2. THE MAIN THEOREMS FOR THE UNIDIMENSIONAL CASE

Tests are functions φ from the set R × R₊ to the interval [0,1] (φ = 0 means that we accept the null; φ = 1 means that we reject the null; and in between we randomize).

The power function of a test φ is the function that attaches to each λ ∈ R the value ∫φdQ_λ. The concept of a power function enables us to compare tests. A test φ is better than a test ψ if

So a better test dominates the worse test for all possible values of the parameter. In Ploberger (2004), it was established that the set C of tests consisting of the test functions defined subsequently is complete (or essentially complete according to Schervish, 1995).

These test functions are described by the following parameters: U,a,b,c and measures μ,ν subject to the following restrictions.

Then define the function

Now let the class of tests be characterized by the following properties: (i) the test should reject the null for all (A,W), so that A < a; (ii) if A > a, then the test should reject if

and accept if

These parameters do not describe the test completely. In particular, we did not define what happens if A = a or if u(A,W) = U. So if these equations do not describe null sets (with respect to all Q_h), our complete class defined previously may contain some additional (not optimal) tests. This, however, is not a contradiction to the definition of a complete class—any larger set of a complete class is a complete class, too!

Remark 1. Because of our restrictions 1 and 2, in particular expressions (7) and (8), we can easily see that every function u(A,W) satisfying (9) can for every β,γ > 0 be written as

where ν′,μ′ are the restrictions of the measure μ + ν on [−β,γ] and R − [−β,γ], respectively.

Remark 2. Because

we can replace the first integral in (9) by

Additionally, we assume that the following three conditions are fulfilled.

Condition C1. For some ε > 0 we have

Condition C2. The support of the measure Q₀ (and hence the support of all the measures Q_λ) is the whole R × R₊ = {(W,A) : A > 0} (i.e., every open set of R × R₊ has positive probability under Q₀).

Condition C3. The distribution function of A is continuous (i.e., for all a Q₀([A = a]) = 0).

THEOREM 1. Suppose restriction 1 and Conditions C1, C2, and C3 hold and assume that ψ is a test that has the same or a better power function than the test φ, which is from our class and is determined by the parameters a,b,c,U,μ,ν. Then ψ is from our class, too, and determined by the same parameters.

Proof. The proof is provided in the Appendix.

It is, however, a nontrivial task to show that tests are not in our complete class. For a proof of this fact, it is not sufficient to show that the corresponding test statistic cannot be written in the form (9). One has to show that the corresponding test (or the critical set) cannot be generated as a level set of any function (9)!

Suppose now that we have given a family Q_λ,λ ∈ R and a test φ (which is a function from R × R₊ into [0,1]) satisfying the following requirements.

Then the following theorem holds.

THEOREM 2. The test φ is not in our class and is therefore inadmissible.

Proof. The assumptions of this theorem are quite similar to those of Ploberger (2004), and only our second assumption here is stronger. Here we only postulate that all neighborhoods of the origin have positive mass, instead of assuming that every open set of the half plan R × R₊ has positive mass. One can easily see that this requirement is sufficient, because only the behavior of the power function for alternatives near the origin is used in the proof. █

In case the likelihoods are given by (1), one consequence of the preceding theorems is that the analogues to the “classical” Lagrange multiplier (LM), likelihood ratio (LR), and Wald tests for testing θ = θ₀ are asymptotically inadmissible. More precisely, let us denote by φ_n such a test. The “classical” tests reject when test statistics exceed the critical values. Moreover, it is an easy exercise—very similar to the standard proof of asymptotic equivalence in the stationary case—to show that differences between suitably normed and transformed test statistics and W_n²/A_n converge to zero under the null. Let us denote the asymptotic critical value for the test statistic W_n²/A_n by K and assume that K is such that the cumulative distribution function of W²/A is continuous at K. This is not a very strong restriction, because a cumulative distribution function can have at most countably many discontinuities. Let us define φ_n by

Then we can easily see that under the null the probability of the LM, LR, and Wald tests and the tests φ_n giving different results converges to zero. (So, with probability converging to one under the null, either all of them accept or all of them reject!) Hence the power functions of φ_n and the classical tests are asymptotically the same under the null.

