Solutions to the Measurement Error Problem

In Section 2, it was shown that in the linear regression model with measurement errors, the OLS estimators are biased and inconsistent. There is no "quick fix" since the inconsistency is due to an identification problem. Essentially, identifica­tion is equivalent to the existence of a consistent estimator; see Bekker, Merckens, and Wansbeek (1994, p. 18), and in a generic case the measurement error model given by (8.1) and (8.2) is not identified if all random variables are jointly norm­ally distributed or attention is confined to the first and second order moments of the observed variables only.

Let the elements of e and the rows of V be iid normal. We consider first the case of a structural model. Then, according to Bekker (1986), в is identified if and only if there does not exist a nonsingular matrix A = (a1, A2) such that E’a1 is distributed normally and independently of ^’A2, where E is a typical row of S. In particular, this implies that if E is normally distributed, в is not identified. Due to a result by Wald (1940) this result applies in the functional case as well (cf. Aigner et al., 1984, p. 1335). This means that, when searching for consistent estim­ators, additional information, instruments, nonnormality, or additional structure (typically through panel data) are desirable. In this section we consider these cases in turn. In a Bayesian context, incidentally, the outlook is different and inference on в is possible without identifying information, see, e. g., Poirier (1998).

1.1 Restrictions on the parameters

Equations (8.4) and (8.5) show that the inconsistency of b and s2e could be re­moved if Q were known. For example, rather than b we would take (Ig – S^Q)- 2b as an estimator of в, and from (8.4) it is clear this estimator is consistent. In general, Q is unknown. If, however, we have a consistent estimator of Q, we can replace Q by its consistent estimate and obtain an estimator of в that by virtue of Slutsky’s theorem is consistent. The resulting statistic is then a least squares estimator that is adjusted to attain consistency.

As a generalization, assume that a system of just identifying restrictions on the unknown parameters в, о Є, and Q is available:
with F a totally differentiable vector-function of order g2. In view of the sym­metry of Q, 2g(g – 1) of the restrictions in (8.10) are of the form Qj – Qji = 0.

If we combine the sample information with the prior information and add hats to indicate estimators we obtain the following system of equations:

(Ig – sx1V)S – b = 0,

(8.11a)

62 + S’Vb – si = 0,

(8.11b)

F(S, 62, V) = 0.

(8.11c)

When F is such that this system admits a unique solution for S, 6^ and V, this solution will be a consistent estimator of p, с2, and Q since, asymptotically, SX tends to EX and the system then represents the relationship between the true parameters on the one hand and plim b and plim si on the other. This solution is called the consistent adjusted least squares (CALS) estimator (Kapteyn and Wansbeek, 1984).

The CALS estimator is easy to implement. One can use a standard regression program to obtain b and s2 and then employ a computer program for the solution of a system of nonlinear equations. In many cases it will be possible to find an explicit solution for (8.11), which then obviates the necessity of using a computer program for the solution of nonlinear equations.

It can be shown that, in both the structural model and the functional model, the CALS estimator has asymptotic distribution

Vn(0cals – 0) -4 N(0, He1HsAH^(He1)’),

0,

and Qg = 2( Igi + Pg), where Pg is the commutation matrix (Wansbeek, 1989).

Let us now consider the case in which inexact information on the measurement error variances is available of the following form:

0 < Q < Q* < Xx,

with Q* given. The motivation behind such a bound is that researchers who have reason to suppose that measurement error is present may not know the actual size of its variance but may have an idea of an upper bound to that variance. This makes it possible to derive a bound on в that can be consistently estimated. If Q > 0, i. e. there is measurement error in all variables, в should satisfy the inequality

І (к + k*)) < і (к* – k)’Xxk,

(в – і(к + k*))’Xx(Q* -1 – Xx1) Хх(в

where к* = (Xx – Q*)-1Xxk is the probability limit of the estimator under the assumption that Q = Q*. This is an ellipsoid with midpoint 2 (к + k*), passing through к and к* and tangent to the hyperplane к’Хх(в – к) = 0. If Q is singular (some variables are measured without error), a similar but more complicated inequality can be derived, see Bekker, Kapteyn, and Wansbeek (1984, 1987). In practical cases such ellipsoid bounds will be hard to use and bounds on the elements of в separately will be of more interest. Let a be an arbitrary g-vector. The ellipsoid bound implies for the linear combination а’в that

2 а'(к + k*) – I-Jc < а’в < -1 а'(к + k*) + j4C,

with с = (к* – k)’Xxk • a’F*a and F* = (Xx – Q*)-1 – Xx1. Bounds on separate

elements of в are obtained when a is set equal to any of the g unit vectors. Erickson (1993) derived similar results by bounding the error correlation matrix, rather than the error covariance matrix. Bounds on quantities that are not identi­fied are not often considered in econometrics and run against conventional wis­dom. Yet, their usefulness has been recently advocated in the monograph by Manski (1995).

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