2.1. Model

We consider models specified by a finite number of moment restrictions. Let {z_i : i = 1,…,n} be R^l-valued data and, for each n ∈ N, g_n : G × Θ → R^k a given function, where G ⊂ R^l and Θ ⊂ R^p denotes the parameter space. The model has a true parameter θ₀ for which the moment condition

is satisfied. For g_n(z_i,θ) we will usually write g_i(θ).

Example 1 (i.i.d. linear IV regression)

Guggenberger (2003, Ch. 1) discusses in detail GEL estimation and testing for this model under weak identification. The structural form (SF) equation is given by

and the reduced form (RF) for Y by

where y,u ∈ Rⁿ, Y,V ∈ R^n×p, Z ∈ R^n×k, and Π ∈ R^k×p. The matrix Y may contain both exogenous and endogenous variables, Y = (X,W) say, where X ∈ R^n×p_X and W ∈ R^n×p_W denote the respective observation matrices of exogenous and endogenous variables. The variables Z = (X,Z_W) constitute a set of instruments for the endogenous variables W. The first p_X columns of Π equal the first p_X columns of I_k, and the first p_X columns of V are 0. Denote by Y_i, V_i, Z_i,…(i = 1,…,n) the ith row of the matrix Y, V, Z,… written as a column vector. Assuming the instruments and the structural error are uncorrelated, Eu_i Z_i = 0, it follows that Eg_i(θ₀) = 0, where for each i = 1,…,n, g_i(θ) := (y_i − Y_i′θ)Z_i. Note that in this example g_i(θ) depends on n if the RF coefficient matrix Π is modeled to depend on n (see Staiger and Stock, 1997), where Π_n = n^−1/2C for a fixed matrix C.

Example 2 (conditional moment restrictions)

As in Stock and Wright (2000) the moment conditions may result from conditional moment restrictions. Assume E [h(Y_i,θ₀)|F_i] = 0, where h : H × Θ → R^k₁, H ⊂ R^k₂, and F_i is the information set at time i. Let Z_i be a k₃-dimensional vector of instruments contained in F_i. If g_i(θ) := h(Y_i,θ) [otimes ] Z_i, then Eg_i(θ₀) = 0 follows by taking iterated expectations. In (2.1), k = k₁ k₃ and l = k₂ + k₃.

2.2. Assumptions

This section is concerned with the asymptotic distribution of the GEL estimator for θ when some components of θ₀ = (α₀′,β₀′)′, α₀ say, α₀ ∈ A, A ⊂ R^p_A, are only weakly identified. Intuitively, this means that the moment condition (2.1) is not very informative about α₀. For parameter vectors θ = (α′,β₀′)′, Eg_n(z_i,θ) may be very close to zero, not only for α close to α₀ but also when α is far from α₀. In that case, the restriction Eg_n(z_i,θ₀) = 0 is not very helpful for making inference on α₀. Assumption ID, which follows, provides a theoretical asymptotic framework for this phenomenon, which is taken from Assumption C in Stock and Wright (2000, p. 1061). We refer the reader to Stock and Wright (2000, pp. 1060–1061), which provides substantial detailed motivation for this assumption and an explanation of why it models α₀ as weakly and β₀ as strongly identified.

To describe the moment and distributional assumptions, we require some additional notation:

where, if defined, G_i(θ) := (∂g_i /∂θ)(θ) ∈ R^k×p. For notational convenience, a subscript n has been omitted in certain expressions. Define the k × k matrices¹

Let θ = (α′,β′)′, where α ∈ A, A ⊂ R^p_A, β ∈ B, B ⊂ R^p_B, and p_A + p_B = p. Also let

denote an open neighborhood of β₀.

Assumption Θ. The true parameter θ₀ = (α₀′,β₀′)′ is in the interior of the compact space Θ = A × B.

Assumption ID.

Next we detail the necessary moment assumptions.²

Assumption M.

