2. MODEL SELECTION IN THE PRESENCE OF UNDERLYING PARAMETER INSTABILITY

2.1. Heuristics

To gain some intuition about the results in this paper, consider a simple example where the data generating process (DGP) is as follows and the time of the break is known:

If the researcher is interested in testing whether the parameter β_t is constant over time and equal to a specific value β₀, a possible test statistic would be

where

are the sample averages of y_t in the two subsamples. By adding and subtracting the full sample average

inside the square of the first addend on the numerator, (2) can be rewritten as

where

Thus, the test is decomposed in two components: the one on the left is a test on β, and the one on the right is the standard Chow test for structural break. Hence, the test achieves power in detecting deviations from β₀ by adding to the traditional test for structural break a component that is variant to constant shifts in the mean. The asymptotic distribution of the test can easily be found in this case because the two components are independent.¹

In fact, the standard Chow test can be rewritten as a Wald test:

are independent. To see why, note that

.

Let B₁(.) denote a scalar Brownian motion and BB₁(.) denote a scalar Brownian bridge and π = [τ/T]. Note that the first component on the right-hand side in (3) is asymptotically the square of a standardized normal (B₁(1)²), whereas the distribution of the second component is known from Andrews (1993) to be BB₁(π)²/π(1 − π). As a result, the asymptotic distribution of this modified Chow test will be

Thus, the first component, which makes the test powerful in detecting constant shifts in the mean, adds a chi-square component to the limiting distribution of a standard Chow test for parameter instability. This example provides an easy and intuitive explanation of the asymptotic distribution of the tests considered in this paper.

2.2. Framework

This section describes the class of models considered in this paper and the assumptions under which the results are valid. The parametric model applies to a stationary and ergodic time series process.

Assumption 1. For each T, the sequence {x_t,T}_t=1^T consists of the first T elements of an r-dimensional stationary and ergodic process. The parameter space Θ is a compact subset of R^k. For notational simplicity, x_t will be used to denote x_t,T.

The class of local alternatives allows both for structural changes and for nonlinear hypotheses on the parameters.

Assumption 2 (Local alternatives). The local alternatives (H_AT) are specified as

where g(γ,π,s), for s ∈ [0,1], is a k-dimensional step function, γ ∈ Rⁱ, π ∈ (0,1)^j denotes the times of the structural changes as fractions of the sample size (j being the number of such breaks), a(θ*) = 0 is a possibly nonlinear restriction that identifies the true parameter value under the null hypothesis when there is no structural change, and a denotes its local alternative.

Hence, the parameter θ is unknown and possibly time-varying. The class of estimators considered here are extremum estimators that minimize the objective function Q_T(θ), which depends on both the data and the sample size. The focus will be on the restricted estimator

:

where

is the sample analogue of E(f (x_t,θ)), the moment condition that is equal to zero at the true parameter value, and E(.) is the expected value function. The moment condition is such that f : R^r × R^k → R^m and W_T is a (sequence of) positive semidefinite matrices. Note that our framework allows for both exactly and overidentified generalized method of moments (GMM).

The next assumptions are sufficient to ensure consistency and identification of the estimator (see Sowell, 1996, p. 1100; see also Andrews, 1993).²

The assumptions used in this paper are stronger than necessary, and the results are expected to hold if the assumptions are relaxed as in Andrews (1993). I thank G. Elliott and U. Muller for pointing this out.

Furthermore, the class of estimators is restricted to efficient GMM estimators, and Assumption 6 provides a sufficient condition for efficiency.

Assumption 3 (Identification). lim_T→∞ E [f (x,θ)] = 0 only if θ = θ*.

