2. STRUCTURAL ESTIMATION WITH AN IDENTITY

The model considered here is based on the simplest Keynesian model with a single structural equation involving two measured endogeneous variables y_t, x_t, a stochastic disturbance u_t, and a parameterized structural identity involv-ing an observed IV z_t that is assumed to be exogenous. The system is

where the (spending propensity) parameter β is assumed to satisfy β ≠ 1, so that an equilibrium solution exists. Of course, this condition is needed for the existence of a reduced form and a proper data generating mechanism for the sample data {y_t,x_t : t = 1,…, n}. The parameter γ controls the relevance of the instrument z_t in the system and is convenient to use as a scale coefficient in this equation, but it could readily be absorbed into z_t and its effects measured in terms of the signal from the instrument. When γ → 0, the instrument z_t becomes irrelevant to the determination of y_t and x_t, and we end up with the identity x_t = y_t in place of (2). On the other hand, when γ → ∞, the system is dominated by the signal from z_t. In view of the identity (2) and the exogeneity of z_t, the degree of endogeneity as measured by the correlation coefficient of x_t and u_t is unity, so that there is strong endogeneity in the system.

Sometimes it is convenient to extend the model by an array formulation and index one or the other of the parameters γ and β by the sample size n. Use of an indexed sequence γ_n for γ opens up the study of weak instrument cases, where γ_n may be passed to zero at certain rates, and use of β_n for β enables the model to be analyzed for spending propensities in the vicinity of unity, where β_n → 1.

For a finite-sample development, the errors {u_t : t = 1,…, n} in (1) may be assumed to be iid N(0,σ²), although Gaussianity is unnecessary for the asymptotics. Phillips and Wickens, like Bergstrom (1962), who first considered finite-sample distributions in this system, allow for an intercept in (1), which is inconsequential, and they did not parameterize (2), setting γ = 1.

Define

. The reduced form is

and β is identified by

The IV or LIML estimator of β is

, where

are the reduced form least squares estimates. Analogous to Bergstrom (1962), Phillips and Wickens (1978) gave the exact density of

:

where λ_n = γ²s_zz is the noncentrality parameter. Note that λ_n depends on the sample size through the sample moment s_zz but also through the parameter γ when γ = γ_n depends on n. The exact density (6) is the same as that studied in Nelson and Startz (1990), after notational translation. This is because the model studied by Nelson and Startz (1990) is observationally equivalent to a structural equation with a parameterized identity under the conditioning and zero covariance assumptions that are made in that paper.¹

As shown in Phillips and Wickens (1978), the density pdf(b) is continuous and has a zero at b = 1, the same result later being given in Lemma 2 of Nelson and Startz (1990). Rather obviously, the tails are O(b⁻²), or Cauchy-like (as pointed out in Sargan, 1970/1988, and Phillips, 1983, 1984, 1985, 1986, for structural full information maximum likelihood [FIML] and LIML estimators), and the distribution therefore has modes on either side of the zero at b = 1. Simple calculations reveal that there are two modes located at

where the last two expressions are asymptotic expansions of the modal locations as λ_n → 0 and λ_n → ∞, respectively. For small λ_n, the modes are placed nearly symmetrically on either side of unity at

, and at these points the density is of O(λ_n⁻¹) as λ_n → 0. For large λ_n, one mode is located near β around the value β + 2σ²(1 − β)/λ_n, and the second mode is located around 2 − β + (λ_n /2σ²)(1 − β). At this second mode, the density is O(λ_n⁻²) as λ_n → ∞, and so it is negligible in magnitude for large λ_n.

These properties hold for β ≠ 1 and for all sample sizes n, or all values of the noncentrality parameter λ_n provided γ ≠ 0. The bimodality disappears asymptotically when λ_n → ∞ (as happens when γ is fixed and nonzero and n → ∞), because then the distribution of

is asymptotically Gaussian. In this event we have the limit theory

which we can write in standardized form as

which covers the case where β = β_n → 1. In the latter case, it is apparent that the presence of a spending propensity in the vicinity of unity raises the convergence rate above

, which is explained by the fact that both the structural equation and the identity work to attract the estimator to unity. Nelson and Startz (1990) argue that the usual Gaussian limit theory is often a very poor approximation to the finite-sample distribution and that it is particularly bad when the instrument is weak. This follows earlier arguments made in Phillips (1989) for the case of irrelevant instruments. However, as we see later, there is an alternative limiting inverse normal theory in this case that provides a very satisfactory approximation to the finite-sample distribution, including its bimodality.