Assumption M(i) adapts Assumption 1(d) of Newey and Smith (2004), E sup_β∈B∥g_i(β)∥^ξ < ∞ for some ξ > 2, from the i.i.d. setting with strong identification (p_A = 0 and thus θ = β and Θ = B) to the weakly identified setup considered here. A sufficient condition for M(i) in the time series context and under ID is given by

Indeed, a simple application of the Markov inequality shows that (2.4) implies max_1≤i≤n sup_θ∈Θ∥g_i(θ)∥ = O_p(n^1/ξ) = o_p(n^1/2). See the Appendix for a proof. Assumption M(ii), which adapts Assumption 1(e) of Newey and Smith to the weakly identified setup, ensures that

is nonsingular for

. Assumption M(iii) is essentially the “high-level” Assumption B of Stock and Wright (2000, p. 1059) that states that Ψ_n obeys a functional central limit theorem. In Assumption B′, Stock and Wright provide primitive sufficient conditions for their Assumption B that can also be found in Andrews (1994). Note that the definition of weak convergence [Andrews (1994, p. 2250)] and M(iii) imply that sup_θ∈Θ∥Ψ_n(θ)∥ →_d sup_θ∈Θ∥Ψ(θ)∥ and, thus, also that

. In the proof of Theorem 2 we require

bounded in probability.

It is interesting to note that for i.i.d. data an application of the Borel–Cantelli lemma shows that M(i) is implied by Assumption 1(d) of Newey and Smith (2004) even if ξ = 2; see Owen (1990, Lemma 3) for a proof. Hence, using Lemmas 7–9 given subsequently, their Assumption 1(d) can be weakened to ξ ≥ 2 for the consistency and asymptotic normality of the GEL estimator under strong identification with i.i.d. data (see their Theorems 3.1 and 3.2). Therefore, for i.i.d. data, identical assumptions guarantee consistency and asymptotic normality for both GEL and two-step efficient GMM estimators (Hansen, 1982).

Example 1 (continued)

See Guggenberger (2003). For the linear IV model (2.2) Assumption ID can be expressed as the following assumption.

Assumption ID′. Π = Π_n = (Π_An,Π_B) ∈ R^k×(p_A+p_B), where p_A + p_B = p. For a fixed matrix C_A ∈ R^k×p_A, Π_An = n^−1/2C_A and Π_B has full column rank.

Under Assumption ID′, i.i.d. data, and instrument exogeneity it follows that

which implies that in the notation of ID(i), m_1n(θ) = m₁(θ) = E(Z_i Z_i′) C_A(α₀ − α) and m₂(β) = E(Z_i Z_i′)Π_B(β₀ − β). Also, note that Assumption ID′ includes the partially identified model of Phillips (1989). In particular, choosing p_A and setting C_A = 0, one obtains a model in which Π may have any desired (less than full) rank.

We now give simple sufficient conditions that imply Assumption M. Let U := (u,V).

Assumption M′.

Assumptions M′(i) and (ii) state that errors and exogenous variables are jointly i.i.d. and the standard instrument exogeneity assumption is satisfied, whereas M′(iii) and (iv) are technical conditions.

The following lemma shows that Assumption M′ in the linear model implies Assumption M.

LEMMA 1. Suppose that Assumptions ID′, M′, and Θ hold in the linear IV model (2.2). Then Assumptions ID and M hold.

Therefore the various technical conditions of Assumption M reduce to very simple moment conditions in the linear model. Note that M′ implies E [sup_θ∈Θ∥g_i(θ)∥^ξ] < ∞ for ξ = 2. However, we do not need the assumption E [sup_θ∈Θ∥g_i(θ)∥^ξ] < ∞ for a ξ > 2 to prove n^1/2-consistency of the GEL estimator of the strongly identified parameters.

Assumption HOM (conditional homoskedasticity). E(U_iU_i′|Z_i) = Σ_U > 0.

HOM, which is used in Staiger and Stock (1997), is sufficient for Assumption M′(iv). That is, Assumptions M′(i)–(iii) and HOM imply M′(iv) under ID′. This follows from Ω(θ) = Q_ZZ v_α′Σ_uVA v_α for θ ∈ A × {β₀}, where v_α′ := (1,(α₀ − α)′) and Σ_uVA is the (1 + p_A) × (1 + p_A) upper left submatrix of Σ_U. However, M′ is more general than HOM because it allows for conditional heteroskedasticity. For example, u_i = ∥Z_i∥ζ_i, where ζ_i ∼ N(0,1) is independent of Z_i ∼ N(0,I_k), is compatible with M′.

