How to avoid the zero-power trap in testing for correlation

How to avoid the zero-power trap in testing for correlation

Introduction

Testing whether the errors in a regression model are uncorrelated is a standard problem in econometrics. For many forms of correlation under the alternative there are well-established tests available. Two prominent examples are the Durbin-Watson test for serial autocorrelation, and the Cliff-Ord test for spatial autocorrelation. Nevertheless, this type of testing problem is not completely solved, not even in the Gaussian case. This is partly due to the fact that tests for correlation, including the well-established tests mentioned before, do not always behave as they ideally should in finite samples: Whereas the size of most tests can be easily controlled, at least under suitable distributional assumptions such as Gaussianity, their power function can attain very small values in regions of the alternative where the correlation is very strong. This, however, does not match with the intuition that strong correlations should be easily detectable from the data, i.e., that the power of a test for correlation should be close to one if the degree of correlation in the errors is very strong.

That the power function of a test for correlation can drop to zero as the correlation increases was first formally established in kramer85, who considered the power function of the Durbin-Watson test in testing for serial autocorrelation. The results in kramer85 were extended in later work by zeisel1989, KramerZeisel1990 and lobus2000. kleiber2005 obtained similar results for the Durbin-Watson test when the disturbances are fractionally integrated. kramer2005finite proved related results for Cliff-Ord-type tests in case the regression errors are spatially autocorrelated. A unifying general theory that neither relies on the specific form of correlation nor on very special structural properties of the tests was developed recently in mart10 and PP17. We refer the interested reader to the latter articles for formal results and a thorough discussion of the literature.

The major practical value of the just mentioned articles is of a diagnostic nature: they provide conditions which depend on observable quantities only and which let a user decide whether a particular test is subject to the zero-power trap, i.e., whether its power function drops to zero as the correlation increases. This is important, because if it turns out that an initial test is subject to this trap, one may want to use another test. However, one is then confronted with the problem of finding a test that avoids the zero-power trap. One complication is as follows: Typically, the initial test was chosen for a reason, i.e., for its “optimal” power properties in certain regions of the parameter space (think of a locally best invariant test). In such situations, one would not just like to use some other test that avoids the zero-power trap. Much more likely, one would prefer to slightly modify the initial test in such a way that its optimality properties are preserved, at least approximately, but such that its modified version does not suffer from the zero-power trap. Compared to the amount of literature that concentrates on deriving diagnostic tools for detecting the zero-power trap, the attention that has been paid to the question how one can construct tests which do not suffer from the zero-power trap is much less. Furthermore, it is not clear how to obtain said “optimality-preserving” modifications. The main contribution of the present article is to fill this gap. In the following paragraphs we provide an overview of the article's structure together with a more detailed summary of our contributions.

In Section (ref) we introduce the framework: the model and the testing problem, some notational conventions and an important class of tests. In Section (ref) we formally define the zero-power trap phenomenon, obtain some sufficient conditions for it from results in PP17, and then consider in our general framework the question how often, i.e., for “how many” design matrices, the zero-power trap actually arises. We answer this question in Propositions (ref) and (ref). The former proposition proves (and generalizes) an observation already made in the discussion section of kramer85. The latter proposition is obtained by generalizing an argument in martellosio2012testing, who considered the same question in a spatial autoregressive setting. Essentially, these two propositions show (for the tests based on the specific family of test statistics and the corresponding critical values considered) respectively that (i) the zero-power trap arises for generic design matrices (i.e., up to a Lebesgue null set of exceptional matrices) for small enough critical values; and (ii) for any critical value that leads to a size in $(0, 1)$ there exists an open set of design matrices for which the zero-power trap arises.

In Section (ref) we present three ways to avoid the zero-power trap: In Section (ref) we briefly discuss a test for which PP17 have shown that it does not suffer from the zero-power trap. This test typically does not have very favorable power properties, apart from the fact that it avoids the zero-power trap. We shall mainly use it later as a building block in our construction of “optimality-preserving” tests. In Section (ref) we discuss tests that incorporate artificial regressors to avoid the zero-power trap. The suggestion of adding artificial regressors to the regression and to use “optimal” tests in this expanded model is present already in kramer85, who observed numerically that adding the intercept to a regression without intercept helps to avoid the zero-power trap for the Durbin-Watson test. Our theoretical results in Section (ref) exploit results in PP17, and are related to the methods in PP13 and preinerstorfer2017, who considered the construction of tests with good size and power properties for testing restrictions on the regression coefficient vector. While the tests in Section (ref) are “optimality-preserving” to some extent (more specifically they often have the same optimality property as initial tests, but within a smaller class of tests), it turns out that this solution to the zero-power trap is not ideal. For example, the power function of these tests does not increase to one as the strength of the correlation increases (which is the case for the approach outlined in Section (ref)).