Assumption 4 (Smoothness and boundedness). (i) θ* ∈ interior(Θ); (ii) f (x,θ) is continuously partially differentiable in a neighborhood ϒ of θ*, ∀θ ∈ Θ; (iii) the functions f (x,θ) and ∇_θ f (x,θ) ≡ (∂/∂θ) f (x,θ) are measurable functions of x ∀θ ∈ Θ and E [∥ f (x,θ*)∥²] is finite; (iv) E [f (x_t,θ*)] = 0,E [f (x_t,θ*)′f (x_t,θ*)] < ∞, and sup_θ∈Θ∥ f (x_t,θ)∥ < ∞ ∀t = 1,…,T and T = 1,2,…; each element of f (x_t,θ_t,T) is uniformly square integrable ∀t = 1,…,T and T = 1,2,…; (v) M = lim_T→∞ E [∇_θ f (x,θ*)] ∈ R^m×k has full column rank, where ∇_θ f (x,θ*) = (∂/∂θ) f (x,θ)|_θ=θ* and M′W_T M is nonsingular; (vi) {x_t} is strong mixing with strong mixing coefficients {α(n)^1−2/β} < ∞ with β > 2, and the individual elements of f (x_t,θ_t,T) have finite absolute moments E [| f⁽ⁱ⁾(x_t,θ_t,T)|^β] for i = 1,…,m.

Assumption 5 (Constraints). a(θ) is continuously partially differentiable in a neighborhood ϒ of θ*, ∀θ ∈ Θ; A ≡ ∇_θ a(θ*) ∈ R^r×k has rank r ≤ k.

Assumption 6 (Efficiency in the class of GMM estimators). The asymptotic variance of the GMM estimator is efficient in the class of GMM estimators:

.

When the alternative hypothesis of interest is either H_AT⁽¹⁾ or H_AT⁽²⁾ then optimal tests are available. In the former case, an optimal test when the break date is known is the Chow (1960) test, and when the break date is unknown, a class of tests with optimal weighted average power is that of Andrews and Ploberger (1994); although Andrews' Sup-LR test (see Andrews, 1993) is not a member of that class, Andrews and Ploberger, 1993, show its asymptotic admissibility against alternatives that are sufficiently distant from the null hypothesis). In case the alternative is H_AT⁽²⁾ only, the likelihood ratio test (and the asymptotically equivalent Wald and Lagrange multiplier [LM] tests) is asymptotically locally most powerful among all invariant tests, and, hence, it is optimal (see Engle, 1984).

However, when both hypotheses are of interest then considering separately tests for parameter instability and likelihood ratio tests is not sufficient anymore. This paper identifies a class of tests that are optimal, in the sense of having the highest asymptotic local power function for some specified alternatives. This class of tests is discussed in Section 2.3.

2.3. Optimal Tests

We are interested in constructing a LM test statistic for testing jointly alternatives H_AT⁽¹⁾ and H_AT⁽²⁾. The test builds on partial sums of the form

where the partial sums are evaluated at the restricted estimator vector,

. When Assumptions 1–6 are satisfied, the asymptotic distribution of the partial sums of sample moments under the null and the alternative hypotheses is stated in Results 1 and 2, which follow. For notational convenience, let M ≡ Σ^−1/2M ∈ R^m×k and partition it as M = (M_β, M_δ). Also, let ⇒ denote weak convergence to the relevant stochastic process and

denote convergence in probability.

RESULT 1 (Distribution under the alternative hypothesis). If Assumptions 1–6 are satisfied, then

where B_m(·) is an m-dimensional standard Brownian motion, D′ ≡ D⁻¹A′ ×(AD⁻¹A′)⁻¹, D ≡ M′M, H ≡ I_k − D^−1/2A′(AD⁻¹A′)⁻¹AD^−1/2, I_k is a k-dimensional identity matrix, H ≡ MD^−1/2HD^−1/2M′, and both H and H are idempotent with rank equal to (k − r).

See Appendix A for proofs. Result 2 shows the asymptotic distribution of the standardized moment condition under the null hypothesis that there is no parameter instability in any of the coefficients and that a subset of parameters satisfies some restriction condition as follows.

