2. ON AR(1) MODELS

The model favored by Bhargava (1986) exploits an elegant manner of adding a constant to an AR(1) model:

Here ε_t ∼ NID(0,σ²) (normally independently distributed), σ² > 0, and x₀ = 0. The Cox–Reid (1993) notation is used, so the nuisance parameters or the constant and the innovation variance are denoted by φ = [γ σ²]′ and the parameter of interest or the autoregressive (AR) coefficient by ψ. The focus is on the unit-root value ψ = 1, but values |ψ| < 1 are considered, too. This unobserved components model is labeled AR^B(1). An intriguing feature of the model is that the unconditional mean of the process is γ, so the interpretation of the constant does not depend on whether the popular null hypothesis of a unit root is valid or not. This is a major difference from the conventional model described next (cf. (3)).

Handling of the first observation differentiates the AR^B(1) model from that of the more usual formulation:

where y₀ is a fixed initial value. The complication here is that the constant (α) becomes a drift under ψ = 1, making the customary Dickey–Fuller (1979) unit-root tests, based on (2), invalid. To induce a nondrifted random walk under ψ = 1, the extra assumption of α = 0 under ψ = 1 should be made. Model (2) will carry the tag AR^μ(1).

The first model corresponds to the process

instead of

which results from the latter model. The initial value x₀ occurs in (1) and (3) to clarify the potential role of x₀ and for the invariance and similarity results in Appendix C. The MLE for ψ should in general take x₀ into account if it were allowed to be nonzero. Empirically x₀ would be likely to be unobserved and difficult to estimate (Schmidt and Phillips, 1992). Hence x₀ = 0 is assumed, except that the invariance and similarity results entail that the unit-root fractiles in Section 7 apply also when x₀ ≠ 0.

The first process is assumed to start from the neighborhood of its long-run mean (γ, assuming that σ² is not large relative to γ). The assumption is evident if the process is nonstationary (ψ = 1, in which case the process shrinks to a random walk that has started from γ). A similar assumption is often also made when a time series has been in operation for some time and the process is known to be stable (|ψ| < 1). The model gains credibility when the process truly originates at t = 1 as opposed to this being the moment to begin recording of the time series.

Models (1) and (2) yield the same process in two circumstances:

The AR^B(1) process lies in a sense between the AR^μ(1) and the stationary AR(1) process with constant where the first observation already follows the stationary distribution. A feature shared with the former is the nonconstancy of the variance of y_t and the constancy of the mean with the latter. If γ = α/(1 − ψ) and |ψ| < 1, then the differences between the AR^B(1) and AR^μ(1) processes, if any, diminish for large T, and the processes unify with the stationary process. The AR^B(1) process shares perhaps more characteristics with the AR^μ(1) than with the stationary process: the latter cannot accommodate ψ = 1, but the AR^μ(1) and AR^B(1) processes can. The variance of y_t (σ²(1 − ψ^2t)/(1 − ψ²) or σ²t if ψ = 1) is the same under the AR^μ(1) and AR^B(1) processes. If x₀ ≠ 0 were allowed for in (1), the mean of y_t would exhibit start-up effects and the common feature of the AR^B(1) and the stationary process would disappear. As a whole the AR^B(1) process can be comprehended as an AR(1) process with constant γ(1 − ψ) and start-up effects altering the variance. Furthermore, if γ = 0 then the process shrinks to the AR(1) without a constant

or (2) with α = y₀ = 0. This model, but allowing for y₀ ≠ 0, will be denoted by AR⁰(1). However, under this model the MLE of ψ will be calculated conditional on y₁ for reasons explained next.

