Weighted simulated integrated conditional moment tests for parametric conditional distributions of stationary time series processes

ABSTRACT

In this article, we propose a weighted simulated integrated conditional moment (WSICM) test of the validity of parametric specifications of conditional distribution models for stationary time series data, by combining the weighted integrated conditional moment (ICM) test of Bierens (1984) for time series regression models with the simulated ICM test of Bierens and Wang (2012) of conditional distribution models for cross-section data. To the best of our knowledge, no other consistent test for parametric conditional time series distributions has been proposed yet in the literature, despite consistency claims made by some authors.

KEYWORDS:

• Characteristic functions
• consistent tests
• parametric conditional distributions
• simulated integrated method of moments
• stationary time series

JEL CLASSIFICATION:

• C12
• C15
• C22
• C32
• C52

Acknowledgment

The helpful comments of the Editor, an Associate Editor, and two referees, leading to substantial improvement of the paper over previous versions, are gratefully acknowledged.

Notes

Here and in the sequel, i denotes the complex number

See also Bierens (2014b) for extensions and corrections of Bierens and Wang (2012).

See for example Billingsley (1968) or van der Vaart and Wellner (1996).

For example, ∂(x′b)/∂x′ = b, ∂(x′b)/∂x = ∂(b′x)/∂x = b′.

Of course, there are many more tests for the validity of conditional distributions, but none of these tests are consistent.

See for example McLeish (1974).

Together with the measurability conditions in Bierens (2004, Theorem 7.8(b), condition (a)), which we will not make explicit.

See for example Billingsley (1968) or van der Vaart and Wellner (1996).

See for example Bierens (2004, Theorem 7.8(a), p. 187).

Recall that the matrix norm ||.|| in (8.1) is the maximum absolute value of the elements of the matrix involved.

This Monte Carlo analysis took several weeks to conduct on a PC.

Agilent Technologies produces hi-tech measurement equipments. See http://www.home.agilent.com/

The value c = 5 has been chosen on the basis of the simulation results in Bierens and Wang (2012).

The GARCH estimation and WSICM test have been conducted via the latest version of the free econometric software package EasyReg International (See Bierens, 2014c).

In principle, we could repeat this simulation procedure 1,000 times, for example, to determine the size properties of the WSICM test. However, the computation of the WSICM test and its bootstrap p-value takes about half an hour on a 32-bit Windows XP PC, so that in this case a full Monte Carlo simulation would take about 20 days!

See the reference Bierens and Wang (2012).