Assumption 7 (Null hypothesis). Under the null hypothesis (H₀):

where θ* satisfies a(θ*) = 0.

RESULT 2. (Distribution under the null hypothesis). If Assumptions 1 and 3–7 are satisfied then

for an orthonormal matrix C such that

. Here BB_k−r(s) is a (k − r)-dimensional Brownian bridge and [B_r(s)′,B_m−k(s)′]′ is an (m − k + r)-dimensional vector Brownian motion. The Brownian motions and the Brownian bridges are independent.

Note that, under the null hypothesis, the asymptotic distribution of the standardized partial sum of moment conditions is composed by both Brownian bridges and Brownian motions. The (k − r)-dimensional Brownian bridge component derives from the parameters that are not specified under the null. In fact, this component is a partial sum of mean zero moment conditions, where the zero mean is obtained by estimating the drift.³

For example, when the process is univariate and such that y_t = β₀ + ε_t, ε_t ∼ i.i.d. N(0,1), β₀ = 0 (as in the introductory example at the beginning of the paper) then the partial sum of moments is

and the origin of the Brownian bridge is evident. If there were no restrictions under the null hypothesis, then the asymptotic distribution of

would be (BB_q(s)′ B_m−q(s)′)′, which is the Sowell (1996) result. When there are restrictions on a subset of p parameters under the null hypothesis, these will show up as p-Brownian motions, in addition to the previous components. These are Brownian motions because they are the limiting distribution of a partial sum of mean zero moment conditions, where the zero mean is obtained by imposing, rather than estimating, the drift. In the previous example, in this case the partial sum of moments is

and the origin of the Brownian motion is clear. The B_m−q(s) component corresponds to the overidentified moment restrictions.

The alternative hypothesis will add drift components to the moment conditions, as Result 1 shows. In particular, the drift components originate both from deviations from the parameter stability hypothesis and from deviations from the specified null hypothesis on the value of the parameters. For the local alternatives considered in this paper, the normalized partial sum of the sample moments evaluated under the null hypothesis converges to a stochastic process denoted by Z(s). Under the local alternative, Z(s) satisfies the following stochastic differential equation:

where

. Under the null hypothesis, the same expression holds with v(s) = 0. To get some insight, rearrange (7):

⁴

The result follows because MD^−1/2HD^1/2 = MD^−1/2HD^−1/2M′M = HM, HMD′ = 0 and (I − H)MD′ = MD′, which can be verified by direct calculations.

so that

where C⁽¹⁾ and C⁽²⁾ are, respectively, the first (k − r) and the last (m − k + r) rows of C. Thus, the null hypothesis puts restrictions on both the Brownian motions and the Brownian bridge components. In fact, it is a joint hypothesis on parameter instability (affecting the Brownian bridge component) and on the parameters (affecting the Brownian motion component). This differs from the Sowell (1996) case (see the discussion following his eqn. (3), p. 1091), where the alternative only places restrictions over Brownian bridges. However, Sowell (1996) derived optimal tests in terms of the Radon–Nikodym derivative of the measure implied by the null hypothesis for both the Brownian motion and the Brownian bridge components (see the proof of his Thm. 3), so we can apply a similar argument. Thus, the test with the greatest weighted average power, according to some weighting functions R(η,π) (on η for every π) and J(π) (on π), rejects the joint null hypothesis of no structural break and a(θ*) = 0 if

η ≡ [a,γ′]′ ∈ R^2p×1, and k_α is defined so that the test has size α.

3. SPECIAL TESTS⁵

I consider only two-sided alternatives here; one may generalize the argument to one-sided alternatives.