Even though the conditions for the processes (1), (2), and (5) to agree are met, there are aspects to take account of in choosing the model to use. If the time series is due to the AR^B(1) process (with a potentially nonzero constant) then the AR⁰(1) or AR^μ(1) model could be applied, too. With them the analysis would have to be conducted conditional on the first observation, so one would have one observation less for estimation. If the data follow an AR⁰(1) or AR^μ(1) process, the AR^B(1) model would be valid (for y₁,…,y_T leaving y₀ aside) with an equal number of observations available for estimation. However, if y₀ is known to be the long-run mean, so that the processes can match in general, then one can simply deduct it from the time series and apply the parsimonious AR⁰(1) model. Therefore later, when the models are compared by simulation, the data are generated by the AR^B(1) model and ψ is estimated conditional on y₁ when the AR^μ(1) or AR⁰(1) model is used.

The potential benefit from using the AR^B(1) model is not confined to avoiding losing an observation for conditioning. The AR⁰(1) and AR^B(1) models have much in common in terms of asymptotics as shown next.

3. PROFILE LIKELIHOOD ANALYSIS AND UNIT-ROOT ASYMPTOTICS FOR THE AR^B(1) MODEL

Pantula et al. (1994) refer to unpublished proofs for the asymptotic unit-root distributions of this section. Proofs introducing formulas needed for the APL and demonstrating the convenience of the PL are given subsequently.

The first observation is treated separately in the log-likelihood under the AR^B(1) model:

The PL has the standard appearance at the outset:

but

and

The formula for

reveals the particular handling of the first observation. In the unit-root case, it alone carries information on γ. Indeed,

equals y₁ when evaluated at ψ = 1, so the MLE of

is a kind of dummy that is in general inconsistent for γ under ψ = 1 (consistency of the MLE of ψ—denoted

for this model—is shown subsequently). The model has some resemblance to the Neyman–Scott (1948) incidental parameters model, to which APL has been successfully applied (Cox and Reid, 1993): in both models a sole observation carries all of the information on a nuisance parameter, yielding an inconsistent MLE. By contrast, the analogous estimate for α equals (y_T − y₀)/T, or a drift is sought under model (2) and ψ = 1.

The likelihood equation for ψ,

is equivalent to a fifth-order polynomial in ψ, so analytical solutions do not necessarily exist. The formula for

is given in Appendix A. The equation could still be solved or the PL maximized numerically to find

. The equation has at least one real root because the order of the polynomial is odd.

The MLE for a known γ is

The PL is helpful in proving that

tends in probability to y₁ or is O_p(1) when ψ = 1 (Appendix A). The O_p(1)ness of

and the well-known orders of magnitude and convergencies of expressions such as

suffice to prove that

under ψ = 1 and regardless of γ. Here ⇒ signifies weak convergence, B(r) is standard Wiener process with variance r, and ∫ denotes integral over (0,1).

Remarkably, the AR^B(1) model and the assumption ψ = 1 alone—with no restrictions on the constant—yield a (standardized) MLE with the same asymptotic distribution as

or the MLE of ψ for the AR⁰(1) model (e.g., Dickey and Fuller, 1979). Heuristically, the MLE picks ψ = 1 and γ(1 − ψ) = 0 or the random-walk process as T increases. This is in sharp contrast with the AR^μ(1) model where α remains a nuisance even asymptotically. The corresponding asymptotic distribution, under ψ = 1 and α = 0, is then (11) with the integrands replaced by B(r) − ∫B(s) ds (see (24)) (Phillips and Perron, 1988). This MLE is far more biased and scattered than the MLE(s) associated with the distribution in (11) (e.g., Pere, 2000). Moreover, if α is not zero then the asymptotics changes (Dickey, 1976, pp. 58–61; West, 1988). A conjecture is that the bias and variance of

remain smaller for a finite but reasonable T. Despite the common unit-root asymptotics of

, it does not follow that the two MLEs could not take very different values (Section 7).

Parallel results can be derived for the (signed square root of the) Wald statistic for the null hypothesis ψ = ψ₀:

(Seber and Wild, 1989, pp. 39–42). An analytic expression of

cannot be given, the analytic form of the MLE being unknown. However, the numerical value, for a specific sample, can be exposed straightforwardly after substituting the numerical value of the MLE into the formulas defining

. The details are in Appendix A.