2.3. The GEL Estimator

This section provides a formal definition of the GEL estimator of θ₀.

Let ρ be a real-valued function Q → R, where Q is an open interval of the real line that contains 0 and

If defined, let ρ_j(v) := (∂^jρ/∂v^j)(v) and ρ_j := ρ_j(0) for nonnegative integers j.

The GEL estimator is the solution to a saddle point problem³

For compact Θ, continuous ρ, and g_i (i = 1,…,n), the existence of an argmin

may be shown. In fact,

, viewed as a function in θ, can be shown to be lower semicontinuous (ls). A function f (x) is ls at x₀ if, for each real number c such that c < f (x₀), there exists an open neighborhood U of x₀ such that c < f (x) holds for all x ∈ U. The function f is said to be ls if it is ls at each x₀ of its domain. It is easily shown that ls functions on compact sets take on their minimum. Uniqueness of

, however, is not implied. As a simple example, consider the i.i.d. linear IV model in (2.2) when p = 2 and let the two components Y_ij, (j = 1,2), of Y_i be independent Bernoulli random variables. Then, for each n, the probability that Y_i1 = Y_i2 for every i = 1,…,n is positive. If Y_i1 = Y_i2 for every

is an argmin of

, then each θ ∈ Θ with

is also. To uniquely define

, we could, for example, do the following. From the set of all vectors θ ∈ Θ that minimize

, let

be the vector that has the smallest first component. (If that does not pin down

uniquely, choose from the remaining vectors according to the second component, and so on.)

where

Assumption ρ.

The definition of the GEL estimator

is adopted from Newey and Smith (2004). We slightly modify their definition of

by recentering and rescaling, which simplifies the presentation. We usually write

.

The most popular GEL estimators are the CUE, the EL, and the ET estimator, which correspond to ρ(v) = −(1 + v)²/2, ρ(v) = ln(1 − v), and ρ(v) = −exp v, respectively. The EL estimator was introduced by Imbens (1997), Owen (1988, 1990), and Qin and Lawless (1994) and the ET estimator by Imbens et al. (1998) and Kitamura and Stutzer (1997). For a recent survey of GEL methods see Imbens (2002).⁴

A choice of

as the weighting matrix W_T(θ_T(θ)) in Stock and Wright (2000, equation (2.2), p. 1058), i.e.,

, results in the CUE which is the GEL estimator based on ρ(v) = −(1 + v)²/2; see Newey and Smith (2004, Theorem 2.1). Hansen, Heaton, and Yaron (1996) and Pakes and Pollard (1989) define the (GMM) CUE using the centered weighting matrix

. However, as shown in Newey and Smith (2004, footnote 2), both versions of the CUE are numerically identical.

2.4. First-Order Equivalence

This section obtains the asymptotic distribution of the GEL estimator

under Assumption ID. Theorem 2 shows that the weakly identified parameters of θ₀ are estimated inconsistently and their GEL estimator has a nonstandard limiting distribution whereas the GEL estimator of the strongly identified parameters is n^1/2-consistent but no longer asymptotically normal. Analogous results are available for LIML or more generally for GMM; see Phillips (1984) and Stock and Wright (2000, Theorem 1). The rather surprising finding is that the first-order asymptotic theory under ID is identical for all GEL estimators

, as long as ρ satisfies Assumption ρ.

The proof of Theorem 2 uses a second-order Taylor expansion of

in λ about 0 in which the only impact of ρ asymptotically is through ρ₁ and ρ₂, which are both −1.

If defined, let

For θ = (α′,β′)′ ∈ Θ and b ∈ R^p_B let

The next theorem establishes the asymptotic behavior of

under Assumption ID.

THEOREM 2. Suppose Assumptions Θ, ID, M, and ρ are satisfied. Then

Remark 1. Theorem 2(ii) is analogous to Theorem 1 in Stock and Wright (2000, p. 1062) for GMM. Note that from (A.5) in the Appendix

. Moreover, using the proof of Theorem 2 it can be shown that

Therefore, like

, although n^1/2-consistent,

admits a nonstandard asymptotic distribution (see also Caner, 2003). If p_A = 0, where all parameters are strongly identified,

, where M₂ := M₂(β₀), Ω := Ω(β₀), and Δ := Δ(β₀). The covariance matrix reduces to Ω⁻¹M_M2(Ω) in the i.i.d. case.