In Section (ref) we construct optimality-preserving modifications avoiding the zero-power-trap out of an initial test that suffers from the zero-power trap. Our approach overcomes the limitations of the approaches discussed in Sections (ref) and (ref). In particular, our method leads to tests that have approximately the same power properties as the initial test. Furthermore, their power converges to one as the strength of the correlation increases. The construction is inspired by the power enhancement principle of fan2015power in the formulation used in Section 3 of kock2017power. The basic idea of this principle is to improve the asymptotic power of an initial test by using another test, a power enhancement component, which has better asymptotic power properties than the initial test in certain regions of the alternative. Since the theory in fan2015power and kock2017power is asymptotic, and the present article is concerned exclusively with finite sample properties, their results do not apply here. Nevertheless, we can adapt the underlying heuristic to our context: given an initial test that suffers from the zero-power trap, but has favorable power properties in other regions of the alternative, we “combine” this initial test with the test from Section (ref) to obtain an “enhanced” test.

In Section (ref) we compare the approaches for avoiding the zero-power trap discussed in Section (ref) numerically. We reconsider an example in kramer2005finite in which the Cliff-Ord test turns out to suffer from the zero-power trap. Section (ref) concludes. All proofs are collected in Appendices (ref)-(ref).

Framework

In the present section we introduce the model, the testing problem and some notation, and we discuss an important class of tests. Most of the notational conventions and terminology we use are standard, and coincide to a large extent with the ones in PP17. We repeat them here for the convenience of the reader.

Model and testing problem

We consider the linear model

where $X\in \mathbb{R}^{n\times k}$ is a non-stochastic matrix of rank $k$ with $0< k<n$, and where ${\Greekmath 010C} \in \mathbb{R}^{k}$ is the regression coefficient vector. The disturbance vector $\mathbf{u}$ is assumed to be Gaussian with mean zero and covariance matrix ${\Greekmath 011B} ^{2}\Sigma ({\Greekmath 011A} )$. Here $\Sigma (.)$ is a known function from $[0,a)$ to the set of symmetric and positive definite $n\times n$ matrices, and $a$ is a prespecified positive real number. Without loss of generality we assume throughout that $\Sigma (0)$ equals the identity matrix $I_{n}$. The parameters ${\Greekmath 010C} \in \mathbb{R}^k$, ${\Greekmath 011B} \in (0, \infty)$ and ${\Greekmath 011A} \in \lbrack 0,a)$ are unknown.

The Gaussianity assumption could be relaxed considerably. It is imposed mainly to avoid technical conditions that do not deliver deeper insights into the problem. For example, we could replace the Gaussianity assumption by the assumption that the distribution of the error vector $\mathbf{u}$ is elliptically symmetric without changing any of our results. This and other generalizations are discussed in detail in Section 3 of PP17.

Denoting the Gaussian probability measure with mean $X{\Greekmath 010C}$ and covariance matrix ${\Greekmath 011B}^2 \Sigma({\Greekmath 011A})$ by $P_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}$, we see that the model (ref) induces the parametric family of distributions

on the sample space $\mathbb{R}^{n}$ equipped with its Borel ${\Greekmath 011B}$-algebra. The expectation operator with respect to (w.r.t.) $P_{{\Greekmath 010C} ,{\Greekmath 011B} ,{\Greekmath 011A} }$ will be denoted by $E_{{\Greekmath 010C} ,{\Greekmath 011B} ,{\Greekmath 011A} }$. Note that the set of probability measures in the previous display is dominated by Lebesgue measure ${\Greekmath 0116}_{\mathbb{R}^n}$ on the Borel sets of $\mathbb{R}^n$, because $\Sigma({\Greekmath 011A})$ is positive definite for every ${\Greekmath 011A} \in [0, a)$ by assumption.

In the family of distributions (ref) we are interested in the testing problem ${\Greekmath 011A} = 0$ against ${\Greekmath 011A} > 0$. More precisely, the testing problem is

with the implicit understanding that always ${\Greekmath 011A} \in \lbrack 0,a)$. In this testing problem the parameter ${\Greekmath 011A}$ is the target of inference, and the regression coefficient vector ${\Greekmath 010C}$ and the parameter ${\Greekmath 011B}$ are nuisance parameters.

Two specific examples that received a considerable amount of attention in the econometrics literature and which fit into the above framework are testing for positive serial autocorrelation and testing for spatial autocorrelation, cf. Examples 2.1 and 2.2 in PP17 for details and a discussion of related literature. See also Section (ref) below for more information on testing for spatial autocorrelation and related numerical results.

