Testing fractional unit roots with non-linear smooth break approximations using Fourier functions

Abstract

In this paper, we present a testing procedure for fractional orders of integration in the context of non-linear terms approximated by Fourier functions. The test statistic has an asymptotic standard normal distribution and several Monte Carlo experiments conducted in the paper show that it performs well in finite samples. Various applications using real life time series, such as US unemployment rates, US GNP and Purchasing Power Parity (PPP) of G7 countries are presented at the end of the paper.

KEYWORDS:

• Fractional unit root
• Chebyshev polynomial
• Monte Carlo simulation
• nonlinearity
• smooth break
• Fourier transform

JEL Classifications:

• C15
• C12
• C22

Acknowledgements

Comments from the Editor and an anonymous reviewer are also gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This augmentation, however, is allowed in Lobato and Velasco [54] and Lobato and Velasco [55].

2 Becker, Enders and Lee [5], Enders and Lee [24] and Rodrigues and Taylor [62] suggested using a single frequency component for the Fourier function in the detection of smooth breaks since higher frequencies lead to over-filtration.

3 The unit root test proposed by Becker, Enders and Lee [5] and Ender and Lee [24] is used when the break dates and the precise form of the breaks are unknown.

4 Though Tukey [66] suggested using an alternative technique with the Fourier transformation in order to control over-differencing by reducing the periodogram bias. However, as noted by Hurvich and Ray [46] and Deo and Hurvich [17], the use of tapering could strongly inflate the variance estimate of the fractional differencing parameter d, and the efficiency loss might be substantial. Due to this, non-tapered model in (1) is considered.

5 See other papers such as Davies [15], Gallant and Sonza [31] and Bierens [6].

6 Multiple (fractional) cyclical structures [26,63], and Gil-Alana [36] can also be examined if d₁ = d₂ = 0 with j > 3.

7 That means that if the test is directed against local alternatives of the form: H_a: d = d_o + δ/(T)^0.5, the limit distribution is χ²(ω) with a non-centrality parameter ω that is optimal under Gaussianity of u_t.

8 See Gil-Alana [35], Gil-Alana and Moreno [39].

9 For this particular version of the tests of Robinson [61], the spectrum has the singularity at the zero frequency, so, j runs from 1 to T-1.

10 The choice of the values (0.6 and 0.4) is arbitrary. We tried with other values and the results were very similar to those reported in this work. Note, also that when these values were assumed to be zero, the model in (17) becomes the (linear) model of Robinson [61] and the results obtained would be the same as those reported in the tables in that paper.

11 The codes are available from the authors upon request. Smaller sample sizes than those used here are unrealistic since fractional unit root operation is based on the truncated Binomial expansion which requires long lags for initialization of the time process.

12 Once more the choice of these values is completely arbitrary.

13 Using other more classical representations for the I(0) disturbances term, for instance a simple AR(1) structure, it produced confronting results. Thus, for example, in some of the applications conducted below, we observed values of d very close to 0 with the AR(1) model, while the estimated value of d were very close to 1 with the Bloomfield [9] approach. This can be clearly explained by the competition between the two structures (i.e., the fractional one and the autoregressive one) in describing the nonstationarity of the series. In that respect, the application of Bloomfield [9] is clearly preferred.

Additional information

Funding

First named author gratefully acknowledges financial support from the MINEIC-AEI-FEDER ECO2017-85503-R project from ‘Ministerio de Economía, Industria y Competitividad’ (MINEIC), ‘Agencia Estatal de Investigación’ (AEI) Spain and ‘Fondo Europeo de Desarrollo Regional’ (FEDER). Also, he acknowledges support from an internal Project of the Universidad Francisco de Vitoria. The second-named author gratefully acknowledges financial support from the Ministerio de Educación, Cultura y Deporte [grant number ECO2017-85503-R].