Under the null of a unit root,

tends to a Dickey–Fuller distribution:

(Appendix A). This is the asymptotic distribution that arises also when the analogous Wald statistic is calculated from the AR⁰(1) model; again the AR^μ(1) model would yield a different distribution or (24) (Dickey and Fuller, 1979).

A comparable LR statistic is

The asymptotic unit-root distribution of this statistic, after standardization, is

(Appendix A). It can be shown by standard unit-root asymptotic results that this is the distribution of the corresponding LR statistic, T(LR_ml⁰ − 1), under the AR⁰(1) model also (proof omitted). The T(LR_ml⁰ − 1) statistic essentially equals the usual F = t² statistic, with obvious notation, for the null of a unit root. It gives intuition for the fact that the preceding asymptotic distribution of T(LR_ml^B − 1) is the square of that of

.

The agreement of the asymptotic unit-root distributions of the statistics extends to the case of a near unit root (Section 10). By simple arguments, the agreement holds also in the stable case (for the MLE and Wald statistics considered by the author).

4. ADJUSTED PROFILE LIKELIHOOD

In this section, ψ and φ are considered as a general parameter of interest and a vector of nuisance parameters, respectively. The APL of Cox and Reid (1993) is

Here

is the k × k observed information matrix for φ or −∂²l(ψ, φ)/∂φ∂φ′ (with (i,j)th element j_φiφj) evaluated at

, the coefficient c is defined by

its MLE being

or c evaluated at the MLEs of ψ and φ, I_φφ is the Fisher information matrix with (i,j)th component i_φiφj, and i_φψ is the Fisher information vector with (i,1)th element i_φiψ or i_φψ = E [−∂²l(ψ, φ)/∂φ∂ψ] .

If ψ and φ are orthogonal or i_φψ = 0 then the coefficient c, and its MLE

, collapse to zero and, helpfully, only the two first terms of the APL need to be calculated. This will turn out to be the case under the AR^B(1) model. Oddly enough, the coefficient can equal zero even if the parameters are not orthogonal, as in the AR^B(1) model with an unrestricted x₀ in Appendix C. In the AR^μ(1) model the parameters are nonorthogonal and the coefficient has to be calculated (Pere, 1997, 2000). Some intuition for the APL under orthogonality is given in the next section.

The value of ψ that maximizes (14) is the adjusted estimator (AE) of ψ or

. An adjusted Wald statistic for the null hypothesis ψ = 1 is defined as

(Pere, 1997, 2000). An analogous adjusted LR statistic is

where the power is set for convenience only for the application to follow. The statistics and the corresponding asymptotic distributions, under the AR^B(1) model with a unit root, are calculated in the next section.

5. ADJUSTED PROFILE LIKELIHOOD ANALYSIS AND UNIT-ROOT ASYMPTOTICS FOR THE AR^B(1) MODEL

Under the AR^B(1) model ψ is advantageously orthogonal with respect to γ and σ² and the latter to each other:

The following equality was already touched on when (8) was contemplated:

The number 1 in (19) arises from the first observation, whereas the term (T − 1)(1 − ψ)² ensues from the rest of the observations. Hence the first observation remains the most informative on γ as long as ψ ≠ 0. If ψ = 0 then all observations are equally informative.

The final information measure to note is an observed measure:

These calculations yield

and the APL:

The term −½ log[1 + (T − 1)(1 − ψ)²] pushes

to the right of

on the real axis if

, but it shrinks

downward if

. The term is nonpositive and penalizes large values of (1 − ψ)² or small values of ψ when the parameter space is restricted to (−1,1]. Absorbingly, the term diminishes as ψ approaches 1 and vanishes at 1. A divergent maximum of the APL can be excluded because there is a bounded solution for the minimum of

by Section 3 and the potential divergence of the second term of (20) to minus infinity would be boosted if ψ diverged.

A way to interpret the Cox–Reid adjustment in general is that it penalizes values of ψ that provide relatively more information on the nuisance parameters. The phenomenon is quite explicit here: i_γγ is minimized at ψ = 1, and other more informative values are penalized as a simple function of i_γγ.