The leading case of the class of alternatives for structural break is that of alternatives that are linear in the parameters, that is,

. In the case of a single structural break,

, where 1(s ≥ π) is the indicator function, equal to one if s ≥ π and zero otherwise, and G is a (k × p) matrix identifying the p-dimensional vector of time-varying parameters, say, G = [I_p 0_q×p]. Let us define

The optimal test statistic described by (11) becomes ∫∫ exp{η′A(π) − ½η′V(π)η} dR(η,π) dJ(π). As in Sowell (1996), different choices of the weighting function R(η,π) lead to different test statistics. The weighting function considered here is an (r + p)-dimensional multivariate normal distribution with zero mean and covariance U(π). When the time of the break is not known and we are interested in the test statistic that gives equal weight to alternatives that are equally difficult to detect when π is known, so that U(π)⁻¹ = (1/c)V(π), then the test statistic in (11) becomes ∫_Π(exp{½(c/(1 + c))Φ*(π)}) dJ(π) where Π is the support of J(π) and Φ_T*(π) = A(π)′V(π)⁻¹A(π). The latter is a Wald test for the fixed and known π scenario. The test statistic can be estimated as

where

if f_t(.) are independent and identically distributed (i.i.d.); otherwise

is estimated with a Newey–West heteroskedasticity and autocorrelation consistent (HAC) estimator. The limiting distribution of this test statistic under the null hypothesis is described in the following proposition.

PROPOSITION 1. Let Assumptions 1–6 hold. The test statistic for testing a(θ*) = 0 against H_AT⁽¹⁾ and H_AT⁽²⁾ with the greatest average power according to the weighting function R(η,π) ∼ N(0,cV(π)⁻¹), for V(π) defined in (14), is the test statistic defined in (15). Its asymptotic distribution under the null hypothesis is

Proposition 1 shows that TS_c,T^AP* is a weighted average of LM tests. As noted previously, the difference between the asymptotic distribution of the tests defined in this paper and that of the test for structural break only is that the latter does not have the B_r(1)′B_r(1) component. This component arises from testing restrictions on θ over the whole sample. In fact, it corresponds to a centered chi-square with r degrees of freedom, the usual limiting distribution of the Wald test statistic for testing hypotheses on a parameter vector. Appendix A shows that both the tests for structural break and the classical tests obtain as special cases of (24).

Although this paper is mainly concerned about testing a null hypothesis on the parameters in the presence of possible parameter instability, instabilities may also affect other aspects of the model, namely, the overidentifying restrictions (OIRs). Hall and Sen (1999) and Sowell (1996) show that the population moment conditions can be decomposed into two orthogonal components: identifying restrictions—the part used in estimation—and overidentifying restrictions—the part unused in estimation. Hall and Sen (1999) propose tests for the structural stability of the OIRs. Their approach turns out to be useful to discriminate between situations in which the instability is confined to the parameters alone and those in which the instability permeates other aspects of the model. In what follows, we examine the relationship between the tests proposed by Hall and Sen (1999) and those proposed in the present paper, and we show that the Hall and Sen (1999) results do carry over to the tests proposed in this paper.

The tests proposed by Hall and Sen (1999) are as follows:

where

. Hall and Sen (1999) find that their test statistics are asymptotically independent of tests for parameter instability. They also show that their tests have no local power against parameter variation and tests for parameter variation have no local power against instability in the OIRs. This latter result follows from the fact that the components of O_T(π) are orthogonal to the components of LM₂(π). In fact, they show that

(see Hall and Sen, 1999, eqn. (A.4); Andrews, 1991, p. 848, for the more general case of subsets of parameters), where [otimes ] denotes the Kronecker product. Thus, the rescaled moment conditions in O_T(π) become

and it is clear that they are orthogonal to LM₂(π), which instead builds on M (see equation (16)), as M′(I − M(M′M)⁻¹M′) = 0.

Note that all of the preceding results still hold in our framework. Also, note that the components in LM₁ are orthogonal to the components of O_T(π) too, as LM₁, like LM₂(π), builds on M. Thus, the results in Hall and Sen (1999) do carry over to the test statistics QLR_T*, Mean-Wald_T*, and Exp-Wald_T*. More details are provided in Section 4.

