ARCH and GARCH Models White Noise ARCH/GARCH Outline

1 White Noise

2 ARCH/GARCH

Arthur Berg ARCH and GARCH Models 2/ 18 White Noise ARCH/GARCH Not All White Noise Are Created Equal

Two different types of white noise: 1 strict white noise (SWN) — sequence of iid random variables 2 uncorrelated white noise (UWN) — sequence of uncorrelated, but not necessarily independent, white noise

Certainly SWN ⇒ UWN. However SWN is uninformative whereas UWN can be informative.

Arthur Berg ARCH and GARCH Models 3/ 18 White Noise ARCH/GARCH Comparison of IID N(0, 1) with a stationary GARCH(1,1)

Arthur Berg ARCH and GARCH Models 4/ 18 White Noise ARCH/GARCH Comparison of IID N(0, 1) with a stationary GARCH(1,1)

Arthur Berg ARCH and GARCH Models 5/ 18 White Noise ARCH/GARCH Modeling Volatility

Properties of ARCH/GARCH models: Primary interest is in modeling changes in variance Provides improved estimations of the local variance (volatility) Not necessarily concerned with better forecasts Can be integrated into ARMA models Useful in modeling ﬁnancial time series

Arthur Berg ARCH and GARCH Models 6/ 18 White Noise ARCH/GARCH Returns

Let yt = log(xt), then

ut = ∆yt

= yt − yt−1 x = log t xt−1 x − x = log 1 + t t−1 xt−1 x − x ≈ t t−1 xt−1 = return or relative gain

Returns in ﬁnancial time series tend to have highly volatile periods clustered together.

Arthur Berg ARCH and GARCH Models 7/ 18 White Noise ARCH/GARCH S&P 500 Returns

Arthur Berg ARCH and GARCH Models 8/ 18 White Noise ARCH/GARCH Growth Rate of Seasonally Adjusted GNP (1947—2002)

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Consider the following ARCH(1) model:

yt = σtεt 2 2 σt = α0 + α1yt−1

iid where εt ∼ N (0, 1) and 0 < α1 < 1.

The process yt is stationary, but the conditional variance of yt varies in time.

Note that yt given yt−1 has the distribution

2 yt|yt−1 ∼ N 0, α0 + α1yt−1

2 yt resembles an AR(1) process.

Arthur Berg ARCH and GARCH Models 10/ 18 White Noise ARCH/GARCH 2 yt as a Pseudo AR(1)

The deﬁning equations of yt are rewritten as

2 2 2 yt = σt εt 2 2 α0 + α1yt−1 = σt Subtracting gives 2 2 2 2 2 yt − (α0 + α1yt−1) = σt εt − σt which is equivalent to 2 2 yt = α0 + α1yt−1 + vt 2 2 where vt = σt (εt − 1).

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Parameters estimators are typically conditional maximum likelihood estimators. We will use the tseries package in R which contains the garch. garch(x, order = c(1, 1), coef = NULL, itmax = 200, eps = NULL, grad = c("analytical","numerical"), series = NULL, trace = TRUE, ...)

Arthur Berg ARCH and GARCH Models 12/ 18 White Noise ARCH/GARCH Analysis of US GNP

Fit an AR(1) model to the US GNP and test library(tseries) gnp96 = read.table("mydata/gnp96.dat") gnpr = diff(log(gnp96[,2])) gnpr.ar = ar.mle(gnpr, order.max=1) # recall phi1 =.347 y = gnpr.ar$resid[2:length(gnpr)] # first resid is NA arch.y = garch(y,order=c(0,1)) summary(arch.y)

Arthur Berg ARCH and GARCH Models 13/ 18 White Noise ARCH/GARCH Call: garch(x = y, order = c(0, 1))

Model: GARCH(0,1)

Residuals: Min 1Q Median 3Q Max -3.21636 -0.54732 -0.03107 0.59886 3.34757

Coefficient(s): Estimate Std. Error t value Pr(>|t|) a0 7.403e-05 7.275e-06 10.175 < 2e-16 *** a1 1.939e-01 6.781e-02 2.859 0.00425 ** --- Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1

Diagnostic Tests: Jarque Bera Test data: Residuals X-squared = 8.4801, df = 2, p-value = 0.01441

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Box-Ljung test data: Squared.Residuals X-squared = 3e-04, df = 1, p-value = 0.9865 White Noise ARCH/GARCH ARCH(m), GARCH(m,r)

ARCH(m):

yt = σtεt m 2 X 2 σt = α0 + αjyt−j j=1

GARCH(m,r):

yt = σtεt m r 2 X 2 X 2 σt = α0 + αjyt−j + βjσt−j j=1 j=1

Arthur Berg ARCH and GARCH Models 15/ 18 White Noise ARCH/GARCH Analysis of S&P 500 returns

Fit GARCH(1,1) to the data. library(tseries) nyse = scan("mydata/nyse.dat") nyse.g = garch(nyse, order=c(1,1)) summary(nyse.g) u = predict(nyse.g) plot(nyse, type="l", xlab="Time", ylab="NYSE Returns") lines(u[,1], col="blue", lty="dashed") lines(u[,2], col="blue", lty="dashed")

Arthur Berg ARCH and GARCH Models 16/ 18 White Noise ARCH/GARCH Model: GARCH(1,1)

Residuals: Min 1Q Median 3Q Max -8.66460 -0.45228 0.08785 0.59918 4.07508

Coefficient(s): Estimate Std. Error t value Pr(>|t|) a0 6.552e-06 6.761e-07 9.691 <2e-16 *** a1 1.118e-01 4.056e-03 27.554 <2e-16 *** b1 8.086e-01 1.292e-02 62.566 <2e-16 *** --- Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1

Jarque Bera Test data: Residuals X-squared = 3983.873, df = 2, p-value < 2.2e-16

Box-Ljung test data: Squared.Residuals X-squared = 1.5874, df = 1, p-value = 0.2077

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