The proof of Theorem 2 also provides a formula (equation (A.7) in the Appendix) for b*(α) := arg min_b∈R^pB P_αb for α ∈ A. In particular, if p_A = 0, (A.7) shows that

where

The matrix V(β₀) simplifies to (M₂′Ω⁻¹M₂)⁻¹ in the i.i.d. case, and thus the preceding formula coincides with Theorem 3.2 of Newey and Smith (2004). However, the asymptotic variance matrix of

in the time series context is in general different from that in Newey and Smith, and the estimator

as defined previously would thus be inefficient. Block methods as in Kitamura (1997) or kernel-smoothing methods as in Smith (2001) can be used for efficient GEL estimation in a time series context with strong identification. In the case p_A > 0, the fact that the asymptotic distribution of the strongly identified parameter estimates is in general nonnormal is a consequence of the inconsistent estimation of α₀.

Remark 2. Given the nonnormal asymptotic distribution of the GMM and GEL parameter estimates under weak identification (established in Theorem 1 in Stock and Wright, 2000, and our Theorem 2, respectively) the asymptotic distribution of test statistics based on these estimators, such as t- or Wald statistics, will also be nonstandard and non-pivotal. Furthermore, these limiting distributions depend on quantities that cannot be consistently estimated (see Staiger and Stock, 1997, p. 564), which militates against their use for the construction of test statistics or confidence regions for θ₀. The next section introduces alternative approaches that overcome these difficulties.

Example 1 (continued)

The specialization of Theorem 2 to the i.i.d. linear IV model of Example 1 was derived in Guggenberger (2003).

Example 1 (continued)

Guggenberger (2003) derives the results given in Theorems 3 and 4 under Assumptions Θ, ID′, M′, and ρ allowing for alternatives α ∈ A and Pitman drift in the data generating process (DGP) for the strongly identified parameters to assess the asymptotic power properties of the tests; i.e., ID′ holds and for some fixed b ∈ R^p_B, y = Y(θ₀ + n^−1/2(0′,b′)′) + u. To simplify our presentation here we ignore the possibility of Pitman drift. Results for the i.i.d. linear IV model follow directly from the preceding theorems because, as is easily shown, Assumptions ID′, M′, ρ, and V(θ) > 0 imply M_θ for any consistent estimator

. In particular, V(θ) has a simple representation. For θ = (α′,β₀′)′, Ω(θ) = Δ(θ) and Δ_AA(θ) = E(V_iAV_iA′ [otimes ] Z_i Z_i′), where V_iA consists of the first p_A components of V_i in (2.3).

Example 1 (continued)

Guggenberger (2003) derives the corresponding results. Note that Assumptions Θ, ID′, M′, and ρ and also assuming that V^α(θ_aβ0) is full column rank imply Assumption M_α. In the linear model the components of V^α(θ_aβ0) can be easily calculated. For example, Δ_{A1 A1} = E(V_iA1V_iA1′ [otimes ] Z_i Z_i′), where V_iA1 is the subvector of V_i that contains its first p_A1 components. Let Y = (X,W) denote the partition of the included variables of the structural equation into exogenous and endogenous variables. Partition θ₀ = (θ_X0′,θ_W0′)′ and θ = (θ_X′,θ_W′)′ conformably. Valid inference is possible on any subvector of θ_W0 if the appropriate assumptions given previously are fulfilled. Unfortunately, if the dimension of the parameter vector not subject to test is large, then the argmin-sup problem in (4.1) is computationally very involved. Premultiplication of equation (2.2) by M_X should ameliorate this problem through the elimination of the exogenous variables; i.e., M_X y = M_XWθ_W0 + M_Xu. If Assumption M_α holds for θ_W0 = (α_W0,β_W0) and g_i(θ_W) := M_X,i′(y − Wθ_W)Z_i, where M_X,i denotes the ith row of M_X written as a column vector, valid inference may be undertaken on α_W0.

