BerkowitzTest: Berkowitz Density Forecast Likelihood Ratio Test

In rugarch: Univariate GARCH Models

Berkowitz Density Forecast Likelihood Ratio Test

Description

Implements the Berkowitz Density Forecast Likelihood Ratio Test.

Usage

BerkowitzTest(data, lags = 1, significance = 0.05, tail.test = FALSE, alpha = 0.05)

Arguments

data	A univariate vector of standard normal transformed values (see details and example).
lags	The number of autoregressive lags (positive and greater than 0).
significance	The level of significance at which the Null Hypothesis is evaluated.
tail.test	Whether to use the tail test of Berkowitz using a censored likelihood.
alpha	The quantile level for the tail.test cuttoff.

Details

See not below.

Value

A list with the following items:

uLL	The unconditional Log-Likelihood of the maximized values.
rLL	The restricted Log-Likelihood with zero mean, unit variance and zero coefficients in the autoregressive lags.
LR	The Likelihood Ratio Test Statistic.
LRp	The LR test statistic p-value (distributed chisq with 2+lags d.o.f).
H0	The Null Hypothesis.
Test	The test of the Null Hypothesis at the requested level of significance.
mu	The estimated mean of the model.
sigma	The estimated sd of the model.
rho	The estimated autoregressive coefficients of the model (not calculated when tail.test is used).
JB	The Jarque-Bera Test of Normality Statistic (not calculated when tail.test is used).
JBp	The Jarque-Beta Test Statistic p-value (not calculated when tail.test is used).

Note

The data must first be transformed before being submitted to the function as described here. Given a forecast density (d*) at time t, transform the actual(observed) realizations of the data by applying the distribution function of the forecast density (p*). This will result in a set of uniform values (see Rosenblatt (1952)). Transform those value into standard normal variates by applying the standard normal quantile function (qnorm). The example below hopefully clarifies this. The function also returns the Jarque Bera Normality Test statistic as an additional check of the normality assumption which the test does not explicitly account for (see Dowd reference). When tail.test is used, the test of the tail at the “alpha” quantile level is performed using a censored normal likelihood.

Author(s)

Alexios Ghalanos

References

Berkowitz, J. 2001, Testing density forecasts, with applications to risk management, Journal of Business and Economic Statistics, 19(4), 465–474.
Dowd, K. 2004, A modified Berkowitz back-test, RISK Magazine, 17(4), 86–87.
Jarque, C.M. and Bera, A.K. 1987, A test for normality of observations and regression residuals, International Statistical Review, 55(2), 163–172.
Rosenblatt, M. 1952, Remarks on a multivariate transformation, The Annals of Mathematical Statistics, 23(3), 470–472.

Examples

## Not run: 
#  A univariate GARCH model is used with rolling out of sample forecasts.
data(dji30ret)
spec = ugarchspec(mean.model = list(armaOrder = c(6,1), include.mean = TRUE),
variance.model = list(model = "gjrGARCH"), distribution.model = "nig")
fit = ugarchfit(spec, data = dji30ret[, 1, drop = FALSE], out.sample = 1000)
pred = ugarchforecast(fit, n.ahead = 1, n.roll = 999)
dmatrix = cbind(as.numeric(fitted(pred)),as.numeric(sigma(pred)), 
rep(coef(fit)["skew"],1000), rep(coef(fit)["shape"],1000))
colnames(dmatrix) = c("mu", "sigma", "skew", "shape")
# Get Realized (Oberved) Data
obsx = tail(dji30ret[,1], 1000)
# Transform to Uniform
uvector = apply(cbind(obsx,dmatrix), 1, FUN = function(x) pdist("nig", q = x[1],
mu = x[2], sigma = x[3], skew = x[4], shape = x[5]))

# hist(uvector)
# transform to N(0,1)
nvector = qnorm(uvector)
test1 = BerkowitzTest(data = nvector, lags = 1, significance = 0.05)
test2 = BerkowitzTest(data = nvector, alpha = 0.05, significance = 0.05, 
tail.test=TRUE)
test3 = BerkowitzTest(data = nvector, alpha = 0.01, significance = 0.05, 
tail.test=TRUE)

## End(Not run)

rugarch documentation built on Sept. 30, 2024, 9:30 a.m.