2.1. Asymptotic Power Functions

To justify the claim that the

dominate the M-tests, it will prove convenient to develop asymptotic representations for the various statistics. Consider some of the g functions used for robust regression problems:

where the constant k is chosen by the researcher. Because g may not be everywhere differentiable, we cannot approximate the log-likelihood function with Taylor series expansions. Instead of pointwise differentiability, the proofs make use of “stochastic differentiability,” an idea described in Pollard (1985). Application of the idea requires imposition of smoothness conditions on the error density to make up for the lack of smoothness in the objective function.

Assumption 1. (Smoothness of the Error Density). The errors {ε_t}_t=1^T are i.i.d. mean zero with E|ε₁|^2+δ < H for some δ > 0. The term ε₁ has a density function f (z) that is bounded and uniformly continuous.

The g function may have finitely many points of nondifferentiability.

Assumption 2. (Objective Function). g(x) is convex and strictly increasing in |x|, and g(x) is everywhere twice differentiable except for x in P, where P contains the D points p₁,…, p_D. There exists some finite positive H so that |g′′(x)| < H for x not in P. There exists some finite positive h satisfying P ∈ [−h − δ, h + δ] for some δ > 0, so that for all x and y in [−h, h] we have |g′(x)| < H and |g(x) − g(y)| ≤ H|x − y|.

I assume that g is convex because it simplifies the proofs. Assuming convexity allows me to extend several pointwise convergence results to apply uniformly over the parameter space. Convexity also greatly simplifies the demonstration of the rate of convergence of the estimators. This extensive use of convexity is due to results in Pollard (1991) and Hjort and Pollard (1993).

For nondifferentiable g, it is not possible to define an approximate likelihood in terms of the derivatives g′ and g′′. We replace g′ with the derivative-like function ψ.

DEFINITION 1. ψ(x) is equal to g′(x) if g is differentiable at x and ψ(x) = 0 otherwise.

If g is everywhere differentiable then g′ = ψ. For LAD regression ψ(x) = sign(x), and for Huber's function ψ(x) = x1(|x| < k) + k1(|x| ≥ k).

In standard (non–unit root) problems, the second derivative g′′ enters the asymptotic representation though its expectation Eg′′(ε₁). We replace Eg′′(ε₁) with the parameter

. When g is everywhere twice differentiable, ω = Eg′′(ε_t). For LAD regression ω = 2f (0), and for Huber's function ω = Pr[|x| < k] .

In large samples the power functions admit representations in terms of functionals of Brownian motion. Define W(·) to be standard Brownian motion and define W_c(·) to be the Ornstein–Uhlenbeck process

with initial condition W₀(0) = 0. The asymptotic representations make use of the parameters σ_ε² = Var(ε_t), ρ = Corr(ε_t, ψ(ε_t)), and σ_ψ² = Var ψ(ε_t) and also of the stochastic process

where

is standard Brownian motion, independent of W. The following theorem is proved in Appendix A.

THEOREM 1. If Eψ(ε_t) = 0, and if Assumptions 1 and 2 hold, then

Rothenberg and Stock (1997) and Xiao (2001) derive similar representations without assuming convexity of g but do not allow for nondifferentiable functions.

The large sample power function for the Neyman–Pearson test is

Power functions for the other tests may be obtained similarly.

Figure 1 plots envelope power functions and asymptotic power for a variety of tests. The curves for LAD errors (from the double exponential distribution) are given and so are standard normal errors and Huber errors.³

Asymptotic power curves for unit root tests in the model with no time trend (xt = (1,0)). The curves are drawn under the assumption of correct specification, so the g function used to form the test statistics is equal to f, the negative log-density of the errors. (The simulations that appear in this paper were performed by computing stochastic integrals as the realizations of normalized sums of 500 successive draws from a discrete time Gaussian AR(1) process with autoregressive parameter 1 − c/T. There are 100,000 Monte Carlo replications.)

The power curves for nonnormal errors are all substantially higher than the curve for normal errors. The most powerful test for Gaussian errors achieves 50% power at c close to −7.0, and the most powerful test for double exponential errors (corresponding to LAD estimation under correct specification) achieves 50% power at c close to −3.75.

