The GMM Orthogonality Test of Hansen

Description

Implements the GMM Orthogonality Test of Hansen.

Usage

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GMMTest(z, lags = 1, skew=0, kurt=3, conf.level = 0.95)

Arguments

z

A numeric vector the standardized residuals.

lags

The number of lags to test for.

skew

The skewness of the standardized residuals (derived from the estimated model). This can be either a scalar or numeric vector the same size as z.

kurt

The kurtosis (not excess) of the standardized residuals (derived from the estimated model). This can be either a scalar or numeric vector the same size as z.

conf.level

The confidence level at which the Null Hypothesis is evaluated.

Details

This is a mispecification test based on Hansen's GMM procedure. Under a correctly specified model, certain population moment conditions should be satisfied and hold in the sample using the standardized residuals. The moment conditions can be tested both individually using a t-test or jointly using a Wald test (the vignette gives more details). The test returns a matrix (moment.mat) containing the first 4 moments statistics, their standard errors and t-values (2-sided t-test with alternative hypothesis that the value is not equal to zero). The matrix of joint conditions (joint.mat) contains the t-values and critical values of ‘Q2’, ‘Q3’ and ‘Q4’ representing the autocorrelation, given the chosen lags in the second, third and fourth moments and distributed as chi-squared with n.lag d.o.f, and the joint test (‘J’) for all moment conditions distributed chi-squared with 4+(n.lagx3) d.o.f.

Value

A list with the following items:

joint.mat

The matrix of the joint tests.

moment.mat

The matrix of the individual moment tests.

H0

The Null Hypothesis.

Decision

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

Author(s)

Alexios Ghalanos

References

Hansen, L. (1982), Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50(4), 1029–1054.

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)
skew = dskewness("sstd",skew = coef(fit)["skew"], shape= coef(fit)["shape"])
# add back 3 since dkurtosis returns the excess kurtosis
kurt = 3+dkurtosis("sstd",skew = coef(fit)["skew"], shape= coef(fit)["shape"])
print(GMMTest(z, lags = 1, skew=skew, kurt=kurt))

## End(Not run)

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