Some typical shapes of the finite-sample density (6) are shown in Figures 1 and 2. For λ_n = 50, the distribution is close to symmetric, is centered on β, and is well approximated by the Gaussian limit. For λ_n = 5,1 the distributions are decidedly non-Gaussian in form and show substantial bias and asymmetry, with the mode shifting away from β toward unity as λ_n decreases. In Figure 1 the mode on the right side of unity is so small that it cannot be seen on this scale, so the right tail of the distribution is omitted from view.

Figure 2 shows the densities when λ_n = 0.1, 0.01. Both have marked bimodality. As γ and λ_n decrease, the pull of the structural identity becomes stronger, and the primary mode shifts closer to unity. As γ and λ_n decrease further, the secondary mode on the right side of unity becomes more accentuated, and the primary mode shifts even closer to unity. The following intuition explains the bimodality and the location of the modes. Because the IV density is held to zero at b = 1, the secondary mode occurs to the right of unity, much in the same way as a balloon (representing density) is squeezed and a new bubble (mode) arises outside the point of compression to accommodate the density that has been squeezed out elsewhere. Here, the identity serves to provide the point of compression (delivered by the coefficient of the structural identity), and the weak instrumentation provides the energy of compression. Interestingly, the distribution becomes more symmetric again as γ and λ_n decrease (just as it is for large λ_n), but now about the value b = 1. Also, although the modes are more peaked and move closer to unity from the right and left sides, the density still descends to zero at unity for all n.

As is clear from the functional form (6), the density pdf(b) does not tend to a proper probability density as λ_n → 0. In fact, pdf(b) → 0 for all values of b as λ_n → 0, but not uniformly. So the probability mass escapes at infinity in this case as we take the limit of the density. Nonetheless, we still get convergence of

in this case, but to unity, not to β. In fact,

as λ_n → 0. Correspondingly, the modes in the density become more peaked and move closer together.

In the limit where γ = 0, we have π_y = π_x = 0, and the reduced form coefficients contain no information

²

about the parameter β. In this case, the instrument z_t is irrelevant for x_t in estimating β in the structural equation (1). Nonetheless, when γ = 0, the identity (2) becomes x_t = y_t, and it is apparent that in this event

, a.s. for all n. The IV estimator also takes the same value as OLS in this case. This result is different from the distribution in the irrelevant instrument case considered in Phillips (1989), where the IV estimator differs from OLS, is random for all n, and converges to a random variable as n → ∞, unlike OLS, which has a finite nonrandom probability limit.

The result also differs from the limiting case studied in Forchini (2006). Forchini's model is a standard two variable simultaneous system³

without an identity and with its degree of endogeneity measured by the correlation coefficient ρ = −β*/(1 + β*²)^1/2, where β* is the standardized form (see Phillips, 1983) of the structural coefficient β to be estimated by IV. Forchini takes the limit of the finite-sample density of the IV estimator of β* as the concentration parameter λ_n → 0 along the parameterized path λ_n = θ(1 − ρ²) + o(1 − ρ²) = θ(1 + β*²)⁻¹ + o(β*⁻²) for some θ ≥ 0. In this case, the limit of the finite-sample density is a proper probability density, which in the just-identified case has the following form (Forchini, 2006, Thm. 2):

where ₁F₁ denotes the confluent hypergeometric function. Forchini shows this limit distribution to be bimodal when θ > 1.

When standardizing transformations are employed so that n⁻¹Z′Z = I_k, the parameter θ has the form

and may therefore be interpreted as the total concentration coefficient because it represents the total amount of “energy” in the reduced form coefficients (γ and β*γ). Note that λ_n → 0 requires γ′γ → 0 and therefore weak instrumentation. The limiting density (7) includes the totally unidentified case where there is no energy (θ = 0) as in Phillips (1989) and cases where there is some energy (θ > 0) in the coefficients.