Figure 1 also provides power curves for the tests based on the constrained MLE

and the M-estimator

. The

-test is asymptotically admissible. Test power is tangent to the power envelope when envelope power is large. The

-test is not asymptotically admissible and is dominated by the

-test. The power curve for

touches the envelope function only under the null (c = 0) and for alternatives so far from zero that any sensible test would have power equal to 1.

The M-estimator t-test is not asymptotically admissible, whereas the constrained

-test is admissible. Figure 1 shows that the constrained t-test achieves tangency to the power envelope function at power close to 50%. The figure also shows that the constrained

-test is not admissible. As Rothenberg and Stock (1997) show, straightforward manipulations of the asymptotic representations demonstrate that rejecting for large values of

is asymptotically equivalent to rejecting for large

. Because the tests based on

are one-sided and two-sided tests of the same one-sided hypothesis, it is not surprising that the t-test dominates the test based on

.

We have obtained power improvements by imposing the constraint that the intercept estimate

is zero. We can obtain identical results by replacing the requirement that

with the requirement that the estimator

is stochastically bounded. Consider two more estimators.

(1)

, where Θ is a compact set. It is common to assume a bounded parameter space, and this estimator imposes that assumption. The estimator

is obviously stochastically bounded, and the term sup_β1∈Θ|β₁ c/T | disappears from the likelihood function in large samples. Thus

has the same limiting distribution as

.

(2)

. These are the MLES when we assume the initial condition u₀ = 0. In an earlier draft of this paper it was shown that

is stochastically bounded. Thus

fast enough so that

has the same distribution as

.

2.2. Failure of Robustness to Error Misspecification

These power improvements occur as long as g, the estimating function, is equal to f, the true negative log-density of the errors. When g ≠ f,

may behave poorly. Consider the classical regression model y = α₀ + α₁ x + ε. If the errors come from the double exponential distribution, ε has zero median and the maximum likelihood estimates are the LAD estimates

. If the true value of α₀ is zero, then under correct specification we can get a better estimator for α₁ by removing α₀ from the objective function:

. Now suppose that α₀ = 0 and that ε comes from an incorrectly specified, asymmetric error distribution with zero mean but nonzero median. For example take ε = Z² − 1 where Z is standard normal. It is well known that in this case

has a limiting distribution. It is also well known that the distribution of

is not stochastically bounded, even if α₀ = 0. Thus the constant α₀ “recenters” the incorrectly specified errors. In the classical setting we include the constant to protect ourselves from errors with nonzero median.

The same thing happens in the unit root problem. Herce (1996) shows that if the intercept β₁ is zero and the errors have zero median, then

both have limiting distributions. When ε_t has a nonzero median

has a limiting distribution whereas

blows up. This can be seen in Figures 2a–d. Each histogram depicts 5,000 Monte Carlo realizations of

, estimated from simulated data sets of 500 observations from the model with c = 0 (so the null is true), β₁ = 0 and initial condition u₀ = 0. Figures 2a and b show that when the errors come from the zero median Student's t-distribution with four degrees of freedom, both estimators have limiting distributions. Figures 2c and d show that when the errors do not have zero medians (ε_t = Z_t² − 1 where Z_t are i.i.d. standard normal) the distribution of

blows up. So for any fixed critical value q, the probability of rejecting a true null hypothesis converges to

.

Mathematically this can be understood as failure of an identification condition. Consider the classical regression model with no intercept: y = α₁ x + ε. If g is differentiable, the estimator

will solve the first-order condition

. Under the usual assumptions this will deliver a consistent estimator of α₁ if the identification condition Eg′(y − α₁ x)x = 0 holds. Because y − α₁ x = ε and x is not random, this condition is equivalent to Eg′(ε) = 0. Thus the identification condition is that ε has a distribution with the property that Eg′(ε) = 0.

In our unit root problem

solves

, which suggests the identification condition

For the LAD problem ψ(ε_t) = sign(ε_t) and the identification condition is E sign(ε_t) = 0. The condition holds for LAD only if the errors have zero medians.

What assumptions do we need to ensure that Eψ(ε_t) = 0?

(1) Expression (2) demonstrates that Eψ(ε_t) = 0 under correct specification (so f = g).

(2) Eψ(ε_t) equals zero for the Gaussian likelihood, no matter what the distribution of the errors. The Gaussian likelihood has g(x) = x²/2 and ψ(x) = x, so the assumption Eε₁ = 0 insures that Eψ(ε₁) = 0.