From now until the end of this section, we specialize the preceding findings to situations in which the researcher is interested in testing hypotheses on a subset of the parameters. This is discussed in the following corollary.

COROLLARY 1 (Null hypotheses on subsets of parameters). Let the parameter vector θ ∈ R^k be partitioned as θ = [β′,δ′]′, where β ∈ R^p and δ ∈ R^q. Let Assumptions 1 and 3–6 hold. Let Assumption 2 be replaced by Assumption 2′:

. It follows that

We will finally consider special cases of TS_c,T^AP* that have been considered in the literature for tests for structural break only. Each of these special cases has greatest weighted average power against particular forms of parameter instability. We will analyze the form that the optimal test proposed in this paper assumes for these particular forms of parameter instability.

Andrews and Ploberger Test.

Let β_t = β₁(π) for t = 1,2,…,[Tπ] and β_t = β₂(π) for t = [Tπ] + 1,…,T, where [·] denotes the greatest integer function. Also, to simplify notation, let

. Let

be the unrestricted GMM estimator under the hypothesis that there is a break at the fraction [Tπ] of the sample and let

be the constrained estimator. Thus, the Wald test for a fixed and known π can be estimated as either

⁶

For completeness, let us mention that the LM statistic can also be obtained as

where

is a consistent estimator of

. However, the LM formula provided in the main text is easier to calculate.

where notation is in Table 1. The table assumes that f_t(.) consists of mean zero uncorrelated random variables. When f_t(.) consists of mean zero but serially correlated random variables then consistent estimation of

requires a HAC estimator (e.g., see Newey and West, 1987). Note that (22) is particularly easy to calculate. It is simply the sum of the two LM tests to test H_AT⁽¹⁾ and H_AT⁽²⁾ separately. Then, Proposition 2 follows.

Notation related to equations (20), (21), and (22)

PROPOSITION 2. Let Assumptions 1, 2′, and 3–6 hold. The test statistic for testing β = β* against

with the greatest average power according to the weighting function R(η,π) ∼ N(0,cV(π)⁻¹), for V(π) defined in (14) with either (20), (21), or (22). Its asymptotic distribution under the null hypothesis is

As special cases, we have

The special cases that correspond to extreme values of the parameter c are similar to those in Andrews and Ploberger (see also Andrews, Lee, and Ploberger, 1996). When c → ∞ (c → 0), more weight is assigned to alternatives about parameter instability further from (closer to) the null hypothesis.

Andrews (1993) Sup-LR Test.

A test statistic commonly considered in the literature of structural breaks is the Quandt likelihood ratio (QLR) test statistic (or Sup-LR test), which is the supremum (over all possible break dates) of the Chow statistic designed for these alternatives for a fixed break date. Andrews (1993) derived its asymptotic distribution. The modified QLR test statistic for the alternatives specified in this paper can be obtained by letting c/(1 + c) → ∞ in (17), which gives

The limiting distribution of (27) under the null hypothesis is given in the following proposition.

PROPOSITION 3. Let Assumptions 1, 2′, and 3–6 hold. The test statistic for testing β = β* against

with the greatest average power according to the weighting function R(η,π) ∼ N(0,cV(π)⁻¹), for V(π) defined in (14) and c such that c/(1 + c) → ∞, is (27), whose asymptotic distribution under the null hypothesis is

Nyblom (1989) Test.

Another test for parameter instability is that considered by Nyblom (1989) and Nyblom and Mäkeläinen (1983). These authors derive the locally most powerful invariant (to translations and scale transformations) test for constancy of the parameter process against the alternative that the parameters follow a random walk process:⁷

The notation is the same as in Nyblom and Mäkeläinen (1983);

is a known matrix and σ_e² is a scalar. See also King (1980), King and Hillier (1985), and Stock and Watson (1998).

