This document discusses volatility modeling using GARCH models. It provides an overview of the basic GARCH specification and the steps involved in GARCH modeling, including descriptive statistics, testing for ARCH effects, GARCH specification, estimation, evaluation, and inferences. Specific GARCH models discussed include GARCH(1,1), TARCH, and EGARCH. The goal of GARCH modeling is to characterize volatility for applications such as risk analysis and portfolio selection.

1. FINANCIAL
ECONOMETRICS
VOLATILITY MODELING USING GARCH

2. BASIC GARCH SPECIFICATION
 GARCH(p,q)
 Uses
- Measuring of volatility (e.g. exchange rate volatility and its
impact on trade).
- Characterizing volatility for risk analysis and portfolio
selection.
- improving statistical estimation
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i
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3. STEPS IN GARCH MODELING
Descriptive Statistics
Test for ARCH effects
GARCH Specification
Estimation
Evaluation
Inferences

4. DESCRIPTIVE STATISTICS
-.02
-.01
.00
.01
.02
.03
.04
.05
.06
1970 1975 1980 1985 1990 1995 2000
DMYCPI
0
4
8
12
16
20
24
0.000 0.025 0.050
Series: DMYCPI
Sample 1970Q1 2003Q4
Observations 135
Mean 0.009508
Median 0.007813
Maximum 0.056791
Minimum -0.013301
Std. Dev. 0.010447
Skewness 1.748337
Kurtosis 8.207119
Jarque-Bera 221.2921
Probability 0.000000

5. TESTS FOR ARCH EFFECTS
 Examine the correlation of squared errors

6. TESTS FOR ARCH EFFECTS
 FORMAL LM TEST FOR ARCH EFFECT
ARCH Test:
F-statistic 12.24353 Prob. F(1,130) 0.000640
Obs*R-squared 11.36182 Prob. Chi-Square(1) 0.000750
ARCH Test:
F-statistic 7.036496 Prob. F(2,128) 0.001261
Obs*R-squared 12.97616 Prob. Chi-Square(2) 0.001521
ARCH Test:
F-statistic 4.691188 Prob. F(4,124) 0.001469
Obs*R-squared 16.95554 Prob. Chi-Square(4) 0.001972

7. GARCH SPECIFICATION
 Specification of mean equation – if correctly
specified, there should be no autocorrelation of
the standardized residuals. [REFER TO ARIMA
MODELING]
 Variance equation – research has shown that
GARCH(1, 1) is an adequate specification.
However, an information criteria can be used to
choose (p, q) of GARCH(p, q).
 In this case, the squared standardized
residuals should not be autocorrelated or the
standardized residuals do not exhibit additional
ARCH.

8. GARCH SPECIFICATION
 BASIC GARCH(1, 1)
 Squared residuals (ARCH term) – news about
volatility from the period period
 GARCH term (h) – last period’s variance
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−
1
2
1
),0(~|
ϕφεα
ε

9. GARCH SPECIFICATION
 BASIC GARCH(1, 1) – M
 The mean equation has conditional variance as a regressor
(use: risk-return tradeoff, inflation-inflation uncertainty
tradeoff).
 In certain case, the standard deviation is used instead.
 Other variables can also be included in the mean equation.
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2
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),0(~|
ϕφεα
ε

10. GARCH SPECIFICATION
 TARCH (1, 1) – Threshold ARCH
 This is asymmetric GARCH introduced by Zakoian (1990)
and Glosten et al. (1993) to capture the observation that
downward movements in the market are followed by higher
volatilities.
 It capture volatility response to good and bad news.
 If θ > 0, we say that the LEVERAGE EFFECT exists.
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1
2
1
<=
++++= −−−−
−
tt
tttttt
titt
d
uhdh
hI
ε
ϕθεφεα
ε

11. GARCH SPECIFICATION
 EGARCH
 This is asymmetric GARCH introduced proposed
by Nelson (1991)
 If φ < 0, we say that the LEVERAGE EFFECT
exists.
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t
t
t
t
t uhh ++++= −
−
−
−
−
1
1
1
1
1
loglog ϕ
σ
ε
θ
σ
ε
φα

12. GARCH ESTIMATION-
EVALUATION - INFERENCE
 Estimation of GARCH models is done using
Maximum Likelihood Estimation
 Diagnostic tests – most important is
autocorrelation of standardized residuals, squared
standardized residuals, and ARCH tests. This is
done using VIEW, Residual Tests.
 Once satisfied, the results can be interpreted and
measures of volatility can be obtained.
EXERCISE: OBTAIN A GARCH MODEL OF
A VARIABLE OF YOUR INTEREST