We now have to show that the same holds true for the alternatives. Consider local alternatives of the form

In Ploberger (2004), contiguity of the probability measures corresponding to θ₀ and θ_n was established. This means that for any sequence E_n of events such that P_θ0(E_n) → 0 P_θn(E_n) → 0, too. (Contiguity is one of the cornerstones of modern theoretical asymptotic theory. For a more detailed discussion see Le Cam and Yang, 1990; Strasser, 1985.) Because the probability of conflict between the LM, LR, and Wald tests and φ_n converges to zero under the null, the result quoted previously allows us to conclude that the probability of a conflict under the local alternatives converges to zero, too. Hence, the difference of the probabilities of rejection—which is exactly the definition of a power function—must converge to zero for n → ∞. Therefore, the power functions are the same asymptotically. We assumed that (W_n,A_n) → (W,A) in distribution under the null, and so W_n²/A_n → W²/A in distribution, too. Because the cumulative distribution function of the limiting distribution is continuous at K, the power function of the φ_n (and, consequently, the power functions of the LM, LR, and Wald tests) must converge to the power function of our test φ under the null.

To show the convergence of the power functions of the φ_n to the power function of φ at the local alternatives requires only a bit more technical effort. Let us fix the sequence of local alternatives and consider the densities

Then we can easily conclude from (1) that

in distribution. We did assume, however, that

because otherwise Q_C would not be a probability measure. Now let ε > 0 be arbitrary. Then we can find an L so that

Consequently, as [ell ]_n → [ell ] in distribution,

and as ∫[ell ]_n dP_θn = 1,

Hence we have

One can easily see that

Further, because of the convergence of (A_n,W_n) and [ell ]_n,

We have that (when using (17))

As ε + ∫φ min([ell ],L) dQ₀ ≥ ∫([ell ] − min([ell ],L)) dQ₀ + ∫φ min([ell ],L) dQ₀ ≥ ∫φ[ell ] dQ₀,we can conclude that

Because ε was arbitrary, we have shown that

This establishes the fact that the power function of the φ_n converges to the power function of φ for all local alternatives. Because we already know that φ is not within our class, we can find a test ψ with (strictly) better power properties. In Theorem 3 in Section 3 we show that in this situation it is possible to construct tests ψ_n with strictly better asymptotic power properties than the φ_n, which establishes our result, namely, that the φ_n are asymptotically inadmissible.

In view of the preceding results one is tempted to look at other “generalizations” of the classical LR test to situations with stochastic information. One such generalization is the test based on the posterior information criterion (PIC) (cf. Phillips, 1996; Phillips and Ploberger, 1996; Ploberger and Phillips, 2003). Consider tests that reject if

and accept if

where K is a constant determined by the desired significance level. These tests are all admissible. Simply consider our measures μ,ν to be the Lesbesgue measure and choose the other constants appropriately. Then it is a simple but tedious calculation to show that (12) yields a test based on

which is immediately seen to be equivalent to the test determined by (18) and (19).

It should be noted that in the “classical” case of A being constant this test is equivalent to a test based on W²/A, too. So we can view the PIC test as a generalization of the classical tests, too.

3. APPLICATIONS

The result of the previous section establishes a minimal complete class theorem for a rather large class of testing problems. Apart from the three technical conditions we only require the likelihood function to be locally asymptotically quadratic. The main point of interest seems to be that this class of admissible tests is quite different from the one for the classical problem where W is Gaussian and A constant. In this kind of situation the set of all admissible tests is well known even for multivariate parameters (Birnbaum's theorem; cf. Strasser, 1985). Each of these tests is characterized by a convex set (in the univariate case an interval) and rejects if W lies outside the interval. Hence Theorem 1 of the previous section seems to be of considerable theoretical interest. It shows that even a small change in our assumptions makes enormous differences for the class of admissible tests.

In the previous section we only dealt with one-parameter families. This fact represents a significant hurdle for a direct application of the result. However, we can apply the result to prove the inadmissibility of tests. We will show that certain tests—e.g., the Dickey–Fuller τ-test and the pseudo-t-test based on the statistic (5)—can be interpreted as tests on a unidimensional parameter. Our result then shows that there must exist a test with uniformly better power function. The drawback is that the result is nonconstructive. The only information we supply about the better test is that it has a form of (9). Nevertheless, the result may motivate researchers in the field. Efforts to find better tests are now not doomed from the start. We know that a better test exists!

Let us first analyze the Dickey–Fuller test. Unfortunately, the techniques presented here cover only the cases that do not occur that often in practice. In most cases the data contain a drift or a deterministic trend. It should be noted that especially the presence of deterministic trends changes the situation completely. I conjecture that in this case the Dickey–Fuller τ-test is admissible. The results of Müller and Elliot (2001) establish an optimality property. For a detailed analysis also see Elliot et al. (1996), Phillips and Xiao (1998), and Stock (1995). Nevertheless, I think that the results of this paper are important to the theory of testing for unit roots. It shows, in particular, that the situation is radically different from the classical one.

