A wavelet-based variance ratio unit root test for a system of equations

2.1 Variance ratio unit root tests and the wavelet filters

There has been considerable research into testing the random walk and martingale difference hypotheses, mainly in the context of asset prices. Of particular interest is the model where the error term is an uncorrelated process, which is common in financial time series. Consider the Random Walk Model given by,

Variance ratio unit root tests use the fact that, for a unit root time series, the variance of the kth difference of the series is an increasing linear function of the difference, k. The test statistics of these tests are, therefore, based on estimators of the ratio of variances at different lags to that at lag 1.

Let σq2=Var(yt−yt−q)/q and Δyt=(yt−yt−1). The relation between σq2 and the autocorrelation coefficients of Δy_t are given as (see Cochrane (1988)),

where ρ_i is the lag i autocorrelation coefficient of the first differences of the {yt}t=0T series. This type of variance ratio unit root test is essentially a specification test using the null hypothesis,

The variance ratio (see Cochrane 1988) is given as follows,

where σ^2(1) is an unbiased estimate of the variance at lag 1 and f_Δy(0) is the spectral density estimator of Δy_t at the zero frequency. An estimate of f_Δy(0) can be based on the sample autocorrelations of Δy_t. When the time series has a unit root, the expected value of the variance ratio should be close to 1 for all lags k. The variance ratio will be less than 1 when the first differences are correlated, indicating the rejection of the null hypothesis of a serially uncorrelated random walk.

Other variance ratio tests are those suggested by Tanaka (1990) and Kwiatkowski et al. (1992). The test statistic for the variance ratio test given in Kwiatkowski et al. (1992) is,

where Yt=∑i=1tyi is the partial sum of the {yt}t=0T process. ∑t=1Tyt2/T estimates the long-run variance which, in the case of serial dependence, can be estimated using semi-parametric kernel based methods e.g. the Newey-West estimator (Newey and West, 1987). When testing against trend stationary alternatives, the test statistic is for the detrended series,

where the deterended series is given as y~t=(yt−μ^), and μ^ is an estimate of the deterministic component – for example, the sample average in the case of the null being of stationarity about a non-zero mean. The test statistic of Kwiakowski et al. given above tests for stationarity i.e. it has its null hypothesis as stationary. Breitung (2002) reversed the roles of the null and alternative hypotheses and proposed using ϱ^T as a unit root test where the null hypothesis is non-stationarity. Used in this way, its limiting distribution under the null hypothesis (see Breitung (2002), Proposition 3) does not depend on the long-run variance as the long-run variance cancels out in the variance ratio. This removes the need for kernel function selection and tuning parameter optimization necessary for estimating the long-run variances.

In the frequency domain, the use of variance ratios for unit root testing is motivated by the fact that the spectrum of a unit root process peaks at the near zero frequencies, and tails off exponentially. As a consequence, the largest proportion of the variance is found in the lowest frequency bands. Suitable test statistics can, therefore, be based on the relative distribution of the variance with regards to frequency. For this to be feasible, the spectral variance needs to be decomposed in order to obtain the proportions of the variance contributed by the different frequency intervals. The Discrete Wavelet Transform (DWT) is a variance preserving transform, which decomposes the spectral variance on a scale-by-scale basis using filtering operations. The transform outputs two vectors; a vector of the DWT wavelet coefficients, and a vector of its scaling coefficients. The wavelet coefficients describe the changes at each scale, i.e. the details resulting from differences within each scale. The scaling coefficients, on the other hand, describe averages at each scale, i.e. the smooth resulting from averaging at each scale. The scale of the transform, which is inversely related to frequency, refers to the number of the recursive decompositions. Each recursive iteration from the second onwards decomposes the scaling coefficients from the preceding iteration.

The DWT has its filters operate on non-overlapping values, which means that the input time series have to be of dyadic lengths (2k,k=2,3,…). In contrast, the Maximal Overlap DWT (MODWT) has its filters operate on overlapping values, which makes it possible to handle samples of any size. The MODWT, therefore, extracts more information on the local variation of the time series. Unlike wavelet functions with longer and smoother filters, the Haar MODWT does not suffer from boundary effects, i.e. the loss of coefficients which are subject to circular filtering operations at the end of the time series. The transform also provides a better estimator of the wavelet variance (see Percival 1995) compared to the DWT. For these reasons, we use the Haar MODWT in this paper. For more details on wavelet filters and their properties, we refer to texts by Percival and Walden (2000) and Gençay, Selçuk, and Whitcher (2001).

