Econometrics

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
tttY εµ +=
t
q
i
iti
p
i
itit
titt
uhh
hI
+++= ∑∑ =
−
=
−
−
11
2
),0(~|
ϕεφα
ε

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
tttY εµ +=
tttt
titt
uhh
hI
+++= −−
−
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.
tttt hY εθµ ++=
tttt
titt
uhh
hI
+++= −−
−
1
2
1
),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.
ttttY εµ +=
otherwise0and,0if1
),0(~|
11
2
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.
tt
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