Testing for unit roots based on sample autocovariances

We propose a new unit-root test for a stationary null hypothesis $H_0$ against a unit-root alternative $H_1$. Our approach is nonparametric as $H_0$ only assumes that the process concerned is $I(0)$ without specifying any parametric forms. The new test is based on the fact that the sample autocovariance function (ACVF) converges to the finite population ACVF for an $I(0)$ process while it diverges to infinity for a process with unit-roots. Therefore the new test rejects $H_0$ for the large values of the sample ACVF. To address the technical challenge `how large is large', we split the sample and establish an appropriate normal approximation for the null-distribution of the test statistic. The substantial discriminative power of the new test statistic is rooted from the fact that it takes finite value under $H_0$ and diverges to infinity under $H_1$. This allows us to truncate the critical values of the test to make it with the asymptotic power one. It also alleviates the loss of power due to the sample-splitting. The test is implemented in a user-friendly R-function.


Introduction
Models with unit-root are frequently used for modeling nonstationary time series. The importance of the unit-root concept stems from the fact that many economic, financial, business and social-domain data exhibit segmented trend-like or random wandering phenomena. While the random-walk-like behavior of stock prices was notified and recorded much earlier by, for example, Jules Regnault, a French broker, in 1863 and then by Louis Bachelier in his 1900 PhD thesis, the development of statistical inference for unit-roots only started in late 1970s. Nevertheless the literature on unit-root tests by now is immense and diverse. We only state a selection of some important developments below, which naturally leads to a new test to be presented in this paper.
The Dickey-Fuller tests and their variants are based on the regression of a time series on its first lag in which the existence of unit root is postulated as a null hypothesis in the form of the regression coefficient being equal to one. This null hypothesis is tested against a stationary alternative that the regression coefficient is smaller than one. This setting leads to innate indecisive inference for ascertaining the existence of unit-roots, as a statistical test is incapable in accepting null hypothesis. To make the assertion of unit-roots on a firmer ground, Kwiatkowski et al. (1992) adopted a different approach: the proposed KPSS test considers a stationary null hypothesis against a unit-root alternative. It is based on a plausible representation for possible nonstationary time series in which a unit-root is represented as an additive random walk component. Then under null the variance of the random walk component is zero. The KPSS test is the one-sided Lagrange multiplier test for testing the variance to be zero against greater than zero.
In spite of the many exciting developments stated above, testing for the existence of unit roots remains as a challenge in time series analysis, as most available methods suffer from the lack of accurate size control and poor power. In this paper we propose a new test, based on a radically different idea from the existing approaches. Our setting is similar in spirit to the KPSS test as we test for stationary null hypothesis H 0 again a unit-root alternative H 1 . However our approach is nonparametric as H 0 only assumes that the process concerned is I(0) without specifying any parametric forms. The new test is based on the simple fact that under H 0 the sample autocovariance function (ACVF) converges to the finite population ACVF while under H 1 it diverges to infinity. Therefore we can reject H 0 for large (absolute) values of the sample ACVF. To address the technical challenge 'how large is large', we split the sample and establish an appropriate normal approximation for the null-distribution of the test statistic. Note that our sample ACVF based test statistic offers substantial discriminative power as it takes finite value under H 0 or diverges to infinity under H 1 . This allows us to truncate the critical values determined by the normal approximation to make the test with the asymptotic power one. Furthermore, it also alleviates the loss of power due to the sample-splitting as it outperforms the KPSS test in the power comparison in simulation. Another advantage of the new method is that it has a remarkable discriminate power to tell the difference between, for example, a random walk and an AR(1) with the autoregressive coefficient close to (but still smaller than) one, for which most the available unit-root tests, including the KPSS method, suffer from weak discriminate power. Admittedly the new test is technically sophisticated, which, we argue, is inevitable in order to gain improvement over the existing methods. Nevertheless to make it user-friendly, we have developed an R-function ur.test in the package HDTSA which implements the test in an automatic manner.