Let us now assume that a process y_t is defined by (4) and let us furthermore assume that the

with zero expectation and that all of the roots of the polynomial

lie outside the unit circle. Then let us consider local alternatives

Jeganathan (1991, 1995) performed a rigorous asymptotic analysis for these kinds of testing problems. In particular, he established limiting properties of these experiments. Using his results, one can easily show that in many interesting cases the necessary conditions of our theorems are fulfilled. The “elementary” case of the Dickey–Fuller test without a trend is discussed in Ploberger (2004).

In many applications, however, one wants to avoid relatively narrow parametric assumptions such as (20). Standard statistical theory of optimal testing (cf. Strasser, 1985) is based on Neyman–Pearson tests, which critically depend on densities. Hence we cannot show the admissibility of tests. Under certain circumstances, however, we can show that certain tests are not admissible in the sense that there exists a test with a uniformly better power function.

Let us now assume that P_θ is a family of probability measures parametrized by a parameter θ. We assume that the parameter can be split into two parts,

where δ should be one-dimensional, whereas the nuisance parameter β can be of arbitrary dimension. Suppose we want to test the hypothesis δ = 0.

Suppose one has an estimator—say,

—for δ; then one will be tempted to use an analogue to the usual t-test. One will construct some other statistic—say,

—so that

Then it is easy to construct a test. In the “classical” case, where

is asymptotically normal,

will be an appropriately scaled consistent estimator for the asymptotic variance. We are more interested in the nonstandard case, where the limiting distribution of the properly scaled

is nontrivial.

In some cases, we can utilize our approach analyzing the nonoptimality of tests. In the previous section, we were mainly working with the scores and the “information” (the second derivative) of the likelihood function. Here the situation is different. We start with estimators for the parameter and their accuracy.

So let us review the basic assumption of the previous section. For doing so, we must find analogues to the W_n and A_n defined earlier. We started with the model (1):

Then it is easily seen that—in case the logarithm of the likelihood is twice continuously differentiable—the maximum likelihood estimator

can be approximated in the following way:

Moreover, we can easily conclude from our assumptions that

converges in distribution, and so

would be a feasible normalizing estimator.

In our situation, however, we start with the estimators

,

. So let us assume that there exists an appropriate sequence of scaling factors D_n and define (keeping in mind that we want to test δ = δ₀)

Then we assume that the following four assumptions are fulfilled.

Assumption A1. The nuisance parameter β is fixed.

Assumption A2. (A_n,W_n) converges (for

) in distribution to some random variable (A,W) with distribution Q₀, which fulfills the prerequisites of the previous section.

Assumption A3. For local alternatives

(i.e.,

), the distributions of (A_n,W_n) converge to a measure Q_c with

Assumption A4. The mapping c → Q_c is continuous (with respect to the weak topology of measures). This means that c_n → c implies Q_cn → Q_c weakly.

Essentially, we assume that the statistic (A_n,W_n) behaves “the same way” as in the case of a unidimensional parameter. We will now construct some examples that show that these assumptions will be fulfilled in many cases of considerable interest. In particular, it should be noted that we do not require parametric assumptions such as, e.g., Gaussian distributions of the error terms!

1. The generalized augmented Dickey–Fuller (ADF) test with no deterministic trend. Said and Dickey (1984) investigated the ADF test for unit roots for autoregressive moving average (ARMA) processes with independent and identically distributed (i.i.d.) innovations. Lately, in Chang and Park (2002) further generalizations of the model (4) are considered (e.g., u_t are martingale differences with a very general structure, and the index p may depend on the sample size and increase to infinity). So let us consider the model (4) and define

They show that, even under the general circumstances considered in their paper (using our notation), letting

be the OLS estimator for δ and using an estimator

, where

equals the denominator in the t-test (i.e., “variance estimator”), the properly normalized distributions of

and the estimation error jointly converge in distribution to the corresponding distributions for the (simple) Dickey–Fuller test. Thie convergence holds for δ = 0 and also for the local alternatives (22) (cf. Chang and Park, 2002). So let us define the “scores” by

and denote by Q_c the limiting distribution of

The measures Q_c are defined on the set {(A,W) : A ≥ 0} ⊂ R². As mentioned before, the limiting distributions of our statistics are the same as for the Dickey–Fuller test. Hence we can conclude that the measures Q_c are the same as the asymptotic distributions for the simple random walk model, which was discussed in Ploberger (2004). So we may conclude that

Moreover, it can easily be seen that other assumptions of our main theorem are fulfilled, too. Elementary calculations show that the two-sided ADF test rejects when

exceeds the critical value—say, U. Hence the power of the ADF test against a local alternative (22) will converge to

This only holds for all those U for which Q_c({W²/A = U}) = 0. Because, however, the Q_c have densities with respect to Q₀, Q_c({W²/A = U}) > 0 implies that Q₀({W²/A = U}) > 0. But this inequality can only be true for countably many U. Hence we have convergence for all but at most countably many U.