The modified Nyblom test statistic for testing whether β_t is equal to β₀ is

where

and the gradient of the objective function is defined as

Note that (30) is a generalization of the locally best invariant test statistic proposed by Nabeya and Tanaka (1988) for the case in which β is known and equal to β₀. The test proposed in this paper is more general than that of Nabeya and Tanaka, as estimation is not restricted to the ordinary least squares case and β can be a vector. Appendix A shows that the asymptotic distribution of the modified Nyblom statistic under the null hypothesis is as follows.

PROPOSITION 4. Let Assumptions 1, 2′, and 3–6 hold. The test statistic for testing β = β₀ against

with the greatest average power according to the weighting function R(η,π) ∼ N(0,cV(π)⁻¹), for V(π) defined in (14), is (30). Its asymptotic distribution under the null hypothesis is

Tables B1, B2, B3, and B4 in Appendix B report critical values for the optimal tests for J(π) uniformly distributed on [0.15, 0.85].⁸

Trimming values are required. See Andrews (1993).

The significance levels considered in the tables are 10%, 5%, 2.5%, and 1%. The critical values are obtained by simulating the asymptotic distributions described in this section. The number of Monte Carlo replications is 5,000. Notice that all the values in Tables B1, B2, and B3 are higher than those for the corresponding tests for structural break only, the reason being that the optimal tests add the nonnegative component B_p(1)′B_p(1) (see equation (24)).

4. ASYMPTOTIC LOCAL POWER ANALYSIS

The local power properties of the optimal tests derived previously can be compared with those of tests for parameter instability only and those of tests for a(θ*) = 0 only. The comparison can be made both theoretically and by Monte Carlo simulations.

Let us first consider the theoretical local power properties of the various tests. To facilitate a comparison with the tests existing in the literature, we focus on the tests discussed in the second part of Section 3, and, for brevity, we analyze only (25).

. From (22) and (25), and using the notation in Table 1, we have that

Appendix A shows that

See also Appendix A for more details. Note that

.

To verify these insights, we perform some Monte Carlo simulations. A variety of DGPs is considered, paying particular attention to situations where the standard tests fail to detect the alternative hypothesis. For simplicity, only a univariate model is considered:

The likelihood ratio LR₁ tests whether the parameter equals β₀, whereas parameter instability tests check whether β_t,T is constant; optimal tests jointly test the two hypotheses. The parameter instability tests (TVP) considered here are the Andrews and Ploberger exponential Wald tests (Exp-Wald_T), the Nyblom test (Nyblom_T), and the Quandt likelihood ratio (QLR_T). The optimal tests are Exp-Wald_T*, QLR_T*, and Nyblom_T* defined in Section 3. The nominal size is 5%.

We consider the following DGPs.¹¹

Figure 1a shows the asymptotic local power of the tests as a function of β_A. It shows that when the parameter is not time-varying, the likelihood ratio LR₁ is the most powerful test, according to the Neyman and Pearson lemma. The test designed to detect structural break, Exp-Wald_T, has a flat power function around the size of the test, whereas the Exp-Wald_T* test is almost as powerful as the LR₁ test.

Asymptotic power functions of 5% tests for Designs 1–4.

Design 2 involves a single break in the data. This particular alternative is both a deviation of the parameter vector from the null hypothesis and a structural break, so all the tests (the most powerful likelihood ratio test, LR*,¹²

Figure 1c shows that, in this design, the shift in the parameter vector is not detected by a simple likelihood ratio (LR₁) because the statistic on which it is based (the average of the observations) is invariant to it; in fact, notwithstanding the structural break, the average over the whole sample is asymptotically equal to β₀. Although the TVP test is the most powerful, the optimal test is powerful too.

Design 4. β_t = β₀ + β_t−1 + u_t, where u_t ∼ N(0,σ_u²/T²) is independent from ε_t and σ_u² ≥ 0

The asymptotic local power functions for this design are depicted in Figure 1d as functions of the parameter σ_u² ≥ 0. When σ_u² = 0 then β_t is constant, whereas when σ_u² ≠ 0 then β_t is a random walk with no drift. The test designed for this hypothesis is the Nyblom test; the LR₁ test is also powerful. The reason is that LR₁ is detecting deviations from the null hypothesis by comparing the sample average with the null hypothesis and the sample average is not a consistent estimate of the true parameter value. Note that the optimal Nyblom test is powerful too.

