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Chapter 14 Time-Varying Volatility and ARCH Models. Walter R. Paczkowski Rutgers University. Chapter Contents. 14.1 The ARCH Model 14.2 Time-Varying Volatility 14 .3 Testing, Estimating, and Forecasting 14 .4 Extensions. - PowerPoint PPT Presentation
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Principles of Econometrics, 4th Edition Page 1Chapter 14: Time-Varying Volatility and ARCH Models
Chapter 14Time-Varying Volatility and ARCH
Models
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition Page 2Chapter 14: Time-Varying Volatility and ARCH Models
14.1 The ARCH Model14.2 Time-Varying Volatility14.3 Testing, Estimating, and Forecasting14.4 Extensions
Chapter Contents
Principles of Econometrics, 4th Edition Page 3Chapter 14: Time-Varying Volatility and ARCH Models
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 Page 4Chapter 14: Time-Varying Volatility and ARCH Models
14.1 The ARCH Model
Principles of Econometrics, 4th Edition Page 5Chapter 14: Time-Varying Volatility and ARCH Models
Consider a model with an AR(1) error term:
14.1The ARCH Model
Eq. 14.1a t ty e
2~ (0, )t vv N
1ρ , ρ 1t t te e v Eq. 14.1b
Eq. 14.1c
Principles of Econometrics, 4th Edition Page 6Chapter 14: Time-Varying Volatility and ARCH Models
The unconditional mean of the error is:
The conditional mean for the error is:
14.1The ARCH Model
21 2ρ ρ 0t t t tE e E v v v
1 1 1 1| ρ | ρt t t t t tE e I E e I E v e
Principles of Econometrics, 4th Edition Page 7Chapter 14: Time-Varying Volatility and ARCH Models
The unconditional variance of the error is:
The conditional variance for the error is:
14.1The ARCH Model
22 21 2
2 2 2 4 21 2
2 2 4
2
2
0 ρ ρ
ρ ρ
1 ρ ρ
1 ρ
t t t t
t t t
v
v
E e E v v v
E v v v
2 2 21 1 1ρ | |t t t t t vE e e I E v I
Principles of Econometrics, 4th Edition Page 8Chapter 14: Time-Varying Volatility and ARCH Models
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:
– The second equation says the error term is conditionally normal
14.1The ARCH Model
t ty e
1| ~ (0, )t t te I N h
20 1 1 0 1, 0, 0 1t th e
Eq. 14.2a
Eq. 14.2b
Eq. 14.2c
Principles of Econometrics, 4th Edition Page 9Chapter 14: Time-Varying Volatility and ARCH Models
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
14.1The ARCH Model
Principles of Econometrics, 4th Edition Page 10Chapter 14: Time-Varying Volatility and ARCH Models
The standardized errors are standard normal:
–We can write:
14.1The ARCH Model
1 0,1tt
t
eI z N
h
20 1 1
2 2 2 20 1 1 0 1 1
α α
α α α α
t t t
t t t t
E e E z E e
E e E z E e E e
Principles of Econometrics, 4th Edition Page 11Chapter 14: Time-Varying Volatility and ARCH Models
14.2 Time-Varying Volatility
Principles of Econometrics, 4th Edition Page 12Chapter 14: Time-Varying Volatility and ARCH Models
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
14.2Time-Varying
Volatility
Principles of Econometrics, 4th Edition Page 13Chapter 14: Time-Varying Volatility and ARCH Models
14.2Time-Varying
VolatilityFIGURE 14.1 Time series of returns to stock indices
Principles of Econometrics, 4th Edition Page 14Chapter 14: Time-Varying Volatility and ARCH Models
14.2Time-Varying
VolatilityFIGURE 14.2 Histograms of returns to various stock indices
Principles of Econometrics, 4th Edition Page 15Chapter 14: Time-Varying Volatility and ARCH Models
14.2Time-Varying
Volatility FIGURE 14.3 Simulated examples of constant and time-varying variances
Principles of Econometrics, 4th Edition Page 16Chapter 14: Time-Varying Volatility and ARCH Models
14.2Time-Varying
Volatility FIGURE 14.4 Frequency distributions of the simulated models
Principles of Econometrics, 4th Edition Page 17Chapter 14: Time-Varying Volatility and ARCH Models
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
14.2Time-Varying
Volatility
Principles of Econometrics, 4th Edition Page 18Chapter 14: Time-Varying Volatility and ARCH Models
14.3 Testing, Estimating, and Forecasting
Principles of Econometrics, 4th Edition Page 19Chapter 14: Time-Varying Volatility and ARCH Models
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
14.3Testing, Estimating,
and Forecasting
14.3.1Testing for ARCH
Effects
2 20 1 1ˆ ˆt t te e v
0 1 1 1: 0 : 0H H
Eq. 14.3
Principles of Econometrics, 4th Edition Page 20Chapter 14: Time-Varying Volatility and ARCH Models
Consider the returns from buying shares in the hypothetical company Brighten Your Day (BYD) Lighting
14.3Testing, Estimating,
and Forecasting
14.3.1Testing for ARCH
Effects
Principles of Econometrics, 4th Edition Page 21Chapter 14: Time-Varying Volatility and ARCH Models
14.3Testing, Estimating,
and Forecasting
14.3.1Testing for ARCH
Effects
FIGURE 14.5 Time series and histogram of returns for BYD Lighting
Principles of Econometrics, 4th Edition Page 22Chapter 14: Time-Varying Volatility and ARCH Models
The results for an ARCH test are:
– 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.