A power-one test
. An I(d) process is also called a unit-root process with the integration order d. With the observations {Y t } n t=1 , we are interested in testing the hypotheses (1) WriteȲ = n −1 n t=1 Y t and denote the sample ACVF at lag k byγ(k) = n −1 n−k t=1 (Y t+k − Y )(Y t −Ȳ ), which is consistent estimator for γ(k) under H 0 . Proposition 1 indicates thatγ(k) diverges to infinity under H 1 . Thus we can reject H 0 for large values of |γ(k)|. When Y t ∼ I(d), the Wold's decomposition for the purely non-deterministic I(0) process admits with the scalar multi-fold integrated Brownian motion F d−1 (t) defined recursively as F j (t) = t 0 F j−1 (x) dx for any j ≥ 1 and the standard Brownian motion F 0 (t). For any given integer k ≥ 0, as n → ∞, it holds that (i) n −(2d−1)γ (k) → a 2 σ 2

bounded constant only depending on d and k.
By Proposition 1, one may consider to reject H 0 for the large values of T naive = K 0 k=0 |γ(k)| 2 with a prescribed integer K 0 ≥ 0, as T naive converges to K 0 k=0 |γ(k)| 2 < ∞ under H 0 . Unfortunately, there are two obstacles preventing using T naive : (i) to determine the critical values one has to derive the null-distribution of a n {T naive − K 0 k=0 |γ(k)| 2 } with some a n → ∞, (ii) one needs a consistent estimator for K 0 k=0 |γ(k)| 2 under H 0 , which is not readily available as we do not know if H 0 holds or not in practice. To overcome these two obstacles, we implement the idea of 'data splitting' where K 0 ≥ 0 is a prescribed integer which controls the amount of information from different time lags to be used. Although our theory allows K 0 diverging with sample size n, the simulation results reported in §3 indicate that the finite sample performance of the test is robust with respect to the different values of K 0 and it works well even with small K 0 .
Formally we reject H 0 at the significance level φ ∈ (0, 1) if T n > cv φ , where cv φ is the critical value satisfying pr H 0 (T n > cv φ ) → φ. As we will see in (3), {γ 1 (k)} K 0 k=0 are used to determine the critical value cv φ . One obvious concern for splitting the sample into two halves is the loss in testing power. However the fact that T n takes finite values under H 0 and it diverges to infinity under H 1 implies that T n has a strong discriminant power to tell apart H 1 from H 0 , which is enough to sustain the adequate power in comparison to that of, for example, the KPSS test. Our simulation results indicate that the sample-splitting works well even for sample size n = 80. Under H 0 , write The following regularity conditions are now in order. See the supplementary material for the discussion of their validity.

Determining the event T in
To avoid the effect of the innovation variance σ 2 ǫ , we consider the ratio R = {γ(0) +γ(1)}/{γ x (0) +γ x (1)}. Notice that R = O p (1) under H 0 , and pr H 1 (R ≥ C * N 3/5 ) → 1 for any fixed constant C * > 0. We define T in (3) as follows: To use T with finite samples, C * must be specified according to the underlying process.
Proposition 2 shows that R with µ 1 = 0 diverges faster than that with µ 1 = 0. Thus for any given C * > 0 the requirement pr H 1 (T c ) → 1 is satisfied more readily with µ 1 = 0. Hence we focus on the cases with µ 1 = 0 only.
Though the above specification was derived for Y t ∼ I(1), our simulation results indicate that it also works well for I(2) processes. Note that testing I(0) against I(d) with d > 1 is easier than that with d = 1, as the autocovariances are of the order at least n 2d−1 for I(d) processes. So the difference between the values of T n under H 1 and those under H 0 increases as d increases. Andrews (1991) found that the Quadratic Spectral kernel is optimal for such estimation. We suggest using this kernel in practice by calling function lrvar from the R-package sandwich with the default bandwidth specified in the function. To state the required asymptotic property for B 2N −K 0 with general kernels, we need following regularity conditions. Condition 4. The kernel function K(·) : R → [−1, 1] is continuously differentiable on R and satisfies conditions: (i) K(0) = 1, (ii) K(x) = K(−x) for any x ∈ R, and (iii)