5.1. Error Distributions

We examine several distributions for (u,V) to investigate the robustness of the test statistics to potentially different features of the error distribution. All designs are constructed from Design (I) obtained by modifying the distribution of the structural error u.

• Design (I): (u,V)′ ∼ N(0,Σ [otimes ] I_n), where Σ ∈ R^2×2 with diagonal elements unity and off-diagonal elements ρ_uV.
• Design (II): u_i in Design (I) is modified as u_i /(w_i /r)^1/2, where w_i is a χ²(r) random variable independent of u_i and V_i, i.e., u_i is t_r-distributed. We fix r = 2.
• Design (III): modifies Design (I) by exchanging u_i² − 1 for u_i, i.e., u_i is a recentered χ²(1) random variable.
• Design (IV): u_i from Design (I) is replaced by B_i|u_i + 2| − (1 − B_i)|u_i + 2| where B_i is Bernoulli (0.5,0.5) distributed and independent of all other random variables.

Design (II) examines the robustness of the performance of the test statistics to thick-tailed distributions for the structural equation error. Design (III) examines robustness with respect to asymmetric structural error distributions. In Design (IV) the structural error u_i is bimodal with peaks at −2 and +2.

In addition, the impact of conditional heteroskedasticity on the performance of the test statistics is examined. Designs (I_HET)–(IV_HET) modify Designs (I)–(IV), respectively, replacing u_i by u_i = ∥Z_i∥u_i.

5.2. Test Statistics

We calculate three versions of the statistic GELR_ρ(θ) in (3.2), for ρ(v) = −(1 + v)²/2 (CUE), ρ(v) = ln(1 − v) (EL), and ρ(v) = −exp v (ET). We also consider the corresponding versions for each of S_ρ(θ) in (3.5) and LM_ρ(θ) in (3.6) with

replaced by

. As noted previously, for CUE, S_ρ(θ) and LM_ρ(θ) are then numerically identical. Theorems 3 and 4 present the asymptotic null distributions of these statistics.

To calculate GELR_ρ(θ), S_ρ(θ), and LM_ρ(θ) for EL and ET, the globally concave maximization problem

must be solved numerically. To do so we implement a variant of the Newton–Raphson algorithm. We initialize the algorithm by setting λ equal to the zero vector. At each iteration the algorithm tries several shrinking step sizes in the search direction and accepts the first one that increases the function value compared to the previous value for λ. This procedure enforces an “uphill climbing” feature of the algorithm.

Additional statistics considered are the Anderson–Rubin test statistic (AR) (see Anderson and Rubin, 1949), two versions of the K-statistic proposed by Kleibergen (2001, 2002a), one assuming homoskedastic errors K, the other robust to conditional heteroskedasticity K_HET, the conditional likelihood ratio test LR_M of Moreira (2003), and two versions of the two-stage least squares (2SLS) Wald statistic 2SLS (see, e.g., Wooldridge, 2002, pp. 98, 100), one assuming homoskedastic errors (2SLS_HOM) and the other robust to conditional heteroskedasticity (2SLS_HET).⁹

The statistics are defined as follows:

where s_uu(θ) := (y − Yθ)′M_Z(y − Yθ)/(n − k),

where

. The statistic K(θ) (Kleibergen, 2002a), is not robust to conditional heteroskedasticity. However, a version of the K-statistic in Kleibergen (2001, equation (22)) that uses a heteroskedasticity consistent estimator for the covariance matrix of g_i(θ) overcomes this drawback. For model (5.1), the statistic is given by

where

, and

. The statistic K_HET(θ) is identical in structure to LM_CUE(θ) except the centered components

are used in place of g_i(θ) and G_i, respectively. Note that G_i := G_i(θ) does not depend on θ in a linear model. For the LR_M statistic, see Moreira (2003, Sect. 3). Finally, the Wald statistics are given by

where

, and

is a conditional heteroskedasticity robust estimator for the variance of

.