As θ increases, we might expect the density to be somewhat more revealing about β*, even when there is weak instrumentation (γ′γ is small), and this is precisely what happens. As θ becomes large, we can use the following simple approximation:

based on the asymptotic expansion of the confluent hypergeometric function ₁F₁ for large θ (e.g., Lebedev, 1972, p. 271). By a straightforward calculation, the modes of (8) are located at

This approximation is sufficiently accurate, in fact, to capture correctly the criterion for bimodality, namely, θ > 1, given in Forchini (2006).

Figure 3 shows the density (8) for various values of θ. Interestingly, as θ increases, the modes in the density (8) move further away from the origin (the point of attraction provided by the least squares estimate and the mode of the IV estimator in the zero energy case), and they diverge with θ. Thus, the modes are drawn toward the two possible limiting values of the standardized structural coefficient, namely, β* = ±∞, which serve as dual attractors based on the point at infinity β*² = ∞, for which we have ρ² = 1. Correspondingly, the density becomes less peaked and the modes are less well defined as θ increases. Indeed, mass escapes from the density and pdf(b) → 0 for all b as θ → ∞. The situation here is precisely the opposite of what happens in the case of the density (6), where the modes become more accentuated and move toward the value b = 1, a single attractor provided by the coefficient in the structural identity. In that case, weak instrumentation (γ → 0) is the force behind the movement in the modes and the concentration of the distribution. In the case of the density (8), however, the modes move apart and are drawn to points at infinity corresponding to β* = ±∞, mass escapes to infinity, and the IV estimator of β* diverges. In this event, the force behind the movement in the modes and the divergence of the IV estimator is the energy (θ → ∞) in the reduced form coefficients. Thus, even in the presence of weak instrumentation where γ′γ → 0, the degree of endogeneity increases sufficiently fast (i.e., ρ² → 1) that the total energy in the reduced form coefficients diverges (θ → ∞), thereby revealing information about β*.

3. SOME LIMIT THEORY FOR WEAK INSTRUMENTS

Next consider the case where γ_n depends on n and γ_n → 0 as n → ∞. The simplest situation of this kind occurs when a local

to zero sequence

is used, an approach that has become conventional in the study of weak instrumentation. Some recent reviews of this literature are given in Andrews and Stock (2005), Hahn and Hausman (2003), and Stock, Wright, and Yogo (2002). Cases of many weak instruments are studied in Chao and Swanson (2005) and Han and Phillips (2006).

The particular

local to zero sequence has no special significance, and limit results may be obtained for other cases. However, we may usefully start with this case, because of its popularity in the literature, and compare results with what happens when γ_n → 0 at a faster rate. Accordingly, set

, require d ≠ 0, and suppose that the sample second moment of the instrument converges to a positive constant, so that n⁻¹s_zz → m_zz > 0. In this event, the noncentrality parameter tends to a positive constant

and it is apparent that the density (6) has the following limit as n → ∞:

The limit theory in this weak instrument case, analogous to Staiger and Stock (1997), simply reproduces the finite-sample distribution under Gaussianity, a phenomenon that reflects the fact that as n becomes large, the data (and the instrument in particular) become less informative about the true value of the parameter of interest, β. As originally shown in Phillips (1989), it is a straightforward matter to convert the limit result (10) into an invariance principle. All that is required is an appeal to a central limit theorem for the reduced form coefficients on which

depends. In the present case, we have the singular normal limit

and the limit distribution (10) follows in a straightforward manner that is analogous to the finite-sample development. Observe that the limit distribution (10), like the finite-sample distribution, has asymmetric bimodality and Cauchy-like tails.