2. Cointegrated system testing. Another possible application is the testing of restrictions on certain parameters of cointegrated systems. Let us first consider the simplest possible case, namely, a bivariate system in Phillips' triangular form:

For simplicity, let us assume that u_t,v_t are i.i.d. standard normal with zero expectation and the covariance being the identity matrix (which implies that u_t,v_t are uncorrelated). Then we can estimate γ with OLS, and our estimation error will be

Let us now suppose that we want to test γ = γ₀. Then some tedious but elementary calculations show that the asymptotic distribution of

converges under the null to

where W₁,W₂ are independent Wiener processes. Moreover, we can also consider local alternatives of the form

Then

converges in distribution to

3. Some generalizations. We can generalize the ideas of the preceding example for use in a more general context. Suppose we want to test whether a parameter

and we have an estimator

, an estimator for the “information”

, and a sequence of scaling factors D_n, so that the following properties hold true.

For local alternatives

the distributions of

, where

, converge in distribution to a (W,A) with the following properties.

• The distribution of A is independent of c.
• The conditional distribution of W given A is normal with expectation cA and variance A.
• The distribution of A satisfies all the assumptions of our main theorem.

Many popular estimators for the cointegrating relationships have these asymptotically mixed normal properties: It is well established that the standard estimators for the cointegrating relationships are normal only after conditioning: cf. Ahn and Reinsel (1990), Phillips and Hansen (1990), Phillips and Ouliaris (1990), Saikkonen (1991), Johansen (1988, 1991), Stock and Watson (1993), and also Hamilton (1994, Sect. 19.3). Testing linear restrictions on these parameters with Wald-type tests (cf. Davidson, 1998) therefore will be covered by our theory, at least for one-dimensional restrictions.

A testing problem satisfying the preceding conditions is called locally asymptotically mixed normal (LAMN) (cf. Le Cam and Yang, 1990; Jeganathan, 1991, 1995). Apart from the economic applications quoted later, this type of model has also received much attention in the statistical literature (cf. the references cited previously).

4. Other cases. Asymptotically mixed normal families do not only occur in connection with unit root tests or cointegration. Other cases of interest in economics are given in Park and Phillips (2001) and Aït-Sahalia (1999).

So let us now assume that we have given a testing problem satisfying the preceding assumptions. We might be tempted to use the generalized t-test φ_T for our testing problem. This means we reject when the absolute value or, equivalently, the square of

is larger than the critical value. Furthermore, let us assume that this critical value has an absolute value larger than one, i.e.,

Let us additionally assume that the cumulative distribution function of the limiting distribution of our test statistic is continuous at U. This is equivalent to

Hence the cumulative distribution function of the limiting distribution of our test statistic is continuous in U.

THEOREM 3. Suppose Assumptions A1–A4 and also (25) and (26) are satisfied. Then the tests

defined earlier are inadmissible. There exist tests

depending on W_n and A_n alone that have a strictly better asymptotic power function than φ_n. For alternatives (24) we have

where for at least one c ≠ 0 the left-hand side is smaller than (and not equal to) the right-hand side. Moreover, we also have

Remark 3. In many cases, it is not that surprising that the tests φ_n are inadmissible. As an example take, e.g., one of the preceding models with non-Gaussian error terms u_t, v_t, when their distribution is known up to a finite-dimensional parameter. Then estimators based on parametric models for u_t, v_t should be more accurate, and hence the corresponding tests would yield more power. However, it is interesting to observe that there exists a test better than φ_n depending on (W_n,A_n) alone! Therefore there exists some way of computing a test statistic with (W_n,A_n) other than the t-test statistic, which defines a test with a uniformly better asymptotic power function!

Remark 4. Unfortunately, the proof is nonconstructive. I only assert the existence of such a test. The proof does not provide a way to construct this dominating test.

Remark 5. Here we will prove pointwise convergence in (27) and (28). With some technical refinements, the proof can be modified to show that the convergence in (27) is uniform on bounded sets of C.

Proof of Theorem 3. The proof can be found in the Appendix.