Chapter 14 Time-Varying Volatility and ARCH Models

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


14.1 The ARCH Model14.2 Time-Varying Volatility14.3 Testing, Estimating, and Forecasting14.4 Extensions

Chapter Contents


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


14.1 The ARCH Model


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


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


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


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


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


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


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

14.2Time-Varying

Volatility


14.2Time-Varying

VolatilityFIGURE 14.1 Time series of returns to stock indices


14.2Time-Varying

VolatilityFIGURE 14.2 Histograms of returns to various stock indices


14.2Time-Varying

Volatility FIGURE 14.3 Simulated examples of constant and time-varying variances


14.2Time-Varying

Volatility FIGURE 14.4 Frequency distributions of the simulated 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


14.3 Testing, Estimating, and Forecasting


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


Consider the returns from buying shares in the hypothetical company Brighten Your Day (BYD) Lighting


and Forecasting


Effects



and Forecasting


Effects

FIGURE 14.5 Time series and histogram of returns for BYD Lighting


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.


and Forecasting


Effects2 2 2

1ˆ ˆ0.908 0.353 0.124( ) (8.409)t te e Rt


ARCH models are estimated by the maximum likelihood method


and Forecasting

14.3.2Estimating 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


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


For our case study of investing in Brighten Your Day Lighting, the forecast return and volatility are:


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



and Forecasting FIGURE 14.6 Plot of conditional variance

14.3.3Forecasting Volatility


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:

– 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


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


Consider Eq. 14.6 but write it as:

14.4Extensions



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


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




Consider again the returns to our shares in Brighten Your Day Lighting, which we reestimate (by maximum likelihood) under the new model:

14.4Extensions



21 1

ˆ 1.049

ˆ ˆˆ0.401 0.492 0.238( ) (4.834) (2.136)

t

t t t

r

h e ht


14.4Extensions



FIGURE 14.7 Estimated means and variances of 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


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


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


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


Key Words


Keywords

ARCHARCH-in-meanconditional and unconditional forecastsconditionally normal

GARCHGARCH-in-meanT-ARCH and T-GARCHtime-varying variance

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Chapter 14 Time-Varying Volatility and ARCH Models

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