(3) Eψ(ε_t) equals zero when f ≠ g and both functions are symmetric around zero.

(4) When f ≠ g and f is not symmetric, Eψ(ε_t) can be different from zero. For example, for LAD estimation applied to the errors ε_t = Z_t² − 1, Eψ(ε_t) = E sign(Z_t² − 1) ≈ −.3656.

It turns out that the optimal tests are not robust to unknown, asymmetric error distributions. To get the tests to work, we either need to assume that we know the distribution of ε, or we need to assume that ε comes from a symmetric distribution. Thus the optimal tests are not robust to unknown asymmetric error distributions. The Gaussian tests of Elliott, Rothenberg and Stock (1996) are the one exception—those tests are valid under fairly general forms of misspecification, including asymmetric errors.

Figure 2 depicts an example where the optimal tests reject a true null hypothesis too often. This is generally a problem with asymmetric errors.

Histograms of 5,000 Monte Carlo simulations of (on the left) and (on the right) estimated from 500 observations from the model with no trend or intercept. The true value of c is 0. (a) and (b) Here the errors come from the Student's t4 distribution. Because the median is zero is more efficient and has a smaller spread around 0. (c) and (d) Here each error is εt = Zt2 − 1, where Zt are i.i.d. standard normal. Because the median is not zero, has a limiting distribution and blows up.

PROPOSITION 1. Suppose that g is three times differentiable with bounded third derivatives and suppose that the errors satisfy assumption 1. If Eψ(ε_t) ≠ 0 then in large samples tests based on

and the Neyman–Pearson statistic all reject a true null hypothesis with probability approaching .5, no matter what the nominal size of the test. The M-tests based on

have the same limiting representations as in Theorem 1.

The M-tests have the same limiting representations as in Theorem 1, with the nuisance parameters σ_ψ², ρ, and ω replaced by Var[ψ(ε_t − η)], Corr[ε_t, ψ(ε_t − η)] , and

, where η denotes the parameter that solves Eψ(ε_t − η) = 0.

M-tests are robust to asymmetric error densities. Let η denote the parameter that solves the equation Eψ(ε_t − η) = 0. So for LAD estimation, ψ(ε_t − η) = sign(ε_t − η), and η is the median of the errors. The M-estimator objective function can be rewritten

with

. These recentered errors satisfy

. Thompson (2004) shows that if Eψ(ε_t) ≠ 0 then

in probability and

has the same limiting distribution as in Theorem 1, with a slight redefinition of the nuisance parameters (see note 5). Thus estimation of the free parameter a causes a power loss under correct specification but ensures robustness against incorrect specification.

We can avoid these centering problems by assuming that Eψ(ε_t) = 0. For example for LAD estimation we could assume that the median of ε is zero and leave the mean unspecified. The zero mean assumption is essential for nearly integrated models because it identifies the trend. If the mean is not zero, then the trends behave very differently under the unit root null than for stationary alternatives. Because u_t follows the process Δu_t = γu_t−1 + ε_t, we have

with

. If γ = 0, then y_t has both a unit root and a nonstochastic trend. If γ < 0, then y_t is stationary with the long-run mean

. Thus the zero mean assumption is essential if we wish to test for mean reversion around an intercept. Once we assume zero means, adding additional centering assumptions such as zero medians takes us closer to assuming symmetric errors.

Sketch of proof of Proposition 1.

When Eψ(ε_t) ≠ 0, the Neyman–Pearson statistic is not stochastically bounded. To understand why, notice that the statistic T⁻¹[sum ]ψ(ε_t)u_t−1 appearing in the approximation to the Neyman–Pearson statistic is not stochastically bounded. Lemma 3.1 of Phillips (1988) implies that we must divide by T^1/2 to get a limiting distribution:

where σ²(c) = (σ_εEψ(ε_t)/c)²[1 + (e^2c − 1)/(2c) − 2(e^c − 1)/c] . If g is three times differentiable with bounded third derivatives, the Neyman–Pearson statistic must also be divided by T^1/2:

Suppose we form the Neyman–Pearson statistic and use the critical value q(c) constructed under the assumption that Eψ(ε_t) = 0. If Eψ(ε_t) ≠ 0 then under the null hypothesis the probability of rejecting is

where σ²(0) = lim_c→0 σ²(c) = (σ_εEψ(ε_t))²/3. In large samples the Neyman–Pearson test rejects a true null hypothesis 50% of the time, no matter what the nominal size of the test.

