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Chapter 14
An Introduction to Financial Econometrics: Time-Varying Volatility and ARCH Models
Prepared by Vera Tabakova, East Carolina University
Chapter 14: An Introduction to Financial Econometrics: Time-Varying Volatility and ARCH Models
14.1 The ARCH Model
14.2 Time-Varying Volatility
14.3 Testing, Estimating and Forecasting
14.4 Extensions
Slide 14-2Principles of Econometrics, 3rd Edition
14.1 The Arch Model
Slide 14-3Principles of Econometrics, 3rd Edition
(14.1a)
(14.1c)
(14.1b)
0t ty e
2~ (0, )te N
20
14.1 The Arch Model
Slide 14-4Principles of Econometrics, 3rd Edition
(14.2a)
(14.2c)
(14.2b)
0t ty e
1| ~ (0, )t t te I N h
20 1 1 0 1, 0, 0 1t th e
14.1.1 Conditional and Unconditional Forecasts
Conditional forecast
Slide 14-5Principles of Econometrics, 3rd Edition
1
1
2 2 21 1
, | | 1
[ | ]
[ ] [ ]
t t t
t t t
t t t t t
y y e
E y I y
E y y E e
14.1.1 Conditional and Unconditional Forecasts
Unconditional forecast
Slide 14-6Principles of Econometrics, 3rd Edition
21 1 1 0
21 1 1[ ] [ .....] 0
[ ] 0 for all
tt t t t
t t t t
t j
y e e e y
E y E e e e
E e j
14.1.1 Conditional and Unconditional Forecasts
Slide 14-7Principles of Econometrics, 3rd Edition
2 2 21 1 1 1
2 2 2 4 21 1
22 2 4
2
[ 0] var( ) [ ]
= [ ]
= [1 ] = 1
t t t t t
t t t
E y y E e e e
E e e e
14.2 Time Varying Volatility
Figure 14.1 Examples of Returns to Various Stock Indices
Slide 14-8Principles of Econometrics, 3rd Edition
14.2 Time Varying Volatility
Figure 14.2 Histograms of Returns to Various Stock Indices
Slide 14-9Principles of Econometrics, 3rd Edition
14.2 Time Varying Volatility
Figure 14.3 Simulated Examples of Constant and Time-Varying Variances
Slide 14-10Principles of Econometrics, 3rd Edition
14.2 Time Varying Volatility
Figure 14.4 Frequency Distributions of the Simulated Models
Slide 14-11Principles of Econometrics, 3rd Edition
14.3 Testing, Estimating and Forecasting 14.3.1 Testing for ARCH effects
Slide 14-12Principles of Econometrics, 3rd Edition
(14.3)2 20 1 1ˆ ˆt t te e v
0 1
1 1
: 0
: 0
H
H
2 2 21ˆ ˆ0.908 0.353 0.124
( ) (8.409)t te e R
t
14.3 Testing, Estimating and Forecasting
Figure 14.5 Time Series and Histogram of Returns
Slide 14-13Principles of Econometrics, 3rd Edition
14.3.2 Estimating ARCH Models
Slide 14-14Principles of Econometrics, 3rd Edition
(14.4a)
(14.4b)
0ˆˆ 1.063tr
2 20 1 1 1
ˆ ˆ ˆ ˆ ˆ0.642 0.569
( ) (6.877)t t th e e
t
14.3.3 Forecasting Volatility
Slide 14-15Principles of Econometrics, 3rd Edition
(14.5a)
(14.5b)
1 0ˆˆ 1.063tr
2 2
1 0 1 0ˆ ˆˆ ˆ 0.642 0.569 1.063t t th r r
14.3.3 Forecasting Volatility
Figure 14.6 Plot of Conditional Variance
Slide 14-16Principles of Econometrics, 3rd Edition
14.4 Extensions
Slide 14-17Principles of Econometrics, 3rd Edition
(14.6)2 2 2
0 1 1 2 2...t t t q t qh e e e
14.4.1 The GARCH Model - Generalized ARCH
Slide 14-18Principles of Econometrics, 3rd Edition
(14.7)
2 2 2 20 1 1 1 1 2 1 1 3
2 2 20 1 0 1 1 1 0 1 2 1 1 3
2 2 2 21 0 1 2 1 1 3 1 1 4
t t t t
t t t t
t t t t
h e e e
h e e e
h e e e
21 1 1 1t t th e h
14.4.1 The GARCH Model - Generalized ARCH
Slide 14-19Principles of Econometrics, 3rd Edition
21 1
ˆ 1.049
ˆ ˆˆ0.401 0.492 0.238
( ) (6.290) (3.324)
t
t t t
r
h e h
t
14.4.1 The GARCH Model - Generalized ARCH
Figure 14.7 Estimated Means and Variances of Various ARCH Models
Slide 14-20Principles of Econometrics, 3rd Edition
14.4.2 Allowing for an Asymmetric Effect
Slide 14-21Principles of Econometrics, 3rd Edition
(14.8)
2 21 1 1 1 1 1
1 0 (bad news)
0 0 (good news)
t t t t t
t
t
t
h e d e h
ed
e
14.4.2 Allowing for an Asymmetric Effect
Slide 14-22Principles of Econometrics, 3rd Edition
2 21 1 1 1
ˆ 0.994
ˆ ˆˆ ˆ0.356 0.263 0.492 0.287
( ) (3.229) (3.520) (3.681)
t
t t t t t
r
h e d e h
t
14.4.3 GARCH-in-Mean and Time-varying Risk Premium
Slide 14-23Principles of Econometrics, 3rd Edition
(14.9a)
(14.9c)
(14.9b)
0t t ty h e
1| ~ (0, )t t te I N h
21 1 1 1
1 1
,
0, 0 1, 0 1t t th e h
14.4.3 GARCH-in-Mean and Time-varying Risk Premium
Slide 14-24Principles of Econometrics, 3rd Edition
2 21 1 1 1
ˆ 0.818 0.196
( ) (3.448)
ˆ ˆˆ ˆ0.370 0.295 0.321 0.278
( ) (3.383) (2.419) (4.074)
t t
t t t t t
r h
t
h e d e h
t
Keywords
Slide 14-25Principles of Econometrics, 3rd Edition
ARCH Conditional and Unconditional
Forecasts Conditionally normal GARCH ARCH-in-mean and GARCH-in-
mean T-ARCH and T-GARCH Time-varying variance