Notation, invariance and an important class of tests

Notation

All matrices we shall consider are real matrices, the transpose of a matrix $A$ is denoted by $A^{\prime }$, and the space spanned by the columns of $A$ is denoted by $ \limfunc{span}(A)$. Given a linear subspace $L$ of $\mathbb{R}^{n}$, the symbol $\Pi _{L}$ denotes the orthogonal projection onto $L$, and $L^{\bot }$ denotes the orthogonal complement of $L$. Given an $n\times m$ matrix $Z$ of rank $m$ with $0\leq m<n$, we denote by $C_{Z}$ a matrix in $\mathbb{R} ^{(n-m)\times n}$ such that $C_{Z}C_{Z}^{\prime }=I_{n-m}$ and $ C_{Z}^{\prime }C_{Z}=\Pi _{\limfunc{span}(Z)^{\bot }}$ where $I_{r}$ denotes the identity matrix of dimension $r$. We observe that every matrix whose rows form an orthonormal basis of $\limfunc{span}(Z)^{\bot }$ satisfies these two conditions and vice versa. Hence, any two choices for $C_{Z}$ are related by premultiplication by an orthogonal matrix. Let $l$ be a positive integer. If $A$ is an $l\times l$ matrix and ${\Greekmath 0115} \in \mathbb{ R}$ is an eigenvalue of $A$ we denote the corresponding eigenspace by $ \limfunc{Eig}\left( A,{\Greekmath 0115} \right)$. The eigenvalues of a symmetric matrix $B\in \mathbb{R}^{l\times l}$ ordered from smallest to largest and counted with their multiplicities are denoted by ${\Greekmath 0115} _{1}(B),\ldots ,{\Greekmath 0115} _{l}(B)$. We shall sometimes denote ${\Greekmath 0115} _{1}(B)$ by ${\Greekmath 0115}_{\min}(B)$, and ${\Greekmath 0115} _{l}(B)$ by ${\Greekmath 0115}_{\max}(B)$. Lebesgue measure on the Borel ${\Greekmath 011B}$-algebra of $\mathbb{R}^{n \times l}$ shall be denoted by ${\Greekmath 0116} _{\mathbb{R}^{n \times l}}$, and $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{Pr}$ is used as a generic symbol for a probability measure. The Euclidean norm of a vector is denoted by $\|.\|$, a symbol that is also used to denote a matrix norm.

Invariance, an important class of tests, and size-controlling critical values

Given a matrix $Z\in \mathbb{R}^{n\times m}$ with column rank $m$ and where $1\leq m<n$, define the group of bijective transformations (the group action being composition of functions)

where $g_{{\Greekmath 010D} ,{\Greekmath 0112} }: \mathbb{R}^n \to \mathbb{R}^n$ denotes the function $y\mapsto {\Greekmath 010D} y+Z{\Greekmath 0112}$.

Under our distributional assumptions (and if additionally all parameters of the model are identifiable) the testing problem in Equation (ref) is invariant w.r.t. the group $G_X$ (cf. Section 6 in lehmannromano). It thus appears reasonable to consider tests that are $G_X$-invariant, a property shared by most commonly used tests. Recall that a function $f$ defined on the sample space (e.g., a test or a test statistic) is called invariant w.r.t. $G_X$ if and only if for every $y \in \mathbb{R}^n$ and every $g_{{\Greekmath 010D}, {\Greekmath 0112}} \in G_X$ it holds that $f(y) = f(g_{{\Greekmath 010D}, {\Greekmath 0112}}(y))$. A subset $A$ of $\mathbb{R}^n$ will be called invariant w.r.t. $G_X$ if the indicator function $\mathbf{1}_A$ is $G_X$-invariant.

In addition to being $G_X$-invariant, most tests for (ref) used in practice are non-randomized, i.e., they are indicator functions of Borel sets -- their corresponding rejection regions. An important class of such tests is based on rejection regions of the form

where $c \in \mathbb{R}$ is a critical value and the test statistic

Here $B\in \mathbb{R}^{(n-k)\times (n-k)}$ is a symmetric matrix, which typically depends on $X$ and the function $\Sigma$. Recall that the matrix ${C_{X}}$ satisfies ${C_{X}C}^{\prime }{_{X}=I}_{n-k}$ and ${C}^{\prime }{ _{X}C_{X}=\Pi }_{\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{$\limfunc{span}$}(X)^{\bot }}$ (cf. Section (ref)). Clearly, the test statistic $T_{B}$ is $G_{X}$-invariant. Note furthermore that in case ${\Greekmath 0115}_1(B) = {\Greekmath 0115}_{n-k}(B)$ the test statistic $T_B$ is constant everywhere on $\mathbb{R}^n$. Therefore, such a choice of $B$ is uninteresting for practical purposes. Note also that assigning the value ${\Greekmath 0115}_1(B)$ (instead of any other value) to the test statistic on $\limfunc{span}(X)$ has no effect on rejection probabilities, because $P_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}$ is absolutely continuous w.r.t. ${\Greekmath 0116}_{\mathbb{R}^n}$ for every ${\Greekmath 010C} \in \mathbb{R}^k$, ${\Greekmath 011B} \in (0, \infty)$ and ${\Greekmath 011A} \in [0, a)$, and $\limfunc{span}(X)$ being of dimension $k < n$ implies ${\Greekmath 0116}_{\mathbb{R}^n}(\limfunc{span}(X)) = 0$.

The following remark discusses two particularly important choices of $B$:

The main focus of the present article concerns power properties of tests based on a test statistic as in (ref) for the testing problem (ref). Before investigating power properties of a test, one needs to ensure that its size does not exceed a given value of significance ${\Greekmath 010B}$. While this can be a nontrivial problem in general, achieving size control through the choice of a proper critical value turns out to be an easy task here. More specifically, the following lemma shows that exact size control for tests based on a test statistic $T_B$ introduced in Equation (ref) is possible at all levels of significance in the leading case ${\Greekmath 0115}_1(B) < {\Greekmath 0115}_{n-k}(B)$. The subsequent remark discusses numerical aspects.