A kind of degrees-of-freedom correction emerges to the multiplier of

. By numerical experiments, this adjustment may increase or decrease the estimate compared to the case where T − 3 is replaced by the usual T (cf. (6)). An enlargement arises if

, whereas a subtraction results if

. Therefore the correction strengthens the adjustment due to the term −½ log[1 + (T − 1)(1 − ψ)²] .

It can be inferred from (20) that the MLE and AE share the same asymptotic distribution under a unit root: the terms

and [1 + (T − 1)(1 − ψ)²] are of the same magnitude for ψ ∈ (−1,1] (O_p(T) or O_p(1) when evaluated at ψ ∈ (−1,1) or ψ = 1, respectively). Logs are taken of both terms in (20), but the second term becomes negligible for large T because the first term is multiplied by −(T − 3)/2. Hence the first term determines both the MLE and AE for large T.

The adjusted likelihood equation

can be shown to be equivalent to a fifth-order polynomial in ψ, so numerical methods are needed to find

. The equation is guaranteed to possess a real root because the order of the polynomial is odd.

The second derivative of the APL is needed to construct the W_a^B statistic of (16):

The formula for

is supplied in Appendix A. Despite absence of an analytical formula, the numerical value of W_a^B is easily found by substituting

in (16) and (22).

The adjustment does not alter the asymptotic unit-root distribution of the (signed square root of the) Wald statistic:

The convergence to zero of the additive adjustment (T − 1)[1 − (T − 1) × (1 − ψ)²]/{T²[1 + (T − 1)(1 − ψ)²]²}, evaluated at

, is easily reasoned:

is O_p(T⁻¹) under ψ = 1, so the effect is O_p(T) over T². The other convergencies are explained in Appendix A.

The adjusted LR statistic is from (17)

The term

does not play a part in the asymptotic distribution of T(LR_a^B − 1) under a unit root:

converges in probability to zero, so

converges in probability to unity. It follows that T(LR_a^B − 1) shares the same asymptotic unit-root distribution with T(LR_ml^B − 1) or

6. STATISTICS AND UNIT-ROOT ASYMPTOTICS FOR THE AR^μ(1) MODEL

Under the AR^μ(1) model the MLE of ψ and the Wald statistic for the unit-root null hypothesis are

where

, and the formulation of the Wald statistic is akin to (12) (Pere, 2000). The estimator

is also the ordinary least-squares (OLS) estimator. The AE for ψ under the AR^μ(1) model is essentially

where

is (1 − ψ)⁻¹ − (1 − ψ^T)/[T(1 − ψ)²] evaluated at

. The formula for

applies for a large T, but in the simulations that follow the exact definition given in Pere (2000) is employed. The adjusted Wald statistic of (16) is

(Pere, 2000).

The conventional statistics tend to the following well-known unit-root distributions as T increases:

where

(e.g., Pere, 2000).

(Pere, 2000). The LR statistics are not inspected under the AR^μ(1) model.³

Under the AR^μ(1) model with (ψ,α) = (1,0), one might presume that an LR unit-root test for the joint hypothesis (ψ,α) = (1,0) renders a more powerful test than one focusing on ψ only. However, in the simulations of Dickey and Fuller (1981) the test did not yield a gain in power. An explanation is that

, with obvious notation, absorbs changes in the level of the time series due to an altered initial value, say, and is comparable to the in general inessential constant of an ordinary linear regression.

7. FRACTILES FOR THE UNIT-ROOT AR^B(1) MODEL

The small-sample unit-root fractiles of the standardized coefficients (SCs),

, are reported in Tables 1 and 4 (based on simulations reported in Appendix D). These fractiles (and those in Tables 6, 7, 8, and 9 for the Wald and LR statistics) apply irrespective of the nuisance parameters γ and σ², or the initial value x₀, now relaxed to take any finite value, by the invariance and similarity results in Appendix C.

