2. THE VR STATISTIC'S POWER

Given n + 1 observations x₀,x₁,…,x_n of a time series, the VR statistic with a positive integer k(< n) as differencing period is defined as

In the usual fixed k asymptotic treatment, under the null hypothesis that the {x_t} follow a random walk with possible drift, given by

where μ is a real number and {ε_t} is a sequence of zero mean independent random variables, it is possible to show (see, e.g., Lo and MacKinlay, 1988) that

where σ_k² is some simple function of k. This result extends to the case where the {ε_t} are a martingale difference series with conditional heteroskedasticity (see, e.g., Campbell, Lo, and MacKinlay, 1997), though the variance σ_k² has to be adjusted a little. The fixed k asymptotic distribution is well known to fare badly when k is large relative to n (see Lo and MacKinlay, 1989).

As an alternative to the fixed k asymptotics, Richardson and Stock (1989) develop an alternative approach under k → ∞, n → ∞, and k/n → δ > 0 asymptotics. Under the assumption that {x_t} follows (1), where {ε_t} is a conditionally heteroskedastic martingale difference series, they show that

where W(λ) is standard Brownian motion. Thus, whereas the VR statistic converges to one under the fixed k treatment, here it has a nondegenerate limiting distribution. Clearly this distribution is nonnormal and instead can be characterized by a functional of Brownian motions. Through Monte Carlo simulations, Richardson and Stock (1989) demonstrate that this alternate theory provides a much better approximation to the small sample distribution of the VR statistic.

Of course, the problems of generating consistent tests when the statistic converges to a nondegenerate distribution are well known. In fact, as we show in Theorem 1, which follows, under a wide class of mean-reverting models, the VR statistic converges to a positive multiple of the limiting distribution in (2) when k/n → δ > 0. This implies that the resulting test based on the VR statistic will be inconsistent for such alternatives when k is a constant fraction of the sample size n. In practical terms, this means that no matter how large the sample size, the probability of detecting such an alternative is bounded below some number strictly less than one, which is clearly an undesirable property.

THEOREM 1. Let {e_t} and {u_t} be two series of zero mean independent processes with finite variance σ_e² and σ_u², respectively, which are independent of each other. Define the process {y_t} by y_t = λy_t−1 + u_t where |λ| < 1. Let r_t = μ + r_t−1 + e_t and

If k → ∞, n → ∞, and k/n → δ > 0, then

where W(λ) is standard Brownian motion.

Proof of Theorem 1. Let

. Thus,

By (A.6) of Richardson and Stock (1989, p. 346) for α > δ,

where W(α) is standard Brownian motion. Because {y_t} is a stationary process, it is trivially true that

Hence, by (5)–(7).

Furthermore, by the weak law of large numbers,

The same argument as that employed in Richardson and Stock (1989, p. 329) then yields the required result. █

Theorem 1 shows that the test based on the VR statistic when k is a fraction of the sample size will result in an inconsistent test for mean-reverting alternatives of the form (3). Such mean-reverting alternatives are of the kind often imposed in finance applications (see, e.g., Poterba and Summers, 1988; Fama and French, 1988). The variable x_t contains a permanent (i.e., random walk) component r_t and a temporary (i.e., stationary) component y_t. It is also obvious from (4) that when the variance of the innovations of the temporary component, σ_u², is small relative to σ_e² the test based on the VR statistic will have power that is barely larger than the nominal size of the test in detecting the alternative even in infinitely large sample sizes.

A limiting result for VR similar to (4) in Theorem 1 can actually be obtained under much more general conditions on {r_t} and {y_t}. One can allow both the first difference series {r_t − r_t−1} and {y_t} to be arbitrary stationary processes instead of the independent process {e_t} and the autoregressive process of order one, respectively, that was imposed in Theorem 1. Thus, the VR statistic can be shown to be inconsistent for a very wide range of alternatives that might be of interest in detecting mean reversion if k is a fraction of the sample size.