A proof by contradiction shows that error misspecification may also cause the constrained MLE to be stochastically unbounded. Suppose that c = β₁ = 0 and g is three times differentiable with bounded third derivatives. If

is stochastically bounded, the minimized objective function admits the approximation

If

is stochastically bounded, then in large samples

must converge to the minimizer of

. The minimizer [T⁻²[sum ]g′′(ε_t)u_t−1²]⁻¹T⁻¹[sum ]ψ(ε_t)u_t−1 is not stochastically bounded, and we have our contradiction. █

2.3. Some Monte Carlo Results

A Monte Carlo study demonstrates the size distortions that occur under incorrect specification. Table 1 presents rejection frequencies for 10 tests under various assumptions about the true data generating process. The abbreviations in the table are as follows.

(1) ERS—the Dickey–Fuller generalized least squares (DF-GLS) test of Elliott et al. (1996). This test efficiently handles the intercept for Gaussian errors but does not use the information in thick-tailed error distributions.

(2) Adap—the adaptive test of Shin and So (1999). This test adapts to the error distribution but does not efficiently handle the intercept.

(3)

test, LAD—The Thompson (2001) version of the test based on the LAD M-estimator.

The test rejects for small values of

, where

. Thompson (2001) shows that

.

and in some cases has more accurate size. This test does not efficiently handle the intercept.

(4)

test, t3—The Thompson (2001) version of the test based on the Student's t3 M-estimator.

(5) Trend-optimal LAD NP,

—these tests are optimal for a double exponential likelihood. NP denotes the Neyman–Pearson test statistic evaluated at c = −3 and β₁ = 0: L(−3,0) − L(0,0). The three tests efficiently handle the trend and will be more powerful than the DF-GLS test for many thick-tailed error distributions. When the errors come from an asymmetric distribution the tests will overreject a true null hypothesis.

(6) Trend-optimal t3 NP,

—the optimal tests for a Student's t3 likelihood with β₁ = 0. These tests are not robust to asymmetric errors.

Rejection frequencies for selected tests in the model with no time trend

The bold numbers in Table 1 illustrate the size problems with asymmetric errors.

Critical values for first four tests are obtained using the methods described in Elliott et al. (1996), Shin and So (1999), and Thompson (2001). In all cases the errors are i.i.d. and no correction is made for serial correlation. Critical values for the trend-optimal tests are obtained by simulating from the asymptotic distributions in Theorem 1. The representations depend on the nuisance parameters σ_ε, σ_ψ, ρ, and ω. For all four tests the nuisance parameters are estimated using the formulas

, where

is a residual and ε and ψ are sample averages. For the t3 estimator ω = Eψ′(ε_t) is estimated by

. For the LAD test ω = 2f (η), which is estimated by the usual kernel estimator of the density of

evaluated at zero:

, where φ is the density function of a standard normal variable and h is the bandwidth

. For the Neyman–Pearson test c is not estimated and there is no residual, so we use the nuisance parameters computed for the

estimator.

-tests have accurate sizes for the asymmetric distributions.

The results demonstrate that the ERS test is a viable alternative to the robust tests even when the errors are not Gaussian. The ERS test has accurate size and good power for all four error distributions. Somewhat surprisingly, the ERS test even has accurate size for the infinite variance Cauchy distribution. The adaptive test works well for the asymmetric log normal and chi-squared distributions but has poor power for the Cauchy errors. The

-tests have accurate size for all four distributions. For the Cauchy errors the

-tests are very powerful.

3.1. Asymptotic Power Functions

To derive the power functions of the various test statistics, it will prove useful to provide a limiting representation for the objective function. By Lemma 3.1 of Phillips (1988),

where P_{c, φ}(r) = σ_ε(c − c)W_c(r) + φ(1 − cr) is a stochastic process. The following theorem is proved in Appendix A.

THEOREM 3. If Eψ(ε_t) = 0 and Assumptions 1 and 2 hold, then

Appendix B provides a closed-form expression for

in terms of stochastic integrals. Because

is a nonlinear function of c and φ, the asymptotic representation for

does not admit an analytic solution in terms of random integrals. Appendix B provides a method for simulating from the asymptotic distribution of

.