Implementation of the test
Based on §2.2 and §2.3, Algorithm 1 outlines the steps to perform our test which includes two tuning parameters. The algorithm is implemented in an R-function ur.test in the package HDTSA available at CRAN. To perform the test using function ur.test, one merely needs to input time series {Y t } n t=1 and nominal level φ. The package sets the default value c κ = 0.55 and returns the five testing results for K 0 = 0, 1, . . . , 4 respectively. One can also set (c κ , K 0 ) subjectively. We recommend to use c κ ∈ [0.45, 0.65] and K 0 ∈ {0, 1, 2, 3, 4}.
Step 2. Call function lrvar from the R-package sandwich (with the default bandwidth in the function) to compute the long-run variances of {Q t } and {X t }, denoted bỹ V 2N −K 0 andσ 2 L , respectively, whereQ t is defined in §2.3. Putλ =γ x (0)/σ 2 L .
To illustrate the robustness with respect to the choice of (c κ , K 0 ), we apply our test to the 14 US annual economic time series (Nelson & Plosser, 1982) that were often used for testing unit-roots in the literature; leading to the exactly same results with c κ ∈ {0.45, 0.55, 0.65} and K 0 ∈ {0, 1, 2, 3, 4} for each of the 14 time series. See the details in the supplementary material.

Simulation study
We illustrate the finite sample properties of our test T n by simulation with K 0 ∈ {0, 1, 2, 3, 4} and c κ ∈ {0.45, 0.55, 0.65}. We also consider T n with the untruncated critical value cv φ,naive , i.e. c κ = ∞ in (5). Hualde & Robinson (2011) proposed the pseudo MLEd for the integration order d in the ARFIMA models that can be used to construct a t-statisticd/sd(d) for H 0 : d = 0 versus H 1 : d ≥ 1. We call it HR test that rejects N (0, 1). For comparison, we also include the KPSS test (Kwiatkowski et al., 1992) and the HR test in our experiments. We set N = 40, 70, 100 and repeat each setting 2000 times. To examine the rejection probability of the tests under H 0 , we consider three models: To examine the rejection probability of the tests under H 1 , we consider the following models: Unless specified otherwise, we always assume that ǫ t ∼ N (0, σ 2 ǫ ) independently with σ 2 ǫ = 1 or 2, and set the nominal level φ = 5%. The results with different (c κ , K 0 ) are similar; indicating once again that our test is robust with respect to the choice of (c κ , K 0 ). We only list the results with K 0 = 0 and σ 2 ǫ = 1 in Table 1, and report other results in the supplementary material. We also consider the cases ǫ t ∼ t(2) and t(5), and report the results in the supplementary material.
Overall the rejection probabilities of our test under H 0 are close to the nominal level φ = 5% especially with large n (N = 100). The performance of our test is stable across different models with different parameters, different K 0 and different innovation distributions, while that of the KPSS test and the HR test vary and are adequate only for some settings. Table 1 indicates that our test works well for Model 1 with both positive and negative ρ, while the KPSS test and the HR test perform poorly when ρ < 0, and even worse when ρ > 0. The KPSS test and the HR test completely fail when ρ = 0.9, as the rejection probabilities are at least 46.7%. This is due to the fact that when ρ is close to 1, the KPSS test and the HR test have difficulties in distinguishing it from 1 which is unit-root. See also Table 3 of Kwiatkowski et al. (1992). Our test does not suffer from this closeness to 1, as for which the order of the magnitude of ACVF matters. Our test  and the KPSS test work well for Model 2 while the HR test is too conservative. For Model 3, the rejection probabilities of our test and the HR test are close to 5% while the KPSS test does not work as its rejection probabilities range from 16.6% to 26.2%. Our test with c κ = ∞ has no power which shows that the truncation step for the critical value in (3) is necessary. The KPSS test has impressive power due to the fact that it has a tendency to overestimate the rejection probability under H 0 , leading to inflated power. Nevertheless our test displays greater power in most cases in comparison to the KPSS test. The HR test has good power in Models 4 and 5 while it performs poorly in Model 6. The power one property of our test is observable in the simulation as the rejection probability tends to 1 when N increases. Comparing the results of Models 5 and 7, we found that our tests show off the power one property more distinctly as our test statistic has more discriminate power between I(2) and I(0) than that between I(1) and I(0).