The Non-Parametric Density Test of Hong and Li

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Description

Implements the Non-Parametric Density Test of Hong and Li.

Usage

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HLTest(PIT, lags = 4, kernel = "quartic", conf.level = 0.95)

Arguments

PIT

This represents the actual data transformed into a U(0,1) series by applying the distribution function of the estimated model conditional on the parameters.

lags

The number of lags to use for testing the joint hypothesis.

kernel

The kernel to use for the comparison against the PIT series (only the ‘quartic’ currently implemented).

conf.level

The confidence level at which the Null Hypothesis is evaluated.

Details

A novel method to analyze how well a conditional density fits the underlying data is through the probability integral transformation (PIT) discussed in Rosenblatt (1952) and used in the BerkowitzTest. More recently, Hong and Li (2005) introduced a nonparametric portmanteau test, building on the work of Ait-Sahalia (1996), which tests the joint hypothesis of i.i.d and uniformity for a series of PIT transformed data. To achieve this, it tests for misspecification in the conditional moments of the model transformed standardized residuals, and is distributed as N(0, 1) under the Null of a correctly specified model. These moment tests are reported as ‘M(1,1)’ to ‘M(4,4)’ in the output, with ‘M(1,2)’ related to ARCH-in-mean effects, and ‘M(2,1)’ to leverage, while ‘W’ is the Portmanteu type test statistic for general misspecification (using p lags) and also distributed as N(0, 1) under the Null of a correctly specified model. Only upper tail critical values are used in this test. The interested reader is referred to the paper for more details.

Value

A list with the following items:

statistic

The individual moment and joint test statistics.

Decision

Whether to reject or not the Null given the conf.level.

Author(s)

Alexios Ghalanos

References

Ait-Sahalia, Y. (1996), Testing continuous-time models of the spot interest rate, Review of Financial Studies, 9(2), 385–426.
Berkowitz, J. (2001), Testing density forecasts, with applications to risk management, Journal of Business and Economic Statistics, 19(4), 465–474.
Hong, Y., and Li, H. (2005), Nonparametric specification testing for continuous-time models with applications to term structure of interest rates, Review of Financial Studies, 18(1), 37–84.
Rosenblatt, M. (1952), Remarks on a multivariate transformation, The Annals of Mathematical Statistics, 23(3), 470–472.

Examples

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## Not run: 
data(dji30ret)
spec = ugarchspec(mean.model = list(armaOrder = c(1,1), include.mean = TRUE),
variance.model = list(model = "gjrGARCH"), distribution.model = "sstd")
fit = ugarchfit(spec, data = dji30ret[, 1, drop = FALSE])
z = residuals(fit)/sigma(fit)
PIT = pdist("sstd",z, mu = 0, sigma = 1, skew = coef(fit)["skew"], 
shape=coef(fit)["shape"])
print(HLTest(PIT, lags=4))

## End(Not run)

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