rugarch (version 1.0-16)

BerkowitzTest: 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.

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).

Details

See not below.

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

#  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.array(pred)[,2,],as.array(pred)[,1,], coef(fit)["skew"], 
coef(fit)["shape"])
colnames(dmatrix) = c("mu", "sigma", "skew", "shape")

# Get Realized (Oberved) Data
obsx = tail(dji30ret[,1], 1000)
# you can check that this is correct by looking at the dates of the first and 
# last predictions:
as.data.frame(pred, rollframe = 0)
head(tail(dji30ret[,1, drop = FALSE], 1000), 1)

as.data.frame(pred, rollframe = 999)
tail(dji30ret[,1, drop = FALSE], 1)

# 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)