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Testing for misspecification in the short-run component of GARCH-type models

Thomas Chuffart, Emmanuel Flachaire and Anne Péguin-Feissolle


In this article, a misspecification test in conditional volatility and GARCH-type models is presented. We propose a Lagrange Multiplier type test based on a Taylor expansion to distinguish between (G)ARCH models and unknown GARCH-type models. This new test can be seen as a general misspecification test of a large set of GARCH-type univariate models. It focuses on the short-term component of the volatility. We investigate the size and the power of this test through Monte Carlo experiments and we compare it to two other standard Lagrange Multiplier tests, which are more restrictive. We show the usefulness of our test with an illustrative empirical example based on daily exchange rate returns.


We would like to thank the anonymous referee for very useful comments and suggestions that helped us to prepare the final version of our paper, and Marjorie Sweetko for language verification of the manuscript. Finally, the paper has also benefited from the comments of Cyril Dell’Eva on the empirical application and Bilel Sanhaji. Previous versions of this paper have circulated under the title “Testing for Misspecification in GARCH-type models”.


A Computation of the LM statistic

I(θ), given in Equation (10) asI(θ)=E[2lt(θ)θθ], can be written:




Using lt(θ) given by (5), we have


Thus, the elements in (28) can be derived as




Under conditional symmetry of the error term, I11 becomes diagonal. The first derivative of the log-likelihood with respect to the parameters θ, lt(θ)θ, is defined as follows:


Under the null hypothesis, lt(θ)λ=0 and lt(θ)ϕ=0, thus we have:


We can derive the LM statistic:


Using the properties of the partitioned matrix, the statistic is equal to:




htϕ and htB are respectively given by equations (14) and (15). If we note ft=m(xt,λ)λ, ct=1hthtλ, vt=1hthtB and zt=1hthtϕ, the statistic will be computed as




The LM statistic has an asymptotic χ2 distribution with N degrees of freedom (because the dimension of the vector B is N given by (9).)

B On Taylor expansions

In this appendix, we present the Taylor expansion when k is even. We also propose some examples where pa, qa and k take different values.

  • The Taylor expansion expression for k odd differs a bit from Equation (B):


    with k~=k1 and k¯=k+1.

  • Order k = 2 expansion of ht = α0 + G(εt−1, ht−1, ρ):

  • Order k = 3 expansion of ht = α0 + G(εt−1, ht−1, ρ):

  • Order k = 3 expansion of ht = α0 + G(εt−1, εt−2, ht−1, ht−2, ρ):



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Published Online: 2018-07-03

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