Chapter 14
Time-Varying Volatility and ARCH
Models
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 1
Chapter Contents




14.1 The ARCH Model
14.2 Time-Varying Volatility
14.3 Testing, Estimating, and Forecasting
14.4 Extensions
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 2
The nonstationary nature of the variables studied
earlier implied that they had means that change
over time
– Now we are concerned with stationary series,
but with conditional variances that change over
time
– The model is called the autoregressive
conditional heteroskedastic (ARCH) model
• ARCH stands for auto-regressive
conditional heteroskedasticity
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 3
14.1
The ARCH Model
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 4
14.1
The ARCH Model
Consider a model with an AR(1) error term:
Eq. 14.1a
Eq. 14.1b
Eq. 14.1c
Principles of Econometrics, 4th Edition
yt    et
et  ρet 1  vt ,
ρ 1
vt ~ N (0, v2 )
Chapter 14: Time-Varying Volatility and ARCH Models
Page 5
14.1
The ARCH Model
The unconditional mean of the error is:
E  et   E vt  ρvt 1  ρ 2vt  2 
  0
The conditional mean for the error is:
E et | It 1   E ρet 1 | It 1   E vt   ρet 1
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 6
14.1
The ARCH Model
The unconditional variance of the error is:
E  et  0  E vt  ρvt 1  ρ vt  2 
2
2
 E vt2  ρ 2 vt21  ρ 4 vt2 2 
  v2 1  ρ 2  ρ 4 


2


 v2
1  ρ2
The conditional variance for the error is:
2

E  et  ρet 1  | It 1   E vt2 | It 1    v2


Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 7
14.1
The ARCH Model
Suppose that instead of a conditional mean that
changes over time we have a conditional variance
that changes over time
– Consider a variant of the above model:
yt    et
Eq. 14.2a
et | I t 1 ~ N (0, ht )
Eq. 14.2b
ht  0  1et21,
Eq. 14.2c
0  0, 0  1  1
– The second equation says the error term is
conditionally normal
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 8
14.1
The ARCH Model
The name — ARCH — conveys the fact that we
are working with time-varying variances
(heteroskedasticity) that depend on (are
conditional on) lagged effects (autocorrelation)
– This particular example is an ARCH(1) model
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 9
14.1
The ARCH Model
The standardized errors are standard normal:
 et

I t 1   z

 h

t


– We can write:
E  et   E  zt  E
 

  
N  0,1
α 0  α1et21


 
E et2  E zt2 E α 0  α1et21  α 0  α1 E et21
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 10
14.2
Time-Varying Volatility
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 11
14.2
Time-Varying
Volatility
The ARCH model has become a popular one
because its variance specification can capture
commonly observed features of the time series of
financial variables
– It is useful for modeling volatility and
especially changes in volatility over time
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 12
14.2
Time-Varying
Volatility
Principles of Econometrics, 4th Edition
FIGURE 14.1 Time series of returns to stock indices
Chapter 14: Time-Varying Volatility and ARCH Models
Page 13
14.2
Time-Varying
Volatility
FIGURE 14.2 Histograms of returns to various stock indices
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 14
14.2
Time-Varying
Volatility
FIGURE 14.3 Simulated examples of constant and time-varying variances
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 15
14.2
Time-Varying
Volatility
FIGURE 14.4 Frequency distributions of the simulated models
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 16
14.2
Time-Varying
Volatility
The ARCH model is intuitively appealing because
it seems sensible to explain volatility as a function
of the errors et
– These errors are often called ‘‘shocks’’ or
‘‘news’’ by financial analysts
• They represent the unexpected!
– According to the ARCH model, the larger the
shock, the greater the volatility in the series
– This model captures volatility clustering, as big
changes in et are fed into further big changes in
ht via the lagged effect et-1
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 17
14.3
Testing, Estimating, and Forecasting
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 18
14.3
Testing, Estimating,
and Forecasting
14.3.1
Testing for ARCH
Effects
A Lagrange multiplier (LM) test is often used to
test for the presence of ARCH effects
– To perform this test, first estimate the mean
equation
Eq. 14.3
eˆt2  0  1eˆt21  vt
H 0 : 1  0
Principles of Econometrics, 4th Edition
H1 : 1  0
Chapter 14: Time-Varying Volatility and ARCH Models
Page 19
14.3
Testing, Estimating,
and Forecasting
14.3.1
Testing for ARCH
Effects
Consider the returns from buying shares in the
hypothetical company Brighten Your Day (BYD)
Lighting
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 20
14.3
Testing, Estimating,
and Forecasting
FIGURE 14.5 Time series and histogram of returns for BYD Lighting
14.3.1
Testing for ARCH
Effects
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 21
14.3
Testing, Estimating,
and Forecasting
14.3.1
Testing for ARCH
Effects
The results for an ARCH test are:
eˆt2  0.908  0.353eˆt21
R 2  0.124
(t )
(8.409)
– The t-statistic suggests a significant first-order
coefficient
– The sample size is 500, giving LM test value of
(T – q)R2 = 61.876
– Comparing the computed test value to the 5%
critical value of a χ2(1) distribution
(χ2 (0.95, 1)= 3.841) leads to the rejection of the null
hypothesis
• The residuals show the presence of ARCH(1)
effects.
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 22
14.3
Testing, Estimating,
and Forecasting
14.3.2
Estimating ARCH
Models
ARCH models are estimated by the maximum
likelihood method
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 23
14.3
Testing, Estimating,
and Forecasting
14.3.2
Estimating ARCH
Models
Eq. 14.4 shows the results from estimating an
ARCH(1) model applied to the monthly returns
from buying shares in Brighten Your Day Lighting
rˆt  ˆ 0  1.063
Eq. 14.4a
Eq. 14.4b
hˆt  ˆ 0  ˆ 1eˆt21  0.642  0.569eˆt21
(t )
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
(6.877)
Page 24
14.3
Testing, Estimating,
and Forecasting
14.3.3
Forecasting
Volatility
For our case study of investing in Brighten Your
Day Lighting, the forecast return and volatility
are:
rˆt 1  ˆ 0  1.063
Eq. 14.5a
Eq. 14.5b