The results of the simulations suggest the following conclusions. First, the tests that maintain some power across all the designs considered here are the optimal tests. For all the other tests there is at least one design (a particular direction away from the null hypothesis) in which the power is flat around the size of the test. Hence, they are not “robust” across designs, whereas the optimal tests are. Second, let us consider a two-stage testing procedure, where the first stage tests whether there is a structural break (by using either QLR_T or Exp-Wald_T) and the second stage, conditionally on the first stage, tests hypotheses on the parameters (by using the LR₁ test). Let the tests be labeled “Seq.QLR” and “Seq.Exp-Wald,” respectively. In the special cases considered in Section 3 (obtained with particular weighting matrices), the two stages of the test are asymptotically independent. By choosing a size equal to

, the joint significance level will be the desired nominal level, 0.95. Figure 2 shows that there is no clear ranking between the sequential tests and the optimal tests. The power ranking will depend on the direction of the alternative hypothesis. However, by construction, the optimal tests will have the greatest average local power. Two-stage independent tests have advantages and disadvantages. The advantages are that if we reject we know which part of the alternative we reject and that the first-stage test could be used if the researcher is unsure about which elements of the parameter vector are subject to instability. The disadvantage is that they will not have the optimal weighted average power for alternatives that are equally likely; in other words, if we want tests that are invariant to nonsingular linear transformations of the hypothesis, we cannot construct the test as formed by two independent components, as two-stage tests are not invariant to these transformations.

Comparison of asymptotic power functions of 5% selected optimal tests with the naive sequential test (across different designs).

Finally, to investigate the properties of the Hall and Sen (1999) test for OIRs, we consider the following experiment.

Design 5. This design introduces instability in the OIRs. We introduce one OIR by using the following set of moment conditions: f_t = (ε_t,z_t*ε_t)′ where z_t* is an instrument such that z_t* = z_t + ε_t(β_A·1(t ≤ ½T) − β_A·1(t > ½T)) and z_t ∼ N(0,1) is independent of ε_t. Note that when β_A = 0 then the OIR is valid and stable; on the other hand, it becomes unstable when β_A ≠ 0. Figure 3 compares the power functions of the Exp-Wald_T, the Exp-Wald_T*, and the Exp-O_T tests for this design (see Figure 3d). To explore the properties of the Exp-O_T test in the presence of parameter instability or constant shifts in the parameters, the figure also compares these tests in Designs 1–3 (where all the tests in Figure 3 now build on the two-dimensional moment condition). Figure 3 clearly shows that the Exp-Wald_T, Exp-Wald_T*, and Exp-O_T tests have power only against deviations from their specific null hypotheses. In particular, the Exp-Wald_T and Exp-Wald_T* do not have power against instabilities in the OIRs, and the Exp-O_T test does not have power against parameter instability or against the joint hypotheses considered in this paper, (H_AT⁽¹⁾) and (H_AT⁽²⁾).

Comparison of asymptotic power functions of 5% selected optimal, TVP, and OIRs tests (across different designs).

5. SUMMARY COMMENTS

This paper shows that there exists a class of locally most powerful tests for testing the joint hypothesis of model selection between two nested models and parameter stability. This paper introduces this class of tests, states the assumptions under which they are valid, and works out their asymptotic distributions. It also derives some special cases that apply for specific forms of parameter instability. These tests are easy to calculate, and this paper reports their (asymptotic) critical values. Joint tests such as the ones developed in this paper could also be adapted to the case of multiple breaks, along the lines of Bai and Perron (1998) and Elliott and Müller (2003). We leave this issue for future research.