The zero-power trap in testing for correlation

Definition and sufficient conditions

In the sequel, a test ${\Greekmath 0127}:\mathbb{R}^n \to [0, 1]$ (measurable) for testing problem (ref) is said to be subject to (or suffer from) the zero-power trap, if there exist ${\Greekmath 010C} \in \mathbb{R}^k$ and ${\Greekmath 011B} \in (0, \infty)$ such that

that is, if the power function of ${\Greekmath 0127}$ can get arbitrarily close to $0$ as the strength of the correlation in the data, measured in terms of ${\Greekmath 011A}$, increases. Recall from Remark (ref) that if ${\Greekmath 0127}$ is $G_X$-invariant, which is the case for most tests considered in this article, then $E_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}({\Greekmath 0127})$ does not depend on ${\Greekmath 010C}$ and ${\Greekmath 011B}$. In this case, if Equation (ref) holds for some ${\Greekmath 010C} \in \mathbb{R}^k$ and some ${\Greekmath 011B} \in (0, \infty)$, it holds for every ${\Greekmath 010C} \in \mathbb{R}^k$ and every $0 < {\Greekmath 011B} < \infty$.

A set of sufficient conditions that allows one to conclude whether a test ${\Greekmath 0127}$ is subject to the zero-power trap was developed in mart10 and PP17. The underlying effect leading to (ref) described in the latter article is a concentration effect in the (rescaled) distribution $P_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}$ when ${\Greekmath 011A}$ is close to $a$. PP17 obtained their sufficient conditions under the following property of the function $\Sigma$ (cf. also Assumption 1 in PP17 and the discussion there showing that this condition is weaker than the one previously used by mart10):

For the convenience of the reader and for later use, we shall now formally state two immediate consequences of results in PP17. They provide sufficient conditions for the zero-power trap under Assumption (ref). Specializing Theorem 2.7 and Remark 2.8 in PP17 one obtains the following “high-level”-result.

For the test with rejection region $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ as discussed in Section (ref) and where ${\Greekmath 0114}({\Greekmath 010B})$ is defined through Lemma (ref) one obtains the following result from Corollary 2.21 of PP17.

Note that the sufficient conditions for the zero-power trap phenomenon pointed out in Theorems (ref) and (ref) depend on observable quantities only, and that they are thus checkable by the user. Therefore, a researcher interested in testing problem (ref) can use these conditions to check whether or not the given test suffers from the zero-power trap before actually using a test. In particular, one can decide not to use a test that suffers from the zero-power trap. Before addressing the question how to avoid the zero-power trap, which was raised already in the Introduction, we briefly pay some attention to the following question: “how often” does the zero-power trap actually arise? More specifically, in the important class of tests $\Phi_{B, c}$ introduced in Section (ref), and most notably the tests discussed in Remark (ref), the following question arises: For “how many” design matrices $X$ does the zero-power trap arise? Answering this question is the content of the next section.

For “how many” design matrices does the zero-power trap arise?

We shall focus on the class of tests with rejection regions $\Phi_{B(X), c}$ introduced in Section (ref). Since the question in the section title depends on the design matrix $X$, which is otherwise held fixed in this article, we shall make the dependence of $B$ on $X$ explicit by writing $B(X)$. Furthermore, we shall also write $P_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}^X$ to emphasize its dependence on the design matrix $X$. In our first attempt to answer the question under consideration, we shall use the following simple consequence of Lemma (ref) and Theorem (ref), which provides conditions on $X$ under which Equation (ref) holds for all “small” levels ${\Greekmath 010B}$.

For a class of functions $X \mapsto B(X)$ that includes the ones discussed in Remark (ref) we shall now show that condition (ref) is generically satisfied, unless the matrix $B(X)$ has a very exceptional form. The result is established under a restriction concerning the eigenspace corresponding to the largest eigenvalue of $B(X)$.

Proposition (ref) shows that tests based on $T_{B(X)}$ suffer from the zero-power trap for “most” design matrices $X$, at least for small choices of ${\Greekmath 010B}$. The discussion section of kramer85 contains a corresponding statement (without proof) in a special case.

Choosing ${\Greekmath 010B}$ small is not completely uncommon in practice: Due to the fact that testing for correlation is often just one part of the econometric analysis, the actual level ${\Greekmath 010B}$ employed in this test can be quite small. One example is specification testing. Another example is the situation where tests for correlation are “inverted” to build a confidence interval for ${\Greekmath 011A}$, which is then used for a Bonferroni-type construction of a data-dependent critical value of another test (cf. Leeb2017 for further information concerning such critical values).