hˆt 1  ˆ 0  ˆ 1 rt  ˆ 0
Principles of Econometrics, 4th Edition

2
 0.642  0.569  rt  1.063
Chapter 14: Time-Varying Volatility and ARCH Models
Page 25
2
14.3
Testing, Estimating,
and Forecasting
FIGURE 14.6 Plot of conditional variance
14.3.3
Forecasting
Volatility
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 26
14.4
Extensions
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 27
14.4
Extensions
The ARCH(1) model can be extended in a number
of ways
– One obvious extension is to allow for more lags
– An ARCH(q) model would be:
ht   0  1et21   2et22 ...   q et2q
Eq. 14.6
– Testing, estimating, and forecasting, are natural
extensions of the case with one lag
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 28
14.4
Extensions
14.4.1
The GARCH Model
– Generalized
ARCH
One of the shortcomings of an ARCH(q) model is
that there are q + 1 parameters to estimate
– If q is a large number, we may lose accuracy in
the estimation
– The generalized ARCH model, or GARCH, is
an alternative way to capture long lagged
effects with fewer parameters
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 29
14.4
Extensions
14.4.1
The GARCH Model
– Generalized
ARCH
Consider Eq. 14.6 but write it as:
ht   0  1et21  11et2 2  12 1et23 
or
ht    0  1 0   1et21  1   0  1et2 2  11et23 
but, since:
ht 1   0  1et2 2  11et23  12 1et2 4 
we get:
Eq. 14.7
Principles of Econometrics, 4th Edition
ht    1et21  1ht 1
Chapter 14: Time-Varying Volatility and ARCH Models
Page 30

14.4
Extensions
14.4.1
The GARCH Model
– Generalized
ARCH
This generalized ARCH model is denoted as
GARCH(1,1)
– The model is a very popular specification
because it fits many data series well
– It tells us that the volatility changes with lagged
shocks (e2 t-1) but there is also momentum in the
system working via ht-1
– One reason why this model is so popular is that
it can capture long lags in the shocks with only
a few parameters
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 31
14.4
Extensions
14.4.1
The GARCH Model
– Generalized
ARCH
Consider again the returns to our shares in
Brighten Your Day Lighting, which we reestimate
(by maximum likelihood) under the new model:
rˆt  1.049
hˆt  0.401  0.492eˆt21  0.238hˆt 1
(t )
Principles of Econometrics, 4th Edition
(4.834)
(2.136)
Chapter 14: Time-Varying Volatility and ARCH Models
Page 32
14.4
Extensions
FIGURE 14.7 Estimated means and variances of ARCH models
14.4.1
The GARCH Model
– Generalized
ARCH
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 33
14.4
Extensions
14.4.2
Allowing for an
Asymmetric Effect
The threshold ARCH model, or T-ARCH, is one
example where positive and negative news are
treated asymmetrically
– In the T-GARCH version of the model, the
specification of the conditional variance is:
ht    1et21  d t 1et21  1ht 1
Eq. 14.8
Principles of Econometrics, 4th Edition
1 et  0 (bad news)
dt  
0 et  0 (good news)
Chapter 14: Time-Varying Volatility and ARCH Models
Page 34
14.4
Extensions
14.4.2
Allowing for an
Asymmetric Effect
The returns to our shares in Brighten Your Day
Lighting were reestimated with a T-GARCH(1,1)
specification:
rˆt  0.994
hˆt  0.356  0.263eˆt21  0.492d t 1eˆt21  0.287 hˆt 1
(t )
Principles of Econometrics, 4th Edition
(3.267)
(2.405)
(2.488)
Chapter 14: Time-Varying Volatility and ARCH Models
Page 35
14.4
Extensions
14.4.3
GARCH-In-Mean
and Time-Varying
Risk Premium
Another popular extension of the GARCH model
is the ‘‘GARCH-in-mean’’ model
Eq. 14.9a
yt  0  ht  et
Eq. 14.9b
et | I t 1 ~ N (0, ht )
Eq. 14.9c
Principles of Econometrics, 4th Edition
ht    1et21  1ht 1 ,
  0, 0  1  1, 0  1  1
Chapter 14: Time-Varying Volatility and ARCH Models
Page 36
14.4
Extensions
14.4.3
GARCH-In-Mean
and Time-Varying
Risk Premium
The returns to shares in Brighten Your Day
Lighting were reestimated as a GARCH-in-mean
model:
rˆt  0.818  0.196ht
(t )
(2.915)
hˆt  0.370  0.295eˆt21  0.321dt 1eˆt21  0.278hˆt 1
(t )
Principles of Econometrics, 4th Edition
(3.426)
(1.979)
Chapter 14: Time-Varying Volatility and ARCH Models
(2.678)
Page 37
Key Words
Principles of Econometrics, 4th Edition
Chapter 14: Time-Varying Volatility and ARCH Models
Page 38
Keywords
ARCH
ARCH-in-mean
conditional and
unconditional
forecasts
conditionally
normal
Principles of Econometrics, 4th Edition
GARCH
GARCH-inmean
T-ARCH and TGARCH
time-varying
variance
Chapter 14: Time-Varying Volatility and ARCH Models
Page 39