14.3Testing, Estimating,
and Forecasting
14.3.1Testing for ARCH
Effects2 2 2
1ˆ ˆ0.908 0.353 0.124( ) (8.409)t te e Rt
Principles of Econometrics, 4th Edition Page 23Chapter 14: Time-Varying Volatility and ARCH Models
ARCH models are estimated by the maximum likelihood method
14.3Testing, Estimating,
and Forecasting
14.3.2Estimating ARCH
Models
Principles of Econometrics, 4th Edition Page 24Chapter 14: Time-Varying Volatility and 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
14.3Testing, Estimating,
and Forecasting
14.3.2Estimating ARCH
Models
0ˆˆ 1.063tr
2 20 1 1 1
ˆ ˆ ˆ ˆ ˆ0.642 0.569( ) (6.877)t t th e et
Eq. 14.4a
Eq. 14.4b
Principles of Econometrics, 4th Edition Page 25Chapter 14: Time-Varying Volatility and ARCH Models
For our case study of investing in Brighten Your Day Lighting, the forecast return and volatility are:
14.3Testing, Estimating,
and Forecasting
14.3.3Forecasting Volatility
Eq. 14.5a
Eq. 14.5b
1 0ˆˆ 1.063tr
2 2
1 0 1 0ˆ ˆˆ ˆ 0.642 0.569 1.063t t th r r
Principles of Econometrics, 4th Edition Page 26Chapter 14: Time-Varying Volatility and ARCH Models
14.3Testing, Estimating,
and Forecasting FIGURE 14.6 Plot of conditional variance
14.3.3Forecasting Volatility
Principles of Econometrics, 4th Edition Page 27Chapter 14: Time-Varying Volatility and ARCH Models
14.4 Extensions
Principles of Econometrics, 4th Edition Page 28Chapter 14: Time-Varying Volatility and ARCH Models
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:
– Testing, estimating, and forecasting, are natural extensions of the case with one lag
14.4Extensions
Eq. 14.62 2 2
0 1 1 2 2...t t t q t qh e e e
Principles of Econometrics, 4th Edition Page 29Chapter 14: Time-Varying Volatility and ARCH Models
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
14.4Extensions
14.4.1The GARCH Model
– Generalized ARCH
Principles of Econometrics, 4th Edition Page 30Chapter 14: Time-Varying Volatility and ARCH Models
Consider Eq. 14.6 but write it as:
14.4Extensions
14.4.1The GARCH Model
– Generalized ARCH
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
or
but, since:
we get:
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 Eq. 14.7
Principles of Econometrics, 4th Edition Page 31Chapter 14: Time-Varying Volatility and ARCH Models
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
14.4Extensions
14.4.1The GARCH Model
– Generalized ARCH
Principles of Econometrics, 4th Edition Page 32Chapter 14: Time-Varying Volatility and ARCH Models
Consider again the returns to our shares in Brighten Your Day Lighting, which we reestimate (by maximum likelihood) under the new model:
14.4Extensions
14.4.1The GARCH Model
– Generalized ARCH
21 1
ˆ 1.049
ˆ ˆˆ0.401 0.492 0.238( ) (4.834) (2.136)
t
t t t
r
h e ht
Principles of Econometrics, 4th Edition Page 33Chapter 14: Time-Varying Volatility and ARCH Models
14.4Extensions
14.4.1The GARCH Model
– Generalized ARCH
FIGURE 14.7 Estimated means and variances of ARCH models
Principles of Econometrics, 4th Edition Page 34Chapter 14: Time-Varying Volatility and ARCH Models
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:
14.4Extensions
14.4.2Allowing for an
Asymmetric Effect
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
Eq. 14.8
Principles of Econometrics, 4th Edition Page 35Chapter 14: Time-Varying Volatility and ARCH Models
The returns to our shares in Brighten Your Day Lighting were reestimated with a T-GARCH(1,1) specification:
14.4Extensions
14.4.2Allowing for an
Asymmetric Effect
2 21 1 1 1
ˆ 0.994
ˆ ˆˆ ˆ0.356 0.263 0.492 0.287( ) (3.267) (2.405) (2.488)
t
t t t t t
r
h e d e ht
Principles of Econometrics, 4th Edition Page 36Chapter 14: Time-Varying Volatility and ARCH Models
Another popular extension of the GARCH model is the ‘‘GARCH-in-mean’’ model
14.4Extensions
14.4.3GARCH-In-Mean and Time-Varying
Risk Premium
0t t ty h e
1| ~ (0, )t t te I N h
21 1 1 1
1 1
, 0, 0 1, 0 1
t t th e h
Eq. 14.9a
Eq. 14.9b
Eq. 14.9c
Principles of Econometrics, 4th Edition Page 37Chapter 14: Time-Varying Volatility and ARCH Models
The returns to shares in Brighten Your Day Lighting were reestimated as a GARCH-in-mean model:
14.4Extensions
14.4.3GARCH-In-Mean and Time-Varying
Risk Premium
2 21 1 1 1
ˆ 0.818 0.196( ) (2.915)
ˆ ˆˆ ˆ0.370 0.295 0.321 0.278( ) (3.426) (1.979) (2.678)
t t
t t t t t
r ht
h e d e ht
Principles of Econometrics, 4th Edition Page 38Chapter 14: Time-Varying Volatility and ARCH Models
Key Words
Principles of Econometrics, 4th Edition Page 39Chapter 14: Time-Varying Volatility and ARCH Models
Keywords
ARCHARCH-in-meanconditional and unconditional forecastsconditionally normal
GARCHGARCH-in-meanT-ARCH and T-GARCHtime-varying variance