Nevertheless, the question remains as to how “large” the set $\mathscr{X}({\Greekmath 010B}; B)$ actually is for a fixed ${\Greekmath 010B}$, such as the conventional ${\Greekmath 010B} = .05$ or ${\Greekmath 010B} = .01$. For example, Proposition (ref) does not tell us whether or not the set of design matrices $\mathscr{X}(.01; B)$ is empty. Similarly, one can ask if $\mathscr{X}(.01; B)$ contains an open set, or if it has positive ${\Greekmath 0116}_{\mathbb{R}^{n \times k}}$ measure? The latter questions have already been considered in detail in the main results of martellosio2012testing for point-optimal invariant and locally best invariant tests in the important context of spatial autoregressive regression models. Adopting his proof strategy, we establish the following proposition. The argument requires a different assumption on $B$ than the one used in Proposition (ref). First, the condition used now concerns the eigenspace of $B(X)$ corresponding to its smallest eigenvalue (as opposed to the condition on the largest eigenvalue used in Proposition (ref)). Second, continuity conditions are imposed, which are required for limiting arguments in the proof. As discussed in Remark (ref) below, the assumptions are again satisfied in the leading choices for $B$ discussed in Remark (ref).

From the results in the present section we learn that for tests that satisfy certain structural properties, the zero power trap arises for generic design matrices for ${\Greekmath 010B}$ small enough. Furthermore, for every ${\Greekmath 010B}$ there exists (under suitable assumptions) a nonempty open set of design matrices every element of which suffers from the zero-power trap. We would like to emphasize, however, that these results do not rule out the possibility that for a given $X$ the actual level ${\Greekmath 010B}$ needed such that the zero-power trap arises can be low (far outside the commonly used range of levels), or that given ${\Greekmath 010B}$ the open set of design matrices for which the zero-power trap occurs is “small”. Numerical results that illustrate the “practical severity” of the zero-power trap in spatial regression models are provided in Section 3 of kramer2005finite, in particular his Table 1 is very interesting in this context, and further discussion and examples can be found in mart10 and martellosio2012testing. These results seem to suggest that the zero-power trap occurs frequently for commonly used levels of significance in case $n-k$ is “small”, i.e., in “high-dimensional” scenarios, whereas if $n-k$ is large the zero-power trap does not appear that frequently. However, this also depends on the dependence structure.

Avoiding the zero-power trap

Having provided some context and motivation, we now discuss three ways to avoid the zero-power trap: In Section (ref) we expand on the observation just made in Remark (ref). The strategy discussed in Section (ref) is based on an idea involving artificial regressors. The method we recommend, however, builds on Section (ref) and is introduced in Section (ref). Our suggestion tries to overcome sub-optimality properties of the other methods. As discussed in the Introduction, the idea underlying our approach can be interpreted as a finite sample variant of the power enhancement principle of fan2015power.

Tests based on $T_B$ with $B = C_X ee' C_X'$

As discussed in Remark (ref), tests based on the test statistic $T_B$ with $B = C_X ee' C_X'$ do not satisfy the assumptions underlying Propositions (ref) and (ref). Hence, these two propositions do not let us conclude anything concerning the question “how often” the zero-power trap occurs for such tests. It turns out that these tests do not suffer from the zero-power trap for any ${\Greekmath 010B} \in (0, 1)$ in case the additional condition $e \notin \limfunc{span}(X)$ holds (note that if $e \in \limfunc{span}(X)$ holds, the test statistic $T_B$ with $B = C_X ee' C_X'$ is useless as it equals $0$ for every $y \in \mathbb{R}^n$). As pointed out in Remark (ref), this was already noted in PP17. For later use in Section (ref) we state a corresponding result (which is an immediate consequence of Part 1 of Proposition 2.26 in PP17 together with $G_X$-invariance of $T_B$ and our Lemma (ref)):

From this result we conclude that in case $e \notin \limfunc{span}(X)$ and whenever a test ${\Greekmath 0127}$ with size ${\Greekmath 010B}$ is subject to the zero-power trap, one can alternatively use the test with rejection region $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$ instead, which does not suffer from the zero-power trap. Moreover, the power of the test $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$ even increases to $1$ as ${\Greekmath 011A} \to a$. This is a desirable property as it matches the intuition that strong correlations should be easily detectable from the data.

While avoiding the zero-power trap problem, the test $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$ suffers from one major disadvantage: the power function of $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$ can be, and often will be, quite low for values ${\Greekmath 011A} \in (0, a)$ distant from $a$. If the initial test ${\Greekmath 0127}$, which was dismissed because it is subject to the zero-power trap, was chosen because of its good power properties in this region of the alternative, the test $\Phi_{C_X ee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$ will then not constitute a convincing alternative. This is illustrated in the example discussed in Section (ref). A method that tries to take optimality properties of the initial test into account, at least for the classes of tests discussed in Remark (ref), is discussed next.

Tests based on artificial regressors

The sufficient condition for the zero-power trap in Theorem (ref) requires that the vector $e$ from Assumption (ref) is not an element of $\limfunc{span}(X)$. While this of course does not prove that the zero-power trap does not arise if $e \in \limfunc{span}(X)$, this indeed turns out to be the case under an additional assumption (cf. Corollary 2.22 in PP17). In this section we shall exploit this fact. The method of avoiding the zero-power trap we discuss in this section “enforces” the condition $e \in \limfunc{span}(X)$. More specifically, it is based on adding the vector $e$ from Assumption (ref) as an “artificial” regressor to the design matrix (if it is not already an element of $\limfunc{span}(X)$), and from then constructing tests as if this artificially expanded design matrix was the true one. As discussed in the Introduction, the idea underlying the construction in the present section can be traced back to kramer85.