The Haar MODWT wavelet filter (hj,l:l=0,…Lj−1j=1,2,…) is given as,

Lj=2j for this wavelet, and is the length of filter at scale j. The scaling filter is given as,

The Haar wavelet filter, therefore, approximates a band-pass filter with the nominal pass-band [2^−(j+1), 2^−j] and the Haar scaling filter approximates an ideal low-pass filter with the nominal pass-band [0, 2^−(j+1)].

The jth level wavelet and scaling coefficients are defined as follows respectively:

A useful property of the Haar MODWT transform is,

where J₀ is an arbitrary scale less than or equal to the maximum resolution of the time series. This property implies that the time series itself (not only its variance) can be additively decomposed into its wavelet and scaling coefficients.

For univariate time series, the wavelet variance ratio unit root test introduced by Fan and Gençay (2010) use a normalized version of the test statistic given below,

The numerator is the contribution to the variance from the first level scaling coefficients, and the denominator is the total variance partitioned into the parts contributed by the scaling and wavelet coefficients of the first scale, respectively. The variances of the scaling and wavelet coefficients are given as,

respectively.

Under the unit root null hypothesis, it can be seen that ∑t=0T−1W~t,12=Op(T) and it is shown (see Fan and Gençay, 2010) that ∑t=0T−1V~t,12=Op(T2). The variance ratio therefore takes the form,

The limiting distribution of the test statistic, which is a normalized version of S~T,1, is non-standard and the critical values are obtained using Monte Carlo simulations.

Under the alternative hypothesis both ∑t=0T−1W~t,12 and ∑t=0T−1V~t,12 are stationary and the ratio S~T,1 will be less than 1.

Li and Shukur (2013) proposed using S^T,1 in a panel data setting. Their test statistic, S¯NT, is based on averaging the variance ratios of the individual panel units, i.e. S¯NT=N−1∑i=1NS^T,1i. The test was conducted on cross-sectionally correlated time series as well as series that were decorrelated by wavestrapping. Monte Carlo simulation results showed that the wavelet-based panel unit root test is more powerful than the IPS test in the presence of correlation among the panel units. The test is also more robust to size distortions resulting from cross-sectional dependency compared to the IPS test, but still over-sized.

2.2 The wavelet variance ratio unit root test for a system of equations

Consider the system of equations without deterministic terms for simplicity,

where y_it is the time series of interest and u_it is a zero mean weakly stationary error term, i.e. uit=∑j=0∞ψjLjεit (with finite long-run non-zero variance σεit2ψ(1)2<∞, and ψ(1) ≠ 0), i indexes the individual equation, and t indexes time. Also Cov(uit,ukt)=0 for i ≠ k, and ε_it is an iid, zero mean process with variance σεit2. ψ(L) is the lag polynomial that that relates the response of u_it to ε_it.

The unit root hypothesis for the system,

and the alternative hypothesis is,

Let the matrix of time series be denoted by,

so that the Haar MODWT scaling and wavelet coefficients for the first scale decomposition are given by,

where v_i is the vector of the scaling coefficients of the series y_i (i = 1, 2, … , N), i.e. Vti,1,…VTi,1 and w_i is the vector of the wavelet coefficients of series y_i (i = 1, 2, … , N), i.e. Wti,1,…WTi,1

A wavelet variance ratio unit root test can be based on the following,

where,

is the total variance of the system and,

is the variance contributed by the first scale wavelet coefficients.

Under the null hypothesis (where all the series in the equation system are I(1)), V^TV is a diagonal matrix with diagonal elements being of order O_p(T²). The diagonal elements of V^TV will dominate those of W^TW which are of order of convergence O_p(T). The test statistic will, therefore, take on larger values under the null hypothesis compared the values under the alternative. For white noise series, for example,

since V^TV = W^TW.

VR is not bounded under the null hypothesis. A suitably normalized test statistic given as follows,

where Γ^ is a diagonal matrix with the diagonals consisting of the weights υ^yi,1/ω^i2, and υ^yi,1 and ω^i2 are consistent estimates of the wavelet and long-run variances respectively. The two variances, which enter the limiting distribution as nuisance parameters, are consistently estimated as is shown in the Appendix.

The limiting distribution of the test statistic under the null hypothesis is shown in the following theorem whose proof is given in the Appendix.

where Wi(r),i=1,2,…,N are independent standard Brownian motions, and N is the number of equations.

Since the test statistic is the sum of variance ratios, a Central Limit Theorem could be invoked by normalizing the sum and letting N → ∞, but this is not pursued in this paper as we are mainly interested in the limit only where T → ∞.