3. POWER IMPLICATIONS

The theorem in Section 2 shows that the VR statistic is inconsistent against meaningful alternatives as k → ∞, n → ∞, and k/n → δ > 0. Moreover, equation (4) provides the form for the asymptotic power function. In this section, we explore the implications of these results for the VR statistic's power.

As a first step, Table 1 reports the small sample power of the VR statistic for the relevant case in which its power is consistent. In particular, we consider the alternative model described in Theorem 1 under the assumption that σ_e² = 0, that is, there is no permanent component to x_t. As seen from the table, as n increases, the power approaches 100%. Interestingly, this increase is greater for smaller δ, which is consistent with Theorem 1's main implication that, for consistency, k/n → 0. Intuitively, in small samples, even consistent alternatives are hard to distinguish, and the large k problem is attenuated.

Small sample power of the VR statistic for the consistent power case (in %)

Table 2 reports the relevant power results for a variety of alternatives described by Theorem 1. Several observations are in order. First, the table reports small sample power for n ranging from 60 through 2,880 observations and also the asymptotic power level. As Theorem 1 shows, and Table 2 demonstrates, the small sample power of the VR statistic is clearly limited, supporting its inconsistency as a test statistic. Second, even though there is some increase in power as n increases, the gains are very limited for alternative specifications with substantial permanent components. For example, when the share of the variance captured by the mean-reverting component, γ, is only 0.25 and 0.5, respectively, the power is 14% and 38% as n → ∞. Third, these power gains are worse for larger δ, which is consistent with the requirement that k/n → 0. As an illustration, for γ equal to 0.25, 0.5, and 0.75, respectively, the power is 14%, 38%, and 86% for δ = 1/6 versus only 11%, 27%, and 62% for δ = 1/3. To get a more complete picture of this point, Figure 1 graphs the three-dimensional relation between power, γ, and δ. As borne out by the figure, lower δ's increase power, with varying impact depending on the permanent component's importance (i.e., level of γ). For a fixed value of γ, the drop-off in the power, as δ increases, is quite dramatic. Finally, note that the mean-reversion parameter, λ, has no effect upon the asymptotic distribution as the VR statistic is consistent against pure mean-reversion alternatives (e.g., see Table 1). Table 2, however, shows that λ is important in small samples.

Relevant power results for alternatives described by Theorem 1 (in %)

Asymptotic power versus γ and δ. Asymptotic power of the VR statistic is shown for a test of the random walk null against a specific alternative model at the 5% level of significance for a range of δ's and γ's, where δ represents the ratio of the variance horizon to the data length and γ represents the proportion of the variance due to the mean-reverting component. In particular, the alternative model is xt = rt + yt where rt = rt−1 + et, yt = λyt−1 + ut, and γ = 2σu2/((1 + λ)σe2 + 2σu2).

On the one hand, these results are fairly discouraging for developing tests against slow mean-reverting alternatives. Given a fixed δ, the VR statistic will have limited success. Within the Richardson and Stock (1989) framework, however, Perron and Vodounou (2001) do manage to characterize the maximal possible power by taking a continuous-time limit given a fixed data span, n. On the other hand, and in contrast to the Perron and Vodounou (2001) result, Theorem 1 suggests an alternative approach in which k is allowed to increase with n but at a slower rate. This guarantees consistency against a wide range of alternatives. Practically, this means that, for power purposes, k cannot be too large in small samples.

The practical issue of how to then adjust the size of the statistics for a given k in small samples remains an important question. Nevertheless, Chen and Deo (2001) provide a possible solution to this issue. They show that one way to alleviate the small sample skewness of linear combinations of positive random variables is to use power transformations. Because VR statistics can be characterized as such a linear combination, their approach allows the researcher to apply the variance ratio in small samples across a wide array of unknown distributions. The transformed VR statistic will maintain its consistent power, yet it approximates the small sample distribution much more closely. This particular problem deserves future analysis.