To formally describe the artificial regressor based method in our general setting, consider a situation where a researcher initially wants to use the test $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ as in Section (ref) with ${\Greekmath 0115}_1(B) < {\Greekmath 0115}_{n-k}(B)$, but discovers (e.g., by checking the sufficient conditions in Theorem (ref)) that $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ suffers from the zero-power trap. Suppose further that the initial test $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ has certain optimality properties (cf. Remark (ref)). The researcher does not want to completely sacrifice the optimality properties of the initial test, which prevents him from using the test just discussed in Section (ref). Assume further that $e \notin \limfunc{span}(X)$.

The trick now is to work with the design matrix $\bar{X} = (X, e)$ in the construction of a test statistic, assuming that $k+1 < n$. More precisely, let $\bar{B}$ be a symmetric $(n - k - 1) \times (n - k - 1)$ matrix (cf. Remark (ref) below), and define the adjusted test statistic

Under the additional assumption that ${\Greekmath 0115}_1(\bar{B}) < {\Greekmath 0115}_{n-k-1}(\bar{B})$, one obtains\footnote{To obtain this statement one needs to apply Lemma (ref) to model (ref) but with design matrix $\bar{X}$ instead of $X$. Note that this leads to an “enlarged” model that encompasses the true model as a submodel; and that the distributions satisfying the null hypothesis in the true model also satisfy the null hypothesis in the enlarged model.} from Lemma (ref) for every ${\Greekmath 010B} \in (0, 1)$ the existence and uniqueness of a critical value $\bar{{\Greekmath 0114}}({\Greekmath 010B}) \in ({\Greekmath 0115}_1(\bar{B}), {\Greekmath 0115}_{n-k-1}(\bar{B}))$, say, such that for every ${\Greekmath 010C} \in \mathbb{R}^k$ and every ${\Greekmath 011B} \in (0, \infty)$ it holds that

Finally, define the rejection region

We shall now prove that the test with rejection region (ref) does not suffer from the zero-power trap. The following result requires an additional assumption on $\Sigma(.)$. This is Assumption 4 in PP17 to which we refer the reader for equivalent formulations, examples and further discussion.

The main result concerning artificial regressor based tests is as follows:

Theorem (ref) shows that $\bar{\Phi}_{\bar{B}, \bar{{\Greekmath 0114}}({\Greekmath 010B})}$ is not subject to the zero-power trap. However, its “limiting power” $\lim_{{\Greekmath 011A} \to a} P_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}(\bar{\Phi}_{\bar{B}, \bar{{\Greekmath 0114}}({\Greekmath 010B})}) = \mathrm{Pr}(\bar{T}_{\bar{B}}(\Lambda \mathbf{G}) > \bar{{\Greekmath 0114}}({\Greekmath 010B}))$ can in principle be low. In particular, it is always smaller than one. This is different to the behavior of the test discussed in Section (ref), which has limiting power equal to one. Another limitation of Theorem (ref) is its reliance on the additional Assumption (ref).

Following up on the examples discussed in Remark (ref), an advantage of passing from $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ to $\Phi_{\bar{B}, \bar{{\Greekmath 0114}}({\Greekmath 010B})}$, instead of passing from $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ to the test discussed in Section (ref), is that $\Phi_{\bar{B}, \bar{{\Greekmath 0114}}({\Greekmath 010B})}$ “preserves” in some sense the optimality properties of $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$, but with respect to the larger group $G_{\bar{X}}$. Note, however, that this does not imply that the power functions of $\Phi_{B, {\Greekmath 0114}({\Greekmath 010B})}$ and $\bar{\Phi}_{\bar{B}, \bar{{\Greekmath 0114}}({\Greekmath 010B})}$ are “close”.

Optimality-preserving tests that avoid the zero-power trap

The starting point in this section is an (initial) family of tests ${\Greekmath 0127}_{{\Greekmath 010B}}: \mathbb{R}^n \to [0, 1]$ for the testing problem (ref) indexed by ${\Greekmath 010B} \in (0, 1)$. Given ${\Greekmath 010B} \in (0, 1)$ we interpret ${\Greekmath 0127}_{{\Greekmath 010B}}$ as the (initial) test one would like to use because of some optimality property. That is, the power function of ${\Greekmath 0127}_{{\Greekmath 010B}}$ $$({\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}) \mapsto E_{{\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A}}({\Greekmath 0127}_{{\Greekmath 010B}})$$ is “large” for certain parameter values $({\Greekmath 010C}, {\Greekmath 011B}, {\Greekmath 011A})$ in a given subset pertaining to the alternative hypothesis $\{0\} \times (0, \infty) \times (0, a)$.