Many unit root tests suffer from loss of power when tested against alternatives that are trend stationary. As a consequence of this, efficient detrending methods (see Schmidt and Phillips, 1966) are required to retain power. We use the detrending techniques suggested by Fan and Gençay (2010), and, as in their work, we restrict our scope to the the cases where the models specified under the alternative hypotheses have non-zero means and linear trends only.

The model including deterministic components is given as,

For equation i, the null hypothesis, H₀ : ϕ_i = 1 is the unit root hypothesis while under H_A, ϕ_i < |1| is the hypothesis of stationarity. Following Fan and Gençay (2010), when α = 0, we consider the demeaned series (yit−y¯it) where y¯it=T−1∑t=1Tyit is the individual’s average. Similarly, we consider the detrending (y~it−y~¯i), when α ≠ 0. y~it=T−1∑t=1T(Δyit−Δy¯i), y~¯i is the sample mean of y~it for individual i, Δyit=yit−yi,t−1 and Δy¯i is the sample mean of Δy_it for individual i.

Let the test statistics be denoted by VRMM and VRMd for the cases where α = 0 and α ≠ 0 respectively (see Eqn. (1)). The limiting distributions of these statistics are given by Theorem 2 below. The derivations of these limiting distributions, which are also given in the Appendix, are similar to that given for the distribution of the test statistic given in Theorem 1, except that detrended Brownian motions are used.

Where ⇒ denotes convergence in the associated probability measure, Wμ(r)=W(r)−∫01W(r)dr, is the demeaned Brownian motion, and Vμ(r)=V(r)−∫01V(r)dr is the detrended Brownian motion, with V(r) = W(r) − rW(1).

2.3 Comparison unit root test

The small sample properties of VR_M are compared to CIPS. Pesaran (2007) constructs the CIPS test based the following model:

where the initial values y_i0 are fixed. A single common factor with individual specific factor loadings is specified for the error term,

ϵit,i=1,…,N,t=1,…,T, are zero mean errors with heterogeneous variances σi2. The common factor, f_t, is assumed to be stationary and serially uncorrelated, and without loss of generality, its variance is fixed at 1, i.e. σf2=1. Cross-sectional correlations are introduced by the factor loadings λ_i, themselves random variables. ϵ_it, λ_i and f_t are assumed to be mutually independent.

Pesaran (2007) proposes a test that augments the standard Dickey-Fuller test with the cross-sectional averages, resulting in the following Cross-sectionally Augmented Dickey-Fuller (CADF) estimating equation,

where ȳ_t are the cross-sectional averages, and lags of Δy_it and Δy¯t may be included to whiten the residuals. The cross-sectional averages are used as proxies for the unobservable common factor.

Letting CADF_i represent the CADF statistic for equation i, CIPS is the average of the CADFs over all the equations,

The small sample properties of CIPS are studied using Monte Carlo simulation in Pesaran (2007). The test is shown to be robust to size distortions even, in the presence of strong cross-sectional dependence and serial correlation, and has good power properties for sample sizes of between 50 and 100 for the DGPs considered therein.

In the following section, we examine the performance of VR_M, and make comparisons with that of CIPS for time series of lengths 20–100, using systems of 5 and 10 equations. The size and power of the tests are compared in cases where there is neither cross-sectional dependency nor serial correlation (hereafter called DGP 1), in the presence of weak cross-sectional correlation but no serial correlation (hereafter called DGP 2), in the case where there is strong cross-sectional correlation but no serial correlation (hereafter called DGP 3), and in the case with both strong cross-sectional correlation and strong serial correlation (hereafter called DGP 4). The choice of DGPs follows that of Pesaran (2007).

3.1 Design of the Monte Carlo experiment

Following Pesaran (2007), time series are simulated using the following DGPs

Cross-sectional correlation is introduced using a single common factor denoted by f_t, and represents the unobserved common factor effect.

Table 1 shows the experimental factors and the ranges over which they are varied. The nominal test size is held at 5% as per convention.

The common factor loadings are sampled from the uniform distribution with parameters U[0, 0.2] and U[−1, 3] for weak and strong cross-sectional dependence, respectively. This corresponds to cross-sectional correlations between the equations of 1% and 50% on average, respectively.

4.1 Empirical test sizes and power

Monte Carlo simulations were conducted for two purposes; to study the small sample performance of VR_M by comparing it CIPS, and to study the robustness of VR_M to cross-sectional and serial dependence. We examine these aspects in the cases where testing is against an alternative that has zero mean and no time trend, and in the case where testing is against an alternative that is stationary around a non-zero mean only. The test statistic used in both cases is VRMM since both tests correspond to α = 0 but differ in their specification of μ_i (see Eqn. (1))