Are linear models really unuseful to describe business cycle data?

ABSTRACT

We use first differenced logged quarterly series for the GDP of 29 countries and the euro area to assess the need to use non-linear models to describe business cycle dynamic behaviour. Our approach is model (estimation)-free, based on testing only. We aim to maximize power to detect non-linearities while, simultaneously, avoiding the pitfalls of data mining. The evidence we find does not support some descriptions because the presence of significant non-linearities is observed for two-thirds of the countries only. Linear models cannot be simply dismissed as they are frequently useful. Contrarily to common knowledge, non-linear business cycle variation does not seem to be a universal, undisputable and clearly dominant stylized fact. This finding is particularly surprising for the U.S. case. Some support for non-linear dynamics for some further countries is obtained indirectly, through unit root tests, but this can hardly be invoked to support non-linearity in classical business cycles.

KEYWORDS:

• Business cycles
• non-linear time series models
• testing
• unit roots

JEL CLASSIFICATION:

• C22
• C51
• E32

Acknowledgements

We are grateful to an anonymous referee for helpful comments and suggestions. Obviously, we are responsible for all remaining errors. Previous versions of this article circulated under the title ‘Revisiting non-linearities in business cycles around the world’.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Whom, however, consider that non-linear models are better than linear ones to reproduce some of these features.

² Actually, a great effort has been made recently in topics such as unit root testing against non-linear alternatives or in the estimation of new models. Empirically, much effort has been directed recently towards non-linear modelling of interest rates, exchange rates and public finances.

³ This is the case for China as well, because this test is not robust to the change in the variance of innovations previously detected.

⁴ See Teräsvirta, Tjostheim, and Granger (2010) for a general introduction to non-linear models and, in particular, Sections 3.2 and 3.4 about TAR and STAR models, respectively.

⁵ The fact that testing for the unit root may also be formulated as $H_0:\gamma =0vs.H_1:\gamma >0$, as in KSS, is also a manifestation of this problem; in this case it is the parameter $\varphi$ that is not identified under $H_0$.

⁶ While the GTS $t$-sig method appears to show a slight tendency to over parametrize in relation to the AIC, the MAIC frequently appears to produce lag lengths that are too short.

⁷ A brief introduction to both these models is contained in subsection 3.4 of Teräsvirta, Tjostheim, and Granger (2010).

⁸ Recall that it is defined by ${\hat{\omega }}^2\left\{a_{t,k}\right\}={\hat{\gamma }}_0\left\{a_{t,k}\right\}+2\sum _{j=1}^l\left(1-\frac{j}{l}\right){\hat{\gamma }}_j\left\{a_{t,k}\right\}$, ${\hat{\gamma }}_j\left\{a_{t,k}\right\}=T^{-1}\sum _{t=j+k+1}^T{a}_{t,k}{a}_{t-j,k}$, with $a_{t,k}={\tilde{y}}_y{\tilde{y}}_{t-k}$, $k=(2T{)}^{1/2}$ and $l=12(T/100{)}^{1/4}$ rounded to the nearest integer.

⁹ We adopt the view of the $p$-value as representing a continuous measure of the compatibility between the data and the entire model used to compute it, ranging from 0 to complete incompatibility to 1 for perfect compatibility, and in this sense (it) may be viewed as measuring the fit of the model to the data, Greenland et al. (2016), p. 3.

¹⁰ Notice that we could not find data for several countries of this group: Indonesia, Russia, Saudi Arabia and Turkey.

¹¹ In a recent investigation, Bec, Bouabdallah, and Ferrara (2015) successfully specify and estimate a substantially modified version of Hamilton’s (1989) Markov-Switching model; one of the most important modifications consists of allowing the bounce-back effect to appear only with some delay after the trough, which our conservative testing strategy did not allowed. See also Gadea, Gomez-Loscos, and Perez-Quiros (2017).

¹² However, this is not a completely new finding; Bradley and Jansen (1997) did not find evidence for asymmetry with the CDR test for Canada, France and Japan.

¹³ The complete results are available from the authors upon request.