We shall suppose that the initial test ${\Greekmath 0127}_{{\Greekmath 010B}}$ suffers from the zero-power trap, which one would like to avoid. Ideally a test should have limiting power equal to $1$, a property of the test in Section (ref), but not of the test in Section (ref). Furthermore, we would like to keep, at least approximately, the optimal power properties of ${\Greekmath 0127}_{{\Greekmath 010B}}$, which was the reason why ${\Greekmath 0127}_{{\Greekmath 010B}}$ was considered for use initially. This is a property of the test in Section (ref) (at least to some extent), but not of the test in Section (ref). We shall now present an approach that achieves these two goals.

In what follows, we assume that the family of tests $\{{\Greekmath 0127}_{\Greekmath 010B}\}$ under consideration satisfies Property A, i.e., satisfies the following:

To illustrate the assumption, consider the following important example:

Suppose in the following discussion that $k<n-1$, that Assumption (ref) holds and that $e \notin \limfunc{span}(X)$. Recall from Theorem (ref) that under these conditions the $G_X$-invariant test $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$ does not suffer from the zero-power trap, in fact has limiting power one, at all levels ${\Greekmath 010B} \in (0, 1)$. Using this property, we shall now define a $G_X$-invariant test that has approximately the same power properties of ${\Greekmath 0127}_{{\Greekmath 010B}}$ with the advantage that it has limiting power $1$ just as the test $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 010B})}$.

The basic idea is as follows (precise statements are provided further below): From Property A.3 one obtains that for ${\Greekmath 0122} \in (0, {\Greekmath 010B})$ small, the power functions of ${\Greekmath 0127}_{{\Greekmath 010B}}$ and ${\Greekmath 0127}_{{\Greekmath 010B} - {\Greekmath 0122}}$ are similar. Theorem (ref) tells us that the test with rejection region $\Phi_{C_Xee'C_X', {\Greekmath 0114}({\Greekmath 0122})}$ has limiting power (as ${\Greekmath 011A} \to a$) equal to $1$, and Lemma (ref) shows that this test has size equal to ${\Greekmath 0122}$. Hence, we could use the $G_X$-invariant test

whose power function is similar to ${\Greekmath 0127}_{{\Greekmath 010B}}$ (at least for ${\Greekmath 0122}$ small), but which has limiting power equal to one (for every $0 < {\Greekmath 0122} < {\Greekmath 010B}$). Trivially, this test has size not greater than ${\Greekmath 010B}$, but potentially its size is smaller than ${\Greekmath 010B}$, implying some unnecessary loss in power, which one can try to avoid by decreasing ${\Greekmath 0114}({\Greekmath 0122})$.

More specifically, define the $G_X$-invariant test

where $0 < c({\Greekmath 010B}, {\Greekmath 0122}) \leq {\Greekmath 0114}({\Greekmath 0122})$ is chosen to be the smallest number such that ${\Greekmath 0127}^*_{{\Greekmath 010B}, {\Greekmath 0122}}$ has size equal to ${\Greekmath 010B}$. That such a choice of $c({\Greekmath 010B}, {\Greekmath 0122})$ is indeed possible is the content of the next proposition. Note that ${\Greekmath 0127}^*_{{\Greekmath 010B}, {\Greekmath 0122}}$ is non-randomized if the test ${\Greekmath 0127}_{{\Greekmath 010B} - {\Greekmath 0122}}$ is non-randomized.

Note that the critical value $c({\Greekmath 010B}, {\Greekmath 0122})$ can be easily determined numerically by a simple line search algorithm, cf. also Remark (ref).

Having established that the test in Equation (ref) is actually well-defined, we now prove that it does not suffer from the zero-power trap but has limiting power $1$ for any choice of ${\Greekmath 0122}$. Furthermore, we show that the power function of ${\Greekmath 0127}^*_{{\Greekmath 010B}, {\Greekmath 0122}}$ approximates (even uniformly over suitable subsets of the parameter space) the power function of ${\Greekmath 0127}_{{\Greekmath 010B}}$ as ${\Greekmath 0122}$ converges to $0$. In this sense, choosing ${\Greekmath 0122} > 0$ small, the test ${\Greekmath 0127}^*_{{\Greekmath 010B}, {\Greekmath 0122}}$ preserves “optimal” power properties (such as point-optimal invariance, or locally best invariance, cf. Example (ref) above) from ${\Greekmath 0127}_{{\Greekmath 010B}}$ at least approximately. Furthermore, the degree of approximation can be tuned by the user via ${\Greekmath 0122}$.

Numerical results

In order to illustrate and compare the power properties of the tests introduced in Section (ref), we now consider a simple example from spatial econometrics in which the zero-power trap occurs for a popular test. We focus on a situation where the correlation between the observations is a consequence of their proximity, which might be spatial, but could also be, e.g., social, and which is encoded in the adjacency (“weights”) matrix of a graph.

