Comparison of optimization algorithms for selecting the fractional frequency in Fourier form unit root tests

ABSTRACT

We compare the performance of unit root tests which include flexible Fourier trends in their testing processes. The algorithms considered are those of Broyden, Fletcher, Goldfarb and Shanno (BFGS), Berndt, Hall, Hall and Hausman (BHHH), Simplex, Genetic and grid search (GS). The simulation results indicate that derivative-free methods, such as Genetic and Simplex, have advantages over hill-climbing methods, such as BFGS and BHHH in providing accurate fractional frequencies for fractional frequency flexible Fourier form (FFFFF) unit root test. When the parameters are estimated under the alternative hypothesis of the FFFFF type of unit root test, the grid search and derivative-free methods provide unbiased and efficient estimations. We also provide the asymptotic distribution of the FFFFF unit root test. We extend the FFFFF unit root test to a panel version in order to increase the power of the test. Finally, the empirical analyses of Covid19 unit root test show that derivative-free methods, are better than other methods. However, for big data and accurate estimation of the frequency parameters, the Simplex methodology using the bootstrap process is preferred.

KEYWORDS:

• Nonlinear trend
• deterministic smooth break
• structural change
• estimation methods
• Covid-19

JEL CLASSIFICATION:

• C22
• C23
• C61
• C63

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ See Omay and Emirmahmutoglu (2017) for more details.

² See the appendix for an alternative explanation of Davies (1987) method following Omay and Emirmahmutoglu (2017).

³ For Simplex, Genetic, BFGS and BHHH we do not give explicit explanations, but these are available in any text book.

⁴ In the fractional Fourier type unit root test, we have an opportunity to generate infinitely many critical values, therefore we propose a simple bootstrap algorithm to overcome this problem in the fractional frequency case in the empirical part.

⁵ The results for MSE and RMSE are available upon request.