An extension of the asymmetric causality tests for dealing with deterministic trend components

ABSTRACT

This article extends the asymmetric causality tests, as developed by Hatemi-J (2012), for dealing with deterministic trend parts. It is shown how integrated variables up to three degrees with deterministic trend parts can be transformed into positive and negative cumulative partial components. These cumulative components can be used for implementing the asymmetric causality tests based on a Wald test statistic that is shown to follow a chi-square distribution asymptotically. Each solution is expressed as a proposition and a mathematic proof is provided for each underlying proposition. This issue is important because most economic or financial variables seem to be characterized by both stochastic as well as deterministic trend parts. An empirical application is provided in order to show how the oil prices and the exchange rates as integrated variables with drift and trend can be transformed into cumulative partial sums of positive and negative components. The conducted causality tests reveal that allowing for asymmetry has important repercussions for the underlying causal inference between these two variables.

KEYWORDS:

• Hatemi-J asymmetric causality
• positive changes
• negative changes
• deterministic trend
• bootstrap

JEL CLASSIFICATION:

• C32
• C87
• F31

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Recently, tests for causality within a panel system have also been suggested in the literature. For an asymmetric panel causality test see Hatemi-J (2011).

² Asymmetric causality testing can also be implemented for stationary variables. In that case, positive or negative changes can be used instead of the cumulative sums.

³ It should be mentioned that Hatemi-J (2014) suggested similar solutions, without proofs however, for variables with lower integration order in scalar format for generating asymmetric generalized impulse response functions and asymmetric variance decompositions.

⁴ The VAR model originated from Sims (1980).

⁵ It should be emphasized that additional unrestricted lags need to be included in the VAR model in order to take into account the effect of unit roots as suggested by Toda and Yamamoto (1995). The number of unrestricted lags must be equal to the integration order.

⁶ Note that we assume the p initial values for each variable are available. For details on this assumption, see Lutkepohl (2005).

⁷ This results hold for a corresponding VAR for positive components.

⁸ For a detailed description of this bootstrap algorithm see Hatemi-J (2012) and references therein. For simulations results on power and size properties of the bootstrap approach see Hacker and Hatemi-J (2012).

⁹ The link is http://research.stlouisfed.org/fred2/. For similar applications see Bahmani-Oskooee and Fariditavana (2015) and Bahmani-Oskooee and Saha (2015).

¹⁰ The unit root test results are not presented to save space.

¹¹ In this application we only consider I(1) variables. I(2) variables are rather explosive variables and their existence is rear in reality.

Additional information

Funding

This work was supported by the UAE National Research Fund via the United Arab Emirates University [grant number NF-UPAR 2013-25252].