Figure 3 plots the limiting power functions for the various tests in the model with a time trend. The curves are lower than the corresponding power envelopes for the model with an intercept only. Power rises as the tails of the error distribution become thicker; for Gaussian errors 50% power is achieved at −12.5, and for double exponential errors 50% power is achieved at about −6.0.

Asymptotic power curves for unit root tests in the model with a time trend (xt = (1, t)). The curves are drawn under the assumption of correct specification, so the g function used to form the test statistics is equal to the negative log-density of the errors f. (The simulations that appear in this paper were performed by computing stochastic integrals as the realizations of normalized sums of 500 successive draws from a discrete time Gaussian AR(1) process with autoregressive parameter 1 − c/T. There are 100,000 Monte Carlo replications.)

Figure 3 shows that the constrained

-tests are asymptotically admissible. Careful examination of the figure leads to the conclusion that the constrained

-statistic is not asymptotically admissible. Neither M-test is asymptotically admissible. The

-test is point optimal when power is high, and the

-test is point optimal when power is close to one-half. Rothenberg (1984) describes similar results; he notes that in standard (non–unit root) models with no nuisance parameters, second-order asymptotic theory predicts that estimator-based tests are optimal when power is high and LR tests are optimal when power is close to 50%.

3.2. Failure of Robustness to Error Misspecification

The analysis in the previous section was carried out under the assumption of correct specification, so g = f where g is the function used to form the likelihood function and e^−f is the density of ε_t. As discussed in Section 2.2, correct specification insures that the centering condition Eψ(ε_t) = 0 holds. If g ≠ f then Eψ(ε_t) may not equal zero. When Eψ(ε_t) ≠ 0 the test statistics can behave badly.

PROPOSITION 2. Suppose that g is three times differentiable with bounded third derivatives and suppose that the errors satisfy Assumption 1. If Eψ(ε_t) ≠ 0 then

(1) Under the local alternative

. Here

is O_p(1) but does not have the distribution given in Theorem 3. Therefore power against any local alternative approaches zero.

(2) Under the fixed alternative

. Power against any fixed alternative approaches 1.

(3) If g′′(x) > B > 0 for all x, then under both local and fixed alternatives the best invariant test statistic

, where K is a compact set. Power against any fixed or local alternative approaches zero.

(4) The M-tests based on

have the same limiting distributions as in Theorem 3.

⁹

The M-tests have the same limiting representations as in Theorem 3, with the nuisance parameters redefined as in note 5.

Because the critical values for

are always negative, tests that reject for small

will have size converging to zero and power against any local alternative also converging to zero. The

-test has power equal to size for any local alternative, and its actual size will not match its nominal size, even in large samples. Because the critical values for the best invariant test are also negative, the best invariant test has size and power approaching zero against both fixed and local alternatives. The M-tests are robust to asymmetric errors. No matter what the error distribution, the M-tests have power against local alternatives and are consistent against fixed alternatives.

Thus none of the trend-optimal tests have power against local alternatives, but all except the best invariant test have power approaching 1 against fixed alternatives. In a large sample with a local alternative, the

-tests will be dominated by the robust M-tests. Furthermore, only the M-tests are useful for forming confidence intervals for the local parameter c, because that requires inverting a sequence of tests, each with power against local alternatives (for the Gaussian case, see Elliott and Stock, 2001).

On the other hand, the

have many desirable properties even when ψ(ε_t) ≠ 0: in large samples they reject a true null hypothesis with probability less than any desired size, and they reject a fixed alternative γ < 0 with probability approaching 1. Although the

-test may get the size wrong, because the statistic is stochastically bounded both under the null and alternatives, the size distortions may be small. The magnitude of those distortions is evaluated by Monte Carlo in Section 3.3.

To understand the proposition, recall that the M-estimators

minimize the objective function

where

denotes the parameter that solves Eψ(ε_t − η) = 0. If the condition Eψ(ε_t) = 0 fails to hold then

, and the “recentered” errors

satisfy

. Thus

has the same limiting distribution as in Theorem 3, with a slight redefinition of the nuisance parameters (see note 5). This result is shown by Thompson (2004), and it implies that statement 4 of the proposition will hold. Because there is no free “recentering” parameter in the objective function for

, the parameter on the time trend accomplishes the recentering. If

, no matter what the local alternative c.

Sketch of proof of Proposition 2.