Fractiles of for ψ = 1

The median of the distribution of

is negative and the left tail is the longer, which suggest a biased estimator for all sample sizes. For observations in excess of 50 the fractiles essentially monotonically tend toward zero, or the distribution becomes more compact.

The fractiles are compared with the (unreported) corresponding distribution under the AR⁰(1) model in Table 2. Two sample sizes are printed on most lines of the table (say, 24–25). The larger is the last observed time period and the smaller is the number of observations available for estimation of the AR⁰(1) model (Section 2). (Parallel notation applies to Tables 3, 12, and 13.)

Differences between the fractiles of for ψ = 1

Engrossingly, when the sample size is small the left-tail fractiles under the AR^B(1) model lie far to the left of the corresponding left-tail fractiles under the AR⁰(1) model. It increases the bias and spread of the MLE in comparison to the latter model. Moreover, the distribution under the former converges to the asymptotic fairly slowly. The AR^B(1) model establishes invariance with respect to the initial value at the cost of inaccuracy in estimation in comparison with the AR⁰(1) model and does not yield small-sample distributions that are as dense as the asymptotic distribution suggests. The small-sample left-tail fractiles under the AR^B(1) model fit better with (and lie in general to the right of) those for the AR^μ(1) model (Table 3). In large samples these fractiles naturally differ.

Differences between the fractiles of for ψ = 1

The 0.05th fractiles of

are reported in Pantula et al. (1994) also. These fractiles are smaller than those in Table 1 for samples of 25 and 50 apparently because, by unreported simulations with the OPTMUM module of GAUSS 3.2.17 (1992), the simple algorithm employed in Pantula et al. (1994) yields left-tail fractiles that are too small.

Zivot (1994, p. 564), focusing on the unit-root case, reasons that

takes similar values to

or the MLE of ψ calculated from the AR⁰(1) model treating y₁ as the initial value.

Zivot's formula (37) for S²(ρ,x₀) agrees with (7) here for

after substituting ψ for ρ. Zivot wrote that “S²(μ,x₀) … is minimized near

, the OLS estimate of ρ in the simple AR(1) model without an intercept.” (Here S²(μ,x₀) looks like a misprint for S²(ρ,x₀) because the former was not defined in the paper nor does it have ρ as an argument.)

for ψ = 1. However, unreported simulation experiments reveal that the similarity holds only under a (near) unit root and a fairly large sample size (cf. Table 2). Otherwise the estimators can take very different values.

The adjustment cuts both tails of the distribution of the MLE but especially the left tail (Tables 4 and 5). Moreover, in essence all the fractiles of the AE—except the asymptotic fractiles copied from Table 1—lie closer both to zero and to the asymptotic fractiles than those of the MLE. The AE seems less biased and variable, or more accurate, and follows the common asymptotic distribution better than the MLE.

Fractiles of for ψ = 1

Differences between the fractiles of for ψ = 1

The fractiles of the Wald statistics

are in Tables 6 and 7. The location and shape of the distributions vary less with the sample size than those of the SCs. Yet the distribution of

is more concentrated in general and matches much better with the common asymptotic distribution in small samples. Both statistics take a real value for all draws. However, unreported simulation experiments uncover that

can take a complex value if T is extremely small, such as 5.

Fractiles of for ψ = 1

Fractiles of for ψ = 1

Tables 8 and 9 contain the fractiles for the LR statistics. The adjustment cuts the tails of the small-sample distributions so that LR_a^B follows the common asymptotic distribution far better than LR_ml^B.

Fractiles of T(LRmlB − 1) for ψ = 1

Fractiles of T(LRaB − 1) for ψ = 1

8. FRACTILES FOR THE UNIT-ROOT AR^μ(1) MODEL

Fractiles of the finite-sample and the asymptotic unit-root distributions of

are reported in Tables 10 and 11. We found that

was real in all of the simulations. The fractiles are invariant with respect to the initial value (Pere, 2000). The constant α is fixed to zero, but the magnitude of σ² is irrelevant for the fractiles (proof omitted). The distributions are discussed and the details of the simulations are in Pere (1997, 2000).

