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Outliers and persistence in threshold autoregressive processes

Yamin Ahmad and Luiggi Donayre


This paper uses Monte Carlo simulations to investigate the effects of outlier observations on the properties of linearity tests against threshold autoregressive (TAR) processes. By considering different specifications and levels of persistence for the data-generating processes, we find that additive outliers distort the size of the test and that the distortion increases with the level of persistence. In addition, we also find that larger additive outliers can help to improve the power of the test in the case of persistent TAR processes.

Corresponding author: Luiggi Donayre, Department of Economics, University of Minnesota – Duluth, 1318 Kirby Drive, Duluth, MN 55812, USA, e-mail:


We thank the editors, two anonymous referees, Bruce Hansen, Jun Ma, Dick van Dijk and the participants at the 22nd Symposium of the Society for Nonlinear Dynamics and Econometrics and the 2014 Meetings of the Southern Economic Association for helpful comments and suggestions. All errors are our own.


Fisher’s exact test of independence

To determine whether the size-corrected power of the test is driven by additive outliers generating lower estimated persistence parameters, thus making the identification of regimes by the sup-LR test easier, we conduct Fisher’s exact test of independence.

Let O={t: δt=1}∪{t: δt−1=1} be the set of periods when an AO occurs or of periods after an AO occurs. Let R={t:st>γ^}, be the set of observations estimated to belong to the less persistent regime (or, R={t:stγ^}, depending on the estimated persistence parameters across regimes). If the sets O and R are dependent, then the higher power found in the previous sections could be the result of AOs lowering the estimated degree of persistence.

For a given DGP, each observation t can be categorized as belonging or not to either set, O and R. This type of categorical data that can be arranged in a 2×2 contigency table, as follows:


Fisher (1925) noted that, under the null hypothesis of independence, the distribution of the four cell counts in the 2×2 table is given by the hypergeometric distribution and that, for fixed margins a+b, c+d, a+c, b+d, and a fixed number of observations n=a+b+c+d, the probability of obtaining the specific frequencies {a, b, c, d} is given by:

(1)p=(a+ba)(c+dc)(na+c) (1)

where (nk) is the binomial coefficient. This probability establishes how extreme a given particular combination of cell frequencies is in relation to all other possible 2×2 tables that could have been observed, given fixed margins. The p-value for Fisher’s exact test of independence, therefore, is the sum of a hypergeometric probabilities for outcomes at least as favorable to the alternative hypothesis as the observed outcome.


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Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2014-0058) offers supplementary material, available to authorized users.

Published Online: 2015-8-12
Published in Print: 2016-2-1

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