. (File: excess.txt)
(a) Identify the orders, p and q, of an ARMA(p,q) model. Use AIC for this.
(b) Using the residuals of the identified ARMA model estimation result, test for whiteness of residuals. Use Q-statistics at 5 and 10 lags. Also, check ACF and PACF of residuals.
(c) Using the residuals of the identified ARMA model estimation result, test for ARCH effect. Use 5 lags for this test.
(d) Estimate a GARCH(1, 1) model.
(e) Using the squared residuals of the estimated GARCH model, test for any further ARCH effect. Use Q-statistics at 5 and 10 lags.
(f) Plot the estimated conditional heteroskedasticity.
(g) Find the unconditional mean and the unconditional variance from the ARMA-GARCH model that you have estimated.
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