In place of the conventional weak instrument condition (9), we may consider the very weak instrument case where γ_n = o(n^−1/2) and

In this case, observe that

, so that the systematic part of the reduced form is sufficiently small that the limit theory (11) can be replaced by

where η = (σ/(|1 − β|m_zz^1/2))ξ and ξ is standard normal. We deduce that

so that

is inconsistent and converges to unity in this case. Some further elementary manipulations

⁴

reveal that the limit distribution is given by

the inverse of a central Gaussian variate, and the rate of convergence is given by

. In this very weak instrument case, the systematic part of the reduced form is so small that we do not accumulate information fast enough from the observations (the rate here is still the conventional

rate in view of the behavior of s_zz) to learn enough about π_x and π_y to distinguish these parameters from zero. In effect, what we learn from large samples of data is that

are distributed about the origin. Relative to the rate at which γ_n → 0, we then deduce from the relationship between

in (14) that the estimate

is centered on unity and behaves, after scaling, like the inverse normal variate

.

The asymptotic distribution in the very weak instrument case is shown in Figure 2. As is apparent, the asymptotic distribution provides a good approximation to the finite-sample distribution. Unlike the limit distributions given in (10) and in other weak instrument cases considered in the recent literature, this asymptotic distribution is symmetrically bimodal. It also has zero density at the probability limit of

and continues, of course, to have Cauchy-like tails.

At the point in the parameter space where γ = 0, the data generating mechanism becomes simply y_t = x_t = u_t /(1 − β). This mechanism incorporates the identity (2) and the reduced form equations (3) and (4) with γ = 0. It applies whatever the value of β, and the structural equation (1) is essentially irrelevant to the data generating mechanism. Similarly, when γ_n → 0 fast enough, the same effects occur. The asymptotic distribution of

is centered about unity (the coefficient in the structural identity y_t = x_t), the distribution converges at the rate

, which reflects the manner in which the concentration in the data about the structural identity occurs, and the limit distribution provides no information about β when σ² is unknown.

On the other hand, as remarked earlier, the value of β does affect the reduced form error variance and thereby affects the limit distribution of

about unity. Because this error variance σ_v² = σ²/(1 − β)² → ∞ as β → 1, it follows that 1/η →_p 0, as β → 1. We therefore may write (15) in the standardized form

where ξ is standard normal. This result covers the case of a localizing sequence β_n for which β_n → 1 and then

where the rate of convergence is correspondingly accelerated.

These limit results may be compared with those for the least squares estimator β*, for which standard calculations lead to the following limit theory for the

local to zero sequence

:

and for the very weak instrument case where γ_n = o(n^−1/2):

In both cases β* →_p 1 and the limit distribution is normal. There is a noncentrality in the limit distribution for the

local to zero sequence, and the convergence rate is O(n) in this case. In the very weak instrument case, the limit distribution is central normal and the convergence rate is O(nδ_n), which is faster than O(n). Again, these results extend to the case of a localizing sequence β_n for which β_n → 1, and then (18), for example, becomes

It follows from these results that, although the IV and OLS estimators have the same limit in probability in the very weak instrument case, their limit distributions and rates of convergence are quite different, so the estimators are not asymptotically equivalent, in contrast to the case where the degree of apparent overidentification is large enough to make the estimators equivalent. In the present case, both IV and OLS estimators are attracted to the same limit point but in a very different manner and at different rates.

In weak identification situations, it is sometimes argued that the IV estimator is drawn toward the least squares estimator.

⁵

This can be a misleading representation of the phenomenon, as the present example shows. Rather than IV being drawn to OLS, it is the coefficient in the structural identity in the model (or, equivalently, the regression coefficient implied by the presence of strong endogeneity) that acts as an alternative point of attraction in estimation for both estimators. IV is drawn to this point of attraction from both the left and the right because of its bimodal distribution, in contrast to OLS. In situations of weak identification, both estimators tend to be attracted to this point in the parameter space, and the magnetic force is stronger the weaker the instruments. But, as the preceding results show, the attraction is far greater for OLS than it is for IV, so that the former estimator has a faster rate of convergence and the latter estimator is much more resistant to the attractor. Indeed, for

local to zero sequences like

, OLS converges at an O(n) rate to the attractor, whereas IV converges to a random variable with the bimodal distribution (10), so it puts probability mass away from and on both sides of the attractor. On the other hand, when the instruments are very weak, we have

, so that the IV estimator has a symmetric bimodal limiting distribution about the least squares value and IV is “drawn to OLS” but in an ambivalent way. Clearly, this is very different from the two estimators being the same in the limit or IV simply being biased toward the least squares value.