5.3. Size Comparison

Empirical sizes are calculated using 5% asymptotic critical values for all of the preceding statistics for DGPs (5.1) corresponding to all 54 possible combinations of sample size n = 50, 100, 250, number of instruments k = 1, 5, 10, SF and RF error correlation ρ_uV = 0.0, 0.5, 0.99, and RF coefficient Π₁ = 0.1, 1.0 for Designs (I)–(IV) and (I_HET)–(IV_HET).¹⁰

We use R = 3,000 replications of each DGP. We also use 3,000 realizations each of χ²(1) and χ²(k − 1) random variables to simulate the critical values of Moreira's LR_M statistic. For the results reported in Tables 1 and 2, which follow, we use R = 10,000 replications. We refer to Π₁ = 0.1 and 1.0 as the “weak” and “strong” instrument cases, respectively. The value of ρ_uV allows the degree of endogeneity of Y to be varied. Whereas for ρ_uV = 0, Y is exogenous, Y is strongly endogenous for ρ_uV = 0.99. We include the just-identified case, k = 1, and two overidentified-cases, k = 5 and 10.

Size results for Design (I) at 5% significance level

Size results for Design (IHET) at 5% significance level

We now describe the results for Designs (I) and (I_HET) given in Tables 1 and 2, respectively, which exclude those for GELR_EL, S_ET, LM_ET, AR, and the case n = 100. The qualitative features of the size results for GELR_EL, S_ET, and LM_ET are identical to their ET/EL counterparts. For k = 1, AR coincides with K, and, for k > 1, we find that in most cases K has better size properties than AR. We report K and 2SLS_HOM for the homoskedastic and K_HET and 2SLS_HET for the heteroskedastic design. We now discuss the results for the homoskedastic case of Design (I).

First, we consider the separate effects of Π₁,n, ρ_uV, and k on the size results.

The most important finding is that the empirical sizes of all statistics except 2SLS show little or no dependence on Π₁ (some additional Monte Carlo results show that this even holds true for the completely unidentified case where Π₁ = 0). However, those for 2SLS depend crucially on the strength or weakness of identification. Although for Π₁ = 1.0, 2SLS has reliable size properties for many cases, with weak instruments sizes range over the entire interval, 0% to 100%.

In general, increasing n leads to more accurate size across all statistics. This holds especially true for those that are poor for smaller n. For example, the 2SLS statistics, GELR_ET and S_EL, severely overreject in overidentified and strongly endogenous cases when n = 50. Even though they still overreject for n = 250, the rejection rates are much closer to the 5% significance level.

It is easily shown that the rejection rates under the null hypothesis for AR and GELR_ρ are independent of the value of ρ_uV. The slight dependence of the size results in Table 1 on ρ_uV results from the use of different samples. For all the remaining statistics except for 2SLS, there does not appear to be a clear pattern for how ρ_uV affects their size properties. Moreover, there is little dependence of the results on ρ_uV. However, for 2SLS, increasing ρ_uV leads to severe overrejection when combined with overidentification, especially so in the weak instrument case.

Increasing the number of instruments k usually leads to overrejection for 2SLS, GELR_ET, and S_EL. For 2SLS this is especially true under weak identification and/or strong endogeneity. All the other statistics show little dependence on k.

We now turn to a comparison of performance across statistics. The 2SLS statistics should not be used with weak instruments or in strongly endogenous overidentified situations. In all other cases, 2SLS has competitive size properties. The statistics GELR_ET and S_EL severely overreject in overidentified problems when the sample size is small. Overall, then, the statistics LM_EL, K, and LR_M lead to the best size results. The statistics LM_CUE and GELR_CUE come in only as second winners because they tend to underreject, especially in overidentified situations. Across the 36 experiments in Table 1, the sizes of LM_EL, LM_CUE, GELR_CUE, K, and LR_M are in the intervals [4.0,6.2], [1.6,5.3], [1.3,5.3], [4.8,8.6] and [4.3,10.3], respectively. The statistics K and LR_M usually slightly overreject. In 22 of the 36 cases, the size of LM_EL comes closest to the 5% significance level across all the statistics. The corresponding numbers for LM_CUE, GELR_CUE, K and LR_M are 8, 8, 9, and 7. Based on Design (I), LM_EL seems to have a slight advantage over the remaining statistics.

We now discuss the size results for Design (I_HET) summarized in Table 2. As most findings are similar to those discussed for Design (I), we only describe the new features.