Fractiles of for ψ = 1

Fractiles of for ψ = 1

9. COMPARING ESTIMATION ACCURACY

The bias, standard deviation (SD), and mean-square error (MSE) of the MLE and AE for a random walk under the different AR(1) models are shown in Table 12. The number of observations is 100 for estimation of the AR^B(1) model and one less for the other models (Section 2). The AE and MLE are the same under the AR⁰(1) model (Pere, 1997).

Bias (×100), SD (×100), and MSE (×1,000) of the MLE and AE for a random walk and T = 99–100

The bias, SD, and MSE of the estimators decrease uniformly in absolute value from the top to the bottom row. The AR^μ(1) model is by far the worst with respect to each of the quantities. The statistics for the AR^B(1) model are clearly smaller yet are inferior to the statistics for the AR⁰(1) model. The adjustment improves estimation accuracy in terms of bias and MSE under the AR^μ(1) and AR^B(1) models and also in terms of SD under the latter. Under a unit root this holds even asymptotically under the former model (Pere, 2000). This is possible because the standard regularity conditions behind the usual optimality results for the MLE are not valid under nonstationarity.

Unreported simulation experiments indicate that if the sample size is exceptionally small, say, 10 or less, then the AE is an overwhelmingly better estimator than the MLE under the unit-root AR^B(1) model. This is due to the dominance of the unit-root-favoring term of the APL, −½ log[1 + (T − 1)(1 − ψ)²] , when the sample is uninformative.

10. NEAR UNIT-ROOT ASYMPTOTICS UNDER THE AR^B(1) MODEL

In the case of a near unit-root or ψ = exp(δ/T), δ < 0 (Phillips, 1987) corresponding asymptotic distributions apply:

In the preceding expressions i is an index for ml or a, and

is an Ornstein–Uhlenbeck process. The proofs for i = ml are in Appendix A. The adjusted statistics follow the same distributions as the unadjusted by similar logic as in the unit-root case (Section 5 and Appendix A). Phillips (1987) has discovered that the first two asymptotic distributions apply (and it can be shown that the third distribution also applies) for the corresponding statistics under the AR⁰(1) model with a near unit root. Thus the unit-root and near unit-root distributions of the respective statistics are identical under the AR^B(1) and AR⁰(1) models. It follows that the respective unit-root tests are asymptotically equally powerful against near unit-root alternatives whether or not the adjustment is employed and regardless of which of the two models is used.

11. NEAR UNIT-ROOT ASYMPTOTICS UNDER THE AR^μ(1) MODEL

The asymptotic distributions of

, under ψ = exp(δ/T) and α = 0, are known to be

where

(e.g., Stock, 1994, p. 2772). It is proved in Appendix B that the adjusted statistics follow different distributions under the same assumptions:

The power of the respective unit-root tests against near unit-root alternatives is likely to differ, even asymptotically, under the AR^μ(1) model.

12. UNIT-ROOT TESTS AND POWER COMPARISONS FOR THE AR^B(1) MODEL

The unit-root fractiles of

permit testing of ψ = 1 under the AR^B(1) model for various sample sizes. Large values of the LR statistics and small values of the other statistics suggest rejection of the null.

The tests based on the SCs will be denoted by ψ_ml^B and ψ_a^B with obvious notation. The corresponding Wald and LR tests will carry the labels τ_ml^B, τ_a^B, Φ_ml^B, and Φ_a^B. The simulated powers of the tests are in Table 13. The test size is 5%, and the ranges of ψ and T are 0.4–0.99 and 24–25 to 249–250. The larger sample sizes apply in this section.

Power (×100) of the unit root tests when the size is 5%

The ψ_a^B test is as powerful as or slightly more powerful than the ψ_ml^B test.⁵

The Wald tests are as powerful as or more powerful than the corresponding SC tests when there are 25 observations, contrary to the well-known ranking under the AR^μ(1) model (next section).⁶

The LR tests are never the most powerful and compare unfavorably to the rest of the tests. The Φ_ml^B test can possess more power than a SC or Wald test under T = 25 only, and the weakest test is Φ_a^B. Some intuition is that the LR tests are two-sided, whereas the other tests are one-sided.