One important model in this case is the spatial (autoregressive) error model, which leads to $$\Sigma({\Greekmath 011A}) = [(I-{\Greekmath 011A} W')(I-{\Greekmath 011A} W)]^{-1},$$ for $W$ a fixed weights matrix which is assumed to be (elementwise) nonnegative and irreducible with zero elements on the main diagonal. By the Perron-Frobenius theorem (e.g., HJ1985, Theorem 8.4.4), the matrix $W$ then has a positive (real) eigenvalue ${\Greekmath 0115}_{\max}(W)$, say, with algebraic multiplicity (and thus also geometric multiplicity) equal to 1, such that any other real or complex zero of the characteristic polynomial of $W$ is in absolute value not larger than ${\Greekmath 0115}_{\max}(W)$. We assume that the parameter ${\Greekmath 011A} \in [0, {\Greekmath 0115}_{\max}(W)^{-1})$. For $f_{\max}$ a normalized eigenvector of $W$ w.r.t. ${\Greekmath 0115}_{\max}(W)$ it is not too difficult to see that Assumption (ref) is satisfied (with $e = f_{\max}$), and that Assumption (ref) is satisfied. For details we refer to Section 4.1 in PP17.

The model depends, besides the design matrix $X$, on the specific form of the weights matrix $W$, which encodes the dependence relation of the observations. Subsequently we reconsider a simple example considered in Section 3 of kramer2005finite, who has observed (cf. his Figure 1) that for a weights matrix derived by the Queen criterion from a $4 \times 4$ regular lattice, and for $X = (1, \hdots, 1)' \in \mathbb{R}^{16}$ the Cliff-Ord test suffers from the zero-power trap for ${\Greekmath 010B} = 5\%$. We recall that the Cliff-Ord test is based on a test statistic as in Equation (ref) and with $B = C_X(W + W')C_X'$.

The power function of the Cliff-Ord test and the power functions of the tests described in Section (ref) were obtained numerically (cf. also Remark (ref)), and are shown in Figure (ref). The figure also shows the power envelope in the class of $G_X$-invariant tests. That is, for each alternative $\bar{{\Greekmath 011A}} \in (0, {\Greekmath 0115}_{\max}(W)^{-1})$ Figure (ref) shows the power of the point-optimal $G_X$-invariant level ${\Greekmath 010B} = 5\%$ test against the alternative $\bar{{\Greekmath 011A}}$. Recall from Remark (ref) that the point-optimal invariant test against alternative $\bar{{\Greekmath 011A}}$ is based on a test statistic as in (ref) and with $B = -[C_X\Sigma(\bar{{\Greekmath 011A}})C_X']^{-1}$. In this example the power envelope is not attained by any $G_X$-invariant test, but it serves the purpose of providing an upper bound for comparison.

While Figure (ref) illustrates that the approaches discussed in Sections (ref) and (ref) avoid the zero-power trap, it reveals at the same time that the power functions of these tests are not completely satisfying. On the one hand, even though the test introduced in Section (ref) does not suffer from the zero-power trap, it has low power in a large region of the alternative. On the other hand, the test from Section (ref) based on the Cliff-Ord test (i.e., as in Equation (ref) with $\bar{B} = C_{(X, e)} (W + W') C_{(X, e)}'$) with artificial regressor $e = f_{\max}$ avoids the zero-power trap as well and has a power function that practically coincides with the power envelope for small values of ${\Greekmath 011A}$. But its limiting power is smaller than one (in fact is only $0.619$).

Figure (ref) also contains the power function of some tests corresponding to the procedure outlined in Section (ref) applied to the family ${\Greekmath 0127}_{{\Greekmath 010B}}$ of level-${\Greekmath 010B}$ Cliff-Ord tests (cf. Example (ref)). It shows the power functions corresponding to ${\Greekmath 0122} \in \{.002, .006, .01\}$. These tests have very good power properties. The power functions are practically identical to the one of the Cliff-Ord test (and hence to the power envelope) for small values of ${\Greekmath 011A}$. But for larger values of ${\Greekmath 011A}$ their power function is much closer to the power envelope than the power of the Cliff-Ord test. In particular, by construction, their power converges to $1$ as ${\Greekmath 011A}$ gets close to $a$. One can also observe that smaller values of ${\Greekmath 0122}$ lead to power functions that are closer to the power function of the Cliff-Ord test for ${\Greekmath 011A}$ close to $0$, whereas larger values of ${\Greekmath 0122}$ lead to power functions that are closer to the power envelope for ${\Greekmath 011A}$ close to $a$.

Conclusion

In the present article we have re-considered the zero-power trap phenomenon in testing for correlation in a general framework. Most importantly, we have suggested a way to construct “approximately optimal tests” that avoid the trap. For practical purposes, if an initial test, such as the Cliff-Ord test in the example discussed in Section (ref), turns out to suffer from the zero-power trap, we suggest to use the method introduced in Section (ref) to obtain a modified test with the following properties: (i) it has a similar power function as the initial test, (ii) it does not suffer from the zero-power trap, and (iii) its limiting power equals one. The tuning parameter ${\Greekmath 0122}$ involved in the construction of the modified test can be chosen by graphically comparing the power functions of modified tests corresponding to different values of the tuning parameter with the power envelope and the power function of the initial test. The heuristic underlying our construction can be interpreted as a finite sample variant of the power enhancement principle of fan2015power. The approach, which is not restricted to the testing problem under consideration, might be of some interest in its own right.