The statistics 2SLS_HOM, K, and LR_M perform uniformly worse than in Design (I). Tests based on these statistics severely overreject, especially in the just-identified case. Their performance does not improve when n increases. We therefore report results for the heteroskedasticity robust versions 2SLS_HET and K_HET. Their size properties and those of the statistics based on GEL methods do not appear to be negatively influenced by the presence of conditional heteroskedasticity. This is to be expected from our earlier theoretical discussion of the GEL statistics, which does not assume conditional homoskedasticity. Of course, 2SLS_HET still suffers in weakly identified models, and GELR_ET and S_EL perform poorly in overidentified situations for small n. Rejection rates of the test statistics LM_EL, LM_CUE, GELR_CUE, K_HET, and LR_M across the 36 experiments of Table 2 are in the intervals [3.6,6.4], [1.6,5.1], [1.0,5.1], [4.3,9.2], and [7.8,28.8], respectively. In 21 of the 36 cases, the size of LM_EL comes closest to the 5% significance level across all the statistics. The test statistic K_HET wins in 18 cases.

In summary, the only statistics with accurate size properties across all experiments of Designs (I) and (I_HET) are LM_EL, LM_CUE, GELR_CUE, and K_HET. Based on the preceding results it seems that LM_EL enjoys a slight advantage over the other statistics. From the 72 cases in Tables 1 and 2 the empirical size of LM_EL is closest to the nominal 5% in 43 cases across all statistics.

The qualitative features of the size results for Designs (II)–(IV) and (II_HET)–(IV_HET) are generally very similar to their normal counterparts of Designs (I) and (I_HET). For this reason, we do not include additional tables for these designs. One striking difference however occurs for 2SLS under weak identification with χ²(1) (Design (III)) and bimodal errors (Design (IV)). Rejection rates across these 54 combinations for 2SLS_HOM are in the intervals [0.1,7.1] and [0.0,5.4], respectively. Whereas with normal errors and weak identification 2SLS severely overrejects, with these error distributions it severely underrejects.

To summarize this size study, LM_EL, LM_CUE, GELR_CUE, and K_HET have reliable size properties across all designs that appear independent of both the strength or weakness of identification and possible conditional heteroskedasticity. The test statistic 2SLS performs very poorly in the presence of weak instruments. The LR_M statistic performs well in homoskedastic cases but poorly otherwise.

5.4. Power Comparison

Empirical power curves are calculated for the preceding statistics and DGPs (5.1) corresponding to all 16 possible combinations of sample size n = 100, 250, number of instruments k = 5, 10, SF and RF error correlation ρ_uV = 0.5, 0.99, and RF coefficient Π₁ = 0.1, 1.0 for each of the error distributions of Designs (I)–(III). Except for LR_M, we report size-corrected power curves at the 5% significance level, using critical values calculated in the preceding size comparison. We do so because size correction of LR_M is not straightforward as a result of the conditional construction of LR_M and, as shown before, for Designs (I)–(III), LR_M has empirical size very close to nominal at the 5% significance level.

We use R = 1,000 replications from the DGP (5.1) with various values of the true value θ₀. The null hypothesis under test is again H₀ : θ₀ = 0. For weak identification (Π₁ = 0.1), θ₀ takes values in the interval [−4.0,4.0] whereas, with strong identification (Π₁ = 1.0), θ₀ ∈ [−0.4,0.4]. We use 1,000 realizations each of χ²(1) and χ²(k − 1) random variables to simulate the critical values of LR_M. For those results reported in the figures that follow, we use 10,000 replications from (5.1).

Detailed results are presented only for the statistics LM_EL, K, LR_M, and 2SLS_HET. The statistics LM_CUE, LM_EL, and LM_ET display a very similar performance across almost all scenarios. We therefore only report results for LM_EL. We do not report power results for the statistics S_EL and S_ET because, as seen earlier, their size properties appear to be quite poor for the sample sizes considered here. When k = 1, AR and K are numerically identical. In overidentified cases, K generally performs better than AR. We therefore do not report results for AR (see Kleibergen, 2002a, for a comparison of K and AR). Similarly, GELR_CUE is numerically identical to LM_ρ for k = 1 but leads to a less powerful test for k > 1. Also EL and ET versions of GELR_ρ have rather unreliable size properties for the sample sizes considered here. Therefore we do not report detailed results for GELR_ρ.

