Wavelet energy ratio unit root tests

ABSTRACT

This article uses wavelet theory to propose a frequency domain nonparametric and tuning parameter-free family of unit root tests. The proposed test exploits the wavelet power spectrum of the observed series and its fractional partial sum to construct a test of the unit root based on the ratio of the resulting scaling energies. The proposed statistic enjoys good power properties and is robust to severe size distortions even in the presence of serially correlated MA(1) errors with a highly negative moving average (MA) parameter, as well as in the presence of random additive outliers. Any remaining size distortions are effectively eliminated using a novel wavestrapping algorithm.

KEYWORDS:

• Fractional Brownian motion
• fractional integration
• hypothesis test
• size distortion
• statistical power
• time series
• unit root
• variance ratio statistic
• wavelet energy ratio
• wavestrapping
• wavelets

JEL CLASSIFICATION:

• C01
• C12
• C15
• C21
• C40
• C46
• C63

Acknowledgment

I am grateful to Ramazan Gençay, Morten Nielsen, Asger Lunde, Joon Park, Victoria Zinde-Walsh, Russell Davidson, Burak Alparslan Eroğlu, Taner Yigit, and seminar participants at Aarhus University, Bilkent University, Indiana University, and McGill University.

Notes

¹Borrowed terminology will be referenced throughout the article. The term signal refers to a data source, e.g., a time series.

²MRA was introduced in Mallat (1989).

³The term energy originates from the signal processing literature. It is formalized as ${\int }_{\infty }^{\infty }\left|f\right(t){|}^{2}dt$, for some function f(t). Restricting f(t) to the real plane, energy and variance are effectively synonymous and will henceforth be used interchangeably.

⁴This relationship states that $h_i={(-1)}^i{g}_{l-1-i},g_i={(-1)}^{i+1}{h}_{l-1-i}$ for i = 0,…,l−1.

⁵Both filters exhibit orthogonality to even shifts. Formally, ${\sum }_{i=0}^{l-1}{h}_i{h}_{i+2n}={\sum }_{i=0}^{l-1}{g}_i{g}_{i+2n}={\sum }_{i=0}^{l-1}{g}_i{h}_{i+2n}=0,\forall n\in {\mathbb{Z}}_{+}$.

⁶Limiting series to dyadic lengths is restrictive. Methods such as the maximum overlap discrete wavelet transform (MODWT) otherwise known as the non-decimated DWT overcome this shortcoming.

⁷While W_m and V_m implicitly depend on l, the notation is suppressed for notational brevity.

⁸Since test consistency requires bandwidth parameters to expand at specific rates relative to sample size, both finite sample and asymptotic performance are highly dependent on the tuning parameter choice while the latter is not reflected in the asymptotic distribution. In this regard, the bandwidth choice q is considered a tuning parameter; cf. Nielsen (2009). Moreover, since the bandwidth determines the proportion of information in the covariance structure that is used in the estimate of the long run variance, failing to select the right bandwidth for covariances that dissipate slowly will result in imprecise estimates; cf. Andrews (1991) and Newey and West (1994). Accordingly, the Newey and West (1987) rule of thumb choice q = 4(T∕100)^2∕9 used in Fan and Gençay (2010) may not always be appropriate when the underlying process is highly persistent. As pointed out in Kiefer and Vogelsang (2000), “traditional asymptotics requires the bandwidth to increase with sample size but the fraction of sample autocovariances used goes to zero. Thus, information contained in sample autocovariances must be ignored for the asymptotics to work." This partly explains why simulation results in this article show that the FG test adapt poorly to MA serial correlations when the MA parameter is highly negative.

⁹See Davidson and Hashimzade (2009) for a discussion on type I and type II fractional Brownian motions.

¹⁰Granger (1966) first noticed that it is the ill behaved frequencies near the origin which indicate the presence of a unit root.

¹¹See Davidson and MacKinnon (2000) for details.

¹²Simulations could also have been conducted using the MODWT. Since the MODWT does not suffer the decimation at each scale like the DWT, it may produce further finite sample improvements over the DWT, particularly in terms of power since the MODWT produces series of the same length as the input signal. This is not pursued, however, since the DWT is significantly quicker to compute, requiring O(T) computations vs. O(Tlog₂T) for the MODWT.

¹³Due to excessive size distortion differentials among the NVR, FG, and WSR tests, size adjusted power uses empirical size rather than nominal size α to deflate (inflate) oversized (undersized) tests and calibrate power curves to a common reference point. While this renders different tests directly comparable, the exercise requires Monte Carlo simulations and, therefore, as argued in Horowitz and Savin (2000), is “irrelevant for empirical research.”