To establish statement 1 of the proposition, consider the model with the local alternative γ = c/T. Define φ = (φ₁, φ₂)′ = (T^1/2c,T^1/2(b₂ − β₂ − η))′ and m_t = T^−1/2(−η(t − 1)/T,1)′. The likelihood function is

A Taylor series expansion, combined with the usual asymptotic arguments (see Phillips, 1988, Lemma 3.1), implies that

where

for any compact set K. By the same argument used to prove Lemma 2, φ is stochastically bounded. Therefore, by the argmax continuous mapping theorem of Wellner (1996, p. 286),

converges in probability to the minimizer of the approximating quadratic function, so

Here [sum ] m_t m_t′ converges in probability to a nonrandom matrix, and

converges to a vector of mean zero Gaussian random variables. In large samples

has a mean zero Gaussian distribution, which implies that

.

The distribution of

is obtained by substituting

back into the likelihood function. Under the local alternative γ = c/T, we obtain

This is an O_p(1) variable, but the limiting distribution differs from the one in Theorem 1.

Statement 2 of the proposition says that the

-tests are consistent against any fixed alternative γ < 0. The likelihood function evaluated at (γ, b) may be written

Because u_t−1 is stationary under the fixed alternative, is it straightforward to show using Taylor series–based arguments that

is consistent for γ. Therefore

, and an argument based on a Taylor series expansion demonstrates that

. The proofs are omitted to save space.

To show statement 3 of the proposition, define the parameter η_γ that satisfies Eψ(ε_t + γu_t−1 − η_γ) = 0. For fixed γ < 0, ε_t + γu_t−1 is a stationary random variable, and the expectation exists. By a Taylor series expansion,

where z_t* = −cu_t−1 /T − β₁ c/T + η_γ − (b₂ − β₂)(1 − c((t − 1))/T) and |ε_t* + γu_t−1 − η_γ| ≤ |z_t*|. If b₂ ∈ K then many of the terms are asymptotically negligible. We get the approximation

If c = 0 this expression is minimized at b₂ = β₂ + η_γ, so min_b∈K T⁻¹L(0,0, b) = T⁻¹[sum ]g(ε_t + γu_t−1 − η_γ) + o_p(1). If c ≠ 0, then because g′′(x) ≥ B we have

Therefore T⁻¹{min_b∈K L(c,0, b₂) − min_b∈K L(c,0, b₂)} ≥ (24)⁻²Bη_γ² c² + o_p(1), and the best invariant test converges to +∞ under any fixed alternative. Using the same arguments it is also possible to show that the best invariant test converges to +∞ under any local value γ = c/T (including the null c = 0). The proof is omitted to save space. █

3.3. Some Monte Carlo Results

Table 2 presents rejection frequencies for various tests in the model with a time trend. The tests are the trend versions of the tests that appeared in Table 1, except that the

-test appears in Table 2 in place of the

-test. This substitution was made because the

-test is not asymptotically admissible in the model with a time trend.

As was the case for Table 1, the

tests are the Thompson (2001) versions of the tests. In the model with a time trend the test rejects for small values of

, where r_t is the residual from a least squares regression of y_t−1 on (1, t) and

with

. Thompson (2001) shows that

.

Rejection frequencies for selected tests in the model with a time trend

The power losses from using the trend-optimal

-tests are small for samples of 100 observations but get larger for samples of 1,000. For the asymmetric log normal and chi-squared error distributions, power against the local alternative c = −10 declines as sample size grows. Power against the fixed alternative γ = −0.1 increases with sample size. This can be seen by comparing the results for the samples with (T, c) = (100,−10) against the samples with (T, c) = (1,000,−100). In each case γ = −0.1. These results are consistent with Proposition 2, which states that power against c = −10 converges to zero as the sample grows and power against γ = −0.1 converges to 1.

In the samples of 1,000 observations, asymmetric errors cause the NP test to have low power against both the fixed and local alternatives. This is consistent with Proposition 2, which predicts that power against both kinds of alternatives converges to zero as the sample size grows.

No test dominates the others. The ERS test performs poorly for the Cauchy, log normal, and chi-squared errors. The adaptive test does well for the asymmetric distributions but has very low power with Cauchy errors. The

-tests perform well for all the error distributions and sample sizes but are generally dominated by the trend-optimal

-tests for samples of 100.