We first focus on the separate effects of Π₁,n,ρ_uV, and k on power.

With strong identification all statistics have a U-shaped power curve. With the exception of 2SLS_HET, the lowest point of the power curve is usually achieved at θ₀ = 0. In Designs (I) and (II), 2SLS_HET is usually biased, taking on its lowest value at a negative θ₀ value in the interval [−0.2,0.0]. When θ₀ is weakly identified, the power curves of LM_EL, K, and LR_M are generally very flat across all θ₀ values, often only slightly exceeding the significance level of the test. This is especially true for LM_EL and K but less so for LR_M, which is generally more powerful than the other two statistics in this situation. There is one exception when the power of the three tests is high. In Design (I) with ρ_uV = 0.99, although being flat at about 5% for positive θ₀ values, the power curves reach a sharp peak of almost 100% around θ₀ = −1. The reason for this anomaly is most easily explained in the case k = 1, where

. We have

, which in Design (I) with Π₁ = 0.1 equals 1 + 2θ₀ ρ_uV + (1.01)θ₀². If ρ_uV = 0.99 this expression is minimized at around θ₀ = −0.98 where it equals approximately 0.03. Therefore, this peak is caused by

taking on large values for θ₀ in the neighborhood of −1.

For negative θ₀ values with |θ₀| > 1 power quickly falls, reaching between 20% and 50% across the different designs at θ₀ = −4.

In contrast to the power curves of LM_EL, K, and LR_M, the power curve of 2SLS_HET retains its U-shaped form for Π₁ = 0.1. In many cases, the power curve reaches values close to 100% when |θ₀| is close to 4.

As to be expected the tests are more powerful when n is increased from 100 to 250. This holds uniformly across all statistics and designs with a more pronounced power increase in the strongly identified cases.

There does not seem to be a systematic effect due to ρ_uV as it varies with the specific design. For reasons explained previously, the shape of the power curves can change dramatically in Design (I) when ρ_uV is increased from 0.5 to 0.99 if Π₁ = 0.1.

In most cases, there is only little change in the power functions when k is increased from 5 to 10. In general, if the power function changes, then power is slightly lower for larger k.

We now compare the power functions across statistics. Figures 1a–c display the power curves of the four statistics for Designs (I)–(III) in the case Π₁ = 1.0, n = 250, ρ_uV = 0.5, and k = 5 (the figures for Π₁ = 0.1 and for the other parameter combinations are available upon request). The qualitative comparison for the other parameter combinations is very similar, and we therefore focus on these representative cases.

Power curves, strong instrument. (a) Normal errors, (b) t(2) errors, (c) χ2 errors.

When identification is weak, the test based on LR_M is usually more powerful than those based on LM_EL and K. The power gain of using LR_M is quite substantial for negative θ₀ values but less so for positive θ₀. However, the Wald test 2SLS_HET is by far the most powerful test in all three designs. Except for some small negative θ₀ values its power curve uniformly dominates the power curves of the other tests. Recall though that 2SLS_HET has unreliable size properties under weak identification.

When identification is strong, LM_EL uniformly dominates LR_M and K in Designs (II) and (III) (see Figures 1b and 1c). However, LR_M and K uniformly dominate LM_EL in Design (I) (see Figure 1a). This result is to be expected. On the one hand, the LM_EL test is based on nonparametric GEL methods. On the other hand, LR_M and K are motivated within the normal model framework. Although the power gain of LM_EL is small in Design (III), it is substantial in Design (II). Therefore, LM_EL should be used when errors have thick tails.

With strong identification, the Wald test is the most powerful test for positive θ₀ values. For negative θ₀ values, its performance varies from being most powerful in Design (III) to least powerful in Design (I). These results confirm that the Wald test is a reasonable choice when identification is strong.

Overall, therefore, the power study does not lead to an unambiguous ranking of the different tests considered here. Which test is most appropriate depends on the particular error distribution and degree of identification. We find that with strong identification and errors with thick tails or asymmetric errors, LM_EL seems to be the best choice whereas with normal errors LR_M and K appear preferable. When identification is weak, LR_M generally dominates K and LM_EL in terms of power although as noted previously the size properties of LR_M deteriorate substantially in the presence of heteroskedasticity.