13. UNIT-ROOT TESTS AND POWER COMPARISONS FOR THE AR^μ(1) MODEL

The tests associated with the

statistics will be referred to as ψ_ml^μ, ψ_a^μ, τ_ml^μ, and τ_a^μ. Their powers are in Table 13 also.

The statistics

employ the MLE of σ instead of the degrees-of-freedom corrected estimator as in Dickey and Fuller (1979). The estimator of σ affects the scale of the null distributions, but power is unaffected. The simulation algorithm accepts the null of a unit root if it comes across a noncalculable

(for details, see Pere, 1997, 2000). Another alternative would be to exclude such instances, which would, if anything, increase the power estimates for the τ_a^μ test.

The adjusted versions of the tests tend to be a bit weaker in contrast to the AR^B(1) case. The differences in power are in general larger between the SC than between the Wald tests. As a by-product, Table 13 affirms the Monte Carlo results of Dickey, Bell, and Miller (1986, Table 2), Dickey and Fuller (1979, p. 430), and Schmidt and Phillips (1992): the ψ_ml^μ test is often substantially more powerful than the τ_ml^μ test.

14. POWER COMPARISON BETWEEN THE TWO AR(1) MODELS

Even though the AR^B(1) and AR^μ(1) models agree under the null, the time series, or the underlying processes against which the powers are evaluated, can differ despite a common AR coefficient. The time series match, up to the long-run mean, when x₀ = y₀ − α/(1 − ψ) = 0 (Section 2) or when both time series have started from the respective long-run mean. The power figures for the models can be compared under this assumption meaningfully.

Underlining indicates the most powerful test in Table 13. In case of a tie a number of tests are indicated except on the lines where all tests achieve power of unity.

The ψ_ml^B test is in general more powerful than the ψ_ml^μ test. The difference in power can be very large but is small for 24–25 observations. The latter outcome is intuitively due to the similarity of the small-sample distributions of the associated MLEs (Table 3). The ψ_a^B test is the most powerful of the SC tests for all sample sizes (but see note 5). The τ_ml^B and τ_a^B tests tend to be far more powerful than the corresponding τ_ml^μ and τ_a^μ tests, so that at its best the τ_a^B test can possess more than twice the power of the τ_ml^μ or τ_a^μ test. This is largely because the powers of the Wald tests do not fall short of the powers of the SC tests under the AR^B(1) model as happens under the AR^μ(1) model.

Tests gain power much faster under the AR^B(1) than the AR^μ(1) model as the number of observations increases. For example, the powers (×100) for the ψ_ml^μ test under ψ = 0.95 at sample sizes of 24 and 249 are 7.7 and 62.7. The corresponding figures for the ψ_ml^B test are 7.9 and 85.7. The relative gain is even larger for the Wald tests: the analogous figures for them are 5.7 and 44.6 vs. 7.9 and 84.3.

The AR^B(1) model always yields more powerful tests. The benefit in power (×100) from using the AR^B(1) model is 5.7% units at its best for 24–25 observations (τ_a^B vs. ψ_ml^μ at ψ = 0.6) and swells to 19% units for 99–100 observations (τ_a^B vs. ψ_ml^μ at ψ = 0.90) and to 23.1% units for 249–250 observations (ψ_a^B vs. ψ_ml^μ at ψ = 0.95).

The adjustment does not necessarily increase power. The adjustment decreases power under the AR^μ(1) model and power of LR tests under the AR^B(1) model. Nevertheless, in general the most powerful test employs the adjustment (but see note 5).

The τ_a^B test is, among the inspected tests and values of ψ, the most powerful at sample size of 25 and the most powerful for ψ close to unity at sample size of 99–100. Among the same sets, no test is always the most powerful for sample sizes of 99–100 and 249–250.